View source for Optical path length

{{Short description|Product of geometric length and refractive index}}
In [[optics]], '''optical path length''' ('''OPL''', denoted '''''Λ''''' in equations), also known as '''optical length''' or '''optical distance''', is the length that light needs to travel through a vacuum to create the same phase difference as it would have when traveling through a given medium. It is calculated by taking the product of the [[arc length|geometric length]] of the [[optical path]] followed by [[light]] and the [[refractive index]] of the homogeneous medium through which the [[light ray]] propagates; for inhomogeneous [[optical medium|optical media]], the product above is generalized as a [[Line integral|path integral]] as part of the [[ray tracing (physics)|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 (waves)|phase]] of the light and govern [[Interference (wave propagation)|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

:<math>\mathrm{OPL} = n s .\,</math>

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

:<math>\mathrm{OPL} = \int_C n \mathrm ds,</math>

where ''n'' is the local refractive index as a function of distance along the path ''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 optical path length of ''C''.  Thus, if a [[wave]] is traveling through several different media, then the optical path length of each medium can be added to find the total optical path length.  The optical path difference between the paths taken by two identical waves can then be used to find the phase change.  Finally, using the phase change, the interference between the two waves can be calculated.

[[Fermat's principle]] states that the path light takes between two points is the path that has the minimum optical path length.

== Optical path difference ==

The OPD corresponds to the [[phase shift]] undergone by the light emitted from two previously [[Coherence (physics)|coherent]] sources when passed through mediums of different [[Refractive index|refractive indices]]. For example, a wave passing through air appears to travel a shorter distance than an identical wave traveling the same distance in glass. This is because a larger number of wavelengths fit in the same distance due to the higher [[refractive index]] of the [[glass]].

The OPD can be calculated from the following equation:

:<math>\mathrm{OPD}= d_1 n_1 - d_2 n_2</math>
where ''d''<sub>1</sub> and ''d''<sub>2</sub> are the distances of the [[Optical ray|ray]] passing through medium 1 or 2, ''n''<sub>1</sub> is the greater refractive index (e.g., glass) and ''n''<sub>2</sub> is the smaller refractive index (e.g., air).

==See also==
*[[Air mass (astronomy)]]
*[[Lagrangian optics]]
*[[Hamiltonian optics]]
*[[Fermat's principle]]
*[[Optical depth]]

==References==
*{{FS1037C MS188}}
*{{cite book  | last = Jenkins  | first = F. |author2=White, H.  | title = ''Fundamentals of Optics'' | url = https://archive.org/details/fundamentalsofop0004jenk  | url-access = registration  |edition = 4th  | publisher = McGraw-Hill  | year = 1976  | isbn = 0-07-032330-5 }}

[[Category:Geometrical optics]]
[[Category:Physical optics]]
[[Category:Optical quantities]]