DOI:10.30919/esee8c396

Received: 09 Mar 2020
Revised: 02 Apr 2020
Accepted: 03 Apr 2020
Published online: 11 May 2020

Chiral Absorbers Based on Polarization Conversion and Excitation of Magnetic Polaritons

Xiaohu Wu, Ceji Fu,a,[1] and Zhuomin M. Zhangb,[2]

aLTCS and Dept. of Mechanics & Engineering Science, College of Engineering, Peking University, Beijing 100871, China

bGeorge W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA

 

[1] Corresponding author, Email: cjfu@pku.edu.cn (C.J. Fu)

[2] Corresponding author, Email: zhuomin.zhang@me.gatech.edu (Z.M. Zhang)

 

Abstract

Many chiral absorbers have been proposed to obtain selective absorption for left circularly polarized (LCP) and right circularly polarized (RCP) waves. However, it is difficult to realize tunable chiral absorbers with chiral response for the incidence angle in a wide range. Here we show that strong chiral response can be realized by using a simple structure composed of a uniaxial slab and a silver grating. The numerical results show that the absorptance of the proposed structure for incidence of LCP wave can be close to unity while the absorptance for incidence of RCP wave can stay close to zero at the resonance wavelength of magnetic polaritons (MPs) in the grating grooves. More importantly, the circular dichroism (CD) of the structure is shown to remain larger than 0.7 for all azimuthal angles and for the incidence angle from 0o up to 30o. In addition, this structure can also be engineered to work in a broadband or in different wavelength ranges. Owing to its simple configuration, this device holds promise for polarization-sensitive surface photochemistry and chiral bolometers.

Table of Content

A quarter-wave plate combined with a one-dimensional grating structure is proposed for realizing strong and tunable chiral absorption.

 

 

Keywords: chiral absorber, uniaxial slab, polarization conversion, grating, magnetic polariton


1  Introduction


Achieving perfect absorption of electromagnetic waves is of vital importance in many practical applications, such as energy harvesting, radiation detection, chemical sensing and radiative cooling.[1] Absorption for incidence of linearly polarized waves, such as transverse magnetic (TM) and transverse electric (TE) waves, has been investigated extensively. Taking a radiative cooling device as an example, its working principle is to achieving emission for TM and TE waves as strong as possible in the atmospheric transparency window (8-13 μm), while suppressing absorption as much as possible in the AM1.5 solar spectrum.[2] Absorbers for circularly polarized waves, which can selectively absorb left circularly polarized (LCP) wave or right circularly polarized (RCP) wave, are called chiral absorbers.[3, 4] Chiral absorbers are useful tools in biochemistry for identification of essential elements such as DNA and proteins, which can help reveal the mysteries of nature.[5]

The characteristic of a chiral absorber can be described by its circular dichroism (CD), which is defined as the difference between absorptance for incidence of RCP and LCP waves. Over the past several years, strong chirality has been demonstrated with various metamaterials in different spectral ranges.[6-21] Lee and Chan[6] proposed layer-by-layer photonic crystal structures, which exhibited strong chirality due to the multiple polarization gaps. Wang et al.[7] proposed a design by combining two layers of anisotropic metamaterial structures, which can absorb RCP wave and reflect LCP wave simultaneously. Pan et al.[8] proposed refractory metamaterials composed of molybdenum zigzag arrays, which can work at high temperatures. Ouyang et al.[9] proposed a plasmonic metasurface consisting of a three-layer meta-dielectric-metal structure, which can achieve a maximum CD of 0.7. Khandekar and Jacob[22] proposed a strategy for realizing chiral thermal emission with coupled anisotropic antennas at thermal non equilibrium, from which the thermal photon spin[23,24] was controlled by the nonequilibrium temperature of the antennas. However, all of these structures demand complex fabrication processes. Therefore, it is important to figure out a novel approach to achieve strong chiral response with easy fabrication.
Most of the studies on the chirality are for normal incidence without considering the directional dependence. Dyakov et al.[17] recently demonstrated that the CD can be as high as 0.87 at normal incidence with their proposed chiral metasurface; however, the CD value decreases dramatically as the incidence angle increases even by 5°. The modulation of chiral response has also received some attention. Zhu et al.[19] and Rodrigues et al.[20] demonstrated that the modulation can be achieved by exciting the nonlinear optical response of the material. Their approach, however, requires an incident light with a very high intensity in order to induce nonlinear optical response, which may limit its applications.
Here we propose to achieve giant chiral response by combining a uniaxial slab with a metallic grating. The CD can reach as high as 0.99, while the chiral response can be tailored by controlling the orientation of the optic axis of the uniaxial slab. In addition, the CD can be greater than 0.7 for all azimuthal angles when the incidence angle is smaller than 30°. The uniaxial slab can be easily achieved with widely used polymers and mature mechanical methods in industry,[21,25,26] and the metallic grating can be fabricated by electron-beam lithography (EBL).[27]


2.  Modelling


The structure proposed in this work is shown in Fig. 1, which consists of a one-dimensional (1D) silver (Ag) grating sandwiched by a slab of uniaxial material and an Ag substrate. The permittivity tensor of the uniaxial slab can be expressed in its principal coordinate system oxyz as[28]


 
Fig. 1  Schematic of the proposed structure in this work. The optic axis (OA) of the top uniaxial slab is in the x-y plane of the coordinate system and is tilted off the x-axis by an angle β. The grating grooves are along the y-axis and ϕ denotes the azimuthal angle between the plane of incidence and the x-z plane. θ is the incidence angle.


            
 

when its optic axis (OA) is along the x-axis. In this work, we set "ε"(||)=2.25 and "ε"=2.56 because such kind of material can be easily obtained with widely used polymers and matured mechanical methods in industry.[21,25,26] Besides, the imaginary parts of the permittivity components are very small and will not influence the results too much.[21] Hence, we do not consider optical loss in the uniaxial slab. When the slab is rotated around the z-axis such that its OA forms an angle β with the x-axis in the x-y plane,the permittivity tensor of this slab is changed to[28]


                                     (2)


It is well known that an uniaxial slab can be used to manipulate the polarization state of the light if its permittivity tensor has nonzero off-diagonal elements.[29] As such, the slab in this work should be set with a nonzero value of sin⁡βcos⁡β in order to fulfil the effect of polarization conversion, as can be seen from Eq. (2). In fact, the effect of polarization conversion is the strongest when sin⁡βcos⁡β is maximized.[28] Hence, β is taken as 45^o and in this case, a quarter-wave plate with thickness of λ/(4Δn) can convert a circularly polarized wave into a linearly polarized wave at the central wavelength λ, where and is equal to 0.1 in this work. The function of this uniaxial slab is to convert a circularly polarized wave into a linearly polarized wave.
The Ag grating characterized by period Λ, thickness d2, and filling ratio f can be used to achieve selective absorption for transverse electric (TE) and transverse magnetic (TM) waves through excitation of magnetic polaritons (MPs) in the grating grooves.[30-32] Here, the electric field vector of TE wave is perpendicular to the plane of incidence, while the magnetic field vector of the TM wave is perpendicular to the plane of incidence. The dielectric function of Ag is described using the Drude model as


ε_Ag=ε_∞-(ω_p^2)/(ω^2-jωΓ)            (3)


where ε=3.4, ωp=1.39×10^16 rad/s, and Γ=2.7×10^13 rad/s.[30] The simulation is based on the rigorous coupled-wave analysis (RCWA).[33-35] The conventional RCWA algorithm is focused on incidence of linearly polarized waves, rather than circularly polarized waves. For an aid of easy understanding of the numerical method, a detailed description of the RCWA is presented in the Appendix for incidence of an arbitrarily polarized wave.


3.  Results and discussion


The chiral response originates from the combination effect of polarization conversion and excitation of magnetic polaritons (MPs). Here, we first investigate the absorption property of the Ag grating. The calculated absorptance of the grating for normal incidence of TE and TM waves is shown respectively in Fig. 2(a) for Λ=0.1 μm, f =0.97, d2=0.18 μm and ϕ=0°. As can be clearly seen, the absorptance for incidence of TM wave can reach almost unity at wavelength of 723 nm, while the absorptance at this wavelength for incidence of TE wave is close to zero. The corresponding electromagnetic field distribution in the grating for incidence of TM wave is depicted in Fig. 2(b), where the color contours denote the magnitude of the magnetic field and the arrows indicate the electric field. The outline of the grating is also illustrated in the figure for ease of reading. It can be seen that the magnetic field is greatly enhanced in three confined regions in the grating groove, and the electric field vectors form closed loops around the enhanced magnetic field regions. This is the typical pattern of MPs of multiple orders excited in the grating groove.[31] The enhanced electromagnetic (EM) fields due to excitation of MPs result in the large absorption in the grating at the resonance wavelength. In fact, the resonance wavelength of MPs depends on the geometric parameters such as d2 and f of the grating, thus it can be easily tailored by changing the values of these parameters. Variation of the resonance wavelength with these parameters can be effectively predicted with the LC circuit model.[30-32]


 


Fig. 2  (a) The absorptance of the structure varying with wavelength for normal incidence of TM and TE waves in the x-z plane. (b) The electromagnetic field distribution in the Ag grating indicating excitation of MPs at wavelength 723 nm for incidence of TM wave. The color contours show the relative magnitude of the magnetic field, and the vectors denote the electric field.


Note that the absorptance shown in Fig. 2(a) is only for incidence in the x-z plane, which corresponds to the azimuthal angle ϕ=0^o. The effect of the azimuthal angle ϕ on the absorptance for normal incidence of TM and TE waves at wavelength of 723 nm is shown in Fig. 3. One can see that the absorptance for incidence of TM wave gets its maxima (close to unity) when ϕ equals 0° and 180°, and it reaches its minima (close to zero) at ϕ equal to 90° and 270°. This is because excitation of MPs in the grating grooves requires a time-varying magnetic field along the groove direction. When ϕ equals 0° or 180°, the magnetic field of an incident TM wave is exactly in the groove direction, which then can excite MPs. When ϕ deviates from 0° or 180°, MPs can still be excited by the nonzero component of the magnetic field along the groove direction. But this non-zero component is getting smaller and smaller as ϕ deviates farther and farther from 0° or 180°, which results in weaker and weaker MPs excited in the grooves. Eventually, the magnetic field of an incident TM wave is perpendicular to the groove direction when ϕ equals 90° or 270°, and MPs cannot be excited in this case. Thus, the absorptance is very small. On the contrary, the absorptance for incidence of TE wave gets its minima when ϕ equals 0° or 180°, at which the magnetic field is perpendicular to the groove direction. A nonzero component of the magnetic field parallel to the groove direction appears when ϕ deviates from 0° or 180°, and it becomes the largest at ϕ equal to 90° or 270°, at which the absorptance reaches its maximum due to excitation of MPs.
 


Fig. 3  The absorptance of the Ag grating varying with the azimuthal angle ϕ for normal incidence of TM and TE waves.


At wavelength of 723 nm, the thickness of a quarter-wave plate made of the uniaxial slab is 1.81 µm. We now investigate the transmission property of a single uniaxial slab of thickness 1.81 µm. When a circularly polarized wave passes the uniaxial slab, the transmitted wave is a linearly polarized one which includes both TM and TE components. Accordingly, the transmittance T can be expressed as T=T_TM+T_TE, where the subscripts indicate contribution from the two components. The values of T, TTM and TTE varying with the azimuthal angle ϕ for normal incidence of LCP and RCP waves at wavelength of 723 nm on the quarter-wave plate are shown in Figs. 4(a) and 4(b), respectively. As can be seen that the transmittance T is the same, but TTM and TTE contributed from the TM and TE components are different for incidence of LCP and RCP waves. Particularly, an incident LCP wave will be mostly converted into a TM wave at ϕ equal to 0° and 180°, while it will be mostly converted into a TE wave at ϕ equal to 90° and 270°. In contrast, an incident RCP wave will be mostly converted into a TE wave at ϕ equal to 0° and 180°, while it will be mostly converted into a TM wave at ϕ equal to 90° and 270°. From the results in Figs. 3 and 4, a straightforward way to understand the physical mechanism for chiral absorption with the structure can be outlined. For normal incidence of a LCP wave at azimuthal angle ϕ=0°, it will be mostly converted into a TM wave by the uniaxial slab, which then can excite MPs in the Ag grating. However, a normally incident RCP wave at ϕ=0° will be mostly converted into a TE wave by the uniaxial slab, which then cannot excite MPs in the Ag grating. Consequently, the incident LCP wave is greatly absorbed by the structure while the absorption is small for the incident RCP wave.
 


Fig. 4  The transmittance T, TTM and TTE varying with the azimuthal angle ϕ under normal incidence of (a) a LCP wave and (b) a RCP wave.


In order to justify the above analysis, the absorptance of the whole structure varying with wavelength under normal incidence of LCP and RCP waves in the x-z plane is shown in Fig. 5(a). Clearly, the absorptance for incidence of a LCP wave is different from that for incidence of a RCP wave in the wavelength range between 700 and 800 nm. Especially, the CD, defined as |αLCPRCP) | with αLCP) and αLCP) representing the absorptance for incidence of LCP and RCP waves, is as high as 0.91 at wavelength 739 nm. Based on the above analysis, we attribute the large absorptance for incidence of a LCP wave at this wavelength to excitation of MPs in the grating grooves. However, the resonance wavelength has a redshift from the predicted value 723 nm. The reason for this redshift is that the adjacent media of the Ag grating is the uniaxial slab, instead of air for obtaining the results in Fig. 2. Existence of the uniaxial slab may alter the charge distribution on the groove walls and thus the effective capacitance formed by the groove walls in the LC circuit model,[32] which affects the resonance wavelength of MPs in the grooves. In fact, if the grating is separated from the uniaxial slab by an air gap of width greater than 10 nm, the resonance wavelength of MPs will shift back to 723 nm. This is clearly demonstrated in Fig. 5(b), in which the absorptance of the structure as a function of wavelength for incidence of a LCP wave in the x-z plane for the value of the air gap width ranging from 0.1 nm to 100 nm.
 


Fig. 5  (a) The absorption of the LCP and RCP waves as well as the circular dichroism as a function of the wavelength. (b) The absorption of the LCP wave as a function of wavelength with different distance between the uniaxial slab and the grating.


Most importantly, the chiral absorptance of the structure shown in Fig. 5(a) is independent on the azimuthal angle ϕ. In other words, for the plane of incidence rotated around the z-axis by an arbitrary angle ϕ, as shown in Fig. 1, the chiral absorptance of the structure will remain unchanged and is the same as in Fig. 5(a) for normal incidence of LCP and RCP waves. We show this property below by applying the Jones vector and the Jones matrix. When the plane of incidence is the x-z plane, the polarization state of an incident LCP wave can be described by the Jones vector of the electric field as[26]


         (4)


Likewise, the polarization state of an incident RCP wave can be expressed as
 

         (5)


When the plane of incidence is rotated around the z-axis off the x-z plane by an angle ϕ, the Jones vectors of the incident LCP and RCP waves in Eqs. (4) and (5) should be modified respectively by multiplying the rotation matrix as


                   (6)


and


           (7)


Since the quarter-wave plate is arranged with β=45^o, the Jones matrix of the quarter-wave plate can be written as[26]


         (8)


by assuming perfectly transmission. As a consequence, the Jones vectors of the transmitted waves can be obtained by multiplying the Jones vectors of the incident waves respectively with the Jones matrix of the quarter-wave plate as


            (9)


and


      (10)


Equations (9) and (10) indicate that at any given angle ϕ, the uniaxial slab converts the incident LCP and RCP waves into a linearly polarized wave with the electric field along the x-axis and a linearly polarized wave with the electric field along the y-axis, respectively. In other words, the incident LCP wave is converted into a linearly polarized wave with its magnetic field vector always parallel to the grooves of the grating, while the incident RCP wave is converted into a linearly polarized wave with its magnetic field vector always perpendicular to the grooves of the grating. Therefore, MPs can be excited by an incident LCP wave but cannot be excited by an incident RCP wave, resulting in the chiral absorptance shown in Fig. 5(a).
We have seen from the above analysis that a quarter-wave plate converts a LCP wave or a RCP wave into a linearly polarized wave, which is critical for achieving the chiral absorptance shown in Fig. 5. No chiral absorptance can be achieved by direct incidence of LCP and RCP waves on the grating, without the uniaxial slab. In order to reveal the effect of the uniaxial slab thickness on the chiral absorptance of the structure, we have calculated the absorptance of the structure and the CD as functions of the uniaxial slab thickness d1 for normal incidence of LCP and RCP waves at wavelength 739 nm. The results are shown in Fig. 6(a). It can be clearly seen from the figure that the absorptance varies periodically with the uniaxial slab thickness d1 for incidence of either a LCP wave or a RCP wave. When d1 is zero, i.e., without the uniaxial slab, the absorptance is the same for incidence of both LCP and RCP waves and chiral absorptance does not occur in this case. As the value of d1 is greater than zero, the absorptance for incidence of LCP wave begins to deviate from that for incidence of RCP wave, resulting in chiral absorptance. Note that there are a series of sharp peaks and troughs seen in the absorptance curves, which are caused by the effect of wave interference inside the slab. Because we have figured out that the distance Δd1 between two successive peaks or troughs is about 240 nm, which agrees excellently with the value determined from the relation
[29]  for wave interference inside the slab at wavelength 739 nm. Without wave interference, the absorptance for incidence of LCP wave would increase monotonically to a maximum while the absorptance for incidence of RCP wave would decrease monotonically to a minimum as d1 increases from zero to 1.85 μm. Because the wavelength of the incident LCP and RCP waves is assumed as 739 nm, a uniaxial slab of thickness 1.85 μm is a quarter-wave plate and can convert both the LCP and the RCP waves into linearly polarized waves. The magnetic field vector of the linearly polarized wave converted from the LCP wave is parallel to the grating grooves, by which MPs in the grating grooves can be excited to enhance significantly the absorptance. However, the magnetic field vector of the linearly polarized wave converted from the RCP wave is perpendicular to the grating grooves, by which MPs cannot be excited and the absorptance is small in this case. As a consequence, a large value of CD is obtained. If the value of d1 is smaller than 1.85 μm, the polarization state of the converted wave is generally elliptical, rather than linear. In this case, either a LCP wave or a RCP wave incidence can excite MPs in the grating grooves, since the converted wave (by the uniaxial slab) always has a magnetic field component parallel to the grating grooves. It is interesting to find that the value of CD can be enhanced by the effect of wave interference. For example, as can be seen from Fig. 6(a) that the CD can be as high as 0.99 when the thickness of the uniaxial slab is 1.29 µm, which is comparable to the best result in literature.[7] However, this result is sensitive to the slab thickness and contrasts with the result around d1=1.85 μm. The combined effects of wave interference and polarization conversion result in the oscillation envelope of the absorptance possessing a minimum around d1=1.85 μm, making the absorptance curve appear quite flat around this value of the slab thickness. This explains the large CD obtained at wavelength 739 nm using a uniaxial slab of thickness 1.81 μm, a quarter-wave plate for wavelength 723 nm. The absorptance of the structure for LCP wave incidence will decrease when the slab thickness d1 increases larger than 1.85 μm, while the corresponding absorptance for RCP wave incidence will increase, as shown in Fig. 6(a). Accordingly, the CD will decrease and is equal to zero at d1=3.7 μm, at which the incident LCP wave is converted by the uniaxial slab into a RCP wave, while the incident RCP wave is converted into a LCP wave. The absorptance is the same for incidence of both LCP and RCP waves; thus, the CD is zero in this case. From the results shown in Fig. 6(a), the period of the CD varying with the slab thickness d1 is 3.7 μm, which is half of the period of the absorptance, according to the definition of the CD.


 
Fig. 6  (a) The absorptance and the CD of the structure for normal incidence of LCP and RCP waves as a function of the thickness of the uniaxial slab, (b) the absorptance and the CD of the structure for normal incidence of LCP and RCP waves as a function of the tilting angle.


In addition, since the tilting angle β of the optic axis of the uniaxial slab has great influence on polarization conversion,[25] the influence of the tilting angle β on the CD was studied at wavelength 739 nm when d1 is set at 1.81 μm. The calculated absorptance for incidence of LCP and RCP waves along with the CD varying with the tilting angle β is shown in Fig. 6(b). One can see that there is no chiral absorptance when the tilting angle is 0°, 90°and 180° due to no conversion of polarization. However, the CD obtains its maximum when the tilting angle is equal to 54° and 126°, where the absorptance for LCP wave incidence is at its maximum while that for RCP wave incidence is at its minimum, or vice versa. Interestingly, the values of the tilting angle for maximum CD deviate appreciably from those for maximum conversion of polarization with only one uniaxial slab, i.e., 45° and 135°, when the uniaxial slab is combined with the grating. Nevertheless, these results clearly indicate that chiral absorptance of the structure is tunable via changing the tilting angle. Compared to the modulation of chiral response realized by manipulation of the intensity of the incident light,[19,20] this method does not have restriction on the intensity of the incident light and is more flexible.
Contour plots of the absorptance varying with the azimuthal angle and the incidence angle are shown in Figs. 7(a) and 7(b) for incidence of LCP and RCP waves at wavelength 739 nm, respectively. It is interesting to find that the absorptance for incidence of both LCP and RCP waves is insensitive to the azimuthal angle when the incidence angle is smaller than 30°. Furthermore, the absorptance for incidence of LCP wave stays higher than 0.8 around 0° and 180° of the azimuthal angle ϕ even when the incidence angle θ is greater than 60°. This is because the magnetic field vector of the converted wave by the uniaxial slab is always aligned with the groove direction at ϕ equal to 0° and 180°, regardless of the incidence angle. In contrast, the absorptance for incidence of RCP wave remains lower than 0.2, except when the incidence angle is greater than 80° does the absorptance increase to be higher than 0.2 around 45° and 225° of the azimuthal angle. As a consequence, the CD of the structure is insensitive to the azimuthal angle and is higher than 0.7 when the incidence angle is smaller than 30°, while around 0° and 180° of the azimuthal angle such high CD value can persist for the incidence angle up to 60°, as shown in Fig. 7(c). Such high performance enables the proposed device an ideal light source generating circularly polarized waves.[36, 37]
 


Fig. 7  (a) The absorptance of the structure for incidence of a LCP wave, (b) the absorptance of the structure for incidence of a RCP wave, and (c) the CD of the structure varying with the azimuthal angle and the incidence angle at wavelength 739 nm.


As shown in Fig. 5(a), the strong chiral absorptance cannot occur in a broadband. The main reason is that the strong absorption due to the excitation of MPs in the structure under study is highly dependent on wavelength. In order to broaden the chiral absorptance band, many methods can be employed by exciting multiple MPs with different structures such as pyramid nanostructures,[38] multi-sized structures,[39] and L-shaped structures,[40] etc. Here, we adopt a simple method to achieve this goal with a two-groove grating.[41] As shown in Fig. 8 of the cross section view, the two-groove grating consists of two grooves of different depths in one unit cell, where the distance between the two grooves is D. By exciting MPs at different resonant wavelengths in the two grooves, large absorptance in a broader band can be achieved. For illustrative purpose, the grating period is changed to Λ=400 nm, and the other parameters are taken as b1=b2=5 nm, h1=220 nm, h2=200 nm and D=200 nm. The grating is thick enough for it to be assumed opaque.
 


Fig. 8  The schematic of the two-channel grating, the distance between the two channels is D.


The calculated absorptance of the two-groove grating as a function of wavelength for normal incidence of a TM wave in the x-z plane is shown in Fig. 9(a). Two peaks can be clearly seen in the absorptance curve with one at 2.97 µm and the other at 2.76 µm, which correspond to excitation of MPs in grooves of depths 220 nm and 200 nm, respectively. Because the two peaks are close to and are partially overlapped with each other, the band for high absorptance is broadened compared to the case of only one peak. Adding a uniaxial slab on top of this two-groove grating, chiral absorptance can be obtained by setting properly the thickness of the uniaxial slab according to the resonant wavelength of MPs. The numerical result is shown in Fig. 9(b) for the thickness of the uniaxial slab taken as d=7.15 μm, which corresponds to the quarter-wave plate thickness for the center wavelength between the two peaks shown in Fig. 9(a). It can be seen that the absorptance for incidence of a LCP wave is significantly enhanced in the waveband between the two peaks due to resonance of MPs, though the peak around 2.76 µm is lower than that in Fig. 9(a). In contrast, the absorptance for incidence of a RCP wave is small in the same waveband. This chiral absorptance results in the CD larger than 0.5 in the wavelength range from 2.79 μm to 3.17 µm, which is much broader than the case for only one peak. Based on the same idea, even broader bands for chiral absorption may be achieved with the grating having more grooves of different depths in one-unit cell. However, since the uniaxial slab does not guarantee complete conversion of LCP and RCP waves into linearly polarized waves for wavelength in a broad band, the enhanced absorptance as well as the CD may not be achieved with large values in the whole band.


 
Fig. 9  (a) The absorption of the two channel grating as a function of the wavelength. (b) The absorption of the LCP and RCP waves as well as the circular dichroism as a function of the wavelength for the combining structure of the uniaxial slab with the two channel grating.


It should be noted that a chiral absorber can also be realized with the same structure based on excitation of surface plasmon polaritons (SPPs) at the interface of the uniaxial slab and the grating. However, since resonance of SPPs depends strongly on the incidence angle,[42] a chiral absorber with large CD may not be realized in the range of the incidence angle as shown in Fig. 7. Finally, we emphasize that the prescribed parameters used in the calculations are only for the purpose of demonstrating the mechanisms for achieving large CD with the proposed structure at the given wavelengths. In fact, the chiral absorber can be made to work at different wavelengths by selecting proper values of the geometric parameters as well as proper materials for the structure.


4. Conclusion


A simple chiral absorber structure composed of a uniaxial slab and a metallic grating is proposed. We have shown that very large CD of the chiral absorber can be achieved for incidence of LCP and RCP waves, due to the combined effects of polarization conversion with the uniaxial slab and excitation of MPs in the grating grooves. More importantly, the chiral absorptance is insensitive to the azimuthal angle and the CD can remain higher than 0.7 for incidence angles up to 30°. In addition, we have shown that the band of the chiral absorber can be broadened by using a two-groove grating, where MPs can be excited at different wavelengths. Therefore, the proposed chiral structure may have potential applications in polarization-sensitive surface photochemistry and chiral bolometers.


Acknowledgements
X. Wu and C. Fu were supported by the National Natural Science Foundation of China (51576004) and Z. M. Zhang was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences (DE-SC0018369).


Appendix  The RCWA for arbitrarily polarized wave
Using RCWA to calculate the reflection and transmission of the grating structures has been discussed by several groups.[33-35] Here we would like to give a detailed description about this process for arbitrarily polarized wave in this paper.
A general elliptic polarization wave can be represented by the Jones vector[29]


            (A1)


where ψ is the angle between the electric field vector and the plane of incidence, and η is the phase difference between the electric fields perpendicular and parallel to the plane of incidence. The refractive index of the incident medium and the medium for the transmitted wave are n1 and n2, respectively. The incident electromagnetic fields can be written with reference to the xyz coordinate system as the following form[33]


                   (A2)


where


         (A3)


Here we consider the one layer grating of thickness of d first. The electric field in the incident medium (z<0) and the electric field in the medium for the transmitted wave (z>d) are given by


            (A4)
          (A5)


where R_i is the electric field amplitude of the ith reflected wave in the incident medium, T_i is the electric field amplitude of the ith transmitted wave in the medium for the transmitted wave. The wavevector components are


 

           (A6)
                           (A7)
        (A8)


where k0=2π/λ. In the grating region (0[33]


         (A9)
           (A10)


where Uxi, Uyi, Uzi, Sxi, Syi and Szi are the amplitudes of the i th space-harmonic fields. Substituting Eqs. (A9) and (A10) into the Maxwell equations, we can get the following differential equations


          (A11)


where the coefficient matrix A is


         (A12)


where E(1/εxx) is the matrix formed by the permittivity harmonic components of 1/εxx, with the i, p element being equal to , ,   and E(εzz) are defined by the same way as E(1/εxx). The inverse of the dielectric function is used for the sake of fast and guaranteed convergence according to the Fourier factorization rule.[35] Kx and Ky are diagonal matrices with the diagonal elements  and , respectively. I is the identity matrix and O is the zero matrix. Eq. (A11) can be solved by calculation of the eigenvalues and the eigenvectors associated with the (4n×4n) matrix A, where n is the number of harmonics retained in the field expansion. The space harmonics of the tangential magnetic and electric fields are then given by[33]


       (A13)


           (A14)


         (A15)


          (A16)


where q_i are eigenvalues of matrix A, for 00. w_(i,m) are the elements of the eigenvector matrix W of the matrix A. The quantities c_m^+ and c_m^- are the unknown constants to be determined by matching the tangential component of the electric and magnetic field at each interface.
We calculate the reflection and transmission coefficients by matching the tangential electric and magnetic field components at the boundaries. At z=0, the boundary conditions are


(c+c-)           (A17)


where


    (A18)


and at the bottom surface of the slab


           (A19)
where 
  is the eigenvectors of the matrix A. C^+ and C^- are vectors composed of the unknowns, X and V are diagonal matrices with the diagonal elements    and ,Fs and Fc are diagonal matrices with the diagonal elements sin⁡φi and cos⁡φi,i=1,2,3,...,n,  . and  are diagonal matrices with the diagonal elements  and , respectively. Z1 and Z2 are diagonal matrices with the diagonal elements  and , respectively.


We extend the above analysis to an arbitrary L-layer structure by matching the tangential electric and magnetic field components at each interface. Take incidence wave as TM wave for an example, all the boundary conditions are


        (A20)


          (A21)


    (A22)


where l=2,3,...,L. W_l, X_l and V_l have the same definition as W, X and V described above.
To preempt the numerical instability associated with the inversion of the matrix, we propose to adopt the enhanced transmittance matrix approach.[30] From Eq. (A22), one has

        (A23)


where  , ,


The matrix on the right in the product is well conditioned, and its inversion is numerically stable. Therefore, Eq. (A23) can be rearranged as


           (A24)


where I is the unit matrix and


       (A25)


We adopt the substitution  such that Eq. (A23) becomes


          (A26)


Putting Eq. (A26) into Eq. (A21) for l=L, we have


          (A27)


where


         (A28)


Repeating the above process for all layers, we obtain an equation of the form


           (A29)


We can solve Eq. (A29) for R_S,R_P and T_1, then the transmission coefficient vector T can be obtained as


           (A30)


The diffraction efficiencies are defined as[33]


          (A31)


          (A32)


Supporting Information
Not applicable


Conflict of interest
There are no conflicts to declare.

References
    [1] W. Li and S. H. Fan, Opt. Express, 2018, 26, 15995–16021.    
    [2] A.P. Raman, M.A. Anoma, L.X. Zhu, E. Rephaeli and S.H. Fan, Nature, 2014, 515, 540–544.    
    [3] M. Hentschel, L. Wu, M. Schaferling, P. Bai, E. P. Li and H. Giessen, ACS Nano, 2012, 6, 10355–10365.    
    [4] X. T. Kong, L. K. Khorashad, Z. M. Wang and A. O. Govorov, Nano Lett., 2018, 18, 2001–2008.    
    [5] Z. L. Wu, X. D. Chen, M. S. Wang, J. W. Dong and Y. B. Zheng, ACS Nano, 2018, 12, 5030–5041.    
    [6] J. C. W. Lee and C. T. Chan, Appl. Phys. Lett., 2007, 90, 051912.    
    [7] Z. J. Wang, H. Jia, K. Yao, W. S. Cai, H. S. Chen and Y. M. Liu, ACS Photonics, 2016, 3, 2096–2101.    
    [8] M. Y. Pan, Q. Li, Y. Hong, L. Cai, J. Lu and M. Qiu, Opt. Express, 2018, 26, 17772–17780.    
    [9] L. Ouyang, W. Wang, D. Rosenmann, D. A. Czaplewski, J. Gao and X. Yang, Opt. Express, 2018, 26, 31484–31489.    
    [10] E. Plum, Appl. Phys. Lett., 2016, 108, 241905.    
    [11] B. Tang, Z. Y. Li, E. Palacios, Z. Z. Liu, S. Butun and K. Aydin, IEEE Photonic. Tech. L., 2017, 29, 295–298.    
    [12] L. Q. Jing, Z. J. Wang, Y. H. Yang, B. Zheng, Y. M. Liu and H. S. Chen, Appl. Phys. Lett., 2017, 110, 231103.    
    [13] A. F. Najafabadi and T. Pakizeh, Sci. Rep., 2017, 7, 10251.    
    [14] Q. Wang, E. Plum, Q. L. Yang, X. Q. Zhang, Q. Xu, Y. H. Xu, J. G. Han and W. L. Zhang, Light: Sci. Appl., 2018, 7, 25.    
    [15] M. H. Li, L.Y. Guo, J. F. Dong and H. L. Yang, J. Phys. D: Appl. Phys., 2014, 47, 185102.    
    [16] E. Plum and N. I. Zheludev, Appl. Phys. Lett., 2015, 106, 221901.    
    [17] S. A. Dyakov, V. A. Semenenko, N. A. Gippius and S. G. Tikhodeev, Phys. Rev. B, 2018, 98, 235416.    
    [18] G. C. Zheng, J. Xie, S. L. Wang and E. J. Liang, J. Light Scatt., 2019, 31, 216–228.    
    [19] Y. Zhu, X.Y. Hu, Z. Chai, H. Yang and Q. H. Gong, Appl. Phys. Lett., 2015, 106, 091109.    
    [20] S. P. Rodrigues, S. F. Lan, L. Kang, Y. H. Cui, P. W. Panuski, S. X. Wang, A. M. Urbas and W. S. Cai, Nature Commun., 2017, 8, 14602.    
    [21] Y. R. Qu, Y. C. Shen, K. Z. Yin, Y. Q. Yang, Q. Li, M. Qiu and M. Soljacic, ACS Photonics, 2018, 5, 4125–4131.    
    [22] C. Khandekar and Z. Jacob, Phys. Rev. Applied, 2019, 12, 014053.    
    [23] T. V. Mechelen and Z. Jacob, Optica, 2016, 3, 118–126.    
    [24] C. Khandekar and Z. Jacob, New J. Phys., 2019, 21, 103030.    
    [25] H. W. Ren and S. T. Wu, Appl. Phys. Lett., 2003, 82, 22–24.    
    [26] K. Z. Yin, Z. Zhou, D. E. Schuele, M. Wolak, L. Zhu and E. Baer, ACS Appl. Mater. Interfaces, 2016, 8, 13555–13566.    
    [27] B. Zhao and Z.M. Zhang, J. Quant. Spect. Rad. Trans., 2014, 135, 81–89.    
    [28] X. H. Wu and C. J. Fu, J. Opt., 2018, 20, 075603.    
    [29] P. Yeh Optical Waves in Layered Media, Wiley, New York, 1988.    
    [30] B. Zhao and Z. M. Zhang, J. Quanti. Spectro. Radiat. Transf., 2014, 135, 81–89.    
    [31] L. P. Wang and Z. M. Zhang, Opt. Express, 2011, 19, A126–A135.    
    [31] Y. M. Guo, Y. Shuai and H. P. Tan, Opt. Express, 2019, 27, 21173–21184.    
    [32] M. G. Moharam, E. B. Grann, D. A. Pommet and T. K. Gaylord, J. Opt. Soc. Am. A, 1995, 12, 1068–1076.    
    [33] M. G. Moharam, D. A. Pommet, E. B. Grann and T. K. Gaylord, J. Opt. Soc. Am., 1995, 12, 1077–1086.    
    [34] L. Li, J. Opt. A: Pure Appl. Opt., 2003, 5, 345–355.    
    [35] K. Baek, D. M. Lee, Y. J. Lee, H. Choi, J. Seo, I. Kang, C. J. Yu and J. H. Kim, Light: Sci. Appl., 2019, 8, 120.    
    [36] C. Khandekar, Z. Li and Z. Jacob, arXiv: 1912.07177.    
    [37] Y. B. Liu, J. Qiu, J. M. Zhao and L. H. Liu, Opt. Express, 2017, 25, A980–A989.    
    [38] L. P. Wang and Z. M. Zhang, J. Opt. Soc. Am. B, 2010, 27, 2595–2604.    
    [39] Y. Bai, L. Zhao, D. Q. Ju, Y. Y. Jiang and L. H. Liu, Opt. Express, 2015, 23, 8670–8680.    
    [40] Q. Cheng, P.P. Li, J. Lu, X.J. Yu and H.C. Zhou, J. Quant. Spect. Rad. Trans., 2014, 132, 70–79.    
    [41] J. J. Greffet, R. Carminati, K. Joulain, J. P. Mulet, S. Mainguy and Y. Chen, Nature, 2002, 416, 61–64.    

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