Chirality of optical vortex beams reflected from an air-chiral medium interface

Fuping Wu; Zhiwei Cui; Shenyan Guo; Wanqi Ma; Ju Wang

doi:10.1364/OE.459024

1. Introduction

In recent years, optical vortex beams, or optical vortices, have attracted intensive attention due to their fascinating properties and wide potential applications [1]. It is well established that the complex amplitude of an optical vortex beam comprises a phase term $\exp (il\varphi )$, where $l$ is an integer often called the topological charge and $\varphi$ is the azimuthal angle [2]. Such an optical vortex beam exhibits several unique properties, including helical wavefront, phase singularity, and null intensity in the beam center [3]. More importantly, each photon of an optical vortex beam with topological charge $l$ carries ${\pm} |l |\hbar$ orbital angular momentum (OAM) in the direction of propagation [4], where $\hbar$ is the reduced Planck constant. In addition, a vortex beam propagating with helical wavefront is inherently chiral and possesses a handedness, twisting either to the right or to the left [5–8]. It is because of this unique property that optical vortex beams would produce chiroptical effects during their interactions with chiral matter and may be used to detect, identify, separate, and manipulate chiral substances [9–16]. As is well known, the chiroptical effects are not only related to the chirality of matter, but also closely related to the chirality of light. Chirality, the lack of mirror symmetry, plays an important role in understanding of the interactions between optical vortex beams with chiral matter.

There have been some studies on how the optical vortex beams interact with chiral matter [17–24]. An early theoretical study on the optical vortex interactions with chiral matter was carried out by Andrews et al. [17]. Later, an experimental investigation on the interactions of vortex beams with chiral molecules was conducted by Araoka et al. [18]. Subsequently, Wu et al. presented a T-matrix method to study the plasmon-induced strong interaction between chiral molecules and optical vortex beams, and proposed an experimental scheme to observe such an interaction [19]. Qu et al. examined the interactions of optical vortex beams with chiral spherical particles [20,21]. Forbes et al. systematically studied the interaction mechanism of optical vortex beams with chiral molecules and particles [11–13,22]. Ni et al., Wozniak et al., and Guo et al. investigated the interactions between optical vortex beams with chiral nanostructures and their applications [15,16,23,24]. In addition, Jiang et al. examined the spin Hall effect of optical vortex beams reflected from an air-chiral interface [25]. Their results showed that the intrinsic chiral asymmetry of the medium induce an asymmetric spin splitting of optical vortex beams upon reflection. Now an interesting question arises: how does the chirality of optical vortex beams change if the beams reflected from an air-chiral medium interface? In this paper, we address this question. To this end, we utilize a full vector model to derive the explicit analytical expressions for the electric and magnetic field vectors of the vortex beams reflected at an air-chiral medium interface and apply the results to examine their chirality.

The rest of the paper is organized as follows. In Section 2, the theory and formula for the problem considered are presented. In Section 3, some numerical results are presented and discussed. Finally, Section 4 concludes the paper.

2. Theory and formula

As illustrated in Fig. 1, let us consider the reflection of an optical vortex beam at an interface between air and a chiral medium. We assume that the $z$-axis of the global coordinate system $({x,y,z} )$ is normal to the interface of the chiral medium located at $z = 0$, and use the local coordinate systems $({{x_i},{y_i},{z_i}} )$ and $({{x_r},{y_r},{z_r}} )$ to describe the incident and reflected beams, respectively. It is well known that an optical beam can be decomposed into an infinite series of plane waves, which propagate at different incident angles. Here, we set the incident angle and reflection angle of the central-wave component ${\theta _i}$ and ${\theta _r}$, respectively. It is further assumed that the chiral medium is homogeneous and characterized by the relative permittivity, permeability, and chiral parameter $({{\varepsilon_r},{\mu_r},{\kappa_r}} )$. In such a medium with chiral asymmetry, an optical beam splits into a wave with right-circular polarization(R-CP) and a wave with left-circular polarization (L-CP) having different phase velocities and refraction angles. Following the work of Wang and Zhang [26], the reflection coefficients at the interface between air and a chiral medium are expressed as

(1)$${r_{pp}} ={-} \frac{{({1 - {g^2}} )\cos {\theta _i}({\cos {\theta_ + } + \cos {\theta_ - }} )- 2g({{{\cos }^2}{\theta_i} - \cos {\theta_ + }\cos {\theta_ - }} )}}{{({1 + {g^2}} )\cos {\theta _i}({\cos {\theta_ + } + \cos {\theta_ - }} )+ 2g({{{\cos }^2}{\theta_i} + \cos {\theta_ + }\cos {\theta_ - }} )}},$$

(2)$${r_{ps}} = \frac{{2ig\cos {\theta _i}({\cos {\theta_ + } - \cos {\theta_ - }} )}}{{({1 + {g^2}} )\cos {\theta _i}({\cos {\theta_ + } + \cos {\theta_ - }} )+ 2g({{{\cos }^2}{\theta_i} + \cos {\theta_ + }\cos {\theta_ - }} )}},$$

(3)$${r_{sp}} = \frac{{ - 2ig\cos {\theta _i}({\cos {\theta_\textrm{ + }} - \cos {\theta_ - }} )}}{{({1 + {g^2}} )\cos {\theta _i}({\cos {\theta_\textrm{ + }} + \cos {\theta_ - }} )+ 2g({{{\cos }^2}{\theta_i} + \cos {\theta_\textrm{ + }}\cos {\theta_ - }} )}},$$

(4)$${r_{ss}} = \frac{{({1 - {g^2}} )\cos {\theta _i}({\cos {\theta_ + } + \cos {\theta_ - }} )+ 2g({{{\cos }^2}{\theta_i} - \cos {\theta_ + }\cos {\theta_ - }} )}}{{({1 + {g^2}} )\cos {\theta _i}({\cos {\theta_ + } + \cos {\theta_ - }} )+ 2g({{{\cos }^2}{\theta_i} + \cos {\theta_ + }\cos {\theta_ - }} )}},$$

where ${r_{pp}},\,{r_{ss}},\,{r_{ps}}$, and ${r_{sp}}$ denote the reflection coefficients for parallel, perpendicular and crossing polarizations, respectively. Also, $g\textrm{ = }\sqrt {{\varepsilon _r}/{\mu _r}}$ and $\cos {\theta _ \pm }\textrm{ = }\sqrt {1 - {{\sin }^2}{\theta _i}/n_ \pm ^2}$, with ${\theta _ \pm }$ and ${n_ \pm } = \sqrt {{\varepsilon _r}{\mu _r}} \pm {\kappa _r}$ being the reflection angles and the refractive indices of waves with R-CP and L-CP, respectively.

Fig. 1. Illustration of an optical vortex beam reflected from an air-chiral interface.

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To study the chirality of optical vortex beams reflected from an air-chiral medium interface, it is necessary to carry out a full vector wave analysis of optical vortex beams upon reflection. As stated earlier, optical vortex beams comprise a phase term $\exp (il\varphi )$. A common example of such beams is the Laguerre-Gaussian(LG) beams [27], which are characterized by two mode numbers, denoted as $l$ and $p$, respectively, called the azimuthal and radial indices. Here, we choose the simplest form of LG modes with $p\textrm{ = }0$ as the incident beams. In this case, the angular spectrum of the LG vortex beams takes the form [28]

(5)$${\tilde{u}_i}({{k_{ix}},{k_{iy}}} )\textrm{ = }{\left( {\frac{{{w_0}}}{{\sqrt 2 }}} \right)^{|l |}}{[{ - i{k_{ix}} + \textrm{sign}(l ){k_{iy}}} ]^{|l |}}\frac{{w_0^2}}{{4\pi }}\exp \left[ { - \frac{{w_0^2({k_{ix}^2 + k_{iy}^2} )}}{4}} \right],$$

where ${w_0}$ is the beam waist radius of the beams at ${z_i}\textrm{ = }0,\,\textrm{sign}(l )$, is the sign function, ${k_{ix}}$ and ${k_{iy}}$ are the transverse components of the wave vector ${{\mathbf k}_i}\textrm{ = }{k_i}{\hat{{\mathbf z}}_i}$, and ${k_i}\textrm{ = }{k_0}\textrm{ = }2\pi /\lambda$ is the wave number in the air, with $\lambda$ being the wavelength of incident vortex beams. By employing the two-dimensional Fourier transform, the complex amplitude of the LG vortex beams corresponding to Eq. (5) can be represented as

(6)$${u_i}({{x_i},{y_i},{z_i}} )= {\left( {\frac{{\sqrt 2 }}{{{w_0}}}} \right)^{|l |}}{\left[ {\frac{{{x_i} + i\textrm{sign}(l ){y_i}}}{{1 + i{z_i}/{z_{R,i}}}}} \right]^{|l |}}\frac{1}{{1 + i{z_i}/{z_{R,i}}}}\exp \left[ { - \frac{{({x_i^2 + y_i^2} )/w_0^2}}{{1 + i{z_i}/{z_{R,i}}}}} \right],$$

where ${z_{R,i}}\textrm{ = }{k_i}w_0^2/2$ is the Rayleigh range. It is further assumed that the incident LG vortex beams with polarization parameters $\alpha$ and $\beta$ satisfying the relation ${|\alpha |^2}\textrm{ + }{|\beta |^2}\textrm{ = }1$ propagate parallel to the positive ${z_i}$-axis. The vector potential of such incident beams in coordinate system $({{x_i},{y_i},{z_i}} )$ can be written in the form

(7)$${{\mathbf A}_i} = ({\alpha {{\hat{{\mathbf x}}}_i} + \beta {{\hat{{\mathbf y}}}_i}} ){u_i}({{x_i},{y_i},{z_i}} )\exp ({i{k_i}{z_i}} ).$$

Stemming from the vector Maxwell’s equations and using Lorenz gauge condition, the electric and magnetic field vectors of the incident LG vortex beams under the paraxial approximation can be expressed in terms of ${{\mathbf A}_i}$ as [28]

(8)$${{\mathbf E}_i} = i{k_i}{Z_i}\left[ {{u_i}({\alpha {{\hat{{\mathbf x}}}_i}\textrm{ + }\beta {{\hat{{\mathbf y}}}_i}} )+ \frac{i}{{{k_i}}}\left( {\alpha \frac{{\partial {u_i}}}{{\partial {x_i}}}\textrm{ + }\beta \frac{{\partial {u_i}}}{{\partial {y_i}}}} \right){{\hat{{\mathbf z}}}_i}} \right]\exp ({i{k_i}{z_i}} ),$$

(9)$${{\mathbf H}_i} = i{k_i}\left[ {{u_i}({ - \beta {{\hat{{\mathbf x}}}_i}\textrm{ + }\alpha {{\hat{{\mathbf y}}}_i}} )- \frac{i}{{{k_i}}}\left( {\beta \frac{{\partial {u_i}}}{{\partial {x_i}}} - \alpha \frac{{\partial {u_i}}}{{\partial {y_i}}}} \right){{\hat{{\mathbf z}}}_i}} \right]\exp ({i{k_i}{z_i}} ),$$

where ${Z_i} = \sqrt {{\mu _0}/{\varepsilon _0}} ,\,\partial {u_i}/\partial {x_i}$, and $\partial {u_i}/\partial {y_i}$ are given by

(10)$$\frac{{\partial {u_i}}}{{\partial {x_i}}}\textrm{ = }\left\{ {\frac{{|l |[{{x_i} - i\textrm{sign}(l ){y_i}} ]}}{{x_i^2 + y_i^2}} - \frac{{{k_i}{x_i}}}{{{z_{R,i}}\textrm{ + }i{z_i}}}} \right\}{u_i},$$

(11)$$\frac{{\partial {u_i}}}{{\partial {y_i}}}\textrm{ = }\left\{ {\frac{{|l |[{{y_i} + i\textrm{sign}(l ){x_i}} ]}}{{x_i^2 + y_i^2}} - \frac{{{k_i}{y_i}}}{{{z_{R,i}}\textrm{ + }i{z_i}}}} \right\}{u_i}.$$

After reflection of the incident vortex beams from an air-chiral medium interface, the corresponding vector potential in coordinate system $({{x_r},{y_r},{z_r}} )$ can be expressed as

(12)$${{\mathbf A}_r}\textrm{ = }[{u_r^H({{x_r},{y_r},{z_r}} ){{\hat{{\mathbf x}}}_r}\textrm{ + }u_r^V({{x_r},{y_r},{z_r}} ){{\hat{{\mathbf y}}}_r}} ]\exp ({i{k_r}{z_r}} ),$$

and

(13)$$\left[ {\begin{array}{c} {u_r^H({{x_r},{y_r},{z_r}} )}\\ {u_r^V({{x_r},{y_r},{z_r}} )} \end{array}} \right] = \int\!\!\!\int {\left[ {\begin{array}{c} {\tilde{u}_r^H({{k_{rx}},{k_{ry}}} )}\\ {\tilde{u}_r^V({{k_{rx}},{k_{ry}}} )} \end{array}} \right]} \exp \left[ {i\left( {{k_{rx}}{x_r} + {k_{ry}}{y_r} - \frac{{k_{rx}^2 + k_{ry}^2}}{{2{k_r}}}{z_r}} \right)} \right]d{k_{rx}}d{k_{ry}},$$

where ${k_{rx}}$ and ${k_{ry}}$ are the transverse components of the wave vector ${{\mathbf k}_r}\textrm{ = }{k_r}{\hat{{\mathbf z}}_r}$ with ${k_r}\textrm{ = }{k_0},\,\tilde{u}_r^H$, and $\tilde{u}_r^V$ are the angular spectrums of the reflected beams. Following the work of Yang et al. [29], and introducing the boundary conditions ${k_{ix}}\textrm{ = } - {k_{rx}}$ and ${k_{iy}}\textrm{ = }{k_{ry}}$, we can express $\tilde{u}_r^H$ and $\tilde{u}_r^V$ as

(14)$$\left[ {\begin{array}{cc} {\tilde{u}_r^H}\\ {\tilde{u}_r^V} \end{array}} \right]\textrm{ = }\left[ {\begin{array}{cc} {{r_{pp}} - \frac{{{k_{ry}}({{r_{ps}} - {r_{sp}}} )\cot {\theta_i}}}{{{k_0}}}}&{{r_{ps}}\textrm{ + }\frac{{{k_{ry}}({{r_{pp}}\textrm{ + }{r_{ss}}} )\cot {\theta_i}}}{{{k_0}}}}\\ {{r_{sp}} - \frac{{{k_{ry}}({{r_{pp}}\textrm{ + }{r_{ss}}} )\cot {\theta_i}}}{{{k_0}}}}&{{r_{ss}} - \frac{{{k_{ry}}({{r_{ps}} - {r_{sp}}} )\cot {\theta_i}}}{{{k_0}}}} \end{array}} \right]\left[ {\begin{array}{c} {\tilde{u}_i^H}\\ {\tilde{u}_i^V} \end{array}} \right],$$

where $\tilde{u}_i^H\textrm{ = }\alpha {\tilde{u}_r}$ and $\tilde{u}_i^V\textrm{ = }\beta {\tilde{u}_r}$ with ${\tilde{u}_r} = {\tilde{u}_i}({ - {k_{rx}},{k_{ry}}} )$ being the reflected angular spectrum, which is given by

(15)$${\tilde{u}_r}\textrm{ = }{\left( {\frac{{{w_0}}}{{\sqrt 2 }}} \right)^{|l |}}{[{i{k_{rx}} + \textrm{sign}(l ){k_{ry}}} ]^{|l |}}\frac{{w_0^2}}{{4\pi }}\exp \left[ { - \frac{{w_0^2({k_{rx}^2 + k_{ry}^2} )}}{4}} \right].$$

To obtain the results more precisely, a Taylor series expansion in form of angular spectrum component could be utilized to correct the reflection coefficients. In particular, $({m = p,} s; {n = p,s} )$ can be expanded as [26]

(16)$${r_{mn}}\textrm{ = }{r_{mn}}\left[ {1 - \frac{{{k_{rx}}}}{{{k_0}}}\frac{{\partial \ln {r_{mn}}}}{{\partial {\theta_i}}}} \right],$$

where the boundary condition ${k_{ix}}\textrm{ = } - {k_{rx}}$ has been introduced.

Substituting Eq. (16) into Eq. (14), and neglecting the second order item, we can obtain the corrected expression. Then, by utilizing the Fourier transformation in Eq. (13), the transverse amplitudes of the reflected LG vortex beams can be written as

(17)$$u_r^H\textrm{ = }A_r^H{u_r}, u_r^V\textrm{ = }A_r^V{u_r},$$

where

(18)$$\begin{aligned} A_r^H &=\alpha {r_{pp}} + \beta {r_{ps}} - \left( {\alpha {r_{pp}}\frac{{\partial \ln {r_{pp}}}}{{\partial {\theta_i}}} + \beta {r_{ps}}\frac{{\partial \ln {r_{ps}}}}{{\partial {\theta_i}}}} \right)\frac{1}{{{k_0}}}\left\{ { - \frac{{|l |[{i{x_r} - \textrm{sign}(l ){y_r}} ]}}{{x_r^2 + y_r^2}} + \frac{{i{k_r}{x_r}}}{{{z_{R,r}} + i{z_r}}}} \right\}\\&\quad +[{\alpha ({{r_{sp}} - {r_{ps}}} )+ \beta ({{r_{pp}} + {r_{ss}}} )} ]\frac{{\cot {\theta _i}}}{{{k_0}}}\left\{ { - \frac{{|l |[{i{y_r} + \textrm{sign}(l ){x_r}} ]}}{{x_r^2 + y_r^2}} + \frac{{i{k_r}{y_r}}}{{{z_{R,r}} + i{z_r}}}} \right\}, \end{aligned}$$

(19)$$\begin{aligned} A_r^V &=\alpha {r_{sp}} + \beta {r_{ss}} - \left( {\alpha {r_{sp}}\frac{{\partial \ln {r_{\textrm{s}p}}}}{{\partial {\theta_i}}} + \beta {r_{ss}}\frac{{\partial \ln {r_{ss}}}}{{\partial {\theta_i}}}} \right)\frac{1}{{{k_0}}}\left\{ { - \frac{{|l |[{i{x_r} - \textrm{sign}(l ){y_r}} ]}}{{x_r^2 + y_r^2}} + \frac{{i{k_r}{x_r}}}{{{z_{R,r}} + i{z_r}}}} \right\}\\ &\quad - [{\alpha ({{r_{pp}} + {r_{ss}}} )+ \beta ({{r_{ps}} - {r_{sp}}} )} ]\frac{{\cot {\theta _i}}}{{{k_0}}}\left\{ { - \frac{{|l |[{i{y_r} + \textrm{sign}(l ){x_r}} ]}}{{x_r^2 + y_r^2}} + \frac{{i{k_r}{y_r}}}{{{z_{R,r}} + i{z_r}}}} \right\}, \end{aligned}$$

and

(20)$${u_r}\textrm{ = }{\left( {\frac{{\sqrt 2 }}{{{w_0}}}} \right)^{|l |}}{\left[ {\frac{{ - {x_r} + i\textrm{sign}(l ){y_r}}}{{1 + iz/{z_{R,r}}}}} \right]^{|l |}}\frac{1}{{1 + i{z_r}/{z_{R,r}}}}\exp \left[ { - \frac{{({x_r^2 + y_r^2} )/w_0^2}}{{1 + i{z_r}/{z_{R,r}}}}} \right].$$

Similarly, using Lorenz gauge condition, the electric and magnetic field vectors of the reflected LG vortex beams can be expressed in terms of ${{\mathbf A}_r}$ as

(21)$${{\boldsymbol E}_r}\textrm{ = }i{k_r}{Z_r}\left[ {u_r^H{{\hat{{\mathbf x}}}_r}\textrm{ + }u_r^V{{\hat{{\mathbf y}}}_r}\textrm{ + }\frac{i}{{{k_r}}}\left( {\frac{{\partial u_r^H}}{{\partial {x_r}}} + \frac{{\partial u_r^V}}{{\partial {y_r}}}} \right){{\hat{{\mathbf z}}}_r}} \right]\exp ({i{k_r}{z_r}} ),$$

(22)$${{\mathbf H}_r} = i{k_r}\left[ { - u_r^V{{\hat{{\mathbf x}}}_r} + u_r^H{{\hat{{\mathbf y}}}_r} - \frac{i}{{{k_r}}}\left( {\frac{{\partial u_r^V}}{{\partial {x_r}}} - \frac{{\partial u_r^H}}{{\partial {y_r}}}} \right){{\hat{{\mathbf z}}}_r}} \right]\exp ({i{k_r}{z_r}} ),$$

where ${Z_r} = \sqrt {{\mu _0}/{\varepsilon _0}}$, and

(23)$$\frac{{\partial u_r^H}}{{\partial {x_r}}}\textrm{ = }\left[ {\frac{{\partial A_r^H}}{{\partial {x_r}}}} \right]{u_r}\textrm{ + }A_r^H\left[ {\frac{{\partial {u_r}}}{{\partial {x_r}}}} \right],\frac{{\partial u_r^H}}{{\partial {y_r}}}\textrm{ = }\left[ {\frac{{\partial A_r^H}}{{\partial {y_r}}}} \right]{u_r}\textrm{ + }A_r^H\left[ {\frac{{\partial {u_r}}}{{\partial {y_r}}}} \right],$$

(24)$$\frac{{\partial u_r^V}}{{\partial {x_r}}}\textrm{ = }\left[ {\frac{{\partial A_r^V}}{{\partial {x_r}}}} \right]{u_r}\textrm{ + }A_r^V\left[ {\frac{{\partial {u_r}}}{{\partial {x_r}}}} \right], \frac{{\partial u_r^V}}{{\partial {y_r}}}\textrm{ = }\left[ {\frac{{\partial A_r^V}}{{\partial {y_r}}}} \right]{u_r}\textrm{ + }A_r^V\left[ {\frac{{\partial {u_r}}}{{\partial {y_r}}}} \right],$$

with

(25)$$\begin{aligned} \frac{{\partial A_r^H}}{{\partial {x_r}}}& = - \left( {\alpha {r_{pp}}\frac{{\partial \ln {r_{pp}}}}{{\partial {\theta_i}}} + \beta {r_{ps}}\frac{{\partial \ln {r_{ps}}}}{{\partial {\theta_i}}}} \right)\frac{1}{{{k_0}}}\left\{ {\frac{{i|l |}}{{{{[{{x_r} - i\textrm{sign}(l ){y_r}} ]}^2}}} + \frac{{i{k_r}}}{{{z_{R,r}} + i{z_r}}}} \right\}\\ &\quad + [{\alpha ({{r_{sp}} - {r_{ps}}} )+ \beta ({{r_{pp}} + {r_{ss}}} )} ]\frac{{\cot {\theta _i}}}{{{k_0}}}\left\{ { - \frac{{\textrm{sign}(l )|l |}}{{{{[{{y_r}\textrm{ + }i\textrm{sign}(l ){x_r}} ]}^2}}}} \right\}, \end{aligned}$$

(26)$$\begin{aligned} \frac{{\partial A_r^H}}{{\partial {y_r}}} &={-} \left( {\alpha {r_{pp}}\frac{{\partial \ln {r_{pp}}}}{{\partial {\theta_i}}} + \beta {r_{ps}}\frac{{\partial \ln {r_{ps}}}}{{\partial {\theta_i}}}} \right)\frac{1}{{{k_0}}}\left\{ {\frac{{\textrm{sign}(l )|l |}}{{{{[{{x_r} - i\textrm{sign}(l ){y_r}} ]}^2}}}} \right\}\\ &\quad + [{\alpha ({{r_{sp}} - {r_{ps}}} )+ \beta ({{r_{pp}} + {r_{ss}}} )} ]\frac{{\cot {\theta _i}}}{{{k_0}}}\left\{ {\frac{{i|l |}}{{{{[{{y_r}\textrm{ + }i\textrm{sign}(l ){x_r}} ]}^2}}} + \frac{{i{k_r}}}{{{z_{R,r}} + i{z_r}}}} \right\}, \end{aligned}$$

(27)$$\begin{aligned} \frac{{\partial A_r^V}}{{\partial {x_r}}} &= - \left( {\alpha {r_{sp}}\frac{{\partial \ln {r_{\textrm{s}p}}}}{{\partial {\theta_i}}} + \beta {r_{ss}}\frac{{\partial \ln {r_{ss}}}}{{\partial {\theta_i}}}} \right)\frac{1}{{{k_0}}}\left\{ {\frac{{i|l |}}{{{{[{{x_r} - i\textrm{sign}(l ){y_r}} ]}^2}}} + \frac{{i{k_r}}}{{{z_{R,r}} + i{z_r}}}} \right\}\\ &\quad - [{\alpha ({{r_{pp}} + {r_{ss}}} )+ \beta ({{r_{ps}} - {r_{sp}}} )} ]\frac{{\cot {\theta _i}}}{{{k_0}}}\left\{ { - \frac{{\textrm{sign}(l )|l |}}{{{{[{{y_r}\textrm{ + }i\textrm{sign}(l ){x_r}} ]}^2}}}} \right\}, \end{aligned}$$

(28)$$\begin{aligned} \frac{{\partial A_r^V}}{{\partial {y_r}}} &={-} \left( {\alpha {r_{sp}}\frac{{\partial \ln {r_{\textrm{s}p}}}}{{\partial {\theta_i}}} + \beta {r_{ss}}\frac{{\partial \ln {r_{ss}}}}{{\partial {\theta_i}}}} \right)\frac{1}{{{k_0}}}\left\{ {\frac{{\textrm{sign}(l )|l |}}{{{{[{{x_r} - i\textrm{sign}(l ){y_r}} ]}^2}}}} \right\}\\ &\quad - [{\alpha ({{r_{pp}} + {r_{ss}}} )+ \beta ({{r_{ps}} - {r_{sp}}} )} ]\frac{{\cot {\theta _i}}}{{{k_0}}}\left\{ {\frac{{i|l |}}{{{{[{{y_r}\textrm{ + }i\textrm{sign}(l ){x_r}} ]}^2}}} + \frac{{i{k_r}}}{{{z_{R,r}} + i{z_r}}}} \right\}, \end{aligned}$$

(29)$$\frac{{\partial {u_r}}}{{\partial {x_r}}} = \left\{ {\frac{{|l |[{{x_r} + i\textrm{sign}(l ){y_r}} ]}}{{x_r^2 + y_r^2}} - \frac{{{k_r}{x_r}}}{{{z_{R,r}} + i{z_r}}}} \right\}{u_r},$$

(30)$$\frac{{\partial {u_r}}}{{\partial {y_r}}} = \left\{ {\frac{{|l |[{{y_r} - i\textrm{sign}(l ){x_r}} ]}}{{x_r^2 + y_r^2}} - \frac{{{k_r}{y_r}}}{{{z_{R,r}} + i{z_r}}}} \right\}{u_r}.$$

Once the electric and magnetic field vectors of the incident and reflected vortex beams are obtained, their chirality can be examined. The chirality density of optical beams in the air can be written as [30–32]

(31)$$C = \frac{\omega }{{2{c^2}}}{\mathop{\rm Im}\nolimits} ({{\mathbf E} \cdot {{\mathbf H}^\ast }} ),$$

where $\omega$ is the angular frequency, $c$ is the speed of light in vacuum, and superscript “*” represents the complex conjugate.

3. Results and discussion

In this section, we make some numerical simulations to explore the chirality of optical vortex beams reflected from an air-chiral medium interface. In the simulations, we set the wavelength $\lambda \textrm{ = 632}\textrm{.8nm}$, the beam waist radius ${w_0} = 2.0\lambda$, the topological charge $l = 2$, the incident angle ${\theta _i} = {45^\textrm{o}}$, the polarization parameters $({\alpha ,\beta } ) = ({1,0} )$, i.e., $x$-linear polarization ($x$-LP), the relative permittivity, permeability, and chiral parameter of the chiral medium $({{\varepsilon_r},{\mu_r},{\kappa_r}} ) = ({2.0,1.0,0.7} )$, and the position of observed plane ${z_r} = 1.0\lambda$. Hereafter, the parameters used are the same as mentioned above, unless otherwise stated.

We start with the simulation results for the chirality density distributions of the incident LG vortex beams with different topological charges and polarization states, as illustrated in Fig. 2, where $\sigma = i({\alpha {\beta^ \ast } - {\alpha^ \ast }\beta } )= 0,1$, and $- 1$ respectively denote the beams with $x$-LP, L-CP, and R-CP, corresponding to the polarization parameters $({\alpha ,\beta } ) = ({1,0} ),\,({\alpha ,\beta } ) = ({1,i} )/\sqrt 2$, and $({\alpha ,\beta } ) = ({1, - i} )/\sqrt 2$. All the chirality densities are normalized by their respective maximum values. As we can see, when $l = 0$, the beam (i.e., a fundamental Gaussian beam) with $x$-LP possesses no chirality, while the beams with L-CP and R-CP have chirality. When $l \ne 0$, we observe a non-zero chirality for the linearly polarized vortex beams. It is also observed that there is an on-axis chirality density for $|l |= 1$ when the SAM and OAM are anti-parallel, i.e., $\textrm{sign}(l )={-} \textrm{sign}(\sigma )$, whereas there is no on-axis chirality density for $|l |= 1$ when the SAM and OAM are parallel, i.e., $\textrm{sign}(l )= \textrm{sign}(\sigma )$, as depicted in Fig. 2(c). These effects are spin-orbit interactions of light, the electric and magnetic energy densities of the fields follow the same physics. It should be noted that the effect is not as drastic as the case of linearly polarized light because the circular polarization dominates the chirality density. Such a phenomenon indicates that the chirality of a linearly polarized vortex beam depends on the helical phase structure, while the chirality of a circularly polarized vortex beam originates from the helical front structure that the electric and magnetic field vectors trace out in space when propagating. In addition, the state of polarization changes from the L-CP to R-CP, the chirality densities obviously exhibit a negative value while their amplitude stay the same, which can be explained from the definition of the chirality density.

Fig. 2. Normalized optical chirality density distributions of the incident LG vortex beams with different topological charges and polarization states. (a) Chirality density distributions in the transverse ${x_i}\textrm{ - }{y_i}$ plane, and (b)-(d) chirality density distributions along the ${x_i}$-axis.

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Now, we explore the chirality of vortex beams reflected from an air-chiral medium interface. Figure 3 shows the comparison of chirality density distributions among the incident LG vortex beam, reflected beam from an air-isotropic medium interface, and reflected beam from an air-chiral medium interface. The relative permittivity and permeability of the isotropic medium are set to be the same as that of the chiral medium, i.e., $({{\varepsilon_r},{\mu_r}} ) = ({2.0,1.0} )$. For the case of isotropic medium, the Fresnel reflection coefficients ${r_p}$ and ${r_s}$ take the form given in Ref. [28]. It can be observed that the chirality density distribution of the incident vortex beam is circularly symmetric, while that of the reflected beam loses the circular symmetry. The chirality sign of the reflected beam from an air-isotropic medium interface is opposite to that of the incident beam, which arises from the fact that the boundary conditions ${k_{ix}} = - {k_{rx}}$ and ${k_{iy}} = {k_{ry}}$ are applied. It is worthy to note that the chirality sign of the reflected beam from an air-chiral medium interface is the same as that of the incident beam, which indicating that the intrinsic chiral asymmetry of the medium has a significant effect on the chirality of vortex beam. Further observation, we find that the chirality of the vortex beam reflected from an air-isotropic medium interface is weakened, whereas the chirality of the vortex beam reflected from an air-chiral medium interface is significantly enhanced, which provides an alternative strategy to enhance the chirality of vortex beams. Figure 4 illustrates the normalized optical chirality density distributions of the LG vortex beams reflected from an interface between air and a chiral medium with different relative chiral parameters. Obviously, the intrinsic chiral asymmetry of the medium has a significant influence on the chirality of the reflected vortex beams. As the relative chiral parameter of the medium increases, the maximum absolute value of the chirality density increases. Such a phenomenon provides an important route to control the chirality of optical vortex beams.

Fig. 3. Comparison of optical chirality density distributions among the incident LG vortex beam and reflected beams. (a1) and (a2) incident beam, (b1) and (b2) reflected beam from an air-isotropic medium interface, and (c1) and (c2) reflected beam from an air-chiral medium interface.

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Fig. 4. Normalized optical chirality density distributions of the LG vortex beams reflected from an interface between air and a chiral medium with different relative chiral parameters. (a) ${\kappa _r} = 0.1$, (b) ${\kappa _r} = 0.3$, (c) ${\kappa _r} = 0.5$, (d) ${\kappa _r} = 0.7$, and (e),(f) chirality density distributions along the ${x_r}$-axis and ${y_r}$-axis, respectively.

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It is well known that the reflection characteristics of a light beam impinging on the surface of a medium are closely related not only to the electromagnetic parameters of medium, but also to the parameters of the incident beam, such as the incident angle, polarization state, phase, and so on. Predictably, such parameters of the incident beam also have significant influences on the chirality of the reflected vortex beams for a given chiral medium. Figure 5 shows the normalized chirality density distributions of the vortex beams with different incident angles reflected from an air-chiral medium interface. As can be seen from the figure, the distribution of the chirality density for the reflected beam is sensitive to the incidence angle. The greater the incident angle, the greater the maximum absolute value of the chirality density. Figure 6 shows the normalized chirality density distributions of the vortex beams with different topological charges reflected from an air-chiral medium interface. We can easily find that the chirality density of the reflected beam changes with the change of topological charge. With the increase of the topological charge, the maximum absolute value of the chirality density increases and the chirality density distribution gradually goes away from the center.

Fig. 5. Normalized optical chirality density distributions of the LG vortex beams with different incident angles reflected from an air-chiral medium interface. (a) ${\theta _i} = {20^ \circ }$, (b) ${\theta _i} = {25^ \circ }$, (c) ${\theta _i} = {30^ \circ }$, (d) ${\theta _i} = {45^ \circ }$, and (e),(f) chirality density distributions along the ${x_r}$-axis and ${y_r}$-axis, respectively.

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Fig. 6. Normalized optical chirality density distributions of the LG vortex beams with different topological charges reflected from an air-chiral medium interface. (a) $l = 1$, (b) $l = 2$, (c) $l = 3$, (d) $l = 4$, and (e),(f) chirality density distributions along the ${x_r}$-axis and ${y_r}$-axis, respectively.

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Figure 7 shows the normalized chirality density distributions of the LG vortex beams with different states of circular polarization reflected from an air-chiral medium interface. It is noted that the change of the state of circular polarization leads to a redistribution of the chirality density for the reflected beams, which is different from that of the incident beams. It is also worth noting that circularly polarized vortex beams could increase optical chirality densities when $\textrm{sign}(\sigma )={-} \textrm{sign}(l )$, as depicted in Figs. 7(e) and (f). The above analyses suggest that changing the polarization state offers an alternative way to regulate the local chirality of the vortex beams during the reflection process.

Fig. 7. Normalized optical chirality density distributions of the LG vortex beams with different states of circular polarization reflected from an air-chiral medium interface. (a) $\sigma = 1,l = - 2$, (b) $\sigma = 1,l = 2$, (c) $\sigma = - 1,l = - 2$, (d) $\sigma = - 1,l = 2$, and (e),(f) chirality density distributions along the ${x_r}$-axis and ${y_r}$-axis, respectively.

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4. Conclusion

In conclusion, the chirality of LG vortex beams reflected from an interface between air and a chiral medium are investigated theoretically and numerically. The analytical formulas for the LG vortex beams reflected from an air-chiral medium interface are derived in detail. The effects of the parameters of the chiral medium and incident beams on the chirality of reflected LG vortex beams are numerically simulated and discussed. The results show that the chiral medium can lead to greater local chirality compared with the case of isotropic medium and the intrinsic chiral asymmetry of the medium provides an alternative pathway for controlling the chirality of the reflected vortex beams. Moreover, the incidence angle, topological charge, and polarization state of the incident LG beam have a significant influence on the chirality of the reflected beams. Specifically, the greater the incident angle, the greater the maximum absolute value of the chirality density. With the increase of the topological charge, the maximum absolute value of the chirality density increases. The change of the polarization state leads to a redistribution of the chirality density and the circularly polarized vortex beams could increase chirality densities when $\textrm{sign}(\sigma )={-} \textrm{sign}(l )$. These findings are valuable in understanding the interactions between optical vortex beams with chiral matter and provide an alternative pathway for controlling the local chirality of the optical vortex beams.

Funding

Fundamental Research Funds for the Central Universities (QTZX22042); Natural Science Foundation of Guangdong Province (2022A1515011138).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Chirality of optical vortex beams reflected from an air-chiral medium interface

Abstract

1. Introduction

2. Theory and formula

3. Results and discussion

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (7)

Equations (31)

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