Theoretical analysis of nonlinear surface wave absorbing metasurfaces

Y. Luo; S. Kim; A. Li; D. Sievenpiper

doi:10.1364/OE.26.000797

1. Introduction

Surface wave absorbers are usually designed by patterning metallic surfaces that can be either sub-wavelength scale periodical patterns (for metamaterials) or conventional metallic patterns on a substrate with a ground layer, to absorb the power of incident EM (electromagnetic) waves in the range of a resonant frequency band [1–3]. However, the performance of conventional planar microwave absorbers is limited by substrate thickness, permittivity, and permeability [4,5]. For instance, in the high-impedance surface, the bandwidth is given by B = 2πµ_rh/λ₀, where µ_r is the substrate permeability, h is the substrate thickness, and λ₀ is the center-frequency wavelength [6,7]. To achieve wider bandwidth, many approaches have been used, such as stacking multilayer structures, using multi-frequency resonators and using switches to reconfigure frequency bands [8–13]. But the fundamental limitations of thickness to bandwidth still cannot be exceeded by linear absorbers. In other examples [14], metamaterials absorbers are explored, but in term of bandwidth, there is no fundamental difference in absorption between metamaterial absorbers and absorbers based on other ordinary materials. In a brief summary, for a fixed substrate thickness, we can either obtain higher attenuation magnitude with narrower bandwidth, or achieve wider bandwidth but with lower attenuation.

Another possible approach is to seek advantages by using nonlinear metasurface absorbers, as nonlinear metasurfaces have already demonstrated their merits in super-resolution imaging [15], efficient frequency conversion and optical control [16], giant nonlinear response in metamaterials [17], and plasmonic effects [18,19]. There are several nonlinear absorber examples which indeed show the advantages of using nonlinearity in some applications, by engineering metasurfaces with embedded electric circuits. For example, in [20], it is proposed to use a nonlinear absorbing surface to absorb high-power pulse waves, but with minimal disturbance to low-power surface currents, and in [21], it is proposed to use a circuit with diodes to produce the nonlinear property that the metasuface absorption is dependent on the waveform of pulse. However, the potential advantages of using nonlinearity to increase bandwidth and absorption magnitude are still unknown and the optimum absorption mechanism is not clear.

In this paper, we first propose the advantages of nonlinear absorbing metasurfaces using rectification, and theoretically prove that by rectifying the surface current, converting most of the power to a static or DC term, and thus dissipating it in a resistive sheet, nonlinear metasurface absorbers can have significant advantages over the traditional linear surface wave absorbers in both attenuation and bandwidth. We study the nonlinear metasurface absorption with full-wave rectification. We find that 81% of the power is converted to DC, thus absorbed by the parallel resistance & capacitance (RC) sheet, and consequently significant advantages in both attenuation magnitude and bandwidth are demonstrated in thenonlinear absorbing metasurface. Even for substrates as thin as 0.35 mm which is λ₀/143 for center frequency 6 GHz, the nonlinear absorbing metasurface can achieve 60% relative bandwidth with the magnitude 25.5/m, significantly exceeding the performance of linear metasurface, in which the relative bandwidth is 20% and attenuation magnitude is 2.3/m.

2. The approach of applying nonlinear terms to EM calculation

The proposed concept is extended from the linear surface wave absorber theoretical model in [22,23], which is configured as a very thin RC sheet (with parallel resistance and capacitance) coated over a lossy slab (region 2) with a perfect metal ground layer (region 1). In our nonlinear theoretical analysis, we follow the model in [22,23], but assume the thin top RC sheet is embedded with ideal diode rectifiers. As in Fig. 1, the TM (transverse magnetic) microwave surface mode propagates in the parallel direction (along Z-axis) and the electromagnetic (EM) field is assumed as independent of the width (X-axis direction), meanwhile the height of the air space (region 3) is considered as infinity (Y-axis direction). As the surface current is induced in the top RC sheet, it is rectified through full-wave rectification (Fig. 1(b)), as illustrated in the Eq. (1),

| \sin (ω t) | = \frac{2}{π} - \frac{4}{π} \sum_{n = 2, 4, 6}^{\infty} \frac{\cos (n ω t)}{n^{2} - 1},

by which some of the field is converted to DC, and some is converted to other harmonics (2f₀, and 4f₀, …). By using the Fourier Transform, we find in the full-wave rectification case (as shown in Fig. 1(c)), the DC component contains 81% (magnitude 2/π) of the power, and there is no fundamental component (f₀). We only need to consider the 2nd and 4th harmonic components that contain 18% (magnitude 4/(3π)) and 0.73% (magnitude 4/(15π)) of the power respectively, because these three components (DC, 2f₀, and 4f₀) represent 99.73% of the power, and the remaining harmonic components are negligible.

Fig. 1 (a) The concept of the nonlinear metesurface by assuming ideal diodes embedded in the top RC sheet to rectify induced currents to produce nonlinear harmonics including DC, 2f₀ term, and 4f₀ term. (b) Full-wave rectification case, in which a large proportion of power is converted to DC, and the DC term is confined and absorbed by the RC sheet. (c) The Fourier transform of full-wave rectification, which determines magnitude coefficient of nonlinear terms.

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Table 1 describes the magnitude of nonlinear harmonics including DC, and their spatial distributions, while in the linear surface wave absorber as theoretically analyzed in [22,23], there is only the fundamental mode (f₀) in the air, substrate, and RC sheet. For nonlinear metasurfaces as illustrated in the side view of Fig. 2, as microwave energy with frequency f₀ is incident on the surface, it induces current on the RC sheet and the induced current is rectified to nonlinear harmonics, spreading the electromagnetic field of harmonics (2f₀, and 4f₀, …) to the substrate and air, while the DC term is confined and absorbed in the RC sheet. For each nonlinear term, for example the term 2f₀, it propagates in the three channels the same way as in the linear absorbing metasurfaces, which means the absorption of each nonlinear term can be calculated the same way as surface wave absorber in [22,23].

Table 1. The power proportion and spatial distribution of nonlinear and linear absorbing metasurfaces

View Table

Fig. 2 Side view of the three channels which contain nonlinear terms. When rectifying, the Fourier Transform determines magnitude coefficient of each nonlinear term, and each mode is self-consistent by matching the boundary conditions above/below the parallel RC sheet [22,23].

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In a brief summary, to calculate the absorption of diode-rectifier-based nonlinearabsorbers, we study the magnitude coefficient of nonlinear terms and their spatial distributions, thus we can calculate the power absorption of each nonlinear term (DC, 2f₀, and 4f₀), and combine all of them through the rectification-magnitude coefficient (A_DC, A_2f0, A_4f0). In such way we can determine the overall absorption of the nonlinear metasurfaces for the incoming wave f₀. We will take the power concentration calculation in the substrate channel as an example first, then we can easily extend it to the air channel, and at last we will illustrate the power loss calculations in the substrate channel and RC sheet channel.

3. Power concentration in the nonlinear metasurface substrate

For nonlinear metasurface absorbers, the substrate power concentration contains nonlinear term 2f₀ and 4f₀, as illustrated in Eq. (2),

P_{s u b}_{N L} (f_{0}) = \sum_{n = 1}^{\infty} P_{s u b}^{'} (2 n f_{0}) \approx P_{s u b}^{'} (2 f_{0}) + P_{s u b}^{'} (4 f_{0}),

where P^’_sub (2f₀) and P^’_sub (4f₀) are the substrate power of nonlinear term 2f₀ and 4f₀. As other higher modes are negligible (as in Table 1), we use an approximation in Eq. (2).

Secondly, as each nonlinear harmonic term propagates in the three channels the same way as in the microwave absorber in [22,23], we can follow the method of linear theoretical analysis for one term, and extend it to other nonlinear harmonics. In [22,23], the power in the substrate can be expressed as Eq. (3) (written from Eq. (12) of [22])

P_{S u b L} = K_{Z}^{2} \cdot \frac{1}{4 k_{2}} \frac{β}{ω ε_{2}} [k_{2} a + \frac{1}{2} \sin (2 k_{2} a)],

where k₂ is the wave number in the substrate and air, β is the propagation constant, ϵ₂ is the permittivity of the substrate, and ω, a are the angular frequency and the substrate thickness respectively. K_z is the z-directed current density of the x-direction in the ground layer (y = 0 in Fig. 1(a)), which is written as the Eq. (4),

K_{Z} = {H_{X 2} |}_{y = 0} = - A \cdot \frac{ω ε_{2}}{k_{2}},

where H_x2 (Eq. (15) in [23]) is the tangential magnetic field (in the x direction) at the boundary of the substrate and the perfect ground layer, and A is a constant value. Because in the nonlinear metasurface absorber model, the induced surface current flows and is rectified on the RC sheet in the top surface, and the power converted to DC is absorbed there as well, we need to modify the Eqs. by using the top-layer surface current magnitude K_sz as Eq. (5), to replace the ground-layer current K_z.

K_{S Z} = {(H_{X 3} - H_{X 2}) |}_{y = a} = - A \cdot \frac{ω ε_{3}}{k_{3}} \sin (k_{2} a) + A \cdot \frac{ω ε_{2}}{k_{2}} \cos (k_{2} a)

where H_x3 (Eq. (28) in [23]), and H_x2 are the tangential magnetic fields (in x direction) above and below the RC sheet boundary. k₃, ϵ₃ are the wave number and permittivity in air respectively. Combining Eqs. (4) and (5), we obtain the relationship between K_z and K_sz as

K_{Z} = \frac{K_{S Z}}{[\cos (k_{2} a) - \frac{ε_{3}}{ε_{2}} \cdot \frac{k_{2}}{k_{3}} \sin (k_{2} a)]} .

Consequently, by using K_sz, the substrate power concentration Eq. (3) can be modified as

P_{S u b L} = \frac{K_{S Z}^{2}}{{[\cos (k_{2} a) - \frac{ε_{3}}{ε_{2}} \cdot \frac{k_{2}}{k_{3}} \sin (k_{2} a)]}^{2}} \cdot \frac{1}{4 k_{2}} \frac{β}{ω ε_{2}} [k_{2} a + \frac{1}{2} \sin (2 k_{2} a)] .

Equation (7) gives the substrate power concentration in terms of top-layer RC sheet current K_sz, which is rectified to nonlinear terms 2f₀ and 4f₀ with the magnitude coefficient A_2f0 and A_4f0, respectively. That means that by applying the magnitude coefficient to Eq. (7), we can obtain the power concentration for each nonlinear harmonic term. For nonlinear term 2f₀ and 4f₀, the substrate power of nonlinear terms can be written as

\begin{array}{l} P_{s u b}^{'} (n f_{0}) = \frac{A_{n f_{0}} \cdot K_{S Z}^{2}}{{[\cos (k_{2_n f_{0}} a) - \frac{ε_{3}}{ε_{2}} \frac{k_{2_n f_{0}}}{k_{3_n f_{0}}} \sin (k_{2_n f_{0}} a)]}^{2}} \\ \cdot \frac{1}{4 k_{2_n f_{0}}} \frac{β_{n f_{0}}}{(n ω) ε_{2}} [k_{2_n f_{0}} a + \frac{1}{2} \sin (2 k_{2_n f_{0}} a)], \end{array}

where n = 2 is for the nonlinear term 2f₀, and n = 4 for the nonlinear term 4f₀. For nonlinear term 2f₀, k_2-2f0, k_3-2f0, and β_2f0 are the wave number in the substrate, in air, and the propagation constant, respectively. Correspondingly, k_2-4f0 and k_3-4f0, β_4f0 are the parameters for nonlinear term 4f₀.

Thirdly, by inserting Eq. (8), in which we obtain the substrate power of nonlinear term P^’_sub (2f₀), and P^’_sub (4f₀), into Eq. (2), we can achieve the complete Eq. to describe the substrate power concentration P_{sub NL} (f₀) in diode-rectifier-based nonlinear absorbing metasurfaces.

In a brief summary, in order to apply nonlinear terms to EM calculations, we follow the linear absorber model, but use the top-surface current K_sz rather than the ground-surface current K_z, because currents are rectified there, and the converted power is absorbed there as well. As the power concentration is expressed with surface current K_sz, which is compatible to diode-rectifier-based nonlinear metasurfaces, we use the rectification-magnitude coefficient (A_2f0 and A_4f0) to calculate the power concentration of nonlinear terms P^’_sub (2f₀), and P^’_sub (4f₀). At last we combine power of all the nonlinear terms with Eq. (2) to achieve the power concentration P_{sub NL} (f₀) in nonlinear absorbers for the incident wave f₀. This approach is used to nonlinear absorbers for air power concentration in section 4, power loss in section 5 and 6 as well.

4. Power concentration in air

In this section, we extend the same approach to calculate power in the air channel. For the nonlinear metasurface, power concentration in the air channel includes nonlinear terms as

P_{a i r}_{N L} (f_{0}) \approx P_{a i r}^{'} (2 f_{0}) + P_{a i r}^{'} (4 f_{0}),

while

P_{a i r}^{'} (n f_{0}) = \frac{A_{n f_{0}} \cdot K_{S Z}^{2}}{{[\cos (k_{2_n f_{0}} a) - \frac{ε_{3}}{ε_{2}} \frac{k_{2_n f_{0}}}{k_{3_n f_{0}}} \sin (k_{2_n f_{0}} a)]}^{2}} \cdot \frac{1}{4} \frac{ε_{3}}{ε_{2}} \frac{k_{2_n f_{0}}^{2}}{k_{3_n f_{0}}^{3}} \frac{β_{n f_{0}}}{(n ω) ε_{2}} \sin^{2} (k_{2_n f_{0}} a),

where n = 2 and n = 4 are for nonlinear terms 2f₀, and 4f₀ respectively. Equation 10 is modified from the linear case Eq. (11) (Eq. (13) in [22]), by using top-surface current K_sz to replace the ground layer current K_z

P_{a i r L} = K_{Z}^{2} \cdot \frac{1}{4} \frac{ε_{3}}{ε_{2}} \frac{k_{2}^{2}}{k_{3}^{3}} \frac{β}{ω ε_{2}} \sin^{2} (k_{2} a) .

By inserting Eq. (10) to (9), we finally achieve the nonlinear power concentration P_{air NL} in the air region.

5. Power loss in the nonlinear metasurface substrate

In the same way, we can calculate the power loss in the nonlinear absorber substrate. Substrate loss of the nonlinear metasurface absorber consists nonlinear terms as

P_{s u b_l o s s}^{}_{N L} (f_{0}) \approx P_{s u b_l o s s}^{'} (2 f_{0}) + P_{s u b_l o s s}^{'} (4 f_{0}),

while

\begin{array}{l} P_{s u b_l o s s}^{'} (n f_{0}) = \frac{A_{n f_{0}} \cdot K_{S Z}^{2}}{{[\cos (k_{2_n f_{0}} a) - \frac{ε_{3}}{ε_{2}} \cdot \frac{k_{2_n f_{0}} \sin (k_{2_n f_{0}} a)}{k_{3_n f_{0}}}]}^{2}} \\ \cdot \frac{(n ω) ε "}{4 {(n ω ε)}^{2}} \cdot [(β_{n f_{0}}^{2} + k_{2_n f_{0}}^{2}) a + \frac{(β_{n f_{0}}^{2} - k_{2_n f_{0}}^{2}) \cdot \sin (2 k_{2_n f_{0}} a)}{2 k_{2_n f_{0}}}], \end{array}

where n = 2 and n = 4 are for nonlinear terms 2f₀, and 4f₀ respectively. Equation (13) is derived from Eq. (14) (Eq. (14) of [22]), which describes substrate loss in linear absorbers as

P_{S u b_l o s s L} = K_{Z}^{2} \cdot \frac{ω ε "}{4 {(ω ε)}^{2}} \cdot [(β^{2} + k_{2}^{2}) a + \frac{(β^{2} - k_{2}^{2}) \cdot \sin (2 k_{2} a)}{2 k_{2}}] .

Combining Eq. (13) with (12), we finally obtain the substrate loss P_{sub-loss NL} in nonlinear case.

6. Power loss in the RC sheet

The power loss in RC sheet is the key point for nonlinear advantages, because current is rectified there, and the power converted to DC is dissipated there as well. For parallel RC sheet, the electric field E_z2 (in region 2) along RC sheet (when y = a in Fig. 1) produces the loss that can be written as Eq. (15), by using top-surface current K_sz,

P_{S h e e t_L o s s}_{L} = {\frac{1}{2 R} \cdot E_{Z 2}^{2} |}_{y = a} = \frac{1}{2 R} \cdot \frac{K_{S Z}^{2}}{{[\cos (k_{2} a) - \frac{ε_{3}}{ε_{2}} \cdot \frac{k_{2}}{k_{3}} \sin (k_{2} a)]}^{2}} \cdot {(\frac{k_{2}}{ω ε_{2}} \sin (k_{2} a))}^{2},

where E_z2 is written as Eq. (1) in [22], R is the resistance in the RC sheet surface. But for nonlinear absorbers, we have nonlinear terms 2f₀ and 4f₀, and the DC component that is confined and absorbed completely in the RC sheet as well. The dissipated power of nonlinear case can be expressed as

P_{S h e e t_L o s s}_{N L} = P_{S h e e t_L o s s}^{'} (D C) + P_{S h e e t_L o s s}^{'} (2 f_{0}) + P_{S h e e t_L o s s}^{'} (4 f_{0}),

and based on Eq. (15), each term can be written as

P_{S h e e t_L o s s}^{'} (D C) = \frac{1}{R} \cdot \frac{A_{D C} \cdot K_{S Z}^{2}}{{[\cos (k_{2} a) - \frac{ε_{3}}{ε_{2}} \cdot \frac{k_{2}}{k_{3}} \sin (k_{2} a)]}^{2}} \cdot {(\frac{k_{2}}{ω ε_{2}} \sin (k_{2} a))}^{2},

P_{S h e e t_L o s s}^{'} (n f_{0}) = \frac{1}{2 R} \cdot \frac{A_{n f_{0}} \cdot K_{S Z}^{2}}{{[\cos (k_{2_n f_{0}} a) - \frac{ε_{3}}{ε_{2}} \cdot \frac{k_{2_n f_{0}}}{k_{3_n f_{0}}} \sin (k_{2_n f_{0}} a)]}^{2}} \cdot {(\frac{k_{2_n f_{0}}}{(n ω) ε_{2}} \sin (k_{2_n f_{0}} a))}^{2},

where n = 2 and n = 4 are for nonlinear terms 2f₀, and 4f₀ respectively. Inserting Eq. (17), and (18) to Eq. (16), we finally obtain the nonlinear case sheet loss P_{Sheet-Loss NL}. Furthermore, we can calculate the attenuation for nonlinear absorbing metasurfaces as

α_{N L} = \frac{P_{S u b_L o s s}_{N L} + P_{S h e e t_L o s s}_{N L}}{2 (P_{a i r}_{N L} + P_{s u b}_{N L})},

while for linear metasurfaces

α_{L} = \frac{P_{S u b_L o s s}_{L} + P_{S h e e t_L o s s}_{L}}{2 (P_{a i r}_{L} + P_{s u b}_{L})},

where P_{sub L}, P_{air L}, P_{Sub-loss L}, and P_{Sheet-loss L} are written in Eqs. (3), (11), (14), and (15), respectively.

7. Boundary conditions

After figuring out the above Eqs. for the nonlinear absorbing metasurface absorption, we need to determine the parameters k₂, k₃, and β for each nonlinear term by boundary conditions. As demonstrated in Fig. 2, the current of the incoming wave (frequency f₀) is rectified on the RC sheet to nonlinear harmonics as DC, 2f₀ term, and 4f₀ term, and these nonlinear terms propagate in the substrate channel, air channel, and RC sheet channel. Boundary conditions enable each harmonic term to match below and above the RC sheet, allowing us to obtain Eqs. to determine parameters k₂, k₃, and β for each nonlinear term. For example, for nonlinear term 2f₀, we can obtain boundary Eqs. in the substrate channel and air channel as

β_{2 f_{0}}^{2} = {(2 ω)}^{2} ε_{2} μ_{2} - k_{2_2 f_{0}}^{2},

- k_{3_2 f_{0}}^{2} = {(2 ω)}^{2} ε_{3} μ_{3} - β_{2 f_{0}}^{2} .

Specifically, in the RC sheet surface, we follow to use the TRM (Transverse Resonance Method) [24,25] to construct the RC sheet boundary Eq. as

k_{2_2 f_{0}}^{2} + {(\frac{(2 ω) \cdot ε_{3}}{\frac{(2 ω) \cdot ε_{2}}{k_{2_2 f_{0}}} \cot (a k_{2_2 f_{0}}) - ((2 ω) \cdot C + \frac{1}{R^{2} \cdot (2 ω) \cdot C})})}^{2} = {(2 ω)}^{2} (ε_{2} μ_{2} - ε_{3} μ_{3}) .

Similarly, for nonlinear term 4f₀, we can obtain the same Eq. form as Eqs. (21)-(23) by replacing k₂__2f0, k₃__2f0, β_2f0, 2ω with k₂__4f0, k₃__4f0, β_4f0, 4ω respectively.

Equations from (21) to (23) indicate that the dielectric constant, substrate thickness, frequency and the resistance and capacitance in the RC sheet determine the numerical values of k₂, k₃, and β. Given the parameters such as frequency ω, ϵ₃, ϵ₂, tanδ, µ₂, R, C and the substrate thickness a, we can calculate the power concentration and loss in nonlinear metasurface absorbers, so as to obtain the nonlinear attenuation.

8. Nonlinear advantages

Based on these Eqs. and the given parameters (we use regular values as ϵ₂ = 3, tanδ = 0.0013, µ₂ = 1, thickness a = 1.524 mm, R = 377 Ω/Square, C = 2.5 Pf/Square (for linear metasurfaces),and C = 2.5/4 Pf/Square (for nonlinear metasurfaces)), we obtain several advantages of using nonlinearity in absorbing surfaces as demonstrated in Fig. 3-6.

Fig. 3 For all figures, linear and nonlinear absorbers have these parameters: ϵ₂ = 3, tanδ = 0.0013, µ₂ = 1, thickness = 1.524 mm, R = 377 Ω/Square, C = 2.5/4 Pf/Square (for nonlinear case), C = 2.5 Pf/Square (for linear case) and air box is infinite. Variable parameters are indicated in each figure. (a) Different capacitance values of RC sheet lead to different resonant frequencies, and nonlinear advantages show good frequency flexibility. (b) Different resistance values of the RC sheet lead to different attenuation magnitudes, and nonlinear advantages show good resistance flexibility.

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Fig. 4 For all figures, the nonlinear absorber has these parameters: ϵ₂ = 3, tanδ = 0.0013, µ₂ = 1, thickness a = 1.524 mm, C = 2.5 Pf/Square for linear metasurfaces, C = 2.5/4 Pf/Square for nonlinear metasurfaces, and air box is infinite. Resistance value is the variable parameter that is indicated in each figure. (a) & (b) the nonlinear advantages in magnitude and bandwidth with resistance R as the variable parameter. The attenuation bandwidth is calculated as the 3 dB normalized power loss (1-e-2αz where α is the attenuation, and z is the distance).

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Fig. 5 For all figures, the nonlinear absorber has these parameters: ϵ₂ = 3, tanδ = 0.0013, µ₂ = 1, R = 377 Ω/Square, C = 2.5 Pf/Square for linear metasurfaces, C = 2.5/4 Pf/Square for nonlinear metasurfaces, and air box is infinite. Thickness a is the variable parameter that is indicated in each figure. (a) & (b) with different substrate thickness, nonlinear absorbing metasurfaces demonstrate significant advantages in attenuation and the relative bandwidth.

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Fig. 6 (a) Multiple peaks exist in the nonlinear metasurface, while linear case has single peak. It is because rectification produces nonlinear terms and the nonlinearity leads to multi-resonance. (b) shows an example by comparing the power concentration (including in air and substrate) of linear and nonlinear case, and two bottoms are marked that leads to the two peaks, while in linear case, there is only one. (c) First frequency bands of thin nonlinear metasurfaces.

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Firstly, as shown in Fig. 3, whatever RC sheet we use, nonlinear metasurfaces demonstrate significant advantages over linear metasurfaces. Figure 3(a) is regarding the influence of capacitance. Because we use a parallel RC sheet in the top layer of metasurface absorber, the capacitance affects the resonant frequency and larger capacitance values lead to lower resonant frequencies. For instance, for RC sheet with C = 12.5 pf/Square (R = 377 Ω/Square), the nonlinear metasurface resonates in 1.25 GHz, while C = 2.5 pf/Square, it resonates in 3 GHz. However, whatever frequency band, the nonlinear metasurface possessesboth magnitude and bandwidth advantages, indicating frequency flexibility and size scalability. In Fig. 3(b), we take examples of using resistance of a relative low value 100Ω/Square and a relative high value 800 Ω/Square to show the influence of resistance onthe nonlinear magnitude, but both of them demonstrate that attenuations in nonlinearmetasurfaces are at least ten times over that in linear metasurfaces.

Secondly, in Fig. 4(a), as the RC sheet varies from low resistance values to high resistance values, attenuation magnitude of the linear metasurface decreases gradually in the range below 10/m, while for the nonlinear metasurface, the magnitude varies in the range of 120/m to 20/m, demonstrating the significant nonlinear advantage. The nonlinear advantage in bandwidth is shown in Fig. 4(b), in which nonlinear metasurfaces with high resistance RC sheet show significant merits. For example, for an RC sheet with R = 1200 Ω/Square, thickness a = 1.524 mm and C = 2.5/4 pf/Square, the nonlinear metasurface bandwidth is 60%, almost as three times the bandwidth of the linear metasurface.

Thirdly, another interesting and significant point is the potential to exceed the relationship between the thickness and the bandwidth for linear absorbers. Conventionally, linear metasurface absorbers with very thin substrate have very narrow bandwidth. However, in our proposed diode-rectifier-based nonlinear metasurface absorbers, a large proportion of power is converted to DC, thus absorbed by the RC sheet. That means that bandwidth of nonlinear metasurfaces does not primarily depend on thickness, but is strongly associated with the power dissipation to the RC sheet. Therefore, even in very thin nonlinear metasurfaces, we can still achieve high absorption. As illustrated in Fig. 5(a), for example, for a substrate as thin as 0.35 mm (λ₀/143 for the center frequency of 6 GHz), the attenuation magnitude for linear metasurfaces is 2.3/m, while for the nonlinear metasurface is 25.5/m, more than ten times that of the linear metasurface. In terms of relative bandwidth as shown in Fig. 5(b), with the thin substrate of λ₀/143, the nonlinear relative bandwidth is 60%, almost three times of that in the linear case. This enables substantial bandwidth enhancement using a very thin metasurface with diode rectification.

Fourthly, in nonlinear case, there are multiple peaks (as shown in Fig. 6(a)) and that’s due to the full-wave rectification that produce nonlinear terms DC, 2f₀, and 4f₀, and in the absorption calculation, for instance in the air and substrate power concentration calculation, the summary of these harmonics lead to multiple peaks. Based on Eqs. (7) to (11), we calculate and compare the both linear and nonlinear concentrated power, which includes the power in the substrate and in air. As compared in Fig. 6(b), nonlinear case possesses two bottoms in the frequency range from 1 GHz to 5 GHz, leading to two peaks as shown in Fig. 6(a), while in the linear case, it has only one bottom and one peak. For nonlinear metasurfaces, the key property is the current rectification to DC on the top-layer RC sheet, and the DCabsorption is thickness-independent. That indicates that this kind of nonlinear absorber is more likely to apply for absorbing the first frequency band, due to this frequency band has the largest wavelength and the thin diode-rectifier-based nonlinear absorber can take its advantages best.

9. Discussion

In practice, we establish the full-wave-rectifier cell by applying diodes to the periodic metamaterial units as shown in Figs. 7(a) and 7(b). The nonlinear network layout is derived from the conventional full-wave circuit model. The diodes are arranged so that during each half-cycle of the RF wave, patches A and C will be charged positively or negatively respectively. Specifically, as in Fig. 7(a), as patch A and C are induced with opposite polarities, the current flows in the path from A₁ to C₂, being rectified by the diodes, and passing through patch B. In the next half-cycle the current is rectified through the path C₁ through B to A₂. Figure 7(b) is another schematic of the top layer and has the same mechanism. We implement the full-wave simulation using ideal diodes with nonlinear network of Fig. 7(b) as the metasurface unit cell, while for linear case, we remove diodes. The nonlinear absorber, as shown in Fig. 7(c), demonstrates advantages in all of the resonance frequency bands. The complete comparisons between theoretical analysis and full-wave simulation of both linear and nonlinear case are shown in Fig. 8, from which we have sereral conclusions: (1) with periodic units, the nonlinear metasurface demonstrates advantages as shown in the attenuation for a given resonance frequency, over a wide range of values. Especially, by inducing nonlinearity, the attenuation value exceed both the full-wave simulation and the theoretical linear case as shown by line 5 to line 3 in Figs. 8(a) and 8(c). (2) However, the advantages of the fullwave-basedsimulations are still below the nonlinear theoretical case as line 4 in Fig. 8(a) and line 4’ in Fig. 8(b), which is far more above all others. The same trend is shown in Figs. 8(c) and 8(d), in which thickness is studied as the parameter. This is likely becauses that we need to use periodic patterns to insert diodes for building the nonlinear cells, and this induces additional parasitic caspacitance and multiple resonances. The result is increased attenuation but decreased bandwidth (as shown in Fig. 9(a)). In the theoretical nonlinear case, however, we assume the diodes are uniformly distributed and are considered as infinitely small in the metasurface, and all the induced waves are rectified. In practice, we have to design the full-wave-rectifier unit with a finite size, resulting in deviation from the ideal theoretical result.

Fig. 7 The layout of diode-rectifier-based nonlinear network. (a) & (b) are the unit cells of periodical metasurfaces. We use the layout of (b) as the full-wave simulation model with the dimension are as: a = 7.6 mm, b = 4.7 mm, c = 30.85 mm, d = 4.5 mm, f = 3.1 mm, and other parameters are set as: ϵ₂ = 3, tanδ = 0.0013, µ2 = 1, R = 377 Ω/Square, C = 2.5 Pf/Square (for E, G), and C = 2.5/4 Pf/Square (for F). The air box is 25 mm height. With full-wave simulations, (c) demonstrates the nonlinear advantages in all of the resonances.

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Fig. 8 Nonlinear advantages are demonstrated in both attenuation (a) & (c) and bandwidth (b) & (d). Comparisons include theoretical analysis in both linear and nonlinear case, full-wave simulation with uniform pattern and periodical pattern (as Fig. 6(b)), and the nonlinear full-wave simulation using periodical pattern as Fig. 6(b). The parameters are the same as in Fig. 7, and the air box height is kept as 25 mm when the substrate thickness is increased.

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Fig. 9 To implement full-wave nonlinear simulations, periodic pattern is used due to it is compatible to insert diodes in the split gaps. With full-wave simulation, as shown in (a), the difference of periodic-pattern-based and uniform-pattern-based metasurface is that periodic case has higher attenuation magnitude but decreased bandwidth. (b)Compared with using ideal diode model, using real diode model in the simulation shifts down the resonant frequency, decreases the attenuation magnitude and expands the bandwidth.

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Another point is that in the full-wave simulations we use a real diode model, which affects the results. One effect is that the parasitic capacitance in the real diodes shifts the resonant frequency lower (shown in Fig. 9(b)), and the resistance decreases the Q factor, leading to wider bandwidth and lower magnitude than with ideal diodes, as compared by curve 1 to 2 in Fig. 9(b). With both the ideal and real diodes, the nonlinear advantages over the linear case are similar in magnitude (as compared by line 2 to 3 in Fig. 9(b)), so these effects do not qualitatively change the conclusions.

10. Conclusion

We propose a diode-rectifier-based nonlinear absorbing metasurface, and analyze it by applying nonlinear terms to the absorption calculation. We find that ideal diodes in the RC sheet rectify the currents to DC, and other higher order harmonics, and each mode is self-consistent by matching the boundary conditions above/below the parallel RC sheet. A large proportion of the power is converted to a DC term that can be contained and absorbed by the RC sheet, leading to the advantages of nonlinear absorbers in bandwidth and magnitude. Especially, because a large portion of the power is confined and dissipated in the RC sheet, the nonlinear absorbing metasurface can exceed the performance of linear absorbers. It can be applied to ultra-thin metasurfaces, because the key difference between the linear and nonlinear metasurface is based on the current rectification to DC, and the DC absorption is thickness-independent. Full-wave simulations with ideal diode model are proposed, and the nonlinear full-wave simulations, though perform not as excellent as theoretical nonlinear case, demonstrate the advantages over linear metasurfaces, especially exceeding the linear limitation in the attenuation magnitude. Meanwhile, the differences of the nonlinear theoretical case and the practical case are discussed and we conclude that: (1) in the fair condition, such as the theoretical nonlinear to theoretical linear case and nonlinear-full-wave-periodical case to linear-full-wave-periodical case, the advantages are significant in both attenuation magnitude and bandwidth. (2) While in practice, we compare the nonlinear-full-wave-periodical case to linear theoretical case, attenuation magnitude in nonlinear case still can exceed the linear limitation. This proposed nonlinear absorbing metasurface analysis provides the key to achieve nonlinear microwave metasurface absorbers with enhanced performance both in attenuation and bandwidth, and can be used for very thin metasurfaces to mitigate high power surface currents or protect against destructive high power interference.

Funding

This work was supported by the Office Naval Research under Grant No. N00014–15–1–2062.

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Terms	Diode-rectifier-based nonlinear metasurfaces			Linear metasurfaces
Terms	Coefficient (I²)	Power proportion	Distribution area	Coefficient (I²)	Distribution area
DC	${(\frac{2}{π})}^{2}$ (A_DC)	81%	RC sheet
f₀				1	Air RC sheet Substrate
2f₀	${(\frac{4}{3 π})}^{2}$ (A_2f0)	18%	Air RC sheet Substrate
4f₀	${(\frac{4}{15 π})}^{2}$ (A_4f0)	0.73%	Air RC sheet Substrate
Totally power proportion		99.73%		100%

Theoretical analysis of nonlinear surface wave absorbing metasurfaces

Abstract

1. Introduction

2. The approach of applying nonlinear terms to EM calculation

3. Power concentration in the nonlinear metasurface substrate

4. Power concentration in air

5. Power loss in the nonlinear metasurface substrate

6. Power loss in the RC sheet

7. Boundary conditions

8. Nonlinear advantages

9. Discussion

10. Conclusion

Funding

References and links

Cited By

Figures (9)

Tables (1)

Equations (23)

Optics Express