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Spectrally resolved Hong–Ou–Mandel interferometry for quantum-optical coherence tomography

Pablo Yepiz-Graciano, Alí Michel Angulo Martínez, Dorilian Lopez-Mago, Hector Cruz-Ramirez, and Alfred B. U’Ren

Open Access Open Access

Abstract

In this paper, we revisit the well-known Hong–Ou–Mandel (HOM) effect in which two photons, which meet at a beamsplitter, can interfere destructively, leading to null in coincidence counts. In a standard HOM measurement, the coincidence counts across the two output ports of the beamsplitter are monitored as the temporal delay between the two photons prior to the beamsplitter is varied, resulting in the well-known HOM dip. We show, both theoretically and experimentally, that by leaving the delay fixed at a particular value while relying on spectrally resolved coincidence photon counting, we can reconstruct the HOM dip, which would have been obtained through a standard delay-scanning, non-spectrally resolved HOM measurement. We show that our numerical reconstruction procedure exhibits a novel dispersion cancellation effect, to all orders. We discuss how our present work can lead to a drastic reduction in the time required to acquire a HOM interferogram, and specifically discuss how this could be of particular importance for the implementation of efficient quantum-optical coherence tomography devices.

© 2020 Chinese Laser Press

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Figures (9)
Fig. 1.
Fig. 1. (a) Simulation of frequency-delay interferogram r c ( τ , Ω ) . (b) Result of integrating the interferogram over Ω , yielding the HOM interferogram. (c) Result of integrating the interferogram over τ , yielding a HOM-like dip in the frequency variable Ω . (d) Fourier transform of (a), so as to yield the time-domain interferogram r ˜ c ( τ , T ) . (e) Evaluation of r ˜ c ( τ , T ) at T = 0 , yielding the HOM interferogram. (f) Evaluation of r ˜ c ( τ , T ) at τ = − 1   ps . (g) Function A ( Ω ) . (h) Function B ( Ω ) .

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Fig. 2.
Fig. 2. (a) Frequency-delay interferogram r c ( τ , Ω ) for two-interface sample (borosilicate coverslip of 170 μm thickness). (b) Result of integrating the interferogram over Ω , yielding the HOM interferogram. (c) Fourier transform of (a) yielding the time-domain interferogram r ˜ c ( τ , T ) .

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Fig. 3.
Fig. 3. (a) and (d) Simulation of the temporal-domain interferogram | r ˜ c ( τ , T ) | for a three-layer sample (intermediate layer at 40% of the sample thickness, in addition to the two extremal interfaces); in (a) we show the case of an SPDC source centered at 775 nm with a narrowband pump (0.1 nm), while in (d) we increase the pump bandwidth to 10 nm. (b) and (e) Evaluation of | r ˜ c ( τ , T ) | at τ 0 = − 1.7   ps ; while (b) corresponds to a narrow pump bandwidth (0.1 nm), (e) shows the effect of increasing the bandwidth to 10 nm. (c) and (f) HOM interferogram resulting for the above two cases; (c) for a narrow pump bandwidth (0.1 nm) and (f) for a pump bandwidth of 10 nm.

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Fig. 4.
Fig. 4. Experimental setup. Ti:Sa, titanium–sapphire laser; TC, temperature controller; L, plano-convex spherical lens; PPLN, periodically poled lithium niobate nonlinear crystal; SF, set of bandpass and long-pass filters; MPC, manual fiber polarization controller; PMC, polarization-maintaining optical circulator; FC, compensating fiber; S, sample; RM, reference mirror; BS, beamsplitter; FSs, fiber spools; TDC, time-to-digital converter; APD, avalanche photodetectors.

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Fig. 5.
Fig. 5. (a) Experimental measurement of the delay-frequency interferogram r c ( τ , Ω ) , for a single-layer sample (plain mirror). (b) Result of integrating the interferogram over Ω , yielding the HOM interferogram. (c) Result of integrating the interferogram over τ , yielding a HOM-like dip in the frequency variable Ω . (d) Numerical Fourier transform of (a), yielding the time-domain interferogram | r ˜ c ( τ , T ) | . (e) Evaluation of | r ˜ c ( τ , T ) | at T = 0 , yielding the HOM interferogram. (f) Evaluation of | r ˜ c ( τ , T ) | at τ = − 1   ps .

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Fig. 6.
Fig. 6. Reconstruction procedure for the functions A ( Ω ) and B ( Ω ) , from which we can compute the HOM interferogram through Eq. (6). (a) Evaluation of the delay-frequency interferogram r c ( τ , Ω ) at τ = − 1   ps . (b) Numerical Fourier transform of (a), yielding | r ˜ c ( τ 0 , T ) | with τ 0 = − 1   ps . (c) and (d) Peaks 1 and 2 isolated from | r ˜ c ( τ 0 , T ) | by restricting the T variable to the two windows indicated in panel (b). (e) Function A ( Ω ) obtained as the inverse Fourier transform of peak 1. (f) Function B ( Ω ) obtained as the inverse Fourier transform of peak 2, multiplied by the phase exp ( i Ω τ 0 ) ; both amplitude and phase are shown.

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Fig. 7.
Fig. 7. Reconstructed HOM dip (red line) and conventional HOM dip obtained through scanning the delay with non-frequency-resolved coincidence counting (black dots).

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Fig. 8.
Fig. 8. (a) Experimental measurement of the delay-frequency interferogram r c ( τ , Ω ) for a two-layer sample (borosilicate glass coverslip of 170 μm thickness). (b) Result of integrating the interferogram over Ω , yielding the QOCT interferogram. (c) Numerical Fourier transform of (a), yielding the time-domain interferogram | r ˜ c ( τ , T ) | .

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Fig. 9.
Fig. 9. Reconstruction of the sample morphology and QOCT interferogram for a two-layer sample (borosilicate glass coverslip of 170 μm thickness). (a) Experimental measurement of the function r c ( τ 0 , Ω ) at a fixed delay τ 0 = − 0.363   ps . (b) Numerical Fourier transform of (a), yielding | r ˜ c ( τ 0 , T ) | ; here we have labeled five of the resulting peaks with the numbers 1–5. (c) Reconstructed QOCT interferogram (red line) and conventional delay-scanning, non-spectrally resolved HOM measurement (black points).

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Equations (16)

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| ψ ⟩ = | 0 ⟩ s | 0 ⟩ i + η ∫ − ∞ ∞ d Ω f ( Ω ) | ω 0 + Ω ⟩ s | ω 0 − Ω ⟩ i ,
f ( Ω ) = f 0   sinc [ L 2 Δ k ( Ω ) ] exp [ i L 2 Δ k ( Ω ) ] F f ( Ω ) ,
R c ( τ ) = R 0 2 ∫ − ∞ ∞ d Ω | f ( Ω ) − f ( − Ω ) e i Ω τ | 2 .
r c ( τ , Ω ) = 1 2 | f ( Ω ) − f ( − Ω ) e i Ω τ | 2 ,
R c ( τ ) = R 0 ∫ − ∞ ∞ d Ω r c ( τ , Ω ) .
r c ( τ , Ω ) = 1 2 [ A ( Ω ) + B ( Ω ) e − i Ω τ + B * ( Ω ) e i Ω τ ] ,
A ( Ω ) = | f ( Ω ) | 2 + | f ( − Ω ) | 2 ,
B ( Ω ) = − f ( Ω ) f * ( − Ω ) .
r ˜ c ( τ , T ) = 1 2 π ∫ − ∞ ∞ d Ω r c ( τ , Ω ) e i T Ω ,
r ˜ c ( τ , T ) = 1 2 [ A ˜ ( T ) + B ˜ ( T − τ ) + B ˜ ( − T − τ ) ] .
| τ | < π δ ω .
N = 2 L n M c Δ τ .
r ˜ c ( τ , T ) = F ˜ ( T ) − 1 2 F ˜ ( T − τ ) + cos ( ω 0 T s ) F ˜ ( T − T s / 2 ) − cos ( ω 0 T s ) F ˜ [ T − ( τ − T s / 2 ) ] − 1 2 F ˜ [ T − ( τ − T s ) ] − 1 2 F ˜ ( T + τ ) + cos ( ω 0 T s ) F ˜ ( T + T s / 2 ) − cos ( ω 0 T s ) F ˜ [ T + ( τ − T s / 2 ) ] − 1 2 F ˜ [ T + ( τ − T s ) ] .
R c ( τ ) = R 0 2 ∫ − ∞ ∞ d Ω | f ( Ω ) | 2 | H ( Ω ) − H ( − Ω ) e i Ω τ | 2 ,
H ( Ω ) = ∑ j = 0 N − 1 r ( j ) e i ( ω 0 + Ω ) T s ( j ) = r ( 0 ) + r ( 1 ) e i ( ω 0 + Ω ) T s ( 1 ) + … ,
H ( Ω ) = ∑ j = 0 N − 1 r ( j ) e i [ ( ω 0 + Ω ) T s ( j ) + Φ j ( Ω ) ] .
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