Optical path length

In optics, optical path length (OPL, denoted Λ in equations), also known as optical length or optical distance, is the vacuum length that light travels over the same time taken to travel through a given medium length. For a homogeneous medium through which the light ray propagates, it is calculated as taking the product of the geometric length of the optical path followed by light and the refractive index of the medium. For inhomogeneous optical media, the product above is generalized as a path integral as part of the ray tracing procedure. A difference in OPL between two paths is often called the optical path difference (OPD). OPL and OPD are important because they determine the phase of the light and govern interference and diffraction of light as it propagates.

In a medium of constant refractive index, n, the OPL for a path of geometrical length s is just

${\displaystyle \Lambda =ns.}$

If the refractive index varies along the path, the OPL is given by a line integral

${\displaystyle \Lambda =\int _{C}n\mathrm {d} s,}$

where n is the local refractive index as a function of position along the path C. This can be re-written as ${\textstyle \Lambda ={\bar {n}}\left|C\right|}$ where ${\textstyle {\bar {n}}={\frac {\int _{C}n\mathrm {d} s}{\left|C\right|}}}$ is the average refractive index over the path C which geometric length is |C|.

An electromagnetic wave propagating along a path C has the phase shift over C as if it was propagating a path in a vacuum, length of which is equal to the OPL of C. For a single frequency light, the phase shift over C is ${\textstyle \Delta \varphi =k_{0}\Lambda =k_{0}\int _{C}n\mathrm {d} s}$ where k₀ is the vacuum angular wavenumber.^{[note 1]} Thus, if a wave is traveling through several different media, then the OPL of each medium can be added to find the total OPL. In wave interference, the difference between OPLs (OPD) taken by two coherent waves (e.g., a laser beam split into the two paths by a beam splitter) results in the difference between phase shifts over the corresponding geometric paths. The phase difference at the end of the paths reaching the common destination (like a sensor) contributes the interference between the two waves at this location.

For a single frequency wave emitting from a point source, OPLs from the source to each point of a wavefront are the same by the definition of the wavefront; it is a surface where the phase of the wave is the same. (So ${\textstyle \Delta \varphi =k_{0}\Lambda _{i}}$ where ${\textstyle \Lambda _{i}}$ is the OPL from the source to the i^th point on the wavefront, is the same for all points on the wavefront.)

Fermat's principle states that the path light takes between two points is the path that has the minimum OPL.