Optical path length

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

${\displaystyle \mathrm{\Lambda }=ns.}$

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

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

where n is the local refractive index as a function of position along the path C. This can be re-written as $\mathrm{\Lambda }=\overline{n}\left|C\right|$ where $\overline{n}=\frac{{\int }_{C}n\mathrm{d}s}{\left|C\right|}$ is the average refractive index over the path C of geometric length |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, the length of which is equal to the OPL of C. For single frequency (monochromatic) light, the phase shift over C is $\mathrm{\Delta }\phi =k_0\mathrm{\Lambda }=k_0{\int }_{C}n\mathrm{d}s$ where k₀ is the vacuum angular wavenumber. Thus, if a wave travels through several different media, the optical path lengths of the individual segments may be added to obtain the total OPL. In wave interference, the difference between the optical path lengths of two coherent waves (for example, a laser beam split into two paths by a beam splitter) determines the corresponding phase difference at their common destination, and thus the corresponding interference patterns.

For a monochromatic wave emitted from a point source, a wavefront is a surface of constant phase. In geometrical optics, this means that the optical path length from the source to each point on a given wavefront is the same, up to an integer multiple of the wavelength.

Fermat's principle states the physical ray path is one for which the optical path length is stationary with respect to nearby paths. In many elementary cases, this means that the path light takes between two points is the path that has the minimum OPL.

Optical path difference

The optical path difference (OPD) is the difference between the optical path lengths of two rays or beams reaching a common point. For monochromatic light, the OPD determines the corresponding phase difference through

${\displaystyle \mathrm{\Delta }\phi =k_0\mathrm{O}\mathrm{P}\mathrm{D},}$

where ${\displaystyle k_0}$ is the vacuum angular wavenumber. For example, over the same geometric distance, light traveling in glass has a larger optical path length than light traveling in air because glass has a larger refractive index.

In general, if two rays follow paths ${\displaystyle C_1}$ and ${\displaystyle C_2}$, then

${\displaystyle \mathrm{O}\mathrm{P}\mathrm{D}={\mathrm{\Lambda }}_1-{\mathrm{\Lambda }}_2={\int }_{C_1}nds-{\int }_{C_2}nds.}$

In the special case where each ray travels through a homogeneous medium of constant refractive index, this reduces to

${\displaystyle \mathrm{O}\mathrm{P}\mathrm{D}=d_1{n}_1-d_2{n}_2,}$

where d₁ and d₂ are the geometric lengths of the two paths and n₁, n₂ are the corresponding refractive indices.