Metasurface enabled quantum edge detection

Remotely switchable edge imaging is demonstrated on the basis of the metasurface illuminated by polarization-entangled photons.

A basic principle of a metasurface enabled "solid cat" and an edge enhanced "outlined cat", as shown in Fig. S1. A transmissive input image, "Schrödinger's cat" as an object, is illuminated by a plane wave. As shown in Fig. S1, the first lens (L1) computes the Fourier transform of the electric field at the object plane. The metasurface is located at the Fourier plane of the 4f system, where the Fourier spectrum is formed. The second lens (L2) takes the inverse Fourier transform of the spectrum modified by the metasurface and creates the output edge information at its back focal plane. To illustrate the calculation process, assume the incident light is x-polarized, the input electric field of the object is defined as . After propagating through L1, the electric field right in front of the metasurface is given as: , where is the Fourier transform operator. Considering the spatial differentiation function of the metasurface, the electric field right behind the metasurface is Here, is the period of the metasurface; these two terms, and are the PB phase achieved by metasurface for LCP and RCP components; for terms, and , these are Jones vectors for LCP and RCP components.
To achieve the edge information, two orthogonal polarizers are employed in our system. Therefore, the electric field could be further given as .
After the propagation through L2, the electric field at the image plane is derived from . The light intensity of the output image is given as . As shown in Fig. S2C, the edges end up with an "outlined cat".
It should be noted that for our imaging system, if two orthogonal polarizers are changed to co-polarized state or the second polarizer is removed, the mentioned electric field behind the metasurface will be modified as . After propagation through L2, the electric field at the image plane is written as . The final intensity distribution is given as . We can expect that a complete "solid cat" with the half-intensity of the edges will be obtained as demonstrated in Fig. S2D.
The above discussion all focuses on the classical circumstance. If we replace the linearly polarized illumination source with unknown states of photons from polarization entanglement and remove the P1, the image will be a superposition state, a "Schrödinger's cat" state consisting of both "outlined cat" and "solid cat", which will be discussed specifically in the introduction of the manuscript. In both cases, equivalent experimental conditions are used. For demonstrating the antinoise capacity of quantum edge detection, the ambient light keeps at roughly the same light level as the signal, which means the noise and signal could at the same order of magnitude.

Note S2. The measurement details
Experimentally, we kept the laboratory in dark except the ambient light scattered by a display screen. The total illumination photon counts before the object are roughly 2.4× /s and the total average environmental noise counts detected per pixel is roughly Hz when operates at continuous exposure. It should be noted that the noise source is primarily from the background, not from dark counts of ICCD in our experiment. Guided by the official specification of the camera, dark signal, a charge usually expressed as a number of electrons N, is produced by the flow of dark current. According to the specification sheet, the dark current per second per pixel under the 30° C air cooling condition is 0.1Hz electron. The shot noise is the square root of N, which is 0.32 Hz electron, roughly equivalent to 6.3× Hz photon, which is several orders of magnitude smaller than the ambient light. The calculating equation of the total effective exposure time is given below: Where is the coincidence counts per second, which is roughly 4.4× /s. is 2s exposure time of each frame. is 4 ns gate width and is the number of frames, which is 300 times. Finally, we got an effective exposure time where 0.11 s in both cases. At the final stage, the acquired raw data from ICCD is extracted pixel by pixel using the software of Mathematica and then all the performed data is imported to the software of Tecplot for color rendering.

Theory
To analyze the signal-to-noise ratio (SNR) of image performance, we now have a brief theoretical analysis and calculation of the quantum heralded source characteristic. For the sake of simplicity, three reasonable hypotheses are put forward. First, optical scattering during the imaging process is negligible. Second, the ICCD camera is regarded as a single-pixel detector. Third, the probability of multi-photon (>2) emission events is neglected due to stimulated emission via low SPDC pump power. We start by discussing the SNR of our proposed quantum edge detection scheme. The coincidence probability of correlated photons is given by , where is the photon-pair generated probability and denotes the total efficiency in the signal and idler arm, respectively, including the quantum efficiencies of the instruments, the transmission efficiency, as well as the collection efficiency of fibers. The count probabilities of a single event at the signal path detected by ICCD or the idler path detected by SPAD are given below: where denotes the probability of environmental noise in signal-channel and idler-channel, respectively. is the dark count probability caused by ICCD and SPAD, respectively.
The effective signals ( ) of the image in proportion to the probability of coincidences ( ) can be expressed as . Likewise, the unwanted background light and sensor noise ( ) in proportion to the probability of accidental events (A) can be expressed as . Finally, the SNR in a coincidence image using heralded single photon imaging modality is derived below: (S1) Then we characterize the SNR of conventional classical images with direct measurements.
Herein, first order intensity measurement is carried out by a single detector (i.e. camera sensor).
The effective signal intensity is given by , and the total noise including background light and sensor noise is given by . Therefore, the image SNR using direct imaging modality can be written as From equations (S1) and (S2), we can easily see the distinct working mechanisms of these two types of imaging modalities. For quantum heralded imaging, one effective photon event can only be obtained by second order intensity joint detection using two independent detectors (i.e. SPAD and camera sensor). Besides, the probability of a coincidence noise event is also significantly reduced via joint-detections, which could benefit from the preferential rejection of individual noise events. For direct imaging, first order intensity measurement is performed by using only a single detector (i.e. camera sensor). This modality makes no distinct detection between signals and noise, finally causing a pretty lower SNR. For a better understanding, the corresponding simulations are also given in the following section.

Calculation
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