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Abstract
Quantum imaging has non-negligible advantages in terms of sensitivity, signal-to-noise ratio, and novel imaging schemes. Based on metasurfaces, the information density and stability of the quantum imaging system can be further improved. Here we experimentally demonstrate that two patterns, simultaneously and independently superimposed on a high-efficiency dielectric metasurface, can be remotely switched via polarization-entangled photon pairs. Furthermore, using the time-correlated property of entangled photon pairs, the information carried by quantum light can be remarkably discriminated from background noise. This work confirms that the phase manipulation of quantum light with metasurfaces has a huge potential in the field of quantum imaging, quantum state tomography, and also promises real-world quantum metasurface devices.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Quantum imaging, employing the quantum nature of light, enables various applications, such as quantum ghost imaging [1], undetected photons imaging [2], sub-shot-noise imaging [3], and so on. With the help of entanglement, ghost imaging was firstly demonstrated [1]. Initially, ghost imaging was deemed as an intrinsic quantum mechanical effect. Subsequently, it has been theoretically and experimentally confirmed that ghost imaging is also possible with only classical light sources [4,5]. The key points for quantum ghost imaging and classical ghost imaging are quantum entanglement and classical correlation, respectively [5]. Compared with the classical cases, imaging with quantum states of light can provide improved sensitivity [6], finer resolution [7–10], higher signal-to-noise ratio [11,12], and novel imaging schemes [13–18].
Metasurfaces, two-dimensional arrays of nanoparticles, can implement fascinating functionalities with simplified fabrication processes [19]. Because of its rich phase manipulation capabilities, metasurfaces can produce different states of light, and degrees of freedom, such as polarization [20–22], orbital angular momentum [23–25], wavelength [26,27], incident [28–30] or output angle [31] have been multiplexed. Furthermore, metasurfaces can provide new prospects for quantum optics [32], including quantum sources [33,34], quantum state manipulation [35–38], quantum applications [39–42], and quantum vacuum engineering [43–45]. In particular, remote control of quantum edge detection [42] and quantum imaging enabled by metasurfaces [46] have been experimentally demonstrated. The aforementioned studies are all based on heralded single-photon imaging techniques, in which the operation in the signal arm can be remotely controlled by the heralding arm without changing anything in the signal arm. And to the best of our knowledge, remote controllable image displaying for any pair of orthogonal polarizations based on metasurfaces has not jet been reported.
Here, we show the scanning-free, full-field, switchable imaging displaying enabled by polarization-entangled source and high-efficiency polarization-multiplexing metasurfaces, it is seen that the polarization state measured at the heralding arm determines the corresponding pattern at the signal arm. When linear polarization is measured, it is demonstrated that the metasurface at the signal arm should align with the electric field direction of incident light to avoid crosstalk between orthogonal polarizations. On the contrary, when circular polarization is measured, the crosstalk is found to be nearly zero and insensitive to the rotation angle of the metasurface. In contrast with classical imaging, the heralded single-photon imaging is robust even in the presence of background noise within a certain intensity range.
2. Metasurface details
As shown in Fig. 1(a), this is the diagram of the unit structure in the metasurface. The period is 350 nm and the substrate is silica, on which are amorphous silicon(a-Si) pillars with a height of 390 nm. In this paper, commercial FDTD software (Lumerical FDTD solutions) is used to obtain the transmission functions of meta-atoms. Periodic boundary conditions are applied along the x- and y-direction and PML conditions along the z-direction. By sweeping the length and width, the propagation phase and transmission in the horizontal and vertical direction are achieved. From this information, not only the transmission function for linear polarization but also the one for circular polarization can be deduced. Figure 1(b), (c) displays the linear polarization transmittances and circular polarization interconversion efficiencies of the pillars used in metasurface I and II, respectively, which are mostly close to $1$. The structural unit with high transmittance not only ensures high utilization efficiency of light energy, but also reduces the coupling effect of neighboring structural units. The principle from the literature [20], which uses the interference of $2*2$ meta-atom arrangement as shown in Fig. 1(d) to control the intensity of light through a super-pixel, is employed to design two metasurfaces for linear polarization and circular polarization, respectively. The metasurface I and metasurface II both have a size of $229.6*306.6\,\mathrm{\mu}\textrm{m}$ which consists of $328*438$ pixels. For metasurface I, the near-field pattern is the character H(V) when the H(V) polarization is incident. For metasurface II, when L(R) polarization is incident, the near-field pattern is the character L(R). Then the target phase distribution is replaced by the real transmission functions of pillars, and simulated by using the angular spectrum-based wave-propagation method, which can quickly and accurately simulate the metasurface’s manipulation of the light field. Figure 1(e), (f) shows the measured near field electric intensity distribution for the metasurface I with incident horizontal and vertical polarization, respectively. Figure 1(g), (h) shows the measured near field electric intensity distribution for the metasurface II with incident left- and right-handed circular polarization, respectively. Obviously, the crosstalk between orthogonal polarization states induced by the initial metasurface design is nearly zero.
Fig. 1. Metasurface details. (a) Perspective view of the metasurface unit-cell. (b) Calculated transmission of the 7 meta-atoms in metasurface I. (c) Calculated cross-polarization efficiencies of the 7 meta-atoms in metasurface II. (d) 2*2 meta-atom arrangement at a super-pixel. Meta-atom 1 and 2 possess different structural parameters. (e),(f) Measured near field electric intensity distribution (e) for the metasurface I with incident horizontal polarization, (f) for the metasurface I with incident vertical polarization, (g) for the metasurface II with incident left-handed circular polarization, and (h) for the metasurface II with incident right-handed circular polarization.
3. Theoretical discussion
In our experiment, the density matrix of the quantum state that is produced is
In the signal arm, when the metasurface is not rotated, the operator for a single super-pixel on a metasurface can be written as
Then, the coincidence measurement between the heralding arm and the signal arm is given by
It is worth mentioning that for the metasurface I irradiated by H/V polarization light, it can be considered that ${{\xi }_{xy}}={{\xi }_{yx}}=0$. When H(V) polarization is post-selected in the heralding arm, ${{\theta }_{h}}=0(\pi )$, the probability of a signal photon passed through the super-pixel on the metasurface is
From Eq. (9) and Eq. (3), it is clear that when the metasurface is not rotated, $\xi _{xx},\xi _{yy}$ mainly determines the displayed pattern in the signal arm with the condition of $\left |V\right \rangle,\left |H\right \rangle$ post-selected in the heralding arm, and the condition that the visibility of the H/V basis is not 1 will induce crosstalk between the information encoded by different polarizations, smaller $\epsilon$ will induce less crosstalk, this conclusion is also applicable to other polarization basis. When the metasurface is rotated, there exists cross-talk between orthogonal polarizations.
Similarly, for the metasurface II irradiated by L/R polarization light , $\theta _{h}=\pi /2,\phi _{h}=\pm \pi /2$ determines that the post-selected quantum state is L/R polarization. The probability of a signal photon passed through the super-pixel on the metasurface is
It can be seen from Eq. (10) that when the $\theta _{s}$ changes, the value of a single pixel basically does not change, only the position of the super-pixel is changed, but the none-zero and small value $\epsilon$ will cause slight changes.
4. Experimental setup
The experimental setup is shown in Fig. 2(a). The ultraviolet laser pump power is 60mw, 3nm and 10nm narrow band interference filters are placed at the heralding arm end and the signal arm end, respectively. The heralding photon passes through a set of QHP combination (quarter-wave plate, half-wave plate, polarizing beam splitter) for polarization post-selection and is then detected with a single-photon avalanche diode (SPAD). The SPAD is in free-running mode and outputs a pulse to the intensified charge-coupled device (ICCD, Andor iStar A-DH334T-18 U-73) for triggering image acquisition whenever a heralding photon is detected. The ICCD operates at an external trigger mode and a Digital Delay Generator (DDG) gate mode with electrical delay on the order of nanoseconds, so the signal photon needs to pass through a 20m single-mode fiber which functions as optical delay. Due to the polarization dephasing effect of optical fiber, a set of half-wave plate and quarter-wave plate is put behind the single-mode fiber. Then, to improve the utilization efficiency of signal photons, a pair of beam-reducing lenses whose focal lengths are 200 mm and 50 mm reduces the spot diameter by a factor of 4. Finally, the imaging microscopy system, containing an objective (Mitutoyo, NA=0.26) and a lens of focal length 150 mm, images the metasurface plane onto the ICCD sensor plane at $7.5\times$ magnification. The SEM images of metasurface I and metasurface II are shown in Fig. 2(b) and (c), respectively.
Fig. 2. (a) The abbreviations of the components are BBO, $\beta$-barium borate crystals; THWP, true-zero-order half-wave plate; HWP, half-wave plate; QWP, quarter-wave plate; FC, fiber coupler; PBS, polarizing beam splitter. The 390 nm ultraviolet pulse laser passes through the sandwich-like combination of a true-zero-order half-wave plate and two beamlike BBO crystals. The true-zero-order half-wave plate only works at the wavelength 780 nm rather than 390 nm. The focal length of the lenses before (after) BBO crystals is 150 mm (125 mm). LiNbO3 crystal with thicknesses of 1 mm (3.2 mm) in path 2 (1) are used for spatial compensation. Their optic axes lie in the horizontal plane and are $45^\circ$ away from their surface normal direction. YVO4 crystal with thicknesses of 0.60 mm (0.42 mm) in path 2 (1) is used for temporal compensation. (b) SEM image of the metasurface I. (c) SEM image of the metasurface II.
5. Experimental results and discussion
Bright and high fidelity polarization entangled source is indispensable for high-quality image. To measure the polarization interference fringes, the HWP angle in the signal arm varies in a step of $10^\circ$ when the one in the heralding arm is fixed at $0^\circ$ or $22.5^\circ$. As can be seen in Fig. 3(a), the measured coincidence counts in 1s fits well with sinusoidal function. The visibilities of interference fringes $V={\left ( {{C}_{\max }}-{{C}_{\min }} \right )}/{\left ( {{C}_{\max }}+{{C}_{\min }} \right )}$ are $92.10\pm 0.24\%$ in the $+{{45}^{\circ }}/-{{45}^{\circ }}$ basis, and $95.52\pm 0.14\%$ in the $H/V$ basis, respectively. The visibilities both exceed $71\%$, the bound of violating Bell’s inequality [47]. As for the L/R basis, the visibility is measured to be $9\text {0}.79\pm 0.09\%$, which also meets this requirement. In addition, we measured the coincidence-to-accidental ratio(CAR) of the entangled source under different pump powers. The CAR is defined as $CAR=\left (N_{c}-N_{ac}\right )/N_{ac}$, where $N_{c}$ is the coincidence counts between the signal arm and heralding arm, and $N_{ac}$ is the accidental coincidence count. As shown in Fig. 3(b), as the pump power increases, the CAR gradually decreases. When the pump light power is $10$mw, the accidental coincidence number is too small and the standard deviation is large. When the power is $60$mw, the CAR is reduced to $2866$, which shows that the influence of accidental coincidence is negligible during the experiment. To obtain a comprehensive evaluation of the prepared quantum states, quantum tomography has been carried out. Figure 3(c) and 3(d) show the real and imaginary parts of the reconstructed density matrix, respectively. According to the definition of the fidelity $F=\left \langle \Psi \right |\hat {\rho }\left | \Psi \right \rangle$, where $\Psi$ is the target quantum state, and $\hat {\rho }$ is the density matrix reconstructed from experimental data, the fidelity is $95.49\pm 0.31\%$, and the target quantum state is ${\left ( \left | HV \right \rangle +\left | VH \right \rangle \right )}/{\sqrt {2}}\;$. The single-channel count rate in heralding/signal arm is $90/230$ kHz. Although the coincidence efficiency is about $18.75\%$, the effective heralding efficiency is about $11.94\%$ after taking the loss in the signal arm into consideration. Smaller single-channel count rate at the heralding arm is beneficial for blocking noise photons.
Fig. 3. (a) Coincidence counts in 1s as a function of the HWP angle in signal arm when the HWP angle in the heralding arm is fixed at $0^\circ$(black line) and $22.5^\circ$(red line), the solid lines are sinusoidal fits to the data. (b) The coincidence-to-accidental ratio(CAR) under different pump levels. (c),(d) The real and imaginary parts of the reconstructed density matrix of the entangled source.
After illustrating the feasibility of the well fabricated metasurfaces and produced high-quality polarization entangled source, polarization-multiplexing heralded single-photon imaging is implemented. From the point of view of classical physics, the light passing through adjacent pixels is coherent. However, the behavior of photons essentially obeys quantum physics, i.e., only one photon passes through the metasurface. When the photons passing through adjacent pixels are indistinguishable, the wave functions of different pixels interfere, and the square of the wave function, i.e., the probability distribution of a single photon, is detected at the imaging plane. The position of a single photon is intrinsically uncertain, so if the acquisition time is short, the shot noise dominates and the image is not clearly visible. In order to suppress the shot noise, the intensity of light source used in this paper is bright, about $30$k pairs/s, and the collection efficiency of the imaging system is $40.54\%$, so about $12.2$kHz single photons are collected per second. A clear pattern can already be seen when the acquisition time is only $10$s. From Fig. 4(a), the measured polarization state in the heralding arm precisely determines the corresponding image shown in the signal arm. The calculated image contrast values of images in Fig. 4(a) is shown in Fig. 4(c), which is defined as $V={\left ( {{N}_{\max }}-{{N}_{\min }} \right )}/{\left ( {{N}_{\max }}+{{N}_{\min }} \right )}$, where $N_{\max },N_{\min }$ are the maximum and minimum counts, respectively. Obviously, as the ICCD acquisition time becomes longer, the image contrast is higher, and it will tend to be saturated when the acquisition time increases to a certain value. In fact, decoupling near-field intensity for any pair of orthogonal polarizations is feasible by using metasurfaces [20], here only the linear polarization and circular polarization are selected as the demonstration in this work. Otherwise, From Fig. 4(b), when the metasurface is rotated, the image heralded by the linear polarization changes. The crosstalk is maximum when the rotation angle is $45^\circ$ or $135^\circ$, the image display is completely interchanged when the rotation angle is $90^\circ$, and the image is rotated $180^\circ$ when the rotation angle is $180^\circ$. However, for circular polarization, when the metasurface is rotated, the image is only rotated as a whole. The experimental results are in good agreement with the theoretical discussion above. The reason is that the metasurface designed here is not circularly symmetric, so circular polarization has rotational symmetry, but linear polarization does not. It is worth mentioning that the two patterns of circular polarization can also be interchanged when the metasurface is flipped. The calculated image contrast values of images in Fig. 4(b) is shown in Fig. 4(d), it is shown that there is little relationship between the image contrast and the rotation angle of the metasurface II. Particularly, when calculating the image contrast of metasurface I at rotation angles of $45^\circ$ and $135^\circ$, the minimum counts is selected from the part where the two patterns do not overlap, and the image contrast is significantly less than 1, further confirming there indeed exists crosstalk. When the calculated metasurface I rotation angles are $0^\circ,90^\circ$ and $180^\circ$, there is no obvious change in the image contrast. The cancellation of crosstalk enabled by using circular polarization makes the quantum imaging system more stable and lowers the requirement for rotating metasurface to get the correct image for specific heralding polarization state.
Fig. 4. (a) Mapping of ICCD imaging results vs signal arm integration time and trigger arm post-selected polarization state. (b) Mapping of ICCD imaging results vs signal arm metasurface rotation angle and trigger arm polarization state. (c) and (d) Image contrast values of imgaes in (a) and (b), respectively.
The photon pairs produced by the BBO crystal are not only entangled in polarization but also in energy-time. Uncorrelated photons can be greatly blocked by utilizing single-photon heralded imaging techniques. The uncorrelated photons are supplied by the 780 nm laser, as shown in Fig. 2(a), which directly irradiate onto the metasurface and only a fraction of them are reflected into the imaging system. As shown in Fig. 5, when the ICCD operates at a DDG mode, in which the exposure time is $60$s, the detection of heralding photons triggers the acquisition. The higher the noise level, the more unpaired photons falling in the effective acquisition time windows, leading to lower image contrast. If the noise level is smaller than $80$ counts per second per pixel, the image contrast is almost unchanged. And if the noise level continues to increase and amounts to $28000$ counts per second per pixel, the image contrast is decreased rapidly. In order to more realistically reflect the anti-noise ability of single-photon heralded imaging, a control experiment is conducted, in which ICCD operates at a Fire Only (FO) mode with exposure time of $60$s, and unpaired photons are filtered out. Apparently, the background noise severely damages the image contrast with a noise level of only $160$ counts per second per pixel. In addition, another control experiment has been conducted where the acquisition time in FO mode is set to be $27$ms, which is the total effective acquisition time used in DDG mode, the single photon counts are so small that the shot noise dominates, and the image is not clearly displayed. Furthermore, from comparing Fig. 5(a) and (b), it can be seen that when the background noise is $16000$ counts per second per pixel, the basic outline of the pattern can still be seen with H/V polarization, while the pattern is no longer visible with L/R polarization. This is because the contrast of H/V polarization is higher than that of L/R polarization during our experiment. As mentioned earlier, the effect of accidental coincidence is negligible in the experiment, so the visibility under the two sets of polarization basis plays a dominant role, and the theoretical model has given the fact that the smaller $\epsilon$ will result in less crosstalk between orthogonal polarizations.
Fig. 5. (a) and (b) Comparison between quantum heralded imaging and classical direct imaging when noise level changes in H/V basis and L/R basis, respectively. DDG: Digital Delay Generator, FO: Fire Only.
6. Conclusion
In summary, we have experimentally demonstrated the possibility of using polarization-multiplexing metasurfaces for quantum imaging. Based on metasurfaces, the decoupling of the near-field patterns for any set of orthogonal polarization basis can be realized. Different polarization state which is selected at the heralding arm remotely control the signal arm. Compared to the linear polarization case, the circular polarization case meets less sample placement constraint and can eliminate crosstalk between orthogonal polarization states. At the same time, the heralded-imaging scheme can greatly reduce noise and ensure the signal-to-noise ratio of the target patterns. For a single photon, its phase is intrinsically uncertain, but adding the propagation phase or geometric phase imparted by the metasurface to this uncertain phase, the photon can still be manipulated. Our work further improves the stability and functional richness of the quantum imaging system, fully confirming the feasibility of practical application of quantum metasurfaces. The abundant phase control capabilities of metasurfaces may provide a new driving force for the research of new quantum optical effects, and the development of quantum technologies.
Funding
Fundamental Research Funds for the Central Universities (020414380175); National Natural Science Foundation of China (11621091, 11774162, 11774164, 11822406, 11834007, 62288101); National Program on Key Basic Research Project of China (2017YFA0303700).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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