double-refracting crystal

double-refracting crystal

1. What Double Refraction (Birefringence) Is

Double refraction / birefringence = a single incident ray splits into two refracted rays with orthogonal polarizations and different speeds/directions inside an anisotropic crystal (e.g., calcite $CaCO_3$, quartz).

• Ordinary ray ($o$): obeys Snell's law, speed $v_o = c/μ_o$ same all directions; wavefront spherical; refractive index $μ_o$ constant.
• Extraordinary ray ($e$): does not obey ordinary Snell's (direction depends on crystal orientation), speed $v_e(θ)$ varies with angle to optic axis; wavefront ellipsoidal; index $μ_e(θ)$ varies between $μ_o$ and $μ_e$.

2. Optic Axis and Principal Section

• Optic axis: direction in crystal where $o$ and $e$ travel with same speed (no double refraction along this axis). In calcite, optic axis is diagonal of rhombohedron.
• Principal section: plane containing optic axis and incident ray's normal — $o$ and $e$ both lie in this plane (for calcite).

3. Huygens Construction

Two wavelets: sphere ($o$) radius $v_o t$ and ellipsoid ($e$) with semi-axes $v_o t$ along optic axis and $v_e t$ perpendicular. Tangent envelope gives refracted wavefront.

For negative crystal (calcite, $v_e > v_o$): $e$-ellipsoid outside $o$-sphere except along optic axis. For positive (quartz, $v_e < v_o$): inside.

4. Worked Illustration

Incident ray at 45° to calcite surface, optic axis inclined: splits into o (μo≈1.658) and e (μe≈1.486) with separation visible as double image through calcite slab (text appears doubled).

Along optic axis: μe = μo ⇒ no separation (single image).

Through Nicol prism? Uses birefringence + total internal reflection of o-ray (next topic).

5. Uses

Wave plates (quarter/half-wave use $μ_e-μ_o$ to introduce phase $Δ = (2π/λ)(μ_e-μ_o)t$), compensators, polarizing prisms.

6. Exam Points

• Define birefringence, $o$ vs $e$ ray differences — 5 marks, very frequent.
• Sketch optic axis, spherical vs ellipsoidal wavefronts — 5 marks, frequent.
• Negative vs positive crystal — 3-5 marks, frequent.
• Simple Snell's for $o$ vs angle-dependent for $e$ — 3 marks.