optical length
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In
optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultrav ...
, optical path length (OPL, denoted ''Λ'' in equations), also known as optical length or optical distance, is the product of the geometric length of the
optical path Optical path (OP) is the trajectory that a light ray follows as it propagates through an optical medium. The geometrical optical-path length or simply geometrical path length (GPD) is the length of a segment in a given OP, i.e., the Euclidean dis ...
followed by
light Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 te ...
and the
refractive index In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...
of
homogeneous medium In physics, a homogeneous material or system has the same properties at every point; it is uniform without irregularities. (accessed November 16, 2009). Tanton, James. "homogeneous." Encyclopedia of Mathematics. New York: Facts On File, Inc., 2 ...
through which a
light ray In optics a ray is an idealized geometrical model of light, obtained by choosing a curve that is perpendicular to the ''wavefronts'' of the actual light, and that points in the direction of energy flow. Rays are used to model the propagation o ...
propagates; for inhomogeneous
optical media In computing and optical disc recording technologies, an optical disc (OD) is a flat, usually circular disc that encodes binary data ( bits) in the form of pits and lands on a special material, often aluminum, on one of its flat surfaces ...
, 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 governs
interference Interference is the act of interfering, invading, or poaching. Interference may also refer to: Communications * Interference (communication), anything which alters, modifies, or disrupts a message * Adjacent-channel interference, caused by extr ...
and diffraction of light as it propagates.


Formulation

In a medium of constant refractive index, ''n'', the OPL for a path of geometrical length ''s'' is just :\mathrm = n s .\, If the refractive index varies along the path, the OPL is given by a
line integral In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integral'' is used as well, al ...
:\mathrm = \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 In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visib ...
propagating along a path ''C'' has the
phase shift In physics and mathematics, the phase of a periodic function F of some real variable t (such as time) is an angle-like quantity representing the fraction of the cycle covered up to t. It is denoted \phi(t) and expressed in such a scale that it ...
over ''C'' as if it was propagating a path in a
vacuum A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or " void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often di ...
, length of which, is equal to the optical path length of ''C''. Thus, if a
wave In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (re ...
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 Fermat's principle, also known as the principle of least time, is the link between ray optics and wave optics. In its original "strong" form, Fermat's principle states that the path taken by a ray between two given points is the pat ...
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 In physics and mathematics, the phase of a periodic function F of some real variable t (such as time) is an angle-like quantity representing the fraction of the cycle covered up to t. It is denoted \phi(t) and expressed in such a scale that it ...
undergone by the light emitted from two previously coherent sources when passed through mediums of different
refractive indices In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...
. For example, a wave passing through air appears to travel a shorter distance than an identical wave in glass. This is because the source in the glass experiences a smaller number of wavelengths due to the higher
refractive index In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...
of the
glass Glass is a non-crystalline, often transparent, amorphous solid that has widespread practical, technological, and decorative use in, for example, window panes, tableware, and optics. Glass is most often formed by rapid cooling ( quenching ...
. The OPD can be calculated from the following equation: :\mathrm= d_1 n_1 - d_2 n_2 where ''d''1 and ''d''2 are the distances of the ray passing through medium 1 or 2, ''n''1 is the greater refractive index (e.g., glass) and ''n''2 is the smaller refractive index (e.g., air).


See also

*
Air mass (astronomy) In astronomy, air mass or airmass is a measure of the amount of air along the line of sight when observing a star or other celestial source from below Earth's atmosphere ( Green 1992). It is formulated as the integral of air density along the lig ...
*
Lagrangian optics Hamiltonian opticsH. A. Buchdahl, ''An Introduction to Hamiltonian Optics'', Dover Publications, 1993, . and Lagrangian opticsVasudevan Lakshminarayanan et al., ''Lagrangian Optics'', Springer Netherlands, 2011, . are two formulations of geometrical ...
*
Hamiltonian optics Hamiltonian opticsH. A. Buchdahl, ''An Introduction to Hamiltonian Optics'', Dover Publications, 1993, . and Lagrangian opticsVasudevan Lakshminarayanan et al., ''Lagrangian Optics'', Springer Netherlands, 2011, . are two formulations of geometrical ...
*
Fermat's principle Fermat's principle, also known as the principle of least time, is the link between ray optics and wave optics. In its original "strong" form, Fermat's principle states that the path taken by a ray between two given points is the pat ...
*
Optical depth In physics, optical depth or optical thickness is the natural logarithm of the ratio of incident to ''transmitted'' radiant power through a material. Thus, the larger the optical depth, the smaller the amount of transmitted radiant power throu ...


References

* *{{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 Geometrical optics Physical optics