Question bank — Birefringence — ordinary and extraordinary rays

False — the o-ray is defined by obeying Snell's law, not by going straight; at oblique incidence it refracts like any normal ray, and only happens to go straight at normal incidence.

True — the split is by polarization, not wavelength, so even perfectly monochromatic light produces both an o-ray and an e-ray.

False — bare $n_e$ is only the principal value at $\theta =90^{\circ }$; the real index is the direction-dependent $n_e(\theta )$ from the ellipsoid relation.

True — at $\theta =0$ the formula gives $n_e(0)=n_o$, so both waves share one speed and never separate.

True — the o-ray's $\vec{E}$ is perpendicular to the principal plane while the e-ray's $\vec{E}$ lies in it, so the two are mutually orthogonal (see Polaroid and Nicol prism for how this lets a Nicol prism reject one).

False — calcite is negative ($n_e<n_o$); the sign is set by whether $n_e$ exceeds $n_o$, not by how large the splitting looks.

True — for negative crystals like calcite $\Delta n=1.486-1.658=-0.172$; the sign just records which ray is faster.

True in practice — squeezing or heating glass makes its "springs" direction-dependent (stress birefringence), turning an isotropic material temporarily anisotropic (Dielectric tensor and anisotropic media).

True — the o-ray's $\vec{E}$ has no component along the optic axis ($z$), so the ${\epsilon }_{zz}=n_e^2$ term never acts on it, leaving it with a constant $n_o$.

True for a given crystal and wavelength — thickness sets the phase difference $\Delta \varphi =\frac{2\pi }{\lambda }|\Delta n|d$; $\pi /2$ gives a quarter-wave, $\pi$ a half-wave plate (Wave plates (quarter and half wave)).

False — birefringence gives two linearly polarized rays at different speeds; Optical activity rotates the plane of a single polarization and comes from a different (chiral) mechanism.

The error is using bare $n_e$ — the e-ray's actual index is $n_e(\theta )$, so the local splitting is $n_e(\theta )-n_o$ and vanishes at $\theta =0$.

We can — the e-ray still obeys the wave equation; it just uses the direction-dependent $n_e(\theta )$, so a modified (index-varying) refraction analysis predicts it exactly.

The sine and cosine are swapped — the ${\cos }^2\theta$ must pair with $n_o^2$ so that $\theta =0$ yields $n_o$, matching the "no splitting along the axis" boundary condition.

No — at $\theta =0$ the two indices coincide, the two waves are degenerate, and there is no shift at all; the crystal behaves isotropically for that direction.

The relation is $n=\sqrt{{\epsilon }_r}$, not $n={\epsilon }_r$; the index scales with the square root of the relative permittivity.

That's a true property, but it's because its $\vec{D}$ and $\vec{E}$ stay parallel (isotropic response); the e-ray's $\vec{D}$ and $\vec{E}$ tilt apart, which is why its ray direction separates from its wavevector.

The o-ray's $\vec{E}$ always lies perpendicular to the optic axis, so it only ever samples the $n_o$ "springs"; the e-ray's $\vec{E}$ tilts toward the axis by an amount that depends on direction, sampling a mix of $n_o$ and $n_e$.

Because the geometry of slicing the ellipsoid gives the semi-axis through the relation on $1/n^2$; the displacement direction resolves into $\cos \theta$ and $\sin \theta$ pieces whose $1/n^2$ contributions add, like combining springs "in parallel."

A single number encodes only an isotropic response; a tensor lets the material respond differently along different axes, which is exactly the anisotropy that produces two indices (Dielectric tensor and anisotropic media).

There the e-ray's $\vec{E}$ lies fully perpendicular to the optic axis's... correction: at $\theta =90^{\circ }$ the propagation is perpendicular to the axis so the e-ray's field lies fully along the axis, giving $n_e(90^{\circ })=n_e$, the value farthest from $n_o$.

Only the difference in speed between the two rays builds up the phase gap $\Delta \varphi =\frac{2\pi }{\lambda }|\Delta n|d$; the common part of the index just delays both rays equally and cancels out.

Isotropic glass has one index, so $\Delta n=0$ and no relative phase ever accumulates between polarizations — there is nothing to retard.

Because inside the crystal the two rays travel at different speeds and generally in different directions, so a single incident ray refracts into two — the doubling is of the refracted rays, not the incident one.

One — the light travels along the optic axis ($\theta =0$), where $n_e(0)=n_o$, so there is no splitting.

The o- and e-waves travel at different speeds but along the same straight line, so they overlap spatially yet accumulate a phase difference — exactly the wave-plate geometry.

It equals $n_o$ for every $\theta$ — the material is effectively isotropic (no birefringence), which is the degenerate limit of the formula.

Unpolarized light is an equal mix of all polarizations, so on average the power splits roughly evenly between o- and e-rays; a preceding polarizer (Polaroid and Nicol prism) can send it all into one.

Faster — $n_e<n_o$ means a smaller index and $v=c/n$, so the e-ray outruns the o-ray (the opposite holds for positive crystals).

Its $\vec{E}$ tilts progressively into the principal plane; at $\theta =0$ it is indistinguishable from the o-ray, and at $\theta =90^{\circ }$ it lies fully along the optic axis, feeling the extreme index $n_e$.