Optical Phase Conjugation with Less Than a Photon per Degree of Freedom

We demonstrate experimentally that optical phase conjugation can be used to focus light through strongly scattering media even when far less than a photon per optical degree of freedom is detected. We found that the best achievable intensity contrast is equal to the total number of detected photons, as long as the resolution of the system is high enough. Our results demonstrate that phase conjugation can be used even when the photon budget is extremely low, such as in high-speed focusing through dynamic media or imaging deep inside tissue.

Figure 1

Experimental setup. (a) In the recording step, the input beam (denoted as “INP beam”), which is spatially filtered through a single-mode fiber (SMF), propagates through the scattering media (S). The scattered input beam is then collected by the planoconvex lens (L, focal $\text{length}=5\mathrm{cm}$) and relayed to the sCMOS sensor through a 1X telescope system (1X TS), interfering with the collimated reference beam (denoted as “REF beam”). The complex field distribution of the signal beam is then measured using a four-step phase-stepping method. (b) In the playback step, the collimated playback beam (denoted as “OPC beam”) reflects off the spatial light modulator (SLM) on which the conjugated phase of the measured wave front is displayed. The APD measures the fraction of power in the phase conjugate beam that is coupled back to the SMF, and the CCD camera directly captures the transmitted intensity distribution. BS, $50/50$ beam splitter; LP, linear polarizer.

Figure 2

Effect of shot noise on time-reversal fidelity. (a) The intensity distribution of the phase conjugate beam was directly visualized by the CCD sensor. Left and middle: With the phase conjugate beams generated at ${\overline{n}}_s/M$ of $\sim 2000$ and $\sim 0.004$. Right: With an unshaped incident beam (i.e., plane pattern on SLM). (b) Experimental and theoretical enhancement versus the total number of input photons ${\overline{n}}_s$. The solid curve and dotted curve, respectively, represent the theoretical and experimental enhancement.

Figure 3

Effect of shot noise on phase measurement accuracy. The phase error distribution $P_{\varphi }(\mathrm{\Delta }\varphi )$ was obtained by comparing the phase map measured at a low-photon limit (red curve, ${\overline{n}}_s/M$$\sim 20$; orange curve, ${\overline{n}}_s/M$$\sim 0.4$; green curve, ${\overline{n}}_s/M$$\sim 0.07$; cyan curve, ${\overline{n}}_s/M$$\sim 0.004$) with the reference phase map measured at a negligible shot noise level (${\overline{n}}_s/M\sim 2000$). The number of bins is 101, uniformly distributed from $-\pi$ to $\pi$. The solid curve and dotted curve, respectively, represent the theoretical and experimental phase error distribution. All curves are normalized such that the maximum value is unity.