### I. Introduction

*λ*/4) apart [6]. These original structures, consisting of purely resistive sheets, a dielectric slab, and a ground plane, are referred to as Jaumann absorbers. A systematic design method of Jaumann absorbers to enhance the bandwidth at the cost of a lowered absorption rate was investigated in [7].

*λ*/8 at 3 GHz).

*ω*

_{0}. In Section II, the design equations are derived to maximize the flatness of reflection based on an equivalent circuit. Section III presents an easy guideline to finalize the dimensions of the used dipole structure without complicated optimization processes. In Section IV, a wideband microwave absorber at 10 GHz is designed and fabricated using the AgNW resistive film. The theoretical bandwidth of the presented absorber is compared with the measured one. The conclusion is given in Section V.

### II. Theory Based on an Equivalent Circuit

*l*is usually a quarter wavelength [4]. To realize a reactive screen, this work chooses a dipole-type structure. Specifically, a crossed-dipole pattern is chosen to make the absorption characteristics almost insensitive to the polarizations of incident electromagnetic waves, as will be seen in Section IV. The terminal impedance of the reactive screen can then be modelled as a lumped-series RLC resonator, as shown in Fig. 2. To optimize the pattern to have wideband absorption, we first analytically find the representative circuit values that should result in the

*maximum flatness*of the reflection and wide bandwidth in this section. Based on the RLC values obtained from this section, the exact dimension of the pattern structure will be determined through electromagnetic (EM) simulations in Section III.

### 1. Circuit Modelling of the Reactive Screen

*η*

_{0}(=377 Ω) and

*η*

_{1}are the intrinsic impedances of free space and the spacer, respectively, and

*Z*

_{0}is the impedance of the resonant circuit consisting of

*R*

_{0},

*L*

_{0}, and

*C*

_{0}. With a quarter-wavelength spacer

*l*, the electrical length

*βl*is

*π*/2 at the design center frequency

*ω*

_{0}.

*L*

_{0}and

*C*

_{0}are chosen to resonate at the design center frequency

*ω*

_{0}in this work.

*Y*

*at the front of the screen is*

_{in}*A*are given as

*Z*

_{0}=

*η*

_{0}= 377 Ω, and thus match

*Y*

*(*

_{in}*ω*) = 1/

*η*

_{0}at the design center frequency

*ω*=

*ω*

_{0}. This results in perfect absorption

*A*(

*ω*

_{0}) =

*A*

_{0}= 1 at the center frequency, but the bandwidth is relatively narrow.

### 2. Design Choices for Maximum Flatness

#### 2.1 Match factor

*A*

_{0}at the center frequency now deviates from unity as

*A*

_{0}, the match factor

*m*can either be

*A*

_{0}is unity, we have

*m*

_{1}=

*m*

_{2}= 1. However, when a slight mismatch is introduced, we have two options to determine

*R*

_{0};

*m*

_{1}< 1 and

*m*

_{2}> 1.

#### 2.2 Choice of L_{0} and C_{0} for maximum flatness

*L*

_{0}and

*C*

_{0}can be determined to enhance the bandwidth. Reminiscent of how the Butterworth filter has a maximum flat response [21], the bandwidth can be widened by enforcing

*n*as possible. Intuitively, if (8) is satisfied for every non-negative integer

*n*, Γ is simply zero for an infinite bandwidth. In this work, we have only one independent energy storage element

*L*

_{0}. Once

*L*

_{0}is determined,

*C*

_{0}is determined accordingly to resonate with

*L*

_{0}at the design center frequency

*ω*

_{0}. With one independent energy storage element, the number of derivatives that can be set to zero in (8) is at most one in general. The condition of

*ω*)| must be close to zero in any practical absorber design, the second derivative in (8) will automatically be largely satisfied (refer to the Appendix for the derivation). This assures the design to have the maximum

*flatness*of the response with a limited number of independent energy storage elements.

*L*

_{0}and

*C*

_{0}that yield maximum flatness, satisfying (9).

*G*

*and the input susceptance*

_{in}*B*

*should be zero at*

_{in}*ω*=

*ω*

_{0}. From the expression in (1), the derivative of the input conductance

*G*

*is zero at*

_{in}*ω*=

*ω*

_{0}. Therefore, the requirements for

*L*

_{0}and

*C*

_{0}are obtained from the condition

*B*

*is given by*

_{in}*m*.

#### 2.3 Choice of match factor

*m*

_{1}and

*m*

_{2}is a better choice, the circuit simulation of

*B*

*in Fig. 2 is run, and the results are compared.*

_{in}*A*

_{0}= 0.9 for example, the match factor can either be

*m*

_{1}= 0.5195 or

*m*

_{2}= 1.925. For each case, we have an (

*R*

_{0},

*ω*

_{0}

*L*

_{0}) of (196 Ω, 80 Ω) and (725.2 Ω, 1,096.4 Ω), respectively. Fig. 3 shows the susceptances of the reactive screen

*B*

_{0}, short-terminated quarter-wavelength transmission line

*B*

_{1}, and their sum

*B*

*for these two cases. Clearly, for both cases, the susceptance and its slope are zero at the design center frequency, as expected.*

_{in}*m*

_{1}in Fig. 3(a) leads to a wider resonant bandwidth than the choice of

*m*

_{2}in Fig. 3(b). The reason is that the two zero-derivatives of

*B*

_{0}are farther apart with

*m*

_{1}than with

*m*

_{2}. For example, in Fig. 3, the distance between the two zero-derivatives of

*B*

_{0}is about 1.5

*ω*

_{0}(= 1.9

*ω*

_{0}− 0.4

*ω*

_{0}) for

*m*

_{1}, which is much greater than 0.45

*ω*

_{0}(= 1.3

*ω*

_{0}− 0.85

*ω*

_{0}) for

*m*

_{2}. From (10), it is straightforward to show that this holds generally true for any choice of absorption

*A*

_{0}; i.e., the zero-derivatives of

*B*

_{0}are split farther apart for smaller match factors. The wide resonant bandwidth of

*B*

*naturally broadens the absorption bandwidth. Therefore, in this work, the smaller match factor*

_{in}*m*

_{1}of (6) is chosen to determine

*R*

_{0},

*L*

_{0}, and

*C*

_{0}in an effort to enhance the absorption bandwidth.

### 3. Final Design Adjustment through Circuit Simulation

*A*

_{0}= 1 and 0.9 are compared. For simplicity, the relative permittivity of the spacer layer is assumed to be unity. The used circuit components (

*R*

_{0},

*ω*

_{0}

*L*

_{0}) of the resistive Salisbury screen, the reactive one with

*A*

_{0}= 1, and the reactive one with

*A*

_{0}= 0.9 are (377 Ω, 0 Ω), (377 Ω, 296 Ω), and (196 Ω, 80 Ω), respectively.

*A*

_{0}= 1 to

*A*

_{0}= 0.9, as shown in Fig. 4(a). As a result, when

*A*

_{0}= 0.9, the 90% absorption bandwidth reaches up to 119% from 0.35

*ω*

_{0}to 1.54

*ω*

_{0}(Fig. 4(b)).

*L*

_{0}and

*C*

_{0}can be made to further enhance the bandwidth according to specific bandwidth criteria. For example, let us assume that we are interested in maximizing the 90% absorption bandwidth. In Fig. 4(b), the maximum flatness design of

*A*

_{0}= 0.9 suffers asymmetry around the design center frequency

*ω*

_{0}. This scenario is better visualized using the Smith chart in Fig. 4(c). The reflection coefficient Γ of the maximum flatness design quickly moves to the short impedance above the design center frequency. This results in the absorption bandwidth being considerably narrow in the upper range,

*ω*

_{0}–1.54

*ω*

_{0}, compared with 0.35

*ω*

_{0}–

*ω*

_{0}in the lower range.

*ω*

_{0}

*L*

_{0}, creating an additional resonant frequency beyond the design center frequency, as shown in the solid curve in Fig. 4(c). The appearance of an additional resonant frequency is not a coincidence. The maximum flatness design places multiple roots of the resonance at the design center frequency

*ω*

_{0}because

*ω*

_{0}

*L*

_{0}is chosen to be 120 Ω to further increase the 90% absorption bandwidth up to 124% (Fig. 5). The capacitance

*C*

_{0}is again determined accordingly by

*A*

_{0}is desired at

*ω*

_{0}, the match factor

*m*

_{1}< 1 is obtained using (6). Then

*R*

_{0},

*L*

_{0}, and

*C*

_{0}are first determined to maximize the flatness of the reflection at the design center frequency following (4), (11), and (12). The maximum flatness essentially widens the bandwidth of the absorption. To increase the bandwidth specifically for certain criteria,

*L*

_{0}and

*C*

_{0}can be tuned to maximize the bandwidth through circuit simulation while maintaining

### III. Crossed-Dipole Structure to Synthesize the Circuit Parameters

*a*is the side length of the absorber square unit,

*w*and

*h*are the width and height of the crossed-dipole made of a resistive film, respectively,

*R*

*is the resistance per square of the film material, and*

_{s}*g*is the gap distance.

*R*

_{0},

*L*

_{0}, and

*C*

_{0}of the unit cell are monotonous functions of

*R*

_{s}*h*/

*w*,

*h*/

*w*, and

*w*/

*g*, respectively. Therefore, finding a dimension to synthesize the required circuit components requires only a few iterations of electromagnetic simulations. In other words, by separating the work of optimization and realization each done by the circuit and electromagnetic simulations, the design process of the structure is less expensive in terms of computation than other optimization approaches, which heavily rely on electromagnetic simulations [13, 19, 20].

### IV. Fabrication and Measurement

*A*

_{0}= 1 at the design center frequency of 10 GHz is fabricated and measured.

### 1. Design and Simulation

*ɛ*

*= 1.03, we first attached the film on an acrylic layer with*

_{r}*ɛ*

*= 2.56, which was later mounted on the Styrofoam layer. The acrylic layer with a thickness*

_{r}*l*

_{1}of 1.5 mm was used. The thickness of the Styrofoam

*l*

_{2}was determined to be 4 mm to ensure a zero susceptance of

*Y*

_{1}at 10 GHz. Based on the same condition, the effective permittivity

*ɛ*

*and the intrinsic impedance*

_{eff}*η*

_{1}of an effective single-layer spacer were 1.86 and 276.5 Ω, respectively.

*m*= 1,

*η*

_{0}= 377 Ω, and

*η*

_{1}= 276.5 Ω, the circuit values for the maximum flatness were (

*R*

_{0},

*L*

_{0},

*C*

_{0}) = (377 Ω, 6.42 nH, 39.5 fF). The fine adjustments of

*L*

_{0}and

*C*

_{0}for further bandwidth enhancement were omitted in this fabrication example. After all, the dimensions of the unit structure to realize the above circuit values were determined through electromagnetic simulations and are summarized in Table 3.

*Z*

*of the structure in Table 3 was first simulated using HFSS in Fig. 9(a) and then compared with the input impedance of the circuits in Fig. 2 with (*

_{in}*R*

_{0},

*L*

_{0},

*C*

_{0}) = (377 Ω, 6.42 nH, 39.5 fF) for the maximum flatness. The crossed-dipole structure realized the impedances for the maximum flatness across a wide frequency range. Therefore, the absorption of the crossed-dipole structure was also in good agreement with that of the circuit in Fig. 9(b), showing the wide bandwidth. The EM-simulated 99% and 90% absorption bandwidths were about 42.7% and 78.1%, respectively.

*φ*. Fig. 10 shows the electromagnetically simulated absorptions depending on the polarization angles of the normally incident electric fields from 0° to 45°. The polarization angle exhibited negligible effects on the absorption due to the employed crossed-dipole structure. For the degradation effects with oblique TE and TM incidence cases, we can apply similar techniques given in [8], but these are not included in this work.

### 2. Fabrication and Measurement

*S*

_{11}|

^{2}on the absorber plane based on the reflection coefficient measured by the horn antenna. In the measurement, only the reflection on the absorber plane was considered, not the horn antenna itself. To calibrate the self-reflection of the horn antenna, the horn antenna in free space is used as the match load in the calibration process instead of the typical 50 Ω load. In addition, the reflection coefficient in front of the reversed absorber with the same size as the absorber is used as the reference to compensate for the energy leakage through the space. A more detailed measurement method using a single antenna can be referenced in [13]. For example, the 90% absorption bandwidths of each one are 75.7%, 78.1%, and 66.4%, respectively. The difference between the simulation and the measurement could be attributed to the non-uniformity of the resistivity in the AgNW film and the limited size of the absorber in the measurement.