Optical path length: Difference between revisions
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{{Short description|Product of geometric length and refractive index}} | {{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 | In [[optics]], '''optical path length''' ('''OPL''', denoted '''''Λ''''' in equations), also known as '''optical length''' or '''optical distance''', is the vacuum length that light travels over the same time taken to travel through a given medium length. For a homogeneous medium through which the [[light ray]] propagates, it is calculated as taking the product of the [[arc length|geometric length]] of the [[optical path]] followed by [[light]] and the [[refractive index]] of the medium. 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 | In a medium of constant refractive index, ''n'', the OPL for a path of geometrical length ''s'' is just | ||
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where ''n'' is the local refractive index as a function of distance along the path ''C''. | 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 | 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''.<ref group="note">For example, a single wavelength light traveling in a medium with the refractive index ''n'' is often expressed in a simplified formula <math display="inline">E\left( t,x \right) = A \cos \left( \omega t - kx + \varphi \right)</math>, where <math display="inline">\omega</math> and <math display="inline">k = k_0 n</math> are the angular frequency and the wavenumber of the light respectively (<math display="inline">k_0 = \frac{2\pi}{\lambda_0}</math> is the vacuum wavenumber), and <math display="inline">-k</math> indicates that the light travels towards +<math display="inline">\infty</math> in the ''x''-axis. ... in cos(...) is the phase of the light. | ||
The phase shift over the path ''C'' starting at ''x''<sub>0</sub>, that is the phase difference between ''x''<sub>0</sub> and ''x''<sub>0</sub> + ''C'' at the same time ''t'', is <math display="inline">kC = \left[ k_0 n \right] C = k_0 \left[ nC \right]</math> where <math display="inline">nC</math> is the OPL (Optical Path Length) of ''C'', indicating that the same phase shift is obtained by treating the light traveling a vacuum over the distance of OPL.</ref> 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 paths taken by two identical waves can then be used to find the difference between phase shifts over the paths. Finally, using this phase difference, the interference between the two waves at the end of the paths 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. | [[Fermat's principle]] states that the path light takes between two points is the path that has the minimum optical path length. | ||
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== Optical path difference == | == Optical path difference == | ||
The '''optical path difference''' (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 '''optical path difference''' (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 optical distance (the refractive index ''n''<sub>2</sub> ~ 1) than an identical wave traveling the same geometric distance in glass (''n''<sub>1</sub> > 1). This is because a larger number of wavelengths fit in the same geometric distance due to the higher [[refractive index]] of the [[glass]]. | ||
The OPD can be calculated from the following equation: | The OPD can be calculated from the following equation: | ||
:<math>\mathrm{OPD}= d_1 n_1 - d_2 n_2</math> | :<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). | where ''d''<sub>1</sub> and ''d''<sub>2</sub> are the geometric 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). | ||
==Note== | |||
{{reflist|group = "note"}} | |||
==See also== | ==See also== | ||
Latest revision as of 05:01, 27 December 2025
Template:Short description In optics, optical path length (OPL, denoted Λ in equations), also known as optical length or optical distance, is the vacuum length that light travels over the same time taken to travel through a given medium length. For a homogeneous medium through which the light ray propagates, it is calculated as taking the product of the geometric length of the optical path followed by light and the refractive index of the medium. For inhomogeneous optical media, 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 govern 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
If the refractive index varies along the path, the OPL is given by a line integral
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.[note 1] 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 paths taken by two identical waves can then be used to find the difference between phase shifts over the paths. Finally, using this phase difference, the interference between the two waves at the end of the paths 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 optical path difference (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 passing through air appears to travel a shorter optical distance (the refractive index n2 ~ 1) than an identical wave traveling the same geometric distance in glass (n1 > 1). This is because a larger number of wavelengths fit in the same geometric distance due to the higher refractive index of the glass.
The OPD can be calculated from the following equation:
where d1 and d2 are the geometric 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).
Note
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- ↑ For example, a single wavelength light traveling in a medium with the refractive index n is often expressed in a simplified formula , where and are the angular frequency and the wavenumber of the light respectively ( is the vacuum wavenumber), and indicates that the light travels towards + in the x-axis. ... in cos(...) is the phase of the light. The phase shift over the path C starting at x0, that is the phase difference between x0 and x0 + C at the same time t, is where is the OPL (Optical Path Length) of C, indicating that the same phase shift is obtained by treating the light traveling a vacuum over the distance of OPL.
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See also
References
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