Broadband electromagnetic wave absorption

Abstract

Embodiments can include, for example: a conductor-backed dielectric; and circuitry connected in parallel with the conductor-backed dielectric, wherein the circuitry connected in parallel with the conductor‑backed substrate exhibits an admittance with an imaginary component having a negative slope within a design absorption frequency range of the absorber structure.

Description

BROADBAND ELECTROMAGNETIC WAVE ABSORPTION CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority to U.S. Patent Application No.63 / 646,311, filed May 13, 2024, entitled, “BROADBAND ELECTROMAGNETIC WAVE ABSORPTION,” which is incorporated herein by reference in its entirety. This application also claims priority to U.S. Patent Application No.63 / 693,061, filed September 10, 2024, entitled, “BROADBAND ELECTROMAGNETIC WAVE ABSORPTION,” which is incorporated herein by reference in its entirety. BACKGROUND

[0002] Embodiments herein relate generally to electromagnetic waves and specifically to broadband absorption of electromagnetic waves.

[0003] Numerous applications from radio to optical frequencies can benefit from structures, often in layer form, capable of absorbing electromagnetic waves. From a practical point of view, there is an ever-growing demand for thinner absorbing layers that provide higher absorption bandwidths. However, there is an upper bound on the bandwidth-to-thickness ratio of these absorbers. The thinner these absorbers get, the narrower absorption bandwidth they provide. In one aspect, absorbers developed to date, irrespective of their operational frequency range or material thickness, significantly underperform when compared to this upper limit, failing to exploit the full potential that linear time-invariant systems can provide. BRIEF DESCRIPTION

[0004] Embodiments herein can include, for example, a conductor-backed substrate; and circuitry connected in parallel with the conductor-backed substrate, wherein the circuitry connected in parallel with the conductor-backed substrate exhibits an admittance with an imaginary component having a negative slope within a design absorption frequency range of the absorber structure. Page 1 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)

[0005] In one particular embodiment, the conductor-backed substrate can include a conductor- backed dielectric.

[0006] In one particular embodiment, the conductor-backed substrate can include a conductor- backed magnetic substrate.

[0007] In one particular embodiment, the circuitry connected in parallel with the conductor- backed substrate can include, for example, a capacitor filled with a dispersive dielectric.

[0008] In one particular embodiment, the circuitry connected in parallel with the conductor- backed substrate can include, for example, a capacitor connected in parallel with a series connected circuit path, the series connected circuit path including a series connected resistor, a series connected capacitor, and a series connected inductor.

[0009] In one particular embodiment, the circuitry connected in parallel with the conductor- backed substrate can include, for example, a capacitor in series with first and second circuit paths that are connected in parallel to one another, the first circuit path having a second capacitor, and the second circuit path having an inductor in series with a resistor.

[0010] Embodiments herein can include, for example, a conductor-backed substrate; and a capacitor filled with dispersive material connected in parallel with the conductor-backed substrate, wherein the capacitor filled with dispersive material exhibits an admittance with an imaginary component having a negative slope within a design absorption frequency range of the absorber structure.

[0011] Embodiments herein can include, for example, configuring circuitry connected in parallel with a conductor-backed substrate, wherein the configuring includes performing the configuring so that the circuitry connected in parallel with the conductor-backed substrate exhibits an admittance with an imaginary component having a negative slope within a design absorption frequency range of the absorber structure.

[0012] Embodiments herein can include, for example, a conductor-backed dielectric; and circuitry connected in parallel with the conductor-backed dielectric, wherein the circuitry connected in parallel with the conductor-backed dielectric exhibits an admittance with an Page 2 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)imaginary component having a negative slope within a frequency range encompassing a design frequency of the absorber structure.

[0013] In one particular embodiment, the circuitry connected in parallel with the conductor- backed dielectric can include, for example, a capacitor filled with a dispersive dielectric.

[0014] In one particular embodiment, the circuitry connected in parallel with the conductor- backed dielectric can include, for example, a capacitor connected in parallel with a series connected circuit path, the series connected circuit path including a series connected resistor, a series connected capacitor, and a series connected inductor.

[0015] In one particular embodiment, the circuitry connected in parallel with the conductor- backed dielectric can include, for example, a capacitor in series with first and second circuit paths that are connected in parallel to one another, the first circuit path having a second capacitor, and the second circuit path having an inductor in series with a resistor.

[0016] Embodiments herein can include, for example, a conductor-backed dielectric; and a capacitor filled with dispersive material connected in parallel with the conductor-backed dielectric, wherein the capacitor filled with dispersive material exhibits an admittance with an imaginary component having a negative slope within a frequency range encompassing a design frequency of the absorber structure.

[0017] Embodiments herein can include, for example, configuring circuitry connected in parallel with a conductor-backed dielectric, wherein the configuring includes performing the configuring so that the circuitry connected in parallel with the conductor-backed dielectric exhibits an admittance with an imaginary component having a negative slope within a frequency range encompassing a design frequency of the absorber structure.

[0018] Additional features are realized through the techniques set forth herein. Other embodiments and aspects, including but not limited to methods, computer program product and system, are described in detail herein and are considered a part of the claimed invention. BRIEF DESCRIPTION OF THE DRAWINGS Page 3 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)

[0019] One or more aspects of the present invention are particularly pointed out and distinctly claimed as examples in the claims at the conclusion of the specification. The foregoing and other objects, features, and advantages of the invention are apparent from the following detailed description taken in conjunction with the accompanying drawings in which:

[0020] Fig.1 depicts as follows: (a) Illustrative schematic of the normal wave incidence onto an absorber composed of an impedance grid with grid admittance ^^^^^^^^located with a distance d from a ground plane. The space between the grid and the ground plane is free space. The middle panel demonstrates the circuit model for this scenario. The right panel is the simplified version of thecircuit model when ^^^^ ≪ ^^^^. (b) The conventional way of achieving perfect matching using acapacitive grid. (c) Profiles of the imaginary parts of admittance provided by the capacitive grid ^^^^^^^^(^^^^^^^^), negative inductor grid ^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^{^^^^^^^^} and required grid admittance ^^^^^^^^{^^^^^^^^} (d) Matching network employing negative inductor.

[0021] Fig.2 depicts as follows: (a) Matching network using Lorentz dispersive dielectric-filled capacitor whose grid admittance is ^^^^^^^^^^^^^^^^^^^^^^^^(b) Lorentz dispersive dielectric-filled capacitor emulating negative slope around resonant frequency ω0. In other words, negative inductive characteristics can be seen within a finite band around ω0. Δ^^^^^^^^^^^^^^^^is the bandwidth with negative slope. (c) One of the circuit representations of Lorentz dispersive dielectric-filled capacitor.

[0022] Fig.3 depicts as follows: (a) Normalized admittance profiles of Case I and Case II dispersion engineered ^^^^^^^^^^^^^^^^^^^^^^^^. The shaded areas represent the bandwidth of the negative slope observed in the imaginary part of the grid admittance. (b) The reflection and absorption spectrum of a λ0 / 30 thick absorber designed based on Case I and Case II dispersion engineering (section entitled Design of Electrically Thin Wideband Absorbers) with maximum allowable reflection ρ0 = 0.1.

[0023] Fig.4 depicts as follows: (a) Several shunt resonators made of single order Lorentz dispersive dielectric capacitors to form higher order matching grid. (b) An equivalent circuit model of higher order Lorentz dispersive grid ^^^^^^^^^^^,^^^^^^^^^^^^^^^^^can be abstracted as an RLC resonator shunted with the capacitor. The design parameters for first order can be calculated using closed form solution given in the section entitled Design of Electrically Thin Wideband Absorbers. The Page 4 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)extracted circuit parameters for a second order grid are as follows : ^^^^0,2= 0.11 ^^^^^^^^, ^^^^2= 0.013^^^^^^^^, ^^^^2 = 16.2 ^^^^^^^^,^^^^2 = 279.1Ω, ^^^^0,1 = 0.09 ^^^^^^^^,^^^^1 = 0.018 ^^^^^^^^, ^^^^1 = 16.9 ^^^^^^^^,^^^^1 = 259.8 Ω ,circuit parameters for third order grid are as follows: ^^^^0,3 = 0.076 ^^^^^^^^,^^^^3 = 0.01 ^^^^^^^^, ^^^^3 = 27.8^^^^^^^^,^^^^ = 3738.331Ω, ^^^^0,2 = 0.064 ^^^^^^^^,^^^^2 = 0.01 ^^^^^^^^, ^^^^2 = 19.34 ^^^^^^^^,^^^^2 = 320.1Ω, ^^^^0,1 = 0.062^^^^^^^^,^^^^1 = 0.011 ^^^^^^^^, ^^^^1 = 28.25 ^^^^^^^^,^^^^1 = 312.8Ω, circuit parameters for fourth order grid are asfollows : ^^^^0,4 = 0.051 ^^^^^^^^,^^^^4 = 0.0075 ^^^^^^^^, ^^^^4 = 31.527 ^^^^^^^^,^^^^4 = 348.431Ω,^^^^0,3 = 0.051 ^^^^^^^^,^^^^3 =0.0054 ^^^^^^^^, ^^^^3 = 67.3 ^^^^^^^^,^^^^3 = 402.9Ω,^^^^0,2 = 0.051 ^^^^^^^^ , ^^^^2 = 0.0086 ^^^^^^^^, ^^^^2 = 34.26^^^^^^^^,^^^^2 =331.657Ω, ^^^^0,1 = 0.051 ^^^^^^^^, ^^^^1 = 0.0086 ^^^^^^^^, ^^^^1 = 34.26 ^^^^^^^^,^^^^1 = 331.65Ω, circuit parametersfor fifth order grid are as follows : ^^^^0,5= 0.041 ^^^^^^^^,^^^^5= 0.0057 ^^^^^^^^, ^^^^5= 37.44 ^^^^^^^^, ^^^^5=365.9 Ω,^^^^0,4 = 0.041 ^^^^^^^^,^^^^4 = 0.0081 ^^^^^^^^, ^^^^4 = 38.78 ^^^^^^^^,^^^^4 = 332.051Ω,^^^^0,3 = 0.041 ^^^^^^^^,^^^^3 =0.0064 ^^^^^^^^, ^^^^3 = 40 ^^^^^^^^,^^^^3 = 372.37Ω, ^^^^0,2 = 0.041 ^^^^^^^^,^^^^2 = 0.0037 ^^^^^^^^, ^^^^2 = 48.41 ^^^^^^^^, ^^^^2 =398.1 Ω, ^^^^0,1 =0.041 ^^^^^^^^,^^^^1 = 0.0036 ^^^^^^^^, ^^^^1 =102.63 ^^^^^^^^, ^^^^1 = 469.5 Ω (c) Real and imaginaryprofiles of admittance, obtained from n = 1 to n = 5 order dispersive grids. There are depictedadmittance profiles for higher order dispersive grid (i.e.,^^^^ = 1 through 5). It can be seen that asthe order of dispersion increases, the bandwidth of the negative slope of Im{Yg} increases, and Re{Yg} becomes flatter around ω0. (d) The reflection spectrum of the absorber based on n = 1 to n = 5 order dispersive grids. It can be seen that bandwidths of absorption increase as the order of dispersion increases. Please note that the maximum allowable reflection from the structure is set to -10 dB = 0.316.

[0024] Fig.5 depicts as follows: (a) Comparison of bandwidth to thickness ratio of the presented absorber, absorber based on conventional RC grid, Rozanov's ultimate limit

[0016] , and upper bounds of absorbers based on artificially engineered surfaces [17, 18] (c) Comparison of the bandwidth-to-thickness ratio of the presented multi-order order dispersive grid-based absorber with Rozanov's limits on metal-backed thin multi-order dispersive dielectric absorber.

[0025] Fig.6 depicts as follows: Bandwidth-to-thickness ratio comparison between the presented concept, Rozanov's limits, and existing broadband absorbers in literature. The y axis is bandwidths normalized to Rozanov's infinite order dispersive dielectric bound and x axis is the thickness normalized with design wavelength. The numbers on the graph indicate the references. Page 5 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)

[0026] Fig.7 depicts as follows: (a) The suggested equivalent RLC model for a dispersive grid with f0 = 10 GHz, d = λ0 / 30, and ρ0 = 0.15. The circuit parameters are as follows : ^^^^0= 0.2016 ^^^^^^^^, ^^^^1= 13.22 ^^^^^^^^, ^^^^1= 308.234 Ω, ^^^^1= 19.151 ^^^^^^^^ , and ^^^^^^^^^^^^= 1.25 ^^^^^^^^ (b) Equivalent circuit after Zobel’s transformation and its circuit parameters are as follows: ^^^^0′= 0.215 ^^^^^^^^, ^^^^1′= 3.2735^^^^^^^^, ^^^^′1 = 1.1687 Ω, ^^^^′1 = 0.072615 ^^^^^^^^ (c) Different layers used in the unit cell design, thethickness of the substrate used is around 0.254 mm ≈ ^^^^0 / 120 and spacing between the substrateand ground plane is around 0.746 mm≈ ^^^^0 / 40 (d) Full-wave simulation results of designed unitcell and comparison with theoretical results (e) The fabricated prototype of the presented dispersive grid and measurement setup including a spot focusing lens and holders (f) Measurement, simulation, and theoretical results of reflection coefficient for the designed and fabricated absorber under normal incident.

[0027] Fig.8 depicts as follows: (a) Initializing the ^^^^^^^^ℎorder grid parameters. (b) Overview of the design methodology for higher-order dispersive grid-based absorbers.

[0028] Fig.9 depicts as follows: (a) The circles on the graphs represent the numerically obtained maximum bandwidth-to-thickness ratios at various reflection threshold levels for ^^^^^^^^ℎorderabsorbers (^^^^ = 2, 3, 4, 5). The solid lines correspond to the empirical equations describing therelationship between the bandwidth-to-thickness ratio and the reflectivity for a given order of dispersion. It can be seen that there is good agreement between the empirical relation with the values extracted through the presented semi-analytical design methodology. (b) The comparison of the derived upper bound on bandwidth-to-thickness ratio of dispersive grid-based absorbers with fundamental limits established by Rozanov.

[0029] Fig.10 depicts a processing circuit according to one embodiment.

[0030] Fig 11 depicts as follows: (a) Illustration of normal wave incidence on magnetic absorber. (b) Equivalent transmission line model of the magnetic absorber. (c) Equivalent circuit model of the magnetic absorber when thickness is very less compared to wavelength. (d) Admittance of electrically thin metal-backed magnetic substrate. (e) Conventional matching method using capacitive grid. (f) Imaginary part of admittance profiles for capacitive grid, required for wideband matching and metal-backed substrate. Page 6 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)

[0031] Fig.12 depicts as follows: (a) Equivalent circuit model of the described dispersive grid based magnetic absorber. (b) Comparison of the imaginary parts of the admittance for the metal- backed magnetic substrate, the required admittance for wideband matching, and the admittance provided by the described dispersive grid. (c) Comparison of the imaginary parts of required admittance for wideband matching, the admittance provided by the described dispersive grid and admittance of capacitive grid. (d) An electrically thin magnetic absorber operating around10^GHz with a total thickness of 1.5^mm, ^^^^^^^^=2 and ^^^^ = 0.4.

[0032] Fig.13 depicts as follows: (a) Equivalent circuit model of the described dispersive grid based magnetic absorber with higher order dispersion. (b) Illustration of the bandwidth enhancement of the designed magnetic absorber with higher order dispersion. (c) Maximum achievable Δ^^^^ / ^^^^ for different orders of dispersion in the grid and substrate. (d) Comparison of the Δ^^^^ / ^^^^ of the existing magnetic absorbers from the literature with that of described bounds. DETAILED DESCRIPTION

[0033] From a practical point of view, there is an ever-growing demand for thinner absorbing layers that provide higher absorption bandwidths. However, a theoretical upper limit exists for the bandwidth-to-thickness ratio of passive, linear time-invariant absorbing layers. Additionally, specific bounds have been established for particular absorber designs, such as those based on high-impedance surfaces. The thinner these absorbers get, the narrower absorption bandwidth they provide. Absorbers developed to date, irrespective of their operational frequency range or material thickness, significantly underperform when compared to the upper theoretical limit, failing to exploit the full potential that linear time-invariant systems can provide. Here, we introduce a new concept for designing passive ultra-thin absorbers that enables absorbing layers with record-high bandwidth-to-thickness ratio which can be several-fold higher compared to the same thickness absorbers designed based on conventional approaches. Absorbers designed based on this concept can provide bandwidth-to-thickness ratio arbitrarily close to the ultimate theoretical limit for passive, linear time-invariant structures. Furthermore, it will be demonstrated that absorbers based on the presented concept can exceed all established bounds for high-impedance-based absorbers. Utilizing this concept, we have designed and experimentally verified an absorber yielding highest bandwidth-to-thickness ratio ever reported in any part of Page 7 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)the frequency spectrum. Utilizing this concept, we have designed and experimentally verified an absorber yielding highest bandwidth-to-thickness ratio ever reported in any part of the frequency spectrum relative to the theoretical upper limit. INTRODUCTION

[0034] An electromagnetic absorber can include a layer that effectively absorbs the incident electromagnetic waves, either by converting them into heat or by storing the energy of these waves for future use. Two main classical types of microwave absorbers include Dallenbach [1] and Salisbury [2] absorbers. A Dallenbach absorber comprises a metal-backed, lossy dielectric layer whose thickness is one-quarter of the wavelength. In contrast, a Salisbury absorber comprises a resistive sheet positioned at a distance of one-quarter of a wavelength from the metallic ground plane. Since the introduction of these two absorbers, extensive research has been conducted on applying these concepts to different realms of microwave engineering, employing a variety of both naturally occurring and artificially engineered materials [3,4]. Although capable of outstanding absorption at a designated frequency, the inherent requirement of thick layers and the resulting narrow bandwidth limit the applicability of these conventional absorbers in many applications. In recent years, employing artificially engineered surfaces, such as high-impedance surfaces (HIS) and metasurfaces, has revitalized the field of ultrathin absorbers, providing a considerable improvement in design flexibility and miniaturization [5-9]. Over the past few decades, these absorbing layers have become indispensable components in various technological applications [10-15]. Despite achieving significantly reduced thicknesses compared to their traditional counterparts, these ultrathin absorbers often suffer from limitations in bandwidth. From a practical perspective, a key objective has been to engineer absorbers with minimal thickness while reducing reflectance across the widest possible bandwidth.

[0035] Nonetheless, Rozanov's investigation established the fundamental limit on the maximum absorption bandwidth-to-thickness ratio for thin, metal-backed dielectric slab absorbers (where the dielectric slab can be composed of any material)

[0016] . As the thickness of absorbers decreases, their effective absorption bandwidth inevitably narrows. Throughout the paper, we will refer to this bound as Rozanov's ultimate limit. Rozanov also introduced a tighter bound for the more practically relevant case of thin, metal-backed dielectric absorbers with generalized Page 8 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)resonance dispersion law of permittivity

[0016] . This bound is lower than his ultimate limit and will be referred to as Rozanov's ultimate bound for dispersive dielectric absorbers. Subsequently, a comparable upper bound on the absorption bandwidth for thin absorbers based on artificially engineered surfaces (high-impedance surfaces or metasurfaces) were established [17, 18]. These bounds are lower compared to both of the bounds previously deduced by Rozanov. To date, absorbers - regardless of their operational frequency range, thickness, and complexity - significantly underperform relative to both bounds defined by Rozanov, failing to exploit the full potential linear time-invariant passive systems can provide.

[0036] Here, we introduce a new concept for ultrathin absorbers that can enable thin absorbers with bandwidth-to-thickness ratio arbitrarily close to Rozanov's ultimate limit. In the presented concept, the absorber consists of a meticulously engineered passive non-Foster impedance grid, positioned at a distance from a ground plane. The non-Foster impedance grid is designed within a completely passive platform through the use of dispersion engineering. Moreover, we will demonstrate that absorbers designed according to the presented concept, which incorporate a specific order of dispersion in the impedance grid, can achieve a bandwidth-to-thickness ratio that surpasses Rozanov's established limit for a metal-backed dielectric absorber of the same thickness and identical dispersion order within the dielectric. With the fourth order of Lorentz dispersion in the impedance grid, the bandwidth-to-thickness ratio of the presented design will exceed Rozanov's established limit for a metal-backed dielectric absorber that incorporates an infinite order of dispersion in the dielectric. Furthermore, it will be shown that absorbers designed based on the presented concept, with first-order Lorentz dispersion in the impedance grid, surpass both bounds defined for absorbers based on artificially engineered surfaces, such as high-impedance surfaces and metasurfaces [17, 18] As a proof of principle, we have designed and experimentally measured an absorber based on the presented concept yielding highest bandwidth-to-thickness ratio ever reported for any passive absorber relative to Rozanov’s theoretical limit.

[0037] The text is organized as follows. The section entitled Theory provides an overview of the theory behind the metal-backed dielectric absorbers and introduces the main concept. The section entitled Design of Electrically Thin Wideband Absorbers and the section entitled Multi- Order Lorentzian Dispersion Matching Grids detail the design methodology of absorbers based Page 9 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)on concepts sets forth herein. In the section entitled Comparing the Bandwidth-to-Thickness Ration of the Presented Concept with the Established Limits Found in the Literature, we compare our results with existing work and theoretical upper limits found in the literature. The section entitled Simulation, Measurement and Methodologies discusses numerical simulation and measurement results, and aspects of methodologies that are set forth throughout the disclosure herein. THEORY

[0038] A. Conventional absorbers

[0039] Let us start by reviewing the concept of a generic high-impedance-surface-based absorber, shown in Fig.1a, featuring conductor-backed dielectric 102, which can be in planar, plate form. The described dielectric of conductor-backed dielectric 102 can be provided by air or another dielectric. Under normal wave illumination, this absorbing layer can be modeled by a shorted transmission line shunted with the grid admittance (see Fig.1a). For simplicity, let us assume that the ground plane is a perfect electric conductor and that the spacer between the ground plane and the impedance grid is air (air dielectric). If the thickness of the spacer d is notably smaller than the wavelength at the design frequency i.e., d << λ0, the input admittancelooking towards the ground plane from beneath the grid will be ^^^^2^^^^ ^^^^^^^^ = −^^^^ / �^^^^0 tan�^^^^^^^^ ≈−^^^^ / [^^^^^^^^^^^^^^^^] . Here, η0 is the free space intrinsic impedance and λ0 is tt the design frequency. Conductor-backed dielectric 102 can be, e.g., a metal-backed dielectric, metal alloy- backed dielectric, or other conductor-backed dielectric, e.g., a doped semiconductor-backed dielectric.

[0040] Hence, in our transmission line model, we can readily substitute the thin conductor-backed dielectric 102 with an inductor having an inductance value of ^^^^^^^^^^^^ = ^^^^0^^^^. Now, for theabsorber to be perfectly matched to the free space, the grid should provide the admittance equal to ^^^^^^^^^^^^^^^^^^^^ = 1 / ^^^^0 + ^^^^ / [^^^^0^^^^^^^^]. The conventional method to attain this admittance is by employinga capacitive sheet in parallel with a resistive layer directly above it. In terms of the transmission line model, this can be accomplished using a simple capacitor with a capacitance of ^^^^^^^^= 1 / (^^^^02^^^^^^^^^^^^) along with a resistive sheet characterized by a resistance of ^^^^0, as illustrated in Fig. Page 10 of 83 Δ^^^^ ^^^^0Attorney ^ D^^^o=ck4et^^^^No.3153.149AWO (SU 2024- �1 − ^^^^2024) 0lb

[0019] , This is a commonly employed practice when designing an absorbing layer based on high-impedance surfaces. The maximum bandwidth-to-thickness ratio of these conventional RC grid-based absorbers can be derived as

[0041] (1)

[0042] where pQis the maximum allowable reflection within the absorption bandwidth AT. Although capable of absorbing incoming electromagnetic waves at a single frequency, this scenario fails to provide a wide absorption bandwidth. Next, we will consider elements beneficial to realizing wideband absorbing layers.

[0043] B. Wideband perfect absorption

[0044] To achieve broadband perfect absorption, the impedance matching condition discussed above can be met across a wide frequency range of interest (absorber structure design absorption frequency range), rather than being limited to a single frequency. However, close examination of the imaginary parts of required grid admittance [i.e., Im{Ygeq] = l / (jioda))] and the admittance provided by the capacitive gridreveals that, in this scenario, perfect matching can only happen at a single frequency. This is the case because the required and provided admittance by the capacitive grid exhibit different slope signs, i.e., d[Im{YgReq}} dco = -1 / \jioco2d\ < 0 andd[ / m{}^c}] / dm = Cg> 0 [see Fig. lc], The imaginary part of the required grid admittance consistently decreases as frequency increases (representing a non-Foster behavior), while the imaginary part of the lossy capacitive grid admittance continuously rises with frequency (showing a Foster behavior), eliminating the possibility of achieving broadband matching in this scenario. Embodiments herein set forth to identify an alternative component that can, at least within a limited frequency range, exhibit a trend similar to the required admittance. Analysis of Ygeqreveals that the required admittance is non-Foster, thus enhancing a benefit associated to use of a non-Foster element to fulfill this condition at all frequencies. Embodiments herein recognize that this can be achieved by utilizing a negative inductor Lneg= —p.od instead of a capacitor, ensuring perfect matching across all frequencies (see Fig. lc and Fig. Id). One approach to implementing such a negative inductor is through the use of active circuits, which tend to be complex and power-hungry [20-22], Apart from the challenges in implementation,these active circuits can only emulate the behavior of a negative inductor over a limited bandwidth and are susceptible to stability issues. In what follows, we will introduce a new concept based on dispersion engineering, which enables emulating the characteristics of a negative inductor over a desired bandwidth within a fully passive platform.

[0045] C. Realizing negative inductors using dispersion engineering

[0046] To mimic the characteristics of a negative inductor within a fully passive network, embodiments herein can benefit from a platform that allows for dispersion engineering with a high degree of flexibility. Embodiments herein recognize that one platform to achieve this is by utilizing a parallel-plate capacitor with capacitance Co filled with a highly dispersive material (see Fig. 2a). Similar dispersive platform was used by Yaghjian to nullify the Q energy associated with a matching capacitor in the matching network of electrically small antennas

[0023] , The admittance for such a capacitor can be expressed ase0=— me"(o>)C0+ ja)e’(a))C0where e'(aY) and C'(m) are the real and imaginary parts of the permittivity for the dispersive material. This admittance formula demonstrates that such a configuration has significant potential not only for emulating a negative inductor but also for simultaneously supplying the necessary loss due to its dispersive characteristics, thus eliminating the requirement for an additional resistive layer.

[0047] As a proof of principle, we will initiate our investigation by analyzing a first-order Lorentzian dispersion in the dielectric of the capacitor. In the subsequent sections of the paper, we will explore multi-order Lorentzian dispersions and discuss more complex dispersion profiles. For a Lorentzian dispersion 6(o>) / e0 = 1 + (<up / a>0)2 / [I — (m / m0)2+ j (y / m0)(m / o>0)], where coo represents the resonance frequency, copis the plasma frequency, and y is the damping constant. To achieve a negative inductor, the imaginary component of YglsC(o)) can possess a positive value and exhibit a negative slope at the design frequency. A closer investigation of the admittance of this capacitor reveals that at frequencies much smaller than m0,= C0(l + (a>p / a>0)2) which is consistently positive. On the other side, for frequencies much larger than the resonance frequency, d[ / m{l^DlsC(m)}] / + (mp / m) ) which also shows a positive slope. However, at the resonancefrequency (ω = ω0), the situation is different, where ^^^^� ^^^^^^^^�^^^^^^^^^^^^^^^^^^^^^^^^ (^^^^) / ^^^^^^^^|^^^^=^^^^0 = ^^^^0(1 −2�^^^^^^^^ / ^^^^�2) which can be made negative if ^^^^ < √2^^^^^^^^ hil th im in r m nent of theadmittance remains positive ^^^^^^^^�^^^ ^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^ � > 0. Furthermore, ^^^^� ^^^^^^^^�^^^^^^^^ (^^^^) / ^^^^^^^^|^^^^=^^^^0= 0 ,indicating that this design not only provides the required non-Foster characteristics but also ensures a flat real part, i.e., dispersion-less resistor in parallel to the negative inductor around^^^^ = ^^^^0 (see Fig. 2b). This characteristic benefits ensuring wideband absorption. Designing acapacitor filled with a material that precisely demonstrates the desired dispersion characteristics is understandably a challenging task, if feasible at all. In practice, such a capacitor can be identically realized using a circuit network composed of frequency-independent elements. In Fig. 2a inductor Lmdmodels a conductor-backed dielectric 102 (as shown in Fig.1a) and the described capacitor (having capacitance C0 filled with a highly dispersive material defining circuitry 104) connected in parallel with conductor-backed dielectric 102 can be configured so that an impedance of conductor-backed dielectric 102 modeled by Lmdis matched to the impedance of free space over a design absorption frequency range of the described absorber structure. In reference to Fig.2a there is set forth herein, an absorber structure comprising a conductor-backed dielectric 102 (as depicted in Fig.1a); and circuitry 104 connected in parallel with the conductor-backed dielectric 102, wherein the circuitry 104 connected in parallel with the conductor-backed dielectric 102 exhibits (as shown in Fig.2b) an admittance with an imaginary component having a negative slope within a frequency range (depicted range about ^^^^0) encompassing the frequency (^^^^0) (defining a design frequency) of the absorber structure. In a further aspect (as shown in Fig.2b), the admittance imaginary component can include a positive slope outside of the frequency range (range about ^^^^0) encompassing the frequency (^^^^0) (defining a design frequency) of the absorber structure. In Fig.2a the circuitry 104 connected in parallel with the conductor-backed dielectric 102 is provided by the capacitor with capacitance C0 filled with a dispersive material. However, in view of challenges to the physical realization of the capacitor having capacitance C0filled with a dispersive material, embodiments herein include embodiments wherein the described circuitry 104 connected in parallel with the conductor-backed dielectric 102 and having the described admittance characteristics as shown in Fig.2b is provided by alternative equivalent circuitry with more readily realized circuit components. In each alternatively embodiment of circuitry 104 herein, e.g., including in reference to Fig.2a, Fig.2c, Fig.4a, Fig.4b, Page 13 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)Fig.7a, Fig.7b and Fig.7c circuitry 104 can be configured to exhibit the admittance characteristics as shown in in Fig.2b and can be further configured so that an impedance of conductor-backed dielectric 102 modeled by Lmdis matched to the impedance of free space over the frequency range (depicted range about ^^^^0) encompassing the frequency (^^^^0) (defining a design frequency) of the absorber structure, the frequency range defining a design absorption frequency range of the described absorber structure. The depicted range about ^^^^0in Fig.2b is the range Δ^^^^^^^^^^^^^^^^which can define a design absorption frequency range of the absorber structure in which circuitry 104 exhibits an admittance imaginary component having a negative slope and operates as a negative inductor. The depicted range about ^^^^0is the range Δ^^^^^^^^^^^^^^^^illustrated in Fig.2b in which circuitry 104 operates as and emulates a negative inductor. The depicted range Δ^^^^^^^^^^^^^^^^can define a design absorption frequency range of an absorber structure according to one embodiment.

[0048] We have developed a circuit network that can identically replicate the characteristics of such a capacitor. Fig.2c illustrates one of the various circuit implementations for a capacitor filled with a highly dispersive Lorentzian dielectric material. In this setup, the circuitcomponents can be expressed in terms of the dispersion parameters: ^^^^ = 1^^^^11 ^^^^0 ^^^^^2^^^ , ^^^^ =^^^^0 ^^^^^2^^^, and^^^^ = ^^^^0(^^^^^^^^^^^^0)2. Note that in this circuit implementation, all the compoy- independent and passive. In the following section, we will elaborate on the methodology for applying dispersion engineering on this platform to enable wideband negative inductors, thereby facilitating the development of wideband thin absorbers. In reference to Fig.2c there is set forth herein, an absorber structure comprising a conductor-backed dielectric 102 (depicted in Fig.1a); circuitry 104 connected in parallel with the conductor-backed dielectric 102, wherein the circuitry 104 connected in parallel with the conductor-backed dielectric 102 exhibits (as shown in Fig.2b) an admittance with an imaginary component having a negative slope within a frequency range (depicted range about ^^^^0) encompassing the frequency (^^^^0) (defining a design frequency) of the absorber structure, wherein the circuitry 104 (dashed portion of Fig.2c and Fig.7a) connected in parallel with the conductor-backed dielectric 102 includes a capacitor, e.g., C0; and a series connected circuit path connected in parallel with the capacitor, e.g., C0, wherein the series connected circuit path includes a series connected resistor, e.g., R1, a series connected Page 14 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)capacitor, e.g., Ci, and a series connected inductor, e.g., Li. In a further aspect (as shown in Fig. 2b), the admittance imaginary component can include a positive slope outside of the frequency range (range about coo) encompassing the frequency (coo) (defining a design frequency) of the absorber structure. In each alternatively embodiment of circuitry 104 herein, e.g., including in reference to Fig. 2a, Fig. 2c, Fig. 4a, Fig. 4b,Fig. 7a, Fig. 7b and Fig. 7c circuitry 104 can be configured to exhibit the admittance characteristics as shown in in Fig. 2b and can be further configured so that an impedance of conductor-backed dielectric 102 modeled by Lmd is matched to free space over the frequency range (depicted range about coo) encompassing the frequency (co0) (defining a design frequency) of the absorber structure, the frequency range defining a design absorption frequency range of the described absorber structure.DESIGN OF ELECTRICALLY THIN WIDEBAND ABSORBERS

[0049] When designing absorbers for specific frequency bands, the requirements can vary depending on the application under consideration. In some cases, perfect absorption (i.e., zero reflection and zero transmission) at the central (design) frequency is necessary. Alternatively, in some applications, the goal might be maintaining the reflection within the maximum allowable limits across the absorption frequency band. Here, we explore these two scenarios and develop appropriate dispersive networks to meet these conditions.

[0050] Case I: Perfect absorption at the central frequency of the band

[0051] For the first case, we impose two critical conditions: impedance matching and equal slopes for Im[Ygeq} andat UJ0(design frequency). The former condition ensures perfect absorption, while the latter condition guarantees that / m{^e<7}andexhibit equal negative slopes around coo, thus following same trend. These conditions upon imposing allow us to deduce the values of mP1y, and Co as functions of the design frequency coo and thickness of the absorber d. The resulting design conditions are cop= l / (i]0C0) = y = l / (d / zocOo). To demonstrate the idea, we design an electrically-thin absorber at frequency coo with the thickness d — Ao / 30. As shown in upper panel of Fig. 3a, the obtained 7m{}^DlsC} clearly demonstrates the required negative inductive characteristics over a wide frequency band.Furthermore, Re[YglsC] is equal to 1 / z / o and exhibits a zero slope around coo, which is consistent with how / ?e{y^>tsC]behaves. In terms of the maximum allowable reflection (i.e., po), which happens at the edges of the band for this case, the absorption bandwidth reads Am = a>2— a)1, thiscase, the maximum absorption bandwidth-to-thickness ratio derived as

[0053] The upper panel of Fig. 3b illustrates the reflection and absorption spectra obtained using this approach. Assuming the maximum allowable reflection p0= 0.1, this approach yields an absorption bandwidth-to-thickness ratio that is 2.2 times that of the conventional RC design, as described in Eq. 1.

[0054] Case II: Maximum allowable reflection over the entire frequency band

[0055] Contrary to Case I, which necessitates perfect absorption at a single frequency, in more practical scenarios, the design process for any absorber usually starts by establishing a specific threshold for the maximum permissible level of reflection across the absorption band. In this case, the maximum achievable bandwidth is realized when the maximum permissible reflectivity (i.e., po) occurs at the design frequency coo (Note that this is valid for odd orders of Lorentzian dispersion). As for the second condition, we impose= Im{YglsC] at a>0which results in Be{l^DtsC} = ■ As expected, Re[YglsC] exceeds l / ry at coo. Imposing these specifiedconditions results in Co= The absorption bandwidth in thisscenario is Aco = C02 - coi,(l — pQ~) . In this case, the absolute maximum bandwidth Ammnxhappens whenm„ = y =(2-EE) Interestingly, even thoughHVo M-PoZ this maximum bandwidth was achieved through mathematical derivation, ensuring thatIm{Ygeq] and Im[YglsC] display equal negative slopes around coo, i.e., a[ / m{r;eq] / also leads to the same condition for y. For this case, the maximum achievable bandwidth-to-thickness ratio can be derived as

[0057] By specifying the upper limit of permissible reflection and the central frequency of the band, one can deduce the utmost attainable absorption bandwidths at a given thickness. To illustrate this, we have designed an absorber with the same thickness as for the Case I (i.e., d = Ao / 30 at the same center frequency coo and maximum allowable reflectivity pQ— 0.1. As shown in lower panel of Fig. 3a, Im{YglsC} clearly shows negative inductive characteristics over a wideband. This approach yields a bandwidth-to-thickness ratio that is nearly 3.5 times that of the conventional RC grid (Eq. 1) and 1 .5 times that of Case I (Eq.2). The lower panel of Fig. 3b illustrates the reflection and absorption spectra obtained using this approach. It is evident that Case II offers significantly wider bandwidths compared to Case I. Therefore, due to its higher bandwidth and practical appeal we will focus on Case II for the remainder of the paper.MULTI-ORDER LORENTZIAN DISPERSIVE MATCHING GRIDS

[0058] The presented concept can be expanded to higher-order Lorentzian dispersion cases (see Fig. 4a). Incorporating higher-order Lorentz dispersion into the network can enhance the bandwidth of the negative inductor, thereby enabling higher absorption bandwidths. Admittance of / f'-order Lorentzian dispersion reads as

[0060] where Co iis the capacitance of capacitor with free space between its parallel plates, and a)Q i, Yi are plasma frequency, resonant frequency, and damping frequency of the ithresonator, respectively. The circuit implementation shown in Fig. 4b precisely replicated the higher-order dispersive network shown in Fig. 4a. To illustrate the bandwidth enhancement using higher order Lorentzian dispersion networks, we have designed an electrically thin absorber with dispersion orders 1-5. For this proof-of-concept example, we have considered the maximumallowable reflection to be -10 dB (i.e., p = 0.316) over the absorption band with operational center frequency of ) = 10 GHz. The circuit parameters for these higher-order grids (other than the first order for which we have the closed-form solution) were numerically extracted.

[0061] Fig. 4c demonstrates the increase in the bandwidth of the negative slope of / m Ygridj as we progress from the first order to higher-order dispersive networks, providing higher bandwidth for the negative inductor. Another interesting observation to note is the progressively flatter behavior of the real part of the grid admittance around the resonance frequency coo as we increase the order of dispersion in the grid. This characteristic is extremely desired for the absorber design purposes since it aligns with the required dispersion-less resistive part within the frequency band of the interest

[0062] Consequently, an increase in the order of dispersion will lead to an increase in absorption bandwidth as seen in Fig. 4d. In reference to Fig. 4a, there is set forth herein, an absorber structure comprising a conductor-backed dielectric 102 (depicted in Fig. 7c and modeled by inductor Lmd ofFig. 7a and Fig. 7b); circuitry 104 connected in parallel with the conductor-backed dielectric 102; wherein the circuitry 104 connected in parallel with the conductor-backed dielectric 102 exhibits (as shown inFig. 2b) an admittance with an imaginary component having a negative slope within a frequency range (depicted range about m0) encompassing the frequency (m0) (defining a design frequency) of the absorber structure, and wherein the circuitry 104 connected in parallel with the conductor- backed dielectric 102 defines an / z / Zz-order dispersive network. In reference to Fig. 4b, there is set forth herein, an absorber structure comprising a conductor-backed dielectric 102 (depicted in Fig. 7c and modeled by inductor Lmd of Fig. 7a and Fig. 7b); circuitry 104 connected in parallel with the conductor-backed dielectric 102; wherein the circuitry 104 connected in parallel with the conductor-backed dielectric 102 exhibits (as shown in Fig. 2b) an admittance with an imaginary component having a negative slope within a frequency range (depicted range about m0) encompassing the frequency (m0) (defining a design frequency) of the absorber structure, and wherein the circuitry 104 connected in parallel with the conductor-backed dielectric 102 defines an nth order dispersive network.

[0063] In this scenario, incorporating multi-order Lorentz dispersion into the impedance grid of the electrically thin absorbers achieves two objectives: it brings the structure to resonance by neutralizing the reactive component of the input admittance of a conductor-backed dielectric 102 (with the dielectric being free space in this instance), and it provides the resistance to match the absorber’s impedance to free space over a wide frequency band. Here, we engineer loss and resonance simultaneously, embedding the loss directly within the resonator. It should be noted that this approach differs from the conventional method, where a resonator is created separately to address the reactive impedance of the system, followed by the addition of loss to achieve impedance matching (i.e., high-impedance-based absorbers). At this juncture, embodiments herein pose the following question: why not design a higher order (multi-resonance) reactive grid to nullify the reactance arising from the ground plane across a continuous frequency band, and then incorporate a dispersionless resistive sheet to achieve broadband absorption? To address this question, we can refer to Foster’s reactance theorem, which states that for any passive, purely lossless reactive network, both susceptance and reactance will consistently exhibit a positive slope across the entire frequency spectrum [25, 26]:

[0065] Here Xt and BF are the reactance and susceptance of the network, respectively. However, as demonstrated in previous sections, achieving broadband absorption in this structure requires the impedance grid to emulate the behavior of a negative inductor= l / ( / zod<n) , i.e., a non-Foster component, across the absorption band. The admittance of such a grid will exhibit a monotonically decreasing trend over all frequencies. Therefore, a purely Foster network cannot provide this behavior. Consequently, engineering loss and resonators separately will not facilitate broadband absorption. The only way to achieve the required non-Foster behavior is by introducing either gain or loss into the impedance grid. Our presented method accomplishes this by adding loss in the grid.

[0066] In reference to Eq. 4, we present a semi-analytical design methodology for broadband ultrathin absorbers based on an artificially engineered grid characterized by multi-order Lorentz dispersion. From numerical results, there is set forth herein a straightforward empirical relationship linking the maximum achievable absorptionbandwidth-to-thickness ratio, the threshold maximum allowable reflection coefficient ρ0, and the order of dispersion (n) within the grid. Subsequently, there is set forth herein a new upper bound on the absorption bandwidth-to-thickness ratio as the order of dispersion in the grid approaches infinity. This new upper bound on bandwidth-to-thickness ratio has surpassed all of the existing upper bounds and remarkably reached up to 95% of the theoretical limit established by Rozanov.

[0067] In the case of an nthorder grid, referring to Eq. 4, the total number of circuit parameters to be determined is 4n. Determining them analytically would be exceedingly challenging if not impossible. Hence, in this section, we present a numerical approach aimed at optimizing the circuit (dispersion) parameters of a multi-order Lorentz grid aiming to achieve the highest possible absorption bandwidths for a given thickness, while ensuring that the reflectance remains below a specified threshold allowable reflection (ρ0). To start with the optimization routine, consider the first-order design whose equivalent circuit parameters can be extracted for a given thickness (d), frequency (ω0), and ρ0 using the results of the presented first-order absorber (Case 2 dispersion engineering). The circuit parameters of the multi-order grid are initialized to match the grid admittance of the first-order grid, as depicted in Fig. 8a. Utilizing the ‘fmincon’ routine of MATLAB®, the optimization of these higher-order grid parameters can be carried out by minimizing the “cost function”, which is defined as: 68] cost =∆ 0.5[00 ∑ 2^^^^ ^^^^ (^^^^^^^^ − ^^^^0) − ^^^^� ^^^^� (6)

[0069] Here, Ri represents the magnitude of reflection at the ithlocal maximum of the reflectance spectrum within the absorption band, and ∆λ / d denotes the achieved absorption bandwidth-to-thickness ratio. MATLAB® is a registered trademark of The Math Works, Inc. The reflectance spectrum measures the performance of this grid in matching Lmd to free space. The parameter α acts as a scaling factor to ensure that the first and second terms in the cost function are comparable in magnitude. This cost function aims to increase the bandwidth- to-thickness ratio while simultaneously minimizing the penalty associated with reflectance maxima deviating from ρ0within the absorption band. Specifically, it constrains the reflection maxima to ρ0when Ri> ρ0and adjusts the maxima closer to ρ0when Ri< ρ0, Page 20 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)thereby fully utilizing allowable threshold reflectance ρ0. This cost function is similar to the “figure of merit” that Rozanov used to find the ultimate bandwidth-to-thickness ratio of metal-backed dispersive dielectric absorbers. During the optimization process, the circuit parameters are constrained to positive values to ensure their practical feasibility. To validate the obtained maximum absorption bandwidth-to-thickness ratio, the optimization routine was initiated with various starting points, including different thicknesses d, operating frequency bands, and threshold reflectivity ρ0for different orders of dispersion in the grid. Optimizing absorber performance often involves addressing a multi-extremum optimization problem, leading to the emergence of multiple local solutions. These local solutions become more pronounced when the order of dispersion of the grid exceeds 2, typically when n > 3. To achieve the global solution, embodiments herein may benefit from restarting the optimization routine using the local solution as the initial parameters. A straightforward metric for determining whether the solution provided by the optimization routine is global or local is to examine the number of local maxima and minima of reflectance within the absorption band. The global solution of the nthorder dispersive grid will result in a total of n local maxima in the reflection spectrum, where the magnitude of reflection coefficient, Ri, will be approximately equal to ρ0. Fig.8b summarizes the overall optimization method presented to achieve multi-order Lorentz dispersive grid parameters for a very wideband absorber.

[0070] A method for performance by one or more program running on processing circuit 310 is set forth in reference to the flowchart of Fig.8b. Processing circuit 310 can include, according to one example, one or more processor 3101, memory 3102, and one or more input / output interface 3103. One or more processor 3101, memory 3102 and one or more input / output interface 3103 can be connected via system bus 3104. Memory 3102 can include a combination of system memory and storage memory. Memory 3102, according to one example, can store one or more program for facilitating processes that are set forth herein. One or more processor 3101 can run one or more program stored in memory 3102 to facilitate and perform methods as is set forth herein. Memory 3102 can define a computer readable medium. At block 8002, processing circuit 310 can receive from a user, e.g., using a user interface selections regarding thickness d operating band, e.g., the design frequency and the reflectance threshold. Completion of block 8002, processing circuit 310 can proceed to block 8004. At block 8004, processing circuit 310 Page 21 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)can extract first-order parameters. On completion of block 8004, processing circuit 310 can proceed to block 8006. At block 8006, processing circuit 310 can initialize the nth order gridparameters so that the admittance of the ^^^^^^^^ℎ order grid (∑^^^^ ^^^^^^^^^^^^^^^^^^^^=1 ^^^^^^^^,^^^^ ) will be same as that of firstorder grid (^^^^^^^^^^^^^^^^^^^^^^^^) as set forth at block 8006. At block 8008 processing circuit 310 can impose acost function in accordance with Eq.6. On completion of block 8008, processing circuit 310 can proceed to block 8010. At block 8010, processing circuit 310 can optimize the grid parameters using an optimization problem solver tool, e.g., the ‘fmincon’ function in MATLAB®. MATLAB® is a registered trademark of The Math Works, Inc. On completion of block 8010, processing circuit 310 can proceed to decision block 8012. At decision block 8012, processing circuit 310 can ascertain whether local maxima of reflectance equals n, and a value at local at a local maxima is proximate the reflectance threshold ρ0. If no, processing circuit 310 can proceed to block 8014. At block 8014, processing circuit 310 can update grid parameters and can return to block 8010. Processing circuit 310 can iteratively perform the loop of blocks 8010 to 8014 until at in iteration of block 8012 processing circuit 310 determines that a local maxima of reflectance equals n and a value at a local maxima is proximate the reflectance threshold ρ0. On the determination at block 8012, processing circuit 310, that on the determination at block 8012 the local maxima reflectance equals n in the value at a local maxima is proximate ρ0, processing circuit 310 can proceed to block 8016. At block 8016, processing circuit 310 can extract determine grid parameters and can, e.g., present to grid parameters to the user interface used by the user in making the selection referred to at block 8002.

[0071] There is set forth herein, in reference to Eqs.6-10 a simple analytical relationship between the maximum achievable bandwidth-to-thickness ratio and the threshold reflectance ρ0for a given order of dispersion in the grid. The maximum achieved bandwidth-to-thickness ratio at different threshold reflectance levels is shown in Fig.9a for different orders of dispersion in the grids. The numerical data obtained for second order dispersive grid can be fitted to function: 072] ^^^1 / 3[0 ∆^^^^ ^0^^^^ = 3^^^^ 1−^^^^2 / 3(7)

[0073] For the third-order grid, the obtained results is fitted with Page 22 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)∆^^^^12^^^^1 / 4

[0074] ^^^^= 5^^^^ 01−^^^^01 / 2(8) expression fostr bandwidth-to-thickness ratio for 1 order grid and Eqs.7, 8, the maximum achievable bandwidth-to-thickness ratio can be generalized to case of arbitrary order n as ∆^^^^3^^^^01 / (^^^^+1)

[0076] ^^^^ (^^^^0,^^^^) = 4^^^^�^^^^+2� 1−^^^^02 / (^^^^+1)(9)

[0077] , ained for the 4thand 5thorders align very closely with Eq.9. The validity of this empirical expression, established in Eq.9, has been confirmed for different orders ranging from 1 to 8 at various reflection levels of ρ0and thickness d. The above- established analytical relations have the same form as those for metal-backed dielectric absorbers developed by Rozanov. Additionally, it should be noted that any physically realizable dispersivegrid can be defined using Eq. 4 when ^^^^ → ∞. Therefore, the ultimate bandwidth-to-thicknessratio of highly dispersive grid-based absorbers can be expressed as: ∆^^^^ 3

[0078] ^^^^1 / (^^^^+1)6^^^^^^ l^^i→m∞(^^^^ ,^^^^) = ^^ l^^i→m∞4^^^ 0^^^^0 ^�^^^^+2� 1−^^^^02 / (^^^^+1)=|ln (^^^^0)|(10)

[0079] Remarkably, the above-derived upper bound has surpassed all other upper bounds defined in the literature and has reached 95.5% of the fundamental limit established by Rozanov. The final comparison of the derived bandwidth-to-thickness ratio with that of the fundamental bounds established by Rozanov is presented in Fig.9b.

[0080] There is set forth herein a novel design methodology for ultra-thin absorbers that can achieve an absorption bandwidth-to-thickness ratio approaching the theoretical limit described in Rozanov's work

[0016] . The presented approach features an absorber with a precisely engineered passive non-Foster impedance grid, optimally positioned relative to a ground plane. The non- Foster impedance grid is implemented in a fully passive platform through dispersion engineering. Our findings show that absorbers based on this concept, which integrate a specified order of dispersion within the impedance grid, achieve a bandwidth-to-thickness ratio that exceeds Rozanov's established bounds for metal-backed dielectric absorbers of equivalent Page 23 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)thickness and dispersion order. Additionally, the integration of fourth-order Lorentz dispersion within the impedance grid yielded a substantial enhancement, enabling our design to surpass Rozanov's established limit for metal-backed dielectric absorbers with an infinite order of dispersion. Remarkably, absorbers utilizing our presented dispersive grid with a single order of Lorentz dispersion achieve a significantly higher bandwidth-to-thickness ratio compared to absorbers based on conventional designs. Moreover, these single-order dispersive grid absorbers surpass previously established upper bounds on the bandwidth-to-thickness ratio of absorbers using advanced engineered surfaces such as high-impedance surfaces and metasurfaces [17, 18]. Finally, as a proof of principle, by applying this innovative concept, we designed, fabricated, and experimentally validated an absorber that demonstrates the highest bandwidth-to-thickness ratio reported across the full frequency spectrum relative to Rozanov’s theoretical limit. COMPARING THE BANDWIDTH-TO-THICKNESS RATIO OF THE PRESENTED CONCEPT WITH THE ESTABLISHED LIMITS FOUND IN THE LITERATURE

[0081] Understanding the full potential offered by the presented concept necessitates a thorough comparison of the ultimate achievable absorption bandwidth-to-thickness ratio for designs grounded in this concept against the bounds for the bandwidth-to-thickness ratio of electrically thin absorbers as documented in existing literature. To achieve this, we will begin by briefly reviewing the primary limits found in the literature, followed by the comparison.

[0082] A. Rozanov’s bounds

[0083] In 2000, Rozanov established the ultimate absorption bandwidth-to-thickness of metal- backed dielectric slab absorbers, assuming conditions of passivity, linearity, and time-invariance

[0016] . Starting with the analytical properties of the reflection coefficient from a general metal- backed slab, Rozanov formulated a relation connecting the integral of reflectance over wavelength to the thickness and static permeability of the slab:

[0084] �∫∞0 ^^^^^^^^|^^^^(^^^^)|^^^^^^^^� ≤ 2^^^^2^^^^^^^^^^^^ (11)

[0085] Then, by setting the maximum permissible reflection at ^^^^0, he deduced the fundamental theoretical limit on the absorption bandwidth-to-thickness ratio for these structures as Page 24 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)

[0086] Δ^^^^< 2^^^^2^^^^^^^^^^^^|^^^^^^^^^^^^0|(12) s the static permeability of the slab. However, this theoretical limit chievable with any physically realizable passive linear time-invariantsystems as this would require designing the wavelength dependence of the reflection coefficient as a piecewise linear function

[0016] . Therefore, he deduced a tighter bound for a more practically relevant case where he considered a Dallenbach-type absorber (composed of a metal-backed thin dielectric slab) possessing nthorder of Lorentzian dispersion Δ^^^32 3 ^^^^1 / (^^^^+1)

[0088] ^^^^^≤^^^^�^^^^+2� 01− ^^^^2 / (^^^^+1)(13) 0persion in dielectric, this bound simplifies to Δ^^^^

[0090] 4^^ l^^i→m∞ ^^^^= 8^^^^|^^^^^^^^^^^^0|(14)

[0091] whicrepresen s e u mate bandwidth-to-thickness ratio of an absorber made of a metal-backed dispersive dielectric with infinite order of Lorentzian dispersion.

[0092] B. Bandwidth limits of absorbers based on artificially engineered surfaces (high impedance surfaces or metasurfaces).

[0093] Building on Rozanov's research, which established the absolute upper limit for the bandwidth-to-thickness ratio in electrically thin, linear, time-invariant, passive absorbers, subsequent studies have sought to define comparable bounds for absorbers based on artificially engineered surfaces, including those utilizing high-impedance surfaces and metasurfaces [17, 18] Specifically, in

[0017] , the authors introduced a limit for thin absorbers based on metasurfaces, under the assumption that the impedance grid is non-dispersive: 4^^^^^^^^ ^^^^′

[0094] Δ^^^^^^^^≤ 0 ^^^^^^^^^^^^1−^^^^2(15)

[0095] where ^^^^^′^^^^^^^^^^^is the maximum value of the real part of the complex relative permeability of the substrate in the band of interest ∆λ. Similarly, following Foster's reactance theorem (asserting that the slope of Im{YHIS} remains positive over the whole frequency spectrum), an upper bound Page 25 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)on the absorption bandwidth-to-thickness ratio for ultrathin high-impedance-surface-based absorbers was established in

[0018] :

[0096] Δ^^^^8^^^^^^^^0 ^^^^^′^^^^^^^^≤ ^^^^^^^1−^^^^02(16)idths

[0098] Fig.5a presents a comparison between the absorption bandwidth-to-thickness ratio of the presented design, conventional matching method using a capacitive grid, and the aforementioned established bounds from the literature, across a range of maximum allowable reflection levels. Fig.5a demonstrates that the bandwidth-to-thickness ratio achieved by our presented design (i.e., Eq.3) which utilizes a first-order Lorentz dispersive grid, exceeds the bound for the conventional way of matching using a capacitive grid (i.e., Eq.1) and the ultimate limits established for absorbers based on high-impedance surfaces or metasurfaces, as detailed in [17, 18] (i.e., Eqs.15 and 16). Naturally, the achieved bandwidth-to-thickness ratio in the presented concept falls below the theoretical limit presented by Rozanov (i.e., Eq.12), which is not achievable with any physically realizable passive linear time-invariant systems. However, as it will be shown shortly, by adding higher-order Lorentz dispersion, one can get arbitrarily closer to this theoretical limit.

[0099] In this study, we concentrated on absorbers composed of dispersive impedance grid located with a distance from a ground plane and demonstrated that incorporating Lorentz dispersion into the impedance grid facilitates wideband absorption. However, it would be of immense interest to compare these results with those from thin metal-backed dielectric absorbers where Lorentz dispersion is added to the dielectric (Eq.13). This comparison can provide valuable insights into the differential impacts of dispersion on these two absorber types. Fig.5b compares these two types of absorbers with the same thickness (i.e., d) and the same order of Lorentz dispersion. As illustrated in this Figure, for any given order of Lorentz dispersion, the presented concept achieves a significantly higher bandwidth-to-thickness ratio compared to thin metal-backed dielectric absorbers. It is quite fascinating that the fourth-order Lorentz dispersion in the absorber based on the presented concept surpasses the infinite-order Lorentz dispersion in the thin metal-backed dielectric absorber. Moreover, adding higher orders of dispersion in grid will bring us closer to the theoretical limit on bandwidth-to-thickness ratio

[0016] . This result Page 26 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)underscores the superiority of the presented concept. This breakthrough marks a crucial advancement in the design and functionality of electromagnetic absorbers, showcasing their capability of optimum performance through the implementation of finite-order dispersion. Moreover, the synthesis of the absorber utilizing the presented design concept is notably simpler compared to metal-backed dielectric absorbers, where even first-order Lorentz dispersion presents practical hurdles. This simplicity in realization adds a practical advantage to our presented design, enhancing its feasibility and effectiveness in real-world applications, including its adaptability to conformal surfaces. Embodiments herein are particularly well-suited for use with curved surfaces with gradual bending relative to a size of a unit cell.

[0100] So far, we have compared the bandwidth-to-thickness ratio of absorbers designed based on our presented concept with the fundamental bounds in the literature. However, it is essential to also evaluate the performance of our design against existing wideband absorbers. Fig.6 provides a comparison of the absorption bandwidth-to-thickness ratios of some of the best- performing wideband absorbers across different parts of the frequency spectrum. It is important to note that these absorbers typically employ complex design processes, including brute-force optimization techniques. As demonstrated in this Figure, to the best of our knowledge, our first- order design achieves the highest absorption bandwidth-to-thickness ratio ever reported in the literature. Furthermore, while all other absorbers drastically underperform in exploiting all the potential that passive, linear time-invariant structures can provide (achieving less than 60% of the theoretical limit in the best-case scenario), absorbers based on the presented concept can get arbitrarily close to the upper theoretical limit in bandwidth-to-thickness ratio of these structures. SIMULATION, MEASUREMENT, AND METHODOLOGIES

[0101] To demonstrate the presented design concept an absorber structure is designed at ^^^0^ = 10GHz with the thickness of the spacer ^^^^ = ^^^^030= 1mm, and maximum allowable reflection ^^^^0 =0.15. Employing the equations delineatedin the section entitled Design of Electrically ThinWideband Absorbers, part B, the circuit parameters of the structure are as follows: ^^^^0 = 0.2016pF, ^^^^1 = 13.22 fF, ^^^^1 = 308.234 Ω, ^^^^1 = 19.151 nH, and ^^^^^^^^^^^^ = 1.25 nH (refer to Fig. 7a).The high-value series inductance, combined with an extremely low-value series capacitance, increase the structure’s sensitivity during fabrication and pose challenges for practical Page 27 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)implementation. To address this issue, Zobel’s transformation was applied to the matching network to achieve an equivalent circuit providing more implementation-friendly values for the components

[0041] , i.e., a larger value of capacitance and a lower value of inductance. For the newly transformed equivalent circuit shown in Fig. 7b the component values read CQ = 0.215 pF, C] = 3.2735 pF, R\ = 1.1687 Q, and L\ = 0.072615 nH. Fig. 7c illustrates the unit cell for the impedance-sheet implementation of this circuit model. The gap g in the patch, aligned with the electric field direction along the branch, act as capacitive elements, the thin copper track, W2 functions as the inductive component, and a lumped resistor component is placed in W2 track to provide the required loss. We derived all design parameters assuming that the gap between the dispersive grid and the ground plane is filled with air. However, fabrication of a suspended metal structure or procuring a substrate with a dielectric constant similar to that of air is practically unattainable, leading to the employing of a non-air substrate defined by solid material being beneficial. The design process benefited from a thin substrate characterized by the lowest attainable dielectric constant and minimal losses to minimize the substrate's influence on the system. However, diminishing the thickness beyond a certain threshold poses the risk of bending or damaging the board during fabrication, soldering the resistive components, and testing, potentially compromising the accuracy of the results. Taking these limitations and considerations into account, ROGERS™ RT5880 material with a thickness of 0.254mm ~ Ao / 120 was chosen for its low dielectric constant (i.e., er= 2.2) and low loss (i.e., tan(<5) = 0.009). ROGERS™ is a trademark of Rogers Corporation. Additionally, copper cladding with a thickness of 36^m was selected to enhance the board’s physical robustness and prevent it from bending due to the thin dielectric. To model the dispersive grid on the thin substrate discussed earlier, a unit cell was designed using Ansys HFSS. The unit cell size was set at 14 x 14mm, smaller than half the wavelength at the center frequency of 10GHz. The optimized geometrical dimensions are as follows: wo = 14 mm, lo= 14 mm, wi = 13.4 mm, h= 13.8 mm, W2= 0.65 mm, h= 0.3 mm, ws= 10.05 mm, g = 0.4 mm, and the resistor value is R ’i= 150 Q (see Fig. 7c). Fig. 7d demonstrates the perfect agreement between simulated results acquired via full-wave analysis in Ansys HFSS and the presented theory.

[0102] In reference to Fig. 7b there is set forth herein, an absorber structure comprising a conductor-backed dielectric 102 (depicted in Fig. 7c and modeled by inductor Lmd of Fig. 7b);circuitry 104 connected in parallel with the conductor-backed dielectric 102; wherein the circuitry 104 connected in parallel with the conductor-backed dielectric 102 exhibits (as shown in Fig.2b) an admittance with an imaginary component having a negative slope within a frequency range (depicted range about ^^^^0) encompassing the frequency (^^^^0) (defining a design frequency) of the absorber structure, and wherein the circuitry 104 connected in parallel with the conductor-backed dielectric 102 includes a capacitor, e.g., C’0 in series with first and second circuit paths that are connected in parallel to one another, the first circuit path having a capacitor, e.g., C’1, and the second circuit path having an inductor, e.g., L’1in series with a resistor, e.g., R’1. In a further aspect (as shown in Fig.2b), the admittance imaginary component can include a positive slope outside of the frequency range (range about ^^^^0) encompassing the frequency (^^^^0) (defining a design frequency) of the absorber structure. In each alternatively embodiment of circuitry 104 herein, e.g., including in reference to Fig.2a, Fig.2c, Fig.4a, Fig.4b, Fig.7a, Fig. 7b and Fig.7c circuitry 104 can be configured to exhibit the admittance characteristics as shown in in Fig.2b and can be further configured so that an impedance of conductor-backed dielectric 102 modeled by Lmdis matched to free space over the frequency range (depicted range about ^^^^0) encompassing the frequency (^^^^0) (defining a design frequency) of the absorber structure, defining a design absorption frequency range of the described absorber structure.

[0103] Embodiments herein recognize, in reference to Fig.7b, that circuitry 104 of Fig.7b connected in parallel with conductor-backed dielectric 102 (depicted in Fig.7c and modeled by inductor Lmdof Fig.7b) can be expanded in the manner of circuitry 104 of Fig.4a and circuitry 104 of Fig.4b to define an nth order dispersive network.

[0104] To verify the simulation results, an array of 12 × 12 elements was fabricated. Generally,increasing the board size helps minimize edge diffraction, enabling a more precise assessment of the absorber's performance. For the resistive components, 144 small resistors with an 0402- footprint dimension and proper frequency response at 10 GHz (VISHAY® CH0402 series) were placed and soldered onto the final fabricated prototype of the presented grid structure shown in Fig.7e. VISHAY® is a registered trademark of Vishay Intertechnology. Page 29 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)

[0105] A copper ground plane, matching the dimensions of the grid structure, was positioned beneath the structure, maintaining a distance of 1 mm from the matching grid. Given the thin, flexible nature of the grid structure, prone to bending and curving, a holder structure was employed atop the grid to maintain a flat configuration. Thin plastic spacers and edge-covering holders were fabricated using a 3D printer, that ensured consistent spacing while simultaneously preserving the absorber's frequency response.

[0106] For frequency response measurement, the fabricated structure was meticulously positioned at the focal length of a spot-focusing lens antenna, possessing a focal length of 100 mm and generating a 3dB spot size of 25 mm at 10 GHz. The experimental setup for measuring the S-parameters of the absorber structure is illustrated in Fig.7e. Experimenting outside an anechoic chamber was feasible due to the focused beam of the measurement antenna. The thin absorber surface was placed at the focal length of the lens antenna and reflection parameters were measured using a KEYSIGHT™ N5544B network analyzer. KEYSIGHT™ is a trademark of Keysight Technologies. To mitigate the impact of additional scatterers in the setup, such as the holder and antenna legs, a time-gating concept was implemented for calibration.

[0107] Fig.7f compares the measured reflection spectrum of the absorber with simulation and theoretical results. The designed absorber exhibits wideband absorption characteristics, consistently maintaining levels below approximately -15 dB across a broad frequency range. The measurement response unveils a bandwidth of 1.9 GHz centered at 10 GHz, which closely aligns with the theoretical prediction and simulation results showing a bandwidth of 2.1 GHz. However, minor disparities between measurement and simulation data may stem from fabrication precision, resistor tolerance, and measurement inaccuracies attributed to the use of PLA holders and the incidence angle in the setup. The agreement between simulation and measurement within the specified bandwidth and central frequency validates the effectiveness of the designed absorber.

[0108] In each alternatively embodiment of circuitry 104 herein, e.g., including in reference to Fig.2a, Fig.2c, Fig.4a, Fig.4b, Fig.7a, Fig.7b and Fig.7c circuitry 104 defining an impedance grid can be configured to exhibit the admittance characteristics as shown in in Fig.2b and can be further configured so that an input impedance of conductor-backed dielectric 102 modeled by Page 30 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)Lmd is matched to the impedance of free space over the frequency range (depicted range about ^^^^0) encompassing the frequency (^^^^0) o (defining a design frequency) of the absorber structure, the frequency range defining a design absorption frequency range of the described absorber structure.

[0109] Embodiments herein recognize that, since free space includes a real (resistive) impedance component, configuring circuitry 104 so that an input impedance of conductor-backed dielectric 102 modeled by Lmdis matched to the impedance of free space over the design absorption frequency range can include configuring circuitry 104 to include a resistive component. Embodiments herein recognize that a traditional approach for designing an absorber can include providing a resonator to separately address the reactive impedance of the system, followed by the addition of resistive loss to achieve impedance matching (i.e., high-impedance-based absorbers). Approaches herein can include configuring circuitry 104 to define circuitry equivalent to a capacitor filled with a dispersive dielectric material. Accordingly, embodiments herein recognize that since dispersive material can exhibit resistive properties, and since a dispersive material filled capacitor can exhibits an admittance with an imaginary component having a negative slope as set forth in reference to Fig.2b, configuring circuitry 104 to define circuitry equivalent to a capacitor filled with a dispersive material can simultaneously address admittance and resistive loss design elements of the impedance grid defining circuitry 104, wherein resistive loss design elements of circuitry 104 are defined by the resistor(s) of the embodiments of circuitry depicted in Fig.2c, Fig.4a, Fig.4b, Fig.7a, Fig.7b and Fig.7c . Embodiments herein therefore can simplify and expedite design processes for designing an impedance grid configured so that an input impedance of conductor-backed dielectric 102 modeled by Lmd is matched to the impedance of free space over the design absorption frequency range.

[0110] Various approaches are set forth herein for configuring circuitry 104 so that a design absorption frequency range is tuned to a target design frequency range. As set forth in reference to Figs.3a-3b and the accompanying description, adjusting the design absorption frequency range can include adjusting a reflectance threshold parameter ρ0that specifies a maximum allowable reflection. Page 31 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)

[0111] In reference to Figs.4a-4d and the accompanying description, adjusting a design absorption frequency range can include adjusting a dispersive grid order of circuitry 104. Circuitry 104 as shown in Fig.2a, 2c, 7a, 7b, and 7c depicts a first order dispersive grid (first order dispersive network), whereas circuitry 104 as shown in Fig.4a and 4b depicts an nth order dispersive grid (nth order dispersive network). As set forth in reference to Fig.4a and 4b, an nth order dispersive grid defining circuitry 104 can be provided by modeling and producing an equivalent circuit of n shunt resonators connected in parallel across conductor-backed dielectric 102, wherein the respective ones of the n shunt resonators are provided by dispersive dielectric filled capacitors, as set forth herein.

[0112] As described with reference to Fig.4c and the accompanying description, a design absorption frequency range of an absorber structure can be tuned by adjusting a dispersive grid order of a dispersive grid defined by circuitry 104. In one aspect, a design absorption frequency range of an absorber structure can be increased by increasing a dispersive grid order of a dispersive grid defined by circuitry 104.

[0113] Embodiments herein include features for expedited designing of an absorber structure. According to one example, there is set forth herein, in reference to Eq.6, a method for determining parameters of an nth order dispersive grid that includes employing a cost function, and, in reference to Fig.8b and Eq.6-10 there is set forth herein a method employing the cost function of Eq.6 for determining an achievable absorber structure bandwidth (design absorption frequency range) for a given absorber structure dispersion grid order, thickness and reflectance level. DISPERSIVE GRID BASED WIDEBAND MAGNETIC ABSORBERS

[0114] Until now, our analysis has focused on dielectric absorbers with substrates that exhibit no frequency dispersion in permeability. However, with reference to Figs.11-13, a closer examination of Rozanov’s bound indicates that the upper limit on the bandwidth-to-thickness ratio increases linearly with the static permeability of the substrate. This suggests that absorbers employing magnetic substrates can achieve significantly enhanced bandwidth-to-thickness performance. In this section, we extend our design framework to encompass thin, conductor- backed magnetic absorbers, where permeability exhibits specific frequency dispersion. Page 32 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)

[0115] Consider an electrically thin absorbing layer composed of an admittance grid placed atop a conductor-backed magnetic substrate as shown in Fig.11(a), where the substrate’s permeability is frequency-dispersive and follows a Lorentzian dispersion model given by: ^^^^^^^^^^^^^^^^ 0�1+ ^^�

[0116] ^^^^(^^^^) = ^^^^ ^^^^ (^^ )^^0,^^^^0 ^^^^ ^^ = ^^^^0�1 + ^^^^^^^^^^^^2^^^^ 2� (17)ility and ^^^^0,^^^^is the resonance frequency of the magnetic substrate. It is important to note that such Lorentzian dispersion in permeability arises naturally in ferromagnetic materials operating in the microwave frequency range. Under normal incidence, the absorbing layer can be equivalently modeled as a short-circuited transmission line with characteristic impedance ^^^^^^^^^^^^ =�^^^^(^^^^) / ^^^^(^^^^) =^^^^0�^^^^^^^^(^^^^) / ^^^^^^^^(^^^^), in parallel with the grid admittance as sFig.11(b). Here,^^^^(^^^^) = ^^^^0^^^^^^^^(^^^^) is assumed to be the frequency-dependent permittivity of the magneticsubstrate. However, as will be shown later, when the substrate is electrically thin, the matching condition becomes effectively independent of the permittivity. The input admittance of the metal-backed magnetic substrate, when viewed towards the ground plane from beneath the grid,is given by ^^^^^^^^^^^^ = −^^^^ / [^^^^^^^^^^^^tan (^^^^^^^^^^^^^^^^)], where ^^^^^^^^^^^^ =2^^^^ ^^^^ √^^^^^^^^^^^^^^^^is the wavenumber in thesubstrate, ^^^^ is the wavelength of the incident wave and ^^^^ is the thickness of the metal-backedsubstrate. When ^^^^^^^^^^^^^^^^ ≪ 1, or equivalently ^^^^ << ^^^^ / (2^^^^�|^^^^^^^^^^^^^^^^|) , the admittance of the metal-backed substrate can be approximated as ^^^^^^^^ ^^^^^^^^ ≈ −[^^^^^^^^(^^^^)^^^^]. In our equivalent circuit model, this metal-backed magnetic substrate can be represenn inductor with inductance ^^^^^^^^^^^^= ^^^^0^^^^^^^^^^^^(^^^^) which is frequency dispersive as shown in Fig.11(c). The admittance of this dispersive inductor has several interesting properties. A close examination of ^^^^^^^^{^^^^^^^^^^^^} shows � hat around the frequency ^^^^(^^^^0+2)√^^^^2+1−^^^^2−1t ^^^^ = ^^^^0,^^^^ ^^^^2+1 , the slope of ^^^^^^^^{^^^^^^^^^^^^}will be zero,indicating that the metal-backed magnetictrate exhibits dispersion less loss around ^^^^^^^^as can be seen in the Fig.11(d). Furthermore, for frequencies much larger or much smaller than ^^^^^^^^,the slope of the imaginary part of the admittance remains consistently positive, ^^^^[^^^^^^^^{^^^^^^^^^^^^}] / ^^^^^^^^| > 0, indicating a Foster behavior. Nonetheless, in the vicinity of ^^^^^^^^, ^^^^^^^^{^^^^^^^^^^^^} exhibits aPage 33 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)different behavior whereinmay become negative for specific values of / 0and a whilethe imaginary part of the admittance remains negative. As a result, this dispersive component effectively mimics the behavior of a negative capacitor around mr, while exhibiting a positive inductive response at other frequencies, as illustrated in Fig. 11(d). Now for this dispersive metal-backed substrate to be effectively matched to free space, the admittance grid that is placed atop should provide an admittance equal to= - — Re{Yms} ~ jlm{Yms}. By analyzing the imaginary part of the required grid admittance Im^Yg^], reveals that capacitive behavior is required around a>rand conventionally this is achieved by employing a capacitive sheet of admittance Yg as shown in the Fig. 11(e). However, a comparison between imaginary parts of the required grid admittance lrn.{Yg^} and that of admittance provided by the capacitive gridshows that, here, effective matching can only happen within a limited frequency band around (refer to Fig. 11(f)). This is the case because the admittance required for wideband matching lm{Yg^} and that provided by capacitive grid Im{Yg ] exhibits opposite slopes outside this frequency band eliminating the possibility of achieving broadband matching. To be more precise, Im{Yg ^} demonstrates non-Foster, negative inductive behavior beyond the capacitive region, deviating from the behavior typical of conventional capacitive grids.

[0118] To overcome this limitation, we are proposing to utilize a Lorentz dispersive dielectric- filled capacitivegrid, as it can simultaneously provide both capacitive and negative inductive behaviors across different frequency bands as can be readily verified from our analysis for dielectric absorbers. As a result, the described design concept for magnetic absorbers enables broadband impedance matching over a significantly wider range than traditional methods, potentially extending the bandwidth by several times.

[0119] Design methodology for the dispersive grid based magnetic absorber

[0120] Fig. 12(a) demonstrates an equivalent circuit diagram for the described dispersive grid based magnetic absorber. To illustrate the described concept, consider Fig. 12(b, c) showing the imaginary part of the admittance for the first-order Lorentz dielectric-filled capacitoralongside the required grid admittance Im[Yg ^}. As discussed earlier, a dispersive dielectriccapacitor can emulate negative inductive behavior around m = a)0when y <where m0, y and a>pare Lorentz dispersion parameters of the described dispersive grid. Now, the key idea is to precisely engineer these parameters so that the non-F oster (negative inductive) and Foster (capacitive) regions of Im{Ygalign with the corresponding regions of Im{Yg^} as demonstrated in Fig. 12(b,c). However, determining the optimal grid parameters to maximize the absorption bandwidth-to-thickness ratio is highly challenging if not infeasible, since the combined dispersion of the grid and substrate causes the system to behave as a second-order network. To address this, we adopt a semi-analytical design approach previously developed for higher-order dielectric absorbers, where the grid parameters are optimized to achieve the maximum attainable absorption bandwidth-to-thickness ratio for a specified reflection constraint. We begin by analyzing the admittance of the metal-backed magnetic substrate. The initial design parameters of the dispersive grid are then selected to emulate negative inductive behavior at a frequency offset from mr. Specifically, the resonance frequency of the dispersive grid m0should be either higher or lower than a)r. Simultaneously, the values of Co, y, a)pmust be chosen to satisfy the condition / m{}^DtsC(a)0)} + / m{Tms(m0)} = 0 , with the constraint y < V2mp. This ensures that at a> = m0, the substrate behaves inductively while the grid exhibits a negative inductive response, such that the total imaginary part of the input admittance becomes zero at that frequency. These grid parameters are subsequently optimized to maximize the absorption bandwidth-to-thickness ratio for a specified reflection threshold p0. To demonstrate the described concept, we designed an electrically thin magnetic absorber operating around 10 GHz with a total thickness of 1.5 mm, %0=2 and a — 0.4. As shown in Fig. 12(d), this absorber achieves a remarkably wide absorption bandwidth.

[0121] The conductor-backed dielectric 102 illustrated in Figs. 1-10 can be more generally referred to as a conductor-backed substrate. This conductor-backed substrate is not limited to a dielectric-backed structure and can be implemented using alternative materials and configurations. For example, the conductor-backed substrate can be provided by the conductor- backed dielectric 102 as described in connection with the designs of Figs. 1—10. Alternatively, the conductor-backed substrate can be provided by a conductor-backed magnetic substrate 103, as introduced in the embodiment of Figs. 11-13. Embodiments herein recognize that where a conductor-backed substrate includes a conductor-backed dielectric 102 defined by a conductor-backed dielectric substrate, the conductor-backed dielectric substrate can be defined by one or more air dielectric layer.

[0122] In both embodiments, circuitry 104 can be electrically connected across the conductor- backed substrate, whether provided by conductor-backed dielectric 102 or conductor-backed magnetic substrate 103. This generalized approach allows circuitry 104 to be used in a highly flexible design framework to optimize absorber performance for different substrate materials and operating environments. The ability to interchangeably use different substrate types while maintaining a common circuit topology simplifies design considerations and enhances the adaptability of absorber structures to meet diverse performance requirements.

[0123] In the dielectric-based embodiments of Figs.1–10, the conductor-backed dielectric 102 can be electrically modeled as an inductor across all operating frequencies, as shown in Figs.1c and Fig.2b. In this configuration, circuitry 104, implemented as a dispersive capacitor or an equivalent circuit, can operate within an operating frequency region as depicted in Fig.2(b). Within this region, circuitry 104 can emulate the behavior of a negative inductor to achieve the desired impedance characteristics and broadband absorption. The design absorption frequency range in this embodiment can be defined as the depicted range about the frequency ω₀, within which circuitry 104 exhibits negative inductive behavior, as illustrated in Fig.2b.

[0124] The magnetic substrate embodiment of Figs.11–13 introduces additional operating behavior. The conductor-backed magnetic substrate 103 can exhibit dual frequency-dependent behaviors. Within a first operating frequency region, the magnetic substrate can behave inductively, and circuitry 104 can operate to emulate a negative inductor, similar to the dielectric case. However, beyond this first region, in a second operating frequency region, the conductor- backed magnetic substrate can exhibit behavior equivalent to a negative capacitor. In this second region, circuitry 104 can transition to operate as a conventional capacitor, providing the required capacitive behavior for continued impedance control, as illustrated in Fig.12(a).

[0125] The design absorption frequency range in the embodiment of Fig.11-13 can be defined as the range extending from a first frequency at which circuitry 104 commences operation as a negative inductor to a second frequency at which the conductor-backed magnetic substrate 103 ceases to exhibit behavior as a negative capacitor. This range spans both the first and second Page 36 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)operating frequency regions, enabling circuitry 104 to emulate the behavior of a negative inductor below the design border frequency ωB and to transition to capacitive behavior above ωB, as illustrated in Fig.12b. Referring to Fig.12b, the design absorption frequency range in the magnetic substrate spacer embodiment can extend from a lower frequency ω₁, where circuitry 104 can commence negative inductive operation, to an upper frequency ω₂, where the conductor- backed magnetic substrate can cease to exhibit negative capacitive behavior. This range can span both inductive and capacitive operating regimes, enabling circuitry 104 to operate as and emulate negative inductive behavior below the critical design border frequency ωB and to transition seamlessly to capacitive behavior above ωB, as illustrated in Fig.12b. This configuration can provide a flexible design option for achieving broad operational bandwidths.

[0126] In the embodiment of Figs.11–13, circuitry 104 can emulate the behavior of an ideal target matching circuit across both the first and second operating frequency regions. In the first region, circuitry 104 can operate as a negative inductor, and in the second region, it can operate as a capacitor. This behavior allows circuitry 104 to effectively match impedance across a broader range of frequencies, supporting improved absorption performance over the combined first and second operating frequency regions.

[0127] This distinction between the two embodiments can also result in a fundamental difference in how the design frequency is defined. In the dielectric embodiment of Figs.1–10, the design frequency can be ω₀, the frequency about which circuitry 104 provides negative inductive behavior. This design frequency can directly correspond to the frequency at which impedance matching is optimized using negative inductance emulation to achieve maximum absorption at a targeted frequency.

[0128] In the magnetic substrate embodiment of Figs.11–13, the design frequency can instead be defined by a border frequency ωB. This frequency serves as the transition point between the two distinct operating regions of circuitry 104. Below ωB, circuitry 104 can operate as and emulate a negative inductor within a limited band that can be made as large as desired by exploiting multi-order design, and above ωB, it can transition to operate as and emulate a capacitor. This dual-region behavior can enable the absorber to extend effective impedance control over a wider frequency range, making ωBa critical design parameter that governs the Page 37 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)transition between negative inductive and capacitive behaviors within circuitry 104. In this way, the design frequency in the magnetic substrate embodiment can function as a boundary between two operational regimes rather than as a simple central resonance point. This approach can enable broader absorption bandwidth and more flexible frequency response characteristics than the dielectric-based design.

[0129] In the embodiment of Figs.11–13, the conductor-backed magnetic substrate 103 and circuitry 104 can assume the physical configuration illustrated in Fig.7c. Both the conductor- backed substrate and circuitry 104 can be implemented in planar plate form and arranged physically parallel to one another while also being connected electrically in parallel. This parallel arrangement can facilitate compact and planar absorber structures that are well-suited for integration into practical devices and systems requiring thin-profile, broadband absorption characteristics. Such configurations can simplify manufacturing processes and enable efficient integration into layered electronic assemblies.

[0130] The use of a conductor-backed magnetic substrate 103 can enable a significant extension of the absorption bandwidth beyond what can be achieved with a conductor-backed dielectric substrate alone. This extension can be accomplished through the combined effect of the dispersive magnetic properties of the substrate and the engineered dispersion of circuitry 104. This combination can allow for effective impedance control over both inductive and capacitive operating regimes. As a result, the magnetic substrate embodiment can achieve a higher bandwidth-to-thickness ratio and improved absorption performance across a wider frequency range compared to the dielectric-based approach. The enhanced performance characteristics of the magnetic substrate embodiment make it particularly advantageous for modern electromagnetic absorption applications.

[0131] Higher order designs and upper bounds

[0132] Similar to dielectric absorbers, the described concept can be extended to higher-order Lorentz dispersion cases, where introducing additional dispersion orders in the grid further enhances the achievable bandwidth. The circuit model of the higher order grid based magnetic absorber is shown in Fig.13a The design of higher-order dispersive grid-based magnetic absorbers begins with the first-order grid, whose equivalent circuit parameters are extracted Page 38 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)based on the specified magnetic substrate, reflection threshold, and thickness, as previously described. The multi-order grid is then initialized to emulate the admittance of the first-order grid. Subsequently, the same optimization routine used for higher-order dielectric absorbers is applied to optimize the grid parameters, maximizing the absorption bandwidth-to-thickness ratio. Furthermore, unlike conventional dielectric absorbers where higher-order dispersion is typically confined to the grid, this approach allows multi-order dispersion to be incorporated in both the grid and the substrate, offering greater flexibility and enhanced performance. It is a well-known fact that any realistic frequency of permeability of a magnetic substrate can be modelled as an ^^^^^^^^ℎorder Lorentz dispersion as shown below:

[0133] ^^^^(^^^^)(^^^^) = ^^^^ (^^^^)^^^^0,^^^^�1+ ^^^^^^^^^^^^^^^^^^^^� 0,^^^^,^^^0^^^^^^^^ (^^^^) = ^^^^0�1 + ∑^^^^^^^^^=1 ^^^^^^^^ ^^^^2^^^^2� (18)[ ]0,^^^^,^^^^ 0,^^^^,^^^^p y, p g r and resonance frequency of the ^^^^^^^^ℎLorentzian order. The admittance of thin metal-backed magnetic substrate with ^^^^^^^^ℎdispersion in permeability will be

[0135] ^^^^(^^^^) ^^^^^^^^^^^^ = ^^^^^^^^(^^^^)(^^^^)^^^^(19)

[0136] By incorporating a order dispersion in the substrate and ^^^^^^^^ℎorder dispersion in the grid, the bandwidth-to-thickness ratio can be further enhanced as the overall system effectivelybehaves as an (^^^^ + ^^^^)^^^^ℎ order network. This can be seen in Fig. 13b where a higher orderdispersion in the magnetic substrate is effectively increasing the absorption bandwidth for given thickness.

[0137] Building on this analysis, two complementary design approaches naturally emerge. In the first, the magnetic substrate is assumed to have fixed dispersion characteristics, and the grid is tailored potentially with high-order dispersion to achieve optimal performance. That means for any given magnetic substrate we can design absorber that provides maximum bandwidth-to- thickness ratio based on this described approach. In the second, the substrate and grid are treated as a unified dispersive system, and their parameters are co-optimized to extract the ideal combination that yields the maximum achievable bandwidth-to-thickness ratio. Here we can Page 39 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)extract the ideal dispersion parameters of the grid as well that of substrate together to get maximum bandwidth-to-thickness ratio. These two strategies offer design flexibility depending on practical constraints and available magnetic materials.

[0138] With the theoretical foundation established and a robust methodology developed for extracting design parameters that maximize the bandwidth-to-thickness ratio under specific conditions, we now extend this framework to explore the fundamental performance limits of highly dispersive magnetic absorbers. Specifically, we apply the established design methodology to determine upper bounds on the absorption bandwidth-to-thickness ratio for any given combination of dispersion order and static permeability, similar to the bounds previously derived for dielectric absorbers. To this end, we compute the maximum achievable bandwidth-to- thickness ratios across various combinations of grid and substrate dispersion orders, under different reflection thresholds, for any given static magnetic permeability. By examining the extracted numerical data across different dispersion orders, ^^^^0and ^^^^^^^^, we have formulated a generalized expression for the maximum achievable bandwidth-to-thickness ratio as follows: 1 Δ^^3^^^^^^^^+^^^^+1

[0139] ^^(^^^^ , ) 0^^^^0 ^^^^,^^^^, ^^^^^^^^ ≤ 4^^^^^^^^^^^^�^^^^ ^^^^� 2(20)

[0140] where is the order of dispersion in the grid, ^^ is the order of dispersion in the substrate, and ^^^^^^^^is the static permeability of the substrate, defined as

[0141] ^^^^^^^^ = 1 + ∑^^^^^^^^=1 ^^^^^^^^,^^^^ (21)

[0142] Fig.13c illustrates the numerically obtained maximum bandwidth-to-thickness ratios for various dispersion order combinations and can be seen that the described generalized upper bound is well-fitted to the data.

[0143] Comparison of bandwidth-to-thickness ratios with existing wideband magnetic absorbers

[0144] Fig.13d compares the absorption bandwidth-to-thickness ratios of several high- performance wideband magnetic absorbers reported in the literature. While most existing designs fall significantly short of leveraging the full potential permitted by passive, linear, and time- invariant structures reaching at best only about 60% of the ultimate theoretical bound, not a Page 40 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)single design surpasses the performance of our described minimum-order design (first-order dispersion in both grid and substrate, total dispersion order being 2). This highlights the remarkable potential of our design methodology, which enables the realization of magnetic absorbers with the highest bandwidth-to-thickness ratios ever reported and offers a viable pathway to approach arbitrarily close to the ultimate Rozanov bound.

[0145] In each alternatively embodiment of circuitry 104 herein, e.g., including in reference to Fig.2a, Fig.2c, Fig.4a, Fig.4b, Fig.7a, Fig.7b, Fig.7c, Fig.11c, Fig.12a, Fig.13a circuitry 104 defining an impedance grid can be configured to exhibit the admittance characteristics as shown in Fig.2b or Fig.12b and can be further configured so that an input impedance of conductor-backed substrate (e.g., conductor-backed dielectric or conductor-backed magnetic substrate) is matched to the impedance of free space over the design absorption frequency range of the described absorber structure.

[0146] In each alternative embodiment of circuitry 104 herein, including but not limited to those shown in Fig.2a, Fig.2c, Fig.4a, Fig.4b, Fig.7a, Fig.7b, Fig.7c, Fig.11c, Fig.12a, and Fig. 13a, circuitry 104 defining an impedance grid can be configured to exhibit the admittance characteristics as illustrated in Fig.2b or Fig.12b. Circuitry 104 can further be configured so that the input impedance of the conductor-backed substrate—whether provided by a conductor- backed dielectric or a conductor-backed magnetic substrate—is matched to the impedance of free space over the design absorption frequency range of the described absorber structure.

[0147] In each embodiment, circuitry 104 can achieve this matching condition by being designed to present an admittance having an imaginary component that is the inverse of the imaginary component of the input admittance of the conductor-backed substrate across the relevant frequency range. Equivalently, from an impedance perspective, circuitry 104 can present a reactive impedance with an imaginary component that is equal in magnitude and opposite in sign to the reactive impedance of the conductor-backed substrate over the design absorption frequency range. This relationship ensures that the combined parallel admittance of circuitry 104 and the conductor-backed substrate achieves a real-valued input admittance corresponding to the free-space characteristic admittance, thereby minimizing reflection and maximizing absorption. Page 41 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)

[0148] Through precise control of the dispersive properties of circuitry 104, including the use of engineered Lorentzian dispersion profiles or equivalent circuit implementations, this inverse reactive behavior can be tailored to span both the negative inductive and capacitive operating regions as required for broadband impedance matching

[0149] Embodiments herein recognize that the nth-order expansion method described in reference with Fig.8a, originally applied to the conductor-backed dielectric substrate embodiment, also applies directly to the second embodiment involving a conductor-backed magnetic substrate as described in Figs.11–13. This is based on the general applicability of the expansion method, which models the substrate’s frequency-dependent admittance independently of its specific material properties.

[0150] In both embodiments, the substrate admittance exhibits frequency-dependent reactive behavior that must be accurately modeled and compensated to achieve broadband impedance matching. The nth-order expansion method provides a systematic approach for expressing the substrate admittance as a power series in frequency, enabling the design of circuitry 104 to introduce inverse reactive components that cancel the substrate’s reactive effects across a broad frequency range.

[0151] For the dielectric case, the substrate admittance is expressed as: Y_sub(ω) ≈ -j / (ω * μ₀ * d)

[0152] For the magnetic substrate case, the admittance includes the frequency-dependent permeability term μr(ω): Y_sub(ω) ≈ -j / [ω * μ₀ * d * μr(ω)]

[0153] Despite the added dispersion in the magnetic case, the admittance remains suitable for series expansion, often requiring higher-order terms to accurately model the Lorentzian dispersion of μr(ω). Using the nth-order expansion:Y_sub(ω) ≈ Σ (a_n * ω^n) for n = 1 to N

[0154] Circuitry 104 can then be configured to present an admittance that cancels each corresponding order of the substrate admittance, satisfying: Y_grid(ω) + Y_sub(ω) = 1 / η₀

[0155] where η₀ is the free-space impedance. Page 42 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)

[0156] This method enables broadband matching even in the presence of complex, frequency- dispersive substrate behavior. Embodiments herein further recognize that the required component values for the synthesized network—such as inductors, capacitors, and resistors— will vary based on the selected magnetic material properties, including the static susceptibility χ₀, damping factor σ, and resonance frequency ω₀,m. These parameters influence the dispersion characteristics of the substrate and, therefore, the required order and parameters of the higher order network needed to accurately model and cancel the substrate’s reactive components over the desired frequency range.

[0157] As these material-dependent parameters directly affect the slope and curvature of the substrate’s admittance profile, higher values of static susceptibility χ₀ or lower damping factors σ can increase the prominence of higher-order reactive effects. This can enhance the benefit of providing a higher-order dispersive network and corresponding adjustments to the synthesized network’s component values. The described material-dependent tuning ensures that circuitry 104 maintains precise impedance matching across the design absorption frequency range, thereby optimizing absorption performance and maximizing bandwidth efficiency.

[0158] Accordingly, while the nth-order dispersive network was originally developed for dielectric substrates, it remains fully applicable to conductor-backed magnetic substrates and often delivers even greater advantages in achieving wideband absorption and improved bandwidth-to-thickness ratios.

[0159] Embodiments herein recognize that increasing the dispersion order of circuitry 104 directly enhances absorption bandwidth for both the conductor-backed dielectric substrate and conductor-backed magnetic substrate embodiments. Higher dispersion orders provide greater flexibility in shaping the frequency-dependent admittance of circuitry 104, allowing it to more precisely counteract the complex reactive behavior of the substrate across a broader frequency range.

[0160] In both embodiments, broadband absorption is achieved by satisfying the condition: Y_grid(ω) + Y_sub(ω) = 1 / η₀ where Y_grid(ω) is the admittance of circuitry 104, Y_sub(ω) is the substrate admittance, and η₀ is the free-space impedance. Increasing the dispersion order Page 43 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)introduces additional degrees of freedom, enabling circuitry 104 to cancel reactive components at multiple frequencies and broaden the effective absorption bandwidth.

[0161] In the dielectric embodiment, the substrate admittance has a relatively simple inductive dependence, but higher dispersion orders become necessary as the target absorption bandwidth increases. In the magnetic substrate embodiment, the substrate admittance exhibits complex Lorentzian dispersion due to frequency-dependent permeability μr(ω), producing inductive behavior in one region and negative capacitive behavior in another.

[0162] Embodiments herein recognize that for the magnetic embodiment, the minimum required dispersion order starts at second order. This is because the minimum order in the circuitry 104 is n=1 and the minimum dispersion order in the magnetic substrate is m=1, making the total order n+m=2.

[0163] As the desired absorption bandwidth increases, higher-order dispersion becomes even more critical, allowing circuitry 104 to present the appropriate inverse reactive response across a broader spectral range. This ensures wideband impedance matching and maximizes absorber performance in both embodiments.

[0164] Embodiments herein further recognize that configuring circuitry 104 to define circuitry equivalent to a capacitor filled with a dispersive dielectric material can provide the same functional behavior and advantages in the embodiment of Figs.11–13, which employs a conductor-backed magnetic substrate, as it does in the embodiment of Figs.1–10, which employs a conductor-backed dielectric substrate. This is the case because, in both embodiments, the function of circuitry 104 is to supply a frequency-dependent admittance that counteracts the reactive behavior of the underlying conductor-backed substrate over a defined design absorption frequency range. In particular, circuitry 104 can be designed to exhibit an admittance with an imaginary component that is equal in magnitude and opposite in sign to the imaginary component of the admittance presented by the conductor-backed substrate at each frequency within the operating range. This inverse reactive relationship ensures that the combined admittance of the substrate and circuitry 104 forms a broadband matched condition to the free-space admittance, supporting minimal reflection and maximal absorption. Equivalently, from an impedance perspective, circuitry 104 can be configured to present a reactive impedance Page 44 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)that is the inverse of the substrate’s reactive impedance, thereby achieving impedance matching through cancellation of the total imaginary component.

[0165] Embodiments herein recognize that the same equivalent circuits introduced in connection with the conductor-backed dielectric substrate—such as the circuit shown in Fig.2c and the Zobel-transformed network of Fig.7b—can be applied directly to the magnetic substrate case described in Figs.11–13. This applicability is supported by the common transmission-line-based framework used to model both substrate types, where the metal-backed dielectric and metal- backed magnetic substrates are each represented as reactive elements with frequency-dispersive behavior. In both cases, circuitry 104 can be designed to present a complementary admittance profile that matches the substrate’s behavior over a wide frequency range, regardless of whether the substrate's reactive component arises from permittivity-based inductance or permeability- based dispersion.

[0166] The equivalence of circuit modeling across these embodiments is grounded in the underlying mathematical formulation of the system admittance. Specifically, the admittance of the conductor-backed substrate can be approximated as Y_sub ≈ -j / [ω * L_sub(ω)], where L_sub(ω) represents either the effective inductance of the dielectric-based substrate (e.g., L_md = μ₀ * d) or the frequency-dispersive inductance of the magnetic-based substrate (e.g., L_ms = μ₀ * d * μr(ω)). In either case, the condition for matching to free space can be expressed as Y_grid(ω) + Y_sub(ω) = 1 / η₀, where η₀ is the free-space impedance.

[0167] Because the permeability μr(ω) of magnetic substrates introduces additional frequency-dependent behavior not present in conventional dielectric layers, embodiments herein recognize that the specific circuit parameters of any selected equivalent circuit—such as the values of capacitors, inductors, and resistors—may vary based on the chosen magnetic material and the desired performance targets of the absorber. For example, achieving the correct slope and magnitude of the imaginary component of admittance across the design frequency range may require tuning the plasma frequency ωp, damping factor γ, and baseline capacitance C₀ in the dispersive capacitor model. These parameters must be selected in view of the magnetic substrate’s intrinsic Lorentzian dispersion profile to ensure that the composite response of the system maintains broadband matching characteristics. Consequently, while the overall circuit Page 45 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)topology may remain the same, its component values must be adapted based on the magnetic substrate’s properties and the absorber’s intended application frequency range.

[0168] Embodiments herein further recognize that the method illustrated in Fig.8b, originally described in connection with the conductor-backed dielectric substrate embodiment, can also be directly applied to the embodiment of Figs.11–13, which employs a conductor-backed magnetic substrate. This recognition arises from the fundamental equivalence in how both substrate types can be modeled using transmission-line theory and equivalent circuit transformations. In both cases, the core objective is to achieve broadband impedance matching by synthesizing circuitry 104 to present a frequency-dependent admittance profile that cancels the reactive behavior introduced by the conductor-backed substrate.

[0169] In the case of the conductor-backed magnetic substrate, the substrate admittance exhibits complex frequency-dependent behavior due to the Lorentzian dispersion of its permeability, μr(ω). This dispersion introduces effective inductive behavior in one frequency region and negative capacitive behavior in another, depending on the operating frequency. Despite this added complexity, the approach illustrated in Fig.8b—which involves systematic synthesis of circuit elements to achieve the desired frequency response—remains fully applicable. The methodology enables the design of circuitry 104 to introduce compensating reactive elements that achieve broadband matching across the expanded frequency ranges presented by the dispersive magnetic substrate.

[0170] Specifically, the method of Fig.8b applies because the same mathematical principles govern the admittance matching condition for both substrate types, expressed as: Y_grid(ω) + Y_sub(ω) = 1 / η₀ where Y_grid(ω) is the admittance provided by circuitry 104, Y_sub(ω) is the substrate admittance (whether defined by dielectric or magnetic properties), and η₀ is the free- space impedance.

[0171] In the embodiment of Figs.11–13, the magnetic substrate admittance Y_sub(ω) can be expressed as: Y_sub(ω) ≈ -j / [ω * μ₀ * d * μr(ω)].

[0172] This equation parallels the dielectric case but introduces the frequency-dependent permeability term μr(ω), which must be accounted for in the design of circuitry 104. The method Page 46 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)of Fig.8b is directly applicable because it enables the derivation of optimal values for the components of circuitry 104, such that its admittance profile effectively cancels the reactive components introduced by the frequency-dependent behavior of μr(ω).

[0173] Embodiments herein further recognize that although the overall circuit topologies applied to both substrate types remain the same—whether utilizing the circuit synthesis approach of Fig. 8b or the Zobel transformation shown in Fig.7b—the specific values of the inductors, capacitors, and resistors selected within these circuits will differ. These component values are dependent on the selected magnetic material properties, including its static susceptibility χ₀, damping factor σ, and resonance frequency ω₀,m.

[0174] These material parameters fundamentally determine the frequency-dependent behavior of the substrate’s permeability μr(ω), which directly affects the substrate’s reactive admittance profile across the design absorption frequency range. As a result, the values of the inductors, capacitors, and resistors in circuitry 104 must be carefully selected to ensure that the synthesized network accurately cancels the reactive components introduced by the magnetic substrate over both the first and second operating frequency regions.

[0175] For example, a higher static susceptibility χ₀ typically increases the strength of the magnetic resonance response, requiring adjustments to the circuit element values to properly align the compensating admittance profile with the substrate’s behavior. Similarly, the damping factor σ influences the bandwidth of the substrate’s magnetic resonance, dictating the sharpness of the required transition between negative inductive and capacitive behaviors in circuitry 104.

[0176] The resonance frequency ω₀,m sets the center frequency of the substrate’s Lorentzian dispersion, directly impacting the design frequency or border frequency ωBaround which the circuit can manage the transition between reactive behaviors. Proper selection and tuning of these material-dependent parameters is therefore critical to ensure that circuitry 104 effectively compensates for the substrate’s frequency-dispersive characteristics, maintaining broadband impedance matching and maximizing absorption performance across the desired operating frequency range. Page 47 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)

[0177] Accordingly, embodiments herein recognize that precise tuning of these component values is essential to maintain broadband impedance matching, maximize absorber efficiency, and ensure that the desired bandwidth-to-thickness performance ratio is achieved.

[0178] Benefits derived from adjusting these values can arise because the Lorentzian dispersion of the magnetic substrate alters the slope and magnitude of the substrate's admittance profile across frequency, directly impacting the matching conditions that must be satisfied by circuitry 104. Therefore, while the synthesis and transformation methods remain valid and effective across both embodiments, their implementation requires material-specific tuning to ensure that the desired broadband impedance matching is achieved.

[0179] One general aspect includes an absorber structure that includes a conductor-backed dielectric; and circuitry connected in parallel with the conductor-backed dielectric, where the circuitry connected in parallel with the conductor-backed dielectric exhibits an admittance with an imaginary component having a negative slope within a frequency range encompassing a design frequency of the absorber structure.

[0180] Implementations may include one or more of the following features. The absorber structure where the admittance exhibited by the circuitry connected in parallel with the conductor-backed dielectric includes a dispersion-less real component within the frequency range encompassing the design frequency of the absorber structure. The circuitry is passive circuitry. The circuitry may include passive circuit components. The circuitry may be absent of active circuitry. The circuitry connected in parallel with the conductor-backed dielectric exhibits performance characteristics of a capacitor filled with a dispersive dielectric. The circuitry connected in parallel with the conductor-backed dielectric includes a capacitor connected in parallel with a series connected circuit path, the series connected circuit path including a series connected resistor, a series connected capacitor, and a series connected inductor. The capacitor includes a certain frequency-independent capacitance, where a resistance of the series connected resistor is a function of the certain capacitance, where a capacitance of the series connected capacitor is a function of the certain capacitance, and where an inductance of the series connected inductor is a function of the certain capacitance, and where the resistance of the series connected resistor, the capacitance of the series connected capacitor and the inductance of the Page 48 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)series connected inductor are independent of a frequency of an input signal input to the absorber structure. The circuitry connected in parallel with the conductor-backed dielectric includes a capacitor in series with first and second circuit paths that are connected in parallel to one another, the first circuit path having a second capacitor, and the second circuit path having an inductor in series with a resistor. The capacitor includes a certain frequency-independent capacitance, where a capacitance of the second capacitor is a function of the certain capacitance, where an inductance of the inductor is a function of the certain capacitance, where a resistance of the resistor is a function of the certain capacitance, where the capacitance of the second capacitor, the inductance of the inductor, and the resistance of the resistor are independent of a frequency of an input signal input to the absorber structure. The admittance imaginary component has a positive slope outside of the frequency range encompassing the design frequency of the absorber structure. The admittance imaginary component has a positive slope outside of the frequency range encompassing the design frequency of the absorber structure, the frequency range defining a design absorption frequency range of the absorber structure. The circuitry defines an nth order dispersive network. The circuitry connected in parallel with the conductor-backed dielectric includes first and second capacitors, an inductor, and a resistor. The circuitry is configured so that reflection from the absorber structure remains below a defined maximum level over the frequency range encompassing the design frequency of the absorber structure. The frequency range encompassing the design frequency of the absorber structure is centered on the design frequency of the absorber structure.

[0181] One general aspect includes an absorber structure that includes a conductor-backed dielectric; and a capacitor filled with dispersive material connected in parallel with the conductor-backed dielectric, where the capacitor filled with dispersive material exhibits an admittance with an imaginary component having a negative slope within a frequency range encompassing a design frequency of the absorber structure.

[0182] Implementations may include one or more of the following features. The absorber structure where the admittance exhibited by the capacitor filled with dispersive material includes a dispersion-less real component within the frequency range encompassing the design frequency of the absorber structure. The capacitor filled with dispersive material is configured so that a reflection from the absorber structure remains below a defined maximum level over the Page 49 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)frequency range encompassing the design frequency of the absorber structure. The imaginary component of the admittance exhibited by the capacitor filled with dispersive material, outside of the frequency range encompassing the design frequency of the absorber structure, includes a positive slope. The capacitor filled with dispersive material defines an nth order dispersive network.

[0183] One general aspect includes a method of designing an absorber structure. The method also includes configuring circuitry connected in parallel with a conductor-backed dielectric, where the configuring includes performing the configuring so that the circuitry connected in parallel with the conductor-backed dielectric exhibits an admittance with an imaginary component having a negative slope within a frequency range encompassing a design frequency of the absorber structure.

[0184] Implementations may include one or more of the following features. The method where the configuring includes producing a circuit defining an equivalent circuit of capacitor filled with dispersive material. The configuring includes producing a circuit defining an equivalent circuit of a capacitor filled with dispersive material. The configuring includes producing, with use of an equivalent circuit transformation, a circuit defining an equivalent circuit of a capacitor filled with dispersive material. The configuring includes producing, with use of a Zobel equivalent circuit transformation, a circuit defining an equivalent circuit of a capacitor filled with dispersive material. The configuring includes producing a design for a capacitor filled with a dispersive material, and producing an equivalent circuit of the design for a capacitor filled with a dispersive material. The method includes tuning a design absorption frequency range of the absorber structure, where the tuning includes adjusting a reflectance threshold of the absorber structure. The method includes tuning a design absorption frequency range of the absorber structure, where the tuning includes adjusting a dispersive grid order of the absorber structure. The method includes increasing a design absorption frequency range of the absorber structure, where the increasing the design absorption frequency range of the absorber structure includes adjusting a dispersive grid order of the absorber structure. The configuring includes performing the configuring so that the circuitry connected in parallel with the conductor-backed dielectric defines a multi-order dispersive grid. The configuring is performed so that the circuitry connected in parallel with the conductor-backed dielectric defines a multi-order dispersive grid, Page 50 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)and where determining parameters of the multi-order dispersive grid includes employing a cost function. The method includes determining, for a given dispersion grid order, thickness, and reflectance threshold of the absorber structure, an absorber structure giving the maximum achievable bandwidth to thickness ratio. The method includes determining, with use of a cost function, for a given dispersion grid order, thickness, and reflectance threshold of the absorber structure, an achievable absorber structure design absorption frequency range.

[0185] An absorber structure can include a conductor-backed substrate; and circuitry connected in parallel with the conductor-backed substrate, where the circuitry connected in parallel with the conductor-backed substrate exhibits an admittance with an imaginary component having a negative slope within a design absorption frequency range of the absorber structure. Other embodiments of this aspect include corresponding computer systems, apparatus, and computer programs recorded on one or more computer storage devices, each configured to perform the actions of the methods.

[0186] Implementations may include one or more of the following features. The absorber structure where the conductor-backed substrate includes a conductor-backed dielectric. The conductor-backed substrate includes a conductor-backed magnetic substrate. The design absorption frequency range is defined by a frequency range which the circuitry operates as a negative inductor. The design absorption frequency range extends from a first frequency at which the circuitry commences operation as a negative inductor to a second frequency at which the conductor-backed substrate ceases to exhibit negative capacitive behavior. The design absorption frequency range is defined as a range extending from a first frequency at which the circuitry commences operation as a negative inductor to a second frequency at which the conductor-backed substrate ceases to exhibit behavior as a negative capacitor, the design absorption frequency range encompassing a first operating region and a second operating region of the circuitry, where in the first operating region the circuitry operates as a negative inductor and in the second operating region the circuitry operates as a capacitor. The circuitry connected in parallel with the conductor-backed substrate exhibits an admittance with an imaginary component having a positive slope within the design absorption frequency range of the absorber structure. The circuitry connected in parallel with the conductor-backed substrate exhibits an admittance with an imaginary component having a positive slope within the design absorption Page 51 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)frequency range of the absorber structure, the imaginary component of the admittance having the negative slope within a first region of the design absorption frequency range of the absorber structure, the imaginary component of the admittance having the positive slope within a second region of the design absorption frequency range of the absorber structure. The admittance exhibited by the circuitry connected in parallel with the conductor-backed substrate includes a dispersion-less real component within the design absorption frequency range of the absorber structure. The circuitry is passive circuitry. The circuitry may include of passive circuit components. The circuitry is absent of active circuitry. The circuitry connected in parallel with the conductor-backed substrate exhibits performance characteristics of a capacitor filled with a dispersive dielectric. The circuitry connected in parallel with the conductor-backed substrate includes a capacitor connected in parallel with a series connected circuit path, the series connected circuit path including a series connected resistor, a series connected capacitor, and a series connected inductor. The circuitry connected in parallel with the conductor-backed substrate includes a capacitor connected in parallel with a series connected circuit path, the series connected circuit path including a series connected resistor, a series connected capacitor, and a series connected inductor, where the capacitor includes a certain frequency-independent capacitance, where a resistance of the series connected resistor is a function of the certain capacitance, where a capacitance of the series connected capacitor is a function of the certain capacitance, and where an inductance of the series connected inductor is a function of the certain capacitance, and where the resistance of the series connected resistor, the capacitance of the series connected capacitor, and the inductance of the series connected inductor are independent of a frequency of an input signal input to the absorber structure. The circuitry connected in parallel with the conductor-backed substrate includes a capacitor in series with first and second circuit paths that are connected in parallel to one another, the first circuit path having a second capacitor, and the second circuit path having an inductor in series with a resistor. The circuitry connected in parallel with the conductor-backed substrate includes a capacitor in series with first and second circuit paths that are connected in parallel to one another, the first circuit path having a second capacitor, and the second circuit path having an inductor in series with a resistor, where the capacitor includes a certain frequency independent capacitance, where a capacitance of the second capacitor is a function of the certain capacitance, where an inductance of the inductor is a function of the certain capacitance, where a resistance of the resistor is a function of the certain Page 52 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)capacitance, and where the capacitance of the second capacitor, the inductance of the inductor, and the resistance of the resistor are independent of a frequency of an input signal input to the absorber structure. The admittance imaginary component has a positive slope within the design absorption frequency range of the absorber structure. The admittance imaginary component has a positive slope within the design absorption frequency range of the absorber structure, the admittance imaginary component having the negative slope within a first region of the design absorption frequency range, the admittance imaginary component having the positive slope within a second region of the design absorption frequency range. The circuitry defines an nth order dispersive network. The circuitry connected in parallel with the conductor-backed substrate includes first and second capacitors, an inductor, and a resistor. The circuitry is configured so that reflection from the absorber structure remains below a defined maximum level over the design absorption frequency range. The frequency range encompassing the design frequency of the absorber structure is centered on the design frequency of the absorber structure.

[0187] The absorber structure in one embodiment also includes a conductor-backed substrate. The structure also includes a capacitor filled with dispersive material connected in parallel with the conductor-backed substrate, where the capacitor filled with dispersive material exhibits an admittance with an imaginary component having a negative slope within a design absorption frequency range of the absorber structure.

[0188] Implementations may include one or more of the following features. The absorber structure where the conductor-backed substrate includes a conductor-backed dielectric. The conductor-backed substrate includes a conductor-backed magnetic substrate. The design absorption frequency range is defined by a frequency range in which the circuitry operates as a negative inductor. The design absorption frequency range extends from a first frequency at which the circuitry commences operation as a negative inductor to a second frequency at which the conductor-backed substrate ceases to exhibit negative capacitive behavior. The design absorption frequency range is defined as a range extending from a first frequency at which the circuitry commences operation as a negative inductor to a second frequency at which the conductor-backed substrate ceases to exhibit behavior as a negative capacitor, the design absorption frequency range encompassing a first operating region and a second operating region of the circuitry, where in the first operating region the circuitry operates as a negative inductor Page 53 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)and in the second operating region the circuitry operates as a capacitor. The admittance exhibited by the capacitor filled with dispersive material includes a dispersion-less real component within the design absorption frequency range of the absorber structure. The capacitor filled with dispersive material is configured so that a reflection from the absorber structure remains below a defined maximum level over the design absorption frequency range of the absorber structure. The imaginary component of the admittance exhibited by the capacitor filled with dispersive material, within the design absorption frequency range of the absorber structure, includes a positive slope. The capacitor filled with dispersive material defines an nth order dispersive network.

[0189] One general aspect includes a method of designing an absorber structure. The method also includes configuring circuitry connected in parallel with a conductor-backed substrate, where the configuring includes performing the configuring so that the circuitry connected in parallel with the conductor-backed substrate exhibits an admittance with an imaginary component having a negative slope within a design absorption frequency range of the absorber structure.

[0190] Implementations may include one or more of the following features. The method where the conductor-backed substrate is a conductor-backed dielectric. The conductor-backed substrate is a conductor-backed magnetic substrate. The design absorption frequency range is defined by a frequency range which the circuitry operates as a negative inductor. The design absorption frequency range extends from a first frequency at which the circuitry commences operation as a negative inductor to a second frequency at which the conductor-backed substrate ceases to exhibit negative capacitive behavior. The design absorption frequency range is defined as a range extending from a first frequency at which the circuitry commences operation as a negative inductor to a second frequency at which the conductor-backed substrate ceases to exhibit behavior as a negative capacitor, the design absorption frequency range encompassing a first operating region and a second operating region of the circuitry, where in the first operating region the circuitry operates as a negative inductor and in the second operating region the circuitry operates as a capacitor. The configuring includes producing a circuit defining an equivalent circuit of a capacitor filled with dispersive material. The configuring includes producing a circuit defining an equivalent circuit of a capacitor filled with dispersive material. Page 54 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)The configuring includes producing, with use of an equivalent circuit transformation, a circuit defining an equivalent circuit of a capacitor filled with dispersive material. The configuring includes producing, with use of a Zobel equivalent circuit transformation, a circuit defining an equivalent circuit of a capacitor filled with dispersive material. The configuring includes producing a design for a capacitor filled with a dispersive material, and producing an equivalent circuit of the design for a capacitor filled with a dispersive material. The method includes tuning a design absorption frequency range of the absorber structure, where the tuning includes adjusting a reflectance threshold of the absorber structure. The method includes tuning the design absorption frequency range of the absorber structure, where the tuning includes adjusting a dispersive grid order of the absorber structure. The method includes increasing the design absorption frequency range of the absorber structure, where increasing the design absorption frequency range of the absorber structure includes adjusting a dispersive grid order of the absorber structure. The configuring includes performing the configuring so that the circuitry connected in parallel with the conductor-backed substrate defines a multi-order dispersive grid. The configuring is performed so that the circuitry connected in parallel with the conductor- backed substrate defines a multi-order dispersive grid, and where determining parameters of the multi-order dispersive grid includes employing a cost function. The method includes determining, for a given dispersion grid order, thickness, and reflectance threshold of the absorber structure, an achievable absorber structure design absorption frequency range. The method includes determining, with use of a cost function, for a given dispersion grid order, thickness, and reflectance threshold of the absorber structure, an achievable absorber structure design absorption frequency range.

[0191] The absorber structure also includes in one embodiment a conductor-backed substrate; and circuitry connected in parallel with the conductor-backed substrate, where the circuitry connected in parallel with the conductor-backed substrate performs as negative inductor within a design absorption frequency range of the absorber structure.

[0192] Implementations may include one or more of the following features. The absorber structure where the circuitry connected in parallel with the conductor-backed substrate operates as a negative inductor within a first frequency region of the design absorption frequency range of the absorber structure, and operates as a capacitor within a second frequency region of the design Page 55 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)absorption frequency range of the absorber structure. The design absorption frequency range is defined by a frequency range which the circuitry operates as a negative inductor. The design absorption frequency range extends from a first frequency at which the circuitry commences operation as a negative inductor to a second frequency at which the conductor-backed substrate ceases to exhibit negative capacitive behavior. The design absorption frequency range is defined as a range extending from a first frequency at which the circuitry commences operation as a negative inductor to a second frequency at which the conductor-backed substrate ceases to exhibit behavior as a negative capacitor, the design absorption frequency range encompassing first operating region and a second operating region of the circuitry, where in the first operating region the circuitry operates as a negative inductor and in the second operating region the circuitry operates as a capacitor. The conductor-backed substrate includes a conductor-backed dielectric. The conductor-backed substrate includes a conductor-backed magnetic substrate. Implementations of the described techniques may include hardware, a method or process, or computer software on a computer-accessible medium.

[0193] The absorber structure also includes in one embodiment a conductor-backed magnetic substrate; and circuitry connected in parallel with the conductor-backed magnetic substrate, where the circuitry connected in parallel with the conductor-backed magnetic substrate exhibits an admittance with an imaginary component having a negative slope within a first region of a design absorption frequency range of the absorber structure and a positive slope within a second region of the design absorption frequency range of the absorber structure. Other embodiments of this aspect include corresponding computer systems, apparatus, and computer programs recorded on one or more computer storage devices, each configured to perform the actions of the methods.

[0194] One general aspect includes a conductor-backed magnetic substrate. The absorber structure also includes circuitry connected in parallel with the conductor-backed magnetic substrate, the circuitry may include passive circuit components and being absent of active circuitry, the circuitry connected in parallel with the conductor-backed magnetic substrate exhibiting an admittance with an imaginary component having a negative slope within a design absorption frequency range of the absorber structure, and where: the admittance of the circuitry further includes a dispersion-less real component within the design absorption frequency range. The structure also includes the imaginary component of the admittance has the negative slope Page 56 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)within a first region of the design absorption frequency range and has a positive slope within a second region of the design absorption frequency range. The structure also includes the design absorption frequency range is defined as extending from a first frequency at which the circuitry commences operation as a negative inductor to a second frequency at which the conductor- backed magnetic substrate ceases to exhibit behavior as a negative capacitor, the design absorption frequency range encompassing a first operating region in which the circuitry operates as a negative inductor and a second operating region in which the circuitry operates as a capacitor. The structure also includes the circuitry is configured so that reflection from the absorber structure remains below a defined maximum level over the design absorption frequency range. The structure also includes the circuitry defines an nth order dispersive network that exhibits performance characteristics of a capacitor filled with a dispersive dielectric. The structure also includes and the circuitry may include a capacitor in series with first and second circuit paths that are connected in parallel to one another, the first circuit path may include a second capacitor, and the second circuit path may include an inductor in series with a resistor, where the capacitor has a certain frequency-independent capacitance, and where a capacitance of the second capacitor, an inductance of the inductor, and a resistance of the resistor are each a function of the certain capacitance and are independent of a frequency of an input signal input to the absorber structure.

[0195] One general aspect includes a conductor-backed magnetic substrate. The absorber structure also includes and circuitry connected in parallel with the conductor-backed magnetic substrate, the circuitry may include of passive circuit components and being absent of active circuitry, where the circuitry exhibits an admittance with an imaginary component having a negative slope within a design absorption frequency range of the absorber structure, where the design absorption frequency range is defined by a frequency range in which the circuitry operates as a negative inductor, and where the circuitry may include a capacitor in series with first and second circuit paths that are connected in parallel to one another, the first circuit path may include a second capacitor, and the second circuit path may include an inductor in series with a resistor, where the capacitor has a certain frequency-independent capacitance, and where a capacitance of the second capacitor, an inductance of the inductor, and a resistance of the resistor are each a function of the certain frequency-independent capacitance and are independent of a frequency of an input signal input to the absorber structure. Page 57 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)

[0196] One general aspect includes a method of designing an absorber structure, configuring circuitry connected in parallel with a conductor-backed magnetic substrate, where the configuring includes performing the configuring so that the circuitry exhibits an admittance with an imaginary component having a negative slope within a design absorption frequency range of the absorber structure, where the design absorption frequency range is defined as a range extending from a first frequency at which the circuitry commences operation as a negative inductor to a second frequency at which the conductor-backed magnetic substrate ceases to exhibit behavior as a negative capacitor, where the design absorption frequency range encompasses a first operating region and a second operating region of the circuitry, the circuitry operating as a negative inductor in the first operating region and as a capacitor in the second operating region. The method also includes where the configuring includes producing a circuit defining an equivalent circuit of a capacitor filled with a dispersive material, including use of a Zobel equivalent circuit transformation. The method also includes where the configuring further includes designing the circuitry to define a multi-order dispersive grid, and determining parameters of the multi-order dispersive grid using a cost function. The method also includes and may include tuning the design absorption frequency range by adjusting at least one of: a reflectance threshold of the absorber structure, a dispersive grid order, or a thickness of the absorber structure. The method also includes and may include determining, for a given dispersion grid order, thickness, and reflectance threshold of the absorber structure, an achievable absorber structure design absorption frequency range using a cost function.

[0197] The terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting of the disclosure. As used herein, the singular forms “a”, “an” and “the” are intended to include the plural forms as well, unless the context clearly indicates otherwise. It will be further understood that the terms “comprise” (and any form of comprise, such as “comprises” and “comprising”), “have” (and any form of have, such as “has” and “having”), “include” (and any form of include, such as “includes” and “including”), “contain” (and any form contain, such as “contains” and “containing”), and any other grammatical variant thereof, are open-ended linking verbs. As a result, a method or article that “comprises”, “has”, “includes” or “contains” one or more steps or elements possesses those one or more steps or elements but is not limited to possessing only those one or more steps or elements. Likewise, a step of a method or an element of an article that “comprises”, “has”, “includes” or “contains” one Page 58 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)or more features possesses those one or more features, but is not limited to possessing only those one or more features.

[0198] Terms like “obtainable” or “definable” and “obtained” or “defined” are used interchangeably. This, for example, means that, unless the context clearly dictates otherwise, the term “obtained” does not mean to indicate that, for example, an embodiment must be obtained by, for example, the sequence of steps following the term “obtained” though such a limited understanding is always included by the terms “obtained” or “defined” as a preferred embodiment.

[0199] It should be appreciated that all combinations of the foregoing concepts and additional concepts discussed in greater detail below (provided such concepts are not mutually inconsistent) are contemplated as being part of the subject matter disclosed herein. In particular, all combinations of claims subject matter appearing at the end of this disclosure are contemplated as being part of the subject matter disclosed herein. It should also be appreciated that terminology explicitly employed herein that also may appear in any disclosure incorporated by reference should be accorded a meaning most consistent with the particular concepts disclosed herein.

[0200] This written description uses examples to disclose the subject matter, and also to enable any person skilled in the art to practice the subject matter, including making and using any devices or systems and performing any incorporated methods. The patentable scope of the subject matter is defined by the claims, and may include other examples that occur to those skilled in the art. Such other examples are intended to be within the scope of the claims if they have structural elements that do not differ from the literal language of the claims, or if they include equivalent structural elements with insubstantial differences from the literal languages of the claims.

[0201] It is to be understood that the above description is intended to be illustrative, and not restrictive. For example, the above-described examples (and / or aspects thereof) may be used in combination with each other. In addition, many modifications may be made to adapt a particular situation or material to the teachings of the various examples without departing from their scope. While the dimensions and types of materials described herein are intended to define the parameters of the various examples, they are by no means limiting and are merely exemplary. Page 59 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)Many other examples will be apparent to those of skill in the art upon reviewing the above description. The scope of the various examples should, therefore, be determined with reference to the appended claims, along with the full scope of equivalents to which such claims are entitled. In the appended claims, the terms “including” and “in which” are used as the plain-English equivalents of the respective terms “comprising” and “wherein.” Moreover, in the following claims, the terms “first,” “second,” and “third,” etc. are used merely as labels, and are not intended to impose numerical requirements on their objects. Forms of term “based on” herein encompass relationships where an element is partially based on as well as relationships where an element is entirely based on. Forms of the term “defined” encompass relationships where an element is partially defined as well as relationships where an element is entirely defined. Further, the limitations of the following claims are not written in means-plus-function format and are not intended to be interpreted based on 35 U.S.C. § 112(f) unless and until such claim limitations expressly use the phrase “means for” followed by a statement of function void of further structure. It is to be understood that not necessarily all such objects or advantages described above may be achieved in accordance with any particular example. Thus, for example, those skilled in the art will recognize that the systems and techniques described herein may be embodied or carried out in a manner that achieves or optimizes one advantage or group of advantages as taught herein without necessarily achieving other objects or advantages as may be taught or suggested herein.

[0202] The terms “substantially”, “approximately”, “about”, “relatively”, or other such similar terms that may be used throughout this disclosure, including the claims, are used to describe and account for small fluctuations, such as due to variations in processing, from a reference or parameter. Such small fluctuations include a zero fluctuation from the reference or parameter as well. For example, they can refer to less than or equal to ± 10%, such as less than or equal to ± 5%, such as less than or equal to ± 2%, such as less than or equal to ± 1%, such as less than or equal to ± 0.5%, such as less than or equal to ± 0.2%, such as less than or equal to ± 0.1%, such as less than or equal to ± 0.05%. If used herein, the terms “substantially”, “approximately”, “about”, “relatively,” or other such similar terms may also refer to no fluctuations, that is, ± 0%. It is contemplated that numerical values, as well as other values that are recited herein can be modified by the term “about”, whether Page 60 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)expressly stated or inherently derived by the discussion of the present disclosure. Further, any description of a range herein can encompass all subranges.

[0203] The terms “connect,” “connected,” “contact” “coupled” and / or the like are broadly defined herein to encompass a variety of divergent arrangements and assembly techniques. These arrangements and techniques include, but are not limited to (1) the direct joining of one component and another component with no intervening components therebetween (i.e., the components are in direct physical contact); and (2) the joining of one component and another component with one or more components therebetween, provided that the one component being “connected to” or “contacting” or “coupled to” the other component is somehow in operative communication (e.g., electrically, physically, optically, etc.) with the other component (notwithstanding the presence of one or more additional components therebetween). It is to be understood that some components that are in direct physical contact with one another may or may not be in electrical contact with one another. Moreover, two components that are electrically connected, electrically coupled, optically connected, optically coupled, may or may not be in direct physical contact, and one or more other components may be positioned therebetween.

[0204] While the subject matter has been described in detail in connection with only a limited number of examples, it should be readily understood that the subject matter is not limited to such disclosed examples. Rather, the subject matter can be modified to incorporate any number of variations, alterations, substitutions or equivalent arrangements not heretofore described, but which are commensurate with the spirit and scope of the subject matter. Additionally, while various examples of the subject matter have been described, it is to be understood that aspects of the disclosure may include only some of the described examples. Also, while some examples are described as having a certain number of elements it will be understood that the subject matter can be practiced with less than or greater than the certain number of elements. Accordingly, the subject matter is not to be seen as limited by the foregoing description but is only limited by the scope of the appended claims.

[0205] All publications cited in this specification are herein incorporated by reference as if each individual publication were specifically and individually indicated to be incorporated by reference herein as though fully set forth. Page 61 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)

[0206] Subject matter incorporated by reference is not considered to be an alternative to any claim limitations, unless otherwise explicitly indicated.

[0207] Where one or more ranges are referred to throughout this specification, each range is intended to be a shorthand format for presenting information, where the range is understood to encompass each discrete point within the range as if the same were fully set forth herein.

[0208] While several aspects and embodiments of the present disclosure have been described and depicted herein, alternative aspects and embodiments may be affected by persons having ordinary skill in the art to accomplish the same objectives. Accordingly, this disclosure and the appended claims are intended to cover all such further and alternative aspects and embodiments as fall within the true spirit and scope of the present disclosure.

[0209] The following references are incorporated herein by reference in their entireties and a skilled person is considered to be aware of disclosure of these references. [1] W. Dallenbach and W. Kleinsteuber, “Reflection and absorption of decimeter-waves by plane dielectric layers,” Hochfrequenztechnik und Elektroakustik, vol.51, p.152, 1938. [2] W. W. Salisbury, “Absorbent body of electromagnetic waves,” U.S. Patent No.2,599,944, 10 June 1952. [3] F. Knott, J. F. Shaeffer, and M. T. Tuley, Radar Cross Section, Artech House, London, 1993. [4] A. Munk, Frequency Selective Surfaces: Theory and Design, Wiley, New York, 2000. [5] Sievenpiper, “High-impedance electromagnetic surfaces,” Ph.D. dissertation, Dept. Elect. Eng., Univ. California at Los Angeles, Los Angeles, CA, 1999. [6] L. Holloway, E.F. Kuester, J. A. Gordon, J. O’Hara, J. Booth, and D. R. Smith, “An Overview of the Theory and Applications of Metasurfaces: The Two-Dimensional Equivalents of Metamaterials,”IEEE Antennas Propag. Mag., vol.54, p.10, 2012. [7] N. Yu and F. Capasso, “Flat Optics with Designer Metasurfaces,”Nat. Mater., vol.13, p. 139, 2014. [8] S.A. Tretyakov, “Metasurfaces for general transformations of electromagnetic fields,”Philos. Trans. R. Soc. Lond. A, vol.373, p.20140362, 2015. Page 62 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)[9] S.B. Glybovski, S.A. Tretyakov, P.A. Belov, Y.S. Kivshar, C.R. Simovski, “Metasurfaces: From Microwaves to Visible,” Phys. Rep., vol.634, p.1, 2016.

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Claims

What is claimed is:

1. An absorber structure comprising: a conductor-backed dielectric; and circuitry connected in parallel with the conductor-backed dielectric, wherein the circuitry connected in parallel with the conductor-backed dielectric exhibits an admittance with an imaginary component having a negative slope within a frequency range encompassing a design frequency of the absorber structure.

2. The absorber structure of claim 1, wherein the admittance exhibited by the circuitry connected in parallel with the conductor-backed dielectric includes a dispersion-less real component within the frequency range encompassing the design frequency of the absorber structure.

3. The absorber structure of claim 1, wherein the circuitry is passive circuitry.

4. The absorber of claim 1, wherein the circuitry consists of passive circuit components.

5. The absorber structure of claim 1, wherein the circuitry is absent of active circuitry.

6. The absorber structure of claim 1, wherein the circuitry connected in parallel with the conductor-backed dielectric exhibits performance characteristics of a capacitor filled with a dispersive dielectric.

7. The absorber structure of claim 1, wherein the circuitry connected in parallel with the conductor-backed dielectric includes a capacitor connected in parallel with a series connected circuit path, the series connected circuit path including a series connected resistor, a series connected capacitor, and a series connected inductor.

8. The absorber structure of claim 1, wherein the circuitry connected in parallel with the conductor-backed dielectric includes a capacitor connected in parallel with a series connected circuit path, the series connected circuit path including a series connected resistor, a series connected capacitor, and a series connected inductor, wherein the capacitor includes a certainPage 67 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)frequency-independent capacitance, wherein a resistance of the series connected resistor is a function of the certain capacitance, wherein a capacitance of the series connected capacitor is a function of the certain capacitance, and wherein an inductance of the series connected inductor is a function of the certain capacitance, and wherein the resistance of the series connected resistor, the capacitance of the series connected capacitor and the inductance of the series connected inductor are independent of a frequency of an input signal input to the absorber structure.

9. The absorber structure of claim 1, wherein the circuitry connected in parallel with the conductor-backed dielectric includes a capacitor in series with first and second circuit paths that are connected in parallel to one another, the first circuit path having a second capacitor, and the second circuit path having an inductor in series with a resistor.

10. The absorber structure of claim 1, wherein the circuitry connected in parallel with the conductor-backed dielectric includes a capacitor in series with first and second circuit paths that are connected in parallel to one another, the first circuit path having a second capacitor, and the second circuit path having an inductor in series with a resistor, wherein the capacitor includes a certain frequency-independent capacitance, wherein a capacitance of the second capacitor is a function of the certain capacitance, wherein an inductance of the inductor is a function of the certain capacitance, wherein a resistance of the resistor is a function of the certain capacitance, wherein the capacitance of the second capacitor, the inductance of the inductor, and the resistance of the resistor are independent of a frequency of an input signal input to the absorber structure.

11. The absorber structure of claim 1, wherein the admittance imaginary component has a positive slope outside of the frequency range encompassing the design frequency of the absorber structure.

12. The absorber structure of claim 1, wherein the admittance imaginary component has a positive slope outside of the frequency range encompassing the design frequency of the absorber structure, the frequency range defining a design absorption frequency range of the absorber structure.

13. The absorber structure of claim 1, wherein the circuitry defines an nth order dispersivePage 68 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)network.

14. The absorber structure of claim 1, wherein the circuitry connected in parallel with the conductor-backed dielectric includes first and second capacitors, an inductor, and a resistor.

15. The absorber structure of claim 1, wherein the circuitry is configured so that reflection from the absorber structure remains below a defined maximum level over the frequency range encompassing the design frequency of the absorber structure.

16. The absorber structure of claim 1, wherein the frequency range encompassing the design frequency of the absorber structure is centered on the design frequency of the absorber structure.

17. An absorber structure comprising: a conductor-backed dielectric; and a capacitor filled with dispersive material connected in parallel with the conductor-backed dielectric, wherein the capacitor filled with dispersive material exhibits an admittance with an imaginary component having a negative slope within a frequency range encompassing a design frequency of the absorber structure.

18. The absorber structure of claim 17, wherein the admittance exhibited by the capacitor filled with dispersive material includes a dispersion-less real component within the frequency range encompassing the design frequency of the absorber structure.

19. The absorber structure of claim 17, wherein the capacitor filled with dispersive material is configured so that a reflection from the absorber structure remains below a defined maximum level over the frequency range encompassing the design frequency of the absorber structure.

20. The absorber structure of claim 17, wherein the imaginary component of the admittance exhibited by the capacitor filled with dispersive material, outside of the frequency range encompassing the design frequency of the absorber structure, includes a positive slope.

21. The absorber structure of claim 17, wherein the capacitor filled with dispersive material defines an nth order dispersive network.Page 69 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)22. A method of designing an absorber structure, the method comprising: configuring circuitry connected in parallel with a conductor-backed dielectric, wherein the configuring includes performing the configuring so that the circuitry connected in parallel with the conductor-backed dielectric exhibits an admittance with an imaginary component having a negative slope within a frequency range encompassing a design frequency of the absorber structure.

23. The method of claim 22, wherein the configuring includes producing a circuit defining an equivalent circuit of capacitor filled with dispersive material.

24. The method of claim 22, wherein the configuring includes producing a circuit defining an equivalent circuit of a capacitor filled with dispersive material.

25. The method of claim 22, wherein the configuring includes producing, with use of an equivalent circuit transformation, a circuit defining an equivalent circuit of a capacitor filled with dispersive material.

26. The method of claim 22, wherein the configuring includes producing, with use of a Zobel equivalent circuit transformation, a circuit defining an equivalent circuit of a capacitor filled with dispersive material.

27. The method of claim 22, wherein the configuring includes producing a design for a capacitor filled with a dispersive material, and producing an equivalent circuit of the design for a capacitor filled with a dispersive material.

28. The method of claim 22, wherein the method includes tuning a design absorption frequency range of the absorber structure, wherein the tuning includes adjusting a reflectance threshold of the absorber structure.

29. The method of claim 22, wherein the method includes tuning a design absorption frequency range of the absorber structure, wherein the tuning includes adjusting a dispersive grid order of the absorber structure.

30. The method of claim 22, wherein the method includes increasing a design absorption frequency range of the absorber structure, wherein the increasing the design absorptionPage 70 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)frequency range of the absorber structure includes adjusting a dispersive grid order of the absorber structure.

31. The method of claim 22, wherein the configuring includes performing the configuring so that the circuitry connected in parallel with the conductor-backed dielectric defines a multi- order dispersive grid.

32. The method of claim 22, wherein the configuring is performed so that the circuitry connected in parallel with the conductor-backed dielectric defines a multi-order dispersive grid, and wherein determining parameters of the multi-order dispersive grid includes employing a cost function.

33. The method of claim 22, wherein the method includes determining, for a given dispersion grid order, thickness, and reflectance threshold of the absorber structure, an achievable absorber structure design absorption frequency range.

34. The method of claim 22, wherein the method includes determining, with use of a cost function, for a given dispersion grid order, thickness, and reflectance threshold of the absorber structure, an achievable absorber structure design absorption frequency range.

35. An absorber structure comprising: a conductor-backed substrate; and circuitry connected in parallel with the conductor-backed substrate, wherein the circuitry connected in parallel with the conductor-backed substrate exhibits an admittance with an imaginary component having a negative slope within a design absorption frequency range of the absorber structure.

36. The absorber structure of claim 35, wherein the conductor-backed substrate includes a conductor-backed dielectric.

37. The absorber structure of claim 35, wherein the conductor-backed substrate includes a conductor-backed magnetic substrate.

38. The absorber structure of claim 35, wherein the design absorption frequency range isPage 71 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)defined by a frequency range which the circuitry operates as a negative inductor.

39. The absorber structure of claim 35, wherein the design absorption frequency range extends from a first frequency at which the circuitry commences operation as a negative inductor to a second frequency at which the conductor-backed substrate ceases to exhibit negative capacitive behavior.

40. The absorber structure of claim 35, wherein the design absorption frequency range is defined as a range extending from a first frequency at which the circuitry commences operation as a negative inductor to a second frequency at which the conductor-backed substrate ceases to exhibit behavior as a negative capacitor, the design absorption frequency range encompassing a first operating region and a second operating region of the circuitry, wherein in the first operating region the circuitry operates as a negative inductor and in the second operating region the circuitry operates as a capacitor.

41. The absorber structure of claim 35, wherein the circuitry connected in parallel with the conductor-backed substrate exhibits an admittance with an imaginary component having a positive slope within the design absorption frequency range of the absorber structure.

42. The absorber structure of claim 35, wherein the circuitry connected in parallel with the conductor-backed substrate exhibits an admittance with an imaginary component having a positive slope within the design absorption frequency range of the absorber structure, the imaginary component of the admittance having the negative slope within a first region of the design absorption frequency range of the absorber structure, the imaginary component of the admittance having the positive slope within a second region of the design absorption frequency range of the absorber structure.

43. The absorber structure of claim 35, wherein the admittance exhibited by the circuitry connected in parallel with the conductor-backed substrate includes a dispersion-less real component within the design absorption frequency range of the absorber structure.

44. The absorber structure of claim 35, wherein the circuitry is passive circuitry.

45. The absorber structure of claim 35, wherein the circuitry consists of passive circuit components.Page 72 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)46. The absorber structure of claim 35, wherein the circuitry is absent of active circuitry.

47. The absorber structure of claim 35, wherein the circuitry connected in parallel with the conductor-backed substrate exhibits performance characteristics of a capacitor filled with a dispersive dielectric.

48. The absorber structure of claim 35, wherein the circuitry connected in parallel with the conductor-backed substrate includes a capacitor connected in parallel with a series connected circuit path, the series connected circuit path including a series connected resistor, a series connected capacitor, and a series connected inductor.

49. The absorber structure of claim 35, wherein the circuitry connected in parallel with the conductor-backed substrate includes a capacitor connected in parallel with a series connected circuit path, the series connected circuit path including a series connected resistor, a series connected capacitor, and a series connected inductor, wherein the capacitor includes a certain frequency-independent capacitance, wherein a resistance of the series connected resistor is a function of the certain capacitance, wherein a capacitance of the series connected capacitor is a function of the certain capacitance, and wherein an inductance of the series connected inductor is a function of the certain capacitance, and wherein the resistance of the series connected resistor, the capacitance of the series connected capacitor, and the inductance of the series connected inductor are independent of a frequency of an input signal input to the absorber structure.

50. The absorber structure of claim 35, wherein the circuitry connected in parallel with the conductor-backed substrate includes a capacitor in series with first and second circuit paths that are connected in parallel to one another, the first circuit path having a second capacitor, and the second circuit path having an inductor in series with a resistor.

51. The absorber structure of claim 35, wherein the circuitry connected in parallel with the conductor-backed substrate includes a capacitor in series with first and second circuit paths that are connected in parallel to one another, the first circuit path having a second capacitor, and the second circuit path having an inductor in series with a resistor, wherein the capacitor includes a certain frequency-independent capacitance, wherein a capacitance of the second capacitor is a function of the certain capacitance, wherein an inductance of the inductor is a function of thePage 73 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)certain capacitance, wherein a resistance of the resistor is a function of the certain capacitance, and wherein the capacitance of the second capacitor, the inductance of the inductor, and the resistance of the resistor are independent of a frequency of an input signal input to the absorber structure.

52. The absorber structure of claim 35, wherein the admittance imaginary component has a positive slope within the design absorption frequency range of the absorber structure.

53. The absorber structure of claim 35, wherein the admittance imaginary component has a positive slope within the design absorption frequency range of the absorber structure, the admittance imaginary component having the negative slope within a first region of the design absorption frequency range, the admittance imaginary component having the positive slope within a second region of the design absorption frequency range.

54. The absorber structure of claim 35, wherein the circuitry defines an nth order dispersive network.

55. The absorber structure of claim 35, wherein the circuitry connected in parallel with the conductor-backed substrate includes first and second capacitors, an inductor, and a resistor.

56. The absorber structure of claim 35, wherein the circuitry is configured so that reflection from the absorber structure remains below a defined maximum level over the design absorption frequency range.

57. The absorber structure of claim 35, wherein the frequency range encompassing the design frequency of the absorber structure is centered on the design frequency of the absorber structure.

58. An absorber structure comprising: a conductor-backed substrate; and a capacitor filled with dispersive material connected in parallel with the conductor-backed substrate, wherein the capacitor filled with dispersive material exhibits an admittance with an imaginary component having a negative slope within a design absorption frequency range of the absorber structure.Page 74 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)59. The absorber structure of claim 58, wherein the conductor-backed substrate includes a conductor-backed dielectric.

60. The absorber structure of claim 58, wherein the conductor-backed substrate includes a conductor-backed magnetic substrate.

61. The absorber structure of claim 58, wherein the design absorption frequency range is defined by a frequency range in which the circuitry operates as a negative inductor.

62. The absorber structure of claim 58, wherein the design absorption frequency range extends from a first frequency at which the circuitry commences operation as a negative inductor to a second frequency at which the conductor-backed substrate ceases to exhibit negative capacitive behavior.

63. The absorber structure of claim 58, wherein the design absorption frequency range is defined as a range extending from a first frequency at which the circuitry commences operation as a negative inductor to a second frequency at which the conductor-backed substrate ceases to exhibit behavior as a negative capacitor, the design absorption frequency range encompassing a first operating region and a second operating region of the circuitry, wherein in the first operating region the circuitry operates as a negative inductor and in the second operating region the circuitry operates as a capacitor.

64. The absorber structure of claim 58, wherein the admittance exhibited by the capacitor filled with dispersive material includes a dispersion-less real component within the design absorption frequency range of the absorber structure.

65. The absorber structure of claim 58, wherein the capacitor filled with dispersive material is configured so that a reflection from the absorber structure remains below a defined maximum level over the design absorption frequency range of the absorber structure.

66. The absorber structure of claim 58, wherein the imaginary component of the admittance exhibited by the capacitor filled with dispersive material, within the design absorption frequency range of the absorber structure, includes a positive slope.

67. The absorber structure of claim 58, wherein the capacitor filled with dispersive materialPage 75 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)defines an nth order dispersive network.

68. A method of designing an absorber structure, the method comprising: configuring circuitry connected in parallel with a conductor-backed substrate, wherein the configuring includes performing the configuring so that the circuitry connected in parallel with the conductor-backed substrate exhibits an admittance with an imaginary component having a negative slope within a design absorption frequency range of the absorber structure.

69. The method of claim 68, wherein the conductor-backed substrate is a conductor-backed dielectric.

70. The method of claim 68, wherein the conductor-backed substrate is a conductor-backed magnetic substrate.

71. The method of claim 68, wherein the design absorption frequency range is defined by a frequency range which the circuitry operates as a negative inductor.

72. The method of claim 68, wherein the design absorption frequency range extends from a first frequency at which the circuitry commences operation as a negative inductor to a second frequency at which the conductor-backed substrate ceases to exhibit negative capacitive behavior.

73. The method of claim 68, wherein the design absorption frequency range is defined as a range extending from a first frequency at which the circuitry commences operation as a negative inductor to a second frequency at which the conductor-backed substrate ceases to exhibit behavior as a negative capacitor, the design absorption frequency range encompassing a first operating region and a second operating region of the circuitry, wherein in the first operating region the circuitry operates as a negative inductor and in the second operating region the circuitry operates as a capacitor.

74. The method of claim 68, wherein the configuring includes producing a circuit defining an equivalent circuit of a capacitor filled with dispersive material.

75. The method of claim 68, wherein the configuring includes producing a circuit defining an equivalent circuit of a capacitor filled with dispersive material.Page 76 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)76. The method of claim 68, wherein the configuring includes producing, with use of an equivalent circuit transformation, a circuit defining an equivalent circuit of a capacitor filled with dispersive material.

77. The method of claim 68, wherein the configuring includes producing, with use of a Zobel equivalent circuit transformation, a circuit defining an equivalent circuit of a capacitor filled with dispersive material.

78. The method of claim 68, wherein the configuring includes producing a design for a capacitor filled with a dispersive material, and producing an equivalent circuit of the design for a capacitor filled with a dispersive material.

79. The method of claim 68, wherein the method includes tuning a design absorption frequency range of the absorber structure, wherein the tuning includes adjusting a reflectance threshold of the absorber structure.

80. The method of claim 68, wherein the method includes tuning the design absorption frequency range of the absorber structure, wherein the tuning includes adjusting a dispersive grid order of the absorber structure.

81. The method of claim 68, wherein the method includes increasing the design absorption frequency range of the absorber structure, wherein increasing the design absorption frequency range of the absorber structure includes adjusting a dispersive grid order of the absorber structure.

82. The method of claim 68, wherein the configuring includes performing the configuring so that the circuitry connected in parallel with the conductor-backed substrate defines a multi- order dispersive grid.

83. The method of claim 68, wherein the configuring is performed so that the circuitry connected in parallel with the conductor-backed substrate defines a multi-order dispersive grid, and wherein determining parameters of the multi-order dispersive grid includes employing a cost function.

84. The method of claim 68, wherein the method includes determining, for a given dispersionPage 77 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)grid order, thickness, and reflectance threshold of the absorber structure, an achievable absorber structure design absorption frequency range.

85. The method of claim 68, wherein the method includes determining, with use of a cost function, for a given dispersion grid order, thickness, and reflectance threshold of the absorber structure, an achievable absorber structure design absorption frequency range.

86. An absorber structure comprising: a conductor-backed substrate; and circuitry connected in parallel with the conductor-backed substrate, wherein the circuitry connected in parallel with the conductor-backed substrate performs as negative inductor within a design absorption frequency range of the absorber structure.

87. The absorber structure of claim 86, wherein the circuitry connected in parallel with the conductor-backed substrate operates as a negative inductor within a first frequency region of the design absorption frequency range of the absorber structure, and operates as a capacitor within a second frequency region of the design absorption frequency range of the absorber structure.

88. The absorber structure of claim 86, wherein the design absorption frequency range is defined by a frequency range which the circuitry operates as a negative inductor.

89. The absorber structure of claim 86, wherein the design absorption frequency range extends from a first frequency at which the circuitry commences operation as a negative inductor to a second frequency at which the conductor-backed substrate ceases to exhibit negative capacitive behavior.

90. The absorber structure of claim 86, wherein the design absorption frequency range is defined as a range extending from a first frequency at which the circuitry commences operation as a negative inductor to a second frequency at which the conductor-backed substrate ceases to exhibit behavior as a negative capacitor, the design absorption frequency range encompassing first operating region and a second operating region of the circuitry, wherein in the first operating region the circuitry operates as a negative inductor and in the second operating regionPage 78 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)the circuitry operates as a capacitor.

91. The absorber structure of claim 86, wherein the conductor-backed substrate includes a conductor-backed dielectric.

92. The absorber structure of claim 86, wherein the conductor-backed substrate includes a conductor-backed magnetic substrate.

93. An absorber structure comprising: a conductor-backed magnetic substrate; and circuitry connected in parallel with the conductor-backed magnetic substrate, wherein the circuitry connected in parallel with the conductor-backed magnetic substrate exhibits an admittance with an imaginary component having a negative slope within a first region of a design absorption frequency range of the absorber structure and a positive slope within a second region of the design absorption frequency range of the absorber structure.

94. An absorber structure comprising: a conductor-backed magnetic substrate; and circuitry connected in parallel with the conductor-backed magnetic substrate, the circuitry comprising passive circuit components and being absent of active circuitry, the circuitry connected in parallel with the conductor-backed magnetic substrate exhibiting an admittance with an imaginary component having a negative slope within a design absorption frequency range of the absorber structure, and wherein: the admittance of the circuitry further includes a dispersion-less real component within the design absorption frequency range; the imaginary component of the admittance has the negative slope within a first region of the design absorption frequency range and has a positive slope within a second region of the design absorption frequency range; the design absorption frequency range is defined as extending from a first frequency atPage 79 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)which the circuitry commences operation as a negative inductor to a second frequency at which the conductor-backed magnetic substrate ceases to exhibit behavior as a negative capacitor, the design absorption frequency range encompassing a first operating region in which the circuitry operates as a negative inductor and a second operating region in which the circuitry operates as a capacitor; the circuitry is configured so that reflection from the absorber structure remains below a defined maximum level over the design absorption frequency range; the circuitry defines an nth order dispersive network that exhibits performance characteristics of a capacitor filled with a dispersive dielectric; and the circuitry comprises a capacitor in series with first and second circuit paths that are connected in parallel to one another, the first circuit path comprising a second capacitor, and the second circuit path comprising an inductor in series with a resistor, wherein the capacitor has a certain frequency-independent capacitance, and wherein a capacitance of the second capacitor, an inductance of the inductor, and a resistance of the resistor are each a function of the certain capacitance and are independent of a frequency of an input signal input to the absorber structure.

95. An absorber structure comprising: a conductor-backed magnetic substrate; and circuitry connected in parallel with the conductor-backed magnetic substrate, the circuitry consisting of passive circuit components and being absent of active circuitry, wherein the circuitry exhibits an admittance with an imaginary component having a negative slope within a design absorption frequency range of the absorber structure, wherein the design absorption frequency range is defined by a frequency range in which the circuitry operates as a negative inductor, and wherein the circuitry comprises a capacitor in series with first and second circuit paths that are connected in parallel to one another, the first circuit path comprising a second capacitor, and the second circuit path comprising an inductor in series with a resistor,Page 80 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)wherein the capacitor has a certain frequency-independent capacitance, and wherein a capacitance of the second capacitor, an inductance of the inductor, and a resistance of the resistor are each a function of the certain frequency-independent capacitance and are independent of a frequency of an input signal input to the absorber structure.

96. A method of designing an absorber structure, the method comprising: configuring circuitry connected in parallel with a conductor-backed magnetic substrate, wherein the configuring includes performing the configuring so that the circuitry exhibits an admittance with an imaginary component having a negative slope within a design absorption frequency range of the absorber structure, wherein the design absorption frequency range is defined as a range extending from a first frequency at which the circuitry commences operation as a negative inductor to a second frequency at which the conductor-backed magnetic substrate ceases to exhibit behavior as a negative capacitor, wherein the design absorption frequency range encompasses a first operating region and a second operating region of the circuitry, the circuitry operating as a negative inductor in the first operating region and as a capacitor in the second operating region; wherein the configuring includes producing a circuit defining an equivalent circuit of a capacitor filled with a dispersive material, including use of a Zobel equivalent circuit transformation; wherein the configuring further includes designing the circuitry to define a multi-order dispersive grid, and determining parameters of the multi-order dispersive grid using a cost function; and further comprising tuning the design absorption frequency range by adjusting at least one of: a reflectance threshold of the absorber structure, a dispersive grid order, or a thickness of the absorber structure; and further comprising determining, for a given dispersion grid order, thickness, and reflectance threshold of the absorber structure, an achievable absorber structure designPage 81 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)absorption frequency range using a cost function.Page 82 of 83 Attorney Docket No.3153.149AWO (SU 2024-024)

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