Vortex generation in the spin-orbit interaction of a light beam propagating inside a uniaxial medium: origin and efficiency

Xiaohui Ling; Xiaohui Ling; Xiaohui Ling; Huiling Luo; Huiling Luo; Fuxin Guan; Xinxing Zhou; Hailu Luo; Lei Zhou; Lei Zhou

doi:10.1364/OE.403650

1. Introduction

Just as a mechanical particle, light can also exhibit angular momenta, including both spin angular momentum (SAM) and orbital angular momentum (OAM). The SAM is related to the polarization of light, and a circularly polarized (CP) light with left- or right-handedness carries ${\pm} \hbar$ SAM per photon. Meanwhile, light can carry two distinct types of OAM [1–4], of which one is called the intrinsic OAM dictated by the vortex nature of the wave-front, while another is the extrinsic OAM associated with the off-set trajectory of the beam propagation with respect to a reference point, bearing the same physics as the mechanical angular momentum of a classical particle. The spin-orbit interaction (SOI) of light can generate mutual conversions between SAMs and OAMs inside a light beam [3,4], in many different optical processes [5–18]. Such SOI–related effects have attracted intensive attention recently, due to not only scientific curiosities but also many potential applications such as precision metrology, information storage and processing [19–27].

Among these SOI-induced effects, of particular interest is the conversion from SAM to intrinsic OAM, as it provides an effective way to generate and control an optical vortex beam. The latter can have many applications in practice, such as particle manipulation [28], optical metrology [29], optical communications [30], and super-resolution imaging [31]. Recently, such an intriguing effect has been theoretically predicted and/or experimentally demonstrated in many different optical processes, such as strong focusing of light beam [3,14,15] and light scatterings by certain inhomogeneous anisotropic media [10–13,26] (e.g., Q-plates, S-waveplates, and structured metasurfaces). Specifically, when a CP light beam propagates in a bulk uniaxial crystal along the optical axis direction, a portion of which will undergo a spin-reversal and acquire a vortex phase with a topological charge of ±2, i.e., a SOI process will take place [32–43]. However, while the working mechanisms of the vortex generations have been clearly revealed in the first two optical processes (i.e., focusing [3,14,15] and Q/S-plates [10–13]), the physical origin in the last case (i.e., propagating inside a bulk uniaxial crystal) is rarely discussed in literature and still remains obscure. Without deep understandings on the inherent physics, it is also difficult for researchers to improve the conversion efficiency and utilize such effect in reality.

In this paper, we uncover the physical origin of such fascinating effect and discuss how to improve the conversion efficiency for practical applications. Based on calculations with a rigorous full-wave theory, we show that as a CP light beam with a particular spin (handedness) travel along the principle axis of a uniaxial crystal, the unique polarization-dependent interactions between light and the medium can convert the spin direction of a certain portion of the beam, with efficiency determined by both the optical properties of the medium and the structure of the light beam. Interestingly, such spin-reversed light beam can gain a vortex phase of geometrical nature dictated by the Pancharatnam-Berry mechanism [44–46], which is quite different from that in the Q/S-plates. Understanding the intrinsic mechanism behind such intriguing effect can not only help us improve the efficiency of vortex generation for application purpose, but also shed important light on other SOI-induced effects.

2. Physical origin of the vortex phase in bulk uniaxial crystals

2.1 Full-wave theory of light beams propagating along the optical axis in a uniaxial crystal

We first establish a laboratory Cartesian coordinate system {x, y, z}, and assume that a monochromatic finite-width light beam propagates along the optical axis (parallel to the z-axis) in a lossless and non-magnetic (µ=1) uniaxial crystal (see Fig. 1). The relative dielectric tensor of the uniaxial crystal is

(1)$${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over {\mathrm{\boldsymbol{\varepsilon}}} } } = \left( \begin{array}{l} \begin{array}{ccc} {{ }n_o^2}&0{ }&0\\ 0&n_o^2{ }&0 \\ { }0{ }&0{ }&n_e^2\end{array} \end{array} \right),$$

where n_o and n_e are the ordinary and extraordinary refractive indices, respectively.

Fig. 1. Spin-dependent vortex generation in a bulk uniaxial crystal. A CP light beam propagates along the optical axis direction, a portion of which reverses its spin and acquires a vortex phase with topological charge of ±2. This SOI process can be qualitatively described as ${E_ \pm }{(\mathbf r}_ \bot ^{},0) \to {E_ \pm }{(\mathbf r}_ \bot ^{},z) + \exp ({\pm} i2\varphi ){E_ \mp }{(\mathbf r}_ \bot ^{},z)$. Note that the spin-maintained portion of the beam is not shown in the figure.

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A light beam can be seen as a coherent superposition of many plane waves with slightly different propagation directions. When propagating in a uniaxial crystal, each plane wave has an ordinary mode (o-wave) and an extraordinary mode (e-wave). According to the angular spectrum theory, the electric field of the beam can be written as

(2)$${\mathbf E}({\mathbf r}_ \bot ^{},z) = \int {{{\textrm{d}} ^2}{\mathbf k}_ \bot ^{}} {e^{i{\mathbf k}_ \bot ^{} \cdot {\mathbf r}_ \bot ^{}}}[{\tilde{u}_e^{}({\mathbf k}_ \bot^{},z)\hat{\textbf{v}}_e^{}({\mathbf k}_ \bot^{}) + \tilde{u}_o^{}({\mathbf k}_ \bot^{},z)\hat{\textbf{v}}_o^{}({\mathbf k}_ \bot^{})} ].$$

Here $\textbf{k}_ \bot ^{} = k_x^{}\hat{\textbf{x}} + k_y^{}\hat{\textbf{y}}$ and $\textbf{r}_ \bot ^{} = x\hat{\textbf{x}} + y\hat{\textbf{y}}$ are the transverse wave vector and position vector, respectively, where $\{{\hat{\textbf{x}},\hat{\textbf{y}},\hat{\textbf{z}}} \}$ are the unit vectors corresponding to the Cartesian coordinates {x, y, z}. $\tilde{u}_{e,o}^{}({\mathbf k}_ \bot ^{},z)$ are the amplitudes of each plane wave (e- and o-waves) at the propagation distance z, and we have $\tilde{u}_{e,o}^{}({\mathbf k}_ \bot ^{},z) = \tilde{u}_{e,o}^{}({\mathbf k}_ \bot ^{},0)\exp (i{k_{ez,oz}}z)$ which can be written in a matrix form as

(3)$$\left[ \begin{array}{l} \tilde{u}_{e}^{}({\mathbf k}_ \bot^{},z)\\ \tilde{u}_{o}^{}({\mathbf k}_ \bot^{},z) \end{array} \right] = \left[ {\begin{array}{cc} {\exp (i{k_{ez}}z)}&0\\ 0&{\exp (i{k_{oz}}z)} \end{array}} \right]\left[ \begin{array}{l} \tilde{u}_{e}^{}({\mathbf k}_ \bot^{},0)\\ \tilde{u}_{o}^{}({\mathbf k}_ \bot^{},0) \end{array} \right].$$

Here ${k_{oz}} = {(k_0^2n_o^2 - k_ \bot ^2)^{1/2}}$ and ${k_{ez}} = {(k_0^2n_e^2 - k_ \bot ^2)^{1/2}}{n_o}/{n_e}$ are wave vector components of o- and e-waves in the z-direction [32–34], and k₀=2π/λ with λ being the working wavelength. $\hat{\textbf{v}}_{o,e}^{}({\mathbf k}_ \bot ^{})$ in Eq. (2) are the polarization vectors of o- and e-waves, which can be expressed as [33,47]

(4)$$\hat{\textbf{v}}_o^{}({\mathbf k}_ \bot ^{}) = \frac{{\hat{\textbf{z}} \times {\textbf{k}_o}}}{{|\hat{\textbf{z}} \times {\textbf{k}_o}|}} = \frac{{ - {k_y}\hat{\textbf{x}} + {k_x}\hat{\textbf{y}}}}{{{k_ \bot }}} ={-} \sin \varphi \hat{\textbf{x}} + \cos \varphi \hat{\textbf{y}},$$

(5)$$\begin{array}{l} \hat{\textbf{v}}_e^{}({\mathbf k}_ \bot ^{}) = {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over {\mathrm{\boldsymbol{\varepsilon}}} } }}^{ - 1}} \cdot \left[ {\hat{\textbf{v}}_o^{}({\mathbf k}_ \bot^{}) \times \frac{{{\textbf{k}_e}}}{{{k_e}}}} \right] = \frac{{{k_{ez}}}}{{n_o^2{k_e}}}\frac{{{k_x}\hat{\textbf{x}} + {k_y}\hat{\textbf{y}}}}{{{k_ \bot }}} - \frac{{{k_ \bot }}}{{n_e^2{k_e}}}\hat{\textbf{z}}\\ { } = \cos \theta (\cos \varphi \hat{\textbf{x}} + \sin \varphi \hat{\textbf{y}})/n_o^2 - \sin \theta \hat{\textbf{z}}/n_e^2 \end{array}, $$

where k_o_,e=n_o_,ek₀, and θ and φ are respectively the propagation angle and azimuthal angle of the plane wave components in the spherical coordinate system.

It is assumed that the light beam starts at z=0. In the basis of CP (spin basis), we write Eq. (2) as

(6)$${\mathbf E}({\mathbf r}_ \bot ^{},0) = \int {{d ^2}{\mathbf k}_ \bot ^{}} {e^{i{\mathbf k}_ \bot ^{} \cdot {\mathbf r}_ \bot ^{}}}[{\tilde{u}_{ + }^{}({\mathbf k}_ \bot^{},0)\hat{\textbf{v}}_{ + }^{}({\mathbf k}_ \bot^{}) + \tilde{u}_ -^{}({\mathbf k}_ \bot^{},0)\hat{\textbf{v}}_ -^{}({\mathbf k}_ \bot^{})} ].$$

Here $\hat{\textbf{v}}_ \pm ^{}({\mathbf k}_ \bot ^{}) = [{\hat{\textbf{v}}_e^{}({\mathbf k}_ \bot^{}) \pm i\hat{\textbf{v}}_o^{}({\mathbf k}_ \bot^{})} ]/\sqrt 2$ are unit vectors in spin basis, and the subscripts + and –, respectively, represent left- and right-handed CPs.

We require that the beam at the starting point (z=0) has a uniform CP distribution in the transverse direction, and its amplitude has a cylindrical symmetry (e.g., Gaussian beams and Bessel beams), which can be easily generated in the experiment. Then Eq. (6) must also satisfy

(7)$${\mathbf E}({\mathbf r}_ \bot ^{},0) = \int {{d ^2}{\mathbf k}_ \bot ^{}} {e^{i{\mathbf k}_ \bot ^{} \cdot {\mathbf r}_ \bot ^{}}}[{\tilde{U}_{ + }^{}({\mathbf k}_ \bot^{},0)\hat{\textbf{V}}_{ + }^{} + \tilde{U}_ -^{}({\mathbf k}_ \bot^{},0)\hat{\textbf{V}}_ -^{}} ],$$

where ${\hat{\textbf{V}}_ \pm } = ({\hat{\textbf{x}} \pm i\hat{\textbf{y}}} )/\sqrt 2$ are the unit vector of circular polarization in the transverse direction of the beam propagation, and ${\tilde{U}_ \pm }({\mathbf k}_ \bot ^{},0)$ are the two-dimensional Fourier transform of the transverse field ${{\mathbf E}_ \pm }{(\mathbf r}_ \bot ^{},0)$ at z=0.

According to the above limiting condition and combining Eqs. (6) and (7), we can connect the amplitudes $\tilde{u}_ \pm ^{}({\mathbf k}_ \bot ^{},0)$ of each plane wave and the initial condition ${\tilde{U}_ \pm }({\mathbf k}_ \bot ^{},0)$:

(8)$$\left[ {\begin{array}{c} {{{\tilde{U}}_ + }({\mathbf k}_ \bot^{},0)}\\ {{{\tilde{U}}_ - }({\mathbf k}_ \bot^{},0)} \end{array}} \right] = \hat{\textbf{P}}(0)\left[ \begin{array}{l} \tilde{u}_{ + }^{}({\mathbf k}_ \bot^{},0)\\ \tilde{u}_ -^{}({\mathbf k}_ \bot^{},0) \end{array} \right] = \left( {\begin{array}{cc} {{\hat{\mathbf V}}_ +^ \ast \hat{\textbf{v}}_{ + }^{}}&{{\hat{\mathbf V}}_ +^ \ast \hat{\textbf{v}}_ -^{}}\\ {{\hat{\mathbf V}}_ -^ \ast \hat{\textbf{v}}_{ + }^{}}&{{\hat{\mathbf V}}_ -^ \ast \hat{\textbf{v}}_ -^{}} \end{array}} \right)\left[ \begin{array}{l} \tilde{u}_{ + }^{}({\mathbf k}_ \bot^{},0)\\ \tilde{u}_ -^{}({\mathbf k}_ \bot^{},0) \end{array} \right].$$

Here matrix $\hat{\textbf{P}}$ represents a projection operation$\hat{\textbf{V}}_\sigma ^\ast{\cdot} \hat{\textbf{v}}_{\sigma ^{\prime}}^{}$, where $\sigma ^{\prime},\sigma \in \{ + , - \}$, between spin-vectors of arbitrary plane wave and the central plane wave inside the light beam, and the orthonormal conditions $\hat{\textbf{V}}_\sigma ^\ast{\cdot} \hat{\textbf{V}}_{\sigma ^{\prime}}^{} = {\delta _{\sigma \sigma ^{\prime}}}$ is used. After some algebraic calculation, we have

(9)$$\hat{\textbf{P}}(0) = \frac{1}{2}\left[ {\begin{array}{cc} {({\cos \theta /n_o^2 + 1} )\exp ( - i\varphi )}&{({\cos \theta /n_o^2 - 1} )\exp ( - i\varphi )}\\ {({\cos \theta /n_o^2 - 1} )\exp (i\varphi )}&{({\cos \theta /n_o^2 + 1} )\exp (i\varphi )} \end{array}} \right].$$

At the propagation distance with z≠0, the above equation still applies, i.e., $\hat{\textbf{P}}(z) = \hat{\textbf{P}}(0)$. And we have

(10)$$\left[ {\begin{array}{c} {{{\tilde{U}}_ + }({\mathbf k}_ \bot^{},z)}\\ {{{\tilde{U}}_ - }({\mathbf k}_ \bot^{},z)} \end{array}} \right]{ = }\hat{\textbf{P}}(z)\left[ \begin{array}{l} \tilde{u}_{ + }^{}({\mathbf k}_ \bot^{},z)\\ \tilde{u}_ -^{}({\mathbf k}_ \bot^{},z) \end{array} \right].$$

Meanwhile, Eq. (3) can be rewritten in the spin basis as

(11)$$\left[ \begin{array}{l} \tilde{u}_{ + }^{}({\mathbf k}_ \bot^{},z)\\ \tilde{u}_ -^{}({\mathbf k}_ \bot^{},z) \end{array} \right] = {\hat{\mathbf T}}({\mathbf k}_ \bot ^{},z)\left[ \begin{array}{l} \tilde{u}_{ + }^{}({\mathbf k}_ \bot^{},0)\\ \tilde{u}_ -^{}({\mathbf k}_ \bot^{},0) \end{array} \right] = \left[ {\begin{array}{cc} {{t_{ +{+} }}({\mathbf k}_ \bot^{},z)}&{{t_{ +{-} }}({\mathbf k}_ \bot^{},z)}\\ {{t_{ -{+} }}({\mathbf k}_ \bot^{},z)}&{{t_{ -{-} }}({\mathbf k}_ \bot^{},z)} \end{array}} \right]\left[ \begin{array}{l} \tilde{u}_{ + }^{}({\mathbf k}_ \bot^{},0)\\ \tilde{u}_ -^{}({\mathbf k}_ \bot^{},0) \end{array} \right].$$

Here ${t_{ +{+} }} = {t_{ -{-} }} = [\exp (i{k_{ez}}z) + \exp (i{k_{oz}}z)]/2$ and ${t_{ -{+} }} = {t_{ +{-} }} = [\exp (i{k_{ez}}z) - \exp (i{k_{oz}}z)]/2$.

In combination of Eqs. (8)–(11), we now associate the transverse field distribution at the propagation distance z with the initial field by three cascaded matrices

(12)$$\left[ {\begin{array}{c} {{{\tilde{U}}_ + }({\mathbf k}_ \bot^{},z)}\\ {{{\tilde{U}}_ - }({\mathbf k}_ \bot^{},z)} \end{array}} \right] = \hat{\textbf{P}}(z){\hat{\mathbf T}}({\mathbf k}_ \bot ^{},z){\hat{\textbf{P}}^{ - 1}}(0)\left[ {\begin{array}{c} {{{\tilde{U}}_ + }({\mathbf k}_ \bot^{},0)}\\ {{{\tilde{U}}_ - }({\mathbf k}_ \bot^{},0)} \end{array}} \right].$$

This is one of the central results in this paper.

After some algebra, we solve the matrix in Eq. (12) as

(13)$${\hat{\mathbf M}}(z) = \hat{\textbf{P}}(z){\hat{\mathbf T}}({\mathbf k}_ \bot ^{},z){\hat{\textbf{P}}^{ - 1}}(0) = \left[ {\begin{array}{cc} {{t_{ +{+} }}({\mathbf k}_ \bot^{},z)}&{{t_{ +{-} }}({\mathbf k}_ \bot^{},z)\exp ( - i2\varphi )}\\ {{t_{ -{+} }}({\mathbf k}_ \bot^{},z)\exp (i2\varphi )}&{{t_{ -{-} }}({\mathbf k}_ \bot^{},z)} \end{array}} \right].$$

Note that the two parameters θ and n_o in Eq. (9) are eliminated in the calculation process and do not appear in the final matrix. When the incident beam [Eq. (7)] is a left-handed CP one which requires ${\tilde{U}_ + }({\mathbf k}_ \bot ^{},0) \ne 0$ and ${\tilde{U}_ - }({\mathbf k}_ \bot ^{},0) \equiv 0$, the field at any distance z is

{\tilde{U}_ + }({\mathbf k}_ \bot ^{},z) = {t_{ +{+} }}({\mathbf k}_ \bot ^{},z){\tilde{U}_ + }({\mathbf k}_ \bot ^{},0),

(14)$${\tilde{U}_ - }({\mathbf k}_ \bot ^{},z) = {t_{ - + }}({\mathbf k}_ \bot ^{},z)\exp (i2\varphi ){\tilde{U}_ + }({\mathbf k}_ \bot ^{},0).$$

The fields in real space can be obtained by a two-dimensional Fourier transform, namely,

(15)$${{\mathbf E}_ \pm }({\mathbf r}_ \bot ^{},z) = \int {{d ^2}{\mathbf k}_ \bot ^{}} {e^{i{\mathbf k}_ \bot ^{} \cdot {\mathbf r}_ \bot ^{}}}{\tilde{U}_ \pm }({\mathbf k}_ \bot ^{},z)\hat{\textbf{V}}_ \pm ^{}.$$

Equation (14) means that even if the incident beam is a uniform CP beam, an additional spin-flip component (we call abnormal mode) will generate upon beam propagation, which carries a vortex phase factor exp(i2φ) with a topological charge of 2. And the spin-maintained component is referred to as normal mode.

These results are consistent with that obtained by other methods in the existing work [33–35]. However, our approach connects the initial state and the final state of the beam with three cascaded matrices, which is very concise. What's more, the three matrices clearly show the different physical contributions of the anisotropy of the uniaxial crystal and the beam itself in the SOI process. In fact the vortex phase only comes from the matrix $\hat{\textbf{P}}$, that is, the projection operation of the polarization vectors of each plane wave component and that of the central plane wave in the beam, which is only related to the topological structure of the beam itself. In addition, the anisotropy of the uniaxial crystal determines the amplitude ${t_{ -{+} }}({\mathbf k}_ \bot ^{},z)$ of the abnormal mode, that is, the efficiency of vortex generation, which is related to the propagation distance z and the angular spectrum width of the beam. It can also be seen from Eq. (13) that the matrix $\hat{\textbf{P}}$ contributes to the vortex phase, and the matrix ${\hat{\mathbf T}}$ contributes to the efficiency of vortex generation.

2.2 Berry phase: the physical origin of vortex phase

We now analyze the physical origin of the vortex phase. The angular spectrum of the beam can be represented by infinite k-cones in the k-space [ Fig. 2(a)], in which the polarization vector of any wave vector rotates around k_z direction for one cycle (contour C), resulting in additional geometric phase factor, namely spin-redirection Berry phase [44–46]. Using the CP unit vectors of each wave vector $\hat{\textbf{v}}_\sigma ^{}(\textbf{k}_ \bot ^{}) = [{\hat{\textbf{v}}_e^{}(\textbf{k}_ \bot^{}) \pm i\sigma \hat{\textbf{v}}_o^{}(\textbf{k}_ \bot^{})} ]/\sqrt 2$ on spherical coordinates (θ, φ) in k-space, the Berry connection and Berry curvature are written as

(16)$${\textbf{A}_\sigma }(\textbf{k}_ \bot ^{}) ={-} i{[{\hat{\textbf{v}}_\sigma^{}(\textbf{k}_ \bot^{})} ]^ \ast } \cdot ({\nabla _\textbf{k}})\hat{\textbf{v}}_\sigma ^{}(\textbf{k}_ \bot ^{}) ={-} \sigma \frac{{\cot \theta }}{k}{\hat{\textbf{e}}_\varphi },$$

(17)$${\textbf{F}_\sigma }(\textbf{k}_ \bot ^{}) = {\nabla _\textbf{k}} \times {\textbf{A}_\sigma }(\textbf{k}_ \bot ^{}) = \sigma \frac{\textbf{k}}{{{k^3}}}.$$

Fig. 2. Geometric representation of Berry phase. (a) Schematic of the coupling between the local spin σ_z and coordinate rotation φ in a unit k-direction sphere. The cyan cone represents the angular spectrum of the light beam. The yellow arrows indicate the wave vectors of arbitrary plane waves. The red circles and arrows represent uniform CPs in the transverse plane of the central wave vector (green arrow). (b) A Pancharatnam-Berry phase is gained as a left-handed CP plane wave with k experiences a spin-reversal propagation, which is half of the solid angle Ω of the shaded area enclosed by two paths (k and k’) connecting the north and south poles on the Poincaré sphere.

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Here $\sigma \in [ - 1,1]$ is the local spin of each plane wave, which is a function of k-vector. Only the ${\hat{\textbf{e}}_\varphi }$ component of the Berry connection is non-vanishing. So we have the Berry phase as

(18)$${\mathrm{\boldsymbol{\Phi}}} _\textbf{B}^{} = \int_C {{\textbf{A}_\sigma }(\textbf{k}_ \bot ^{}) \cdot d} \textbf{k} ={-} \sigma \cos \theta \cdot \varphi .$$

Here $\sigma \cos \theta$ is the spin of any plane waves projected in the k_z direction. Since both the normal and abnormal modes in their transverse plane are uniform CP beams (Fig. 2(a)), which requires ${\sigma _z} = \sigma \cos \theta ={\pm} 1$, the Berry phase can also be expressed as

(19)$${\mathrm{\boldsymbol{\Phi}}} _\textbf{B}^{} ={-} {\sigma _z}\varphi ={\mp} \varphi .$$

This phase is only associated with the path of coordinate rotation and manifests as the coupling of spin ${\sigma _z}$ and coordinate rotation φ [46], so it is geometrical. φ is also an azimuthal phase factor, which is related to the intrinsic OAM, therefore ${\mathrm{\boldsymbol{\Phi}}}_\textbf{B}^{}$ also reflects the coupling between the SAM and the intrinsic OAM.

In essence, this Berry phase comes from the topological structure of the beam itself, which is an unobservable quantity. Only the phase difference between the final state and the initial state of the beam is an observable quantity. For the abnormal mode, the phase difference is ±2φ because of the spin-flip; for the normal mode, the phase difference is zero. Therefore the final phase of the abnormal mode and the normal mode are

(20)$${\mathrm{\boldsymbol{\Phi}}}_\textbf{B}^{abn} = 2{\mathrm{\boldsymbol{\Phi}}} _\textbf{B}^{} ={\pm} 2\varphi ,\quad {\mathrm{\boldsymbol{\Phi}}} _\textbf{B}^{nor} = 0. $$

This result is exactly the same as that given by Eq. (13), which is also one of the central results of this paper. The abnormal-mode phase results from the spin reversal, which is of the Pancharatnam-Berry origin, representing the geometric phase acquired by the spin state related to the wave-vector k moving from the north (south) pole to the south (north) pole on the Poincaré sphere along different longitudes tied with k [Fig. 2(b)]. Here each k-vector takes a longitude on the Poincaré sphere. Although this phase is very similar to the Pancharatnam-Berry phase (also vortex phase) in Q-plates to some extent, it has a different physical origin [10–12,44–46].

3. Discussion and comparison

3.1 Efficiency of the vortex generation

This section discusses the efficiency of vortex generation, that is, the efficiency of SOI. Given the field distribution [Eq. (7)] at z=0, and combining Eqs. (12)-(15), the field distribution at any transmission distance can be solved. It is assumed that at the waist plane (z=0), the incident beam is a left-handed CP Gaussian one:

(21)$${{\mathbf E}_ + }({\mathbf r}_ \bot ^{},0) = \exp \left( { - \frac{{r_ \bot^2}}{{w_0^2}}} \right)\hat{\textbf{V}}_ + ^{}, $$

where w₀ is the half width at beam waist. The angular spectrum of the above Gaussian beam is $\tilde{U}_ + ^{}(\textbf{k}_ \bot ^{},0) = \frac{{w_0^2}}{2}\exp [{ - {{({w_0}k_ \bot^{})}^2}/4} ]$. In paraxial approximations (${k_ \bot } \ll {k_0}$), we get ${k_{oz}} = {(k_0^2n_o^2 - k_ \bot ^2)^{1/2}} \approx {k_0}{n_o} - \frac{{k_ \bot ^2}}{{2{k_0}{n_o}}}$ and ${k_{ez}} = \frac{{{n_o}}}{{{n_e}}}{(k_0^2n_e^2 - k_ \bot ^2)^{1/2}} \approx {k_0}{n_o} - \frac{{{n_o}k_ \bot ^2}}{{2{k_0}n_e^2}}$. Then the normal and abnormal modes of the electric field at the transmission distance z can be solved analytically by combining Eqs. (7), (13) and (15):

(22)$$\begin{aligned} {E_ + }({\mathbf r}_ \bot ^{},z) &= \int {{d ^2}{\mathbf k}_ \bot ^{}} {e^{i{\mathbf k}_ \bot ^{} \cdot {\mathbf r}_ \bot ^{}}}{{\tilde{U}}_ + }({\mathbf k}_ \bot ^{},z)\\ & = 2\pi w_0^2\left\{ {\frac{1}{{w_o^2(z)}}\exp \left[ { - \frac{{r_ \bot^2}}{{w_o^2(z)}}} \right] + \frac{1}{{w_e^2(z)}}\exp \left[ { - \frac{{r_ \bot^2}}{{w_e^2(z)}}} \right]} \right\} \end{aligned}$$

(23)$$\begin{aligned} {E_ - }({\mathbf r}_ \bot ^{},z) &= \int {{d ^2}{\mathbf k}_ \bot ^{}} {e^{i{\mathbf k}_ \bot ^{} \cdot {\mathbf r}_ \bot ^{}}}{{\tilde{U}}_ - }({\mathbf k}_ \bot ^{},z)\\ & ={-} {e^{i2\varphi }}2\pi w_0^2\left\{ {\frac{{r_ \bot^2 + w_o^2(z)}}{{r_ \bot^2w_o^2(z)}}\exp \left[ { - \frac{{r_ \bot^2}}{{w_o^2(z)}}} \right] - \frac{{r_ \bot^2 + w_e^2(z)}}{{r_ \bot^2w_e^2(z)}}\exp \left[ { - \frac{{r_ \bot^2}}{{w_e^2(z)}}} \right]} \right\} \end{aligned}$$

Here $w_o^2(z) = w_0^2 + \frac{{i2z}}{{{n_o}{k_0}}}$ and $w_e^2(z) = w_0^2 + \frac{{i2{n_o}z}}{{n_e^2{k_0}}}$.

Next, we can calculate the conversion efficiency of the abnormal mode as [9,34]

(24)$${\eta _\textbf{G}} = \frac{{{{\int {{d^2}{\textbf{k}_ \bot }|{{{\tilde{U}}_ - }(\textbf{k}_ \bot^{},z)} |} }^2}}}{{{{\int {{d^2}{\textbf{k}_ \bot }|{{{\tilde{U}}_ + }({\textbf{k}_ \bot^{},0} )} |} }^2}}} = \frac{{{{\int {{d^2}{\textbf{r}_ \bot }|{{E_ - }(\textbf{r}_ \bot^{},z)} |} }^2}}}{{{{\int {{d^2}{\textbf{r}_ \bot }|{{E_ + }(\textbf{r}_ \bot^{},0)} |} }^2}}}. $$

After a series of algebra, we can get

(25)$${\eta _\textbf{G}} = \frac{1}{2}\left[ {1 - \frac{1}{{1 + {{({z/L} )}^2}}}} \right], $$

where $L = {k_0}{n_o}w_0^2/(n_o^2/n_e^2 - 1)$. This means that the efficiency of the abnormal mode will increase with z, and gradually reach its maximum value 50% [ Fig. 3(a)]. Therefore, for a Gaussian beam, the maximum efficiency of vortex generation is 50%. The smaller the beam half-width w₀, the smaller the propagation distance required to reach the maximum efficiency. This conclusion also agrees with the experimental results [35]. Note that in experiments, the normal and abnormal modes can be separated by a circular polarizer (a quarter-wave plate followed by a linear polarizer) [35].

Fig. 3. The efficiency of vortex generation at two different values of w₀ for (a) Gaussian beams and (b) Bessel beams, respectively. In the calculation, we set λ=633 nm, n_o=1.656 and n_e=1.458.

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Then we consider a left-handed CP Bessel beam $\textbf{E}_ + ^{}({\textbf{r}_ \bot^{},0} )= {A_0}{J_0}({\Delta k{r_ \bot }} )\hat{\textbf{V}}_ + ^{}$, where A₀ is an arbitrary amplitude, $\Delta k = 2\pi /{w_0}$ is the spectral half-width in the transverse direction, and ${J_n}(\xi )$ represents the Bessel function of the first kind of order n. The angular spectrum distribution of a Bessel beam is a delta function: $\tilde{U}_ + ^{}({k_ \bot },0) = \frac{{{A_0}{w_0}}}{{{2^{5/2}}{\pi ^2}}}\delta ({k_ \bot } - \Delta k)$ which is in fact a hollow cone [Fig. 2(a)], with the central axis parallel to the z-direction, and the angle between any plane wave component and the central axis is $\theta = {\sin ^{ - 1}}(\Delta k/{k_0})$.

The normal and abnormal modes of electric field at the propagation distance z are

{{\mathbf E}_ + }({\mathbf r}_ \bot ^{},z) = {t_{ +{+} }}(\Delta k,z){J_0}({\Delta k{r_ \bot }} )\hat{\textbf{V}}_ + ^{},

(26)$${{\mathbf E}_ - }({\mathbf r}_ \bot ^{},z) = - \exp (i2\varphi ){t_{ - + }}(\Delta k,z){J_2}({\Delta k{r_ \bot }} )\hat{\textbf{V}}_ - ^{}.$$

The conversion efficiency can also be solved by substituting the above equation into Eq. (24), and we have

(27)$${\eta _\textbf{B}} = {|{{t_{ -{+} }}(\Delta k,z)} |^2} = \frac{1}{4}{|{\exp (i{k_{ez}}z) - \exp (i{k_{oz}}z)} |^2} = \frac{1}{4}{\left|{\exp \left( { - i\frac{{2{\pi^2}\Delta }}{{{k_0}{n_o}w_0^2}}z} \right) - 1} \right|^2}. $$

Here $\Delta = (n_o^2/n_e^2 - 1)$ shows the anisotropy of the uniaxial crystal. In the above equation, the efficiency oscillates in a sinusoidal form as the propagation distance z increases, with the maximum value 100% and the minimum value zero [Fig. 3(b)]. Therefore, for Bessel beams, 100% efficiency can be achieved. It is noted that the angular spectrum of a Gaussian beam can be decomposed into infinite Bessel k-cones with continuously changing radius, and its efficiency is also their weighted average. With the increase of transmission distance z, the “average” efficiency gradually increases, and finally approaches to 50%.

3.2 Comparison of the vortex generation in other SOI systems

When a CP light beam passes through a Q-plate, a part of which reverses its spin, and acquires a vortex phase factor 2ϕ twice the local optical axis direction ϕ(x,y) of the Q-plate [10–13]. This phase factor is a Pancharatnam-Berry phase, a function of real coordinates (x,y), originating from the inhomogeneous anisotropy of the Q-plate. The Berry phase of the beam propagating in the uniaxial crystal we discussed above is related to (k_x,k_y), which comes from the inhomogeneous anisotropy of o- and e-wave components of different plane waves within the beam. The same thing is that the final Berry phases of the abnormal mode are both related to spin reversal. Therefore, the Berry phase discussed here could also be regarded as a Pancharatnam-Berry phase in this viewpoint.

Our theory can also explain the generation of vortex phase when a light beam is normally impinging upon a sharp interface composed of isotropic materials [8,9]. Upon reflection and refraction, a portion of the incident beam reverses its spin, and obtains a vortex phase with a topological charge of ±2. This process is very similar to that discussed in this paper. When the beam is reflected and refracted at the interface, the incidence angle of each angular spectrum component is different, which makes the Fresnel coefficients of their TM and TE components have different “effective” anisotropy. While beams propagating in a uniaxial crystal does not have reflection and refraction process, and the anisotropy comes from the uniaxial crystal itself.

When a Q-plate has a half-wave phase retardation, the efficiency of vortex generation is 100% [10–13]; but for a general sharp interface, the efficiency is extremely low [8,9]. So it is difficult to observe the vortex generation in experiments for an isotropic interface. Here for a bulk uniaxial crystal, the efficiency of vortex generation increases or oscillates with the beam propagation distance; especially for Bessel beams, the efficiency can reach 100%.

4. Conclusions

We have established a full-wave theory to describe the beam propagation along the optical axis in a uniaxial crystal, and revealed the physical origin of vortex generation. The theory is mainly composed of three cascaded matrices, which clearly shows the different roles played by the beam itself and the anisotropy of the uniaxial crystal. The physical origin of the vortex generation is attributed to the topological nature of the beam itself, dictated by the Berry phase mechanism. The anisotropy of uniaxial crystal results in the spin reversal of a portion of the beam, which, together with the beam width, affects the generation efficiency of the vortex. Our findings provide a deeper perspective for understanding the SOI-induced effects of light and pave the way for possible applications.

Funding

National Natural Science Foundation of China (11604087, 11874142); Natural Science Foundation of Hunan Province (2018JJ1001); National Key Research and Development Program of China (2017YFA0700202); Funding of Key Laboratory of Optoelectronic Control and Detection Technology of the institution of higher learning of Hunan Province; Excellent Talents Program of Hengyang Normal University.

Disclosures

The authors declare no conflicts of interest.

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Vortex generation in the spin-orbit interaction of a light beam propagating inside a uniaxial medium: origin and efficiency

Abstract

1. Introduction

2. Physical origin of the vortex phase in bulk uniaxial crystals

2.1 Full-wave theory of light beams propagating along the optical axis in a uniaxial crystal

2.2 Berry phase: the physical origin of vortex phase

3. Discussion and comparison

3.1 Efficiency of the vortex generation

3.2 Comparison of the vortex generation in other SOI systems

4. Conclusions

Funding

Disclosures

References

Cited By

Figures (3)

Equations (29)

Optics Express