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Sum Rule Bounds Beyond Rozanov Criterion in Linear and Time-Invariant Thin Absorbers

Dallenbach layer is composed of an absorbing magnetic-dielectric layer attached to a perfect electric conductor (PEC) sheet. Under linearity and time invariance (LTI) assumptions Rozanov has established analytically a sum-rule trade-off between the absorption efficacy over a predefined bandwidth and the thickness of the layer, that is the so called Rozanov bound. In recent years several proposals have been introduced to bypass this bound by using non-LTI absorbers. However, in practice, their implementation may be challenging. Here, we introduce several new sum rules for LTI layer absorbers that are not covered by the original Rozanov's criterion, and give rise to more relaxed constraints on the absorption limit. We then, demonstrate practical LTI designs of absorbing thin layers provide absorption beyond the Rozanov's bound. These are based on the introduction of effective magnetic conductivity in the layer, or the introduction of a penetrable impedance sheet that replaces the PEC in the original Dallanbach layer.


I. INTRODUCTION
Wave absorbers have fascinated many researches over the years, since they play a key role in a variety of electromagnetic [1][2][3][4], acoustic [4][5][6][7][8][9] and optical [10][11][12] wave systems throughout the entire frequency spectrum. A canonical type of wave absorber is the Dallenbach layer that is composed of a slab of magneto-dielectric lossy substance backed by a perfect electric conductor (PEC) [1,[13][14][15][16][17][18][19][20]. Usually the absorbing bulk is composed of LTI materials, which may be composed of a continuous matter, laminate composite or periodic lattice of conductive, dielectric or magnetic elements at scales that are substantially smaller than the operating wavelength,i.e., metamaterials. For Dallenbach absorbers, Rozanov has established an analytic bound which relates its absorption efficiency over a predefined bandwidth and the thickness of the layer [20]. A similar relation was later introduced also for the acoustic wave regime at [17].
In recent years, wave engineering using time varying media has gained a lot of attention. Time varying wave devices allow an additional degree of freedom in comparison to traditional, time invariant wave devices. Therefore, they can potentially overcome some fundamental limitations enforced by the LTI requirements [21][22][23][24][25][26]. This approach has been used to design broadband causal reflectionless absorbers [27], to improve reflection bandwidth for quasimonochromatic signals [28] and in [29] to bypass Rozanov absorption bound by allowing the characteristics of the layer to abruptly or gradually vary with time, i.e. the permittivity ( ), permeability (µ) and electric conductivity (σ). However, the realization of such a time-dependent absorber raises several practical challenges, specifically in regard to the ways in which the modulation network operates, its complexity, its intereference with the wave system, and issues of detection of the impinging wave that should be absorbed. While some proposals to solutions do exist, the following question is called for: is it possible to go beyond Rozanov bound using a LTI system?
In this manuscript we provide an affirmative answer to this question. The key is that besides the linearity and time-invariance (LTI) assumptions, Rozanov's bound holds an additional latent assumption. This is hidden in the assumed asymptotic behaviour of the reflection coefficient ρ as λ → ∞, specifically that [20] ρ ∼ −1 + 4πjµ r d/λ, where µ r is the static relative permeability of the layer ('static' -i.e., at the long wavelength limit). Here, we prove that this assumption does not hold for layers with static magnetic conductivity (σ m ). Consequently, we show that by a proper selection of the layer's static magnetic conductivity it is possible to absorb efficiently an ultawideband continuous wave or pulsed wave way beyond what is dictated by Rozanov bound. This concept is demonstrated here both theoretically, and experimentally in a transmission-line model that emulates magnetic conductivity in a metamaterial fashion. The concept of magnetic loss, while may be less practical, opens a venue to other more applicable LTI approaches to go beyond Rozanov bound. Such an approach that uses a Dallanbach screen that is backed by a reactive or resistive impedance layer is demonstrated in the successive sections including a detailed design. The motivation to explore this case is twofold: (1) it generalizes Rozanov's bound and allowing improved absorption performances, and (2) Rozanov's derivation is based on the existence of an unattainable PEC boundary at extremely large wavelengths, which is physically impossible since the skin depth of any conductive metallic sheet is infinitely large at small-scale frequencies. In practice, copper substrates are placed as a boundary with small but not negligible inductance and resistance over extended frequency range. Therefore, our analysis is aimed to fill this gap by analyzing a general penetrable impedance sheet. In fact, similar approach has been already demonstrated for the acoustic wave regime at [9], where the authors considered a Dallenbach absorber with a soft instead of the standard hard boundary condition. The authors emphasized in [9] that a soft boundary can be realized only at the limit of zero frequency (extremely large wavelength). By increasing the frequency, the boundary becomes penetrable (non-opaque), therefore allowing the existence of a transmitted scattered wave. The immediate electromagnetic analog to the soft acoustical bondary is a perfectly magnetic conductor (PMC) boundary, however there is no physical realization of such substance at the limit of zero frequency. In this work we also suggest a few more generalizations with different types of reactive and resistive impedance layers that back the absorber.

II. DESCRIPTION OF THE PROBLEM
A normal incidence plane wave that propagates in vacuum E i (z, t) =xE 0 f (t − z/c) impinges at a Dallenbach layer, as shown in Fig. 1. Here, E 0 is the electric field amplitude, f (t) is the field waveform, and c 0 = 1/ √ µ 0 0 is the speed of light in vacuum, where 0 and µ 0 are the vacuum permittivity and permeability, respectively. The layer thickness is d and it is electromagnetically characterised by relative permittivity, relative permeability, electric conductivity and magnetic conductivity that their static values are ( r , µ r , σ, σ m ), respectively. We consider a linear and time-invariant (LTI) system and aim to takle Rozanov's assumption on the reflection coefficient at large wavelength by considering two cases. In the first case which is described in details in Sec. III and depicted in Fig. 1(a), the Dallenbach layer is composed of a magnetic conductive substance and terminated by a PEC boundary. Therefore, there is no rightward transmitted wave into the vacuum, since the PEC boundary is impenetrable with loading impedance Z L = 0. Later, we explore a more realistic design that is discussed in detail in Sec. IV and shown in Fig. 1(b), the Dallenbach absorbing layer is composed of standard materials, i.e. no magnetic conductance, and terminated by an arbitrary impedance sheet with negligible thickness in comparison to the operating wavelength. Our bound analysis is based on the long wavelength approximation, then due to practical reasoning, the impedance layer has to be transparent. Implying that both reflection and transmission may take place trough the absorbing layer. We denote by Z circ the self impedance of the sheet, therefore the load impedance that is composed of the impedance sheet and the outer medium reads Z L = Z circ η 0 , where η 0 = µ 0 / 0 is the characteristic impedance of the vacuum.
In this work, the impinging wave is of a transverse electromagnetic mode (TEM) with respect to the plane of incidence which allows us to make use of an equivalent TL model [30,31] to calculate the reflected, transmitted and absorbed energies. composed of an absorbing substance with magnetic conductance σ m terminated by a standard PEC boundary condition. Therefore, there is no transmitted wave passing through the boundary. (b) Dallenbach layer is considered with an arbitrary boundary, thus there exists a transmitted wave into +z direction in addition to the back reflected wave.

III. GENERALIZATION OF ROZANOV'S BOUND -MAGNETIC CONDUCTIVE ABSORBER
We consider a Dallenbach layer with magnetic conductive substance terminated by a PEC boundary (see Fig. 1 (a)) and derive analytic bounds on the reflection coefficient. We provide numerical and experimental demonstrations that support our main claim of this section -wave absorption beyond Rozanov's bound can be achieved by considering magnetic conductive substances which are mathematically modeled in a TL to series resistance (as opposed to electric conductive sheets that are modeled as shunt admittance).

III.1. Analytic relation
The reflection coefficient at the interface between the vacuum and the absorbing medium is wavelength depen-dent ρ(λ) and reads, where Z in (λ) = Z 0 (λ) tanh(γ(λ)d) is the input impedance at the interface between the Dallenbach layer and the surroundings, Z 0 (λ) is the characteristic impedance and γ(λ) is the complex propagation term (see Appendix A). Interestingly, for long wavelength, (|λ| → ∞), it asymptotically behaves as where and where = 0 r and µ = µ 0 µ r . The constant A is recognized as the transmission coefficient (T ) into the absorbing layer (which serves as a load) at long wavelength, i.e., A = T (|λ| → ∞). It can be observed (see Appendix A) that by setting the magnetic conductivity σ m to zero, Eq. (3) reduces to Rozanov's long wavelength approximation to the reflection coefficient [20] that is in Eq. (1). While at first glance the approximations in Eq. (3) and Eq. (1) seem similar, the presence of the extra term A in Eq. (3) implies a striking consequence. This is because under these circumstances, | ∞ 0 ln |ρ(λ)| dλ| by itself cannot be bounded from above and therefore the maximal absorption is not limited. Instead, we may readily derive a bound on a modified expression (see Appendix B), Implying that the maximal absorption is intimately connected with the coefficient A. Obviously, it can be observed in Eq. (6) that the static parameters themselves set the bound over the entire real wavelength axis without any additional frequency dependent contributions. Also, by setting σ m → 0 in Eq. (6) Rozanov's bound is obtained. Once again, the presence of σ m implies A = 0 and thus essentially the static magnetic conductance provides an additional design degree of freedom that resolves the fundamental limitation suggested by Rozanov regarding the maximal absorption efficacy for ultrawideband baseband signals. Consequently, careful designs may allow a substantial improvement of the absorption efficiency even without electric conductivity as we show next.

III.2. The effective emulation of static magnetic conductivity
In duality with Ohm's law, J = σ E that connects the electric field E in an electric conductor with electric conductance σ to the electric current density J, magnetic conductance σ m will be defined such that J m = σ m H. Here, J m is the magnetic current density that is induced by a magnetic field H. Obviously, since a magnetic monopole has not been found, also magnetic conductance does not naturally exist. Nevertheless, the concept of magnetic conductivity may be suggested effectively in a metamaterial fashion. In fact, the implementation of PMC was one of the first and prominent metamaterial examples, with substantial practical implications in antenna design [32][33][34], wave absorbers [35][36][37] and in other applications such as suppression of noise [38]. Furthermore, Maxwell's equations for one-dimensional x−polarized plane wave propagation in a homogenous medium with electric and magnetic conductivities reads, where f = c 0 /λ is the frequency. On the other hand, the Telegrapher's equations for propagation on a lossy transmission-line (TL) read, where V and I denote the voltage and current on the TL, L and C are the per-unit-length inductance and capacitance of the TL, G represents shunt conductance between the TL conductors, and R represents the finite conductivity of the TL conductors. By comparing the two sets of equations Eq. (7) and Eq. (8) it is evident that in a metamaterial TL realization, while G takes the role of σ, the magnetic conductivity σ m may effectively be emulated by the introduction of serial per-unit-length resistance R. This approach is followed in the simulations and experimental results below.

III.3. Monte-carlo simulations
The bound described in Eq. (6) can be simplified for Dallenbach layers with no static electric conductivity using Taylor series taking the limit σ → 0 (see Appendix B), (9) Here, we perform numerical simulations to verify the bound in Eq. (9), with the absorbing layer having unit relative permeability, µ r = 1. It can be observed that the right hand side in Eq. (9) is independent of , thus only dependent on σ m . We considered two different scenarios; first, the parameters ( , σ m ) were taken as constant over the entire frequency spectrum, thus equal to the static parameters. For each selection of magnetic conductivity, we generated random positive relative permittivity, ∈ (0, 500) × 0 and evaluated both sides of Eq. (9). Next, we considered a more realistic case where the permittivity has some frequency dependence. We used of the following Lorentzian model [39], where ∞ = 1 is the relative permittivity at extremely large frequency and λ res and λ rel are the resonance wavelength and dumping coefficient, respectively. Here, positive values of {A, λ res , λ rel } ∈ (0.1, 100) were randomly selected. Let us denote the LHS of Eq. (9) with I. The RHS will be termed 'Bound'. Fig. 11 depicts in black line the Bound and the blue dots denote repeated calculations of I. We examined the two cases of constant frequency variation parameters in (a) and the Lorentzian model in (b). It can be observed that the integration in the LHS of Eq. (9) is always above the corresponding calculated lower bound, thus indicating that indeed does not affect the bound.

III.4. Numerical examples of absorbers
Electric conductivity loss mechanism is commonly facilitated in a TL model by a shunt conductance. As opposed to electric conductivity, magnetic conductive substances do not exist in nature, however their effect on wave propagation, scattering and absorbtion can be described by a series resistance in a TL modeling. Here, we use this concept to model and evaluate the scattering and absorption of an ultrawideband pulsed planewave with a temporal pulse-width T p = 1 (ms) and a trapezoidal pulsed waveform with equal rise and fall time T r = T f = 0.4 (ms), propagating normally towards a Dallenbach layer with thickness d = 0.4 (m). Here, we consider the following fictitious permittivity and permeability parameters, 0 = 1 × 10 −6 (F/m) , µ 0 = 2.5 × 10 −3 (H/m), thus the speed of light within the medium is c 0 = 20, 000 (m/s) and the characteristic impedance is η 0 = 50 (Ω). This modification of the standard values of { 0 , µ 0 , c 0 , η 0 } makes it easier to build such discrete TL absorber on a basic breadboard, since the required elements (capacitors, inductors and resistors) acquire standard values and can be readily handcraft fabricated. The absorbing layer consists of magnetic conductance σ m that serves as the loss mechanism. Eq. (11) provides the ratio between the incident (E inc ), absorbed (E abs ) and reflected (E ref ) energies of such a continuous bulk, where E 0 (f ) is the Fourier transform of the incident wave electric field [voltage in a TL] waveform and ρ(f ) is the standard frequency domain reflection coefficient of a short-circuit TL. In order to produce the magnetic conductivity effect we make use of a discrete TL model where the capacitance (C), inductance (L) and series resistance (R) values per unit length are set as (C, L, R) = ( , µ, σ m ) [30,40]. Fig. 3a depicts E abs /E inc for a variety values of σ m (analytical model, blue) and from PSpice [41] simulation of a discrete TL with 20 unit cells (red dots) and the measured data (magenta). Furthermore, Rozanov bound is also depicted (black dashed line). Fig. 3b sketch a single discrete unit cell. Fig. 3c depicts the measurement setup, where we make use of National Instruments (NI) myDAQ [42] to produce and measure the analog signals (both as waveform generator and oscilloscope). It can be observed in Fig. 8 that nearly perfect absorption (> 99%) is obtained for σ m = 125 (Ω/m) where the optimal value that can be achieved by the Rozanov bound with σ m = 0 is about 16.5%. Furthermore, there is an excellent agreement between the analytical electromagnetic simulation and the discrete TL modeling used in PSpice simulation and the measured results. Achieving ultrawideband efficient absorption is not an easy task, since it requires both impedance matching at the boundary and high enough loss within the absorbing layer. In general, these two requirements contradict each other, since changing the loss terms within the absorbing layer imposes some variation on the characteristic impedance and the complex propagation terms which affect on the reflection coefficient at the interface between the absorbing layer and the outer medium. An exception to this rule is a lossy however dispersive layer that was introduced in [43,44]. To ensure both conditions, a specific relation between the electric and magnetic conductivities is required, while the amount of loss within the absorbing medium grows together with {σ, σ m }. Under regular conditions that satisfy Rozanov's bound, i.e., no static magnetic conductivity, wideband low frequency absorption is impossible to achieve. Here, the ultrawideband efficient extreme low frequency absorbtion is possible since the series resistance is dominant over the series inductance and the parallel capacitance. In other words, it is possible to design a TL with effective magnetic conductivity which yields almost pure series resistive behavior. In Appendix C we repeat the design process and simulations with the regular values of the parameters, i.e. Additional possible realizations are considered in Appendices E,F. Specifically, we consider a closed coaxial structure with PEC boundaries which enables the propagation of a TEM mode. The absorbing structure is composed of a discrete coaxial cable absorber where each unit cell is composed of a centered PEC pin attached to a series resistor which plays the role of the magnetic conductivity. These examples emulate the behaviour discussed in the metamaterial TL model above, in a 3D structure. Furthermore, for an incidence plane wave in a free space we discuss a parallel-plate waveguide stacking configuration, where the magnetic conductivity is modeled by a resistivity sheet attached to the upper side of the PEC plates.

IV. GENERALIZATION OF ROZANOV'S BOUND -IMPEDANCE BOUNDARY CONDITION
As opposed to electric conductive substances, natural magnetic conductive materials do not exist in nature and there is no realization of an effective magnetic conductive bulk with finite magnetic conductivity over a broad bandwidth. This technological barrier implies that the description in Sec. III can be readily used for TL absorbers rather than physical 3D wave absorbers. Nevertheless, it provides insights regarding other LTI venues to go beyond the Rozanov absorption bound. Here we describe an alternative approach where the Dallenbach absorber is terminated by an arbitrary non-opaque impedance (see Fig. 1 (b)) with no magnetic conductivity. In such a system there are both reflected and transmitted waves due to the penetrability of the boundary. However, careful design allows an increased absorption performances in comparison to Rozanov's bound due to invalidity of the assumption on the reflection coefficient at large wavelength (see Eq. (1)).

IV.1. Large wavelength approximation of the reflection and transmission coefficients
The impedance sheet (loading circuit) has the following generalized form that can be approximated for long wavelength (λ → ∞), where k denotes the lowest power with non-zero coefficient, i.e. A n = 0 for n < k. The corresponding load impedance (Z L ) at large wavelength reads, According to Eq. (13), the behaviour of the load impedance depends on whether Z circ has a positive (k ≥ 1) or non-positive (k ≤ 0) reactance.
• positive reactance. This case implies k ≥ 1, thus Z L ∼ A k (2jπc/λ) k . The input impedance as observed at the interface between the absorbing material and its surroundings reads, It can be observed by Eq. (14) that for k > 1 the first order approximated input impedance reads Z in ∼ 2jπcµd/λ. Using Eq. (2) (see also Appendices A,B) the reflection and transmission coefficients revert back to the form that was used by Rozanov, i.e.
ρ ∼ −1 + 4πjµ r d/λ and T ∼ O(1/λ 2 ). The physical implication of this result is that for k > 1 it is impossible to design a Dallenbach absorber with improved absorption performances in comparison to Rozanov's bound, i.e. Dallenbach absorber that is terminated by a PEC boundary. Next, the case of k = 1 is considered. The surface inductance L is represented by the first coefficient A 1 L. In this work, the surface inductance represents the total inductance of the loading circuit within a single unit cell which is equivalent to the total inductance of the impedance sheet due to its square dimensions. By performing similar steps to Appendix B, we reveal the long wavelength expressions for the reflection and transmission coefficients (see also Appendix G), By substituting L = 0 into Eq. (15), the inductive boundary degenerates into a PEC, therefore the reflection coefficient takes the original form that was used by Rozanov. The new expression of the reflection coefficient is differ by a single term which is directly depend on the parameter L in the following way: increasing L is equivalent to enlargement of the multiplication between the relative permeability and the thickness of the layer, thus effectively leads to a larger d or µ r . As opposed to Rozanov derivation where impenetrable PEC boundary was considered, here the absorption is affected also by a transmitted field. In the next section we provide numerical details on this effect and demonstrate that careful design allows enhanced absorption performances in comparison to Rozanov's bound.
• non-positive reactance case. This case implies a resistive/capacitive boundary and the long wavelength approximation of the load impedance can be simplified to, At large wavelengths the transmitted wave is no longer negligible, i.e. |ρ| |T |. Therefore, the large wavelength absorption is affected by both the reflection and transmission coefficients. It can be observed in Eq. (16) that the results for high order circuits with negative reactance (k < −1), revert back to the k = 0 case with A 0 → ∞ indicates that they do not contribute to the first order approximation. The input impedance is determined by substituting the loading impedance (Eq. (16)) to the input impedance formulation described in Eq. (14).

IV.2. Inductive boundary condition -absorption bounds and parametric study
Non-opaque inductive impedance sheet allows the existence of both backward reflected (ρ) and a transmitted (T ) waves. Energy conservation implies the following relation between these two coefficients and the power absorption coefficient (Ã) at any wavelength of interest (λ), As opposed to the magnetic conductive case that was described in Sec. III, the zero order approximation of the reflection coefficient reads, ρ(λ) ∼ −1 with no bias term. A generalized absorption bound (sum rule) that is analogous to Rozanov's bound, can be readily derived (similarly to the process in Appendix B),  (17), where the second transition comes from shrinking the argument (that is smaller than 1) of the ln function, |ρ(λ)| 2 +|T (λ)| 2 < 1 and the last inequality is derived similarly to Appendix B. For comparison, Rozanov's bound is given by 1 2 ∞ 0 ln 1 −Ã(λ) dλ ≤ 2π 2 µ r d. It can be observed in Eq. (15) that under the large wavelength approximation |T (λ)| |ρ(λ)|, however, this condition is not necessary applicable at lower wavelengths. In fact, for non-opaque impedance boundaries the transmission coefficient may not be negligible at a certain waveband. Therefore, it arises the following question: how tight is the new absorption bound? We address this issue in Fig. 4 by considering a nonmagnetic Dalllenbach layer (µ r = 1) with thickness d = 0.4 (m). In in Fig. 4(a) the inductive bound in Eq. (18) is presented in red, where a linear behavior is observed. Rozanov's bound is presented in a horizontal blue line due to its independency of the inductance L. In order to verify how tight is the inductive bound, we performed a parametric sweep along the two remaining free parameters of the absorbing layer, i.e. and σ (black dashed line). We stress that in our optimization approach we considered the case where the parameters do not vary with wavelength (non-dispersive) or with respect to the spatial coordinates (as in a stratified medium). By considering these additional degrees of freedom, results may be slightly improved, however the tightness of the bound is mainly affected by the inductance L, where at large values the bound is not tight, since the effect of the transmitted wave on the result is no longer negligible. Mathematically, as L increases, the self impedance of the sheet increases as well (Z circ = 2jπLc/λ), therefore the boundary (Z L = Z circ ||η 0 ) becomes penetrable at larger wavelength, thus loosen the first inequality in Eq. (18). The tightness (τ ) of the bound is defined by, where τ (L) ∈ [0, 1] for L ≥ 0 . Fig. 4(b) describes the behavior of τ as a function of the inductance L, where it can be observed that the bound is relatively tight (τ > 93%) for reasonable values of inductance (L ≤ 10 (µH)). As expected, L = 0 (i.e. Rozanov's bound) imposes a perfectly tight bound (τ = 1), where as L increases the tightness of the bound is monotonically decreases.   We set the relative permeability and permittivity to µ r = 1, r = 1.5 with varying electric conductivity (σ) and inductive impedance (L).  The inductance of the impedance sheet serves as an additional degree of freedom in the design which enables enhanced absorption performance in comparison to Rozanov's bound, i.e. PEC boundary case. It can be observed by Fig. 5 that there exists a wide range of parameters that allow such improvement (values that are grater than 1). On the other hand, using an inductive boundary does not necessary guarantee enhanced results, since there are domains where the ratio is smaller than 1, therefore perform poorly in comparison to an optimal Rozanov's absorber.

IV.3. Practical design -inductive boundary
A practical design of an ultrawideband wave absorber with inductive boundary condition is presented. The surroundings is assumed with { 0 = 8.85 × 10 −12 (F/m) , µ 0 = 1.2566 × 10 −6 (H/m)}. The absorber thickness is d = 0.4 (m) with parameters r = 1.5, µ r = 1, σ = 0.05 (S/m) and terminated by a non-opaque inductive impedance sheet. Fig. 6(a) depicts the analytical reflection (blue) and transmission (red) as a function of the operating frequency, while the corresponding simulated results are presented in purple and green circles, respectively with an excellent agreement. In addition, the optimal Rozanov's behavior of the reflection coefficient, i.e. | ∞ 0 ln |ρ(λ)| dλ| = 2π 2 µ r d, with uniform reflection in the frequency band f ∈ {4, 1600} (MHz) is depicted in dashed black line. Fig. 6(b) described the absorption coefficient as a function of frequency, where it can be observed that the performances of the impedance inductive sheet absorber are enhanced in comparison to the optimal PEC case, i.e. Rozanov's bound. The improvement can be observed at any frequency of operation, i.e. from extreme low frequencies up to higher frequencies. We stress that by increasing the minimal frequency (reducing the maximal wavelength) of operation in Rozanov's bound, the absorption performances may be improved at higher frequencies. Finally, Fig. 6(c) depicts the simulation layout that was used in HFSS [45]. The inductive impedance has been realized by placing a lumped inductor at the back side of the absorber.

IV.4. Resistive and capacitive impedance sheets
Here, we consider the case of a Dallenbach absorber terminated at one end by a non-opaque resistive [capacitive] impedance sheet (k ≤ 0). As previously discussed, such an absorber allows the existence of both backward (reflected) and forward (transmitted) waves. In fact, the resistive [capacitive] case has a unique feature in comparison to the previous cases, when considering extremely large wavelengths the transmission coefficient is no longer negligible (|ρ| |T |) thus cannot be neglected when deriving a sum rule on the absorption as performed in Eq. (18). Therefore, there is no analytic relation that can be derived similarly to Eqs. (6),(9), (18). However, the absorption performances can be analytically or numerically verified by electromagnetic simulations (HFSS [45]) . Fig 7(a) depicts the reflection and transmission coefficients as a function of the operating frequency (analytical and HFSS simulation results) for the resistive case (Z L = η 0 ), where Fig 7(b) depicts the absorption coefficient (Ã) in comparison with these of the optimal Rozanov's bound with uniform reflection in the frequency band f ∈ {4, 1600} (MHz) . Fig 7(c,d) describe the corresponding results for the capacitive case with C = 1[nF], Z circ = 1/(j2πf C).
Interestingly, it can be observed that as the frequency tends to zero (large wavelength approximation), the absorption does not vanishes, thus implying In fact, the absorption is bounded from above such thatÃ(λ → ∞) ≤ 0.5 (see Appendix H) .This non-trivial result can be readily explained by observing the TL equivalence (see Sec. III). At the low frequency limit, the shunt admittance (which is analogous to electric conductivity) is the dominant component within each unit cell, since the inductors impedance vanishes while the impedance of the shunt capacitors is extremely large. Therefore, the lossy TL physically behaves as a lumped resistor that is connected in parallel to the impedance of the free space (η 0 ) implying a non-zero scattering (reflection, transmission) and absorption.  It can be observed in Fig. 7 that at small frequencies the capacitive loading impedance behaves similarly to the resistive loading, since the impedance of the capacitor is dominant with respect to the free space impedance, |Z circ | η 0 . On the other hand, at large frequencies the opposite relation holds (|Z circ | η 0 ), thus the boundary behaves similarly to PEC. It can be observed in Figs. 6-7 that at large frequencies (small wavelengths), the absorption is negligibly affected by the loading impedance, since the thickness of the Dallenbach layer is substantially larger with respect to the operating wavelength. At the extreme case of f → ∞, i.e d/λ → ∞, the absorbing layer is infinitely long, thus rendering the bounding sheet redundant. This physical observation further underline the discussion above that improved low frequency absorption is possible by altering the boundary condition. See similar observation in [9] for an acoustical realization.

V. DALLENBACH WAVE ABSORBERS RANKING
In this manuscript, several types of LTI Dallenbach wave absorbers were introduced. The aim of this section is to provide hierarchy among the differ-ent types of Dallenbach wave absorbers. We stress that our ranking is based on the criterion of maximizing the absorption integral over the free space wavelength, 1 2 ∞ 0 ln 1 −Ã(λ) dλ . This criterion prioritizes the behavior of the absorption coefficient at large wavelengths, i.e. small frequencies. By considering a different criterion such as replacing λ → f within the integral results in prioritizing the large frequencies (small wavelengths), thus may lead to different results. 1. Dallenbach absorber composed of a magnetic conductance substance terminated by PEC boundary -It can be observed by Eqs. (6),(9) that the criterion integral does not converge, i.e. it cannot be bounded from above. Furthermore, by controlling the four parameters { r , µ r , σ, σ m } both essential conditions for a wideband perfect absorption (impedance matching at the interface and sufficient losses) can be achieved independently [43,44] for a wideband frequency domain. This absorber takes the first place since it is possible to design a wave absorber with unity absorption at extremely large wavelength, i.e.Ã = 1 for λ → ∞. The complete discussion on this wave absorber can be found in Sec. III. 2. Dallenbach absorber with no magnetic conductivity terminated by resistive or capacitive impedance sheet -As in the previous case, the criterion integral does not converge, however the absorption performance at large wavelength is limited by non-zero reflection and transmission coefficients, thusÃ ≤ 0.5. The complete discussion can be found in Sec. IV.4. 3. Dallenbach absorber with no magnetic conductivity terminated by an inductive impedance sheet -Here, the criterion integral is bounded by 2π 2 µ r [d + L/ (µ 0 µ r )] andÃ ∼ 0 at large wavelengths. The derivation of this bound relies on the fact that at large wavelengths, the transmission coefficient is negligible in comparison to the reflection coefficient, thus we were allowed to neglect it during the calculations. Therefore, this bound is not strikingly tight at large values of L, however provides decent results for standard inductance values. The complete discussion can be found in Secs. IV.2, IV.3. 4. Dallenbach absorber with no magnetic conductivity terminated by a PEC boundary -The analytical treatment of this case was carried by Rozanov in [20] and turned to be the most influential and important guidance in the research area of wave absorbers. Rozanov proved that the maximal achievable result of the criterion integral is 2π 2 µ r d.

VI. SUMMARY AND CONCLUSION
We derived generalized absorption bounds for Dallenbach layers with magnetic conductance substances terminated by a PEC sheet and for Dallenbach layers with a non-opaque impedance sheet. The new bounds are a generalization of Rozanov's bound that is valid for the case of a Dallenbach layer with no magnetic conductance terminated by a PEC boundary. The bounds were derived similarly to Rozanov's bound by removing a single assumption (no magnetic conductance or a PEC boundary). As a practical implementation for the magnetically conductive layer we designed, simulated and measured a discrete TL where a series resistance plays the role of the magnetic conductance. By a proper selection of the resistance an absorption of more than 99% for an ultra wideband pulsed wave is achieved. An additional example of coaxial cable absorber was introduced, where similarly to the previous design, the reflection coefficient tends to zero at DC. We then investigated the case of an ordinary absorber, i.e. no magnetic conductance terminated by an arbitrary non opaque impedance sheet. In such a wave absorber, both backward (reflected) and forward (transmitted) wave exist. We derived a new sum rule on the absorption for the case of an inductive impedance sheet and demonstrated that by allowing a non-opaque impedance (inductive, resistive or capacitive) enhanced absorption performances can be obtained in comparison to Rozanov's bound. One may wonder how the improved absorption performances can be achieved by replacing an impenetrable boundary by a penetrable one? This can be accommodated by noting that in Rozanov's bound the integration is over the wavelength λ , ∞ 0 ln |ρ(λ)| dλ . This integral can be easily converted to integration over the real frequencies with λ = c 0 /f , thus the bound can be re-written by, It can be observed in Eq. (20) that the bound (often referred as the causal limit) emphasizes low frequencies by the 1/f 2 weighting. We stress that by considering a Dallenbach layer backed by a PEC boundary under a different absorption measure such as straight forward integration over the frequency axis, ∞ 0 ln |ρ(f )| df leads to a non-convergent result which implies that the system is optimal with respect to the criterion. We provided several analytical and numerical examples of wave absorbers that demonstrate our arguments. The input impedance as seen at the surface of the lossy medium towards the absorbing layer is given by Z in (λ) = Z 0 (λ) tanh(γ(λ)d), with γ(λ) = (j/λ) (2πc 0 µ − jλσ m )(2πc 0 − jλσ) as the complex propagation term where c 0 = 1/ √ 0 µ 0 is the free space wave speed and Z 0 (λ) is the characteristic impedance of a lossy TL which is given by [30] with its low frequency approxiation(f → 0), Similarly, approximating tanh(γ(λ)d) by Using, Eqs. (A1) and (A2), gives the long wavelength (low frequency) approximation the input impedance by terms of order λ 0 and λ −1 gives, The reflection coefficient at the interface between the absorbing layer and the surroundings (the semi-infinite TL) is defined as, where η 0 is the wave impedance of the free space (also the characteristic impedance of the semi-infinite TL model). Substituting Eq. (A3) into Eq. (A4) yields the long wavelength approximation of the reflection coefficient, By applying the small argument approximation, tanh( √ σ m σd) ∼ √ σ m σd − (σ m σ) 3/2 d 3 /3 it follows that for σ m → 0, δ ∼ σ m d/η 0 , and therefore ρ(λ) ∼ −1 + 4πjµ r d/λ which identifies with Rozanov's approximation [20]. While for the "no electric conductivity", σ → 0, it follows that δ ∼ σ m d/η 0 1 − d 2 σ m σ/3 and ρ(λ) is approximated by,

(A8)
Appendix B: Reflection coefficient sum rules Here, we assume a time dependence of e j2πf t , thus the reflection coefficient ρ (f ) is an analytic function in the lower half plane of the complex frequency f . As previously described, the long wavelength approximation of the reflection coefficient satisfies ρ(λ) ∼ −1+A+j2πB/λ. Since ρ (λ) − A is analytic in the upper half complex λ plane, it has no poles or other singularities in this domain, however it may have zeros there. Let us denote the zeros of ρ(λ) − A in the upper half plane by λ 1 , λ 2 , ... and define a minimum phase function, where * denotes complex conjugation. It can be observed in Eq. (B1) that ln [ρ (λ)] is also analytic in the upper half plane, since it does not acquire any singularities or zeros there. Integrating over a closed path consisting of the entire real axis, C R and the semi-circle, C ∞ (see Fig. 8), and applying Cauchy's theorem yields, Since the time domain reflection coefficient, ρ(t), is a real function, ρ(λ) is subjected to symmetry relations in the complex f (λ) planes, thus the first term in the right hand side can be calculated as ∞ −∞ ln |ρ(λ) − A| dλ = 2 ∞ 0 ln |ρ(λ) − A| dλ. Moreover, due to the real part , the 2 nd and 4 th integration terms in the right hand side in Eq. (B2) vanish and the integration takes the form of,

Re{
Noting that Imλ j > 0 and discarding the summation in Eq. (B3) provides the generalized relation, where A and B are given in Eq. (A5). Following the same procedure but for the "no electric conductivity" (σ = 0) case gives, whereÂ is given in Eq. In the main text we make use of this observation in a mote-carlo simulation to verify the bound.
Appendix C: Numerical simulations of a discrete TL with standard values of 0 and µ0 The first numerical example presented in the main text describes a wideband TL absorber with 0 = 1 × 10 −6 (F/m) , µ 0 = 2.5 × 10 −3 (H/m). The use of such non-typical parameters makes it easier to build and measure such absorber on a breadboard, since the required capacitors and inductors acquire standard values. Here, we provide an additional numerical example with the standard parameters 0 = 8.85 × 10 −12 (F/m) , µ 0 = 4π × 10 −7 (H/m). We evaluate the absorption of a planewave with an ultrawideband trapezoidal temporal pulsed shape pulse-width T p = 1 (µs). The impinging planewave propagates in vacuum in the normal direction towards a Dallenbach layer with thickness d = 0.4(m) composed of the following parameters { = 0 , µ = µ 0 , σ = 0, σ m }. The magnetic conductivity, σ m serves as the loss mechanism. Fig. 9 depicts the ratio between the absorption and the incident energies for a variety values of σ m (blue) and from OrCAD PSpice simulation of a discrete TL with 20 unit cells (red dots). Furthermore, Rozanov bound is presented (black dashed line). It can be observed in Fig. 9 that nearly perfect absorption (> 99%) is obtained for σ m = 940 (Ω/m) where the optimal value that can be achieved by the Rozanov bound with σ m = 0 and σ = 0 is about 1.6%. Furthermore, there is an excellent agreement between the free-space electromagnetic simulation and the discrete TL modeling used in PSpice simulation. To obtain such agreement, an essential condition has to be fulfilled, ∆x << λ min , where ∆x = 0.02(m) is the size of each cell and λ min ∼ 30(m) is the minimal wavelength of the incident pulse. We stress again that the scenario discussed in this section was not measured, since it requires extremely low values of capacitors and inductors, C = 0.177 (pF) , L = 25.133 (nH).

Appendix D: Explanation of the experiment setup and calculations
Electromagnetic wave propagation, absorption and scattering in a layered media can be modeled into TL problem [30]. Here, we used such analogy to simulate the effect of magnetic conductive Dallenbach absorbers by a lossy TL. Fig. 10(a) describes the TL problem, where the first TL (#1) describes the lossless environment with characteristic impedance R g and phase velocity V p1 , while the second TL (#2) illustrates the lossy media, Dallenbach absorber, with characteristic impedance Z(f ) and phase velocity ,V p2 . From energy dissipation considerations, the system in Fig. 10(a) is equivalent to simpler system described in Fig. 10 (b) which we used for the measurement of the reflected and absorbed energies. Let us denote by V 1 and V 2 as the voltage drop over the source resistor and the voltage drop at the input terminals of the lossy TL, respectively, in Fig. 10(b). In addition we defineṼ s = V s /2, the voltage that would enter to TL #1 in Fig. 10(a) due to voltage division at the source for TL long enough with respect to the pulsewidth (i.e., with no back reflected field there). Note, that realization implies that half of the energy supplied by the source is absorbed at the series resistor. It can be observed by Fig. 10 that the following relations hold, where Z L is the input impedance of the lossy TL (load impedance), V ref is the reflected voltage from the lossy TL at the terminal. It can be observed by Eq. (D1) that |ρ| reads the right expression. To verify our conclusion we used both systems in OrCAD PSpice which yielded exact same results for the reflected and absorbed energies as reported in the manuscript and in the previous sections.
Appendix E: Coaxial cable absorber As an additional example we consider a closed coaxial structure with PEC boundaries which enables the propagation of a TEM mode. The absorbing structure is composed of a discrete coaxial cable absorber (see Fig. 11). Each unit cell is composed of a centered PEC pin with length l wire = 18(mm) where a series resistor is placed at its end with length l R = 2(mm), thus the total length of each cell is l = 20(mm). The pin diameter 2a is set to 2(mm) where the diameter of the outer conducting cover 2b is set to 8(mm), see Fig. 12a.
According to standard textbook, [30], the effective material properties (that exist in the entire space in this example) are 0 = 4.0111 × 10 −11 (F/m) , µ 0 = 2.7726 × 10 −7 (H/m) with no contrast between the absorbing medium to its surroundings { r = 1, µ r = 1}. Fig. 12b depicts the analytical reflection coefficient as a function of the frequency (blue), and the simulated reflection coefficient (dashed red) for R = 4.2(Ω) which corresponds to σ m = 210 (Ω/m). The simulation was performed with HFSS commercial software [45]. An excellent agreement is observed over the entire frequency band of interest. It can be observed that perfect absorption is obtained for extreme low frequencies, where the physical principles for such absorption were described above. Fig. 12c presents the simulation setup that was simulated in HFSS. The discrete coaxial cable is composed of 20 unit cells with total length of d = 0.4(m). Wave port has been set at the edge of the absorbing structure with characteristic impedance of Z 0 = µ 0 / 0 while PEC boundary was set at its other end. . An outer shield of PEC is used as ground plane for the waves within the cylindrical waveguide. (b) comparison between the reflection coefficient obtained analytically and numerically (HFSS simulation) as a function of frequency. High efficient extreme low frequency absorption is obtained. (c) HFSS simulation setup. A coaxial structure that is composed of 20 unit cells is excited by a wave port. Each cell is composed of a collocated PEC pin attached to a series resistor and an outer PEC shield that is used as a ground plane for the waves within the cable.

Appendix F: Conceptual parallel plate absorber
Here, we provide a conceptual design of a parallel plate absorber with effective magnetic conductivity. Parallel plate structure supports the propagation of TEM mode [30,31]. Unlike enclosed waveguides which support TEM modes (such as coaxical cable), parallel plate structure is suitable for absorbing a normal incidence impinging plane wave since the polarization of the electric and magnetic fields are compatible to the TEM waveguide mode. Here, the magnetic conductivity is modeled by a resistivity sheet attached to the upper side of the PEC strip. As previously described in Appendix C, in order to maximize the absorption the effective magnetic conductivity is set to σ m = 940 (Ω/m), which corresponds to R = 18.8 (Ω). Fig. 12(a) depicts the proposed unit cell. Fig. 12(b) depicts the analytical reflection coefficient as a function of the frequency (blue), and the HFSS [45] simulation results of the reflection coefficient (dashed red). The results show an excellent agreement over the entire frequency band of interest and efficient low-frequency absorption is observed. Fig. 12(c) depicts the simulation setup that was simulated in HFSS. The parallel plate waveguide is composed of 20 unit cells with total length of d = 0.4(m) (similar to other designs in this work). Periodic boundary conditions (PBC) were enforced on the 4 wrapping walls of the structure. Floquet port has been set at one end of the periodic structure, where PEC boundary condition has been set on its other end. We stress that although seems promising, the suggested absorber is not easily applicable since the implementation of an ideal opaque resistive sheet is challenging.( evaluation of the reflection coefficient, ρ, is described in details at Appendix B. Here, we use TL theory [30,31] to derive a closed form expression of the large wavelength approximation of the transmission coefficient, T . The TL scheme is presented in Fig. 13, where the first TL represents the semi-infinite TL and the second TL is the Dallenbach absorber with thickness d, characteristic impedance Z(λ) and complex propagation constant γ(λ). Z L is the total loading impedance as seen at the edge of the second TL at z = d − .
FIG. 13: TL analogy. The first TL represents the semiinfinite TL where the excited (incidence) and backward (reflected) waves exist. The second TL is the Dallenbach absorber loaded by a general penetrable boundary with impedance Z L .
For a given load impedance, Z L , the reflection coefficients at the edges of the second TL reads, ρ out = Z L − Z(λ) Z L + Z(λ) , ρ in = ρ out e −2γ(λ)d .
We denote by V + i [V − i ] the steady-state forward [backward] propagating voltage wave in the i th TL. The voltage at z = 0 for both TLs reads, By the voltage continuity at z = 0 we obtain, The voltage at z = d is composed of two wave contributions, V 2 (d) = V + 2 (d) + V − 2 (d) where V + 2 (d) = V + 2 (0)e γd and V − 2 (d) = ρ out V + 2 (d). Using these results and Eqs.(G2),(G3) we obtain the transmission coefficient, T ,