# Optical Path Length

Author: the photonics expert Dr. Rüdiger Paschotta

Definition: product of physical path length and refractive index

Alternative term: optical length

Units: m

Formula symbol: OPL, <$\Lambda$>

Optical Path Length (OPL) is a fundamental concept in the field of optics. It refers to the product of the physical path length that light travels through a medium and the refractive index of that medium. This implies that light on that path acquires the same change in optical phase as it would when traveling that distance (the OPL) in vacuum.

In a simple case with light traveling through a homogeneous, we have the simple equation

$$\Lambda = n \cdot d$$where <$n$> is the refractive index and <$d$> the geometric path length. This can be generalized, for example, to situations where light gets through different media in some sequence (e.g., in a multilayer dielectric coating), or even to cases where the ray path is curved. Note, however, that simply using a line integral involving the refractive index along the path may not be accurate. For example, for light propagation through an optical waveguide (e.g. a fiber), the phase delay per unit distance is *not* governed by the ordinary kind of refractive index, but rather by the effective refractive index, which is a non-local property and also takes into account waveguiding effects. In such a context, where also the application of geometrical optics becomes inaccurate, the naive application of the original concept of optical path length may lead to wrong results concerning the phase delay, which is often the crucial quantity.

Optical path lengths are usually considered only for (quasi-)monochromatic light, i.e., for light with a well-defined wavelength.

According to Fermat's principle, light always takes the path between two points such that the minimum possible optical path length (i.e., the minimum phase delay) results.

## Applications

The concept of optical path length is particularly significant in the following fields:

**Interferometry:**The interference conditions in an interferometer are basically determined by some difference in optical path lengths in the interferometer arms. Therefore, an interferometer can be used to detect very small changes of optical path length.**Optics design:**An optical lens should be designed such that the optical path length difference for different rays is minimized. For example, for a lens used for focusing a collimated beam, the path lengths of rays from a plane before the lens to a plane in the focus (with different distances from the beam axis) should be very similar (differing much less than by one wavelength). In that way, optical aberrations are minimized. Such optimization is important, for example, for obtaining optimal beam focusing or optical image quality in some imaging system.

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