Optics 094 · Waveguides, structured light, and modern optics

Vector Beams & Optical Skyrmions

What makes the polarization pattern across a beam an optical skyrmion? Its normalized Stokes field wraps the Poincaré sphere. An integer degree requires a continuous polarization field and a fixed, compactifiable boundary; a finite observed disk usually gives a noninteger integral. Radial polarization alone does not cover the sphere.

Explore propagated two-component Jones fields with 3D electric ellipses and a separate Stokes diagnostic. Rotate the analyzer, inspect circular and linear polarization, reconstruct local Stokes vectors from six ideal exposures, integrate spherical triangles on a finite disk and freeze an acquisition.

Finite-disk Stokes integral

Q_R

$0.50$

Power through linear analyzer

P_\alpha/P_0

$50\%$

Integrated incident power

P/P_0

$0.00\pi$

Model regime $\text{valid model regime}$

Model checked against analytic results

Vector Beams & Optical Skyrmions

Signed vortex order

m

1

Common Gaussian envelope waist

w_0

0.60\,\mathrm{mm}

Distance from common waist

z/z_R

1

Second circular component power fraction

q

0.50

Relative circular-component phase

\delta

0^{\circ}

Linear analyzer angle from horizontal

\alpha

0^{\circ}

Integration disk radius in beam widths

R/w(z)

3

Rotate a linear analyzer to tell radial and azimuthal polarization apart. Then combine a Gaussian beam with a vortex in the orthogonal circular polarization: the center and outer edge approach opposite poles of the Poincaré sphere. Grow the sampling disk and watch its signed sphere coverage approach the infinite-plane prediction.

The model uses coherent paraxial Laguerre–Gaussian envelopes at 633 nm, with a shared waist and focal plane. Input preparation and analyzer exposures are ideal and noiseless. A finite-disk integral is not rounded to an integer, and a dark axis is left undefined. Stokes arrows are abstract polarization coordinates. High-NA focusing, partial polarization, material interactions and experimental data are not included.

Physics tutorial

Audit the physics of Vector Beams & Optical Skyrmions

BackgroundA vector beam has a polarization ellipse that changes across its cross-section. The full-Poincaré preset combines two orthogonal circular polarizations: $\mathbf E=\sqrt{1-q}\,u_0\mathbf e_++\sqrt q\,e^{i\delta}u_m\mathbf e_-$ . The first spatial mode is Gaussian and the second has a vortex. Both have the same waist and focal plane and propagate as normalized paraxial LG modes. The radial and azimuthal presets instead use opposite vortex orders; at their default settings every illuminated point is linearly polarized.

Why it mattersWhat makes the polarization pattern across a beam an optical skyrmion?

Start with the essentials

Focus question: What makes the polarization pattern across a beam an optical skyrmion?
One-sentence intuition: Compute sphere coverage from the local Stokes vectors: $Q_R=\frac{1}{4\pi}\sum_{t\subset D_R}\Omega_t$ . Each triangle in the sampling disk maps to an oriented spherical triangle. The sum is a finite-window integral, not automatically an integer. An integer degree requires a continuous field and a boundary that can be compactified to a fixed state. A dark axis is left undefined, never patched with an invented polarization.

Core mathematical model

Circular basis and Stokes convention

\begin{aligned}\mathbf e_\pm&=\frac{\hat{\mathbf h}\pm i\hat{\mathbf v}}{\sqrt2}\\S_0&=|E_h|^2+|E_v|^2\\s_1&=(|E_h|^2-|E_v|^2)/S_0\\s_2&=2\Re(E_h^*E_v)/S_0\\s_3&=2\Im(E_h^*E_v)/S_0\end{aligned}

Horizontal, vertical and propagation directions form a right-handed frame. The carrier is the positive propagation phase with negative time phase. The sign of the third Stokes component is defined by this equation, avoiding ambiguous right/left-handed labels. At zero intensity the normalized vector is undefined.

What the analyzer detects

I_\alpha=|E_h\cos\alpha+E_v\sin\alpha|^2

The analyzer passes one transverse electric-field projection. Three complementary pairs of ideal exposures give the total intensity and the three Stokes differences. They are sequential exposures of the same field, not six simultaneous beams whose powers should be added.

Independent boundary comparison

Q_R=\frac{m}{2}[s_3(0)-s_3(R)]

For the axisymmetric Gaussian–vortex family this gives a separate comparison to the triangle calculation. The circular center and the linear-polarization ring come from the same Jones components. With both components present and nonzero vortex order, the ideal infinite-plane degree is the signed order. Extending the disk into extremely weak light is mathematical, not a claim of noise-resistant measurement.

Common difficulties

Does any winding polarization pattern make a skyrmion?

Typical misconceptionA radial vector beam must have a unit Skyrmion number because its electric arrows turn around the center.

Better mental modelNo. The default radial beam samples only the equator of the Poincaré sphere and has a dark axis. A full-Poincaré Gaussian–vortex beam spans latitudes as well as azimuth. Even then a finite disk need not cover a whole sphere.

Run the experiment

01
Scene 1: Radial vector beam
Compare the radial and azimuthal presets, then rotate the linear analyzer. Their unfiltered intensities match but the transmitted lobes turn by a right angle. Change the relative phase to rotate the local polarization.
What to observe: Electric ellipses are physical transverse field orbits. Stokes arrows live in polarization space; their axial component is not a longitudinal electric field.
02
Scene 2: Azimuthal vector beam
Choose the full-Poincaré preset. Sample the center, then click the ellipse plot away from the axis. Find the circular center and the gold linear-polarization ring. Change the mixture and watch the ring radius move.
What to observe: Changing the Gaussian waist changes the physical beam width and Rayleigh range together. The axial and transverse display scales are unequal and explicitly stated.
03
Scene 3: Full-Poincaré beam and finite disk
Change the integration radius and reverse the signed vortex order. Compare the gold numerical point with the cyan boundary curve and the purple infinite-plane limit. Move the detector to see Gouy rotation, then freeze six exposures before changing another control.
What to observe: The Stokes reconstruction uses six ideal intensity arrays. The frozen record keeps those arrays and both complex Jones components even when the controls change.