Optics 075 · Polarization, anisotropy, and modulation

Birefringent Crystal Lab

Why does a birefringent crystal split one ray into two side-by-side rays? In a uniaxial crystal, light splits into two polarizations that see different refractive indices. For the extraordinary one, energy flows at an angle to the wave direction, so it walks off sideways unless the light travels along or straight across the optic axis. The two rays leave parallel but displaced; a thicker crystal gives a bigger offset, not a bigger image.

Light crosses a finite crystal and lands on a screen. The tensor eigenmodes give parallel exit rays, the walk-off between them and two images at equal scale; an ideal biaxial crystal is available for comparison.

Equal-scale image separation

\Delta r

$0.50$

Phase indices, fast / slow

n_1,n_2

$50\%$

Total normal-flux transmission

P_{\rm out}/P_{\rm in}

$0.00\pi$

Model regime $\text{valid model regime}$

Model checked against analytic results

Birefringent Crystal Lab

Physical crystal thickness

t

25\,\mathrm{mm}

Tilt of the third principal axis from the beam

\theta

45^{\circ}

Principal frame azimuth from upward

\phi

20^{\circ}

Frame spin about third axis (uniaxial invariant)

\chi

0^{\circ}

Material: zero calcite, one ideal biaxial

j_M

0

Vacuum wavelength (biaxial indices fixed)

\lambda_0

532\,\mathrm{nm}

Analyzer axis from upward toward image right

\alpha

0^{\circ}

Analyzer: zero removed, one inserted

j_A

0

One pattern goes in and two copies of equal size come out. Rotate the crystal and watch only the extraordinary copy move. Insert an analyzer to remove one copy, then send the light along the optic axis so the two copies coincide. The white line marks the wave normal, and the colored paths follow the energy flow.

Light enters at normal incidence and is traced geometrically, the interfaces do not reflect, and the pattern is unpolarized and collimated. Calcite dispersion comes from manufacturer data; the biaxial indices are made up. The images are not photographs, and diffraction and conoscopic figures are not modeled.

Physics tutorial

Audit the physics of Birefringent Crystal Lab

BackgroundInside an anisotropic crystal, the phase and the energy of a wave need not travel in the same direction. Energy flows along $\mathbf S=\mathbf E\times\mathbf H$ . At normal incidence, both transmitted wavevectors stay perpendicular to the parallel faces, yet the Poynting vector of the extraordinary wave can still tilt. After the second face, both beams travel in the original direction again, but they leave at different positions.

Why it mattersWhy does a birefringent crystal split one ray into two side-by-side rays?

Start with the essentials

Focus question: Why does a birefringent crystal split one ray into two side-by-side rays?
One-sentence intuition: The same energy directions set both where the rays leave the crystal in the 3D view and every pixel on the screen: $\Delta\mathbf r_j=t\,\frac{\mathbf S_{j,\perp}}{S_{j,x}}$ . Every point of the input pattern shifts by the same amount, so there is no separate image magnification. A thicker crystal moves the copy farther; it does not make it bigger.

Core mathematical model

Eigenmodes of the electric displacement

\begin{aligned}\boldsymbol\eta&=\boldsymbol\varepsilon_r^{-1}\\\mathbf P\boldsymbol\eta\mathbf P\,\mathbf D_j&=n_j^{-2}\mathbf D_j\\\mathbf P&=\mathbf I-\widehat{\mathbf k}\widehat{\mathbf k}^{\mathsf T}\\\mathbf E_j&=\boldsymbol\eta\mathbf D_j\\\mathbf H_j&\propto n_j\widehat{\mathbf k}\times\mathbf E_j\end{aligned}

The crystal is nonmagnetic, with a real, symmetric, positive-definite dielectric tensor. The displacement eigenvectors lie in the plane perpendicular to the shared wavevector. The electric field may have a component along the wavevector, and the cross product gives the energy direction. Field units are arbitrary here; only directions are used.

Uniaxial index and the angle convention

\begin{aligned}\frac{1}{n_{\rm ex}^2(\theta)}&=\frac{\cos^2\theta}{n_o^2}+\frac{\sin^2\theta}{n_e^2}\\n_{\rm ex}(0)&=n_o\\n_{\rm ex}(\pi/2)&=n_e\end{aligned}

The angle runs from the wave normal to the optic axis, not from the plate surface. Along the optic axis the two phase indices are equal. Away from it, the extraordinary energy can flow in a different direction from the wave normal.

One set of shifts makes both images

\begin{aligned}I_{\rm out}(\mathbf r)&=\sum_{j=1}^{2}w_j I_{\rm in}(\mathbf r-\Delta\mathbf r_j)\\w_j&=\tfrac12\left|\mathbf a\cdot\widehat{\mathbf D}_j\right|^2\\w_1+w_2&=\tfrac12\end{aligned}

The weights shown apply with the ideal analyzer inserted. Without it, each weight is one half and the total flux is one. At normal incidence the transverse electric field and the displacement point the same way, and the reflectionless interfaces keep those axes on exit. The screen image, the enlarged data and the ray exits all use the same shifts.

Common difficulties

Do two refractive indices mean two exit angles?

Typical misconceptionEven at normal incidence, the outgoing beams should keep diverging because their refractive indices differ.

Better mental modelThe tangential wavevector must be continuous across the face, and that fixes both wave normals inside the crystal. The beams separate because their energy walks off inside the crystal, not because they leave at two Snell angles in air. Because the faces are parallel, the outgoing beams are parallel again. A Wollaston prism has a different geometry and is a different device.

Run the experiment

01
Scene 1: Double-image calcite
Start with calcite and no analyzer. Compare the input with the screen; both use the same scale. Rotate the azimuth: one copy stays put while the extraordinary copy circles around it.
What to observe: The source and screen images are intensity maps computed forward from geometry, not camera photographs. The input is an assumed collimated pattern, not a diffuse object imaged through some unspecified lens.
02
Scene 2: Along the optic axis
Pass only the cyan channel, then only the amber channel. The matching ray and copy remain. Either way the total power halves, because the input is unpolarized.
What to observe: Both images share the same physical field of view, linear normalization and sRGB display mapping. The white source pattern holds two polarization channels that are incoherent with each other, so a phase difference alone cannot produce interference in this experiment.
03
Scene 3: Ideal biaxial comparison
Choose along the optic axis and the copies merge. In the ideal biaxial comparison, both energy paths walk off. Spin the frame about its third principal axis to leave the principal plane; this spin has no effect on uniaxial calcite. The third principal axis is neither of the optic axes.
What to observe: The ellipse is a true-proportion cross-section of the index ellipsoid. It is neither the wavevector surface nor an image reconstructed from detector data. The thickened rays and the axis markers are only visual guides.