Paraxial Skyrmionic beams

TL;DR

This paper explores the topological properties of vector vortex beams, specifically their Skyrmionic nature, and discusses how the Skyrmion number characterizes their polarization and amplitude variations.

Contribution

It introduces the concept of Skyrmion number in vector vortex beams and analyzes its physical significance in describing their topological properties.

Findings

Vector vortex beams exhibit Skyrmionic topological properties.

The Skyrmion number quantifies the beam's polarization and amplitude variations.

Understanding Skyrmion number aids in characterizing complex light fields.

Abstract

Vector vortex beams possess a topological property that derives both from the spatially varying amplitude of the field and also from its varying polarization. This property arises as a consequence of the inherent Skyrmionic nature of such beams and is quantified by the associated Skyrmion number, which embodies a topological property of the beam. We illustrate this idea for some of the simplest vector beams and discuss the physical significance of the Skyrmion number in this context.

Equations

$$|\Psi (\mathbf{r})⟩=u_0(\mathbf{r})|0⟩+e^{i{\theta }_0}{u}_1(\mathbf{r})|1⟩.$$

$$|\psi (\mathbf{r})⟩=\frac{|0⟩+e^{i{\theta }_0}v(\mathbf{r})|1⟩}{\sqrt{1+|v(\mathbf{r}){|}^2}},$$

$$\mathbf{M}=⟨\psi (\mathbf{r})|\mathbf{\sigma }|\psi (\mathbf{r})⟩,$$

$${\Sigma }_i=\frac{1}{2}{ϵ}_{ijk}{ϵ}_{pqr}{M}_p\frac{\partial M_q}{\partial x_j}\frac{\partial M_r}{\partial x_k},$$

$$n=\frac{1}{4\pi }\int {\Sigma }_{z}dxdy,$$

$$u_p^{\ell }(\rho ,\varphi ,z)$$

$$\times L_p^{|\ell |}\left(\frac{2{\rho }^2}{w^2(z)}\right)e^{i\ell \varphi }\exp \left(-i\frac{{\rho }^2}{w^2(z)}\frac{z-z_0}{z_R}\right)$$

$$\times \exp \left[-i(2p+|\ell |+1){\tan }^{-1}\left(\frac{z-z_0}{z_R}\right)\right],$$

$$v(\mathbf{r})=f(\rho ,z)e^{i\theta (\rho ,z)}{e}^{i\Delta \ell \varphi },$$

$$n=\Delta \ell \left(\frac{1}{1+f^2(0,z)}-\frac{1}{1+f^2(\infty ,z)}\right).$$

$${\int }_0^{2\pi }$$

$$-\Delta \ell \left(\frac{1}{1+f^2(R,-z_0)}-\frac{1}{1+f^2(R,z_0)}\right)$$

$$\Delta \ell \left(\frac{1}{1+f^2(R,-z_0)}-\frac{1}{1+f^2(R,z_0)}\right),$$

Full text

Optical skyrmions, polarization, orbital angular momentum, topology

I Introduction

Recent developments have highlighted the growing utility of structured light, that is optical fields in which the spatial variation of the field amplitude and/or the polarization are specifically designed for a given task Nye (1999); Nye and Berry (1974); Zambrini and Barnett (2007); Dennis et al. (2009); Rubinsztein-Dunlop et al. (2016). Important examples include the formation of optical beams carrying orbital angular momentum Allen et al. (1992, 2003); Bekshaev et al. (2008); Franke-Arnold et al. (2008); Yao and Padgett (2011), polarization or helicity patterns Cohen-Tannoudji and Phillips (1990); Dalibard and Cohen-Tannoudji (1989); Cameron et al. (2012, 2014); van Kruining et al. (2018); Kravets et al. (2019) and the vector vortex beams and their relatives Zhan (2009); Piccirillo et al. (2010); Milione et al. (2011, 2012); D’Ambrosio et al. (2012); Ndagano et al. (2016); Alpmann et al. (2017); Radwell et al. (2016); Rosales-Guzmán et al. (2017). We show that there is a Skyrmion field associated specifically with vector vortex beams and that the associated Skyrmion number is readily identified with a simple property of the beam. As such the Skyrmion number provides a natural way to present the variety of possible vector beams. It is noteworthy that this property is explicitly a feature of vector beam: a Skyrmion field exists only if both the polarization and the field amplitude are spatially varying.

Skyrmions were first proposed for the study of mesons Skyrme (1961, 1962), but the idea has since found wide application in many areas of physics including quantum liquids Vollhardt and Wolfle (2013); Volovik and Press (2003); Leggett (2006), magnetic materials Sachdev (2011); Seki and Mochizuki (2015); Dennis (2011), 2D photonic materials Mechelen and Jacob (2019) and in the study of fractional statistics Wilczek (1990). Recently they have been observed in optics by the controlled interference of plasmon polaritons Tsesses et al. (2018); Du et al. (2019). We show here that a wide range of freely propagating optical beams also possess a non-trivial Skyrmion field and with it a Skyrmion number, the value of which is simply related to a topological property of the beam.

II Constructing Skyrmionic beams

We consider a paraxial beam of either light Marcuse (1989); Siegman (1986) or electrons El-Kareh and El-Kareh (2016); Klemperer and Barnett (1971); Hawkes (1972) and express the local polarization or spin direction, respectively, in the form

[TABLE]

Here $|0\rangle$ and $|1\rangle$ represent any two orthogonal polarization (or electron spin) states, while $u_{0}(\bm{r})$ and $u_{1}(\bm{r})$ are two orthogonal spatial modes 111We have used a quantum mechanical notation as this aids the analysis to come, but our results apply both to classical and quantum states of light and also to electron beams. and the global phase difference between the two modes is denoted by $\theta_{0}$. That this decomposition is always possible follows from the Schmidt decomposition Barnett (2009). The Skyrmion field and number depend only on the spatial variation of the polarization or spin direction and for this reason it is convenient to work with a locally-normalized state in the form

[TABLE]

where $v(\bm{r})=u_{1}(\bm{r})/u_{0}(\bm{r})$.

The Skyrmion field is most readily defined in terms of an effective magnetization $\bm{M}$, which is the local direction of the Poincaré vector for light in Fig. 1 or the Bloch vector for an electron beam. In terms of our locally normalized state it is

[TABLE]

where $\bm{\sigma}$ is a vector operator with the Pauli matrices as Cartesian components. For a light beam, the Cartesian components of $\bm{M}$ correspond to the normalized local Stokes parameters $S_{1},S_{2}$ and $S_{3}$ Born and Wolf (2000), and for the electrons to the local directions of the electron spin. The $i$th component of the associated Skyrmion field is

[TABLE]

where $\epsilon_{ijk}$ is the alternating or Levi-Civita symbol and we employ the summation convention. The form of the Skyrmion field ensures that it is transverse $(\bm{\nabla}\cdot\bm{\Sigma}=0)$. This means that there are no sources or sinks for the Skyrmion field and the associated field lines can only form loops or extend to infinity 222This property is familiar from the study of electric and magnetic fields $\bm{E}$ and $\bm{B}$ in free space. We note, however, that $\bm{\Sigma}$ for light is a nonlinear function of $\bm{E}$ and $\bm{B}$ and so its transverse nature is not simply a consequence of $\bm{\nabla}\cdot\bm{E}=0=\bm{\nabla}\cdot\bm{B}$.. It follows that the flux of the Skyrmion field through any closed surface is zero, $\oint\bm{\Sigma}\cdot d\textbf{S}=0$.

We consider a beam propagating in the $z$-direction. In each transverse plane of the beam the polarization or spin pattern can form a Skyrmion reminiscent of those familiar from the study of magnetic Skyrmions. To facilitate this comparison, and also to characterize the variety of Skyrmions, we employ the Skyrmion number

[TABLE]

where the integral runs over the whole of the plane perpendicular to the propagation direction of the beam.

Optical vector vortex beams typically have a spatially varying polarization pattern that originates from the differential orbital angular momentum of the contributing modes Dennis et al. (2009); Piccirillo et al. (2010) and exhibit intriguing topological Freund (2010); Foster et al. (2019); Freund (2011); Bauer et al. (2015) and focussing properties Zhan and Leger (2002); Dorn et al. (2003). We consider the simplest case of such beams in which the two orthogonal modes, with amplitudes $u_{0}(\bm{r})$ and $u_{1}(\bm{r})$, are Laguerre-Gaussian (LG) modes

[TABLE]

familiar from the study of orbital angular momentum Allen et al. (1992, 2003); Bekshaev et al. (2008); Franke-Arnold et al. (2008); Yao and Padgett (2011). Here, we have employed cylindrical polar coordinates ($\rho,\phi$,$z$), $z_{R}=\pi w_{0}^{2}/\lambda$ is the Rayleigh range and $w(z)=w_{0}\sqrt{1+(z-z_{0})^{2}/z^{2}_{R}}$ is the beam width on propagation. We assume that the modes have the same wavelength $\lambda$, but they may differ in the beam parameters $\ell,p,w_{0}$ and the focal position $z_{0}$. These modes have a vortex of strength $\ell$ on the $z$-axis, which is associated with a $z$-component of the orbital angular momentum of $\ell\hbar$ per photon (or electron) Allen et al. (1992, 2003); Bekshaev et al. (2008); Franke-Arnold et al. (2008); Yao and Padgett (2011). Modes with different angular momentum numbers $\ell$ are orthogonal and if we choose two such modes for our two complex amplitudes $u_{0}$ and $u_{1}$ in (1) then the function $v(\bm{r})$ in (2) for the locally normalized state $|\psi(\bm{r})\rangle$ has the general form

[TABLE]

where $\Delta\ell=\ell_{1}-\ell_{0}$, $f$ and $\theta$ are real functions of the coordinates $\rho$ and $z$, and $\theta$ incorporates all phase terms including $\theta_{0}$, the phase difference between the modes. It is then straightforward to calculate the Skyrmion field and from this the Skyrmion number for our vector vortex beam. We find the simple result that for such beams the Skyrmion number is

[TABLE]

The value of $n$ is determined solely by which of the two modes $u_{0}(\bm{r})$ and $u_{1}(\bm{r})$ dominates on the $z$-axis, the location of the vortex, and which dominates as $\rho$ tends to infinity.

The Skyrmionic beams that are simplest to construct comprise a superposition of orthogonal polarization (or spin) states multiplied by $u^{\ell}_{0}$ LG modes with no radial nodes, the same beam width, a common focal point and with orbital angular momentum differing by one. In this case (7) simplifies to $v(\bm{r})=A(z)\rho e^{i\phi}$ (where $A$ is generally complex and includes the overall phase difference $\theta_{0}$) and one polarization dominates at the position of the vortex, with the orthogonal polarization appearing as $\rho\rightarrow\infty$. We provide two examples of such polarization patterns in Figs. 2b & 2d together with the corresponding effective magnetization in Figs. 2c & 2e. The local Bloch vector, representing the local spin direction, is clearly reminiscent of the spiral and hedgehog Skyrmions, familiar from the study of magnetic Skyrmions Seki and Mochizuki (2015); the former arises when $A$ is imaginary and the latter when the amplitude $A$ is real.

We note that the natural propagation of the beam will cause the magnetization or polarization pattern to evolve continuously from one of these forms into the other by virtue of the relative Gouy phase Siegman (1986), which changes as the beam propagates. The Skyrmion number is unchanged, however, taking the value $+1$ at every transverse plane.

There is however a subtle difference in the geometric interpretation between the Poincaré and Bloch sphere. On both spheres, orthogonal states are diametrically opposite. However, for the Poincaré sphere this corresponds to a right angle in the major axes of the polarization, whereas the Bloch vectors of orthogonal states are antiparallel.

We can illustrate the effect of the discrepancy between rotation on the Poincaré sphere and rotation of the polarization ellipse on the geometry of the Skyrmion pattern in a comparision between spiral Skyrmions from superposing LG beams with orbital angular momentum numbers differing by one and two. In Fig. 3 we compare the local polarization ellipse and Bloch vector for a pair of modes with $\Delta\ell=1$ (as in Fig. 2) with a pair of modes for which $\Delta\ell=2$. We see that the polarization ellipses and the Bloch vectors rotate as one traverses a path around the vortex. Moreover, along such a path, the polarization ellipse completes half a rotation when $\Delta\ell=1$, whereas the Bloch vector rotates fully. For $\Delta\ell=2$ the polarization ellipse completes one full rotation and the Bloch vector winds twice for one complete circle around the vortex.

These are examples of a more general result that for a superposition of modes with a difference in orbital angular momentum number of $\Delta\ell$, the Bloch or Poincaré vector rotates $\Delta\ell$ times on a path enclosing the vortex. The corresponding polarization ellipse rotates by only half the amount. This behavior persists when we consider modes with radial indices different from zero, although the polarization structure becomes more intricate because of the additional nodal lines. The resulting Skyrmion number is nevertheless governed by the difference in dominating behavior described in (8).

The corresponding Skyrmion number is $\Delta\ell$ if the spin or polarization states at the vortex position and at infinity are orthogonal but will be zero if they are the same. This dependence of the Skyrmion number on both $\Delta\ell$ and on the position dependence of the polarization clearly demonstrates that the Skyrmion field and number are topological properties of both the spin and orbital angular momenta.

III Conservation of the Skyrmion field

The fact that the Skyrmion field, $\Sigma$, is divergenceless does not mean that the Skyrmion number, defined as the $z$-component of the flux in (5) is necessarily conserved on propagation. Consider a circular-cylindrical surface of radius $R$ centered on the position of the vortex extending from $-z_{0}$ to $z_{0}$. For the Skyrmion field to be divergenceless the flux through all surfaces of this cylinder has to vanish. The radial flux throug the mantle of the cylinder

[TABLE]

is compensated by the flux through the cylinder ends in the $z$-direction at $z=-z_{0}$ and $z=z_{0}$. The expression for these is essentially given by (8), evaluated at $z=-z_{0}$ and $z=z_{0}$ and $\rho=R$ instead of infinity. The two terms evaluated $\rho=0$ for $z=-z_{0}$ and $z=z_{0}$ cancel and the total flux through both ends of the cylinder is given by

[TABLE]

which is the the negative of (III), proving that there is no total flux through the cylinder.

If we now construct a superposition of LG beams such that the radial flux is non-vanishing, the flux along the $z$ direction also needs to be different from zero which indicates a change in the Skyrmion number. The simplest way to demonstrate this is to consider a superposition of LG beams that are focussed at different positions along the $z$-axis. The effect of this is that the polarization behavior at large values of $\rho$ changes as the beam propagates and the Skyrmion number changes from $\Delta\ell$ to [math] (or from [math] to $\Delta\ell$). This behavior is depicted in Fig. 4, where we see that the polarization at large distances from the central vortex changes abruptly at one transverse plane and with it the Skyrmion number. At plane A and B the Skyrmion number is $+1$ and at plane D and E it is equal to zero. The boundary between these two regimes is at plane C, where the Skyrmion field lines escape to $\rho\to\infty$. Clearly, this will give a non-zero value for the radial flux because $f^{2}(R,-z_{0})\neq f^{2}(R,z_{0})$ and hence a change in the Skyrmion number if we allow $R$ to tend to infinity.

IV Conclusions

We have shown that paraxial vector vortex beams, either of light or electrons, possess a topological property that can be identified with a Skyrmion number. The associated Skyrmion field is transverse (or divergenceless) and this means that there are no sinks or sources of this field. The Skyrmion number for a beam can change on free space propagation, however, if Skyrmion field lines escape radially out of the beam towards regions of negligible intensity. Demonstrating these properties requires the preparation of vector vortex beams and measurement of the polarization or spin in planes perpendicular to the beam axis Selyem et al. (2019). We shall report on such experiments elsewhere.

We close by emphasising that the Skyrmionic property of vector beams is distinct from the familiar spin and orbital angular momentum of optical beams Allen et al. (1992, 2003); Bekshaev et al. (2008); Franke-Arnold et al. (2008); Yao and Padgett (2011); Barnett et al. (2016). It is true that the beams we consider here combine optical vortices and polarization, commonly associated with orbital and spin angular momentum respectively, but the Skyrmion number is a topological rather than a mechanical property of the beam. To see this we note that the Skyrmion number is unchanged if we apply a global transformation of the polarization, for example via reflection at a surface or a phase retardiation of the constituent beams. On the other hand we have seen that it is possible for the Skyrmion number to change if the two superimposed modes are focussed at different propagation distances. The total spin and angular momentum passing through each transverse plane, however, remains unchanged.

Acknowledgements.

This work was supported by the Royal Society (RP150122 and RPEA180010), and by the UK Engineering and Physical Sciences Research Council (EP/R008264/1) and by the European Training Network ColOpt, funded by the European Union (EU) Horizon 2020 program under the Marie Sklodowska-Curie Action, Grant Agreement No. 721465.