## Abstract

Black phosphorus (BP), a promising new two-dimensional (2D) material, has drawn a lot of attentions in academia and industry due to its extraordinary physical and chemical properties. In this paper, we theoretically demonstrate a monolayer BP that achieves coherent perfect absorption (CPA) at the THz/infrared band. It is found that quasi-CPA point does exist at the THz/infrared band. The CPA, which has a relative bandwidth of 141.3% and a coherent absorptivity of more than 90%, can be implemented at the quasi-CPA wavelength through a proper phase modulation. Moreover, the coherent absorptivity can be modulated with a high modulation depth by means of the phase difference between the two coherent counter-propagating beams. The angular selectivity of the monolayer BP is also investigated. The CPA wavelength is divided into two wavelength branches for TE and TM polarization at oblique incidence. In addition, the CPA wavelength can be tuned from THz to infrared band by adjusting the electron doping of the BP while maintaining the modulation depth of 10^{4}. Hence, our results may be potentially used for coherent modulations in terahertz/infrared detections and signal processing with 2D materials.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

In the past few years, two dimensional (2D) materials, such as graphene, h-BN, MoS_{2} and so on, have been a popular research area due to its distinctive electrical and optical properties [1]. They are widely applied in chemical and biological sensors, solar cells, optical storage, and optoelectronic devices [1, 2]. Perfect absorber based on 2D materials has also been developed for its extensive applications in photodetectors and photovoltaics [3, 4]. The realizations of the perfect absorption using different kinds of routes have been proposed recently. For example, a graphene-based hyperbolic metamaterial structure was proposed by Xiang *et al*. in which a tunable perfect absorption based on critical coupling phenomenon is demonstrated [5]. Wu *et al.* presented the perfect absorption phenomena in the hexagonal boron nitride (h-BN) crystals by means of critical coupling using one dimensional photonic crystals spaced by the air [6]. Guo *et al.* investigated the enhanced and perfect absorption in a graphene/planar waveguide hybrid structure [7]. Jiang *et al.* studied the perfect absorption in THz in the modified Otto configuration based on graphene surface plasmons [8]. Wang *et al.* designed a graphene-DBR structure to realize tunable and multi-channel terahertz perfect absorber based on Tamm surface plasmons [9]. Fan *et al* proposed tunable terahertz meta-surface with graphene cut-wires to achieve absorption enhancement [10,11]. Sundry structures based on 2D materials have been proposed to achieve perfect absorption. However, its absorptivity can hardly be changed once the structure has been designed. In general, most of the absorbers are based on one input channel only and the perfect absorption occurs under the unilateral incident light. It is preferable to make the absorption tunable without changing the inherent parameters of the structure. One example is to use another beam.

Coherent perfect absorption (CPA), thanks to its destructive interference of two incident lights, has become a popular research topic. It provides an effective and flexible way to manipulate the absorption by changing the phase difference of two counter-propagating beams [12, 13]. CPA was first demonstrated experimentally with a simple silicon-resonator [14], and recently, it has been realized in composite medium, metasurface structure, waveguides, etc [15–18]. Gupta *et al.* studied CPA under the oblique incidence with a two-component metal-dielectric heterogeneous medium structure [19]. Zhu *et al.* designed an all-dielectric metasurface structure made of Ba_{0.5}Sr_{0.5}TiO_{3} (BST) ceramic to realize CPA and manipulated the output signals by adjusting the intensity and phase delay of the two incident beams [20]. For 2D materials whose thickness is much smaller than the wavelength of light, it is difficult to realize CPA as the light-matter coupling at nanoscale is inefficient. Therefore, it is interesting to study the possibility of the perfect absorption in 2D materials. Recently, several typical coherent perfect absorbers based 2D materials have been studied. Fan *et al.* realized tunable terahertz coherent perfect absorption in a monolayer graphene and proposed a coherent perfect absorber composed of non-resonant 2D carbon material-graphene [21, 22]. Shraddha *et al.* realized the application by applying the unstructured multilayer graphene films [23]. Zhang *et al.* studied the manipulation of the coherent absorption in a monolayer nanostructured graphene by adjusting the structural design and the relative phase difference of the input coherent beams [24]. Hu *et al.* exploited a split-ring graphene to realize high efficiency tunable CPA in the terahertz regime [25]. As an alternative of graphene, molybdenum disulfide (MoS_{2}) has also been extensively investigated for its distinctive physical, electronic, and optical properties. Zhu *et al.* proposed a broadband coherent perfect absorber with bulky MoS_{2} crystal slab in terahertz regime [26].

However, since black phosphorus (BP) joined the 2D material family in 2014, it has drawn much attention due to its tunable bandgap and anisotropic properties [27]. Unlike other 2D materials, the monolayer BP has a direct bandgap of ∼2 eV and its bandgap value is closely related to the thickness. Such characteristic makes BP a good candidate for the photodetection. Moreover, the monolayer BP can provide in-plane anisotropic features due to the hexagonal lattice with a puckered structure formed by phosphorus atoms [27, 28]. Recently, BP-based perfect absorber has been reported. Xiong *et al.* proposed an anisotropy nanostructure based on BP to realize polarization dependent absorption [29]. However, the CPA of BP has not been investigated thoroughly and the BP-based coherent perfect absorbers may have more potentials in the applications of signal processing and optical devices.

In this paper, the perfect absorption at THz/infrared is realized based on the theory of coherent absorption within a non-resonant monolayer BP. It is shown that the necessary formation condition of the coherent absorption can be well achieved at quasi-CPA wavelength by proper selection of the BP parameters. The CPA can be obtained at the quasi-CPA wavelength with an appropriate initial phase difference between the two coherent beams. It is found that the quasi-CPA wavelength can be manipulated from infrared to THz by changing the doped electron concentration. In addition, the absorptivity can be manipulated continuously by changing the relative phase of the input beams. Moreover, this work shows that the CPA can be realized for both TM and TE polarization with the incident angle varying from 0° to 70°. We believe that this work has potential applications in tunable terahertz/infrared detections and signal modulations.

## 2. Theoretical model and method

It is known that the coherent characteristic of the laser light enables to realize CPA in a sub-wavelength film. In this work, we employ a monolayer BP which has a much thinner thickness than the sub-wavelength film to achieve CPA in terahertz/infrared regime. As shown in Fig. 1, a monolayer BP is placed on the *x-o-y* plane, *I _{+}* and

*I*are the two counter-propagating coherent input beams illuminated on the monolayer BP,

_{-}*O*and

_{+}*O*stand for the output beams respectively. This structure can be considered as a standing wave system. The realization of the CPA depends on the position of the BP in the standing wave system. Since the BP is placed at a node of the standing wave, the amplitude of the electromagnetic field remains zero. Most of the energy will be scattered due to the weak interaction between BP and the incident light. At the antinode, however, perfect absorption can be achieved for the intense interaction of BP and light [23].

_{-}To describe the propagation of light in the proposed structure, a transfer matrix can be derived. The relationship between the complex scattering coefficients (*O* _{±} ) and input beams (*I* _{±} ) can be expressed as [21]

*r*/

*t*)

_{+}and (

*r*/

*t*)

_{-}are scattering elements along the z

_{+}direction for

*I*and along the z

_{+}_{-}direction for

*I*, respectively.

_{-}*A*= |

*I*| and

_{+}*B*= |

*I*|are the amplitude of the forward and backward incident light. For simplification, here we only consider the phase modulation of CPA. As a result, the amplitude of coherent beams can be considered the same (

_{-}*A*=

*B*=

*I*).

*φ*

_{+}and

*φ*

_{-}stand for the phase of the input coherent beams, and ∆

*φ*=

*φ*

_{+}-

*φ*

_{-}is the initial phase difference between the two beams. The scattering matrix can be simplified by assuming

*t*=

*t*and

_{±}*r*=

*r*due to the symmetry of the structure. Finally, the amplitude of the scattering coefficients can be derived as [22]:

_{±}To demonstrate a CPA, the scattering beams have to be eliminated, i.e. |*O _{+}*| = |

*O*| = 0. According to Eq. (2), the necessary condition for CPA phenomenon can be obtained:

_{-}*tIe*= -

^{iφ+}*rIe*[21, 22], namely,

^{iφ-}The complex surface conductivity (*σ _{jj}*) of a monolayer BP can be described using a simple semi-classical Drude model [26, 27]:

*j*=

*x*,

*y*indicates the direction of conductivity,

*ω*and

*ħ*are the frequency of incident light and reduced Planck’s constant, respectively. In this paper, the electron doping and relaxation rate are chosen as

*n*= 4*10

^{13}cm

^{−2}and

*ƞ*= 10 meV.

*D*and

_{j}*m*present the Drude weight and the electron’s effective mass along the

_{j}*x*and

*y*direction, respectively [29,30]. As a result, the electron’s effective mass can be described as:

For a monolayer BP, we choose ∆ = 2eV, *ƞ _{c}* =

*ħ*/(0.4

*m*

_{0}),

*ν*=

_{c}*ħ*/(1.4

*m*

_{0}),

*γ*= 4

*a/π*eVm, where

*m*

_{0}is the static electron mass,

*a*is thickness of the BP and

*π/a*is the width of the Brillouin Zone.

It is well known, the surface current can be excited while the BP is irradiated by an incident light, and it only depends on the dynamic surface conductivity [28–31]. Therefore, the scattering coefficients of a normal incident light can be obtained using the surface conductivities for the negligence of the high-order scatterings in deep-subwavelength BP [22, 32], namely,

where*ƞ*

_{0}is the wave impedance of free space. So far, the scattering coefficients

*r*and

*t*of a monolayer BP have been derived, and hence the necessary condition for CPA in a monolayer BP can be calculated based on Eqs. (3), (6) and (7).

## 3. Results and discussions

To better illustrate the formation condition of the CPA with a monolayer BP, Fig. 2 shows the reflection coefficients and transmission coefficients with a monolayer BP illuminated by a single-propagated beam. Considering the in-plane anisotropic features of BP, the normal incident light should be decomposed into two states in the *x-o-y* plane, corresponding to *x* direction and *y* direction, respectively. In Fig. 2(a), as to the electric vector $(\overrightarrow{E})$ along *x* direction, we can see a cross point at 39μm, namely quasi-CPA point, which indicates the necessary condition of CPA according to Eq. (3). For *y* direction, however, the quasi-CPA does not occur even with the optimal BP parameter is optimized, indicating that realizing CPA in *y*-direction is difficult. Therefore, in the following paragraphs we will only discuss the coherent absorption for the electric vector $(\overrightarrow{E})$ along *x* direction. Once the monolayer BP is irradiated by two coherent input beams, which have the same intensity and no phase difference, the scattering fields can be suppressed due to the enhanced interaction between light and the BP, which leads to an enhanced absorption of the monolayer BP. Furthermore, as shown in Fig. 2(b), the scattering fields will be suppressed completely at the quasi-CPA wavelength after the phase difference is modulated to ∆*φ* = 1.402π, where a CPA is achieved. It can be seen that the CPA wavelength is well matched with the analysis of the formation condition of CPA based on Eq. (3). Moreover, the CPA in monolayer BP remains higher than 90% with the wavelength ranging from 18.5 to 107.5 μm. The relative bandwidth is defined as (*λ*_{max}−*λ*_{min})/*λ*_{0}, where *λ*_{max}, *λ*_{min}, and *λ*_{0} are the maximum, minimal, and center wavelength, respectively. The relative bandwidth of the CPA in a monolayer BP is as high as 141.3%, which is greatly improved as compared to the coherent absorption in a typical metasurface [14, 33]. It is shown that the BP with phase modulation achieves a better absorption performance as compared to the ones without phase modulation. Therefore, it can be concluded that the phase modulation plays an important role in the CPA and can offer great advantages for coherent perfect absorber to modulate the absorptivity with flexibility.

The modulation of the CPA is a fundamental and important property in applications where the intensity of the output signal can be selected with flexibility. It is known that a standing wave system can be formed with two coherent beams illuminating from both sides of a monolayer BP. The enhanced absorption or transmission depends on the position of the BP in the standing wave system, where the enhanced absorption at an antinode and enhanced transmission at a node. In fact, the changing in BP position is another form of phase modulation because the adjustment of the position will lead to a change in optical path difference, which has a direct relationship to phase difference. To further investigate the phase modulation of the CPA with a monolayer BP, the normalized total absorbed intensity spectra as a function of phase difference and wavelength is plotted in Fig. 3(a). It is shown that the CPA has a strong dependence on the phase difference of the two input coherent beams. The normalized total coherent absorption (*A*_{c}) is obtained by *A*_{c} = 1-*SI*, where *SI* is the normalized total output intensity, given by the square of the scattering amplitudes (*SA*) denoted by *SA* = |*r* + *t*|. It can be seen that the CPA phenomenon at the quasi-CPA wavelength (*λ* = 39μm) occurs with a proper phase modulation (∆*φ* = 1.402π). The corresponding normalized absorption at quasi-CPA wavelength as a function of phase modulation (∆*φ*) is shown in Fig. 3(b). A substantial modulation of absorbed intensity can be demonstrated by the phase difference tuned from 0 to 2*π*. The realization of CPA for ∆*φ* = 1.402π indicates that the input beams are prevented to escape from the monolayer BP due to the destructive interference. Normally, the modulation depth is used to weigh the performance of modulation in a CPA, which is defined as *M*(*λ*) = max(*SI*)/min(*SI*) to. According to the result in Fig. 3(b), an excellent modulation can be realized with *M*(*λ* = 39μm) = 10^{5}, where the destructive suppression of the fields is achieved in the monolayer BP.

To further study the coherent absorption performance of the monolayer BP under oblique incidence, the coherent absorption peak and the CPA wavelengths of the monolayer BP at various angles of oblique incidence for both TE and TM polarizations are plotted in Fig. 4. As shown in Fig. 4(a), the CPA wavelengths of the coherent absorption peak show opposite variations and can split into two wavelength branches ranging from THz to infrared for TE and TM polarizations, respectively. The CPA wavelength shows a red shift from 39μm to 107.1μm for TM polarization while the incident angle increases from 0° to 70°, and TE polarization shows a blue shift from 13.75μm to 39μm. This means that the proposed monolayer BP can be utilized for the realization of the broadband coherent modulation with angular tunability. More importantly, it can be seen in Fig. 4(b) that the peak absorptivity for both polarizations keep being more than 96.5% as the incident angle increases. Especially, the coherent absorption reaches up to 99.9% while the incident angle ranging from 0° to 30°. Such feature makes it more efficient in the application of optical switching and signal processing.

As is well known that the charge-carrier density has a significant influence to the conductivity of graphene, and such an impressive characteristic has been widely used to design tunable optoelectronic devices. Similar to graphene, the electron doping, referred in the introduction of the conductivity in a monolayer BP, also has critical effects to the CPA realization. As shown in Fig. 5(a), CPA can also be achieved at various wavelengths with proper phase difference. As the electron doping increases, the CPA wavelength has a blue shift. The range of achieved CPA wavelengths indicates that the proposed BP-based coherent perfect absorber can operate in both THz and infrared band without changing the intrinsic parameters of the BP. In order to assess the modulation capability, Fig. 5(b) shows the corresponding modulation depth spectra for different electron doping. It is seen that the modulation depth remains more than 10^{4} while the electron doping is changing from 3*10^{13}cm^{−2} to 6*10^{13}cm^{−2}. Therefore, it is very convenient to realize CPA with high-efficiency modulation at any wavelengths by changing the electron doping.

## 4. Conclusions

In conclusion, we have analyzed and demonstrated in theoretical level of achieving CPA using a monolayer BP at the THz/infrared band. We have shown that the electromagnetic energy of two counter-propagating waves from both sides of the monolayer BP can be absorbed perfectly, and the coherent absorption can also be modulated with high flexibility by changing the phase difference of the two coherent beams. Meanwhile the angular selectivity of the monolayer BP is studied, which shows the feasibility of the CPA under oblique incidence circumstances. Moreover, it is shown that the CPA wavelengths of the coherent absorption peak exhibit opposite variations and can split into two wavelength branches for TE and TM polarizations. Furthermore, the CPA wavelength can be tuned from THz to infrared band by adjusting the electron doping, and the modulation depth can be maintained as more than 10^{4} at the same time. We believe that this work has potential applications for the coherent modulations in terahertz/infrared detections and signal processing with 2D materials.

## Funding

National Natural Science Foundation of China (Grant Nos. 61505111, 11604216 and 61505117; the Science and Technology Planning Project of Guangdong Province (Grant No. 2016B050501005); the Guangdong Natural Science Foundation (Grant No. 2015A030313549); Science and Technology Innovation Commission of Shenzhen (Grant No. CYJ20170302153323978); Natural Science Foundation of SZU (Grant No. 2017021).

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