Optical axis

The term has two meanings. In a lens or mirror system, the optical axis is the line of rotational symmetry: it passes through the centers of curvature of every surface, and a ray traveling along it crosses each surface at normal incidence and leaves undeviated. In a birefringent crystal, the optic axis is a direction of propagation along which light sees a single refractive index whatever its polarization, so that no birefringence appears. The crystal usage is usually written "optic axis", but the two spellings are mixed in practice and the context decides.

Axis of a lens system

Ray optics measures everything from this axis: object and image distances, focal points and principal planes lie on it, and ray heights and angles are taken relative to it. The paraxial approximation treats rays that stay close to the axis at small angles, replacing $\sin\theta$ by $\theta$; the error is 0.13% at 5°, 0.51% at 10° and 2.1% at 20°. Within that regime imaging is linear and is described by ray-transfer (ABCD) matrices; rays farther from the axis or at steeper angles are what produce third-order aberrations such as spherical aberration, coma and astigmatism, which grow with aperture and field angle.

A real lens also has a mechanical axis, defined by its edge or mount. When the optical axis is displaced or tilted from it, the lens is decentred and deviates a beam that enters along the mechanical axis; centration tolerances on a datasheet bound this. On the bench, a beam is placed on the axis of a lens by centering it on the lens and adjusting tilt until the weak reflections from the two surfaces return along the incoming beam. A sphere has no unique axis, which is why a ball lens needs no angular alignment. In a laser resonator, the line joining the mirrors' centers of curvature plays the same role and is called the cavity axis.

Optic axis of a crystal

In an anisotropic crystal the refractive index depends on the direction of the light's electric field. Uniaxial crystals, including calcite, crystalline quartz, sapphire, lithium niobate and YVO₄, have one optic axis, which coincides with the crystallographic c-axis. Light propagating along it sees the ordinary index $n_o$ for any polarization. For propagation at angle $\theta$ to the axis, the polarization perpendicular to the plane containing the axis and the wave vector is the ordinary wave, with index $n_o$; the other is the extraordinary wave, with index

which runs from $n_o$ along the axis to $n_e$ at 90°. The extraordinary wave's energy also travels at a small angle to its wave vector, the walk-off angle: 6.2° in calcite ($n_o$ = 1.658, $n_e$ = 1.486 at 589 nm) at 45° to the axis. Biaxial crystals such as KTP, LBO and mica have three different principal indices and two optic axes.

Device cuts follow from this geometry. A waveplate is cut with the optic axis in the plane of the plate, so that light at normal incidence sees the full difference $n_e - n_o$; a Pockels cell in KD*P is aligned with its optic axis along the beam, so that the crystal shows no birefringence until a voltage is applied. Viewed in convergent light between crossed polarizers along the optic axis, a uniaxial crystal shows a dark cross with colored rings, the conoscopic figure used to identify it in polarized-light microscopy.

Common questions

What is the difference between the optic axis and the optical axis?

"Optical axis" usually refers to the symmetry axis of a lens or optical system; "optic axis" refers to the direction of no birefringence in a crystal. The terms are often interchanged, and the subject (lenses or crystals) shows which is meant.

Is the optic axis a single line in the crystal?

No: it is a direction. Every line parallel to it through the crystal is equally an optic axis, which is why a crystal plate can be used anywhere across its face.

References: E. Hecht, Optics, 5th ed. (Pearson, 2017); M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999); B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019).