Robust holography of the temporal wave function via second-order interference

Containing the capacity to record both amplitude and phase information of the light field, holography is of great importance in science and technology. Here, we present robust holography via second-order interference, considering both single-photon states and coherent states. Temporal amplitude and phase distribution carried by a weak coherent state are reconstructed experimentally. By comparison between holography using first-order interference and second-order interference, the robustness against phase noise of our method is verified. Besides, information about the state can be directly retrieved, with no complex algorithm required.

Figure 1

Experimental setup. For state preparation, a continuous laser is separated by the beam splitter (BS) into two. One serves as the reference with the overall phase being randomly disturbed by the phase modulator (PM). The other one provides the test states, with temporal distribution in amplitude and phase actively prepared with an intensity modulator (IM) and a PM. To operate for weak coherent sates, two attenuators (ATT) are employed to reduce the average photon number to the single-photon level. Coincidence between two single-photon detectors (SPD) is recorded. By scanning the trigger time of two detectors, the coincidence pattern is obtained. The same setup is also used as first-order holography, for comparison, with the PM in the reference arm providing a fixed phase.

Figure 2

Hologram of a coherent state. Only the phase distribution is introduced into the test states. The coincidence pattern depends on the phase distribution, as shown in Eq. (3). Since the rectangular phase is introduced into the test states, the coincidence pattern appears like a chess board. (a) The ideal coincidence counting (c.c.) pattern from theory. (b) The electrical wave form which drives PM in the test arm. (c) The normalized c.c. pattern obtained by scanning the trigger time of both detectors. The result of phase retrieval is shown in panel (d).

Figure 3

Hologram of a coherent state carrying both phase and amplitude modulation. (a) The ideal coincidence pattern from theory. (b) and (c) The electrical wave forms which drive the IM and PM, respectively. (d) The coincidence pattern from experiment. The retrieved results of amplitude and phase are shown in panels (e) and (f).

Figure 4

Results from first-order holography (left column) and second-order holography (right column). The record time of one pixel is 0.1, 0.2, and 0.5 s, respectively. The results show that the second-order method is more robust against phase noise.