Quantum optical coherence tomography

See also: 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). ^[1] 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. ^[2] The primary advantage of Q-OCT over OCT is insensitivity to even-order dispersion in multi-layered and scattering media. ^{[3] [4] [5]}

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). ^[6] 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. ^[7]

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. ^[8]

Hong-Ou-Mandel interferometer

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 (${\displaystyle c\tau _{q}}$) 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. ^[9]

The twin-photon source is characterized by the frequency-entangled state:

${\displaystyle \left|\psi \right\rangle =\int \,d\Omega \zeta (\Omega )\left|\omega _{0}+\Omega \right\rangle _{1}\left|\omega _{0}-\Omega \right\rangle _{2},}$

where ${\displaystyle \Omega }$ is the angular frequency deviation about the central angular frequency ${\displaystyle \omega _{0}}$ of the twin-photon wave packet, ${\displaystyle \zeta (\Omega )}$ is the spectral probability amplitude.

A reflecting sample is described by a transfer function:

${\displaystyle H(\omega )=\int \limits _{0}^{\infty }\,dzr(z,\omega )e^{i2\phi (z,\omega )},}$

where ${\displaystyle H(\omega )=r(z,\omega )}$ is the complex reflection coefficient from depth ${\displaystyle z}$,

The coincidence rate ${\displaystyle C(\tau _{q})}$ is then given by

A-scan plot of the quantum optical coherence tomography

${\displaystyle C(\tau _{q})\propto \Lambda _{0}-Re{\Lambda (2\tau _{q})},}$

where

${\displaystyle \Lambda _{0}=\int \,d\Omega |H(\omega _{0}+\Omega )|^{2}S(\Omega )}$,

and

${\displaystyle \Lambda (\tau _{q})=\int \,d\Omega H(\omega _{0}+\Omega )H^{\ast }(\omega _{0}-\Omega )S(\Omega )e^{-i\Omega \tau _{q}},}$

represent the constant (self-interference) and varying contributions (cross-interference). ^[10]

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; ^[11]
• intrinsic resolution enhancement by a factor of two for the same source bandwidth; ^[12]
• interferogram components that are insensitive to even-order dispersion of the medium; ^[13]
• interferogram components that are sensitive to the dispersion of the medium ^[14]

Applications

Similar to FD-OCT, Q-OCT can provide 3D imaging of biological samples with a better resolution due to the photon entanglement. ^[15] Q-OCT permits a direct determination of the group-velocity dispersion (GVD) coefficients of the media. ^[16] 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.

References

1. Hong, C. K.; Ou, Z. Y.; Mandel, L. (1987-11-02). "Measurement of subpicosecond time intervals between two photons by interference" . Physical Review Letters. 59 (18): 2044–2046. Bibcode:1987PhRvL..59.2044H. doi:10.1103/PhysRevLett.59.2044. PMID   10035403.
2. Gilgen, H. H.; Novak, R. P.; Salathe, R. P.; Hodel, W.; Beaud, P. (August 1989). "Submillimeter optical reflectometry" . Journal of Lightwave Technology. 7 (8): 1225–1233. Bibcode:1989JLwT....7.1225G. doi:10.1109/50.32387. ISSN   1558-2213.
3. Franson, J. D. (1992-03-01). "Nonlocal cancellation of dispersion" . Physical Review A. 45 (5): 3126–3132. Bibcode:1992PhRvA..45.3126F. doi:10.1103/PhysRevA.45.3126. PMID   9907348. S2CID   36542368.
4. Steinberg, A. M.; Kwiat, P. G.; Chiao, R. Y. (1993-08-02). "Measurement of the single-photon tunneling time" . Physical Review Letters. 71 (5): 708–711. Bibcode:1993PhRvL..71..708S. doi:10.1103/PhysRevLett.71.708. PMID   10055346. S2CID   31009201.
5. Larchuk, Todd S.; Teich, Malvin C.; Saleh, Bahaa E. A. (1995-11-01). "Nonlocal cancellation of dispersive broadening in Mach-Zehnder interferometers" . Physical Review A. 52 (5): 4145–4154. Bibcode:1995PhRvA..52.4145L. doi:10.1103/PhysRevA.52.4145. PMID   9912731.
6. Klyshko, D. N. (1988-01-01). Photons Nonlinear Optics. CRC Press. ISBN   978-2-88124-669-2.
7. Lavoie, J.; Kaltenbaek, R.; Resch, K. J. (2009-03-02). "Quantum-optical coherence tomography with classical light". Optics Express. 17 (5): 3818–3826. arXiv: 0909.0791 . Bibcode:2009OExpr..17.3818L. doi: 10.1364/OE.17.003818 . PMID   19259223. S2CID   8115209.
8. Teich, Malvin Carl; Saleh, Bahaa E. A.; Wong, Franco N. C.; Shapiro, Jeffrey H. (2012-08-01). "Variations on the theme of quantum optical coherence tomography: a review" . Quantum Information Processing. 11 (4): 903–923. doi:10.1007/s11128-011-0266-6. S2CID   254985458.
9. Nasr, Magued B.; Saleh, Bahaa E. A.; Sergienko, Alexander V.; Teich, Malvin C. (2003-08-22). "Demonstration of Dispersion-Canceled Quantum-Optical Coherence Tomography". Physical Review Letters. 91 (8): 083601. arXiv: quant-ph/0304160 . Bibcode:2003PhRvL..91h3601N. doi:10.1103/PhysRevLett.91.083601. PMID   14525237. S2CID   7206765 . Retrieved 2021-04-14.
10. Abouraddy, Ayman F.; Nasr, Magued B.; Saleh, Bahaa E. A.; Sergienko, Alexander V.; Teich, Malvin C. (2002-05-08). "Quantum-optical coherence tomography with dispersion cancellation". Physical Review A. 65 (5): 053817. arXiv: quant-ph/0111140 . Bibcode:2002PhRvA..65e3817A. doi:10.1103/PhysRevA.65.053817. S2CID   15047941.
11. Abouraddy, Ayman F.; Nasr, Magued B.; Saleh, Bahaa E. A.; Sergienko, Alexander V.; Teich, Malvin C. (2002-05-08). "Quantum-optical coherence tomography with dispersion cancellation". Physical Review A. 65 (5): 053817. arXiv: quant-ph/0111140 . Bibcode:2002PhRvA..65e3817A. doi:10.1103/PhysRevA.65.053817. S2CID   15047941.
12. "Quantum optical coherence tomography data collection apparatus and method for processing therefor". 2002-11-26.
13. Nasr, Magued B.; Saleh, Bahaa E. A.; Sergienko, Alexander V.; Teich, Malvin C. (2003-08-22). "Demonstration of Dispersion-Canceled Quantum-Optical Coherence Tomography". Physical Review Letters. 91 (8): 083601. arXiv: quant-ph/0304160 . Bibcode:2003PhRvL..91h3601N. doi:10.1103/PhysRevLett.91.083601. PMID   14525237. S2CID   7206765.
14. Nasr, Magued B.; Saleh, Bahaa E. A.; Sergienko, Alexander V.; Teich, Malvin C. (2004-04-05). "Dispersion-cancelled and dispersion-sensitive quantum optical coherence tomography". Optics Express. 12 (7): 1353–1362. Bibcode:2004OExpr..12.1353N. doi: 10.1364/OPEX.12.001353 . PMID   19474956.
15. Nasr, Magued B.; Goode, Darryl P.; Nguyen, Nam; Rong, Guoxin; Yang, Linglu; Reinhard, Björn M.; Saleh, Bahaa E.A.; Teich, Malvin C. (2009-03-15). "Quantum optical coherence tomography of a biological sample". Optics Communications. 282 (6): 1154–1159. arXiv: 0809.4721 . Bibcode:2009OptCo.282.1154N. doi:10.1016/j.optcom.2008.11.061. S2CID   931548.
16. Nasr, Magued B.; Saleh, Bahaa E. A.; Sergienko, Alexander V.; Teich, Malvin C. (2004-04-05). "Dispersion-cancelled and dispersion-sensitive quantum optical coherence tomography". Optics Express. 12 (7): 1353–1362. Bibcode:2004OExpr..12.1353N. doi: 10.1364/OPEX.12.001353 . PMID   19474956.