A Nonreciprocal Angular Selective Absorber with the Special Multilayer Structure

:Due to the orderly design of the special anti-reflection structure and the absorption structure, the one-dimensional layered periodic structure has a good impedance matching in a certain frequency band and a large-angle range, providing for the realization of an angular selective absorber (ASA) with a high rectangle coefficient. For the sake of obtaining excellent absorptivity, the indium tin oxide (ITO) film is used, and it also acts as a function of tuning the absorption angle range (AAR) of the ASA by adjusting the plasma frequency. The proportional relationship between the thickness of the dielectric layers is also discussed to satisfy a good absorption function. At the same time, the ASA also possesses productive nonreciprocal performance (NP) and can also be controlled by the plasma frequency. The transfer matrix method is used for numerical simulation. Our special tunable ASA with the NP is relatively rare in previous studies, which can be applied to optical communications and military fields. Furthermore, we hope that the design we proposed can provide new possibilities for the development of the ASAs.


Introduction 
The one-dimensional layered periodic structure [1][2] has been widely used in the past decade due to its flexibility in manufacturing, small size, and strong functionality. It plays an indispensable role in the fields of sensors [3][4][5], filters [6][7][8], and absorbers [9][10][11]. Nowadays, angular selective function devices (ASFDs) [12][13][14][15] have been sought after by researchers owing to their common application scenarios in the military and communications fields. In Ref. [16], an angular selective filter based on the Brewster angle mode is presented. Attributed to the inherent limitation of the Brewster angle, this ASFD can only be applied in the TM mode with very small angle selectivity. In Ref. [17], an ASFD based on the generalized Brewster mode has attracted people's attention. The researchers have realized a wide range of angle selection functions by cascading Brewster angles, about 20°. Their structures enlarge the angle range, but they cannot solve the problem affected by the polarization state of electromagnetic (EM) waves. In Ref. [13], a wide-angle ASFD that is not affected by the polarization mode is studied. The structure they proposed shows properties independent of the polarization state in the dispersion model, so the defect layer can make part of the light propagate at a specific frequency in the photonic band gap like a Fabry-Perot resonance. Unfortunately, their proposed filtering structure is not tunable. Benefiting from the efforts of a large number of predecessors, the development of the ASFDs has gradually become fuller. However, previous studies on the ASFDs with productive performance are more about the implementation of filter pieces, and few people can realize the angular selective absorber (ASA) with a high absorption rate and high rectangular coefficient. In addition, the tunability and non-reciprocity of this kind of absorber based on angle selection are also studied.
Optical nonreciprocal devices play a very important role in optical communications [18][19][20], such as photodiodes and optical isolators. A common mechanism for achieving nonreciprocal performance (NP) is to excite the nonreciprocal surface waves. However, the excitation of such surfaces is difficult after all and needs to meet specific excitation conditions. The structure we proposed can avoid this shortcoming through the clever design of the structure. Indium tin oxide (ITO) films are popular with various absorber researches on account of its tunability and excellent absorption performance [21][22][23]. Fortunately, the realization of the ASA put forward in this article also profits from it.
In this research, a tunable ASA with interesting NP is raised. Based on the premise of large impedance matching, we can view from the dispersion relationship graph that the incident EM waves can be passed through at a certain frequency and within a certain angle range, possessing very good angle selection characteristics. The transfer matrix method is used to explore the effects of different thicknesses of the dielectric layers, the ratio between the thickness of different dielectric layers, and the plasma frequency on the generation and performance of the ASA. It is worth mentioning that through the ingenious design of the structure, the NP can also be acquired, and the tunability can be completed by adjusting the plasma frequency. We hope that these very fascinating phenomena will promote the development of the optical communications and military fields to a certain extent. As displayed in Fig.1

Structure design and simulation
(1) Where, εꝏ=3.9 means the high-frequency dielectric constant. ωp=(e 2 ne/ε0/meff) 1/2 is the plasma frequency, ne means the plasma density. ε0 represents the vacuum dielectric constant and e is the electronic power. THz, ωp2=3880 THz, and ωp3=5000 THz. The corresponding plasma collision frequencies are γ1=0.0001ωp1, γ2=0.0001ωp2, and γ3=0.1ωp3. The incident EM wave is the TE wave. θ refers to the angle formed by the incident light and the +x direction. A clockwise direction is a positive number, and a counterclockwise direction is a negative number.
The propagation of the EM waves in the medium is expressed by the total transfer matrix as [25]: The reflection and transmission coefficients are calculated as follows [24]: In which, η0=(ε0/μ0) 1/2 /cosθ , therefore, the corresponding reflectance and transmittance are: Hence, we can get its absorptance A=1-R-T. This paper aims at the theoretical design of a new type of absorber with a high rectangular angle selection function and focuses more on the pursuit of theoretical optimization results. The research focus is not on the specific manufacturing, but it can be manufactured with reference to the scheme in Ref. [26].

Analysis and discussion
In Fig.2, apparently, in the frequency band higher than 514 THz, the absorption effect is excellent (red area in the figure), and has a very good rectangular coefficient (the red area has smooth edges). However, when the frequency is lower than 514 THz, the incident light will be exposed irregularly at large angles, seriously affecting the performance of the absorber, which is undesirable. To observe the absorption performance of the ASA more clearly, we define a variable B to represent the rectangular coefficient. B=G/H, where, G is the angular selective width when the absorptance is -30 dB, while H is that in the case of the absorptance is -3 dB [27]. The highest and lowest values of B are 1 and 0. The larger the value of B, the higher the steepness and the better the absorption property. If the frequency band satisfying the condition that the absorptance is higher than 0.9 and B is over 0.95 is selected as the effective area, then the ASA we put forward can work in the range of 514 THz~548 THz, such a large bandwidth can meet certain requirements in practical applications. Considering the convenience of the explanation, f=530 THz is chosen as the research object. As exhibited in Fig.3, suppose f=530 THz, the energy of the incident light is mainly absorbed or reflected, which is the effective mechanism for satisfying the ASA. Precisely, within the absorption angle range (AAR) is -36.5°~36.5°, the absorptance exceeds 0.9, and B is close to 1, which is almost a vertical drop, greatly compensating for the traditional ASAs' low rectangularity. These favourable performances are mainly due to the artful design of the structure. On account that the structure shown in Fig.1 is structurally symmetrical about the 72nd floor, whose ultra-high plasma frequency and ultra-wide thickness prevent the incident wave from entering the rear structure, for both forward and backward incidences, the effects produced by the structures on both sides are also similar and independent, so we only take the upper half of the black dotted line for an explanation. The first to the 11th layers and the 61st to the 71th layers are actually two anti-reflection structures, so that part of the EM waves can be propagated in the structure. The 72nd layer serves as an absorber. With the help of its larger thickness and higher ωp3, it produces strong absorption. Coupled with the superposition of more periodic structures from the 11st layer to the 61st layer, a high B value will be well guaranteed.
For the sake of expounding the mechanism of the ASA more distinctly, we combine the dispersion theory and impedance matching theory to analyze the structure. From the perspective of impedance matching, the reason for the appearance of the ASA is the good impedance matching of the entire structure and free space. The normalized surface impedance is defined as the ratio of the impedance of the entire structure to the impedance of the vacuum wave [28]: In which, Z0=|E0|/|H0|=(μ0/ε0) 1/2 is the vacuum wave impedance, whose value is about 377 Ω. Zeff =|E|/|H|=(μ/ε) 1/2 is the effective impedance of the entire structure. r infers the reflection coefficient. In order to fully suppress reflection, the designed structure needs to achieve impedance matching. In general, it is believed that the inhibition is relatively sufficient when the reflectivity is less than 0.1, and the reflection coefficient is less than 0.32, in which case, the normalized surface impedance is in the range of 0.51 -1.94. That is to say, when the normalized surface impedance is between 0.51 and 1.94, the structure involved can fully suppress the reflection and the impedance matching is good. In Fig.4, within the range of -36.5°~36.5°, the real part of the normalized surface impedance is around 0.9, which belongs to the range of 0.51-1.94, indicating good impedance matching and fully suppressing reflection, providing conditions for the production of high absorption.  (k represents wave vector and d=d1+dB) is not equal to 0, that is, when it falls in the red area in the picture, the EM waves can be propagated.
In Fig.6, if d1 is set as 220 nm, 225 nm, or 230 nm, the corresponding AAR values are individual -28.4°~28.4°, -36.5°~36.5°, and -43.6°~43.6°. With the expansion of d1, the AAR also tends to increase, broadening by 16.2° and 14.2° in sequence. At the same time, the absorptance of the ASA has maintained a level of more than 0.9, and B is also surpassing 0.95. Visibly, an appropriate amplification in d1 has a positive effect on increasing the working range of the ASA. The reason for this phenomenon is that the enhancement of d1 expands the angle range of impedance matching, thus, the AAR is also enlarged. In Fig.7, as dA differs from 0.46d1 to 0.54d1 with an interval of 0.02d1, the AAR hardly changes, always being -36.5°~36.5°, but the absorptance has been obviously influenced.
Only when dA is equal to 0.5d1, can it show excellent absorption performance. Under other conditions, the absorptance will be significantly lower than 0.9, threatening the filtering performance of the ASA. Therefore, only when the precondition of dA=0.5d1 is satisfied, the effect of the anti-reflection structure can be exerted to the best situation and the optimized absorption effect can be produced. In Fig.8, in case that d2 is 80 nm, its AAR is -32°~32°; if d2 is changed to 85 nm, the AAR becomes -36.5°~36.5°; suppose d2=90 nm, the AAR will be changed to -40.6°~40.6°. The increase of d2 will respectively expand the AAR by 9° and 8.2°, but the intensity of the increase is not as strong as that caused by d1. Similarly, during the change of d2, the absorptance is always higher than 0.9 and the B value is maintained at 0.95. Hence, a proper extension in d2 can also promote the expansion of the working range of the ASA. This can be attributed to that the augment in d2 makes a wider range of impedance matching well, so the AAR is also expanded. In Fig.9, when dB is 0.91d2, 0.93d2, 0.95d2, 0.97d2, or 0.99d2, the impact on the ASA has been investigated, but only when dB=0.95d2, the performance of the ASA is outstanding. Under other conditions, the impedance matching will be misaligned to a certain extent, so that the absorptance will decrease, and it will not completely reach more than 0.9. Consequently, the relationship between dB and d2 is fixed after all and cannot be changed at will.    In Fig.10(a), the augment of ωp1 has a reduced effect on the AAR. In the case of ωp1=3400 THz, the AAR is -45.1°~45.1°. If ωp1 becomes 3600 THz, the AAR will decrease to -41.7°~41.7°. The AAR is reduced by 6.8°. When ωp1 increases to 3800 THz, the AAR will change to -38°~38°, and decrease again by 7.4°. It is worth mentioning that during the change of ωp1, only the AAR has been evidently reduced, the absorptance is still above 0.9, and B is always overtopped. Since the three-dimensional top view can more clearly see the adjustment of the continuous ωp1 to the AAR, Fig.10(b) is shown. Because the amplification of ωp1 lessens the range of impedance matching and also broadens the photonic band gap in the dispersion curve, the AAR meeting the conditions will also decline. On condition that the ωp1 is greater than 4800 THz, the AAR disappears to 0. In this process, the absorptance will always maintain a high value, as revealed in the red area in the figure, and the edge of the red area always keeps a smooth state, so B will also be maintained productively. In Fig.1, the thickness of the 72nd layer of ITO is very large, and ωp3=5000 THz is also very large. This arrangement completely absorbs the incident EM wave, making it unable to propagate into the following structure. Based on this principle, we adopt the same arrangement structure on both sides of the 72nd layer, but the parameter settings are inconsistent, thus realizing the phenomenon of nonreciprocity. In order to highlight the degree of nonreciprocity, D is defined to describe. D=|F-H|, where F and H refer to the absorptance of the forward and backward incidences, respectively. The value of D ranges from 0~1. The larger the value of D, the stronger the NP, and the more obvious the difference between forward and backward incidences. Here we change ωp1 to 3500 THz, which can better emphasize the NP. As revealed in Fig.11(a), the AAR is -43.5°~43.5° for the forward incidence and -20.4°~20.4° for the backward incidence. They all maintain a high absorptance of 0.9 and a good B value of 0.95. The forward and backward incident EM waves are clearly separated in the range of -43.5°~-20.4° and 20.4°~43.5°. Ulteriorly, this NP is tunable. According to the previous analysis, d1, d2, and ωp1 can be utilized to adjust for this phenomenon. But considering the flexibility of actual operation, ωp1 is chosen. In Fig. 11(b), with the increase of ωp1, the area of high D value (the red area in the figure) is obviously decreasing. In the process of the adjustment, the B value is always in a good state, which can be seen from the smooth edge of the red area.

Conclusion
In this paper, we present a one-dimensional periodic multilayer structure containing ITO thin films to analyze the ASA and its NP. The tuning effects of d1, d2, and ωp1 on the ASA and NP are also discussed. The results show that the two conditions dA=0.5d1 and dB=0.95d2 are the prerequisites for ensuring the excellent performance of the ASA. The dispersion theory and impedance matching theory are introduced to explain the reasons. Ingenious structural design is also an important reason to ensure a high absorptance and high B value. In previous studies on the ASFDs, the research on the ASAs based on one-dimensional layered structures like ours is rare, let alone exploring its NP. Consequently, we hope that our proposed method can provide new design ideas for manufacturing new ASAs.