Questions: Electromagnetic Waves in Anisotropic Media

Question 1

A linearly polarized light wave enters a birefringent crystal with its polarization at 45° to the two principal axes. After propagating through the crystal, the polarization state will be:

• A. Still linear at 45°, because the crystal is symmetric about the propagation axis
• B. Rotated by 45° to align with one of the principal axes
• C. Elliptical (or circular), because the two eigenmodes accumulate a phase difference as they travel at different speeds
• D. Unchanged because birefringence only affects waves polarized along a principal axis

Answer: (C)

A wave polarized at 45° decomposes into equal parts of the two eigenmodes (one along each principal axis). Each eigenmode propagates at its own phase velocity, set by the corresponding principal permittivity. After a distance, the two components are out of phase by some amount φ. When φ = 90°, the polarization is circular; at other values it is elliptical. Only when φ = 0° or 180° is the polarization linear again. This is the working principle of wave plates. Option A is wrong because the two principal axes have different ε values — the crystal is not symmetric in the relevant sense. Option B conflates birefringence with optical rotation.

Question 2

In an anisotropic crystal, why are the displacement vector D and the electric field E generally not parallel?

• A. Because D includes the magnetic contribution to the field while E does not
• B. Because the permittivity tensor ε_ij couples different components of E when producing D, so D = ε·E mixes directions
• C. Because D is always perpendicular to the wave's propagation direction while E can have a longitudinal component
• D. This only occurs at interfaces between materials; inside a uniform crystal D and E are always parallel

Answer: (B)

In an isotropic medium, ε is a scalar so D = εE: D and E point the same way. In an anisotropic medium, ε is a tensor: the i-th component of D is Σ_j ε_ij E_j. If ε has off-diagonal components (i.e., the coordinate axes are not the principal axes), then an E field pointing along one direction generates a D with components along multiple directions. The result is D ≠ εE in the scalar sense, and D and E are generally not parallel. Along principal axes, ε is diagonal and D is parallel to E — but only along those special directions.

Question 3

Along a principal axis of a birefringent crystal, a linearly polarized wave with its electric field directed along another principal axis propagates as an eigenmode — its polarization state does not change.

• T. True
• F. False

Answer: True

Principal axes are the directions along which the permittivity tensor is diagonal. For a wave traveling along one principal axis, the transverse directions are the other two principal axes, and for these polarizations D and E are parallel (since ε is diagonal in this frame). Each such polarization is an eigenmode of propagation: it travels at a definite phase velocity without mixing into the other polarization. Birefringence arises when you combine two such eigenmodes — not when you propagate one of them alone.

Question 4

Birefringence means that two polarization eigenmodes propagate at different frequencies through the crystal, so blue light and red light experience the same phase shift.

• T. True
• F. False

Answer: False

Birefringence means the two eigenmodes travel at *different phase velocities* (different refractive indices), not at different frequencies. Both components of a given wavelength propagate at the same frequency — they must, since frequency is set by the source and is conserved at interfaces. What differs is the phase velocity v = c/n, where n depends on both the polarization direction and (due to dispersion) on frequency ω. So blue and red light do experience different phase shifts at the same thickness — but this is because n(ω) depends on frequency, not because the eigenmodes have different frequencies.

Question 5

Explain in physical terms why a wave polarized at 45° to the principal axes of a birefringent crystal does not propagate unchanged, but instead undergoes a change in polarization state.

Model answer: A wave polarized at 45° to the two principal axes is not an eigenmode of the crystal. It decomposes into two eigenmodes — one polarized along each principal axis — each of which propagates at its own phase velocity v₁ = c/n₁ and v₂ = c/n₂. Because n₁ ≠ n₂, the two components travel at different speeds and accumulate a phase difference Δφ = (2π/λ)(n₁ − n₂)L as they travel a distance L. The combined polarization state at any point is determined by the phase difference between the two components: at Δφ = 0 the polarization is linear (same as input); at Δφ = π/2 it is circular; at intermediate values it is elliptical. The polarization state continuously evolves as the wave propagates through the crystal.

This is birefringence in action. The key physical point is that a non-eigenmode polarization cannot maintain its state because its two component eigenmodes are accumulating different phases. Wave plates exploit this deliberately: a quarter-wave plate is tuned to produce Δφ = π/2, converting linear to circular polarization; a half-wave plate produces Δφ = π, rotating linear polarization by 90°.