Editing Optical coherence tomography (section)

===Time domain===

:<math> F _ODT \left ( \nu \right ) =  2S_0 \left ( \nu \right ) K_r \left ( \nu \right ) K_s \left ( \nu \right )  \qquad \quad (3) </math>
In time domain OCT the path length of the reference arm is varied in time (the reference mirror is translated longitudinally). A property of low coherence interferometry is that interference, i.e. the series of dark and bright fringes, is only achieved when the path difference lies within the coherence length of the light source. This interference is called autocorrelation in a symmetric interferometer (both arms have the same reflectivity), or cross-correlation in the common case. The envelope of this modulation changes as path length difference is varied, where the peak of the envelope corresponds to path length matching.

The interference of two partially coherent light beams can be expressed in terms of the source intensity, <math>I_S</math>, as

:<math> I = k_1 I_S + k_2 I_S + 2 \sqrt { \left ( k_1 I_S \right ) \cdot \left ( k_2 I_S \right )} \cdot Re \left [\gamma \left ( \tau \right ) \right] \qquad (1) </math>

where <math>k_1 + k_2 < 1</math> represents the interferometer beam splitting ratio, and <math> \gamma ( \tau ) </math> is called the complex degree of coherence, i.e. the interference envelope and carrier dependent on reference arm scan or time delay <math> \tau </math>, and whose recovery is of interest in OCT. Due to the coherence gating effect of OCT the complex degree of coherence is represented as a Gaussian function expressed as<ref name="Fercher"/>

:<math> \gamma \left ( \tau \right ) = \exp \left [- \left ( \frac{\pi\Delta\nu\tau}{2 \sqrt{\ln 2} } \right )^2 \right] \cdot \exp \left ( -j2\pi\nu_0\tau \right ) \qquad \quad (2) </math>

where <math> \Delta\nu </math> represents the spectral width of the source in the optical frequency domain, and <math> \nu_0 </math> is the centre optical frequency of the source. In equation (2), the Gaussian envelope is amplitude modulated by an optical carrier. The peak of this envelope represents the location of the microstructure of the sample under test, with an amplitude dependent on the reflectivity of the surface. The optical carrier is due to the [[Doppler effect]] resulting from scanning one arm of the interferometer, and the frequency of this modulation is controlled by the speed of scanning. Therefore, translating one arm of the interferometer has two functions; depth scanning and a Doppler-shifted optical carrier are accomplished by pathlength variation. In OCT, the Doppler-shifted optical carrier has a frequency expressed as

:<math> f_{Dopp} = \frac { 2 \cdot \nu_0 \cdot v_s } { c } \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad (3) </math>

where <math> \nu_0 </math> is the central optical frequency of the source, <math> v_s </math> is the scanning velocity of the pathlength variation, and <math> c </math> is the speed of light.

The axial and lateral resolutions of OCT are decoupled from one another; the former being an equivalent to the coherence length of the light source and the latter being a function of the optics. The axial resolution of OCT is defined as
:{|
|-
|<math> \, {l_c} </math>
|<math>=\frac {2 \ln 2} {\pi} \cdot \frac {\lambda_0^2} {\Delta\lambda}</math>
|-
|
|<math>\approx 0.44 \cdot \frac {\lambda_0^2} {\Delta\lambda} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (4) </math>
|}

where <math> \lambda_0 </math> and <math> \Delta\lambda</math> are respectively the central wavelength and the spectral width of the light source.<ref name="ReferenceA">{{cite book| title= Anterior & Posterior Segment OCT: Current Technology & Future Applications, 1st edition |year=2014| vauthors = Garg A }}</ref>