# Quantum optical coherence tomography

Quantum optical coherence tomography (Q-OCT) is an imaging technique that uses nonclassical (quantum) light sources to generate high-resolution images based on the Hong-Ou-Mandel effect (HOM). Q-OCT is similar to conventional OCT but uses a fourth-order interferometer that incorporates two photodetectors rather than a second-order interferometer with a single photodetector. The primary advantage of Q-OCT over OCT is insensitivity to even-order dispersion in multi-layered and scattering media.
Several quantum sources of light have been developed so far. An example of such nonclassical sources is spontaneous parametric down-conversion that generates entangled photon pairs (twin-photon). The entangled photons are emitted in pairs and have stronger-than-classical temporal and spatial correlations. The entangled photons are anti-correlated in frequencies and directions. However, the nonclassical light sources are expensive and limited, several quantum-mimetic light sources are developed by classical light and nonlinear optics, which mimic dispersion cancellation and unique additional benefits.

## Theory

The principle of Q-OCT is fourth-order interferometry. The optical setup is based on a Hong ou Mandel (HOM) interferometer with a nonclassical light source. Twin photons travel into and recombined from reference and sample arm and the coincidence rate is measured with time delay.
The nonlinear crystal is pumped by a laser and generates photon pairs with anti-correlation in frequency. One photon travels through the sample and the other through a delay time before the interferometer. The photon-coincidence rate at the output ports of the beam splitter is measure as a function of length difference ($c\tau _{q}$) by a pair of single-photon-counting detectors and a coincidence counter.
Due to the quantum destructive interference, both photons emerge from the same port when the optical path lengths are equal. The coincidence rate has a sharp dip when the optical path length difference is zero. Such dips are used to monitor the reflectance of the sample as a function of depth.
The twin-photon source is characterized by the frequency-entangled state:
$\left|\psi \right\rangle =\int \,d\Omega \zeta (\Omega )\left|\omega _{0}+\Omega \right\rangle _{1}\left|\omega _{0}-\Omega \right\rangle _{2},$
where $\Omega$ is the angular frequency deviation about the central angular frequency $\omega _{0}$ of the twin-photon wave packet, $\zeta (\Omega )$ is the spectral probability amplitude.
A reflecting sample is described by a transfer function:
$H(\omega )=\int \limits _{0}^{\infty }\,dzr(z,\omega )e^{i2\phi (z,\omega )},$
where $H(\omega )=r(z,\omega )$ is the complex reflection coefficient from depth $z$,
The coincidence rate $C(\tau _{q})$ is then given by
$C(\tau _{q})\propto \Lambda _{0}-Re{\Lambda (2\tau _{q})},$
where
$\Lambda _{0}=\int \,d\Omega |H(\omega _{0}+\Omega )|^{2}S(\Omega )$,
and
$\Lambda (\tau _{q})=\int \,d\Omega H(\omega _{0}+\Omega )H^{\ast }(\omega _{0}-\Omega )S(\Omega )e^{-i\Omega \tau _{q}},$
represent the constant (self-interference) and varying contributions (cross-interference).
Dips in the coincidence rate plot arise from reflections from each of the two surfaces. When two photons have equal overall path lengths, the destructive interference of the two photon-pair probability amplitude occurs.

## Advantages

Compared with conventional OCT, Q-OCT has several advantages:
* greater signal-to-background ratio;
* intrinsic resolution enhancement by a factor of two for the same source bandwidth;
* interferogram components that are insensitive to even-order dispersion of the medium;
* interferogram components that are sensitive to the dispersion of the medium

## Applications

Similar to FD-OCT, Q-OCT can provide 3D imaging of biological samples with a better resolution due to the photon entanglement. Q-OCT permits a direct determination of the group-velocity dispersion (GVD) coefficients of the media. The development of quantum-mimetic light sources offers unique additional benefits to quantum imaging, such as enhanced signal-to-noise ratio, better resolution, and acquisition rate. Although Q-OCT is not expected to replace OCT, it does offer some advantages as a biological imaging paradigm.