Evolution of the Stokes parameters, polarization singularities and optical skyrmion

Here we adopt the well known procedure of the “paraxial approximation”. Let ${\bm{E}}$ be the light field (electric field) which is traveling through the anisotropic medium in $z$-direction. We suppose the modified plane wave LL : ${\bm{E}}(x,y,z)={\bm{f}}(x,y,z)\exp[ikn_{0}z]$ where $k=\frac{{\omega}}{c}$ and $n_{0}(\equiv\sqrt{\epsilon_{0}})$ is the refractive index of the isotropic medium that is just used for reference. We choose the $z$-direction as one of three principal axes of the dielectric tensor $\hat{\epsilon}$ (see below). The amplitude ${\bm{f}}(x,y,z)$ is written in terms of the linear polarization basis: ${\bm{f}}={}^{t}(f_{1},f_{2})=f_{1}{\bm{e}}_{1}+f_{2}{\bm{e}}_{2}$ with the mutual orthogonal basis ${\bm{e}}_{1,2}$. Alternatively, we transform the basis vectors to the circular polarization basis: ${\bm{f}}=\psi_{1}{\bm{e}}_{+}+\psi_{2}{\bm{e}}_{-}$ with ${\bm{e}}_{\pm}=\left(1/{\sqrt{2}}\right)({\bm{e}}_{1}\pm i{\bm{e}}_{2})$. The corresponding two-component wave is given by

Now we introduce the Stokes parameters, for which it is efficient to use the wave (1) written in terms of the circular polarization basis Dennis ; Kakigi in contrast to the linear polarization basis, e.g., Ref. Brosseau . The former is particularly suitable to discuss the polarization singularity as is seen below. Using the Pauli-spin $\hat{\sigma}_{i}$, the Stokes parameters are written as

or explicitly,

The vector ${\bm{S}}~{}(=(S_{1},S_{2},S_{3}))$ satisfies the relation $S_{0}^{2}=S_{1}^{2}+S_{2}^{2}+S_{3}^{2}$, namely, $S_{0}$ represents the field strength, $S_{0}\equiv\left|{\bm{E}}\right|^{2}$. Using the complex polar representation footnote

we obtain the polar form:

This parametrization is used for the pictorial representation of the polarization on the Poincaré sphere, which is given in Fig.1

Singular feature of the Stokes parameters: The expressions (4) and (5) bear the polarization singularity in an apparent way. This is shown by introducing the stereographic coordinate Arecchi

which is the projection from the south pole. From this expression one sees that the phase $\phi$ cannot be defined at the point $w=0$ or $\theta=0$, which corresponds to the Stokes vector $S_{3}=+S_{0}$. We have another form of the stereographic coordinate that means the projection from the north pole:

for which the phase $\phi$ is also indefinite at $w=0$ or $\theta=\pi$, which corresponds to the Stokes vector $S_{3}=-S_{0}$.

The above observation is reminiscent of the fact that the phase loses the meaning at the zero of wave function. This implies that the two circular polarizations reveal the singularities of the polarized light, called the C-point. On the other hand, at the equatorial circle $\theta=\frac{\pi}{2}$, $w=\exp[-i\phi]$ (or $w=\exp[i\phi]$), there do not reveal any singularities, and this circle divides the Poincaré sphere into two hemispheres corresponding to left-handed and right-handed polarizations. So we call the equatorial circle the L-line.

In the paraxial scheme, the Maxwell (Helmholz) equation is written for the electric field ${\bm{E}}$:

where ${\nabla}^{2}$ means the Laplacian in the transverse plane, ${\nabla}^{2}\equiv\frac{{\partial}^{2}}{\partial x^{2}}+\frac{{\partial}^{2}}{\partial y^{2}}$. The dielectric tensor $\hat{\epsilon}$ stands for the $2\times 2$ Hermitian matrix.

First we consider the special case that the Laplacian can be discarded, which means that the modulation of light field ${\bm{E}}$ in the transverse plane $(x,y)$ changes slowly compared with the change in the longitudinal direction $z$. This procedure may be called the single mode approximation, for which the modulated amplitude depends only on the propagation distance $z$, namely, ${\bm{E}}(z)$. Hence we write

By applying the approximation of slowly varying amplitude (alias the envelope approximation) Akhmanov , namely, $\left|\frac{{\partial}^{2}{\bm{f}}}{\partial z^{2}}\right|\ll k\left|\frac{\partial{\bm{f}}}{\partial z}\right|$, we keep only the first derivative term $\frac{\partial{\bm{f}}}{\partial z}$. Writing it in terms of the circular polarization basis, we arrive at two-component nonlinear Schrödinger equation for $\psi$:

Here, $\lambda\left(=\frac{2\pi}{k}\right)$ is the wave length. The transformed Hamiltonian is given by $\hat{H}=-\frac{\pi}{n_{0}}T(\hat{\epsilon}-n_{0}^{2})T^{-1}$. In terms of the Pauli-spin we write

which may depend in general on the field $(\psi^{\dagger},\psi)$ as is shown below. In what follows the wave equation (10) is transcribed to the evolutional equation for the Stokes parameters (see, e.g.,Refs. Sala ; Tratnik ; Kakigi ). The concise way to carry this out is given by using the action principle Kra . That is, we introduce the action function $I=\int L\,dz$ with the “Lagrangian”:

The variational equation $\delta I=0$ leads to

Using the Stokes vector, we have

with the expectation value of the Hamiltonian $H={\psi}^{\dagger}\hat{H}\psi$. The evolution equation (14) determines the orbit on the Poincaré sphere once the parameters $(\alpha,\beta,\gamma)$ are provided.

Following the conventional terminology, the meaning for the parameters appearing in the Hamiltonian is explained as follows: $\alpha$ and $\beta$ represent the linear birefringence, whereas $\gamma$ represents the optical activity or chirality. The crucial point here is that these parameters depend on the spatial coordinate, generally such that $\alpha(x,y.z)$ etc. In what follows, we restrict the argument to the special case in which these parameters have dependence only on the transversal coordinates, such that $\alpha(x,y)$.

This case is realized in such a way that the electric field is applied in the transverse plane: ${\bm{{\mathcal{E}}}}(x,y)=({\mathcal{E}}_{x},{\mathcal{E}}_{y})$, which is arranged to depend on the transversal coordinate $(x,y)$. In this way, the coordinates $(x,y)$ could be treated as if they were constant during manipulation of the evolution of the Stokes vectors.

As the dielectric tensor, we adopt the one coming from an external Kerr effect LL , namely,

Hence the wave equation for the amplitude ${\bm{f}}$ (namely, the linear polarization basis) is written as

with $a=\frac{\pi}{n_{0}}G\left[\frac{{\mathcal{E}}_{x}^{2}-{\mathcal{E}}_{y}^{2}}{2}\right]$ and $b=\frac{\pi}{n_{0}}G{\mathcal{E}}_{x}{\mathcal{E}}_{y}$. We have here omitted the term arising from the conventional Kerr effect, which is proportional to $G|{\mathcal{E}}|^{2}$ with $|{\mathcal{E}}|^{2}={\mathcal{E}}_{x}^{2}+{\mathcal{E}}_{y}^{2}$. In terms of the circular polarization basis, this results in the Hamiltonian:

Now it is instructive to examine the wave equation (16), for which there are two independent solutions $f_{\pm}(z)=\exp\left[\pm i\frac{G}{2n_{0}}|{\mathcal{E}}|^{2}kz\right]$. By combining these with the reference plane wave, one gets the total polarization wave

which correspond to two refractive indices, namely: $n_{\|}=n_{0}+\frac{G}{2n_{0}}|{\mathcal{E}}|^{2}$ and $n_{\perp}=n_{0}-\frac{G}{2n_{0}}|{\mathcal{E}}|^{2}$, where the symbol $\|$ means the polarization parallel to the applied field, while $\perp$ means the polarization perpendicular to the applied field foot . As such the emergence of two refractive indices represents the literal meaning of birefringence, which is contrast to the optical activity.

We now examine the evolution of the Stokes parameters. The corresponding Hamiltonian reads $H=aS_{1}+bS_{2}$, which is transformed to

by using the orthogonal transform:

with $\tan\Theta=\frac{b}{a}$. Here, for the sake of simplicity of notation, we write $c=\sqrt{a^{2}+b^{2}}$. Then the evolution equation becomes

leading to

with the initial phase being chosen to be zero.

We summarize a type of polarization, “quantization rule” and trajectory for each typical value of $S_{3}(z)$ in Table 1.

Thus, noting that $c~{}(\equiv\sqrt{a^{2}+b^{2}})\propto G({\mathcal{E}}_{x}^{2}+{\mathcal{E}}_{y}^{2})$, the above rule forms a surface in $(x,y,z)$ space. From Table 1, for the particular case, $n=0$, we have ${\mathcal{E}}_{x}=0$ and $~{}{\mathcal{E}}_{y}=0$. These two equations determine the point in the transversal plane $(x,y)$. In other words, the zero point of the external electric field forms a C-line in three dimensional space. For nonzero $n$, the corresponding trajectory forms the objects; the C-points become the “C-surface”, whereas the L-line the “L-volume”. If cutting these objects by the plane $z=const.$, we have co-centric pattern according to the value $n$. These represent the L-surface and C-circle, so to speak, which are arranged according to the numbering of $n$.

At this point it is natural to consider the case such that $\mathcal{E}=|\mathcal{E}_{0}|\sqrt{f(r)}\hat{r}$, ($r=\sqrt{x^{2}+y^{2}},~{}\hat{r}=\frac{{\bm{r}}}{r}$), namely, the radial vector in the transverse plane, which is axially symmetric with respect to the $z$ axis. In what follows we demonstrate the simplest choice, $f(r)=r$. The corresponding profile is given in Fig. 2, which represents the rotational body around $z$-axis. It is apparent that the $z$ axis (that is $r=0$) forms a C-line.

Furthermore the following point must be mentioned: The configurational structure of polarization singularity can be controlled from an external condition through modulation of an electric field, for example, by allowing the time variation of the amplitude ${\mathcal{E}}_{0}(t)$.

Remarks on the role of optical activity: In the above argument, it is essential that the Hamiltonian does not include the term coming optical activity or chirality governed by $S_{3}$, which causes $S_{3}$ to be invariant under Eq. (20). So one may wonder how about the the effect of chirality or optical activity. It is easy to examine this effect: Instead of Eq. (19) let us consider $H=aS_{1}+\gamma S_{3}=\Gamma S_{1}^{\prime}$ with $\gamma=v{\mathcal{B}}$ (${\mathcal{B}}$ is the magnetic field and $v$ is the Verde constant), $\Gamma=\sqrt{a^{2}+{\gamma}^{2}}$. One has the orthogonal transform corresponding to Eq. (20)

with $\tan\Phi=\frac{\gamma}{a}$. Then the evolution equation becomes $\lambda\frac{dS_{1}^{\prime}}{dz}=0,~{}\lambda\frac{dS_{2}}{dz}=-2\Gamma S_{3}^{\prime},~{}\lambda\frac{dS_{3}^{\prime}}{dz}=2\Gamma S_{2}$, from which one gets $S_{3}(z)=C\sin\Phi+S_{0}\cos\Phi\cos\left[\frac{2}{\lambda}\Gamma z\right]$, where $C$ is the constant of motion $S_{1}^{\prime}=C$. The trajectory of the C-line (or L-surface) is thus derived from

From this expression one sees that the additional term, which depends on the angle $\Phi$ as well as arbitrarily chosen constant of motion $C$, violates a simple quantization rule leading to the polarization singularities. This fact suggests that the optical activity plays a role of obstacles for forming the polarization singularities. The exceptional case is that; $\Phi=0$ (that means $\gamma$ vanishes), for which the problem is reduced to the case of the pure birefringence.

As an extreme case, we consider the pure optical activity, for which the evolution equation reads $\dot{S}_{1}=-\frac{2}{\lambda}\gamma S_{2},~{}\dot{S}_{2}=\frac{2}{\lambda}\gamma S_{1},~{}\dot{S}_{3}=0$. From this we have the constant of motion $S_{3}=C$. Here the points $C=\pm S_{0},~{}0$; the circular and linear polarization should be careful. Namely, the circular polarization consists only of “isolated points” on a Poincaré sphere arbitrarily chosen in 3-dimensional space, whereas the linear polarization describes the equatorial circle on the same sphere.

Turning to the nonlinear Kerr media, it is known that the dielectric tensor is modified to be expressed in terms of the complex components of the spinor LL ; Maker . Namely this is given as

The corresponding Hamiltonian operator becomes

which results in the expectation value: $\psi^{\dagger}\hat{V}_{nl}\psi=-\frac{g}{2}(S_{1}^{2}+S_{2}^{2}-S_{3}^{2})\equiv gS_{3}^{2}$ up to additional constant. By taking account of this term, the Hamiltonian (19) is modified as

By this modified Hamiltonian, the equation of motion turns out to be Sala ; Kra

for which we have the two constants of motion:

One sees that by choosing the energy constant as $E=gS_{0}^{2}$, the equation of motion is derived for the third component $S_{3}$. According to the ratio between the linear birefringence and nonlinear coupling, namely, $\kappa=\frac{gS_{0}^{2}}{c}$ (called moduli parameter), we have two cases of equations for $S_{3}$; (A) $c>gS_{0}^{2}$ and (B) $c<gS_{0}^{2}$ footnote1 . By putting $S_{3}=S_{0}X$ and $\tau=\frac{2cz}{\lambda}$, these are written as

with $\bar{\kappa}=\sqrt{1-\frac{1}{{\kappa}^{2}}}$. Hence the corresponding solution of $S_{3}$ are given by

where ${\rm{cn}}$ and ${\rm{dn}}$ represent the Jacobi elliptic functions.

The former solution indicates $S_{3}$ oscillates between $-S_{0}$ and $S_{0}$, while the latter exhibits the small oscillation in the vicinity of the north (or south) poles; this feature is implied from $\bar{\kappa}S_{0}\leq|S_{3}|\leq S_{0}$. In what follows we give analysis for the trajectory of the C-points and L-lines, $S_{3}(\tau)=\pm S_{0},~{}0$.

For case (A), $\kappa<1$, we summarize a type of polarization, “quantization rule” and trajectory for each typical value of $S_{3}(z)$ in Table 2.

In Table 2, $K(\kappa)$ denotes the period of the ${\rm{cn}}$ function, which is given by the complete elliptic integral:

The contents of the Table 2 correspond to the contents obtained for the case of pure linear birefringence.

The trajectory revealing these singularities is rather complicated to display, because of the highly sophisticated nature of the elliptic functions. Here we note that $n=0$ is not allowed, because it yields $c\propto G\left({\mathcal{E}}_{x}^{2}+{\mathcal{E}}_{y}^{2}\right)=0$, but this contradicts to the constraint $c>gS_{0}^{2}>0$. In other words, there does not appear the case that ${\bm{\mathcal{E}}}=({\mathcal{E}}_{x},{\mathcal{E}}_{y})$ vanishes leading to the C-line. This feature is in contrast to the case of the linear birefringence.

We discuss the trajectory of left-handed circular polarization by adopting the same profile as the linear birefringence for the applied external field, namely $|{\mathcal{E}}|^{2}=|{\mathcal{E}}_{0}|^{2}r$. We take up the case $n=1$, that is, the period $4K(\kappa)$. Then, we write

On the other hand, the trajectory of left-handed circular polarization is also written as

With the aid of these two relations, one can eliminate the moduli parameter $\kappa$ to get a trajectory in the $(\bar{r},\bar{z})$ plane (see Fig. 3). Note that in the limit $\kappa\rightarrow 0$ ($g\rightarrow 0$) we recover the trajectory for the linear birefringence.

Now turning to the case (B): $\kappa>1$, we see that there appear two orbits, for which we give a brief sketch. These are determined by ${\rm{dn}}=\pm 1$, which turn out to be $\frac{2cz}{\lambda}=4K(1/\kappa)$ for $S_{3}=+S_{0}$ and $\frac{2cz}{\lambda}=2K(1/\kappa)$ for $S_{3}=-S_{0}$. There are no orbits for the linear polarization, $S_{3}=0$, because of the ${\rm{dn}}$ function takes never zero.

Remarks: If the term coming from the optical activity is added to the Hamiltonian, the equation of motion is not reduced to a simple form that is described by the third component of the Stokes parameters. This implies that one has the same problem as in the linear birefringence case; that is, the optical activity destroys a simple quantization rule to extract the configurational structure revealing the polarization singularities.