Optical path length

Light slows down in a medium of refractive index $n$, so a length $L$ of glass takes as long to cross as a length $nL$ of vacuum. That product is the optical path length. Along a path through regions of varying index it is

and the phase a wave accumulates is $2\pi\,\mathrm{OPL}/\lambda$, with $\lambda$ the vacuum wavelength. A millimetre of N-BK7, with an index of 1.5168 at 587.6 nm, is 1.517 mm of optical path; a metre of standard fiber, whose guided mode has an effective index of about 1.447 at 1550 nm, is 1.447 m.

Optical path difference

Interference depends on the optical path difference (OPD) between two beams, not on their geometric lengths. In an interferometer, a change in OPD of one wavelength moves the pattern by one fringe, so a Mach-Zehnder arm that changes its index by $\Delta n$ over a length $L$ shifts the phase by $2\pi L\,\Delta n/\lambda$; this is how thermo-optic and electro-optic phase shifters work. Wavefront error is an OPD map across a beam. Fermat's principle states the same quantity from the ray side: light between two points follows a path whose optical path length is stationary, from which the laws of reflection and refraction follow, and a perfect lens is one for which every ray from an object point to its image has the same optical path length.

Phase and group path

For continuous-wave interference the phase index $n$ is the right one. For pulses, and for the arrival time of any modulated signal, the relevant length is the group path $\int n_g\,ds$ with the group index $n_g$, which is larger in normal dispersion. Low-coherence interferometry, as in optical coherence tomography, balances the group path, and white-light fringes appear only when the group paths of the two arms match to within the coherence length. Mixing up the two is a common source of error in estimating fiber lengths from timing or waveguide lengths from fringe spacing.

Measurement

Optical path length is measured interferometrically: counting fringes while one arm is changed, reading the phase of a heterodyne signal, or, for a transparent plate, from the fringe shift when it is inserted into one arm of an interferometer. Absolute path lengths of fibers and waveguides are measured with reflectometers, which report group path, or from the spacing of Fabry-Perot fringes, $\Delta\lambda = \lambda^2/(2 n_g L)$, again a group quantity.

References: M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), Ch. 3; E. Hecht, Optics, 5th ed. (Pearson, 2017), Ch. 4.