Optical path length

In optics, optical path length (OPL) or optical distance is the product of the geometric length of the path followed by light through a given system, and the refractive index of the medium through which it propagates. In many textbooks, it is symbolically written as Λ. A difference in optical path length 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 governs interference and diffraction of light as it propagates.

Formulation

Optical path length

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

\mathrm {OPL} =ns.\,

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

\mathrm {OPL} =\int _{C}n\mathrm {d} s,\quad

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 coherent sources when passed through mediums of different refractive indices. For example, a wave passed through glass will appear to travel a greater distance than an identical wave in air. This is because the source in the glass will have experienced a greater number of wavelengths due to the higher refractive index of the glass.

The OPD can be calculated from the following equation:

\mathrm {OPD} =d_{1}n_{1}-d_{2}n_{2}

where d₁ and d₂ are the distances of the ray passing through medium 1 or 2, n₁ is the greater refractive index (e.g., glass) and n₂ is the smaller refractive index (e.g., air).

References

This article incorporates public domain material from the General Services Administration document: "Federal Standard 1037C". (in support of MIL-STD-188)
Jenkins, F.; White, H. (1976). Fundamentals of Optics (4th ed.). McGraw-Hill. ISBN 0-07-032330-5.

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Optical path length

Formulation