## Abstract

Using the transfer matrix method, a unidirectional absorber with an ultrabroadband absorption bandwidth and angular stability is realized in the graphene-embedded photonic crystals (GPCs) arranged by the cascading structure formed with the periodic sequence and the quasi-periodic Octonacci sequence in the terahertz regime. As a result, the surface conductivity of the graphene sheet can be modulated via the chemical potential, and the characteristics of the proposed absorber are tunable. Compared to the structure spliced by the diverse periodic sequences, the relative absorption bandwidth of the proposed composite construction is up to 94.53%, which far outweighs that of the periodic one. We compare the Octonacci sequence, the Fibonacci sequence, and the Thue–Morse sequence, and the calculated results reveal the advantage of the Octonacci sequence in the expansion of the absorption bandwidth. Under the optimization of the related parameters, the incident wave is primarily reflected in the forward propagation and absorbed in a wide range of $\theta$ under the TM mode in backward propagation, which shows the splendid unidirectionality and angular stability. The impacts of the chemical potential, structural thicknesses, and stack numbers on the absorption properties are also investigated in detail. Additionally, the impedance match theory and the interference field theory are introduced to explain the intrinsic absorption mechanism of the presented GPCs. In short, the unidirectional broadband and angle-insensitive absorber has extensive application prospects in optical sensing, optical filtering, photodetection, and solar energy collection.

© 2020 Optical Society of America

## 1. INTRODUCTION

As a two-dimensional (2D) ultrathin dispersive material, graphene has potential in the manufacture of saturable absorbers [1], optical modulators [2], transparent electrodes [3,4], and solar cells [5] because of its tunable chemical potential that is modulated by the impressed voltage. It is interesting to note that 1D photonic crystals (PCs) embedded in the graphene sheets can exhibit some extraordinary phenomena such as multichannel [6,7], single-frequency [8–10], and single-peak absorption [11–14]. In addition, some advanced algorithms, such as the transfer matrix method [15] and the neural network [16], are introduced to analyze the optical responses of the graphene-embedded photonic crystals (GPCs). However, related research showed that the absorption capacity of the GPCs is at an inferior and uniform level that is not advantageous in optical detector design [17]. Therefore, an improvement of GPC absorption in [18,19] and the graphene monolayer [20–22] have become hot topics. Ning *et al*. [23] surveyed the wide-angle broadband absorber formed with isotropic dielectrics and graphene layers that operates in the mid-infrared region. The effective group index of the graphene can be tuned by regulating the chemical potential and thicknesses of different materials, and it is susceptible to the incident angle and polarization modes. Nevertheless, the mechanism of angular insensitivity and the unidirectional transmission are not discussed in their report. The absorption enhancement of the graphene sheet placed at the top of 1D PCs was studied by Liu *et al*. [24], which proved that the photonic localization in the optical microcavity on the surface of 1D PCs can improve the absorption properties of graphene. However, the absorptance of the structure in their analysis is far less than the ideal standard. Fan *et al*. [25] discovered that regulating the propagation bands of the PCs is an efficient approach to achieve the cracking absorptance of graphene that can be ascribed to the proper terahertz operating range for the electromagnetic (EM) wave and the diversiform resonances caused by multiple scatterings. The influences of the characteristics frequency and lattice constants on the averaged absorption factor were also observed. In spite of all this research, we believe the effective absorption bandwidth still must be strengthened.

Up until now, there has not been much research on the expansion of the one-way absorption region based on the quasi-periodic cascading structure. In this paper, a one-way and angle-insensitive absorber consisting of graphene sheets and isotropic dielectrics is realized via the resonant cavities produced by the cascading structures. The mechanisms of the one-way absorption, angular insensitivity, and the effects of the chemical potential and the structural parameters on the features of absorption are meticulously analyzed. To present the advantages of the proposed absorber, Table 1 offers some comparisons of the absorption properties of the graphene-embedded absorbers.

## 2. SIMULATION MODEL AND CALCULATION METHOD

The construction of the 1D GPCs cascaded by the periodic and quasi-periodic sequences is displayed in Fig. 1(a). The periodic stack obeys the ${({\rm ABG})^N}$ pattern while the Octonacci sequence has its relatively symmetric recursive regulation, which is expressed as ${S_M} = {S_{M - 1}}{S_{M - 2}}{S_{M - 1}}$, $M \mathbin{\lower.3ex\hbox{$\buildrel \gt \over{\smash{\scriptstyle=}\vphantom{_x}}$}} {2}$ with the initial parameters ${S_1} = { \{\rm GBG\}}$, ${S_0} = { \{\rm BGB\}}$, where $G$ represents the graphene monolayer. [See the substitution rules of the colored blocks in Fig. 1(a).] Hence, one attains ${S_2} = { \{\rm GBGBGBGBG\}}$, and so forth. For simplicity, the stack numbers of these sequences are defined as $N = 2$, $M = 4$. The entire cascading structure is exposed to air and the EM wave is incident from the $xoy$ plane with an incident angle $\theta$. As depicted in Fig. 1, the forward (along the ${+}{z}$ axis) and backward (along the ${-}z$ axis) propagating directions are denoted by arrows. The thicknesses and refractive indices of the dielectric $ A $ (the artificial synthetic material) and the dielectric $B$ (air layer) are described as ${d_A} = 1.5\;{\unicode{x00B5}{\rm m}}$, ${d_B} = 12.5\;{\unicode{x00B5}{\rm m}}$, ${n_A} = 6.5$, ${n_B} = 1$. The thickness of the graphene sheet is taken as ${d_G} = 0.34\;{\rm nm}$ and the ambient temperature $ T $ is fixed at 300 K. The feed pattern of the graphene layers is displayed in Fig. 1(b). The gold electrode situated at the top of GPC structure contacts directly with the graphene layers, and the positive gate bias voltage is applied to the gold electrode. The negative voltage is impressed to the silicon dioxide substrate. Hence, the chemical potential of the graphene layers can be tuned effectively by the applied direct-current bias voltage [26].

In accordance with the Kubo formula [27,28], the frequency-dependent surface conductivity of the honeycomb-like graphene sheet can be given as

where $\sigma _g^{{\mathop{\rm inter}}}$ is the interband conductivity and $\sigma _g^{{\mathop{\rm intra}}}$ is the intraband conductivity of graphene, and their specific expressions are [27,28]Assuming that the electronic band structure of the graphene monolayer is unaffected by the surroundings, the effective permittivity of the graphene sheet is written as

where ${\varepsilon _0}$ is the permittivity of the free space and ${d_G}$ is the thickness of the graphene monolayer.Hence, the frequency-dependent complex refractive index of the graphene sheet can be worked out as

To explore the optical characteristics of the proposed GPCs, we consider all the materials as nonmagnetic ($\mu = 1$), in utilization of the transfer matrix method, and the transfer matrix of the dielectrics and graphene monolayer can be signified as [15]

The relationship between the electric and magnetic fields in the 1D GPCs with $ L $ layers can be obtained as

In the periodic layout with the arrangement ${({\rm ABG})^N}$, the overall transfer matrix is

Likewise, the transfer matrix of the structure constructed by the Octonacci sequence [29] can be written as

Hence, the complete transfer matrix of the proposed GPCs is

Based on the expressions mentioned above, the reflection and transmission coefficients can be written as

Therefore, the absorptance $A(\omega)$ of the presented structure ought to be

To express the extent of the one-way absorption more vividly, the variable $ P $ is introduced and has a form of

where ${A_{{\rm backward}}}$ and ${A_{{\rm forward}}}$ are the absorptance in backward and forward directions, respectively.## 3. RESULTS AND DISCUSSION

From Figs. 2(a)–2(d), one can perceive that the quasi-periodic Octonacci sequence has superiority in the expansion of the one-way absorption bandwidth that runs at 3.186–8.896 THz with the relative bandwidth (RBW) 94.53%. As for the periodic one, the one-way absorption scope ranges at 4.34–9.344 THz with the corresponding RBW 72.99%, which is far less than the quasi-periodic sequence. In the forward propagation, the EM wave is mainly reflected by the combination of the conventional dielectrics with high and low refractive indices. This phenomenon leads to the intense interference of the reflected wave caused by Bragg scattering. In backward propagation, the energy of the EM wave is confined in the GPCs due to the surface current induced by the conductivity of the graphene layers. From Figs. 2(a)–2(d), it is obvious that the reflective characteristics of both the quasi-periodic and periodic constructions are similar, while the absorption feature of the former is superior to that of the latter. The difference of the absorption quality in these two structures can be ascribed to the appropriate destruction of the space-reversal symmetry in the quasi-periodic configuration and the irregular arrangement of the graphene layer, which strengthens the absorption capacity of the proposed GPCs.

To highlight the advantage of the Octonacci sequence, some comparisons of the unidirectional absorption bandwidths of the Fibonacci sequence and Thue–Morse sequence are given. As we know, the Fibonacci sequence has its recursive regulation ${S_F} = {S_{F - 1}}{S_{F - 2}}$ for the order $F \mathbin{\lower.3ex\hbox{$\buildrel \gt \over{\smash{\scriptstyle=}\vphantom{_x}}$}} {2}$; herein, the initial parameters are set as ${S_1} = { \{\rm GBG\}}$, ${S_0} = { \{\rm BGB\}}$. The next parameter is ${S_2} = { \{\rm GBGBGB\}}$; herein, $ F $ is set as 7. In addition, the Thue–Morse sequence can be formed via the recursive law ${S_p} = {S_{p - 1}}{\bar S_{p - 1}}$ for the order $p \mathbin{\lower.3ex\hbox{$\buildrel \gt \over{\smash{\scriptstyle=}\vphantom{_x}}$}} {1}$, where ${\bar S_{p - 1}}$ is the complement of ${S_{p - 1}}$. When the original conditions are taken as ${S_0} = { \{\rm C\}}$, ${\bar S_0} = { \{\rm D\}}$, one obtains ${S_1} = { \{\rm CD\}}$, ${S_2} = { \{\rm CDDC\}}$. The substitution laws of the letters C and D can be found in Fig. 1(a), where $ p $ is taken as 4. As shown in Figs. 2(e)–2(h), the Fibonacci sequence and the Thue–Morse sequence with similar total thicknesses are introduced to construct the 1D GPCs. The numerical results show that their one-way absorption bandwidths are inferior to that of the Octonacci sequence. As for the Fibonacci sequence, the one-way absorption region is 3.433–8.874 THz with its RBW 88.42%. In addition, for the GPCs containing the Thue–Morse sequence, there are three inconsecutive one-way absorption scopes that run about 4.091–5.926 THz, 6.264–6.878 THz, and 7.175–8.580 THz with the related RBWs 36.63%, 9.34%, and 17.84%, respectively. Table 2 lists the maximal continuous absorption regions of the structures and includes different sequences. Clearly, because of its better geometric asymmetry and the higher structural complexity, the Octonacci sequence possesses strength in the realization of the unidirectional absorption compared to the other quasi-periodic sequences.

The fundamental reason for the perfect absorption of the proposed GPCs can be ascribed to the fantastic impedance match between the whole structure and the free space. The normalized surface impedance is the ratio between the effective surface impedance of the structure and the wave impedance in vacuum and can be expressed by [30]

Hence, we lay our emphasis on the normalized effective impedance other than the effective surface impedance of the structure for the explicit analysis of the absorption features. When the normalized effective surface impedance is close to one, it means the effective impedance of the structure matches well with that of the free space. In this case, the reflection coefficient $ r $ is ideally up to zero, which means that the reflectance of the materials is approximately negligible and the EM wave is limited in the structure to the utmost extent and consumed thoroughly, thus generating favorable absorption effects. As shown in Fig. 3(a), it is apparent that the normalized surface resistance of the proposed GPCs in the backward propagation is fluctuating up and down around 1 in the mid and high frequencies. The normalized surface reactance is close to zero in the absorption area, as mentioned in Fig. 2(a), which realizes the impedance match in this direction. Meanwhile, as for the forward propagation, the surface resistance is almost zero at the nonabsorbing region ranging from 3.2 THz to 10.5 THz, which brings about the impedance mismatch and thus generate the intense reflection. Therefore, from the calculated results shown in Figs. 3(a) and 3(b), the distinguished one-way absorption of the presented 1D GPCs is embodied fully.

The optical resonant cavities produced in the proposed cascading structure promote the one-way absorption to some extent. From Fig. 4(a), it can be found that the unidirectional absorption is extremely weak, existing only in the region of 3.50–10.12 THz in the periodic structure ${({\rm ABG})^{19}}$. As depicted in Fig. 4(b), the structure formed by the Octonacci sequence presents stronger absorption in middle frequencies in both forward and backward orientations, but there exists no unidirectionality. Considering the drawbacks of the single structure, the cascading compound structure is adopted where the one-way absorption performance is strengthened greatly, as shown in Fig. 2(a). It can be explained in physics that the interaction between the incident EM wave and the dissipative graphene layers accompanied by the reflective dielectrics brings forth the transformation of energy in the passband, thus realizing the absorption resonance. Moreover, in the cascading structure, the amount of resonant cavities has been increased and the mutual coupling appeared, thus apparently widening the absorption band.

As shown in Figs. 5(a) and 5(b), the characteristics of absorption and reflection in the presented GPCs are angle-insensitive, but polarization-dependent. In terms of the backward absorption, as shown in Fig. 5(a), the TM wave has better angular stability than the TE wave, which maintains a perfect absorption below 70º at 4.11 THz. In the higher frequencies, the common absorption region of the TE and TM waves under 30º roughly starts from 5.29 THz to 8.90 THz with a RBW 50.88%. The reason for the blue-shifted phenomenon in the absorption spectrum can be attributed to the rise of $\theta$, the normal wave vector in the medium shrinks rapidly and, for obtaining the ideal absorption, the resonant propagation must be kept at the cost of enlarging the wave vector, thus aggrandizing the operating frequencies of the EM wave. From Fig. 5(b), one can find that the forward propagation of the TE wave is unresponsive to the variation of $\theta$, which gives rise to the angle-insensitive band gap at the 3.02–10.38 THz range with the RBW 109.85%. For the TM wave, to acquire the reflection bandwidth as broad as possible, the maximum $\theta$ sustained under the TM mode is approximately 50º with the corresponding reflection scope 7.97–8.67 THz. With the augment of $\theta$, the reflective region also has a blue-shifted tendency.

The mechanism of the wide-angle absorption under the different polarization modes is analyzed by the interference field theory. The total complex amplitude of the reflected wave’s composite field can be obtained by superimposing the various field amplitudes generated by the reflected wave, which has a relationship with the phase difference and the reflectance. The same is true for the transmitted wave’s composite field. The total complex amplitudes of the reflected and transmitted waves’ synthetic fields (${A^{(r)}}$ and ${A^{(t)}}$) are worked out as

Hence, the intensity distribution functions of the reflected and transmitted waves can be determined from the product of the total complex amplitude and its conjugate amplitude. Based on the intensity expression form of the interference field, the intensity distributions of the reflected and transmitted waves are the function of the reflectance, transmittance, and phase differences versus the incident wave, which can be expressed as

To attain the splendid angular stability in absorption features, the phase differences of the reflected and transmitted waves should satisfy the interference cancellation condition. As for the reflected wave, if the phase difference is an even multiple of $\pi$, the intensity of the reflected wave is zero, which signifies the lowest level of reflection. Also, for the transmitted wave, if the phase difference is an odd multiple of $\pi$, the intensity of the transmitted wave is zero, which presents little transmission. Therefore, the interference cancellation condition in regard to the phase differences of the reflected and transmitted waves is the solid foundation of the angle-insensitive absorption, which can be written as

For the interference state of the TE wave at 7.48 THz, as shown in Fig. 6(a), one can know that when $\theta$ is below 50º, the phase differences of the reflected (${\delta _r}$) and transmitted (${\delta _t}$) waves approximately meet with the interference cancellation condition mentioned above. There appears a sudden plunge in ${\delta _t}$ when $\theta$ is around 55º, which causes a decline in absorption. When $\theta$ exceeds 63º, the quality of absorption deteriorates because of the constructive interference of the transmitted wave. As plotted in Fig. 6(b), for the TM wave at 4.11 THz, the perfect absorption does not vanish until $\theta$ is larger than 70º, and the absorption capacity degenerates promptly when ${\delta _t}$ is falling to zero. In short, the angle-insensitive absorption of the presented GPCs in the TM mode is superior to that in the TE mode.

#### A. Effects of ${\mu _c}$ on the Absorption Features

The dynamic adjustment of the chemical potential ${\mu _c}$ in the graphene sheet is the foundation of the optical tunability. From Figs. 7(a) and 7(b), one can see that with an increase of ${\mu _c}$, the ideal absorption region is blue-shifted with its ideal absorption bandwidth shrinking gradually, and the proper range of ${\mu _c}$ is around 0.3–0.4 eV. It is known that the quality of absorption features in the graphene sheet is mainly dependent on its conductivity, which is affected by the chemical potential. As depicted in Figs. 8(a) and 8(b), the real component of the conductivity in the graphene sheet is approximately zero in the higher frequencies while with ${\mu _c}$ rising ceaselessly, it is getting enhanced overtly in the lower frequencies with the faster changing rate. Likewise, as for the imaginary part of ${\sigma _g}$, its fluctuation scope expands pronouncedly with the growth of ${\mu _c}$. From the analysis mentioned above, one can draw the conclusion that in a smaller variation range of the real and imaginary parts of ${\sigma _g}$ (that is, in the appropriate changing area of ${\mu _c}$ from 0.3 to 0.4 eV), there hardly occurs a distinction in the absorptance of the proposed GPCs. The real and imaginary parts of ${\sigma _g}$ decline steeply once ${\mu _c}$ surpasses 0.4 eV, which diminishes the induced current on the surface of the graphene sheet and deteriorates the absorption properties. All in all, the insufficient or excessive value of ${\mu _c}$ is adverse to the absorption features of the proposed GPCs, and the moderate value of ${\mu _c}$ is suitable for the promotion of absorption.

The various circumstances of the unidirectional absorption parameter $ P $ concerning the chemical potential under different polarization modes at diverse incident angles are plotted in Figs. 9(a)–9(d). It can be derived that at a lower incident angle, the conditions of $ P $ under the TE and TM modes are analogous and both exhibit the blue-shifted phenomena as an augment to $ \mu _c$. At the larger incident angle, the variation tendency of $ P $ under the TM mode is much more distinct than that under the TE mode; when $\theta$ is up to 75º, there is no the unidirectional absorption region under the TM wave. In conclusion, it is clear that the one-way absorption ability of the proposed GPCs is decaying as $\theta$ increases. When ${\theta}$ increases, the angle between the magnetic field vector and the boundary direction gets larger, which leads to the abatement of the magnetic field intensity along with the circulation, thus making the induced current poorer and worsening the absorption features. Additionally, from the computed results, one can find that the polarization sensitivity of the one-way absorption in the presented GPCs is susceptible to the incident angle since the photonic band gaps of the TE and TM waves in the forward propagation are disparate. The one-way absorption similarity between the TE and TM waves is evident at the smaller incident angle, while it reduces with the increase of $\theta$.

#### B. Effects of Dielectric Thicknesses on the Absorption Features

As plotted in Figs. 10(a)–(d), the thicknesses of different dielectrics play a vital role in the modulation of the absorption regions. The dielectric $ A $ with the higher refractive index has nearly nothing to do with the backward absorption, but does affect the forward absorption peaks. With the enlargement of ${d_A}$, the quantity of the absorption peaks in the forward propagation is getting larger and its full width at half-maximum (FWHM) diminishes rapidly, thus allowing the more and narrower one-way absorption areas to emerge. The absorption abilities of the proposed GPCs in the two propagating directions are also affected by the thickness of the dielectric $B$. As for the backward absorption, which contributes a lot to the unidirectionality, the number of the standing waves generated by the resonances of the confined traveling waves increases in the higher frequencies. Moreover, as ${d_B}$ becomes thicker, the amount of the discrete absorption areas get larger. The same is true for the forward absorption peaks, which present the tapering FWHM with an augment of the frequency. It is worth noting that the volume of the one-way absorption bands can be adjusted by the thicknesses of the conventional dielectrics. This regulating effect is caused by the constructive interference of the EM wave, which is transmitting in the proposed GPCs. More specifically, the even multiple of the phase differences in the reflected and transmitted waves resulting from the variation in thicknesses can produce the superposition enhancement effects, thus giving rise to new absorption regions.

The spectra of $ P $ versus the thicknesses of the conventional dielectrics are further depicted in Figs. 11(a) and 11(b). It is worth noticing that an increasing number of the unidirectional absorption bands with narrower bandwidths emerges with the rise of ${d_A}$ and ${d_B}$, which are coincident with the phenomena, as drawn in Figs. 10(a)–(d). Therefore, as shown in Figs. 11(a)–(b), the optimal thicknesses of dielectric $A$ and dielectric $B$ are around 1.5 µm and 12.5 m, respectively, which provides beneficial suggestions for the structural optimization in the achievement of better one-way absorption.

#### C. Effects of the Stack Numbers $ N $ and $ M $ on the Absorption Features

Through the simulation, we find that the absorption level of the GPCs in the forward propagation is almost unaffected by the recursive number $ M $ while it is largely dependent on $ M $ in the contrary propagation direction. Hence, the emphasis is laid on the effects of the stack numbers $ N $ and $ M $ on the forward and backward absorption features, respectively. As presented in Fig. 12(a), with an increase of $ N $, a growing number of absorption peaks with stronger amplitudes in the high frequencies exists, which destroys the Bragg scattering condition of the reflective frequency bands in the forward propagation and are disadvantageous to the one-way absorption characteristics. For stack number $ M $, one can derive from Fig. 12(b) that the increase of $ M $ is somehow beneficial to the improvement of the backward absorption, especially in the higher frequencies. However, once $ M $ surpasses 4, the total number of the layers becomes too large which is adverse to the subsequent manufacturing process. Hence, in the appropriate range, taking a larger value of the stack number is of great significance to the absorption level. It can be deduced that the interior mechanism of the optical path would be varied and reach the perfect resonance state with a reasonable increase in the number of graphene sheets.

## 4. CONCLUSION

In summary, this paper investigates a unidirectional ultrawide terahertz absorber with the splendid angular stability arranged by a cascading structure consisting of conventional dielectrics and graphene sheets. Comparatively speaking, the cascading structure has superiority in realizing the one-way absorption owing to the broken space-reversal symmetry and the introduction of the graphene sheets. The proposed 1D GPCs improve the absorption performances greatly, which make up for the limited single-pass absorptance of 2.3% in the single undoped graphene sheet. In the perfect one-way absorption region ranging from 3.186 THz to 8.896 THz (RBW is up to 94.53%) at 300 K, the incident EM wave is nearly reflected in the forward propagation and absorbed in the reverse direction. The angular stability of absorption in the proposed GPCs performs better under the TM mode and the intrinsic mechanism is analyzed via the interference field theory. The influences of the chemical potential, the structural thicknesses, and the stack numbers on the one-way absorption features also are surveyed. We believe this kind of unidirectional ultrawideband THz absorber has great potential for application in the design of the tunable optoelectronic devices such as optical filters and the solar cells.

## Funding

NUPTSF (NY217131); Open Research Program in China’s State Key Laboratory of Millimeter Waves (K201927).

## Disclosures

The authors declare no conflicts of interest.

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