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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication</title>  <meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport"/> <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/css/bootstrap.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv-fonts.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/latexml_styles.css" rel="stylesheet" type="text/css"/> <script src="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/js/bootstrap.bundle.min.js"></script> <script src="https://cdnjs.cloudflare.com/ajax/libs/html2canvas/1.3.3/html2canvas.min.js"></script> <script src="/static/browse/0.3.4/js/addons_new.js"></script> <script src="/static/browse/0.3.4/js/feedbackOverlay.js"></script> <meta content=" metasurface, backscatter communications, transformation optics, luneburg lens, reflectarray, reconfigurable intelligent surface. " lang="en" name="keywords"/> <base href="/html/2503.16366v1/"/></head> <body> <nav class="ltx_page_navbar"> <nav class="ltx_TOC"> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#S1" title="In Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">I </span><span class="ltx_text ltx_font_smallcaps">Introduction</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#S2" title="In Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">II </span><span class="ltx_text ltx_font_smallcaps">Methods</span></span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#S2.SS1" title="In II Methods ‣ Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">II-A</span> </span><span class="ltx_text ltx_font_italic">Luneburg Lens Design</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#S2.SS2" title="In II Methods ‣ Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">II-B</span> </span><span class="ltx_text ltx_font_italic">Luneburg Lens Fabrication</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#S2.SS3" title="In II Methods ‣ Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">II-C</span> </span><span class="ltx_text ltx_font_italic">Metasurface Design</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#S2.SS4" title="In II Methods ‣ Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">II-D</span> </span><span class="ltx_text ltx_font_italic">Metasurface-backed Luneburg Lens</span></span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#S3" title="In Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">III </span><span class="ltx_text ltx_font_smallcaps">Results</span></span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#S3.SS1" title="In III Results ‣ Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">III-A</span> </span><span class="ltx_text ltx_font_italic">Full phase control</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#S3.SS2" title="In III Results ‣ Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">III-B</span> </span><span class="ltx_text ltx_font_italic">Multiplexed Communications</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#S3.SS3" title="In III Results ‣ Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref"><span class="ltx_text">III-C</span> </span><span class="ltx_text ltx_font_italic">Preventing eavesdropping</span></span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#S4" title="In Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">IV </span><span class="ltx_text ltx_font_smallcaps">Conclusion</span></span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#A1" title="In Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">A </span>Optimization</span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#A2" title="In Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">B </span>Hemispherical Luneburg Lens</span></a></li> <li class="ltx_tocentry ltx_tocentry_appendix"><a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#A3" title="In Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">C </span>Random masks distribution</span></a></li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"> <div class="ltx_para" id="p1"> <p class="ltx_p" id="p1.1"><a class="ltx_ref" href="https:/doi.org/" title=""></a></p> </div> <div class="ltx_para" id="p2"> <span class="ltx_ERROR undefined" id="p2.1">\tfootnote</span> <p class="ltx_p" id="p2.2">This project was supported by independent research and development funding from the Johns Hopkins Applied Physics Laboratory.</p> </div> <div class="ltx_para" id="p3"> <span class="ltx_ERROR undefined" id="p3.1">\corresp</span> <p class="ltx_p" id="p3.2">Corresponding author: Samuel Kim (e-mail: samuel.kim@jhuapl.edu).</p> </div> <h1 class="ltx_title ltx_title_document">Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication </h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Samuel Kim1 </span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname"> Tim Sleasman1 </span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname"> Avrami Rakovsky1 </span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname"> Ra’id Awadallah1 </span></span> <span class="ltx_author_before"> </span><span class="ltx_creator ltx_role_author"> <span class="ltx_personname"> and David B. Shrekenhamer1 </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_address">Johns Hopkins University Applied Physics Laboratory, Laurel, MD 20723 USA </span></span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract</h6> <p class="ltx_p" id="id6.id1">Backscatter communications is attractive for its low power requirements due to the lack of actively radiating components; however, commonly used devices are typically limited in range and functionality. Here, we design and demonstrate a flattened Luneburg lens combined with a spatially-tunable dynamic metasurface to create a low-power backscatter communicator. The Luneburg lens is a spherically-symmetric lens that focuses a collimated beam from any direction, enabling a wide field-of-view with no aberrations. By applying quasi-conformal transformation optics (QCTO), we design a flattened Luneburg lens to facilitate its seamless interface with the planar metasurface. The gradient index of the Luneburg lens is realized through additive manufacturing. We show that the flattened Luneburg lens with a reflective surface at the flattened focal plane is able to achieve diffraction-limited retroreflection, enabling long-range backscatter communication. When an interrogator transmits towards the metasurface-backed Luneburg lens, the device can modulate the reflected signal phase across a wide field of regard to communicate data. We experimentally show that the spatial control over the metasurface allows different bit streams to be simultaneously communicated in different directions. Additionally, we show that the device is able to prevent eavesdroppers from receiving information, thus securing communications.</p> </div> <div class="ltx_keywords"> <h6 class="ltx_title ltx_title_keywords">Index Terms: </h6> metasurface, backscatter communications, transformation optics, luneburg lens, reflectarray, reconfigurable intelligent surface. </div> <div class="ltx_para" id="p4"> <span class="ltx_ERROR undefined" id="p4.1">\titlepgskip</span> <p class="ltx_p" id="p4.2">=-21pt</p> </div> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">I </span><span class="ltx_text ltx_font_smallcaps" id="S1.1.1">Introduction</span> </h2> <figure class="ltx_figure" id="S1.F1"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="230" id="S1.F1.g1" src="x1.png" width="830"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">Figure 1: </span> (a) Schematic of backscatter communications using the metasurface-backed Luneburg lens (MBLL). The MBLL is able to send separate backscatter signals simultaneously to the desired interrogators. The device is also able to secure the communications from the eavesdropper by either attenuating the reflected signal or randomly modulating the reflected signal with noise. (b) The MBLL retroreflects the signal by placing the reflective metasurface at the focal plane of the flattened Luneburg lens, enabling backscatter communications. The Luneburg lens focuses the incoming plane wave onto a diffraction-limited spot size on the focal plane. The metasurface can be modulated with spatial masks, enabling the MBLL to multiplex different signal to different interrogators. Additionally, the MBLL is able to mask information from undesired directions (red signal), thus preventing eavesdropping and securing communications.</figcaption> </figure> <div class="ltx_para" id="S1.p1"> <p class="ltx_p" id="S1.p1.1">Wireless communications devices, networks, and infrastructure have exploded exponentially over the past few decades, driving a variety of advancements in both hardware and software. 5G networks have become ubiquitous, taking advantage of novel technologies to provide the data rates, connectivity, and latency demanded by modern users <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib1" title="">1</a>]</cite>. The next generation of network technology such as 6G will require even more breakthroughs to provide data rates and capacity that are multiple orders of magnitude greater than before and new functionality such as multi-sensory communications (e.g., augmented reality and telepresence) and Internet of Smart Things (IoST) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib2" title="">2</a>]</cite>. It is estimated that nearly 16 billion devices were connected to the Internet of Things (IoT) in 2023 <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib3" title="">3</a>]</cite>; the number of devices as well as the requirements (e.g., bandwidth, energy, range) will only continue to grow as connectivity and computer intelligence expands in applications such as smart grids, healthcare, transportation, and unmanned vehicles <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib4" title="">4</a>]</cite>.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.1">One particularly exciting technology that can address requirements for low power consumption and low cost is backscatter communications, which does not require any radiative radio frequency (RF) components <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib5" title="">5</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib6" title="">6</a>]</cite>. Instead, backscatter communications takes an incident signal from a remote signal source and modulates the reflected signal, typically by modulating the impedance of an antenna. This simplifies communication platforms by eradicating active RF circuitry, thus decreasing power consumption, weight, and size requirements potentially by orders of magnitude. Radio Frequency IDentification (RFID) devices are a common approach for backscatter communications and consist of a compact, low-frequency antenna; limited circuitry to process and modulate the signal; and in some cases, energy storage in the form of a battery or capacitor <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib7" title="">7</a>]</cite>. While tremendously successful, many limitations prevent this technology from being used in settings with high data rates and long distances. For example, the fixed antenna configuration typically results in a tradeoff between field-of-view (FOV) and gain (corresponding to range). Additionally, RFID is subject to multipath fading in closed or cluttered environments, reducing data rates.</p> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.1">Alternatively, reconfigurable intelligent surfaces (RIS) have attracted significant attention due to its flexibility arising from its large number of dynamically reconfigurable reflecting elements, enabling beamforming (including beam steering, multiple beams, and shaped beams) and spatial modulation <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib8" title="">8</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib9" title="">9</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib6" title="">6</a>]</cite>. RISs are often implemented using reflectarrays or metasurfaces, which consist of artificially structured surfaces containing a periodic arrangement of sub-wavelength elements, where the elements can be dynamically tuned to control their electromagnetic properties. Progress in RF dynamic metasurfaces has been substantial such that they are a variety of explorations into different tuning components <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib10" title="">10</a>]</cite> (e.g., graphene <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib11" title="">11</a>]</cite>, field-effect transistor <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib12" title="">12</a>]</cite>, varactors <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib13" title="">13</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib14" title="">14</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib15" title="">15</a>]</cite>, PIN diodes <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib16" title="">16</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib17" title="">17</a>]</cite>, piezoelectric actuation <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib18" title="">18</a>]</cite>), not to mention the nearly unlimited degrees of freedom in geometries. The degree of control over the metasurface elements range from simple binary amplitude modulation to full grayscale control over both phase and magnitude <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib15" title="">15</a>]</cite>.</p> </div> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.1">Here, we focus on the scenario where the transmitter and receiver are co-located, i.e., monostatic backscatter. Additionally, we may wish to send different signals in different backscatter directions simultaneously or prevent eavesdropping from a listener at a different location, as shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#S1.F1" title="Figure 1 ‣ I Introduction ‣ Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_tag">1</span></a>. Reflectarrays and metasurfaces can be designed or configured to retroreflect for the monostatic backscatter scenario simply by adding a linear spatial grading to the element phases. More generally, they can generate arbitrary beam shapes and have been demonstrated to produce multiple beams simultaneously (i.e., multibeam antennas), often relying on holographic theory to calculate the desired phase corresponding to the interference of the beams at the aperture <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib19" title="">19</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib20" title="">20</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib21" title="">21</a>]</cite>. However, to our knowledge, multibeam antennas have not been demonstrated in multiplexing or securing communications. Furthermore, this complicates metasurface design and limits the number of possible communication bands, as each additional frequency or spatial channel negatively impacts the gain of the other channels. On the other hand, safeguarding backscatter communications against eavesdropping has been studied for RFID <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib12" title="">12</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib22" title="">22</a>]</cite> and RIS <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib23" title="">23</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib24" title="">24</a>]</cite> systems. However, these often assume a known transfer function between the transmitter, backscatter device, and receiver in order to optimize the signal reaching the desired user versus the eavesdropper. A more elegant approach would be to add a separate component to provide the desired spatial behavior for retroreflection to enable spatial multiplexing and secure communications. </p> </div> <figure class="ltx_figure" id="S1.F2"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="331" id="S1.F2.g1" src="x2.png" width="705"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 2: </span>Flattened Luneburg lens and simulation. (a) Coordinate transformation for flattening the LL calculated by quasi-conformal transformation optics. Red and blue lines represent constant <math alttext="x^{\prime}" class="ltx_Math" display="inline" id="S1.F2.4.m1.1"><semantics id="S1.F2.4.m1.1b"><msup id="S1.F2.4.m1.1.1" xref="S1.F2.4.m1.1.1.cmml"><mi id="S1.F2.4.m1.1.1.2" xref="S1.F2.4.m1.1.1.2.cmml">x</mi><mo id="S1.F2.4.m1.1.1.3" xref="S1.F2.4.m1.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S1.F2.4.m1.1c"><apply id="S1.F2.4.m1.1.1.cmml" xref="S1.F2.4.m1.1.1"><csymbol cd="ambiguous" id="S1.F2.4.m1.1.1.1.cmml" xref="S1.F2.4.m1.1.1">superscript</csymbol><ci id="S1.F2.4.m1.1.1.2.cmml" xref="S1.F2.4.m1.1.1.2">𝑥</ci><ci id="S1.F2.4.m1.1.1.3.cmml" xref="S1.F2.4.m1.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.F2.4.m1.1d">x^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S1.F2.4.m1.1e">italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="y^{\prime}" class="ltx_Math" display="inline" id="S1.F2.5.m2.1"><semantics id="S1.F2.5.m2.1b"><msup id="S1.F2.5.m2.1.1" xref="S1.F2.5.m2.1.1.cmml"><mi id="S1.F2.5.m2.1.1.2" xref="S1.F2.5.m2.1.1.2.cmml">y</mi><mo id="S1.F2.5.m2.1.1.3" xref="S1.F2.5.m2.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S1.F2.5.m2.1c"><apply id="S1.F2.5.m2.1.1.cmml" xref="S1.F2.5.m2.1.1"><csymbol cd="ambiguous" id="S1.F2.5.m2.1.1.1.cmml" xref="S1.F2.5.m2.1.1">superscript</csymbol><ci id="S1.F2.5.m2.1.1.2.cmml" xref="S1.F2.5.m2.1.1.2">𝑦</ci><ci id="S1.F2.5.m2.1.1.3.cmml" xref="S1.F2.5.m2.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.F2.5.m2.1d">y^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S1.F2.5.m2.1e">italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> contours in the real space plotted into the virtual space. Dotted circle represents the outline of the original LL in the virtual space. The flattened portion represents a <math alttext="50\degree" class="ltx_Math" display="inline" id="S1.F2.6.m3.1"><semantics id="S1.F2.6.m3.1b"><mrow id="S1.F2.6.m3.1.1" xref="S1.F2.6.m3.1.1.cmml"><mn id="S1.F2.6.m3.1.1.2" xref="S1.F2.6.m3.1.1.2.cmml">50</mn><mo id="S1.F2.6.m3.1.1.1" xref="S1.F2.6.m3.1.1.1.cmml">⁢</mo><mi id="S1.F2.6.m3.1.1.3" mathvariant="normal" xref="S1.F2.6.m3.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.F2.6.m3.1c"><apply id="S1.F2.6.m3.1.1.cmml" xref="S1.F2.6.m3.1.1"><times id="S1.F2.6.m3.1.1.1.cmml" xref="S1.F2.6.m3.1.1.1"></times><cn id="S1.F2.6.m3.1.1.2.cmml" type="integer" xref="S1.F2.6.m3.1.1.2">50</cn><ci id="S1.F2.6.m3.1.1.3.cmml" xref="S1.F2.6.m3.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.F2.6.m3.1d">50\degree</annotation><annotation encoding="application/x-llamapun" id="S1.F2.6.m3.1e">50 °</annotation></semantics></math> FOV. (b) Resulting permittivity of the flattened LL after applying fabrication-constrained approximations. (c) Simulated electric field profile of a plane wave entering the flattened LL and focusing at the focal plane. (d, e) Simulated far field pattern for the flattened Luneburg lens with a perfect electric conductor (PEC) placed at the focal plane for (d) fixed angle and various frequencies, and (e) a fixed frequency and various angles. Far field patterns are normalized to the peak power in the respective plots.</figcaption> </figure> <div class="ltx_para" id="S1.p5"> <p class="ltx_p" id="S1.p5.1">To this end, we propose a device for low-power backscatter communications that consists of a dynamic metasurface-backed Luneburg lens (MBLL) as shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#S1.F1" title="Figure 1 ‣ I Introduction ‣ Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_tag">1</span></a>. The Luneburg lens can be constructed using low-loss dielectric materials, thus enabling passive and broadband retroreflection. The retroreflection enhances the gain towards the interrogator, enabling backscatter communications over long distances. Because the lens focuses signals onto a diffraction-limited spot on the metasurface, the device naturally supports spatial multiplexing and the prevention of eavesdropping without extensive optimization. While Luneburg lenses have been combined with modulators at the focal plane <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib25" title="">25</a>]</cite>, the lack of spatial control over the modulator preclude the possibility of spatial multiplexing. Here, the dynamic metasurface is able to modulate the phase of the reflected signal and the spatial independence of its elements enables communicating multiple data streams simultaneously.</p> </div> <div class="ltx_para" id="S1.p6"> <p class="ltx_p" id="S1.p6.1">Specifically, we have experimentally fabricated and measured an MBLL to demonstrate (1) full phase coverage over a backscattered signal with improved gain and phase coverage compared to a uniform modulator, (2) phase modulation of the backscatter at different angles with different signals simultaneously, and (3) modulation of the backscatter in one direction while sending random noise in another direction, preventing an eavesdropper from intercepting the data. The backscatter communicator can be connected to a sensor, can be affected by environmental effects (acting as a sensor), or can serve as a hub in a network where multiple users employ the device as a sensor for the sake of positioning, tracking, and timing.</p> </div> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">II </span><span class="ltx_text ltx_font_smallcaps" id="S2.1.1">Methods</span> </h2> <section class="ltx_subsection" id="S2.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S2.SS1.5.1.1">II-A</span> </span><span class="ltx_text ltx_font_italic" id="S2.SS1.6.2">Luneburg Lens Design</span> </h3> <div class="ltx_para" id="S2.SS1.p1"> <p class="ltx_p" id="S2.SS1.p1.1">Luneburg lenses (LLs) are spherical gradient-index lenses that focus a collimated beam incident from any angle onto the lens surface. Its spherical symmetry and effectively <math alttext="360\degree" class="ltx_Math" display="inline" id="S2.SS1.p1.1.m1.1"><semantics id="S2.SS1.p1.1.m1.1a"><mrow id="S2.SS1.p1.1.m1.1.1" xref="S2.SS1.p1.1.m1.1.1.cmml"><mn id="S2.SS1.p1.1.m1.1.1.2" xref="S2.SS1.p1.1.m1.1.1.2.cmml">360</mn><mo id="S2.SS1.p1.1.m1.1.1.1" xref="S2.SS1.p1.1.m1.1.1.1.cmml">⁢</mo><mi id="S2.SS1.p1.1.m1.1.1.3" mathvariant="normal" xref="S2.SS1.p1.1.m1.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.1.m1.1b"><apply id="S2.SS1.p1.1.m1.1.1.cmml" xref="S2.SS1.p1.1.m1.1.1"><times id="S2.SS1.p1.1.m1.1.1.1.cmml" xref="S2.SS1.p1.1.m1.1.1.1"></times><cn id="S2.SS1.p1.1.m1.1.1.2.cmml" type="integer" xref="S2.SS1.p1.1.m1.1.1.2">360</cn><ci id="S2.SS1.p1.1.m1.1.1.3.cmml" xref="S2.SS1.p1.1.m1.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.1.m1.1c">360\degree</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.1.m1.1d">360 °</annotation></semantics></math> field of view (FOV) makes it attractive for wide-angle applications. For example, a receiver or transmitter can be placed on the focal surface to create a wide-angle beam steering device or antenna without off-axis aberrations. Alternatively, in the case where a reflective surface (such as a metallic layer) is instead placed on the focal surface, the device backscatters with a diffraction-limited beam directly towards the source (i.e., retroreflection) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib26" title="">26</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib27" title="">27</a>]</cite>.</p> </div> <div class="ltx_para" id="S2.SS1.p2"> <p class="ltx_p" id="S2.SS1.p2.1">However, the curved nature of the LL focal surface makes it difficult to conformally incorporate a metasurface, which are typically fabricated using a printed circuit board (PCB) and are thus flat and rigid. Electronics further complicate this picture, and compatibility with a non-planar form factor restrict the metasurface design. To overcome this, we flatten the focal surface of the LL using transformation optics (TO), which is a mathematical method that has been used to design optical devices including invisibility cloaks and wave rotators. We briefly review TO here, but more details can be found in the literature <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib28" title="">28</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib29" title="">29</a>]</cite>.</p> </div> <div class="ltx_para ltx_noindent" id="S2.SS1.p3"> <p class="ltx_p" id="S2.SS1.p3.8">In TO, suppose we have a medium of permittivity <math alttext="\varepsilon" class="ltx_Math" display="inline" id="S2.SS1.p3.1.m1.1"><semantics id="S2.SS1.p3.1.m1.1a"><mi id="S2.SS1.p3.1.m1.1.1" xref="S2.SS1.p3.1.m1.1.1.cmml">ε</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.1.m1.1b"><ci id="S2.SS1.p3.1.m1.1.1.cmml" xref="S2.SS1.p3.1.m1.1.1">𝜀</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.1.m1.1c">\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.1.m1.1d">italic_ε</annotation></semantics></math> and permeability <math alttext="\mu" class="ltx_Math" display="inline" id="S2.SS1.p3.2.m2.1"><semantics id="S2.SS1.p3.2.m2.1a"><mi id="S2.SS1.p3.2.m2.1.1" xref="S2.SS1.p3.2.m2.1.1.cmml">μ</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.2.m2.1b"><ci id="S2.SS1.p3.2.m2.1.1.cmml" xref="S2.SS1.p3.2.m2.1.1">𝜇</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.2.m2.1c">\mu</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.2.m2.1d">italic_μ</annotation></semantics></math> that are defined in the virtual space with Cartesian coordinates <math alttext="(x,y,z)" class="ltx_Math" display="inline" id="S2.SS1.p3.3.m3.3"><semantics id="S2.SS1.p3.3.m3.3a"><mrow id="S2.SS1.p3.3.m3.3.4.2" xref="S2.SS1.p3.3.m3.3.4.1.cmml"><mo id="S2.SS1.p3.3.m3.3.4.2.1" stretchy="false" xref="S2.SS1.p3.3.m3.3.4.1.cmml">(</mo><mi id="S2.SS1.p3.3.m3.1.1" xref="S2.SS1.p3.3.m3.1.1.cmml">x</mi><mo id="S2.SS1.p3.3.m3.3.4.2.2" xref="S2.SS1.p3.3.m3.3.4.1.cmml">,</mo><mi id="S2.SS1.p3.3.m3.2.2" xref="S2.SS1.p3.3.m3.2.2.cmml">y</mi><mo id="S2.SS1.p3.3.m3.3.4.2.3" xref="S2.SS1.p3.3.m3.3.4.1.cmml">,</mo><mi id="S2.SS1.p3.3.m3.3.3" xref="S2.SS1.p3.3.m3.3.3.cmml">z</mi><mo id="S2.SS1.p3.3.m3.3.4.2.4" stretchy="false" xref="S2.SS1.p3.3.m3.3.4.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.3.m3.3b"><vector id="S2.SS1.p3.3.m3.3.4.1.cmml" xref="S2.SS1.p3.3.m3.3.4.2"><ci id="S2.SS1.p3.3.m3.1.1.cmml" xref="S2.SS1.p3.3.m3.1.1">𝑥</ci><ci id="S2.SS1.p3.3.m3.2.2.cmml" xref="S2.SS1.p3.3.m3.2.2">𝑦</ci><ci id="S2.SS1.p3.3.m3.3.3.cmml" xref="S2.SS1.p3.3.m3.3.3">𝑧</ci></vector></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.3.m3.3c">(x,y,z)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.3.m3.3d">( italic_x , italic_y , italic_z )</annotation></semantics></math>. This typically represents the device of known behavior and is often analytically designed, such as a straight waveguide, a conventional refractive lens, or the original spherical Luneburg lens. The real space defined by coordinates <math alttext="(x^{\prime},y^{\prime},z^{\prime})" class="ltx_Math" display="inline" id="S2.SS1.p3.4.m4.3"><semantics id="S2.SS1.p3.4.m4.3a"><mrow id="S2.SS1.p3.4.m4.3.3.3" xref="S2.SS1.p3.4.m4.3.3.4.cmml"><mo id="S2.SS1.p3.4.m4.3.3.3.4" stretchy="false" xref="S2.SS1.p3.4.m4.3.3.4.cmml">(</mo><msup id="S2.SS1.p3.4.m4.1.1.1.1" xref="S2.SS1.p3.4.m4.1.1.1.1.cmml"><mi id="S2.SS1.p3.4.m4.1.1.1.1.2" xref="S2.SS1.p3.4.m4.1.1.1.1.2.cmml">x</mi><mo id="S2.SS1.p3.4.m4.1.1.1.1.3" xref="S2.SS1.p3.4.m4.1.1.1.1.3.cmml">′</mo></msup><mo id="S2.SS1.p3.4.m4.3.3.3.5" xref="S2.SS1.p3.4.m4.3.3.4.cmml">,</mo><msup id="S2.SS1.p3.4.m4.2.2.2.2" xref="S2.SS1.p3.4.m4.2.2.2.2.cmml"><mi id="S2.SS1.p3.4.m4.2.2.2.2.2" xref="S2.SS1.p3.4.m4.2.2.2.2.2.cmml">y</mi><mo id="S2.SS1.p3.4.m4.2.2.2.2.3" xref="S2.SS1.p3.4.m4.2.2.2.2.3.cmml">′</mo></msup><mo id="S2.SS1.p3.4.m4.3.3.3.6" xref="S2.SS1.p3.4.m4.3.3.4.cmml">,</mo><msup id="S2.SS1.p3.4.m4.3.3.3.3" xref="S2.SS1.p3.4.m4.3.3.3.3.cmml"><mi id="S2.SS1.p3.4.m4.3.3.3.3.2" xref="S2.SS1.p3.4.m4.3.3.3.3.2.cmml">z</mi><mo id="S2.SS1.p3.4.m4.3.3.3.3.3" xref="S2.SS1.p3.4.m4.3.3.3.3.3.cmml">′</mo></msup><mo id="S2.SS1.p3.4.m4.3.3.3.7" stretchy="false" xref="S2.SS1.p3.4.m4.3.3.4.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.4.m4.3b"><vector id="S2.SS1.p3.4.m4.3.3.4.cmml" xref="S2.SS1.p3.4.m4.3.3.3"><apply id="S2.SS1.p3.4.m4.1.1.1.1.cmml" xref="S2.SS1.p3.4.m4.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p3.4.m4.1.1.1.1.1.cmml" xref="S2.SS1.p3.4.m4.1.1.1.1">superscript</csymbol><ci id="S2.SS1.p3.4.m4.1.1.1.1.2.cmml" xref="S2.SS1.p3.4.m4.1.1.1.1.2">𝑥</ci><ci id="S2.SS1.p3.4.m4.1.1.1.1.3.cmml" xref="S2.SS1.p3.4.m4.1.1.1.1.3">′</ci></apply><apply id="S2.SS1.p3.4.m4.2.2.2.2.cmml" xref="S2.SS1.p3.4.m4.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS1.p3.4.m4.2.2.2.2.1.cmml" xref="S2.SS1.p3.4.m4.2.2.2.2">superscript</csymbol><ci id="S2.SS1.p3.4.m4.2.2.2.2.2.cmml" xref="S2.SS1.p3.4.m4.2.2.2.2.2">𝑦</ci><ci id="S2.SS1.p3.4.m4.2.2.2.2.3.cmml" xref="S2.SS1.p3.4.m4.2.2.2.2.3">′</ci></apply><apply id="S2.SS1.p3.4.m4.3.3.3.3.cmml" xref="S2.SS1.p3.4.m4.3.3.3.3"><csymbol cd="ambiguous" id="S2.SS1.p3.4.m4.3.3.3.3.1.cmml" xref="S2.SS1.p3.4.m4.3.3.3.3">superscript</csymbol><ci id="S2.SS1.p3.4.m4.3.3.3.3.2.cmml" xref="S2.SS1.p3.4.m4.3.3.3.3.2">𝑧</ci><ci id="S2.SS1.p3.4.m4.3.3.3.3.3.cmml" xref="S2.SS1.p3.4.m4.3.3.3.3.3">′</ci></apply></vector></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.4.m4.3c">(x^{\prime},y^{\prime},z^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.4.m4.3d">( italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> represents the device we wish to fabricate with permittivity <math alttext="\varepsilon^{\prime}" class="ltx_Math" display="inline" id="S2.SS1.p3.5.m5.1"><semantics id="S2.SS1.p3.5.m5.1a"><msup id="S2.SS1.p3.5.m5.1.1" xref="S2.SS1.p3.5.m5.1.1.cmml"><mi id="S2.SS1.p3.5.m5.1.1.2" xref="S2.SS1.p3.5.m5.1.1.2.cmml">ε</mi><mo id="S2.SS1.p3.5.m5.1.1.3" xref="S2.SS1.p3.5.m5.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.5.m5.1b"><apply id="S2.SS1.p3.5.m5.1.1.cmml" xref="S2.SS1.p3.5.m5.1.1"><csymbol cd="ambiguous" id="S2.SS1.p3.5.m5.1.1.1.cmml" xref="S2.SS1.p3.5.m5.1.1">superscript</csymbol><ci id="S2.SS1.p3.5.m5.1.1.2.cmml" xref="S2.SS1.p3.5.m5.1.1.2">𝜀</ci><ci id="S2.SS1.p3.5.m5.1.1.3.cmml" xref="S2.SS1.p3.5.m5.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.5.m5.1c">\varepsilon^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.5.m5.1d">italic_ε start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math> and permeability <math alttext="\mu^{\prime}" class="ltx_Math" display="inline" id="S2.SS1.p3.6.m6.1"><semantics id="S2.SS1.p3.6.m6.1a"><msup id="S2.SS1.p3.6.m6.1.1" xref="S2.SS1.p3.6.m6.1.1.cmml"><mi id="S2.SS1.p3.6.m6.1.1.2" xref="S2.SS1.p3.6.m6.1.1.2.cmml">μ</mi><mo id="S2.SS1.p3.6.m6.1.1.3" xref="S2.SS1.p3.6.m6.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.6.m6.1b"><apply id="S2.SS1.p3.6.m6.1.1.cmml" xref="S2.SS1.p3.6.m6.1.1"><csymbol cd="ambiguous" id="S2.SS1.p3.6.m6.1.1.1.cmml" xref="S2.SS1.p3.6.m6.1.1">superscript</csymbol><ci id="S2.SS1.p3.6.m6.1.1.2.cmml" xref="S2.SS1.p3.6.m6.1.1.2">𝜇</ci><ci id="S2.SS1.p3.6.m6.1.1.3.cmml" xref="S2.SS1.p3.6.m6.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.6.m6.1c">\mu^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.6.m6.1d">italic_μ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math>. In other words, we wish to distort the geometry from <math alttext="(x,y,z)" class="ltx_Math" display="inline" id="S2.SS1.p3.7.m7.3"><semantics id="S2.SS1.p3.7.m7.3a"><mrow id="S2.SS1.p3.7.m7.3.4.2" xref="S2.SS1.p3.7.m7.3.4.1.cmml"><mo id="S2.SS1.p3.7.m7.3.4.2.1" stretchy="false" xref="S2.SS1.p3.7.m7.3.4.1.cmml">(</mo><mi id="S2.SS1.p3.7.m7.1.1" xref="S2.SS1.p3.7.m7.1.1.cmml">x</mi><mo id="S2.SS1.p3.7.m7.3.4.2.2" xref="S2.SS1.p3.7.m7.3.4.1.cmml">,</mo><mi id="S2.SS1.p3.7.m7.2.2" xref="S2.SS1.p3.7.m7.2.2.cmml">y</mi><mo id="S2.SS1.p3.7.m7.3.4.2.3" xref="S2.SS1.p3.7.m7.3.4.1.cmml">,</mo><mi id="S2.SS1.p3.7.m7.3.3" xref="S2.SS1.p3.7.m7.3.3.cmml">z</mi><mo id="S2.SS1.p3.7.m7.3.4.2.4" stretchy="false" xref="S2.SS1.p3.7.m7.3.4.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.7.m7.3b"><vector id="S2.SS1.p3.7.m7.3.4.1.cmml" xref="S2.SS1.p3.7.m7.3.4.2"><ci id="S2.SS1.p3.7.m7.1.1.cmml" xref="S2.SS1.p3.7.m7.1.1">𝑥</ci><ci id="S2.SS1.p3.7.m7.2.2.cmml" xref="S2.SS1.p3.7.m7.2.2">𝑦</ci><ci id="S2.SS1.p3.7.m7.3.3.cmml" xref="S2.SS1.p3.7.m7.3.3">𝑧</ci></vector></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.7.m7.3c">(x,y,z)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.7.m7.3d">( italic_x , italic_y , italic_z )</annotation></semantics></math> to <math alttext="(x^{\prime},y^{\prime},z^{\prime})" class="ltx_Math" display="inline" id="S2.SS1.p3.8.m8.3"><semantics id="S2.SS1.p3.8.m8.3a"><mrow id="S2.SS1.p3.8.m8.3.3.3" xref="S2.SS1.p3.8.m8.3.3.4.cmml"><mo id="S2.SS1.p3.8.m8.3.3.3.4" stretchy="false" xref="S2.SS1.p3.8.m8.3.3.4.cmml">(</mo><msup id="S2.SS1.p3.8.m8.1.1.1.1" xref="S2.SS1.p3.8.m8.1.1.1.1.cmml"><mi id="S2.SS1.p3.8.m8.1.1.1.1.2" xref="S2.SS1.p3.8.m8.1.1.1.1.2.cmml">x</mi><mo id="S2.SS1.p3.8.m8.1.1.1.1.3" xref="S2.SS1.p3.8.m8.1.1.1.1.3.cmml">′</mo></msup><mo id="S2.SS1.p3.8.m8.3.3.3.5" xref="S2.SS1.p3.8.m8.3.3.4.cmml">,</mo><msup id="S2.SS1.p3.8.m8.2.2.2.2" xref="S2.SS1.p3.8.m8.2.2.2.2.cmml"><mi id="S2.SS1.p3.8.m8.2.2.2.2.2" xref="S2.SS1.p3.8.m8.2.2.2.2.2.cmml">y</mi><mo id="S2.SS1.p3.8.m8.2.2.2.2.3" xref="S2.SS1.p3.8.m8.2.2.2.2.3.cmml">′</mo></msup><mo id="S2.SS1.p3.8.m8.3.3.3.6" xref="S2.SS1.p3.8.m8.3.3.4.cmml">,</mo><msup id="S2.SS1.p3.8.m8.3.3.3.3" xref="S2.SS1.p3.8.m8.3.3.3.3.cmml"><mi id="S2.SS1.p3.8.m8.3.3.3.3.2" xref="S2.SS1.p3.8.m8.3.3.3.3.2.cmml">z</mi><mo id="S2.SS1.p3.8.m8.3.3.3.3.3" xref="S2.SS1.p3.8.m8.3.3.3.3.3.cmml">′</mo></msup><mo id="S2.SS1.p3.8.m8.3.3.3.7" stretchy="false" xref="S2.SS1.p3.8.m8.3.3.4.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.8.m8.3b"><vector id="S2.SS1.p3.8.m8.3.3.4.cmml" xref="S2.SS1.p3.8.m8.3.3.3"><apply id="S2.SS1.p3.8.m8.1.1.1.1.cmml" xref="S2.SS1.p3.8.m8.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p3.8.m8.1.1.1.1.1.cmml" xref="S2.SS1.p3.8.m8.1.1.1.1">superscript</csymbol><ci id="S2.SS1.p3.8.m8.1.1.1.1.2.cmml" xref="S2.SS1.p3.8.m8.1.1.1.1.2">𝑥</ci><ci id="S2.SS1.p3.8.m8.1.1.1.1.3.cmml" xref="S2.SS1.p3.8.m8.1.1.1.1.3">′</ci></apply><apply id="S2.SS1.p3.8.m8.2.2.2.2.cmml" xref="S2.SS1.p3.8.m8.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS1.p3.8.m8.2.2.2.2.1.cmml" xref="S2.SS1.p3.8.m8.2.2.2.2">superscript</csymbol><ci id="S2.SS1.p3.8.m8.2.2.2.2.2.cmml" xref="S2.SS1.p3.8.m8.2.2.2.2.2">𝑦</ci><ci id="S2.SS1.p3.8.m8.2.2.2.2.3.cmml" xref="S2.SS1.p3.8.m8.2.2.2.2.3">′</ci></apply><apply id="S2.SS1.p3.8.m8.3.3.3.3.cmml" xref="S2.SS1.p3.8.m8.3.3.3.3"><csymbol cd="ambiguous" id="S2.SS1.p3.8.m8.3.3.3.3.1.cmml" xref="S2.SS1.p3.8.m8.3.3.3.3">superscript</csymbol><ci id="S2.SS1.p3.8.m8.3.3.3.3.2.cmml" xref="S2.SS1.p3.8.m8.3.3.3.3.2">𝑧</ci><ci id="S2.SS1.p3.8.m8.3.3.3.3.3.cmml" xref="S2.SS1.p3.8.m8.3.3.3.3.3">′</ci></apply></vector></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.8.m8.3c">(x^{\prime},y^{\prime},z^{\prime})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.8.m8.3d">( italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT )</annotation></semantics></math> such that the device in the real space behaves as it had in the virtual space. The Jacobian of the coordinate transformation is defined as:</p> <table class="ltx_equation ltx_eqn_table" id="S2.E1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="A=\begin{bmatrix}\frac{\partial x^{\prime}}{\partial x}&\frac{\partial x^{% \prime}}{\partial y}&\frac{\partial x^{\prime}}{\partial z}\\ \frac{\partial y^{\prime}}{\partial x}&\frac{\partial y^{\prime}}{\partial y}&% \frac{\partial y^{\prime}}{\partial z}\\ \frac{\partial z^{\prime}}{\partial x}&\frac{\partial z^{\prime}}{\partial y}&% \frac{\partial z^{\prime}}{\partial z}\end{bmatrix}." class="ltx_Math" display="block" id="S2.E1.m1.2"><semantics id="S2.E1.m1.2a"><mrow id="S2.E1.m1.2.2.1" xref="S2.E1.m1.2.2.1.1.cmml"><mrow id="S2.E1.m1.2.2.1.1" xref="S2.E1.m1.2.2.1.1.cmml"><mi id="S2.E1.m1.2.2.1.1.2" xref="S2.E1.m1.2.2.1.1.2.cmml">A</mi><mo id="S2.E1.m1.2.2.1.1.1" 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id="S2.E1.m1.2c">A=\begin{bmatrix}\frac{\partial x^{\prime}}{\partial x}&\frac{\partial x^{% \prime}}{\partial y}&\frac{\partial x^{\prime}}{\partial z}\\ \frac{\partial y^{\prime}}{\partial x}&\frac{\partial y^{\prime}}{\partial y}&% \frac{\partial y^{\prime}}{\partial z}\\ \frac{\partial z^{\prime}}{\partial x}&\frac{\partial z^{\prime}}{\partial y}&% \frac{\partial z^{\prime}}{\partial z}\end{bmatrix}.</annotation><annotation encoding="application/x-llamapun" id="S2.E1.m1.2d">italic_A = [ start_ARG start_ROW start_CELL divide start_ARG ∂ italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG start_ARG ∂ italic_x end_ARG end_CELL start_CELL divide start_ARG ∂ italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG start_ARG ∂ italic_y end_ARG end_CELL start_CELL divide start_ARG ∂ italic_x start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG start_ARG ∂ italic_z end_ARG end_CELL end_ROW start_ROW start_CELL divide start_ARG ∂ italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG start_ARG ∂ italic_x end_ARG end_CELL start_CELL divide start_ARG ∂ italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG start_ARG ∂ italic_y end_ARG end_CELL start_CELL divide start_ARG ∂ italic_y start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG start_ARG ∂ italic_z end_ARG end_CELL end_ROW start_ROW start_CELL divide start_ARG ∂ italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG start_ARG ∂ italic_x end_ARG end_CELL start_CELL divide start_ARG ∂ italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG start_ARG ∂ italic_y end_ARG end_CELL start_CELL divide start_ARG ∂ italic_z start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG start_ARG ∂ italic_z end_ARG end_CELL end_ROW end_ARG ] .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(1)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p3.9">TO then gives the electromagnetic parameters in the real space as:</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="Ax1.EGx1"> <tbody id="S2.E2"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\varepsilon^{\prime}" class="ltx_Math" display="inline" id="S2.E2.m1.1"><semantics id="S2.E2.m1.1a"><msup id="S2.E2.m1.1.1" xref="S2.E2.m1.1.1.cmml"><mi id="S2.E2.m1.1.1.2" xref="S2.E2.m1.1.1.2.cmml">ε</mi><mo id="S2.E2.m1.1.1.3" xref="S2.E2.m1.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S2.E2.m1.1b"><apply id="S2.E2.m1.1.1.cmml" xref="S2.E2.m1.1.1"><csymbol cd="ambiguous" id="S2.E2.m1.1.1.1.cmml" xref="S2.E2.m1.1.1">superscript</csymbol><ci id="S2.E2.m1.1.1.2.cmml" xref="S2.E2.m1.1.1.2">𝜀</ci><ci id="S2.E2.m1.1.1.3.cmml" xref="S2.E2.m1.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E2.m1.1c">\displaystyle\varepsilon^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.E2.m1.1d">italic_ε start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\frac{A\varepsilon A^{\intercal}}{\text{det}(A)}" class="ltx_Math" display="inline" id="S2.E2.m2.1"><semantics id="S2.E2.m2.1a"><mrow id="S2.E2.m2.1.2" xref="S2.E2.m2.1.2.cmml"><mi id="S2.E2.m2.1.2.2" xref="S2.E2.m2.1.2.2.cmml"></mi><mo id="S2.E2.m2.1.2.1" xref="S2.E2.m2.1.2.1.cmml">=</mo><mstyle displaystyle="true" id="S2.E2.m2.1.1" xref="S2.E2.m2.1.1.cmml"><mfrac id="S2.E2.m2.1.1a" xref="S2.E2.m2.1.1.cmml"><mrow id="S2.E2.m2.1.1.3" xref="S2.E2.m2.1.1.3.cmml"><mi id="S2.E2.m2.1.1.3.2" xref="S2.E2.m2.1.1.3.2.cmml">A</mi><mo id="S2.E2.m2.1.1.3.1" xref="S2.E2.m2.1.1.3.1.cmml">⁢</mo><mi id="S2.E2.m2.1.1.3.3" xref="S2.E2.m2.1.1.3.3.cmml">ε</mi><mo id="S2.E2.m2.1.1.3.1a" xref="S2.E2.m2.1.1.3.1.cmml">⁢</mo><msup id="S2.E2.m2.1.1.3.4" xref="S2.E2.m2.1.1.3.4.cmml"><mi id="S2.E2.m2.1.1.3.4.2" xref="S2.E2.m2.1.1.3.4.2.cmml">A</mi><mo id="S2.E2.m2.1.1.3.4.3" xref="S2.E2.m2.1.1.3.4.3.cmml">⊺</mo></msup></mrow><mrow id="S2.E2.m2.1.1.1" xref="S2.E2.m2.1.1.1.cmml"><mtext id="S2.E2.m2.1.1.1.3" xref="S2.E2.m2.1.1.1.3a.cmml">det</mtext><mo id="S2.E2.m2.1.1.1.2" xref="S2.E2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S2.E2.m2.1.1.1.4.2" xref="S2.E2.m2.1.1.1.cmml"><mo id="S2.E2.m2.1.1.1.4.2.1" stretchy="false" xref="S2.E2.m2.1.1.1.cmml">(</mo><mi id="S2.E2.m2.1.1.1.1" xref="S2.E2.m2.1.1.1.1.cmml">A</mi><mo id="S2.E2.m2.1.1.1.4.2.2" stretchy="false" xref="S2.E2.m2.1.1.1.cmml">)</mo></mrow></mrow></mfrac></mstyle></mrow><annotation-xml encoding="MathML-Content" id="S2.E2.m2.1b"><apply id="S2.E2.m2.1.2.cmml" xref="S2.E2.m2.1.2"><eq id="S2.E2.m2.1.2.1.cmml" xref="S2.E2.m2.1.2.1"></eq><csymbol cd="latexml" 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xref="S2.E2.m2.1.1.1.1">𝐴</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E2.m2.1c">\displaystyle=\frac{A\varepsilon A^{\intercal}}{\text{det}(A)}</annotation><annotation encoding="application/x-llamapun" id="S2.E2.m2.1d">= divide start_ARG italic_A italic_ε italic_A start_POSTSUPERSCRIPT ⊺ end_POSTSUPERSCRIPT end_ARG start_ARG det ( italic_A ) end_ARG</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2)</span></td> </tr></tbody> <tbody id="S2.E3"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mu^{\prime}" class="ltx_Math" display="inline" id="S2.E3.m1.1"><semantics id="S2.E3.m1.1a"><msup id="S2.E3.m1.1.1" xref="S2.E3.m1.1.1.cmml"><mi id="S2.E3.m1.1.1.2" xref="S2.E3.m1.1.1.2.cmml">μ</mi><mo id="S2.E3.m1.1.1.3" xref="S2.E3.m1.1.1.3.cmml">′</mo></msup><annotation-xml encoding="MathML-Content" id="S2.E3.m1.1b"><apply id="S2.E3.m1.1.1.cmml" xref="S2.E3.m1.1.1"><csymbol cd="ambiguous" id="S2.E3.m1.1.1.1.cmml" xref="S2.E3.m1.1.1">superscript</csymbol><ci id="S2.E3.m1.1.1.2.cmml" xref="S2.E3.m1.1.1.2">𝜇</ci><ci id="S2.E3.m1.1.1.3.cmml" xref="S2.E3.m1.1.1.3">′</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E3.m1.1c">\displaystyle\mu^{\prime}</annotation><annotation encoding="application/x-llamapun" id="S2.E3.m1.1d">italic_μ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\frac{A\mu A^{\intercal}}{\text{det}(A)}." class="ltx_Math" display="inline" id="S2.E3.m2.2"><semantics id="S2.E3.m2.2a"><mrow id="S2.E3.m2.2.2.1" xref="S2.E3.m2.2.2.1.1.cmml"><mrow 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xref="S2.E3.m2.1.1.3.3">𝜇</ci><apply id="S2.E3.m2.1.1.3.4.cmml" xref="S2.E3.m2.1.1.3.4"><csymbol cd="ambiguous" id="S2.E3.m2.1.1.3.4.1.cmml" xref="S2.E3.m2.1.1.3.4">superscript</csymbol><ci id="S2.E3.m2.1.1.3.4.2.cmml" xref="S2.E3.m2.1.1.3.4.2">𝐴</ci><ci id="S2.E3.m2.1.1.3.4.3.cmml" xref="S2.E3.m2.1.1.3.4.3">⊺</ci></apply></apply><apply id="S2.E3.m2.1.1.1.cmml" xref="S2.E3.m2.1.1.1"><times id="S2.E3.m2.1.1.1.2.cmml" xref="S2.E3.m2.1.1.1.2"></times><ci id="S2.E3.m2.1.1.1.3a.cmml" xref="S2.E3.m2.1.1.1.3"><mtext id="S2.E3.m2.1.1.1.3.cmml" xref="S2.E3.m2.1.1.1.3">det</mtext></ci><ci id="S2.E3.m2.1.1.1.1.cmml" xref="S2.E3.m2.1.1.1.1">𝐴</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E3.m2.2c">\displaystyle=\frac{A\mu A^{\intercal}}{\text{det}(A)}.</annotation><annotation encoding="application/x-llamapun" id="S2.E3.m2.2d">= divide start_ARG italic_A italic_μ italic_A start_POSTSUPERSCRIPT ⊺ end_POSTSUPERSCRIPT end_ARG start_ARG det ( italic_A ) end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(3)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p3.10">The resulting parameters thus propagate electromagnetic fields the way they had in the virtual space but with the desired geometric distortion.</p> </div> <div class="ltx_para" id="S2.SS1.p4"> <p class="ltx_p" id="S2.SS1.p4.1">Note that in general, the transformed permittivity and permeability are complex tensors and cannot be implemented with naturally occurring materials. Metamaterials, or artificially structured materials, offer a way to achieve such material profiles; however, we would still like to simplify the permittivity and permeability such that they can be achieved with isotropic dielectric metamaterials for low loss and low frequency dispersion. To this end, we use quasi-conformal TO (QCTO) to ensure that the device is able to be fabricated using non-magnetic, dielectric materials. Quasi-conformal mappings are a type of coordinate transformation that preserve local angles, and when used for TO, result in the transformed permittivity tensor being close to scalar (i.e. isotropic) and the permeability being close to unity (i.e. non-magnetic) (assuming that the respective tensors are also scalar and unity in the virtual space).</p> </div> <div class="ltx_para" id="S2.SS1.p5"> <p class="ltx_p" id="S2.SS1.p5.4">Concretely, we flatten a portion of the Luneburg lens such that the flattened focal plane represents a <math alttext="50\degree" class="ltx_Math" display="inline" id="S2.SS1.p5.1.m1.1"><semantics id="S2.SS1.p5.1.m1.1a"><mrow id="S2.SS1.p5.1.m1.1.1" xref="S2.SS1.p5.1.m1.1.1.cmml"><mn id="S2.SS1.p5.1.m1.1.1.2" xref="S2.SS1.p5.1.m1.1.1.2.cmml">50</mn><mo id="S2.SS1.p5.1.m1.1.1.1" xref="S2.SS1.p5.1.m1.1.1.1.cmml">⁢</mo><mi id="S2.SS1.p5.1.m1.1.1.3" mathvariant="normal" xref="S2.SS1.p5.1.m1.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.1.m1.1b"><apply id="S2.SS1.p5.1.m1.1.1.cmml" xref="S2.SS1.p5.1.m1.1.1"><times id="S2.SS1.p5.1.m1.1.1.1.cmml" xref="S2.SS1.p5.1.m1.1.1.1"></times><cn id="S2.SS1.p5.1.m1.1.1.2.cmml" type="integer" xref="S2.SS1.p5.1.m1.1.1.2">50</cn><ci id="S2.SS1.p5.1.m1.1.1.3.cmml" xref="S2.SS1.p5.1.m1.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.1.m1.1c">50\degree</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.1.m1.1d">50 °</annotation></semantics></math> FOV. The Schwarz-Christoffel mapping is used to calculate the coordinate transformation <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib30" title="">30</a>]</cite>, which is shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#S1.F2" title="Figure 2 ‣ I Introduction ‣ Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_tag">2</span></a>(a). The transformation is performed in a two-dimensional (2D) slice of the Luneburg lens, and the resulting permittivity is shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#S1.F2" title="Figure 2 ‣ I Introduction ‣ Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_tag">2</span></a>(b). QCTO for this device results in a slightly anisotropic permittivity, but we make the approximations <math alttext="\varepsilon^{\prime}_{z}=\varepsilon^{\prime}_{r}" class="ltx_Math" display="inline" id="S2.SS1.p5.2.m2.1"><semantics id="S2.SS1.p5.2.m2.1a"><mrow id="S2.SS1.p5.2.m2.1.1" xref="S2.SS1.p5.2.m2.1.1.cmml"><msubsup id="S2.SS1.p5.2.m2.1.1.2" xref="S2.SS1.p5.2.m2.1.1.2.cmml"><mi id="S2.SS1.p5.2.m2.1.1.2.2.2" xref="S2.SS1.p5.2.m2.1.1.2.2.2.cmml">ε</mi><mi id="S2.SS1.p5.2.m2.1.1.2.3" xref="S2.SS1.p5.2.m2.1.1.2.3.cmml">z</mi><mo id="S2.SS1.p5.2.m2.1.1.2.2.3" xref="S2.SS1.p5.2.m2.1.1.2.2.3.cmml">′</mo></msubsup><mo id="S2.SS1.p5.2.m2.1.1.1" xref="S2.SS1.p5.2.m2.1.1.1.cmml">=</mo><msubsup id="S2.SS1.p5.2.m2.1.1.3" xref="S2.SS1.p5.2.m2.1.1.3.cmml"><mi id="S2.SS1.p5.2.m2.1.1.3.2.2" xref="S2.SS1.p5.2.m2.1.1.3.2.2.cmml">ε</mi><mi id="S2.SS1.p5.2.m2.1.1.3.3" xref="S2.SS1.p5.2.m2.1.1.3.3.cmml">r</mi><mo id="S2.SS1.p5.2.m2.1.1.3.2.3" xref="S2.SS1.p5.2.m2.1.1.3.2.3.cmml">′</mo></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.2.m2.1b"><apply id="S2.SS1.p5.2.m2.1.1.cmml" xref="S2.SS1.p5.2.m2.1.1"><eq id="S2.SS1.p5.2.m2.1.1.1.cmml" xref="S2.SS1.p5.2.m2.1.1.1"></eq><apply id="S2.SS1.p5.2.m2.1.1.2.cmml" xref="S2.SS1.p5.2.m2.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p5.2.m2.1.1.2.1.cmml" xref="S2.SS1.p5.2.m2.1.1.2">subscript</csymbol><apply id="S2.SS1.p5.2.m2.1.1.2.2.cmml" xref="S2.SS1.p5.2.m2.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p5.2.m2.1.1.2.2.1.cmml" xref="S2.SS1.p5.2.m2.1.1.2">superscript</csymbol><ci id="S2.SS1.p5.2.m2.1.1.2.2.2.cmml" xref="S2.SS1.p5.2.m2.1.1.2.2.2">𝜀</ci><ci id="S2.SS1.p5.2.m2.1.1.2.2.3.cmml" xref="S2.SS1.p5.2.m2.1.1.2.2.3">′</ci></apply><ci id="S2.SS1.p5.2.m2.1.1.2.3.cmml" xref="S2.SS1.p5.2.m2.1.1.2.3">𝑧</ci></apply><apply id="S2.SS1.p5.2.m2.1.1.3.cmml" xref="S2.SS1.p5.2.m2.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p5.2.m2.1.1.3.1.cmml" xref="S2.SS1.p5.2.m2.1.1.3">subscript</csymbol><apply id="S2.SS1.p5.2.m2.1.1.3.2.cmml" xref="S2.SS1.p5.2.m2.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p5.2.m2.1.1.3.2.1.cmml" xref="S2.SS1.p5.2.m2.1.1.3">superscript</csymbol><ci id="S2.SS1.p5.2.m2.1.1.3.2.2.cmml" xref="S2.SS1.p5.2.m2.1.1.3.2.2">𝜀</ci><ci id="S2.SS1.p5.2.m2.1.1.3.2.3.cmml" xref="S2.SS1.p5.2.m2.1.1.3.2.3">′</ci></apply><ci id="S2.SS1.p5.2.m2.1.1.3.3.cmml" xref="S2.SS1.p5.2.m2.1.1.3.3">𝑟</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.2.m2.1c">\varepsilon^{\prime}_{z}=\varepsilon^{\prime}_{r}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.2.m2.1d">italic_ε start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT = italic_ε start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\mu^{\prime}=1" class="ltx_Math" display="inline" id="S2.SS1.p5.3.m3.1"><semantics id="S2.SS1.p5.3.m3.1a"><mrow id="S2.SS1.p5.3.m3.1.1" xref="S2.SS1.p5.3.m3.1.1.cmml"><msup id="S2.SS1.p5.3.m3.1.1.2" xref="S2.SS1.p5.3.m3.1.1.2.cmml"><mi id="S2.SS1.p5.3.m3.1.1.2.2" xref="S2.SS1.p5.3.m3.1.1.2.2.cmml">μ</mi><mo id="S2.SS1.p5.3.m3.1.1.2.3" xref="S2.SS1.p5.3.m3.1.1.2.3.cmml">′</mo></msup><mo id="S2.SS1.p5.3.m3.1.1.1" xref="S2.SS1.p5.3.m3.1.1.1.cmml">=</mo><mn id="S2.SS1.p5.3.m3.1.1.3" xref="S2.SS1.p5.3.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.3.m3.1b"><apply id="S2.SS1.p5.3.m3.1.1.cmml" xref="S2.SS1.p5.3.m3.1.1"><eq id="S2.SS1.p5.3.m3.1.1.1.cmml" xref="S2.SS1.p5.3.m3.1.1.1"></eq><apply id="S2.SS1.p5.3.m3.1.1.2.cmml" xref="S2.SS1.p5.3.m3.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p5.3.m3.1.1.2.1.cmml" xref="S2.SS1.p5.3.m3.1.1.2">superscript</csymbol><ci id="S2.SS1.p5.3.m3.1.1.2.2.cmml" xref="S2.SS1.p5.3.m3.1.1.2.2">𝜇</ci><ci id="S2.SS1.p5.3.m3.1.1.2.3.cmml" xref="S2.SS1.p5.3.m3.1.1.2.3">′</ci></apply><cn id="S2.SS1.p5.3.m3.1.1.3.cmml" type="integer" xref="S2.SS1.p5.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.3.m3.1c">\mu^{\prime}=1</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.3.m3.1d">italic_μ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = 1</annotation></semantics></math>. Finally, we rotate the permittivity profile around the <math alttext="z" class="ltx_Math" display="inline" id="S2.SS1.p5.4.m4.1"><semantics id="S2.SS1.p5.4.m4.1a"><mi id="S2.SS1.p5.4.m4.1.1" xref="S2.SS1.p5.4.m4.1.1.cmml">z</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.4.m4.1b"><ci id="S2.SS1.p5.4.m4.1.1.cmml" xref="S2.SS1.p5.4.m4.1.1">𝑧</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.4.m4.1c">z</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.4.m4.1d">italic_z</annotation></semantics></math>-axis to achieve the 3D profile.</p> </div> <div class="ltx_para" id="S2.SS1.p6"> <p class="ltx_p" id="S2.SS1.p6.4">To confirm the behavior of the flattened Luneburg lens, simulations are carried out using the finite-difference time-domain (FDTD) full-wave solver in CST Microwave Studio <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib31" title="">31</a>]</cite>. Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#S1.F2" title="Figure 2 ‣ I Introduction ‣ Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_tag">2</span></a>(c) shows the electric field on a cross-section of the lens using a plane wave source, validating the focusing of the flattened Luneburg lens. In subsequent simulations, the Luneburg lens is simulated with a reflective (perfect electrical conductor) backing at the focal plane to confirm the retroreflective behavior of the lens across multiple frequencies and incidence angles, as shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#S1.F2" title="Figure 2 ‣ I Introduction ‣ Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_tag">2</span></a>(d,e). All of the farfield patterns display a main lobe at the incident angle, confirming retroreflection. The <math alttext="15\degree" class="ltx_Math" display="inline" id="S2.SS1.p6.1.m1.1"><semantics id="S2.SS1.p6.1.m1.1a"><mrow id="S2.SS1.p6.1.m1.1.1" xref="S2.SS1.p6.1.m1.1.1.cmml"><mn id="S2.SS1.p6.1.m1.1.1.2" xref="S2.SS1.p6.1.m1.1.1.2.cmml">15</mn><mo id="S2.SS1.p6.1.m1.1.1.1" xref="S2.SS1.p6.1.m1.1.1.1.cmml">⁢</mo><mi id="S2.SS1.p6.1.m1.1.1.3" mathvariant="normal" xref="S2.SS1.p6.1.m1.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p6.1.m1.1b"><apply id="S2.SS1.p6.1.m1.1.1.cmml" xref="S2.SS1.p6.1.m1.1.1"><times id="S2.SS1.p6.1.m1.1.1.1.cmml" xref="S2.SS1.p6.1.m1.1.1.1"></times><cn id="S2.SS1.p6.1.m1.1.1.2.cmml" type="integer" xref="S2.SS1.p6.1.m1.1.1.2">15</cn><ci id="S2.SS1.p6.1.m1.1.1.3.cmml" xref="S2.SS1.p6.1.m1.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p6.1.m1.1c">15\degree</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p6.1.m1.1d">15 °</annotation></semantics></math> incidence angle simulations have main lobes with 3dB beamwidths of <math alttext="21.7\degree" class="ltx_Math" display="inline" id="S2.SS1.p6.2.m2.1"><semantics id="S2.SS1.p6.2.m2.1a"><mrow id="S2.SS1.p6.2.m2.1.1" xref="S2.SS1.p6.2.m2.1.1.cmml"><mn id="S2.SS1.p6.2.m2.1.1.2" xref="S2.SS1.p6.2.m2.1.1.2.cmml">21.7</mn><mo id="S2.SS1.p6.2.m2.1.1.1" xref="S2.SS1.p6.2.m2.1.1.1.cmml">⁢</mo><mi id="S2.SS1.p6.2.m2.1.1.3" mathvariant="normal" xref="S2.SS1.p6.2.m2.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p6.2.m2.1b"><apply id="S2.SS1.p6.2.m2.1.1.cmml" xref="S2.SS1.p6.2.m2.1.1"><times id="S2.SS1.p6.2.m2.1.1.1.cmml" xref="S2.SS1.p6.2.m2.1.1.1"></times><cn id="S2.SS1.p6.2.m2.1.1.2.cmml" type="float" xref="S2.SS1.p6.2.m2.1.1.2">21.7</cn><ci id="S2.SS1.p6.2.m2.1.1.3.cmml" xref="S2.SS1.p6.2.m2.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p6.2.m2.1c">21.7\degree</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p6.2.m2.1d">21.7 °</annotation></semantics></math>, <math alttext="16.5\degree" class="ltx_Math" display="inline" id="S2.SS1.p6.3.m3.1"><semantics id="S2.SS1.p6.3.m3.1a"><mrow id="S2.SS1.p6.3.m3.1.1" xref="S2.SS1.p6.3.m3.1.1.cmml"><mn id="S2.SS1.p6.3.m3.1.1.2" xref="S2.SS1.p6.3.m3.1.1.2.cmml">16.5</mn><mo id="S2.SS1.p6.3.m3.1.1.1" xref="S2.SS1.p6.3.m3.1.1.1.cmml">⁢</mo><mi id="S2.SS1.p6.3.m3.1.1.3" mathvariant="normal" xref="S2.SS1.p6.3.m3.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p6.3.m3.1b"><apply id="S2.SS1.p6.3.m3.1.1.cmml" xref="S2.SS1.p6.3.m3.1.1"><times id="S2.SS1.p6.3.m3.1.1.1.cmml" xref="S2.SS1.p6.3.m3.1.1.1"></times><cn id="S2.SS1.p6.3.m3.1.1.2.cmml" type="float" xref="S2.SS1.p6.3.m3.1.1.2">16.5</cn><ci id="S2.SS1.p6.3.m3.1.1.3.cmml" xref="S2.SS1.p6.3.m3.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p6.3.m3.1c">16.5\degree</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p6.3.m3.1d">16.5 °</annotation></semantics></math>, and <math alttext="11.6\degree" class="ltx_Math" display="inline" id="S2.SS1.p6.4.m4.1"><semantics id="S2.SS1.p6.4.m4.1a"><mrow id="S2.SS1.p6.4.m4.1.1" xref="S2.SS1.p6.4.m4.1.1.cmml"><mn id="S2.SS1.p6.4.m4.1.1.2" xref="S2.SS1.p6.4.m4.1.1.2.cmml">11.6</mn><mo id="S2.SS1.p6.4.m4.1.1.1" xref="S2.SS1.p6.4.m4.1.1.1.cmml">⁢</mo><mi id="S2.SS1.p6.4.m4.1.1.3" mathvariant="normal" xref="S2.SS1.p6.4.m4.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p6.4.m4.1b"><apply id="S2.SS1.p6.4.m4.1.1.cmml" xref="S2.SS1.p6.4.m4.1.1"><times id="S2.SS1.p6.4.m4.1.1.1.cmml" xref="S2.SS1.p6.4.m4.1.1.1"></times><cn id="S2.SS1.p6.4.m4.1.1.2.cmml" type="float" xref="S2.SS1.p6.4.m4.1.1.2">11.6</cn><ci id="S2.SS1.p6.4.m4.1.1.3.cmml" xref="S2.SS1.p6.4.m4.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p6.4.m4.1c">11.6\degree</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p6.4.m4.1d">11.6 °</annotation></semantics></math> at 2, 3, and 4 GHz, respectively. The side lobes levels are -24 dB, -14 dB, and -32 dB for the 3 respective frequencies, which are partially due to finite aperture of the lens and the non-Gaussian source. While the size of the main lobes do not vary much with incidence angle, the side lobe levels tend to increase with increasing incidence angle.</p> </div> <figure class="ltx_figure" id="S2.F3"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="211" id="S2.F3.g1" src="x3.png" width="598"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">Figure 3: </span>Dynamic metasurface for modulating phase of the reflected signal. (a) Image of the fabricated metasurface and schematic of a single unit cell (schematic not to scale). The metasurface consists of a <math alttext="6\times 6" class="ltx_Math" display="inline" id="S2.F3.2.m1.1"><semantics id="S2.F3.2.m1.1b"><mrow id="S2.F3.2.m1.1.1" xref="S2.F3.2.m1.1.1.cmml"><mn id="S2.F3.2.m1.1.1.2" xref="S2.F3.2.m1.1.1.2.cmml">6</mn><mo id="S2.F3.2.m1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S2.F3.2.m1.1.1.1.cmml">×</mo><mn id="S2.F3.2.m1.1.1.3" xref="S2.F3.2.m1.1.1.3.cmml">6</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.F3.2.m1.1c"><apply id="S2.F3.2.m1.1.1.cmml" xref="S2.F3.2.m1.1.1"><times id="S2.F3.2.m1.1.1.1.cmml" xref="S2.F3.2.m1.1.1.1"></times><cn id="S2.F3.2.m1.1.1.2.cmml" type="integer" xref="S2.F3.2.m1.1.1.2">6</cn><cn id="S2.F3.2.m1.1.1.3.cmml" type="integer" xref="S2.F3.2.m1.1.1.3">6</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F3.2.m1.1d">6\times 6</annotation><annotation encoding="application/x-llamapun" id="S2.F3.2.m1.1e">6 × 6</annotation></semantics></math> array of unit cell excluding the corners. The unit cell consists of a square metal patch over a metal-backed ground plane. The patch is connected to varactors (blue triangles) which are connected by through-hole vias to the ground plane. (b) Simulated and experimental results of the reflection amplitude and phase at normal incidence. Experimental results are normalized to the maximum magnitude measured for each voltage setting. Phase is manually corrected to account for the propagation distance, and is normalized to 0 phase at 2.8 GHz to facilitate visualization. </figcaption> </figure> </section> <section class="ltx_subsection" id="S2.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S2.SS2.5.1.1">II-B</span> </span><span class="ltx_text ltx_font_italic" id="S2.SS2.6.2">Luneburg Lens Fabrication</span> </h3> <div class="ltx_para" id="S2.SS2.p1"> <p class="ltx_p" id="S2.SS2.p1.1">The Luneburg lens has a gradient-index profile and is thus difficult to implement with naturally occurring materials; we turn to dielectric metamaterials to serve as an effective medium. Metamaterials, which are artificially structured materials to achieve properties not available in bulk materials, have been used to realize Luneburg lenses in both 2D and 3D <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib32" title="">32</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib33" title="">33</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib34" title="">34</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib35" title="">35</a>]</cite>. Here we use a metamaterial unit cell consisting of a rectangular strut lattice where the strut width may vary in each unit cell to achieve the desired permittivity. The Maxwell-Garnett approximation is used to calculate the width of the struts. Note that as a matter of terminology, the Luneburg lens is physically realized using a 3D meta<span class="ltx_text ltx_font_italic" id="S2.SS2.p1.1.1">material</span>, which differs from the dynamic meta<span class="ltx_text ltx_font_italic" id="S2.SS2.p1.1.2">surface</span> that we place on the focal surface of the Luneburg lens.</p> </div> <div class="ltx_para" id="S2.SS2.p2"> <p class="ltx_p" id="S2.SS2.p2.1">The resulting metamaterial-based lens can then be realized with additive manufacturing. We use a powder-bed printer (EOS P395) using selective laser sintering (SLS) to manufacture the LL with EOS PA3200 silica-loaded nylon, which has a permittivity of 2.8. The strut widths range from 1 mm to 3.095 mm and the metamaterial unit cell size is set to 5 mm. This allow the metamaterial to achieve a continuous range of permittivities from 1.12 to 2.02. The diameter of the lens before applying QCTO is 24 cm. After flattening and truncating the lens to regions with permittivities that are achievable by the metamaterial, the size of the realized lens is 21.59 cm in diameter and 21.71 cm in height.</p> </div> </section> <section class="ltx_subsection" id="S2.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S2.SS3.5.1.1">II-C</span> </span><span class="ltx_text ltx_font_italic" id="S2.SS3.6.2">Metasurface Design</span> </h3> <div class="ltx_para" id="S2.SS3.p1"> <p class="ltx_p" id="S2.SS3.p1.1">The metasurface unit cell consists of a square metal patch on a printed circuit board (PCB) loaded with MAV-000120-1411 varactors, as shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#S2.F3" title="Figure 3 ‣ II-A Luneburg Lens Design ‣ II Methods ‣ Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_tag">3</span></a>(a). The unit cell design is targeted for S-band operation (2–4 GHz) and wide-angle operation so that the focusing of the LL onto the metasurface does not distort the phase response. The square patches measure 11.7 mm across while the center-to-center distance is 25.2 mm. The center of the patch is connected to a control voltage on the back of the metasurface through a via, where the control voltage can vary between unit cells. The varactors connect between the patch and the ground plane through a via.</p> </div> <div class="ltx_para" id="S2.SS3.p2"> <p class="ltx_p" id="S2.SS3.p2.1">The unit cells are arrayed in a 6x6 grid with the corners removed as shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#S2.F3" title="Figure 3 ‣ II-A Luneburg Lens Design ‣ II Methods ‣ Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_tag">3</span></a>(a), which is sufficient to cover the flattened focal plane of the LL. A controller (MCC USB-3114) with 16 distinct voltage levels is used to tune the elements. The central 16 elements in the array each have a distinct voltage, and the boundary elements are connected to their nearest neighbor. This allows for independent grayscale tuning of the patches, which results in control over the reflection phase (and some magnitude variation based on the element’s resonance).</p> </div> <div class="ltx_para" id="S2.SS3.p3"> <p class="ltx_p" id="S2.SS3.p3.1">A comparison of the experimental and simulation results of the metasurface are shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#S2.F3" title="Figure 3 ‣ II-A Luneburg Lens Design ‣ II Methods ‣ Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_tag">3</span></a>(b-e). The simulations are carried out in the Frequency Domain (FD) solver in CST Microwave Studio <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib31" title="">31</a>]</cite> and consist of a single unit cell with periodic boundary conditions. The experimental setup is described in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#S3" title="III Results ‣ Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_tag">III</span></a>. The frequency of the resonances match closely between the experimental and simulations results. However, the reflection amplitude dips of the experimental results at resonance are markedly larger than those of the simulation results, signifying additional losses that are not accounted for. We suspect that this extra loss is due to the electroless nickel immersion gold (ENIG) surface finish that was used to coat the copper, which was not modeled in the results shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#S2.F3" title="Figure 3 ‣ II-A Luneburg Lens Design ‣ II Methods ‣ Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_tag">3</span></a>(d-e). Indeed, when a nickel coating is added to the simulations, the loss increases to a similar magnitude as the experimental results. Future work will use lower loss surface finishes. We note that the same metasurface is used in the results going forward and will similarly show the increased loss. However, we still compare with the low-loss simulation results to demonstrate the potential performance of the device.</p> </div> <figure class="ltx_figure" id="S2.F4"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="643" id="S2.F4.g1" src="x4.png" width="830"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">Figure 4: </span> (a) Back view of the MBLL and the voltage controller. (b) Experimental setup of the MBLL in the anechoic chamber. (c) Backscatter (retroreflection) measurements for the MBLL as well as a metal plate for comparison. Simulation of the MBLL uses a perfect electric conductor (PEC) plate at the LL focal plane to reduce computational complexity. (d) MBLL backscatter field magnitude and phase as a function of metasurface tuning, which is uniform across the metasurface. The metasurface achieves almost a full <math alttext="2\pi" class="ltx_Math" display="inline" id="S2.F4.4.m1.1"><semantics id="S2.F4.4.m1.1b"><mrow id="S2.F4.4.m1.1.1" xref="S2.F4.4.m1.1.1.cmml"><mn id="S2.F4.4.m1.1.1.2" xref="S2.F4.4.m1.1.1.2.cmml">2</mn><mo id="S2.F4.4.m1.1.1.1" xref="S2.F4.4.m1.1.1.1.cmml">⁢</mo><mi id="S2.F4.4.m1.1.1.3" xref="S2.F4.4.m1.1.1.3.cmml">π</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.F4.4.m1.1c"><apply id="S2.F4.4.m1.1.1.cmml" xref="S2.F4.4.m1.1.1"><times id="S2.F4.4.m1.1.1.1.cmml" xref="S2.F4.4.m1.1.1.1"></times><cn id="S2.F4.4.m1.1.1.2.cmml" type="integer" xref="S2.F4.4.m1.1.1.2">2</cn><ci id="S2.F4.4.m1.1.1.3.cmml" xref="S2.F4.4.m1.1.1.3">𝜋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F4.4.m1.1d">2\pi</annotation><annotation encoding="application/x-llamapun" id="S2.F4.4.m1.1e">2 italic_π</annotation></semantics></math> phase control up to <math alttext="30\degree" class="ltx_Math" display="inline" id="S2.F4.5.m2.1"><semantics id="S2.F4.5.m2.1b"><mrow id="S2.F4.5.m2.1.1" xref="S2.F4.5.m2.1.1.cmml"><mn id="S2.F4.5.m2.1.1.2" xref="S2.F4.5.m2.1.1.2.cmml">30</mn><mo id="S2.F4.5.m2.1.1.1" xref="S2.F4.5.m2.1.1.1.cmml">⁢</mo><mi id="S2.F4.5.m2.1.1.3" mathvariant="normal" xref="S2.F4.5.m2.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.F4.5.m2.1c"><apply id="S2.F4.5.m2.1.1.cmml" xref="S2.F4.5.m2.1.1"><times id="S2.F4.5.m2.1.1.1.cmml" xref="S2.F4.5.m2.1.1.1"></times><cn id="S2.F4.5.m2.1.1.2.cmml" type="integer" xref="S2.F4.5.m2.1.1.2">30</cn><ci id="S2.F4.5.m2.1.1.3.cmml" xref="S2.F4.5.m2.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F4.5.m2.1d">30\degree</annotation><annotation encoding="application/x-llamapun" id="S2.F4.5.m2.1e">30 °</annotation></semantics></math> incidence angle, although the magnitude starts to degrade above <math alttext="15\degree" class="ltx_Math" display="inline" id="S2.F4.6.m3.1"><semantics id="S2.F4.6.m3.1b"><mrow id="S2.F4.6.m3.1.1" xref="S2.F4.6.m3.1.1.cmml"><mn id="S2.F4.6.m3.1.1.2" xref="S2.F4.6.m3.1.1.2.cmml">15</mn><mo id="S2.F4.6.m3.1.1.1" xref="S2.F4.6.m3.1.1.1.cmml">⁢</mo><mi id="S2.F4.6.m3.1.1.3" mathvariant="normal" xref="S2.F4.6.m3.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.F4.6.m3.1c"><apply id="S2.F4.6.m3.1.1.cmml" xref="S2.F4.6.m3.1.1"><times id="S2.F4.6.m3.1.1.1.cmml" xref="S2.F4.6.m3.1.1.1"></times><cn id="S2.F4.6.m3.1.1.2.cmml" type="integer" xref="S2.F4.6.m3.1.1.2">15</cn><ci id="S2.F4.6.m3.1.1.3.cmml" xref="S2.F4.6.m3.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.F4.6.m3.1d">15\degree</annotation><annotation encoding="application/x-llamapun" id="S2.F4.6.m3.1e">15 °</annotation></semantics></math>. Dotted lines refer to simulations and solid lines refer to experimental data. </figcaption> </figure> </section> <section class="ltx_subsection" id="S2.SS4"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S2.SS4.5.1.1">II-D</span> </span><span class="ltx_text ltx_font_italic" id="S2.SS4.6.2">Metasurface-backed Luneburg Lens</span> </h3> <div class="ltx_para" id="S2.SS4.p1"> <p class="ltx_p" id="S2.SS4.p1.1">The metasurface is attached to the flattened focal plane of the Luneburg lens (LL) as shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#S2.F4" title="Figure 4 ‣ II-C Metasurface Design ‣ II Methods ‣ Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_tag">4</span></a>(a). A solid ring is added to the perimeter of the LL for structural support and to allow the metasurface to be attached using screws. The combination of the LL with the dynamic metasurface enables the retroreflected signal to be modulated, thus enabling backscattering data communication. Of particular note is the fact that the focusing plane (where the metasurface abuts the LL) has a non-uniform permittivity, which can be compensated for by the spatially-tunable dynamic metasurface. Additionally, because the LL focuses incoming radiation to a diffraction-limited spot, the spatial variation of the metasurface allows for different simultaneous signals to be modulated independently. The MBLL features the most useful aspects of its respective components and enables a sensor to embed data in the backscattered signal.</p> </div> <div class="ltx_para" id="S2.SS4.p2"> <p class="ltx_p" id="S2.SS4.p2.1">We note that we found improved performance in simulation in terms of loss and angular dispersion when a small gradient-index impedance-matching layer of 2 mm thickness is inserted at the focal plane of the Luneburg lens. Because the index at the focal surface of the lens is above unity, this can change the behavior of the metasurface which was designed for a surrounding medium of unity index. In practice, because this impedance-matching layer is smaller than the unit cell size of the metamaterial used to construct the Luneburg lens, we simply add a standoff distance of 1mm between the lens and the metasurface. The standoff distance does not need to be on the order of the wavelength since the metasurface mode is largely confined to the metal and dielectric.</p> </div> <div class="ltx_para" id="S2.SS4.p3"> <p class="ltx_p" id="S2.SS4.p3.1">Other works have used an anti-reflective coating between the flattened portion of the Luneburg lens and the surrounding medium, which is typically on the order of the wavelength in free space to minimize internal reflections <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib36" title="">36</a>]</cite>. However, we have found the dip in transmission to be fairly minimal, likely due to the relatively small perturbation in permittivity when applying transformation optics in our design. Future work expanding the FOV of the system may require a more significant coordinate transformation, and thus a larger anti-reflective coating to minimize losses.</p> </div> </section> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">III </span><span class="ltx_text ltx_font_smallcaps" id="S3.1.1">Results</span> </h2> <div class="ltx_para" id="S3.p1"> <p class="ltx_p" id="S3.p1.1">Measurements are taken with a vector network analyzer (VNA) in an anechoic chamber, where a single directional horn is used as the source and receiver in a monostatic configuration. The MBLL is placed on its side on a turntable to assess angular performance, such that the turntable rotates the lens along its elevation angle relative to the horn. This setup is shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#S2.F4" title="Figure 4 ‣ II-C Metasurface Design ‣ II Methods ‣ Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_tag">4</span></a>(b). Another copy of the metasurface without the LL is also fabricated and measured to separately characterize the behavior of the LL and the metasurface. Undesired signals are time-gated to isolate the response from the MBLL.</p> </div> <div class="ltx_para" id="S3.p2"> <p class="ltx_p" id="S3.p2.3">Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#S2.F4" title="Figure 4 ‣ II-C Metasurface Design ‣ II Methods ‣ Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_tag">4</span></a>(c) shows the backscatter of a rectangular metal plate which serves as a reference, and the MBLL where the metasurface is tuned off-resonance such that it acts as a conductive metal sheet. We confirm that the LL provides retroreflection across a wide field-of-view (FOV), matching the simulated results. The measured 3-dB FOV is <math alttext="36\degree" class="ltx_Math" display="inline" id="S3.p2.1.m1.1"><semantics id="S3.p2.1.m1.1a"><mrow id="S3.p2.1.m1.1.1" xref="S3.p2.1.m1.1.1.cmml"><mn id="S3.p2.1.m1.1.1.2" xref="S3.p2.1.m1.1.1.2.cmml">36</mn><mo id="S3.p2.1.m1.1.1.1" xref="S3.p2.1.m1.1.1.1.cmml">⁢</mo><mi id="S3.p2.1.m1.1.1.3" mathvariant="normal" xref="S3.p2.1.m1.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.1.m1.1b"><apply id="S3.p2.1.m1.1.1.cmml" xref="S3.p2.1.m1.1.1"><times id="S3.p2.1.m1.1.1.1.cmml" xref="S3.p2.1.m1.1.1.1"></times><cn id="S3.p2.1.m1.1.1.2.cmml" type="integer" xref="S3.p2.1.m1.1.1.2">36</cn><ci id="S3.p2.1.m1.1.1.3.cmml" xref="S3.p2.1.m1.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.1.m1.1c">36\degree</annotation><annotation encoding="application/x-llamapun" id="S3.p2.1.m1.1d">36 °</annotation></semantics></math>, which is lower than the simulated FOV of over <math alttext="60\degree" class="ltx_Math" display="inline" id="S3.p2.2.m2.1"><semantics id="S3.p2.2.m2.1a"><mrow id="S3.p2.2.m2.1.1" xref="S3.p2.2.m2.1.1.cmml"><mn id="S3.p2.2.m2.1.1.2" xref="S3.p2.2.m2.1.1.2.cmml">60</mn><mo id="S3.p2.2.m2.1.1.1" xref="S3.p2.2.m2.1.1.1.cmml">⁢</mo><mi id="S3.p2.2.m2.1.1.3" mathvariant="normal" xref="S3.p2.2.m2.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.2.m2.1b"><apply id="S3.p2.2.m2.1.1.cmml" xref="S3.p2.2.m2.1.1"><times id="S3.p2.2.m2.1.1.1.cmml" xref="S3.p2.2.m2.1.1.1"></times><cn id="S3.p2.2.m2.1.1.2.cmml" type="integer" xref="S3.p2.2.m2.1.1.2">60</cn><ci id="S3.p2.2.m2.1.1.3.cmml" xref="S3.p2.2.m2.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.2.m2.1c">60\degree</annotation><annotation encoding="application/x-llamapun" id="S3.p2.2.m2.1d">60 °</annotation></semantics></math>; this may be due to the additional support structures that were added to the region around the focal plane of the LL to attach it to the metasurface. Future work can optimize the support structures so that it does not interfere with the performance of the lens. Furthermore, we note that the backscatter has a slight dip at <math alttext="0\degree" class="ltx_Math" display="inline" id="S3.p2.3.m3.1"><semantics id="S3.p2.3.m3.1a"><mrow id="S3.p2.3.m3.1.1" xref="S3.p2.3.m3.1.1.cmml"><mn id="S3.p2.3.m3.1.1.2" xref="S3.p2.3.m3.1.1.2.cmml">0</mn><mo id="S3.p2.3.m3.1.1.1" xref="S3.p2.3.m3.1.1.1.cmml">⁢</mo><mi id="S3.p2.3.m3.1.1.3" mathvariant="normal" xref="S3.p2.3.m3.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.3.m3.1b"><apply id="S3.p2.3.m3.1.1.cmml" xref="S3.p2.3.m3.1.1"><times id="S3.p2.3.m3.1.1.1.cmml" xref="S3.p2.3.m3.1.1.1"></times><cn id="S3.p2.3.m3.1.1.2.cmml" type="integer" xref="S3.p2.3.m3.1.1.2">0</cn><ci id="S3.p2.3.m3.1.1.3.cmml" xref="S3.p2.3.m3.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.3.m3.1c">0\degree</annotation><annotation encoding="application/x-llamapun" id="S3.p2.3.m3.1d">0 °</annotation></semantics></math> which we also confirm in simulation at certain frequencies; we conjecture that this may be due to the higher index of the Luneburg lens at the center of the focal plane which may result in higher back-reflections.</p> </div> <figure class="ltx_figure" id="S3.F5"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="716" id="S3.F5.g1" src="x5.png" width="788"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">Figure 5: </span>Constellation diagrams for MBLL backscatter at different frequencies and incidence angles. Black points represent the spatially uniform masks and the blue points represent the optimized masks to maximize return signal and fill out the phase. Each combination of frequencies and incidence angles were optimized separately. Axes are arbitrarily normalized.</figcaption> </figure> <section class="ltx_subsection" id="S3.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S3.SS1.5.1.1">III-A</span> </span><span class="ltx_text ltx_font_italic" id="S3.SS1.6.2">Full phase control</span> </h3> <div class="ltx_para" id="S3.SS1.p1"> <p class="ltx_p" id="S3.SS1.p1.2">By tuning the varactor voltage across the metasurface, we can modulate the backscattered phase across a wide FOV and range of frequencies, as shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#S2.F4" title="Figure 4 ‣ II-C Metasurface Design ‣ II Methods ‣ Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_tag">4</span></a>(d). Here, the varactor voltage is uniform across the metasurface unit cells. The simulations, which are carried out using the CST Microwave Studio FDTD solver, consist of a 2.5-dimensional slice of the MBLL rather than the full 3-dimensional MBLL due to computational expense arising from the need for fine meshing of the metasurface. The experimental data is also represented as a constellation plot in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#S3.F5" title="Figure 5 ‣ III Results ‣ Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_tag">5</span></a> (black points). The backscattered signal for the MBLL achieves a phase coverage of <math alttext=">300\degree" class="ltx_Math" display="inline" id="S3.SS1.p1.1.m1.1"><semantics id="S3.SS1.p1.1.m1.1a"><mrow id="S3.SS1.p1.1.m1.1.1" xref="S3.SS1.p1.1.m1.1.1.cmml"><mi id="S3.SS1.p1.1.m1.1.1.2" xref="S3.SS1.p1.1.m1.1.1.2.cmml"></mi><mo id="S3.SS1.p1.1.m1.1.1.1" xref="S3.SS1.p1.1.m1.1.1.1.cmml">></mo><mrow id="S3.SS1.p1.1.m1.1.1.3" xref="S3.SS1.p1.1.m1.1.1.3.cmml"><mn id="S3.SS1.p1.1.m1.1.1.3.2" xref="S3.SS1.p1.1.m1.1.1.3.2.cmml">300</mn><mo id="S3.SS1.p1.1.m1.1.1.3.1" xref="S3.SS1.p1.1.m1.1.1.3.1.cmml">⁢</mo><mi id="S3.SS1.p1.1.m1.1.1.3.3" mathvariant="normal" xref="S3.SS1.p1.1.m1.1.1.3.3.cmml">°</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.1.m1.1b"><apply id="S3.SS1.p1.1.m1.1.1.cmml" xref="S3.SS1.p1.1.m1.1.1"><gt id="S3.SS1.p1.1.m1.1.1.1.cmml" xref="S3.SS1.p1.1.m1.1.1.1"></gt><csymbol cd="latexml" id="S3.SS1.p1.1.m1.1.1.2.cmml" xref="S3.SS1.p1.1.m1.1.1.2">absent</csymbol><apply id="S3.SS1.p1.1.m1.1.1.3.cmml" xref="S3.SS1.p1.1.m1.1.1.3"><times id="S3.SS1.p1.1.m1.1.1.3.1.cmml" xref="S3.SS1.p1.1.m1.1.1.3.1"></times><cn id="S3.SS1.p1.1.m1.1.1.3.2.cmml" type="integer" xref="S3.SS1.p1.1.m1.1.1.3.2">300</cn><ci id="S3.SS1.p1.1.m1.1.1.3.3.cmml" xref="S3.SS1.p1.1.m1.1.1.3.3">°</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.1.m1.1c">>300\degree</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.1.m1.1d">> 300 °</annotation></semantics></math> for an angular swath exceeding <math alttext="\pm 30\degree" class="ltx_Math" display="inline" id="S3.SS1.p1.2.m2.1"><semantics id="S3.SS1.p1.2.m2.1a"><mrow id="S3.SS1.p1.2.m2.1.1" xref="S3.SS1.p1.2.m2.1.1.cmml"><mo id="S3.SS1.p1.2.m2.1.1a" xref="S3.SS1.p1.2.m2.1.1.cmml">±</mo><mrow id="S3.SS1.p1.2.m2.1.1.2" xref="S3.SS1.p1.2.m2.1.1.2.cmml"><mn id="S3.SS1.p1.2.m2.1.1.2.2" xref="S3.SS1.p1.2.m2.1.1.2.2.cmml">30</mn><mo id="S3.SS1.p1.2.m2.1.1.2.1" xref="S3.SS1.p1.2.m2.1.1.2.1.cmml">⁢</mo><mi id="S3.SS1.p1.2.m2.1.1.2.3" mathvariant="normal" xref="S3.SS1.p1.2.m2.1.1.2.3.cmml">°</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p1.2.m2.1b"><apply id="S3.SS1.p1.2.m2.1.1.cmml" xref="S3.SS1.p1.2.m2.1.1"><csymbol cd="latexml" id="S3.SS1.p1.2.m2.1.1.1.cmml" xref="S3.SS1.p1.2.m2.1.1">plus-or-minus</csymbol><apply id="S3.SS1.p1.2.m2.1.1.2.cmml" xref="S3.SS1.p1.2.m2.1.1.2"><times id="S3.SS1.p1.2.m2.1.1.2.1.cmml" xref="S3.SS1.p1.2.m2.1.1.2.1"></times><cn id="S3.SS1.p1.2.m2.1.1.2.2.cmml" type="integer" xref="S3.SS1.p1.2.m2.1.1.2.2">30</cn><ci id="S3.SS1.p1.2.m2.1.1.2.3.cmml" xref="S3.SS1.p1.2.m2.1.1.2.3">°</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p1.2.m2.1c">\pm 30\degree</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p1.2.m2.1d">± 30 °</annotation></semantics></math>, which is consistent with the reflection profile of the bare metasurface without the LL in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#S2.F3" title="Figure 3 ‣ II-A Luneburg Lens Design ‣ II Methods ‣ Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_tag">3</span></a>(b-e). Thus, the LL does not negatively impact the performance of the metasurface.</p> </div> <div class="ltx_para" id="S3.SS1.p2"> <p class="ltx_p" id="S3.SS1.p2.4">To fully exploit the capabilities of the MBLL, we can apply different voltages to each unit cell of the metasurface to achieve a spatially-varying phase response across the metasurface—which we will refer to as spatial masks—and assess how the masks change the reflected backscatter (both magnitude and phase). While we have shown that sweeping a uniform voltage of the varactors across the metasurface can achieve a <math alttext="300\degree" class="ltx_Math" display="inline" id="S3.SS1.p2.1.m1.1"><semantics id="S3.SS1.p2.1.m1.1a"><mrow id="S3.SS1.p2.1.m1.1.1" xref="S3.SS1.p2.1.m1.1.1.cmml"><mn id="S3.SS1.p2.1.m1.1.1.2" xref="S3.SS1.p2.1.m1.1.1.2.cmml">300</mn><mo id="S3.SS1.p2.1.m1.1.1.1" xref="S3.SS1.p2.1.m1.1.1.1.cmml">⁢</mo><mi id="S3.SS1.p2.1.m1.1.1.3" mathvariant="normal" xref="S3.SS1.p2.1.m1.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.1.m1.1b"><apply id="S3.SS1.p2.1.m1.1.1.cmml" xref="S3.SS1.p2.1.m1.1.1"><times id="S3.SS1.p2.1.m1.1.1.1.cmml" xref="S3.SS1.p2.1.m1.1.1.1"></times><cn id="S3.SS1.p2.1.m1.1.1.2.cmml" type="integer" xref="S3.SS1.p2.1.m1.1.1.2">300</cn><ci id="S3.SS1.p2.1.m1.1.1.3.cmml" xref="S3.SS1.p2.1.m1.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.1.m1.1c">300\degree</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.1.m1.1d">300 °</annotation></semantics></math> phase coverage in the backscatter, it is possible that the phase range can be improved further using non-uniform voltage masks due to the interaction between the LL and the metasurface as well as the finite edge effects of the metasurface. Thus in the first experiment, we optimize the non-uniform spatial masks to the metasurface to optimize the backscatter for full magnitude and phase control. We apply a custom optimization algorithm based on multi-objective genetic algorithms to maximize the magnitude of the signal for a set of discretely sampled phases from <math alttext="0\degree" class="ltx_Math" display="inline" id="S3.SS1.p2.2.m2.1"><semantics id="S3.SS1.p2.2.m2.1a"><mrow id="S3.SS1.p2.2.m2.1.1" xref="S3.SS1.p2.2.m2.1.1.cmml"><mn id="S3.SS1.p2.2.m2.1.1.2" xref="S3.SS1.p2.2.m2.1.1.2.cmml">0</mn><mo id="S3.SS1.p2.2.m2.1.1.1" xref="S3.SS1.p2.2.m2.1.1.1.cmml">⁢</mo><mi id="S3.SS1.p2.2.m2.1.1.3" mathvariant="normal" xref="S3.SS1.p2.2.m2.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.2.m2.1b"><apply id="S3.SS1.p2.2.m2.1.1.cmml" xref="S3.SS1.p2.2.m2.1.1"><times id="S3.SS1.p2.2.m2.1.1.1.cmml" xref="S3.SS1.p2.2.m2.1.1.1"></times><cn id="S3.SS1.p2.2.m2.1.1.2.cmml" type="integer" xref="S3.SS1.p2.2.m2.1.1.2">0</cn><ci id="S3.SS1.p2.2.m2.1.1.3.cmml" xref="S3.SS1.p2.2.m2.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.2.m2.1c">0\degree</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.2.m2.1d">0 °</annotation></semantics></math> to <math alttext="360\degree" class="ltx_Math" display="inline" id="S3.SS1.p2.3.m3.1"><semantics id="S3.SS1.p2.3.m3.1a"><mrow id="S3.SS1.p2.3.m3.1.1" xref="S3.SS1.p2.3.m3.1.1.cmml"><mn id="S3.SS1.p2.3.m3.1.1.2" xref="S3.SS1.p2.3.m3.1.1.2.cmml">360</mn><mo id="S3.SS1.p2.3.m3.1.1.1" xref="S3.SS1.p2.3.m3.1.1.1.cmml">⁢</mo><mi id="S3.SS1.p2.3.m3.1.1.3" mathvariant="normal" xref="S3.SS1.p2.3.m3.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.3.m3.1b"><apply id="S3.SS1.p2.3.m3.1.1.cmml" xref="S3.SS1.p2.3.m3.1.1"><times id="S3.SS1.p2.3.m3.1.1.1.cmml" xref="S3.SS1.p2.3.m3.1.1.1"></times><cn id="S3.SS1.p2.3.m3.1.1.2.cmml" type="integer" xref="S3.SS1.p2.3.m3.1.1.2">360</cn><ci id="S3.SS1.p2.3.m3.1.1.3.cmml" xref="S3.SS1.p2.3.m3.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.3.m3.1c">360\degree</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.3.m3.1d">360 °</annotation></semantics></math>. The results are shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#S3.F5" title="Figure 5 ‣ III Results ‣ Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_tag">5</span></a> and more details on the optimization can be found in Appendix <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#A1" title="Appendix A Optimization ‣ Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_tag">A</span></a>. The optimized spatial masks are indeed able to not only find patterns with greater magnitude than the uniform patterns, but are also able to fill in the gap and achieve a full <math alttext="360\degree" class="ltx_Math" display="inline" id="S3.SS1.p2.4.m4.1"><semantics id="S3.SS1.p2.4.m4.1a"><mrow id="S3.SS1.p2.4.m4.1.1" xref="S3.SS1.p2.4.m4.1.1.cmml"><mn id="S3.SS1.p2.4.m4.1.1.2" xref="S3.SS1.p2.4.m4.1.1.2.cmml">360</mn><mo id="S3.SS1.p2.4.m4.1.1.1" xref="S3.SS1.p2.4.m4.1.1.1.cmml">⁢</mo><mi id="S3.SS1.p2.4.m4.1.1.3" mathvariant="normal" xref="S3.SS1.p2.4.m4.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.4.m4.1b"><apply id="S3.SS1.p2.4.m4.1.1.cmml" xref="S3.SS1.p2.4.m4.1.1"><times id="S3.SS1.p2.4.m4.1.1.1.cmml" xref="S3.SS1.p2.4.m4.1.1.1"></times><cn id="S3.SS1.p2.4.m4.1.1.2.cmml" type="integer" xref="S3.SS1.p2.4.m4.1.1.2">360</cn><ci id="S3.SS1.p2.4.m4.1.1.3.cmml" xref="S3.SS1.p2.4.m4.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.4.m4.1c">360\degree</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.4.m4.1d">360 °</annotation></semantics></math> phase response.</p> </div> </section> <section class="ltx_subsection" id="S3.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S3.SS2.5.1.1">III-B</span> </span><span class="ltx_text ltx_font_italic" id="S3.SS2.6.2">Multiplexed Communications</span> </h3> <figure class="ltx_figure" id="S3.F6"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_square" height="713" id="S3.F6.g1" src="x6.png" width="705"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 6: </span>Demonstration of multiplexed communications to two different users—Alice (at <math alttext="0\degree" class="ltx_Math" display="inline" id="S3.F6.4.m1.1"><semantics id="S3.F6.4.m1.1b"><mrow id="S3.F6.4.m1.1.1" xref="S3.F6.4.m1.1.1.cmml"><mn id="S3.F6.4.m1.1.1.2" xref="S3.F6.4.m1.1.1.2.cmml">0</mn><mo id="S3.F6.4.m1.1.1.1" xref="S3.F6.4.m1.1.1.1.cmml">⁢</mo><mi id="S3.F6.4.m1.1.1.3" mathvariant="normal" xref="S3.F6.4.m1.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.F6.4.m1.1c"><apply id="S3.F6.4.m1.1.1.cmml" xref="S3.F6.4.m1.1.1"><times id="S3.F6.4.m1.1.1.1.cmml" xref="S3.F6.4.m1.1.1.1"></times><cn id="S3.F6.4.m1.1.1.2.cmml" type="integer" xref="S3.F6.4.m1.1.1.2">0</cn><ci id="S3.F6.4.m1.1.1.3.cmml" xref="S3.F6.4.m1.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F6.4.m1.1d">0\degree</annotation><annotation encoding="application/x-llamapun" id="S3.F6.4.m1.1e">0 °</annotation></semantics></math>) and Bob (at <math alttext="15\degree" class="ltx_Math" display="inline" id="S3.F6.5.m2.1"><semantics id="S3.F6.5.m2.1b"><mrow id="S3.F6.5.m2.1.1" xref="S3.F6.5.m2.1.1.cmml"><mn id="S3.F6.5.m2.1.1.2" xref="S3.F6.5.m2.1.1.2.cmml">15</mn><mo id="S3.F6.5.m2.1.1.1" xref="S3.F6.5.m2.1.1.1.cmml">⁢</mo><mi id="S3.F6.5.m2.1.1.3" mathvariant="normal" xref="S3.F6.5.m2.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.F6.5.m2.1c"><apply id="S3.F6.5.m2.1.1.cmml" xref="S3.F6.5.m2.1.1"><times id="S3.F6.5.m2.1.1.1.cmml" xref="S3.F6.5.m2.1.1.1"></times><cn id="S3.F6.5.m2.1.1.2.cmml" type="integer" xref="S3.F6.5.m2.1.1.2">15</cn><ci id="S3.F6.5.m2.1.1.3.cmml" xref="S3.F6.5.m2.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F6.5.m2.1d">15\degree</annotation><annotation encoding="application/x-llamapun" id="S3.F6.5.m2.1e">15 °</annotation></semantics></math>). (a) Constellation diagram of 4 spatial masks that is able to achieve every combination of 0 and <math alttext="\pi" class="ltx_Math" display="inline" id="S3.F6.6.m3.1"><semantics id="S3.F6.6.m3.1b"><mi id="S3.F6.6.m3.1.1" xref="S3.F6.6.m3.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S3.F6.6.m3.1c"><ci id="S3.F6.6.m3.1.1.cmml" xref="S3.F6.6.m3.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F6.6.m3.1d">\pi</annotation><annotation encoding="application/x-llamapun" id="S3.F6.6.m3.1e">italic_π</annotation></semantics></math> phases (corresponding to “0” and “1” bits for BPSK) to Alice and Bob. Each color/marker shape represents a unique spatial mask for the metasurface. Mask search is performed at 3.76 GHz. (b) Constellation diagram of 16 spatial masks that is able to achieve 4 different phases to Alice and Bob, thus enabling multiplexed QPSK. Each combination of marker face colors and line colors represents a unique spatial mask for the metasurface (e.g., blue line with red marker represents the same mask for Alice and Bob). Mask search is performed at 3.8 GHz. </figcaption> </figure> <div class="ltx_para" id="S3.SS2.p1"> <p class="ltx_p" id="S3.SS2.p1.1">The MBLL enables spatial multiplexing of backscatter signals since the LL focuses electromagnetic waves incident from different angles onto different spatial locations on the metasurface, which can be modulated separately.</p> </div> <div class="ltx_para" id="S3.SS2.p2"> <p class="ltx_p" id="S3.SS2.p2.2">While optimization algorithms will likely deliver ideal results, here we relegate our analysis to a simpler subset of masks of 5000 randomly sampled metasurface spatial patterns due to the limitations of the experimental setup. The reflection of the MBLL with the 5000 spatial masks is measured at <math alttext="0\degree" class="ltx_Math" display="inline" id="S3.SS2.p2.1.m1.1"><semantics id="S3.SS2.p2.1.m1.1a"><mrow id="S3.SS2.p2.1.m1.1.1" xref="S3.SS2.p2.1.m1.1.1.cmml"><mn id="S3.SS2.p2.1.m1.1.1.2" xref="S3.SS2.p2.1.m1.1.1.2.cmml">0</mn><mo id="S3.SS2.p2.1.m1.1.1.1" xref="S3.SS2.p2.1.m1.1.1.1.cmml">⁢</mo><mi id="S3.SS2.p2.1.m1.1.1.3" mathvariant="normal" xref="S3.SS2.p2.1.m1.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.1.m1.1b"><apply id="S3.SS2.p2.1.m1.1.1.cmml" xref="S3.SS2.p2.1.m1.1.1"><times id="S3.SS2.p2.1.m1.1.1.1.cmml" xref="S3.SS2.p2.1.m1.1.1.1"></times><cn id="S3.SS2.p2.1.m1.1.1.2.cmml" type="integer" xref="S3.SS2.p2.1.m1.1.1.2">0</cn><ci id="S3.SS2.p2.1.m1.1.1.3.cmml" xref="S3.SS2.p2.1.m1.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.1.m1.1c">0\degree</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.1.m1.1d">0 °</annotation></semantics></math> incidence (which we name Alice) and <math alttext="15\degree" class="ltx_Math" display="inline" id="S3.SS2.p2.2.m2.1"><semantics id="S3.SS2.p2.2.m2.1a"><mrow id="S3.SS2.p2.2.m2.1.1" xref="S3.SS2.p2.2.m2.1.1.cmml"><mn id="S3.SS2.p2.2.m2.1.1.2" xref="S3.SS2.p2.2.m2.1.1.2.cmml">15</mn><mo id="S3.SS2.p2.2.m2.1.1.1" xref="S3.SS2.p2.2.m2.1.1.1.cmml">⁢</mo><mi id="S3.SS2.p2.2.m2.1.1.3" mathvariant="normal" xref="S3.SS2.p2.2.m2.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.2.m2.1b"><apply id="S3.SS2.p2.2.m2.1.1.cmml" xref="S3.SS2.p2.2.m2.1.1"><times id="S3.SS2.p2.2.m2.1.1.1.cmml" xref="S3.SS2.p2.2.m2.1.1.1"></times><cn id="S3.SS2.p2.2.m2.1.1.2.cmml" type="integer" xref="S3.SS2.p2.2.m2.1.1.2">15</cn><ci id="S3.SS2.p2.2.m2.1.1.3.cmml" xref="S3.SS2.p2.2.m2.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.2.m2.1c">15\degree</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.2.m2.1d">15 °</annotation></semantics></math> incidence (which we name Bob). We wish to find spatial masks such that Alice and Bob receive different bit streams from the MBLL, similar to the case in Ref. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib37" title="">37</a>]</cite>. Importantly, note that communication is not taking place between Alice and Bob, but rather that the MBLL communicates back separate information to each users when simultaneously interrogated by the two users.</p> </div> <div class="ltx_para" id="S3.SS2.p3"> <p class="ltx_p" id="S3.SS2.p3.1">In a binary phase-shift keying (BPSK) implementation each user needs to receive two phases—0 and <math alttext="\pi" class="ltx_Math" display="inline" id="S3.SS2.p3.1.m1.1"><semantics id="S3.SS2.p3.1.m1.1a"><mi id="S3.SS2.p3.1.m1.1.1" xref="S3.SS2.p3.1.m1.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.1.m1.1b"><ci id="S3.SS2.p3.1.m1.1.1.cmml" xref="S3.SS2.p3.1.m1.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.1.m1.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.1.m1.1d">italic_π</annotation></semantics></math> corresponding to bits 0 and 1. Since there are two users and two phases, we aim to find four masks which can pass the different data permutations. We perform a simple search over the 5000 masks to four masks that minimizes the following loss function:</p> <table class="ltx_equation ltx_eqn_table" id="S3.E4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{L}=|\alpha_{A}(\mathbf{x}_{1})-\phi_{A,0}|+|\alpha_{B}(\mathbf{x}_{1}% )-\phi_{B,0}|+\\ |\alpha_{A}(\mathbf{x}_{2})-\phi_{A,0}|+|\alpha_{B}(\mathbf{x}_{2})-\phi_{B,1}% |+\\ |\alpha_{A}(\mathbf{x}_{3})-\phi_{A,1}|+|\alpha_{B}(\mathbf{x}_{3})-\phi_{B,0}% |+\\ |\alpha_{A}(\mathbf{x}_{4})-\phi_{A,1}|+|\alpha_{B}(\mathbf{x}_{4})-\phi_{B,1}|" class="ltx_Math" display="block" id="S3.E4.m1.113"><semantics id="S3.E4.m1.113a"><mtable displaystyle="true" id="S3.E4.m1.113.113.16" rowspacing="0pt" xref="S3.E4.m1.105.105.8.cmml"><mtr id="S3.E4.m1.113.113.16a" xref="S3.E4.m1.105.105.8.cmml"><mtd class="ltx_align_left" columnalign="left" 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id="S3.E4.m1.105.105.8.8.8.1.1.1.3.cmml" xref="S3.E4.m1.113.113.16"><csymbol cd="ambiguous" id="S3.E4.m1.105.105.8.8.8.1.1.1.3.1.cmml" xref="S3.E4.m1.113.113.16">subscript</csymbol><ci id="S3.E4.m1.88.88.88.14.14.14.cmml" xref="S3.E4.m1.88.88.88.14.14.14">𝛼</ci><ci id="S3.E4.m1.89.89.89.15.15.15.1.cmml" xref="S3.E4.m1.89.89.89.15.15.15.1">𝐵</ci></apply><apply id="S3.E4.m1.105.105.8.8.8.1.1.1.1.1.1.cmml" xref="S3.E4.m1.113.113.16"><csymbol cd="ambiguous" id="S3.E4.m1.105.105.8.8.8.1.1.1.1.1.1.1.cmml" xref="S3.E4.m1.113.113.16">subscript</csymbol><ci id="S3.E4.m1.91.91.91.17.17.17.cmml" xref="S3.E4.m1.91.91.91.17.17.17">𝐱</ci><cn id="S3.E4.m1.92.92.92.18.18.18.1.cmml" type="integer" xref="S3.E4.m1.92.92.92.18.18.18.1">4</cn></apply></apply><apply id="S3.E4.m1.105.105.8.8.8.1.1.3.cmml" xref="S3.E4.m1.113.113.16"><csymbol cd="ambiguous" id="S3.E4.m1.105.105.8.8.8.1.1.3.1.cmml" xref="S3.E4.m1.113.113.16">subscript</csymbol><ci id="S3.E4.m1.95.95.95.21.21.21.cmml" xref="S3.E4.m1.95.95.95.21.21.21">italic-ϕ</ci><list id="S3.E4.m1.96.96.96.22.22.22.1.3.cmml" xref="S3.E4.m1.96.96.96.22.22.22.1.4"><ci id="S3.E4.m1.96.96.96.22.22.22.1.1.cmml" xref="S3.E4.m1.96.96.96.22.22.22.1.1">𝐵</ci><cn id="S3.E4.m1.96.96.96.22.22.22.1.2.cmml" type="integer" xref="S3.E4.m1.96.96.96.22.22.22.1.2">1</cn></list></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E4.m1.113c">\mathcal{L}=|\alpha_{A}(\mathbf{x}_{1})-\phi_{A,0}|+|\alpha_{B}(\mathbf{x}_{1}% )-\phi_{B,0}|+\\ |\alpha_{A}(\mathbf{x}_{2})-\phi_{A,0}|+|\alpha_{B}(\mathbf{x}_{2})-\phi_{B,1}% |+\\ |\alpha_{A}(\mathbf{x}_{3})-\phi_{A,1}|+|\alpha_{B}(\mathbf{x}_{3})-\phi_{B,0}% |+\\ |\alpha_{A}(\mathbf{x}_{4})-\phi_{A,1}|+|\alpha_{B}(\mathbf{x}_{4})-\phi_{B,1}|</annotation><annotation encoding="application/x-llamapun" id="S3.E4.m1.113d">start_ROW start_CELL caligraphic_L = | italic_α start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) - italic_ϕ start_POSTSUBSCRIPT italic_A , 0 end_POSTSUBSCRIPT | + | italic_α start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) - italic_ϕ start_POSTSUBSCRIPT italic_B , 0 end_POSTSUBSCRIPT | + end_CELL end_ROW start_ROW start_CELL | italic_α start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) - italic_ϕ start_POSTSUBSCRIPT italic_A , 0 end_POSTSUBSCRIPT | + | italic_α start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) - italic_ϕ start_POSTSUBSCRIPT italic_B , 1 end_POSTSUBSCRIPT | + end_CELL end_ROW start_ROW start_CELL | italic_α start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) - italic_ϕ start_POSTSUBSCRIPT italic_A , 1 end_POSTSUBSCRIPT | + | italic_α start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) - italic_ϕ start_POSTSUBSCRIPT italic_B , 0 end_POSTSUBSCRIPT | + end_CELL end_ROW start_ROW start_CELL | italic_α start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ) - italic_ϕ start_POSTSUBSCRIPT italic_A , 1 end_POSTSUBSCRIPT | + | italic_α start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( bold_x start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ) - italic_ϕ start_POSTSUBSCRIPT italic_B , 1 end_POSTSUBSCRIPT | end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(4)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS2.p3.11">where <math alttext="\mathbf{x}_{i}" class="ltx_Math" display="inline" id="S3.SS2.p3.2.m1.1"><semantics id="S3.SS2.p3.2.m1.1a"><msub id="S3.SS2.p3.2.m1.1.1" xref="S3.SS2.p3.2.m1.1.1.cmml"><mi id="S3.SS2.p3.2.m1.1.1.2" xref="S3.SS2.p3.2.m1.1.1.2.cmml">𝐱</mi><mi id="S3.SS2.p3.2.m1.1.1.3" xref="S3.SS2.p3.2.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.2.m1.1b"><apply id="S3.SS2.p3.2.m1.1.1.cmml" xref="S3.SS2.p3.2.m1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p3.2.m1.1.1.1.cmml" xref="S3.SS2.p3.2.m1.1.1">subscript</csymbol><ci id="S3.SS2.p3.2.m1.1.1.2.cmml" xref="S3.SS2.p3.2.m1.1.1.2">𝐱</ci><ci id="S3.SS2.p3.2.m1.1.1.3.cmml" xref="S3.SS2.p3.2.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.2.m1.1c">\mathbf{x}_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.2.m1.1d">bold_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> represents the four different spatial masks, <math alttext="\alpha" class="ltx_Math" display="inline" id="S3.SS2.p3.3.m2.1"><semantics id="S3.SS2.p3.3.m2.1a"><mi id="S3.SS2.p3.3.m2.1.1" xref="S3.SS2.p3.3.m2.1.1.cmml">α</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.3.m2.1b"><ci id="S3.SS2.p3.3.m2.1.1.cmml" xref="S3.SS2.p3.3.m2.1.1">𝛼</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.3.m2.1c">\alpha</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.3.m2.1d">italic_α</annotation></semantics></math> represents the backscatter phase, <math alttext="\phi_{\{0,1\}}" class="ltx_Math" display="inline" id="S3.SS2.p3.4.m3.2"><semantics id="S3.SS2.p3.4.m3.2a"><msub id="S3.SS2.p3.4.m3.2.3" xref="S3.SS2.p3.4.m3.2.3.cmml"><mi id="S3.SS2.p3.4.m3.2.3.2" xref="S3.SS2.p3.4.m3.2.3.2.cmml">ϕ</mi><mrow id="S3.SS2.p3.4.m3.2.2.2.4" xref="S3.SS2.p3.4.m3.2.2.2.3.cmml"><mo id="S3.SS2.p3.4.m3.2.2.2.4.1" stretchy="false" xref="S3.SS2.p3.4.m3.2.2.2.3.cmml">{</mo><mn id="S3.SS2.p3.4.m3.1.1.1.1" xref="S3.SS2.p3.4.m3.1.1.1.1.cmml">0</mn><mo id="S3.SS2.p3.4.m3.2.2.2.4.2" xref="S3.SS2.p3.4.m3.2.2.2.3.cmml">,</mo><mn id="S3.SS2.p3.4.m3.2.2.2.2" xref="S3.SS2.p3.4.m3.2.2.2.2.cmml">1</mn><mo id="S3.SS2.p3.4.m3.2.2.2.4.3" stretchy="false" xref="S3.SS2.p3.4.m3.2.2.2.3.cmml">}</mo></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.4.m3.2b"><apply id="S3.SS2.p3.4.m3.2.3.cmml" xref="S3.SS2.p3.4.m3.2.3"><csymbol cd="ambiguous" id="S3.SS2.p3.4.m3.2.3.1.cmml" xref="S3.SS2.p3.4.m3.2.3">subscript</csymbol><ci id="S3.SS2.p3.4.m3.2.3.2.cmml" xref="S3.SS2.p3.4.m3.2.3.2">italic-ϕ</ci><set id="S3.SS2.p3.4.m3.2.2.2.3.cmml" xref="S3.SS2.p3.4.m3.2.2.2.4"><cn id="S3.SS2.p3.4.m3.1.1.1.1.cmml" type="integer" xref="S3.SS2.p3.4.m3.1.1.1.1">0</cn><cn id="S3.SS2.p3.4.m3.2.2.2.2.cmml" type="integer" xref="S3.SS2.p3.4.m3.2.2.2.2">1</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.4.m3.2c">\phi_{\{0,1\}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.4.m3.2d">italic_ϕ start_POSTSUBSCRIPT { 0 , 1 } end_POSTSUBSCRIPT</annotation></semantics></math> is the desired phase for bits 0 and 1, and the subscripts <math alttext="A" class="ltx_Math" display="inline" id="S3.SS2.p3.5.m4.1"><semantics id="S3.SS2.p3.5.m4.1a"><mi id="S3.SS2.p3.5.m4.1.1" xref="S3.SS2.p3.5.m4.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.5.m4.1b"><ci id="S3.SS2.p3.5.m4.1.1.cmml" xref="S3.SS2.p3.5.m4.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.5.m4.1c">A</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.5.m4.1d">italic_A</annotation></semantics></math> and <math alttext="B" class="ltx_Math" display="inline" id="S3.SS2.p3.6.m5.1"><semantics id="S3.SS2.p3.6.m5.1a"><mi id="S3.SS2.p3.6.m5.1.1" xref="S3.SS2.p3.6.m5.1.1.cmml">B</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.6.m5.1b"><ci id="S3.SS2.p3.6.m5.1.1.cmml" xref="S3.SS2.p3.6.m5.1.1">𝐵</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.6.m5.1c">B</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.6.m5.1d">italic_B</annotation></semantics></math> represent Alice and Bob, respectively. In BPSK, the bits should be <math alttext="\pi" class="ltx_Math" display="inline" id="S3.SS2.p3.7.m6.1"><semantics id="S3.SS2.p3.7.m6.1a"><mi id="S3.SS2.p3.7.m6.1.1" xref="S3.SS2.p3.7.m6.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.7.m6.1b"><ci id="S3.SS2.p3.7.m6.1.1.cmml" xref="S3.SS2.p3.7.m6.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.7.m6.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.7.m6.1d">italic_π</annotation></semantics></math> phase apart such that <math alttext="\phi_{1}=\phi_{0}+\pi" class="ltx_Math" display="inline" id="S3.SS2.p3.8.m7.1"><semantics id="S3.SS2.p3.8.m7.1a"><mrow id="S3.SS2.p3.8.m7.1.1" xref="S3.SS2.p3.8.m7.1.1.cmml"><msub id="S3.SS2.p3.8.m7.1.1.2" xref="S3.SS2.p3.8.m7.1.1.2.cmml"><mi id="S3.SS2.p3.8.m7.1.1.2.2" xref="S3.SS2.p3.8.m7.1.1.2.2.cmml">ϕ</mi><mn id="S3.SS2.p3.8.m7.1.1.2.3" xref="S3.SS2.p3.8.m7.1.1.2.3.cmml">1</mn></msub><mo id="S3.SS2.p3.8.m7.1.1.1" xref="S3.SS2.p3.8.m7.1.1.1.cmml">=</mo><mrow id="S3.SS2.p3.8.m7.1.1.3" xref="S3.SS2.p3.8.m7.1.1.3.cmml"><msub id="S3.SS2.p3.8.m7.1.1.3.2" xref="S3.SS2.p3.8.m7.1.1.3.2.cmml"><mi id="S3.SS2.p3.8.m7.1.1.3.2.2" xref="S3.SS2.p3.8.m7.1.1.3.2.2.cmml">ϕ</mi><mn id="S3.SS2.p3.8.m7.1.1.3.2.3" xref="S3.SS2.p3.8.m7.1.1.3.2.3.cmml">0</mn></msub><mo id="S3.SS2.p3.8.m7.1.1.3.1" xref="S3.SS2.p3.8.m7.1.1.3.1.cmml">+</mo><mi id="S3.SS2.p3.8.m7.1.1.3.3" xref="S3.SS2.p3.8.m7.1.1.3.3.cmml">π</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.8.m7.1b"><apply id="S3.SS2.p3.8.m7.1.1.cmml" xref="S3.SS2.p3.8.m7.1.1"><eq id="S3.SS2.p3.8.m7.1.1.1.cmml" xref="S3.SS2.p3.8.m7.1.1.1"></eq><apply id="S3.SS2.p3.8.m7.1.1.2.cmml" xref="S3.SS2.p3.8.m7.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.p3.8.m7.1.1.2.1.cmml" xref="S3.SS2.p3.8.m7.1.1.2">subscript</csymbol><ci id="S3.SS2.p3.8.m7.1.1.2.2.cmml" xref="S3.SS2.p3.8.m7.1.1.2.2">italic-ϕ</ci><cn id="S3.SS2.p3.8.m7.1.1.2.3.cmml" type="integer" xref="S3.SS2.p3.8.m7.1.1.2.3">1</cn></apply><apply id="S3.SS2.p3.8.m7.1.1.3.cmml" xref="S3.SS2.p3.8.m7.1.1.3"><plus id="S3.SS2.p3.8.m7.1.1.3.1.cmml" xref="S3.SS2.p3.8.m7.1.1.3.1"></plus><apply id="S3.SS2.p3.8.m7.1.1.3.2.cmml" xref="S3.SS2.p3.8.m7.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS2.p3.8.m7.1.1.3.2.1.cmml" xref="S3.SS2.p3.8.m7.1.1.3.2">subscript</csymbol><ci id="S3.SS2.p3.8.m7.1.1.3.2.2.cmml" xref="S3.SS2.p3.8.m7.1.1.3.2.2">italic-ϕ</ci><cn id="S3.SS2.p3.8.m7.1.1.3.2.3.cmml" type="integer" xref="S3.SS2.p3.8.m7.1.1.3.2.3">0</cn></apply><ci id="S3.SS2.p3.8.m7.1.1.3.3.cmml" xref="S3.SS2.p3.8.m7.1.1.3.3">𝜋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.8.m7.1c">\phi_{1}=\phi_{0}+\pi</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.8.m7.1d">italic_ϕ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_ϕ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_π</annotation></semantics></math>. Note that in general, <math alttext="\phi_{A}" class="ltx_Math" display="inline" id="S3.SS2.p3.9.m8.1"><semantics id="S3.SS2.p3.9.m8.1a"><msub id="S3.SS2.p3.9.m8.1.1" xref="S3.SS2.p3.9.m8.1.1.cmml"><mi id="S3.SS2.p3.9.m8.1.1.2" xref="S3.SS2.p3.9.m8.1.1.2.cmml">ϕ</mi><mi id="S3.SS2.p3.9.m8.1.1.3" xref="S3.SS2.p3.9.m8.1.1.3.cmml">A</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.9.m8.1b"><apply id="S3.SS2.p3.9.m8.1.1.cmml" xref="S3.SS2.p3.9.m8.1.1"><csymbol cd="ambiguous" id="S3.SS2.p3.9.m8.1.1.1.cmml" xref="S3.SS2.p3.9.m8.1.1">subscript</csymbol><ci id="S3.SS2.p3.9.m8.1.1.2.cmml" xref="S3.SS2.p3.9.m8.1.1.2">italic-ϕ</ci><ci id="S3.SS2.p3.9.m8.1.1.3.cmml" xref="S3.SS2.p3.9.m8.1.1.3">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.9.m8.1c">\phi_{A}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.9.m8.1d">italic_ϕ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT</annotation></semantics></math> does not necessarily equal <math alttext="\phi_{B}" class="ltx_Math" display="inline" id="S3.SS2.p3.10.m9.1"><semantics id="S3.SS2.p3.10.m9.1a"><msub id="S3.SS2.p3.10.m9.1.1" xref="S3.SS2.p3.10.m9.1.1.cmml"><mi id="S3.SS2.p3.10.m9.1.1.2" xref="S3.SS2.p3.10.m9.1.1.2.cmml">ϕ</mi><mi id="S3.SS2.p3.10.m9.1.1.3" xref="S3.SS2.p3.10.m9.1.1.3.cmml">B</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.10.m9.1b"><apply id="S3.SS2.p3.10.m9.1.1.cmml" xref="S3.SS2.p3.10.m9.1.1"><csymbol cd="ambiguous" id="S3.SS2.p3.10.m9.1.1.1.cmml" xref="S3.SS2.p3.10.m9.1.1">subscript</csymbol><ci id="S3.SS2.p3.10.m9.1.1.2.cmml" xref="S3.SS2.p3.10.m9.1.1.2">italic-ϕ</ci><ci id="S3.SS2.p3.10.m9.1.1.3.cmml" xref="S3.SS2.p3.10.m9.1.1.3">𝐵</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.10.m9.1c">\phi_{B}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.10.m9.1d">italic_ϕ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT</annotation></semantics></math> since these are separate communication channels. Additionally, the desired phase <math alttext="\phi_{0}" class="ltx_Math" display="inline" id="S3.SS2.p3.11.m10.1"><semantics id="S3.SS2.p3.11.m10.1a"><msub id="S3.SS2.p3.11.m10.1.1" xref="S3.SS2.p3.11.m10.1.1.cmml"><mi id="S3.SS2.p3.11.m10.1.1.2" xref="S3.SS2.p3.11.m10.1.1.2.cmml">ϕ</mi><mn id="S3.SS2.p3.11.m10.1.1.3" xref="S3.SS2.p3.11.m10.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.11.m10.1b"><apply id="S3.SS2.p3.11.m10.1.1.cmml" xref="S3.SS2.p3.11.m10.1.1"><csymbol cd="ambiguous" id="S3.SS2.p3.11.m10.1.1.1.cmml" xref="S3.SS2.p3.11.m10.1.1">subscript</csymbol><ci id="S3.SS2.p3.11.m10.1.1.2.cmml" xref="S3.SS2.p3.11.m10.1.1.2">italic-ϕ</ci><cn id="S3.SS2.p3.11.m10.1.1.3.cmml" type="integer" xref="S3.SS2.p3.11.m10.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.11.m10.1c">\phi_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.11.m10.1d">italic_ϕ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is also a free parameter.</p> </div> <div class="ltx_para" id="S3.SS2.p4"> <p class="ltx_p" id="S3.SS2.p4.5">A constellation diagram of a candidate set of masks at 3.76 GHz is shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#S3.F6" title="Figure 6 ‣ III-B Multiplexed Communications ‣ III Results ‣ Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_tag">6</span></a>(a) where the different marker colors or shapes correspond to different spatial masks, <math alttext="x_{i}" class="ltx_Math" display="inline" id="S3.SS2.p4.1.m1.1"><semantics id="S3.SS2.p4.1.m1.1a"><msub id="S3.SS2.p4.1.m1.1.1" xref="S3.SS2.p4.1.m1.1.1.cmml"><mi id="S3.SS2.p4.1.m1.1.1.2" xref="S3.SS2.p4.1.m1.1.1.2.cmml">x</mi><mi id="S3.SS2.p4.1.m1.1.1.3" xref="S3.SS2.p4.1.m1.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.1.m1.1b"><apply id="S3.SS2.p4.1.m1.1.1.cmml" xref="S3.SS2.p4.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p4.1.m1.1.1.1.cmml" xref="S3.SS2.p4.1.m1.1.1">subscript</csymbol><ci id="S3.SS2.p4.1.m1.1.1.2.cmml" xref="S3.SS2.p4.1.m1.1.1.2">𝑥</ci><ci id="S3.SS2.p4.1.m1.1.1.3.cmml" xref="S3.SS2.p4.1.m1.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.1.m1.1c">x_{i}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.1.m1.1d">italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>. Each user clearly has bits that are <math alttext="\pi" class="ltx_Math" display="inline" id="S3.SS2.p4.2.m2.1"><semantics id="S3.SS2.p4.2.m2.1a"><mi id="S3.SS2.p4.2.m2.1.1" xref="S3.SS2.p4.2.m2.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.2.m2.1b"><ci id="S3.SS2.p4.2.m2.1.1.cmml" xref="S3.SS2.p4.2.m2.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.2.m2.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.2.m2.1d">italic_π</annotation></semantics></math> phase apart, and there exists a mask for every combination of bit pairs to Alice and Bob. For example, if the green triangle represents <math alttext="(A=0,B=0)" class="ltx_Math" display="inline" id="S3.SS2.p4.3.m3.1"><semantics id="S3.SS2.p4.3.m3.1a"><mrow id="S3.SS2.p4.3.m3.1.1.1"><mo id="S3.SS2.p4.3.m3.1.1.1.2" stretchy="false">(</mo><mrow id="S3.SS2.p4.3.m3.1.1.1.1.2" xref="S3.SS2.p4.3.m3.1.1.1.1.3.cmml"><mrow id="S3.SS2.p4.3.m3.1.1.1.1.1.1" xref="S3.SS2.p4.3.m3.1.1.1.1.1.1.cmml"><mi id="S3.SS2.p4.3.m3.1.1.1.1.1.1.2" xref="S3.SS2.p4.3.m3.1.1.1.1.1.1.2.cmml">A</mi><mo id="S3.SS2.p4.3.m3.1.1.1.1.1.1.1" xref="S3.SS2.p4.3.m3.1.1.1.1.1.1.1.cmml">=</mo><mn id="S3.SS2.p4.3.m3.1.1.1.1.1.1.3" xref="S3.SS2.p4.3.m3.1.1.1.1.1.1.3.cmml">0</mn></mrow><mo id="S3.SS2.p4.3.m3.1.1.1.1.2.3" xref="S3.SS2.p4.3.m3.1.1.1.1.3a.cmml">,</mo><mrow id="S3.SS2.p4.3.m3.1.1.1.1.2.2" xref="S3.SS2.p4.3.m3.1.1.1.1.2.2.cmml"><mi id="S3.SS2.p4.3.m3.1.1.1.1.2.2.2" xref="S3.SS2.p4.3.m3.1.1.1.1.2.2.2.cmml">B</mi><mo id="S3.SS2.p4.3.m3.1.1.1.1.2.2.1" xref="S3.SS2.p4.3.m3.1.1.1.1.2.2.1.cmml">=</mo><mn id="S3.SS2.p4.3.m3.1.1.1.1.2.2.3" xref="S3.SS2.p4.3.m3.1.1.1.1.2.2.3.cmml">0</mn></mrow></mrow><mo id="S3.SS2.p4.3.m3.1.1.1.3" stretchy="false">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.3.m3.1b"><apply id="S3.SS2.p4.3.m3.1.1.1.1.3.cmml" xref="S3.SS2.p4.3.m3.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.p4.3.m3.1.1.1.1.3a.cmml" xref="S3.SS2.p4.3.m3.1.1.1.1.2.3">formulae-sequence</csymbol><apply id="S3.SS2.p4.3.m3.1.1.1.1.1.1.cmml" xref="S3.SS2.p4.3.m3.1.1.1.1.1.1"><eq id="S3.SS2.p4.3.m3.1.1.1.1.1.1.1.cmml" xref="S3.SS2.p4.3.m3.1.1.1.1.1.1.1"></eq><ci id="S3.SS2.p4.3.m3.1.1.1.1.1.1.2.cmml" xref="S3.SS2.p4.3.m3.1.1.1.1.1.1.2">𝐴</ci><cn id="S3.SS2.p4.3.m3.1.1.1.1.1.1.3.cmml" type="integer" xref="S3.SS2.p4.3.m3.1.1.1.1.1.1.3">0</cn></apply><apply id="S3.SS2.p4.3.m3.1.1.1.1.2.2.cmml" xref="S3.SS2.p4.3.m3.1.1.1.1.2.2"><eq id="S3.SS2.p4.3.m3.1.1.1.1.2.2.1.cmml" xref="S3.SS2.p4.3.m3.1.1.1.1.2.2.1"></eq><ci id="S3.SS2.p4.3.m3.1.1.1.1.2.2.2.cmml" xref="S3.SS2.p4.3.m3.1.1.1.1.2.2.2">𝐵</ci><cn id="S3.SS2.p4.3.m3.1.1.1.1.2.2.3.cmml" type="integer" xref="S3.SS2.p4.3.m3.1.1.1.1.2.2.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.3.m3.1c">(A=0,B=0)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.3.m3.1d">( italic_A = 0 , italic_B = 0 )</annotation></semantics></math>, then the red diamond represents <math alttext="(A=0,B=1)" class="ltx_Math" display="inline" id="S3.SS2.p4.4.m4.1"><semantics id="S3.SS2.p4.4.m4.1a"><mrow id="S3.SS2.p4.4.m4.1.1.1"><mo id="S3.SS2.p4.4.m4.1.1.1.2" stretchy="false">(</mo><mrow id="S3.SS2.p4.4.m4.1.1.1.1.2" xref="S3.SS2.p4.4.m4.1.1.1.1.3.cmml"><mrow id="S3.SS2.p4.4.m4.1.1.1.1.1.1" xref="S3.SS2.p4.4.m4.1.1.1.1.1.1.cmml"><mi id="S3.SS2.p4.4.m4.1.1.1.1.1.1.2" xref="S3.SS2.p4.4.m4.1.1.1.1.1.1.2.cmml">A</mi><mo id="S3.SS2.p4.4.m4.1.1.1.1.1.1.1" xref="S3.SS2.p4.4.m4.1.1.1.1.1.1.1.cmml">=</mo><mn id="S3.SS2.p4.4.m4.1.1.1.1.1.1.3" xref="S3.SS2.p4.4.m4.1.1.1.1.1.1.3.cmml">0</mn></mrow><mo id="S3.SS2.p4.4.m4.1.1.1.1.2.3" xref="S3.SS2.p4.4.m4.1.1.1.1.3a.cmml">,</mo><mrow id="S3.SS2.p4.4.m4.1.1.1.1.2.2" xref="S3.SS2.p4.4.m4.1.1.1.1.2.2.cmml"><mi id="S3.SS2.p4.4.m4.1.1.1.1.2.2.2" xref="S3.SS2.p4.4.m4.1.1.1.1.2.2.2.cmml">B</mi><mo id="S3.SS2.p4.4.m4.1.1.1.1.2.2.1" xref="S3.SS2.p4.4.m4.1.1.1.1.2.2.1.cmml">=</mo><mn id="S3.SS2.p4.4.m4.1.1.1.1.2.2.3" xref="S3.SS2.p4.4.m4.1.1.1.1.2.2.3.cmml">1</mn></mrow></mrow><mo id="S3.SS2.p4.4.m4.1.1.1.3" stretchy="false">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p4.4.m4.1b"><apply id="S3.SS2.p4.4.m4.1.1.1.1.3.cmml" xref="S3.SS2.p4.4.m4.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.p4.4.m4.1.1.1.1.3a.cmml" xref="S3.SS2.p4.4.m4.1.1.1.1.2.3">formulae-sequence</csymbol><apply id="S3.SS2.p4.4.m4.1.1.1.1.1.1.cmml" xref="S3.SS2.p4.4.m4.1.1.1.1.1.1"><eq id="S3.SS2.p4.4.m4.1.1.1.1.1.1.1.cmml" xref="S3.SS2.p4.4.m4.1.1.1.1.1.1.1"></eq><ci 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id="S3.SS2.p4.5.m5.1b"><apply id="S3.SS2.p4.5.m5.1.1.1.1.3.cmml" xref="S3.SS2.p4.5.m5.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.p4.5.m5.1.1.1.1.3a.cmml" xref="S3.SS2.p4.5.m5.1.1.1.1.2.3">formulae-sequence</csymbol><apply id="S3.SS2.p4.5.m5.1.1.1.1.1.1.cmml" xref="S3.SS2.p4.5.m5.1.1.1.1.1.1"><eq id="S3.SS2.p4.5.m5.1.1.1.1.1.1.1.cmml" xref="S3.SS2.p4.5.m5.1.1.1.1.1.1.1"></eq><ci id="S3.SS2.p4.5.m5.1.1.1.1.1.1.2.cmml" xref="S3.SS2.p4.5.m5.1.1.1.1.1.1.2">𝐴</ci><cn id="S3.SS2.p4.5.m5.1.1.1.1.1.1.3.cmml" type="integer" xref="S3.SS2.p4.5.m5.1.1.1.1.1.1.3">1</cn></apply><apply id="S3.SS2.p4.5.m5.1.1.1.1.2.2.cmml" xref="S3.SS2.p4.5.m5.1.1.1.1.2.2"><eq id="S3.SS2.p4.5.m5.1.1.1.1.2.2.1.cmml" xref="S3.SS2.p4.5.m5.1.1.1.1.2.2.1"></eq><ci id="S3.SS2.p4.5.m5.1.1.1.1.2.2.2.cmml" xref="S3.SS2.p4.5.m5.1.1.1.1.2.2.2">𝐵</ci><cn id="S3.SS2.p4.5.m5.1.1.1.1.2.2.3.cmml" type="integer" xref="S3.SS2.p4.5.m5.1.1.1.1.2.2.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p4.5.m5.1c">(A=1,B=0)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p4.5.m5.1d">( italic_A = 1 , italic_B = 0 )</annotation></semantics></math>. Note that the relative phase of Alice and Bob is arbitrary. This result demonstrates that the MBLL is able to communicate two separate bit streams for the two users.</p> </div> <div class="ltx_para" id="S3.SS2.p5"> <p class="ltx_p" id="S3.SS2.p5.1">Extending upon this, we also demonstrate the potential for quadrature phase-shift keying (QPSK) in which the communications relies on four points equispaced around a circle in the constellation diagram. Each point in the constellation diagram encodes two bits, thus doubling the data rate of the signal for a constant bandwidth. Using a similar optimization scheme as before over the 5000 random masks, we find 16 different masks that can communicate every possible pair of constellation points to Alice and Bob. The resulting constellation plots are shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#S3.F6" title="Figure 6 ‣ III-B Multiplexed Communications ‣ III Results ‣ Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_tag">6</span></a>(b). Here, each pair of marker face color and line color represents a unique mask. For example, the red line with a green marker face represents the same mask for Alice and Bob. We see that all of the 4 points with a red line represent the same bit for Alice, and each of these 4 points corresponds to a unique pair of bits for Bob. Similarly, the 4 points with green face color represents every possible pair of bits for Alice, and yet corresponds to the same bit pair for Bob. Thus, these 16 spatial masks can be used to encode QPSK for Alice and Bob simultaneously.</p> </div> <div class="ltx_para" id="S3.SS2.p6"> <p class="ltx_p" id="S3.SS2.p6.1">Again, the results here are based on randomly sampled metasurface masks, but more sophisticated sampling or optimization methods may be applied to further improve the results. For example, the MBLL can be likely be extended to higher-order phase modulation techniques for even higher data rates, or multiplex to more than two users. In a simplified model of the MBLL, we can imagine modulating the unit cells only at the focal spot, and thus enabling spatial multiplexing with the number of channels on the order of (metasurface area) <math alttext="/" class="ltx_Math" display="inline" id="S3.SS2.p6.1.m1.1"><semantics id="S3.SS2.p6.1.m1.1a"><mo id="S3.SS2.p6.1.m1.1.1" xref="S3.SS2.p6.1.m1.1.1.cmml">/</mo><annotation-xml encoding="MathML-Content" id="S3.SS2.p6.1.m1.1b"><divide id="S3.SS2.p6.1.m1.1.1.cmml" xref="S3.SS2.p6.1.m1.1.1"></divide></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p6.1.m1.1c">/</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p6.1.m1.1d">/</annotation></semantics></math> (focal spot size). As we have seen from the results in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#S3.SS1" title="III-A Full phase control ‣ III Results ‣ Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_tag"><span class="ltx_text">III-A</span></span></a>, this is indeed a simplified picture as the performance can be improved through the use of spatially varying metasurface masks.</p> </div> <figure class="ltx_figure" id="S3.F7"><img alt="Refer to caption" class="ltx_graphics ltx_centering ltx_img_landscape" height="282" id="S3.F7.g1" src="x7.png" width="830"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 7: </span>Demonstration of the MBLL for secure communications (to Alice) in the presence of an eavesdropper (Eve). (a) Constellation diagram corresponding to a pair of masks that maximize the phase variation for Alice while minimizing the backscatter signal for Eve. In the first configuration, Alice and Eve are at <math alttext="15\degree" class="ltx_Math" display="inline" id="S3.F7.9.m1.1"><semantics id="S3.F7.9.m1.1b"><mrow id="S3.F7.9.m1.1.1" xref="S3.F7.9.m1.1.1.cmml"><mn id="S3.F7.9.m1.1.1.2" xref="S3.F7.9.m1.1.1.2.cmml">15</mn><mo id="S3.F7.9.m1.1.1.1" xref="S3.F7.9.m1.1.1.1.cmml">⁢</mo><mi id="S3.F7.9.m1.1.1.3" mathvariant="normal" xref="S3.F7.9.m1.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.F7.9.m1.1c"><apply id="S3.F7.9.m1.1.1.cmml" xref="S3.F7.9.m1.1.1"><times id="S3.F7.9.m1.1.1.1.cmml" xref="S3.F7.9.m1.1.1.1"></times><cn id="S3.F7.9.m1.1.1.2.cmml" type="integer" xref="S3.F7.9.m1.1.1.2">15</cn><ci id="S3.F7.9.m1.1.1.3.cmml" xref="S3.F7.9.m1.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F7.9.m1.1d">15\degree</annotation><annotation encoding="application/x-llamapun" id="S3.F7.9.m1.1e">15 °</annotation></semantics></math> and <math alttext="0\degree" class="ltx_Math" display="inline" id="S3.F7.10.m2.1"><semantics id="S3.F7.10.m2.1b"><mrow id="S3.F7.10.m2.1.1" xref="S3.F7.10.m2.1.1.cmml"><mn id="S3.F7.10.m2.1.1.2" xref="S3.F7.10.m2.1.1.2.cmml">0</mn><mo id="S3.F7.10.m2.1.1.1" xref="S3.F7.10.m2.1.1.1.cmml">⁢</mo><mi id="S3.F7.10.m2.1.1.3" mathvariant="normal" xref="S3.F7.10.m2.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.F7.10.m2.1c"><apply id="S3.F7.10.m2.1.1.cmml" xref="S3.F7.10.m2.1.1"><times id="S3.F7.10.m2.1.1.1.cmml" xref="S3.F7.10.m2.1.1.1"></times><cn id="S3.F7.10.m2.1.1.2.cmml" type="integer" xref="S3.F7.10.m2.1.1.2">0</cn><ci id="S3.F7.10.m2.1.1.3.cmml" xref="S3.F7.10.m2.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F7.10.m2.1d">0\degree</annotation><annotation encoding="application/x-llamapun" id="S3.F7.10.m2.1e">0 °</annotation></semantics></math> incidence, respectively. The masks are found from searching through 5000 random masks at 3.75 GHz. The smaller of Alice’s symbols and larger of Eve’s symbols differ by 32.2dB. In the first configuration, Alice and Eve are at <math alttext="20\degree" class="ltx_Math" display="inline" id="S3.F7.11.m3.1"><semantics id="S3.F7.11.m3.1b"><mrow id="S3.F7.11.m3.1.1" xref="S3.F7.11.m3.1.1.cmml"><mn id="S3.F7.11.m3.1.1.2" xref="S3.F7.11.m3.1.1.2.cmml">20</mn><mo id="S3.F7.11.m3.1.1.1" xref="S3.F7.11.m3.1.1.1.cmml">⁢</mo><mi id="S3.F7.11.m3.1.1.3" mathvariant="normal" xref="S3.F7.11.m3.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.F7.11.m3.1c"><apply id="S3.F7.11.m3.1.1.cmml" xref="S3.F7.11.m3.1.1"><times id="S3.F7.11.m3.1.1.1.cmml" xref="S3.F7.11.m3.1.1.1"></times><cn id="S3.F7.11.m3.1.1.2.cmml" type="integer" xref="S3.F7.11.m3.1.1.2">20</cn><ci id="S3.F7.11.m3.1.1.3.cmml" xref="S3.F7.11.m3.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F7.11.m3.1d">20\degree</annotation><annotation encoding="application/x-llamapun" id="S3.F7.11.m3.1e">20 °</annotation></semantics></math> and <math alttext="7\degree" class="ltx_Math" display="inline" id="S3.F7.12.m4.1"><semantics id="S3.F7.12.m4.1b"><mrow id="S3.F7.12.m4.1.1" xref="S3.F7.12.m4.1.1.cmml"><mn id="S3.F7.12.m4.1.1.2" xref="S3.F7.12.m4.1.1.2.cmml">7</mn><mo id="S3.F7.12.m4.1.1.1" xref="S3.F7.12.m4.1.1.1.cmml">⁢</mo><mi id="S3.F7.12.m4.1.1.3" mathvariant="normal" xref="S3.F7.12.m4.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.F7.12.m4.1c"><apply id="S3.F7.12.m4.1.1.cmml" xref="S3.F7.12.m4.1.1"><times id="S3.F7.12.m4.1.1.1.cmml" xref="S3.F7.12.m4.1.1.1"></times><cn id="S3.F7.12.m4.1.1.2.cmml" type="integer" xref="S3.F7.12.m4.1.1.2">7</cn><ci id="S3.F7.12.m4.1.1.3.cmml" xref="S3.F7.12.m4.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F7.12.m4.1d">7\degree</annotation><annotation encoding="application/x-llamapun" id="S3.F7.12.m4.1e">7 °</annotation></semantics></math> incidence, respectively. The masks are found from searching through 500 random masks at 3.67 GHz. The smaller of Alice’s symbols and larger of Eve’s symbols differ by 32.2dB. (b) Backscatter phase from 500 random spatial metasurface patterns at various angles. <math alttext="x" class="ltx_Math" display="inline" id="S3.F7.13.m5.1"><semantics id="S3.F7.13.m5.1b"><mi id="S3.F7.13.m5.1.1" xref="S3.F7.13.m5.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.F7.13.m5.1c"><ci id="S3.F7.13.m5.1.1.cmml" xref="S3.F7.13.m5.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F7.13.m5.1d">x</annotation><annotation encoding="application/x-llamapun" id="S3.F7.13.m5.1e">italic_x</annotation></semantics></math>-axis corresponds to Alice and <math alttext="y" class="ltx_Math" display="inline" id="S3.F7.14.m6.1"><semantics id="S3.F7.14.m6.1b"><mi id="S3.F7.14.m6.1.1" xref="S3.F7.14.m6.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S3.F7.14.m6.1c"><ci id="S3.F7.14.m6.1.1.cmml" xref="S3.F7.14.m6.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.F7.14.m6.1d">y</annotation><annotation encoding="application/x-llamapun" id="S3.F7.14.m6.1e">italic_y</annotation></semantics></math>-axis corresponds to Eve, allowing us to see the correlation between the two interrogators. Points are colored according to signal magnitude at Eve. (c) Constellation diagram corresponding to a set of masks that can be used for secure communications, where Alice is at <math alttext="0\degree" class="ltx_Math" display="inline" id="S3.F7.15.m7.1"><semantics id="S3.F7.15.m7.1b"><mrow id="S3.F7.15.m7.1.1" xref="S3.F7.15.m7.1.1.cmml"><mn id="S3.F7.15.m7.1.1.2" xref="S3.F7.15.m7.1.1.2.cmml">0</mn><mo id="S3.F7.15.m7.1.1.1" xref="S3.F7.15.m7.1.1.1.cmml">⁢</mo><mi id="S3.F7.15.m7.1.1.3" mathvariant="normal" xref="S3.F7.15.m7.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.F7.15.m7.1c"><apply id="S3.F7.15.m7.1.1.cmml" xref="S3.F7.15.m7.1.1"><times id="S3.F7.15.m7.1.1.1.cmml" xref="S3.F7.15.m7.1.1.1"></times><cn id="S3.F7.15.m7.1.1.2.cmml" type="integer" xref="S3.F7.15.m7.1.1.2">0</cn><ci id="S3.F7.15.m7.1.1.3.cmml" xref="S3.F7.15.m7.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F7.15.m7.1d">0\degree</annotation><annotation encoding="application/x-llamapun" id="S3.F7.15.m7.1e">0 °</annotation></semantics></math> and Eve is at <math alttext="15\degree" class="ltx_Math" display="inline" id="S3.F7.16.m8.1"><semantics id="S3.F7.16.m8.1b"><mrow id="S3.F7.16.m8.1.1" xref="S3.F7.16.m8.1.1.cmml"><mn id="S3.F7.16.m8.1.1.2" xref="S3.F7.16.m8.1.1.2.cmml">15</mn><mo id="S3.F7.16.m8.1.1.1" xref="S3.F7.16.m8.1.1.1.cmml">⁢</mo><mi id="S3.F7.16.m8.1.1.3" mathvariant="normal" xref="S3.F7.16.m8.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.F7.16.m8.1c"><apply id="S3.F7.16.m8.1.1.cmml" xref="S3.F7.16.m8.1.1"><times id="S3.F7.16.m8.1.1.1.cmml" xref="S3.F7.16.m8.1.1.1"></times><cn id="S3.F7.16.m8.1.1.2.cmml" type="integer" xref="S3.F7.16.m8.1.1.2">15</cn><ci id="S3.F7.16.m8.1.1.3.cmml" xref="S3.F7.16.m8.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.F7.16.m8.1d">15\degree</annotation><annotation encoding="application/x-llamapun" id="S3.F7.16.m8.1e">15 °</annotation></semantics></math>. </figcaption> </figure> </section> <section class="ltx_subsection" id="S3.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection"><span class="ltx_text" id="S3.SS3.5.1.1">III-C</span> </span><span class="ltx_text ltx_font_italic" id="S3.SS3.6.2">Preventing eavesdropping</span> </h3> <div class="ltx_para" id="S3.SS3.p1"> <p class="ltx_p" id="S3.SS3.p1.1">Finally, we consider the scenario where an eavesdropper (Eve) attempts to capture the communication between Alice and the MBLL. Specifically, suppose Alice is interrogating the MBLL from some incidence angle and an eavesdropper is attempting to capture the communication between the MBLL and Alice from a different angular position by detecting the signal from the MBLL towards Eve. We consider two possible methods for eavesdropping: passive and active.</p> </div> <div class="ltx_para" id="S3.SS3.p2"> <p class="ltx_p" id="S3.SS3.p2.3">In passive eavesdropping, Eve is not interrogating the MBLL, but is rather attempting to capture stray reflections from Alice interrogating the MBLL. Although the Luneburg lens ideally provides retroreflection towards the interrogator, there may be sidelobes away from the interrogator due to the finite aperture of the lens. As discussed in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#S2.SS1" title="II-A Luneburg Lens Design ‣ II Methods ‣ Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_tag"><span class="ltx_text">II-A</span></span></a>, the simulated sidelobes for Alice at <math alttext="15\degree" class="ltx_Math" display="inline" id="S3.SS3.p2.1.m1.1"><semantics id="S3.SS3.p2.1.m1.1a"><mrow id="S3.SS3.p2.1.m1.1.1" xref="S3.SS3.p2.1.m1.1.1.cmml"><mn id="S3.SS3.p2.1.m1.1.1.2" xref="S3.SS3.p2.1.m1.1.1.2.cmml">15</mn><mo id="S3.SS3.p2.1.m1.1.1.1" xref="S3.SS3.p2.1.m1.1.1.1.cmml">⁢</mo><mi id="S3.SS3.p2.1.m1.1.1.3" mathvariant="normal" xref="S3.SS3.p2.1.m1.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.1.m1.1b"><apply id="S3.SS3.p2.1.m1.1.1.cmml" xref="S3.SS3.p2.1.m1.1.1"><times id="S3.SS3.p2.1.m1.1.1.1.cmml" xref="S3.SS3.p2.1.m1.1.1.1"></times><cn id="S3.SS3.p2.1.m1.1.1.2.cmml" type="integer" xref="S3.SS3.p2.1.m1.1.1.2">15</cn><ci id="S3.SS3.p2.1.m1.1.1.3.cmml" xref="S3.SS3.p2.1.m1.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.1.m1.1c">15\degree</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.1.m1.1d">15 °</annotation></semantics></math> incidence are generally below <math alttext="-14" class="ltx_Math" display="inline" id="S3.SS3.p2.2.m2.1"><semantics id="S3.SS3.p2.2.m2.1a"><mrow id="S3.SS3.p2.2.m2.1.1" xref="S3.SS3.p2.2.m2.1.1.cmml"><mo id="S3.SS3.p2.2.m2.1.1a" xref="S3.SS3.p2.2.m2.1.1.cmml">−</mo><mn id="S3.SS3.p2.2.m2.1.1.2" xref="S3.SS3.p2.2.m2.1.1.2.cmml">14</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.2.m2.1b"><apply id="S3.SS3.p2.2.m2.1.1.cmml" xref="S3.SS3.p2.2.m2.1.1"><minus id="S3.SS3.p2.2.m2.1.1.1.cmml" xref="S3.SS3.p2.2.m2.1.1"></minus><cn id="S3.SS3.p2.2.m2.1.1.2.cmml" type="integer" xref="S3.SS3.p2.2.m2.1.1.2">14</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.2.m2.1c">-14</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.2.m2.1d">- 14</annotation></semantics></math> dB and reach as low as <math alttext="-32" class="ltx_Math" display="inline" id="S3.SS3.p2.3.m3.1"><semantics id="S3.SS3.p2.3.m3.1a"><mrow id="S3.SS3.p2.3.m3.1.1" xref="S3.SS3.p2.3.m3.1.1.cmml"><mo id="S3.SS3.p2.3.m3.1.1a" xref="S3.SS3.p2.3.m3.1.1.cmml">−</mo><mn id="S3.SS3.p2.3.m3.1.1.2" xref="S3.SS3.p2.3.m3.1.1.2.cmml">32</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p2.3.m3.1b"><apply id="S3.SS3.p2.3.m3.1.1.cmml" xref="S3.SS3.p2.3.m3.1.1"><minus id="S3.SS3.p2.3.m3.1.1.1.cmml" xref="S3.SS3.p2.3.m3.1.1"></minus><cn id="S3.SS3.p2.3.m3.1.1.2.cmml" type="integer" xref="S3.SS3.p2.3.m3.1.1.2">32</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p2.3.m3.1c">-32</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p2.3.m3.1d">- 32</annotation></semantics></math> dB. If the background noise is greater than those sidelobe levels, the eavesdropper will not be able to differentiate the signal from the noise. The sidelobe levels can likely be further suppressed by optimizing the metasurface spatial masks using different objective functions, which we leave for future work.</p> </div> <div class="ltx_para" id="S3.SS3.p3"> <p class="ltx_p" id="S3.SS3.p3.5">In active eavesdropping, Eve is interrogating the MBLL simultaneously with Alice. Since the metasurface is being reconfigured to modulate the backscatter signal to Alice, the reconfiguration may also imprint a modulation on the backscatter to Eve. One possible strategy to secure communications is to find spatial masks that maximize the signal for Alice while minimizing the signal for Eve such that the signal is below the noise threshold. For this experiment, we sample random spatial patterns for the metasurface and measure the backscatter at various angles. In the first configuration, we suppose that Alice is at <math alttext="15\degree" class="ltx_Math" display="inline" id="S3.SS3.p3.1.m1.1"><semantics id="S3.SS3.p3.1.m1.1a"><mrow id="S3.SS3.p3.1.m1.1.1" xref="S3.SS3.p3.1.m1.1.1.cmml"><mn id="S3.SS3.p3.1.m1.1.1.2" xref="S3.SS3.p3.1.m1.1.1.2.cmml">15</mn><mo id="S3.SS3.p3.1.m1.1.1.1" xref="S3.SS3.p3.1.m1.1.1.1.cmml">⁢</mo><mi id="S3.SS3.p3.1.m1.1.1.3" mathvariant="normal" xref="S3.SS3.p3.1.m1.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.1.m1.1b"><apply id="S3.SS3.p3.1.m1.1.1.cmml" xref="S3.SS3.p3.1.m1.1.1"><times id="S3.SS3.p3.1.m1.1.1.1.cmml" xref="S3.SS3.p3.1.m1.1.1.1"></times><cn id="S3.SS3.p3.1.m1.1.1.2.cmml" type="integer" xref="S3.SS3.p3.1.m1.1.1.2">15</cn><ci id="S3.SS3.p3.1.m1.1.1.3.cmml" xref="S3.SS3.p3.1.m1.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.1.m1.1c">15\degree</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.1.m1.1d">15 °</annotation></semantics></math> incidence angle and Eve is at <math alttext="0\degree" class="ltx_Math" display="inline" id="S3.SS3.p3.2.m2.1"><semantics id="S3.SS3.p3.2.m2.1a"><mrow id="S3.SS3.p3.2.m2.1.1" xref="S3.SS3.p3.2.m2.1.1.cmml"><mn id="S3.SS3.p3.2.m2.1.1.2" xref="S3.SS3.p3.2.m2.1.1.2.cmml">0</mn><mo id="S3.SS3.p3.2.m2.1.1.1" xref="S3.SS3.p3.2.m2.1.1.1.cmml">⁢</mo><mi id="S3.SS3.p3.2.m2.1.1.3" mathvariant="normal" xref="S3.SS3.p3.2.m2.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.2.m2.1b"><apply id="S3.SS3.p3.2.m2.1.1.cmml" xref="S3.SS3.p3.2.m2.1.1"><times id="S3.SS3.p3.2.m2.1.1.1.cmml" xref="S3.SS3.p3.2.m2.1.1.1"></times><cn id="S3.SS3.p3.2.m2.1.1.2.cmml" type="integer" xref="S3.SS3.p3.2.m2.1.1.2">0</cn><ci id="S3.SS3.p3.2.m2.1.1.3.cmml" xref="S3.SS3.p3.2.m2.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.2.m2.1c">0\degree</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.2.m2.1d">0 °</annotation></semantics></math> incidence, and search for a pair of masks from 5000 random masks at 3.75 GHz that maximize the signal for Alice—which in the case of BPSK would be two masks with <math alttext="\pi" class="ltx_Math" display="inline" id="S3.SS3.p3.3.m3.1"><semantics id="S3.SS3.p3.3.m3.1a"><mi id="S3.SS3.p3.3.m3.1.1" xref="S3.SS3.p3.3.m3.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.3.m3.1b"><ci id="S3.SS3.p3.3.m3.1.1.cmml" xref="S3.SS3.p3.3.m3.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.3.m3.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.3.m3.1d">italic_π</annotation></semantics></math> phase difference—while maximizing the power ratio of Alice’s backscatter to Eve’s backscatter signal. In the second configuration, we suppose that Alice is at <math alttext="25\degree" class="ltx_Math" display="inline" id="S3.SS3.p3.4.m4.1"><semantics id="S3.SS3.p3.4.m4.1a"><mrow id="S3.SS3.p3.4.m4.1.1" xref="S3.SS3.p3.4.m4.1.1.cmml"><mn id="S3.SS3.p3.4.m4.1.1.2" xref="S3.SS3.p3.4.m4.1.1.2.cmml">25</mn><mo id="S3.SS3.p3.4.m4.1.1.1" xref="S3.SS3.p3.4.m4.1.1.1.cmml">⁢</mo><mi id="S3.SS3.p3.4.m4.1.1.3" mathvariant="normal" xref="S3.SS3.p3.4.m4.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.4.m4.1b"><apply id="S3.SS3.p3.4.m4.1.1.cmml" xref="S3.SS3.p3.4.m4.1.1"><times id="S3.SS3.p3.4.m4.1.1.1.cmml" xref="S3.SS3.p3.4.m4.1.1.1"></times><cn id="S3.SS3.p3.4.m4.1.1.2.cmml" type="integer" xref="S3.SS3.p3.4.m4.1.1.2">25</cn><ci id="S3.SS3.p3.4.m4.1.1.3.cmml" xref="S3.SS3.p3.4.m4.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.4.m4.1c">25\degree</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.4.m4.1d">25 °</annotation></semantics></math> incidence and Eve is at <math alttext="7\deg" class="ltx_Math" display="inline" id="S3.SS3.p3.5.m5.1"><semantics id="S3.SS3.p3.5.m5.1a"><mrow id="S3.SS3.p3.5.m5.1.1" xref="S3.SS3.p3.5.m5.1.1.cmml"><mn id="S3.SS3.p3.5.m5.1.1.2" xref="S3.SS3.p3.5.m5.1.1.2.cmml">7</mn><mo id="S3.SS3.p3.5.m5.1.1.1" lspace="0.167em" xref="S3.SS3.p3.5.m5.1.1.1.cmml">⁢</mo><mi id="S3.SS3.p3.5.m5.1.1.3" xref="S3.SS3.p3.5.m5.1.1.3.cmml">deg</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p3.5.m5.1b"><apply id="S3.SS3.p3.5.m5.1.1.cmml" xref="S3.SS3.p3.5.m5.1.1"><times id="S3.SS3.p3.5.m5.1.1.1.cmml" xref="S3.SS3.p3.5.m5.1.1.1"></times><cn id="S3.SS3.p3.5.m5.1.1.2.cmml" type="integer" xref="S3.SS3.p3.5.m5.1.1.2">7</cn><csymbol cd="latexml" id="S3.SS3.p3.5.m5.1.1.3.cmml" xref="S3.SS3.p3.5.m5.1.1.3">degree</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p3.5.m5.1c">7\deg</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p3.5.m5.1d">7 roman_deg</annotation></semantics></math> incidence, and again search for a pair of masks, this time from 500 random masks sampled at 3.67 GHz. A constellation diagram for the backscatter of one such pair of masks in each configuration is shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#S3.F7" title="Figure 7 ‣ III-B Multiplexed Communications ‣ III Results ‣ Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_tag">7</span></a>(a). The two masks are sufficiently separated in phase space for Alice to implement a code such as BPSK. Additionally, the power of the greatest signal for Eve (purple dot) is 32.2 dB and 24.6 dB lower than the power of the smallest signal for Alice (orange dot) in the two configurations, respectively Thus, if the MBLL is configured such that the interrogator signal is less than this signal difference above the noise floor, then Eve will not be able to eavesdrop on the communication with Alice.</p> </div> <div class="ltx_para" id="S3.SS3.p4"> <p class="ltx_p" id="S3.SS3.p4.1">Alternatively, it is conceivable that with a sufficient amount of power in the transmitter and/or a sufficiently powerful receiver, Eve could detect the remnant phase variations in the backscatter signal which would be correlated with Alice’s phase variations, thus successfully capturing the communications. To counter this, we also consider an alternate strategy of finding multiple masks for each symbol such that the masks present a large enough variation for Eve so as to hide the original symbol. For communicating a particular symbol to Alice, the mask can be randomly chosen from the set of suitable candidates. The backscatter signal at Eve, however, should appear uncorrelated with the backscattered signal at Alice. Thus, even though the signal for Eve may be above the noise floor, the symbols will be random and appear as noise, thus securing communications.</p> </div> <div class="ltx_para" id="S3.SS3.p5"> <p class="ltx_p" id="S3.SS3.p5.12">For this experiment, we sample 500 random spatial patterns for the metasurface and measure the backscatter at several angles: <math alttext="0\degree" class="ltx_Math" display="inline" id="S3.SS3.p5.1.m1.1"><semantics id="S3.SS3.p5.1.m1.1a"><mrow id="S3.SS3.p5.1.m1.1.1" xref="S3.SS3.p5.1.m1.1.1.cmml"><mn id="S3.SS3.p5.1.m1.1.1.2" xref="S3.SS3.p5.1.m1.1.1.2.cmml">0</mn><mo id="S3.SS3.p5.1.m1.1.1.1" xref="S3.SS3.p5.1.m1.1.1.1.cmml">⁢</mo><mi id="S3.SS3.p5.1.m1.1.1.3" mathvariant="normal" xref="S3.SS3.p5.1.m1.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p5.1.m1.1b"><apply id="S3.SS3.p5.1.m1.1.1.cmml" xref="S3.SS3.p5.1.m1.1.1"><times id="S3.SS3.p5.1.m1.1.1.1.cmml" xref="S3.SS3.p5.1.m1.1.1.1"></times><cn id="S3.SS3.p5.1.m1.1.1.2.cmml" type="integer" xref="S3.SS3.p5.1.m1.1.1.2">0</cn><ci id="S3.SS3.p5.1.m1.1.1.3.cmml" xref="S3.SS3.p5.1.m1.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p5.1.m1.1c">0\degree</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p5.1.m1.1d">0 °</annotation></semantics></math>, <math alttext="7\degree" class="ltx_Math" display="inline" id="S3.SS3.p5.2.m2.1"><semantics id="S3.SS3.p5.2.m2.1a"><mrow id="S3.SS3.p5.2.m2.1.1" xref="S3.SS3.p5.2.m2.1.1.cmml"><mn id="S3.SS3.p5.2.m2.1.1.2" xref="S3.SS3.p5.2.m2.1.1.2.cmml">7</mn><mo id="S3.SS3.p5.2.m2.1.1.1" xref="S3.SS3.p5.2.m2.1.1.1.cmml">⁢</mo><mi id="S3.SS3.p5.2.m2.1.1.3" mathvariant="normal" xref="S3.SS3.p5.2.m2.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p5.2.m2.1b"><apply id="S3.SS3.p5.2.m2.1.1.cmml" xref="S3.SS3.p5.2.m2.1.1"><times id="S3.SS3.p5.2.m2.1.1.1.cmml" xref="S3.SS3.p5.2.m2.1.1.1"></times><cn id="S3.SS3.p5.2.m2.1.1.2.cmml" type="integer" xref="S3.SS3.p5.2.m2.1.1.2">7</cn><ci id="S3.SS3.p5.2.m2.1.1.3.cmml" xref="S3.SS3.p5.2.m2.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p5.2.m2.1c">7\degree</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p5.2.m2.1d">7 °</annotation></semantics></math>, <math alttext="15\degree" class="ltx_Math" display="inline" id="S3.SS3.p5.3.m3.1"><semantics id="S3.SS3.p5.3.m3.1a"><mrow id="S3.SS3.p5.3.m3.1.1" xref="S3.SS3.p5.3.m3.1.1.cmml"><mn id="S3.SS3.p5.3.m3.1.1.2" xref="S3.SS3.p5.3.m3.1.1.2.cmml">15</mn><mo id="S3.SS3.p5.3.m3.1.1.1" xref="S3.SS3.p5.3.m3.1.1.1.cmml">⁢</mo><mi id="S3.SS3.p5.3.m3.1.1.3" mathvariant="normal" xref="S3.SS3.p5.3.m3.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p5.3.m3.1b"><apply id="S3.SS3.p5.3.m3.1.1.cmml" xref="S3.SS3.p5.3.m3.1.1"><times id="S3.SS3.p5.3.m3.1.1.1.cmml" xref="S3.SS3.p5.3.m3.1.1.1"></times><cn id="S3.SS3.p5.3.m3.1.1.2.cmml" type="integer" xref="S3.SS3.p5.3.m3.1.1.2">15</cn><ci id="S3.SS3.p5.3.m3.1.1.3.cmml" xref="S3.SS3.p5.3.m3.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p5.3.m3.1c">15\degree</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p5.3.m3.1d">15 °</annotation></semantics></math>, <math alttext="25\degree" class="ltx_Math" display="inline" id="S3.SS3.p5.4.m4.1"><semantics id="S3.SS3.p5.4.m4.1a"><mrow id="S3.SS3.p5.4.m4.1.1" xref="S3.SS3.p5.4.m4.1.1.cmml"><mn id="S3.SS3.p5.4.m4.1.1.2" xref="S3.SS3.p5.4.m4.1.1.2.cmml">25</mn><mo id="S3.SS3.p5.4.m4.1.1.1" xref="S3.SS3.p5.4.m4.1.1.1.cmml">⁢</mo><mi id="S3.SS3.p5.4.m4.1.1.3" mathvariant="normal" xref="S3.SS3.p5.4.m4.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p5.4.m4.1b"><apply id="S3.SS3.p5.4.m4.1.1.cmml" xref="S3.SS3.p5.4.m4.1.1"><times id="S3.SS3.p5.4.m4.1.1.1.cmml" xref="S3.SS3.p5.4.m4.1.1.1"></times><cn id="S3.SS3.p5.4.m4.1.1.2.cmml" type="integer" xref="S3.SS3.p5.4.m4.1.1.2">25</cn><ci id="S3.SS3.p5.4.m4.1.1.3.cmml" xref="S3.SS3.p5.4.m4.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p5.4.m4.1c">25\degree</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p5.4.m4.1d">25 °</annotation></semantics></math>, and <math alttext="30\degree" class="ltx_Math" display="inline" id="S3.SS3.p5.5.m5.1"><semantics id="S3.SS3.p5.5.m5.1a"><mrow id="S3.SS3.p5.5.m5.1.1" xref="S3.SS3.p5.5.m5.1.1.cmml"><mn id="S3.SS3.p5.5.m5.1.1.2" xref="S3.SS3.p5.5.m5.1.1.2.cmml">30</mn><mo id="S3.SS3.p5.5.m5.1.1.1" xref="S3.SS3.p5.5.m5.1.1.1.cmml">⁢</mo><mi id="S3.SS3.p5.5.m5.1.1.3" mathvariant="normal" xref="S3.SS3.p5.5.m5.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p5.5.m5.1b"><apply id="S3.SS3.p5.5.m5.1.1.cmml" xref="S3.SS3.p5.5.m5.1.1"><times id="S3.SS3.p5.5.m5.1.1.1.cmml" xref="S3.SS3.p5.5.m5.1.1.1"></times><cn id="S3.SS3.p5.5.m5.1.1.2.cmml" type="integer" xref="S3.SS3.p5.5.m5.1.1.2">30</cn><ci id="S3.SS3.p5.5.m5.1.1.3.cmml" xref="S3.SS3.p5.5.m5.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p5.5.m5.1c">30\degree</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p5.5.m5.1d">30 °</annotation></semantics></math>. We assume that Alice is at <math alttext="0\degree" class="ltx_Math" display="inline" id="S3.SS3.p5.6.m6.1"><semantics id="S3.SS3.p5.6.m6.1a"><mrow id="S3.SS3.p5.6.m6.1.1" xref="S3.SS3.p5.6.m6.1.1.cmml"><mn id="S3.SS3.p5.6.m6.1.1.2" xref="S3.SS3.p5.6.m6.1.1.2.cmml">0</mn><mo id="S3.SS3.p5.6.m6.1.1.1" xref="S3.SS3.p5.6.m6.1.1.1.cmml">⁢</mo><mi id="S3.SS3.p5.6.m6.1.1.3" mathvariant="normal" xref="S3.SS3.p5.6.m6.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p5.6.m6.1b"><apply id="S3.SS3.p5.6.m6.1.1.cmml" xref="S3.SS3.p5.6.m6.1.1"><times id="S3.SS3.p5.6.m6.1.1.1.cmml" xref="S3.SS3.p5.6.m6.1.1.1"></times><cn id="S3.SS3.p5.6.m6.1.1.2.cmml" type="integer" xref="S3.SS3.p5.6.m6.1.1.2">0</cn><ci id="S3.SS3.p5.6.m6.1.1.3.cmml" xref="S3.SS3.p5.6.m6.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p5.6.m6.1c">0\degree</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p5.6.m6.1d">0 °</annotation></semantics></math> incidence angle and that Eve is at any other angle. Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#S3.F7" title="Figure 7 ‣ III-B Multiplexed Communications ‣ III Results ‣ Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_tag">7</span></a>(b) plots the measured backscatter phase, where the <math alttext="x" class="ltx_Math" display="inline" id="S3.SS3.p5.7.m7.1"><semantics id="S3.SS3.p5.7.m7.1a"><mi id="S3.SS3.p5.7.m7.1.1" xref="S3.SS3.p5.7.m7.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p5.7.m7.1b"><ci id="S3.SS3.p5.7.m7.1.1.cmml" xref="S3.SS3.p5.7.m7.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p5.7.m7.1c">x</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p5.7.m7.1d">italic_x</annotation></semantics></math>-axis and <math alttext="y" class="ltx_Math" display="inline" id="S3.SS3.p5.8.m8.1"><semantics id="S3.SS3.p5.8.m8.1a"><mi id="S3.SS3.p5.8.m8.1.1" xref="S3.SS3.p5.8.m8.1.1.cmml">y</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p5.8.m8.1b"><ci id="S3.SS3.p5.8.m8.1.1.cmml" xref="S3.SS3.p5.8.m8.1.1">𝑦</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p5.8.m8.1c">y</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p5.8.m8.1d">italic_y</annotation></semantics></math>-axis corresponds to the backscatter phase for Alice and Eve, respectively. We see a loose correlation between the phase at <math alttext="0\degree" class="ltx_Math" display="inline" id="S3.SS3.p5.9.m9.1"><semantics id="S3.SS3.p5.9.m9.1a"><mrow id="S3.SS3.p5.9.m9.1.1" xref="S3.SS3.p5.9.m9.1.1.cmml"><mn id="S3.SS3.p5.9.m9.1.1.2" xref="S3.SS3.p5.9.m9.1.1.2.cmml">0</mn><mo id="S3.SS3.p5.9.m9.1.1.1" xref="S3.SS3.p5.9.m9.1.1.1.cmml">⁢</mo><mi id="S3.SS3.p5.9.m9.1.1.3" mathvariant="normal" xref="S3.SS3.p5.9.m9.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p5.9.m9.1b"><apply id="S3.SS3.p5.9.m9.1.1.cmml" xref="S3.SS3.p5.9.m9.1.1"><times id="S3.SS3.p5.9.m9.1.1.1.cmml" xref="S3.SS3.p5.9.m9.1.1.1"></times><cn id="S3.SS3.p5.9.m9.1.1.2.cmml" type="integer" xref="S3.SS3.p5.9.m9.1.1.2">0</cn><ci id="S3.SS3.p5.9.m9.1.1.3.cmml" xref="S3.SS3.p5.9.m9.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p5.9.m9.1c">0\degree</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p5.9.m9.1d">0 °</annotation></semantics></math> incidence and <math alttext="7\degree" class="ltx_Math" display="inline" id="S3.SS3.p5.10.m10.1"><semantics id="S3.SS3.p5.10.m10.1a"><mrow id="S3.SS3.p5.10.m10.1.1" xref="S3.SS3.p5.10.m10.1.1.cmml"><mn id="S3.SS3.p5.10.m10.1.1.2" xref="S3.SS3.p5.10.m10.1.1.2.cmml">7</mn><mo id="S3.SS3.p5.10.m10.1.1.1" xref="S3.SS3.p5.10.m10.1.1.1.cmml">⁢</mo><mi id="S3.SS3.p5.10.m10.1.1.3" mathvariant="normal" xref="S3.SS3.p5.10.m10.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p5.10.m10.1b"><apply id="S3.SS3.p5.10.m10.1.1.cmml" xref="S3.SS3.p5.10.m10.1.1"><times id="S3.SS3.p5.10.m10.1.1.1.cmml" xref="S3.SS3.p5.10.m10.1.1.1"></times><cn id="S3.SS3.p5.10.m10.1.1.2.cmml" type="integer" xref="S3.SS3.p5.10.m10.1.1.2">7</cn><ci id="S3.SS3.p5.10.m10.1.1.3.cmml" xref="S3.SS3.p5.10.m10.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p5.10.m10.1c">7\degree</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p5.10.m10.1d">7 °</annotation></semantics></math> incidence which is unsurprising due to the strongly overlapping focal spots on the LL focal plane. Notably, the correlation falls off above <math alttext="15\degree" class="ltx_Math" display="inline" id="S3.SS3.p5.11.m11.1"><semantics id="S3.SS3.p5.11.m11.1a"><mrow id="S3.SS3.p5.11.m11.1.1" xref="S3.SS3.p5.11.m11.1.1.cmml"><mn id="S3.SS3.p5.11.m11.1.1.2" xref="S3.SS3.p5.11.m11.1.1.2.cmml">15</mn><mo id="S3.SS3.p5.11.m11.1.1.1" xref="S3.SS3.p5.11.m11.1.1.1.cmml">⁢</mo><mi id="S3.SS3.p5.11.m11.1.1.3" mathvariant="normal" xref="S3.SS3.p5.11.m11.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS3.p5.11.m11.1b"><apply id="S3.SS3.p5.11.m11.1.1.cmml" xref="S3.SS3.p5.11.m11.1.1"><times id="S3.SS3.p5.11.m11.1.1.1.cmml" xref="S3.SS3.p5.11.m11.1.1.1"></times><cn id="S3.SS3.p5.11.m11.1.1.2.cmml" type="integer" xref="S3.SS3.p5.11.m11.1.1.2">15</cn><ci id="S3.SS3.p5.11.m11.1.1.3.cmml" xref="S3.SS3.p5.11.m11.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p5.11.m11.1c">15\degree</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p5.11.m11.1d">15 °</annotation></semantics></math> incidence, suggesting that we can mask Alice’s message from a sufficiently distant Eve. In particular, we can choose a set of masks that are approximately <math alttext="\pi" class="ltx_Math" display="inline" id="S3.SS3.p5.12.m12.1"><semantics id="S3.SS3.p5.12.m12.1a"><mi id="S3.SS3.p5.12.m12.1.1" xref="S3.SS3.p5.12.m12.1.1.cmml">π</mi><annotation-xml encoding="MathML-Content" id="S3.SS3.p5.12.m12.1b"><ci id="S3.SS3.p5.12.m12.1.1.cmml" xref="S3.SS3.p5.12.m12.1.1">𝜋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS3.p5.12.m12.1c">\pi</annotation><annotation encoding="application/x-llamapun" id="S3.SS3.p5.12.m12.1d">italic_π</annotation></semantics></math> phase apart for Alice but where the phase is ambiguous for Eve, as shown in Fig. <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#S3.F7" title="Figure 7 ‣ III-B Multiplexed Communications ‣ III Results ‣ Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_tag">7</span></a>(c). Thus, if we want to send a a “0” or a “1” to Alice, we can choose from any of the masks corresponding to an orange or blue point, respectively. Because distributions of phase are similar to Eve regardless of the bit to Alice, the message is masked from Eve.</p> </div> <div class="ltx_para" id="S3.SS3.p6"> <p class="ltx_p" id="S3.SS3.p6.1">Note that in both experiments, the masks have not been optimized due to experimental complexity. It is likely that these results can be further optimized to maximize the signal to Alice while minimizing the signal for Eve. Future experiments can also investigate the possibility of multiple interrogators or eavesdroppers across locations or frequencies.</p> </div> </section> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">IV </span><span class="ltx_text ltx_font_smallcaps" id="S4.1.1">Conclusion</span> </h2> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.1">We have designed and experimentally demonstrated a MBLL capable of spatially-multiplexed backscatter communications. The use of the LL to achieve retroreflection alleviates the metasurface design and control constraints, enabling low-loss backscatter across multiple frequencies and angles. We have also shown that applying spatial masks to the metasurface not only enables communications with two users simultaneously, but also helps secure communications from an eavesdropper.</p> </div> <div class="ltx_para" id="S4.p2"> <p class="ltx_p" id="S4.p2.1">The MBLL has potential for a wide variety of applications. Reconfigurable intelligent surfaces (RISs) and metasurfaces have long been proposed for backscatter communications, with applications in wireless communications and internet of things (IoT) <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib5" title="">5</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib6" title="">6</a>]</cite>. The high-gain of the retroreflective Luneburg lens enables backscatter communications over long distances. The device may be attached to a sensor or altered by a stimulus to exfiltrate data. For example, a sensor that has voltage sensitivity to temperature/salinity can be used to alter reflections, thus communicating without the MBLL radiating or consuming significant power. Alternatively, the robust control provided by a capable voltage source can provide rich behavior and the MBLL can serve as a sensing point or as a hub in a more complex network. Numerous other possibilities exist for augmenting the MBLL and there remain many applications that motivate continued development of the MBLL.</p> </div> <div class="ltx_para" id="S4.p3"> <p class="ltx_p" id="S4.p3.1">While we choose a simple form for the metasurface unit cell which achieves full <math alttext="2\pi" class="ltx_Math" display="inline" id="S4.p3.1.m1.1"><semantics id="S4.p3.1.m1.1a"><mrow id="S4.p3.1.m1.1.1" xref="S4.p3.1.m1.1.1.cmml"><mn id="S4.p3.1.m1.1.1.2" xref="S4.p3.1.m1.1.1.2.cmml">2</mn><mo id="S4.p3.1.m1.1.1.1" xref="S4.p3.1.m1.1.1.1.cmml">⁢</mo><mi id="S4.p3.1.m1.1.1.3" xref="S4.p3.1.m1.1.1.3.cmml">π</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.p3.1.m1.1b"><apply id="S4.p3.1.m1.1.1.cmml" xref="S4.p3.1.m1.1.1"><times id="S4.p3.1.m1.1.1.1.cmml" xref="S4.p3.1.m1.1.1.1"></times><cn id="S4.p3.1.m1.1.1.2.cmml" type="integer" xref="S4.p3.1.m1.1.1.2">2</cn><ci id="S4.p3.1.m1.1.1.3.cmml" xref="S4.p3.1.m1.1.1.3">𝜋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p3.1.m1.1c">2\pi</annotation><annotation encoding="application/x-llamapun" id="S4.p3.1.m1.1d">2 italic_π</annotation></semantics></math> phase coverage, the LL can easily be combined with other types of metasurfaces with different functionality. For example, Ref. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib15" title="">15</a>]</cite> demonstrate a metasurface for independent control over both reflection amplitude and phase. This can enable even higher bitrate communication protocols compared to phase shift keying (PSK) demonstrated here, such as quadrature amplitude modulation (QAM).</p> </div> <div class="ltx_para" id="S4.p4"> <p class="ltx_p" id="S4.p4.1">Many of the results presented here using spatial masks for the metasurface rely on either simple optimization schemes (genetic algorithms) or random sampling, as more advanced optimization is outside the scope of this work. The simple optimization and sampling approaches taken here already demonstrate impressive results, and there is tremendous potential in applying different optimization approaches to achieve improved behavior from the MBLL that would not be possible otherwise. For example, channel estimation aims to model and optimize the communication channel characteristics between a transmitter and receiver, and has been applied to RISs <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib38" title="">38</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib39" title="">39</a>]</cite>. These techniques can possibly be used to efficiently optimize the metasurface configuration without extensive global optimization algorithms, although they often assume perfect control over the phase response. Machine learning has been applied to optimizing beamforming in RISs <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib40" title="">40</a>]</cite>.</p> </div> </section> <section class="ltx_appendix" id="A1"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix A </span>Optimization</h2> <div class="ltx_para" id="A1.p1"> <p class="ltx_p" id="A1.p1.4">To find a set of spatial masks that can maximize the backscatter magnitude while simultaneously achieving a full range of phase response, we use multi-objective optimization. Rather than minimizing or maximizing a single metric as is done in conventional optimization, multi-objective optimization aims to find a set of solutions called the <span class="ltx_text ltx_font_italic" id="A1.p1.4.1">Pareto front</span> that balance the trade-off between the multiple objectives. In particular, we use the multi-objective genetic algorithm included in MATLAB <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib41" title="">41</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib42" title="">42</a>]</cite>. 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id="A1.E5.m1.1c">\displaystyle\mathcal{L}_{1}</annotation><annotation encoding="application/x-llamapun" id="A1.E5.m1.1d">caligraphic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=|S|" class="ltx_Math" display="inline" id="A1.E5.m2.1"><semantics id="A1.E5.m2.1a"><mrow id="A1.E5.m2.1.2" xref="A1.E5.m2.1.2.cmml"><mi id="A1.E5.m2.1.2.2" xref="A1.E5.m2.1.2.2.cmml"></mi><mo id="A1.E5.m2.1.2.1" xref="A1.E5.m2.1.2.1.cmml">=</mo><mrow id="A1.E5.m2.1.2.3.2" xref="A1.E5.m2.1.2.3.1.cmml"><mo id="A1.E5.m2.1.2.3.2.1" stretchy="false" xref="A1.E5.m2.1.2.3.1.1.cmml">|</mo><mi id="A1.E5.m2.1.1" xref="A1.E5.m2.1.1.cmml">S</mi><mo id="A1.E5.m2.1.2.3.2.2" stretchy="false" xref="A1.E5.m2.1.2.3.1.1.cmml">|</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A1.E5.m2.1b"><apply id="A1.E5.m2.1.2.cmml" xref="A1.E5.m2.1.2"><eq id="A1.E5.m2.1.2.1.cmml" xref="A1.E5.m2.1.2.1"></eq><csymbol cd="latexml" id="A1.E5.m2.1.2.2.cmml" xref="A1.E5.m2.1.2.2">absent</csymbol><apply id="A1.E5.m2.1.2.3.1.cmml" xref="A1.E5.m2.1.2.3.2"><abs id="A1.E5.m2.1.2.3.1.1.cmml" xref="A1.E5.m2.1.2.3.2.1"></abs><ci id="A1.E5.m2.1.1.cmml" xref="A1.E5.m2.1.1">𝑆</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.E5.m2.1c">\displaystyle=|S|</annotation><annotation encoding="application/x-llamapun" id="A1.E5.m2.1d">= | italic_S |</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(5)</span></td> </tr></tbody> <tbody id="A1.E6"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathcal{L}_{2}" class="ltx_Math" display="inline" id="A1.E6.m1.1"><semantics 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ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(6)</span></td> </tr></tbody> </table> <p class="ltx_p" id="A1.p1.3">where <math alttext="\phi_{0}" class="ltx_Math" display="inline" id="A1.p1.1.m1.1"><semantics id="A1.p1.1.m1.1a"><msub id="A1.p1.1.m1.1.1" xref="A1.p1.1.m1.1.1.cmml"><mi id="A1.p1.1.m1.1.1.2" xref="A1.p1.1.m1.1.1.2.cmml">ϕ</mi><mn id="A1.p1.1.m1.1.1.3" xref="A1.p1.1.m1.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="A1.p1.1.m1.1b"><apply id="A1.p1.1.m1.1.1.cmml" xref="A1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="A1.p1.1.m1.1.1.1.cmml" xref="A1.p1.1.m1.1.1">subscript</csymbol><ci id="A1.p1.1.m1.1.1.2.cmml" xref="A1.p1.1.m1.1.1.2">italic-ϕ</ci><cn id="A1.p1.1.m1.1.1.3.cmml" type="integer" xref="A1.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.1.m1.1c">\phi_{0}</annotation><annotation encoding="application/x-llamapun" id="A1.p1.1.m1.1d">italic_ϕ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> is the desired angle, and the division by <math alttext="10\degree" class="ltx_Math" display="inline" id="A1.p1.2.m2.1"><semantics id="A1.p1.2.m2.1a"><mrow id="A1.p1.2.m2.1.1" xref="A1.p1.2.m2.1.1.cmml"><mn id="A1.p1.2.m2.1.1.2" xref="A1.p1.2.m2.1.1.2.cmml">10</mn><mo id="A1.p1.2.m2.1.1.1" xref="A1.p1.2.m2.1.1.1.cmml">⁢</mo><mi id="A1.p1.2.m2.1.1.3" mathvariant="normal" xref="A1.p1.2.m2.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.p1.2.m2.1b"><apply id="A1.p1.2.m2.1.1.cmml" xref="A1.p1.2.m2.1.1"><times id="A1.p1.2.m2.1.1.1.cmml" xref="A1.p1.2.m2.1.1.1"></times><cn id="A1.p1.2.m2.1.1.2.cmml" type="integer" xref="A1.p1.2.m2.1.1.2">10</cn><ci id="A1.p1.2.m2.1.1.3.cmml" xref="A1.p1.2.m2.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.2.m2.1c">10\degree</annotation><annotation encoding="application/x-llamapun" id="A1.p1.2.m2.1d">10 °</annotation></semantics></math> in the second objective is to normalize the two objectives to approximately the same magnitude for purposes of setting meaningful convergence criteria. The genetic algorithm is placed inside a loop that iterates over 16 equally spaced phase targets from 0 to <math alttext="2\pi" class="ltx_Math" display="inline" id="A1.p1.3.m3.1"><semantics id="A1.p1.3.m3.1a"><mrow id="A1.p1.3.m3.1.1" xref="A1.p1.3.m3.1.1.cmml"><mn id="A1.p1.3.m3.1.1.2" xref="A1.p1.3.m3.1.1.2.cmml">2</mn><mo id="A1.p1.3.m3.1.1.1" xref="A1.p1.3.m3.1.1.1.cmml">⁢</mo><mi id="A1.p1.3.m3.1.1.3" xref="A1.p1.3.m3.1.1.3.cmml">π</mi></mrow><annotation-xml encoding="MathML-Content" id="A1.p1.3.m3.1b"><apply id="A1.p1.3.m3.1.1.cmml" xref="A1.p1.3.m3.1.1"><times id="A1.p1.3.m3.1.1.1.cmml" xref="A1.p1.3.m3.1.1.1"></times><cn id="A1.p1.3.m3.1.1.2.cmml" type="integer" xref="A1.p1.3.m3.1.1.2">2</cn><ci id="A1.p1.3.m3.1.1.3.cmml" xref="A1.p1.3.m3.1.1.3">𝜋</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A1.p1.3.m3.1c">2\pi</annotation><annotation encoding="application/x-llamapun" id="A1.p1.3.m3.1d">2 italic_π</annotation></semantics></math> radians. The collective Pareto fronts from all previous optimizations are used as part of the initial population for the next optimization to improve convergence, where the Pareto front is re-calculated for the new objective and a clustering algorithm (k-means) is used to down-select the candidates.</p> </div> <div class="ltx_para" id="A1.p2"> <p class="ltx_p" id="A1.p2.1">Note that a constrained single-objective optimization approach where the phase is set as a constraint may be more meaningful in the case where we wish to optimize for a single angle. However, in the case where we wish to optimize for a collection of angles, the multi-objective optimization approach works reasonably well in practice.</p> </div> <div class="ltx_para" id="A1.p3"> <p class="ltx_p" id="A1.p3.1">Finally, note that this is a simple optimization scheme that ignores the magnitude of the backscatter field. We preprocess the dataset by filtering out the masks with a magnitude below a threshold. In practice, this works to find reasonable solutions; alternatively, more sophisticated methods such as multi-objective optimization or an objective more closely tied to the modulation scheme can be used to maximize signal-to-noise ratio and data rates.</p> </div> </section> <section class="ltx_appendix" id="A2"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix B </span>Hemispherical Luneburg Lens</h2> <div class="ltx_para" id="A2.p1"> <p class="ltx_p" id="A2.p1.5">The coordinate mapping used in transformation optics is not unique. While the quasi-conformal mapping is one of the more commonly used mappings due to its ability to be realized using isotropic, non-magnetic materials, other mappings can be used with various tradeoffs. For example, Xu et al. propose a hemispherical Luneburg lens using an analytical coordinate mapping in order to integrate the lens with an array of feed antennas <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib43" title="">43</a>]</cite>. However, we note that such a transformation does not feature retroreflective behavior when a reflective surface is placed at the flattened surface, as shown in Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#A2.F8" title="Figure 8 ‣ Appendix B Hemispherical Luneburg Lens ‣ Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_tag">8</span></a>. In Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#A2.F8" title="Figure 8 ‣ Appendix B Hemispherical Luneburg Lens ‣ Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_tag">8</span></a>(b), we see the focal point has been shifted significantly off of the flattened surface, in agreement with Ref. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib43" title="">43</a>]</cite>. For the far field results in Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#A2.F8" title="Figure 8 ‣ Appendix B Hemispherical Luneburg Lens ‣ Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_tag">8</span></a>(d), note that for <math alttext="15\degree" class="ltx_Math" display="inline" id="A2.p1.1.m1.1"><semantics id="A2.p1.1.m1.1a"><mrow id="A2.p1.1.m1.1.1" xref="A2.p1.1.m1.1.1.cmml"><mn id="A2.p1.1.m1.1.1.2" xref="A2.p1.1.m1.1.1.2.cmml">15</mn><mo id="A2.p1.1.m1.1.1.1" xref="A2.p1.1.m1.1.1.1.cmml">⁢</mo><mi id="A2.p1.1.m1.1.1.3" mathvariant="normal" xref="A2.p1.1.m1.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.p1.1.m1.1b"><apply id="A2.p1.1.m1.1.1.cmml" xref="A2.p1.1.m1.1.1"><times id="A2.p1.1.m1.1.1.1.cmml" xref="A2.p1.1.m1.1.1.1"></times><cn id="A2.p1.1.m1.1.1.2.cmml" type="integer" xref="A2.p1.1.m1.1.1.2">15</cn><ci id="A2.p1.1.m1.1.1.3.cmml" xref="A2.p1.1.m1.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p1.1.m1.1c">15\degree</annotation><annotation encoding="application/x-llamapun" id="A2.p1.1.m1.1d">15 °</annotation></semantics></math> and <math alttext="45\degree" class="ltx_Math" display="inline" id="A2.p1.2.m2.1"><semantics id="A2.p1.2.m2.1a"><mrow id="A2.p1.2.m2.1.1" xref="A2.p1.2.m2.1.1.cmml"><mn id="A2.p1.2.m2.1.1.2" xref="A2.p1.2.m2.1.1.2.cmml">45</mn><mo id="A2.p1.2.m2.1.1.1" xref="A2.p1.2.m2.1.1.1.cmml">⁢</mo><mi id="A2.p1.2.m2.1.1.3" mathvariant="normal" xref="A2.p1.2.m2.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.p1.2.m2.1b"><apply id="A2.p1.2.m2.1.1.cmml" xref="A2.p1.2.m2.1.1"><times id="A2.p1.2.m2.1.1.1.cmml" xref="A2.p1.2.m2.1.1.1"></times><cn id="A2.p1.2.m2.1.1.2.cmml" type="integer" xref="A2.p1.2.m2.1.1.2">45</cn><ci id="A2.p1.2.m2.1.1.3.cmml" xref="A2.p1.2.m2.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p1.2.m2.1c">45\degree</annotation><annotation encoding="application/x-llamapun" id="A2.p1.2.m2.1d">45 °</annotation></semantics></math> plane wave incidence angles, the RCS peaks at <math alttext="-15\degree" class="ltx_Math" display="inline" id="A2.p1.3.m3.1"><semantics id="A2.p1.3.m3.1a"><mrow id="A2.p1.3.m3.1.1" xref="A2.p1.3.m3.1.1.cmml"><mo id="A2.p1.3.m3.1.1a" xref="A2.p1.3.m3.1.1.cmml">−</mo><mrow id="A2.p1.3.m3.1.1.2" xref="A2.p1.3.m3.1.1.2.cmml"><mn id="A2.p1.3.m3.1.1.2.2" xref="A2.p1.3.m3.1.1.2.2.cmml">15</mn><mo id="A2.p1.3.m3.1.1.2.1" xref="A2.p1.3.m3.1.1.2.1.cmml">⁢</mo><mi id="A2.p1.3.m3.1.1.2.3" mathvariant="normal" xref="A2.p1.3.m3.1.1.2.3.cmml">°</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.p1.3.m3.1b"><apply id="A2.p1.3.m3.1.1.cmml" xref="A2.p1.3.m3.1.1"><minus id="A2.p1.3.m3.1.1.1.cmml" xref="A2.p1.3.m3.1.1"></minus><apply id="A2.p1.3.m3.1.1.2.cmml" xref="A2.p1.3.m3.1.1.2"><times id="A2.p1.3.m3.1.1.2.1.cmml" xref="A2.p1.3.m3.1.1.2.1"></times><cn id="A2.p1.3.m3.1.1.2.2.cmml" type="integer" xref="A2.p1.3.m3.1.1.2.2">15</cn><ci id="A2.p1.3.m3.1.1.2.3.cmml" xref="A2.p1.3.m3.1.1.2.3">°</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p1.3.m3.1c">-15\degree</annotation><annotation encoding="application/x-llamapun" id="A2.p1.3.m3.1d">- 15 °</annotation></semantics></math> and <math alttext="-45\degree" class="ltx_Math" display="inline" id="A2.p1.4.m4.1"><semantics id="A2.p1.4.m4.1a"><mrow id="A2.p1.4.m4.1.1" xref="A2.p1.4.m4.1.1.cmml"><mo id="A2.p1.4.m4.1.1a" xref="A2.p1.4.m4.1.1.cmml">−</mo><mrow id="A2.p1.4.m4.1.1.2" xref="A2.p1.4.m4.1.1.2.cmml"><mn id="A2.p1.4.m4.1.1.2.2" xref="A2.p1.4.m4.1.1.2.2.cmml">45</mn><mo id="A2.p1.4.m4.1.1.2.1" xref="A2.p1.4.m4.1.1.2.1.cmml">⁢</mo><mi id="A2.p1.4.m4.1.1.2.3" mathvariant="normal" xref="A2.p1.4.m4.1.1.2.3.cmml">°</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.p1.4.m4.1b"><apply id="A2.p1.4.m4.1.1.cmml" xref="A2.p1.4.m4.1.1"><minus id="A2.p1.4.m4.1.1.1.cmml" xref="A2.p1.4.m4.1.1"></minus><apply id="A2.p1.4.m4.1.1.2.cmml" xref="A2.p1.4.m4.1.1.2"><times id="A2.p1.4.m4.1.1.2.1.cmml" xref="A2.p1.4.m4.1.1.2.1"></times><cn id="A2.p1.4.m4.1.1.2.2.cmml" type="integer" xref="A2.p1.4.m4.1.1.2.2">45</cn><ci id="A2.p1.4.m4.1.1.2.3.cmml" xref="A2.p1.4.m4.1.1.2.3">°</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p1.4.m4.1c">-45\degree</annotation><annotation encoding="application/x-llamapun" id="A2.p1.4.m4.1d">- 45 °</annotation></semantics></math>, indicating behavior closer to specular reflection. Additionally, the RCS at <math alttext="0\degree" class="ltx_Math" display="inline" id="A2.p1.5.m5.1"><semantics id="A2.p1.5.m5.1a"><mrow id="A2.p1.5.m5.1.1" xref="A2.p1.5.m5.1.1.cmml"><mn id="A2.p1.5.m5.1.1.2" xref="A2.p1.5.m5.1.1.2.cmml">0</mn><mo id="A2.p1.5.m5.1.1.1" xref="A2.p1.5.m5.1.1.1.cmml">⁢</mo><mi id="A2.p1.5.m5.1.1.3" mathvariant="normal" xref="A2.p1.5.m5.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.p1.5.m5.1b"><apply id="A2.p1.5.m5.1.1.cmml" xref="A2.p1.5.m5.1.1"><times id="A2.p1.5.m5.1.1.1.cmml" xref="A2.p1.5.m5.1.1.1"></times><cn id="A2.p1.5.m5.1.1.2.cmml" type="integer" xref="A2.p1.5.m5.1.1.2">0</cn><ci id="A2.p1.5.m5.1.1.3.cmml" xref="A2.p1.5.m5.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p1.5.m5.1c">0\degree</annotation><annotation encoding="application/x-llamapun" id="A2.p1.5.m5.1d">0 °</annotation></semantics></math> is significantly smaller than expected for a flat plate of equivalent aperture. Thus, the hemispherical Luneburg lens does not achieve retroreflection (which we note that it was not designed for). It may be possible to achieve retroreflection with the hemispherical Luneburg lens with an optimized, non-planar reflective surface, although that is beyond the scope of this work.</p> </div> <figure class="ltx_figure" id="A2.F8"><img alt="Refer to caption" class="ltx_graphics ltx_img_landscape" height="308" id="A2.F8.g1" src="x8.png" width="705"/> <figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 8: </span>Simulations of a hemispherical Luneburg lens flattened using the transformation optics mapping proposed in Ref. <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#bib.bib43" title="">43</a>]</cite>. (a) Relative permittivity profile. (b,c) Electric field magnitude calculated using full-wave simulations for plane waves at (b) <math alttext="0\degree" class="ltx_Math" display="inline" id="A2.F8.3.m1.1"><semantics id="A2.F8.3.m1.1b"><mrow id="A2.F8.3.m1.1.1" xref="A2.F8.3.m1.1.1.cmml"><mn id="A2.F8.3.m1.1.1.2" xref="A2.F8.3.m1.1.1.2.cmml">0</mn><mo id="A2.F8.3.m1.1.1.1" xref="A2.F8.3.m1.1.1.1.cmml">⁢</mo><mi id="A2.F8.3.m1.1.1.3" mathvariant="normal" xref="A2.F8.3.m1.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.F8.3.m1.1c"><apply id="A2.F8.3.m1.1.1.cmml" xref="A2.F8.3.m1.1.1"><times id="A2.F8.3.m1.1.1.1.cmml" xref="A2.F8.3.m1.1.1.1"></times><cn id="A2.F8.3.m1.1.1.2.cmml" type="integer" xref="A2.F8.3.m1.1.1.2">0</cn><ci id="A2.F8.3.m1.1.1.3.cmml" xref="A2.F8.3.m1.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.F8.3.m1.1d">0\degree</annotation><annotation encoding="application/x-llamapun" id="A2.F8.3.m1.1e">0 °</annotation></semantics></math> and (c) <math alttext="45\degree" class="ltx_Math" display="inline" id="A2.F8.4.m2.1"><semantics id="A2.F8.4.m2.1b"><mrow id="A2.F8.4.m2.1.1" xref="A2.F8.4.m2.1.1.cmml"><mn id="A2.F8.4.m2.1.1.2" xref="A2.F8.4.m2.1.1.2.cmml">45</mn><mo id="A2.F8.4.m2.1.1.1" xref="A2.F8.4.m2.1.1.1.cmml">⁢</mo><mi id="A2.F8.4.m2.1.1.3" mathvariant="normal" xref="A2.F8.4.m2.1.1.3.cmml">°</mi></mrow><annotation-xml encoding="MathML-Content" id="A2.F8.4.m2.1c"><apply id="A2.F8.4.m2.1.1.cmml" xref="A2.F8.4.m2.1.1"><times id="A2.F8.4.m2.1.1.1.cmml" xref="A2.F8.4.m2.1.1.1"></times><cn id="A2.F8.4.m2.1.1.2.cmml" type="integer" xref="A2.F8.4.m2.1.1.2">45</cn><ci id="A2.F8.4.m2.1.1.3.cmml" xref="A2.F8.4.m2.1.1.3">°</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.F8.4.m2.1d">45\degree</annotation><annotation encoding="application/x-llamapun" id="A2.F8.4.m2.1e">45 °</annotation></semantics></math>. Black box outlines the region of the hemispherical lens. (d) Far field radar cross section (RCS) of the lens when a reflective surface (PEC) is placed at the flattened surface of the lens. </figcaption> </figure> </section> <section class="ltx_appendix" id="A3"> <h2 class="ltx_title ltx_title_appendix"> <span class="ltx_tag ltx_tag_appendix">Appendix C </span>Random masks distribution</h2> <div class="ltx_para" id="A3.p1"> <p class="ltx_p" id="A3.p1.1">As mentioned in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#S3.SS2" title="III-B Multiplexed Communications ‣ III Results ‣ Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_tag"><span class="ltx_text">III-B</span></span></a>, the analysis for multiplexed and secure communications relies on post-processing of randomly sampled spatial masks due to experimental limitations. Figure <a class="ltx_ref" href="https://arxiv.org/html/2503.16366v1#A3.F9" title="Figure 9 ‣ Appendix C Random masks distribution ‣ Dynamic Metasurface-Backed Luneburg Lens for Multiplexed Backscatter Communication"><span class="ltx_text ltx_ref_tag">9</span></a> plots the phases of the randomly sampled spatial masks, where data that differ from the maximum magnitude by over 10dB are filtered out. For most cases, the phases of the backscatter signal are narrowly distributed within a small range. Thus, with a more sophisticated experimental setup and the application of optimization algorithms inside the experimental loop will likely be able to deliver significantly improved results compared to what has been presented here.</p> </div> <figure class="ltx_figure" id="A3.F9"><img alt="Refer to caption" class="ltx_graphics ltx_img_square" height="683" id="A3.F9.g1" src="x9.png" width="830"/> <figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure">Figure 9: </span>Histograms of backscatter phase of random spatial masks applied to the MBLL. Masks are filtered out by magnitude, discarding masks that differ from the maximum magnitude by 10dB.</figcaption> </figure> </section> <section class="ltx_appendix" id="Ax1"> <h2 class="ltx_title ltx_title_appendix">Acknowledgment</h2> </section> <section class="ltx_bibliography" id="bib"> <h2 class="ltx_title ltx_title_bibliography">References</h2> <ul class="ltx_biblist"> <li class="ltx_bibitem" id="bib.bib1"> <span class="ltx_tag ltx_tag_bibitem">[1]</span> <span class="ltx_bibblock"> Anutusha Dogra, Rakesh Kumar Jha, and Shubha Jain. </span> <span class="ltx_bibblock">A survey on beyond 5g network with the advent of 6g: Architecture and emerging technologies. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib1.1.1">IEEE access</span>, 9:67512–67547, 2020. </span> </li> <li class="ltx_bibitem" id="bib.bib2"> <span class="ltx_tag ltx_tag_bibitem">[2]</span> <span class="ltx_bibblock"> Muntadher Alsabah, Marwah Abdulrazzaq Naser, Basheera M Mahmmod, Sadiq H Abdulhussain, Mohammad R Eissa, Ahmed Al-Baidhani, Nor K Noordin, Sadiq M Sait, Khaled A Al-Utaibi, and Fazirul Hashim. </span> <span class="ltx_bibblock">6g wireless communications networks: A comprehensive survey. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib2.1.1">Ieee Access</span>, 9:148191–148243, 2021. </span> </li> <li class="ltx_bibitem" id="bib.bib3"> <span class="ltx_tag ltx_tag_bibitem">[3]</span> <span class="ltx_bibblock"> Lionel Sujay Vailshery. </span> <span class="ltx_bibblock">Number of internet of things (iot) connections worldwide from 2022 to 2023, with forecasts from 2024 to 2033 (in billions). </span> <span class="ltx_bibblock"><a class="ltx_ref ltx_url ltx_font_typewriter" href="https://www.statista.com/statistics/1183457/iot-connected-devices-worldwide/" title="">https://www.statista.com/statistics/1183457/iot-connected-devices-worldwide/</a>, Jun 2024. </span> </li> <li class="ltx_bibitem" id="bib.bib4"> <span class="ltx_tag ltx_tag_bibitem">[4]</span> <span class="ltx_bibblock"> Jameel S Yalli, Mohd H Hasan, and Aisha Badawi. </span> <span class="ltx_bibblock">Internet of things (iot): Origin, embedded technologies, smart applications and its growth in the last decade. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib4.1.1">IEEE Access</span>, 2024. </span> </li> <li class="ltx_bibitem" id="bib.bib5"> <span class="ltx_tag ltx_tag_bibitem">[5]</span> <span class="ltx_bibblock"> Jin-Ping Niu and Geoffrey Ye Li. </span> <span class="ltx_bibblock">An overview on backscatter communications. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib5.1.1">Journal of communications and information networks</span>, 4(2):1–14, 2019. </span> </li> <li class="ltx_bibitem" id="bib.bib6"> <span class="ltx_tag ltx_tag_bibitem">[6]</span> <span class="ltx_bibblock"> Ying-Chang Liang, Qianqian Zhang, Jun Wang, Ruizhe Long, Hu Zhou, and Gang Yang. </span> <span class="ltx_bibblock">Backscatter communication assisted by reconfigurable intelligent surfaces. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib6.1.1">Proceedings of the IEEE</span>, 110(9):1339–1357, 2022. </span> </li> <li class="ltx_bibitem" id="bib.bib7"> <span class="ltx_tag ltx_tag_bibitem">[7]</span> <span class="ltx_bibblock"> Colby Boyer and Sumit Roy. </span> <span class="ltx_bibblock">—invited paper—backscatter communication and rfid: Coding, energy, and mimo analysis. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib7.1.1">IEEE Transactions on communications</span>, 62(3):770–785, 2013. </span> </li> <li class="ltx_bibitem" id="bib.bib8"> <span class="ltx_tag ltx_tag_bibitem">[8]</span> <span class="ltx_bibblock"> Payam Nayeri, Fan Yang, and Atef Z Elsherbeni. </span> <span class="ltx_bibblock">Beam-scanning reflectarray antennas: A technical overview and state of the art. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib8.1.1">IEEE Antennas and Propagation Magazine</span>, 57(4):32–47, 2015. </span> </li> <li class="ltx_bibitem" id="bib.bib9"> <span class="ltx_tag ltx_tag_bibitem">[9]</span> <span class="ltx_bibblock"> Robert L. Schmid, David B. Shrekenhamer, Oscar F. Somerlock, Amanda C. Malone, Timothy A. Sleasman, and Ra’id S. Awadallah. </span> <span class="ltx_bibblock">S-band gaas fet reconfigurable reflectarray for passive communications. </span> <span class="ltx_bibblock">In <span class="ltx_text ltx_font_italic" id="bib.bib9.1.1">2020 IEEE Radio and Wireless Symposium (RWS)</span>, pages 91–93. IEEE, 2020. </span> </li> <li class="ltx_bibitem" id="bib.bib10"> <span class="ltx_tag ltx_tag_bibitem">[10]</span> <span class="ltx_bibblock"> Yasir Saifullah, Yejun He, Amir Boag, Guo-Min Yang, and Feng Xu. </span> <span class="ltx_bibblock">Recent progress in reconfigurable and intelligent metasurfaces: A comprehensive review of tuning mechanisms, hardware designs, and applications. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib10.1.1">Advanced Science</span>, 9(33):2203747, 2022. </span> </li> <li class="ltx_bibitem" id="bib.bib11"> <span class="ltx_tag ltx_tag_bibitem">[11]</span> <span class="ltx_bibblock"> Arjun Singh, Michael Andrello, Ngwe Thawdar, and Josep Miquel Jornet. </span> <span class="ltx_bibblock">Design and operation of a graphene-based plasmonic nano-antenna array for communication in the terahertz band. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib11.1.1">IEEE Journal on Selected Areas in Communications</span>, 38(9):2104–2117, 2020. </span> </li> <li class="ltx_bibitem" id="bib.bib12"> <span class="ltx_tag ltx_tag_bibitem">[12]</span> <span class="ltx_bibblock"> Mohammadreza F. Imani, David R Smith, and Philipp del Hougne. </span> <span class="ltx_bibblock">Perfect absorption in a disordered medium with programmable meta-atom inclusions. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib12.1.1">Advanced Functional Materials</span>, 30(52):2005310, 2020. </span> </li> <li class="ltx_bibitem" id="bib.bib13"> <span class="ltx_tag ltx_tag_bibitem">[13]</span> <span class="ltx_bibblock"> Sean Victor Hum, Michal Okoniewski, and Robert J Davies. </span> <span class="ltx_bibblock">Modeling and design of electronically tunable reflectarrays. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib13.1.1">IEEE transactions on Antennas and Propagation</span>, 55(8):2200–2210, 2007. </span> </li> <li class="ltx_bibitem" id="bib.bib14"> <span class="ltx_tag ltx_tag_bibitem">[14]</span> <span class="ltx_bibblock"> Cheng Huang, Changlei Zhang, Jianing Yang, Bo Sun, Bo Zhao, and Xiangang Luo. </span> <span class="ltx_bibblock">Reconfigurable metasurface for multifunctional control of electromagnetic waves. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib14.1.1">Advanced Optical Materials</span>, 5(22):1700485, 2017. </span> </li> <li class="ltx_bibitem" id="bib.bib15"> <span class="ltx_tag ltx_tag_bibitem">[15]</span> <span class="ltx_bibblock"> Timothy Sleasman, Robert Duggan, Ra’id S Awadallah, and David Shrekenhamer. </span> <span class="ltx_bibblock">Dual-resonance dynamic metasurface for independent magnitude and phase modulation. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib15.1.1">Physical Review Applied</span>, 20(1):014004, 2023. </span> </li> <li class="ltx_bibitem" id="bib.bib16"> <span class="ltx_tag ltx_tag_bibitem">[16]</span> <span class="ltx_bibblock"> Timothy A Sleasman, Mohammadreza F Imani, Aaron V Diebold, Michael Boyarsky, Kenneth P Trofatter, and David R Smith. </span> <span class="ltx_bibblock">Implementation and characterization of a two-dimensional printed circuit dynamic metasurface aperture for computational microwave imaging. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib16.1.1">IEEE Transactions on Antennas and Propagation</span>, 69(4):2151–2164, 2020. </span> </li> <li class="ltx_bibitem" id="bib.bib17"> <span class="ltx_tag ltx_tag_bibitem">[17]</span> <span class="ltx_bibblock"> Xinyun Song, Weixu Yang, Kai Qu, Xudong Bai, Ke Chen, Yijun Feng, and Weiren Zhu. </span> <span class="ltx_bibblock">Switchable metasurface for nearly perfect reflection, transmission, and absorption using pin diodes. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib17.1.1">Optics express</span>, 29(18):29320–29328, 2021. </span> </li> <li class="ltx_bibitem" id="bib.bib18"> <span class="ltx_tag ltx_tag_bibitem">[18]</span> <span class="ltx_bibblock"> Muhammad Saqib Rabbani, James Churm, and Alexandros P Feresidis. </span> <span class="ltx_bibblock">Continuous beam-steering low-loss millimeter-wave antenna based on a piezo-electrically actuated metasurface. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib18.1.1">IEEE Transactions on Antennas and Propagation</span>, 70(4):2439–2449, 2021. </span> </li> <li class="ltx_bibitem" id="bib.bib19"> <span class="ltx_tag ltx_tag_bibitem">[19]</span> <span class="ltx_bibblock"> Okan Yurduseven and David R Smith. </span> <span class="ltx_bibblock">Dual-polarization printed holographic multibeam metasurface antenna. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib19.1.1">IEEE Antennas and Wireless Propagation Letters</span>, 16:2738–2741, 2017. </span> </li> <li class="ltx_bibitem" id="bib.bib20"> <span class="ltx_tag ltx_tag_bibitem">[20]</span> <span class="ltx_bibblock"> Xiaokang Qi, Feng Li, Hangbin Zhao, Zhiwei Xu, and Huan Li. </span> <span class="ltx_bibblock">Steerable simultaneous multibeam antenna based on reconfigurable reflectarray. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib20.1.1">IEEE Antennas and Wireless Propagation Letters</span>, 22(8):2037–2041, 2023. </span> </li> <li class="ltx_bibitem" id="bib.bib21"> <span class="ltx_tag ltx_tag_bibitem">[21]</span> <span class="ltx_bibblock"> Xiaokang Qi, Feng Li, Zhuoqun Li, Kuiwen Xu, Zhiwei Xu, and Huan Li. </span> <span class="ltx_bibblock">2d steerable multi-beam antenna for concurrent wireless communication systems. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib21.1.1">IEEE Transactions on Circuits and Systems II: Express Briefs</span>, 2024. </span> </li> <li class="ltx_bibitem" id="bib.bib22"> <span class="ltx_tag ltx_tag_bibitem">[22]</span> <span class="ltx_bibblock"> Bing-Qing Zhao, Hui-Ming Wang, and Jia-Cheng Jiang. </span> <span class="ltx_bibblock">Safeguarding backscatter rfid communication against proactive eavesdropping. </span> <span class="ltx_bibblock">In <span class="ltx_text ltx_font_italic" id="bib.bib22.1.1">ICC 2020-2020 IEEE International Conference on Communications (ICC)</span>, pages 1–6. IEEE, 2020. </span> </li> <li class="ltx_bibitem" id="bib.bib23"> <span class="ltx_tag ltx_tag_bibitem">[23]</span> <span class="ltx_bibblock"> Miao Cui, Guangchi Zhang, and Rui Zhang. </span> <span class="ltx_bibblock">Secure wireless communication via intelligent reflecting surface. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib23.1.1">IEEE Wireless Communications Letters</span>, 8(5):1410–1414, 2019. </span> </li> <li class="ltx_bibitem" id="bib.bib24"> <span class="ltx_tag ltx_tag_bibitem">[24]</span> <span class="ltx_bibblock"> Jinming Wang, Shiji Wang, Shuai Han, and Cheng Li. </span> <span class="ltx_bibblock">Intelligent reflecting surface secure backscatter communication without eavesdropping csi. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib24.1.1">IEEE Communications Letters</span>, 27(6):1496–1500, 2023. </span> </li> <li class="ltx_bibitem" id="bib.bib25"> <span class="ltx_tag ltx_tag_bibitem">[25]</span> <span class="ltx_bibblock"> Kun Qian, Lulu Yao, Kai Zheng, Xinyu Zhang, and Tse Nga Ng. </span> <span class="ltx_bibblock">Uniscatter: a metamaterial backscatter tag for wideband joint communication and radar sensing. </span> <span class="ltx_bibblock">In <span class="ltx_text ltx_font_italic" id="bib.bib25.1.1">Proceedings of the 29th Annual International Conference on Mobile Computing and Networking</span>, pages 1–16, 2023. </span> </li> <li class="ltx_bibitem" id="bib.bib26"> <span class="ltx_tag ltx_tag_bibitem">[26]</span> <span class="ltx_bibblock"> Ryan A Bahr, Ajibayo O Adeyeye, Samantha Van Rijs, and Manos M Tentzeris. </span> <span class="ltx_bibblock">3d-printed omnidirectional luneburg lens retroreflectors for low-cost mm-wave positioning. </span> <span class="ltx_bibblock">In <span class="ltx_text ltx_font_italic" id="bib.bib26.1.1">2020 IEEE International Conference on RFID (RFID)</span>, pages 1–7. IEEE, 2020. </span> </li> <li class="ltx_bibitem" id="bib.bib27"> <span class="ltx_tag ltx_tag_bibitem">[27]</span> <span class="ltx_bibblock"> Petr Kaděra, Jesús Sánchez-Pastor, Hossein Eskandari, Tomáš Tyc, Masoud Sakaki, Martin Schüßler, Rolf Jakoby, Niels Benson, Alejandro Jiménez-Sáez, and Jaroslav Láčík. </span> <span class="ltx_bibblock">Wide-angle ceramic retroreflective luneburg lens based on quasi-conformal transformation optics for mm-wave indoor localization. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib27.1.1">IEEE Access</span>, 10:41097–41111, 2022. </span> </li> <li class="ltx_bibitem" id="bib.bib28"> <span class="ltx_tag ltx_tag_bibitem">[28]</span> <span class="ltx_bibblock"> Nathan Landy, Yaroslav Urzhumov, and David R. Smith. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib28.1.1">Quasi-Conformal Approaches for Two and Three-Dimensional Transformation Optical Media</span>, pages 1–32. </span> <span class="ltx_bibblock">Springer London, London, 2014. </span> </li> <li class="ltx_bibitem" id="bib.bib29"> <span class="ltx_tag ltx_tag_bibitem">[29]</span> <span class="ltx_bibblock"> Fei Sun, Bin Zheng, Hongsheng Chen, Wei Jiang, Shuwei Guo, Yichao Liu, Yungui Ma, and Sailing He. </span> <span class="ltx_bibblock">Transformation optics: From classic theory and applications to its new branches. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib29.1.1">Laser & Photonics Reviews</span>, 11(6):1700034, 2017. </span> </li> <li class="ltx_bibitem" id="bib.bib30"> <span class="ltx_tag ltx_tag_bibitem">[30]</span> <span class="ltx_bibblock"> Toby A. Driscoll. </span> <span class="ltx_bibblock">Schwarz-christoffel toolbox. </span> <span class="ltx_bibblock"><a class="ltx_ref ltx_url ltx_font_typewriter" href="https://www.mathworks.com/matlabcentral/fileexchange/1316-schwarz-christoffel-toolbox" title="">https://www.mathworks.com/matlabcentral/fileexchange/1316-schwarz-christoffel-toolbox</a>, 2023. </span> </li> <li class="ltx_bibitem" id="bib.bib31"> <span class="ltx_tag ltx_tag_bibitem">[31]</span> <span class="ltx_bibblock"> Simulia. </span> <span class="ltx_bibblock">Cst studio suite version: 2023. </span> </li> <li class="ltx_bibitem" id="bib.bib32"> <span class="ltx_tag ltx_tag_bibitem">[32]</span> <span class="ltx_bibblock"> Samuel Kim, Jamison Sloan, Josué J López, Dave Kharas, Jeffrey Herd, Suraj Bramhavar, Paul Juodawlkis, George Barbastathis, Steven Johnson, Cheryl Sorace-Agaskar, et al. </span> <span class="ltx_bibblock">Luneburg lens for wide-angle chip-scale optical beam steering. </span> <span class="ltx_bibblock">In <span class="ltx_text ltx_font_italic" id="bib.bib32.1.1">CLEO: Science and Innovations</span>, pages SF3N–7. Optica Publishing Group, 2019. </span> </li> <li class="ltx_bibitem" id="bib.bib33"> <span class="ltx_tag ltx_tag_bibitem">[33]</span> <span class="ltx_bibblock"> Zachary Larimore, Sarah Jensen, Austin Good, Aric Lu, John Suarez, and Mark Mirotznik. </span> <span class="ltx_bibblock">Additive manufacturing of luneburg lens antennas using space-filling curves and fused filament fabrication. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib33.1.1">IEEE transactions on antennas and propagation</span>, 66(6):2818–2827, 2018. </span> </li> <li class="ltx_bibitem" id="bib.bib34"> <span class="ltx_tag ltx_tag_bibitem">[34]</span> <span class="ltx_bibblock"> Soumitra Biswas, Aric Lu, Zachary Larimore, Paul Parsons, Austin Good, Nicholas Hudak, Benjamin Garrett, John Suarez, and Mark S Mirotznik. </span> <span class="ltx_bibblock">Realization of modified luneburg lens antenna using quasi-conformal transformation optics and additive manufacturing. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib34.1.1">Microwave and Optical Technology Letters</span>, 61(4):1022–1029, 2019. </span> </li> <li class="ltx_bibitem" id="bib.bib35"> <span class="ltx_tag ltx_tag_bibitem">[35]</span> <span class="ltx_bibblock"> Yujiao Guo, Yujian Li, Junhong Wang, Lei Ge, Zhan Zhang, Meie Chen, Zheng Li, Bo Ai, and Ruisi He. </span> <span class="ltx_bibblock">A 3d printed nearly isotropic luneburg lens antenna for millimeter-wave vehicular networks. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib35.1.1">IEEE Transactions on Vehicular Technology</span>, 71(2):1145–1155, 2021. </span> </li> <li class="ltx_bibitem" id="bib.bib36"> <span class="ltx_tag ltx_tag_bibitem">[36]</span> <span class="ltx_bibblock"> Soumitra Biswas and Mark Mirotznik. </span> <span class="ltx_bibblock">High gain, wide-angle qcto-enabled modified luneburg lens antenna with broadband anti-reflective layer. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib36.1.1">Scientific reports</span>, 10(1):12646, 2020. </span> </li> <li class="ltx_bibitem" id="bib.bib37"> <span class="ltx_tag ltx_tag_bibitem">[37]</span> <span class="ltx_bibblock"> Hui Xu, Jun Wei Wu, Zheng Xing Wang, Rui Wen Shao, Han Qing Yang, and Tie Jun Cui. </span> <span class="ltx_bibblock">Two-dimensional and high-order directional information modulations for secure communications based on programmable metasurface. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib37.1.1">Nature Communications</span>, 15(1):6140, 2024. </span> </li> <li class="ltx_bibitem" id="bib.bib38"> <span class="ltx_tag ltx_tag_bibitem">[38]</span> <span class="ltx_bibblock"> A Lee Swindlehurst, Gui Zhou, Rang Liu, Cunhua Pan, and Ming Li. </span> <span class="ltx_bibblock">Channel estimation with reconfigurable intelligent surfaces—a general framework. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib38.1.1">Proceedings of the IEEE</span>, 110(9):1312–1338, 2022. </span> </li> <li class="ltx_bibitem" id="bib.bib39"> <span class="ltx_tag ltx_tag_bibitem">[39]</span> <span class="ltx_bibblock"> Haiyang Zhang, Nir Shlezinger, George C Alexandropoulos, Avner Shultzman, Idban Alamzadeh, Mohammadreza F Imani, and Yonina C Eldar. </span> <span class="ltx_bibblock">Channel estimation with hybrid reconfigurable intelligent metasurfaces. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib39.1.1">IEEE transactions on communications</span>, 71(4):2441–2456, 2023. </span> </li> <li class="ltx_bibitem" id="bib.bib40"> <span class="ltx_tag ltx_tag_bibitem">[40]</span> <span class="ltx_bibblock"> KM Faisal and Wooyeol Choi. </span> <span class="ltx_bibblock">Machine learning approaches for reconfigurable intelligent surfaces: A survey. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib40.1.1">IEEE Access</span>, 10:27343–27367, 2022. </span> </li> <li class="ltx_bibitem" id="bib.bib41"> <span class="ltx_tag ltx_tag_bibitem">[41]</span> <span class="ltx_bibblock"> Dilip Datta, Kalyanmoy Deb, and Carlos M Fonseca. </span> <span class="ltx_bibblock">Multi-objective evolutionary algorithms for resource allocation problems. </span> <span class="ltx_bibblock">In <span class="ltx_text ltx_font_italic" id="bib.bib41.1.1">Evolutionary Multi-Criterion Optimization: 4th International Conference, EMO 2007, Matsushima, Japan, March 5-8, 2007. Proceedings 4</span>, pages 401–416. Springer, 2007. </span> </li> <li class="ltx_bibitem" id="bib.bib42"> <span class="ltx_tag ltx_tag_bibitem">[42]</span> <span class="ltx_bibblock"> The MathWorks Inc. </span> <span class="ltx_bibblock">Matlab version: R2024b. </span> </li> <li class="ltx_bibitem" id="bib.bib43"> <span class="ltx_tag ltx_tag_bibitem">[43]</span> <span class="ltx_bibblock"> Ruolei Xu and Zhi Ning Chen. </span> <span class="ltx_bibblock">A hemispherical wide-angle beamsteering near-surface focal-plane metamaterial luneburg lens antenna using transformation-optics. </span> <span class="ltx_bibblock"><span class="ltx_text ltx_font_italic" id="bib.bib43.1.1">IEEE Transactions on Antennas and Propagation</span>, 70(6):4224–4233, 2022. </span> </li> </ul> </section> <figure class="ltx_float biography" id="id1"> <table class="ltx_tabular" id="id1.1"> <tr class="ltx_tr" id="id1.1.1"> <td class="ltx_td" id="id1.1.1.1"><img alt="[Uncaptioned image]" class="ltx_graphics ltx_img_square" height="100" id="id1.1.1.1.g1" src="extracted/6295025/author_pics/photo_sam.jpg" width="83"/></td> <td class="ltx_td" id="id1.1.1.2"> <span class="ltx_inline-block" id="id1.1.1.2.1"> <span class="ltx_p" id="id1.1.1.2.1.1"><span class="ltx_text ltx_font_bold" id="id1.1.1.2.1.1.1">Samuel Kim</span> received the A.B. degree in physics from Harvard University, Cambridge, MA in 2015 and the Ph.D. degree in electrical engineering and computer science from the Massachusetts Institute of Technology in 2023.</span> <span class="ltx_p" id="id1.1.1.2.1.2">His graduate work focused on the intersection of physics and machine learning, developing algorithms for scientific discovery, Bayesian optimization, and photonics. In 2021, he co-founded Kyber Photonics, a spinoff from his research on silicon photonics devices for optical beam steering. He is currently a research scientist at the Johns Hopkins Applied Physics Laboratory where he focuses on advanced optimization algorithms and computational electromagnetics with applications in metamaterials and integrated photonics.</span> <span class="ltx_p" id="id1.1.1.2.1.3">Dr. Kim was a recipient of the National Defense Science and Engineering Graduate (NDSEG) Fellowship in 2019.</span> </span> </td> </tr> </table> </figure> <figure class="ltx_float biography" id="id2"> <table class="ltx_tabular" id="id2.1"> <tr class="ltx_tr" id="id2.1.1"> <td class="ltx_td" id="id2.1.1.1"><img alt="[Uncaptioned image]" class="ltx_graphics ltx_img_square" height="100" id="id2.1.1.1.g1" src="extracted/6295025/author_pics/photo_tim.jpg" width="100"/></td> <td class="ltx_td" id="id2.1.1.2"> <span class="ltx_inline-block" id="id2.1.1.2.1"> <span class="ltx_p" id="id2.1.1.2.1.1"><span class="ltx_text ltx_font_bold" id="id2.1.1.2.1.1.1">Timothy A. Sleasman</span> (Member, IEEE) received the B.S. degree in mathematics and physics from Boston College, Chestnut Hill, MA, USA, in 2013 and the Ph.D. degree from the Department of Electrical and Computer Engineering, Duke University, Durham, NC, USA, in 2018.</span> <span class="ltx_p" id="id2.1.1.2.1.2">From 2013 to 2018, he was with the Center for Metamaterials and Integrated Plasmonics, Duke University. He is currently a Senior Researcher with the John Hopkins University Applied Physics Lab, Laurel, MD, USA. His current research interests include computational imaging, dynamically tunable metasurfaces, and novel platforms for generating tailored electromagnetic wavefronts.</span> </span> </td> </tr> </table> </figure> <figure class="ltx_float biography" id="id3"> <table class="ltx_tabular" id="id3.1"> <tr class="ltx_tr" id="id3.1.1"> <td class="ltx_td" id="id3.1.1.1"><img alt="[Uncaptioned image]" class="ltx_graphics ltx_img_square" height="100" id="id3.1.1.1.g1" src="extracted/6295025/author_pics/photo_avrami.jpg" width="100"/></td> <td class="ltx_td" id="id3.1.1.2"> <span class="ltx_inline-block" id="id3.1.1.2.1"> <span class="ltx_p" id="id3.1.1.2.1.1"><span class="ltx_text ltx_font_bold" id="id3.1.1.2.1.1.1">Avrami Rakovsky</span> is a Mechanical Engineer at the Johns Hopkins University Applied Physics Laboratory (JHU/APL). Avrami received a B.S. in Industrial Design from the New Jersey Institute of Technology in 2012, providing a foundation in human-centered design principles. Building upon this, he earned a M.S. in Mechanical Engineering from Johns Hopkins University, in 2022.</span> <span class="ltx_p" id="id3.1.1.2.1.2">Driven by a passion for solving real-world challenges, Avrami specializes in translating design concepts into tangible, functional products. His expertise encompasses product design, material development, and the application of specialized software, utilizing a multidisciplinary approach to innovation. He focuses on advanced materials and fabrication methods, with his current research exploring the dynamic relationship between product design and advanced material development.</span> </span> </td> </tr> </table> </figure> <figure class="ltx_float biography" id="id4"> <table class="ltx_tabular" id="id4.1"> <tr class="ltx_tr" id="id4.1.1"> <td class="ltx_td" id="id4.1.1.1"><img alt="[Uncaptioned image]" class="ltx_graphics ltx_img_square" height="100" id="id4.1.1.1.g1" src="extracted/6295025/author_pics/photo_raid.jpg" width="100"/></td> <td class="ltx_td" id="id4.1.1.2"> <span class="ltx_inline-block" id="id4.1.1.2.1"> <span class="ltx_p" id="id4.1.1.2.1.1"><span class="ltx_text ltx_font_bold" id="id4.1.1.2.1.1.1">Ra’id Awadallah</span> received his Ph.D. in Electrical Engineering from Virginia Tech in 1998. He joined Johns Hopkins University Applied Physics Laboratory in the same year and he is currently a member of the principal professional staff. Over the last 27 years, he has led a team of researchers developing efficient numerical models for tropospheric propagation, electromagnetic scattering from randomly rough surfaces, radar cross-section of complex targets, pulsed propagation in complex urban structures, and design and characterization of thin-film metamaterials for various applications. Dr. Awadallah is a member of the IEEE and Commission F or URSI. He is also a lecturer at the Johns Hopkins University Engineering Program for professionals where he teaches courses on basic and applied electromagnetism.</span> </span> </td> </tr> </table> </figure> <figure class="ltx_float biography" id="id5"> <table class="ltx_tabular" id="id5.1"> <tr class="ltx_tr" id="id5.1.1"> <td class="ltx_td" id="id5.1.1.1"><img alt="[Uncaptioned image]" class="ltx_graphics ltx_img_square" height="100" id="id5.1.1.1.g1" src="extracted/6295025/author_pics/photo_david.jpg" width="100"/></td> <td class="ltx_td" id="id5.1.1.2"> <span class="ltx_inline-block" id="id5.1.1.2.1"> <span class="ltx_p" id="id5.1.1.2.1.1"><span class="ltx_text ltx_font_bold" id="id5.1.1.2.1.1.1">David Shrekenhamer</span> is a Staff Scientist at the Johns Hopkins University Applied Physics Laboratory (JHU/APL) and has a secondary research scientist position in the Department of Electrical and Computer Engineering at Johns Hopkins University. Dr. Shrekenhamer received a B.S. degree in physics from University of California San Diego, La Jolla, CA in 2006 and the Ph.D. degree in physics and from Boston College, Chestnut Hill, MA in 2013.</span> <span class="ltx_p" id="id5.1.1.2.1.2">His research interests span numerous areas of electromagnetics, and he has publications and expertise developing concepts that utilize metamaterials for use from DC to light. At APL, he specializes in leading interdisciplinary teams designing custom electromagnetic solutions for a range of complex problem spaces e.g., communications, signature science, sensing and imaging. While his work is largely focused on electromagnetic-based problems, he also has substantial expertise in developing novel materials (e.g. optical phase change materials), fabrication techniques (e.g. nanofabrication and incorporation within novel host materials and form factors), electronic circuitry and signal processing, and incorporating and accounting for multi-physics effects such as electromagnetic heating, structural heating, and more exotic behavior like nonlinear effects, chirality, and bi-anisotropy.</span> </span> </td> </tr> </table> </figure> <div class="ltx_pagination ltx_role_newpage"></div> <div class="ltx_para" id="p5"> <span class="ltx_ERROR undefined" id="p5.1">\EOD</span> </div> </article> </div> <footer class="ltx_page_footer"> <div class="ltx_page_logo">Generated on Thu Mar 20 17:14:24 2025 by <a class="ltx_LaTeXML_logo" href="http://dlmf.nist.gov/LaTeXML/"><span style="letter-spacing:-0.2em; margin-right:0.1em;">L<span class="ltx_font_smallcaps" style="position:relative; bottom:2.2pt;">a</span>T<span class="ltx_font_smallcaps" style="font-size:120%;position:relative; bottom:-0.2ex;">e</span></span><span style="font-size:90%; position:relative; bottom:-0.2ex;">XML</span><img alt="Mascot Sammy" src="data:image/png;base64,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"/></a> </div></footer> </div> </body> </html>