Phase reconstruction via metasurface-integrated quantum analog operation

Introduction

With the unique advantages in label-free imaging and material characterization¹⁻⁶, the phase reconstruction of light field provides an important scientific basis for the detection of biological samples and disease diagnosis⁷. Since the oscillation frequency of the optical wavefront is much higher than the response frequency of the human eye or detector, the phase of the optical wavefront is difficult to be directly detected⁸. With several typical branches, the combination of digital holography and optical imaging seems to solve this difficulty, including optical diffraction tomography⁹, holographic multiplexing¹⁰, and deep learning¹¹. However, complex devices and multiple measurements are usually required in some schemes, which inevitably affect the imaging quality¹²⁻¹⁵. Therefore, how to integrate devices and reduce unnecessary operations remains a significant challenge to achieve accurate quantitative phase reconstruction.

The metasurface, which constitutes a compact nano-photonics platform for effective light field manipulation, is a suitable candidate to solve the above challenge¹⁶⁻²². By customizing tailored photonic response in metasurfaces, the amplitude, phase, and polarization of light can be accurately tuned²³⁻²⁹, which simplify the complex optical information processing to an optical analog operation³⁰⁻³⁷. The introduction of the optical analog operation on metasurface into the quantitative phase reconstruction provides a possible scheme for obtaining high-contrast images of transparent objects³⁸⁻⁴². However, for some photosensitive samples, such as yeast and neuron cells, the imaging system needs to be operated under low signal light conditions⁴³⁻⁴⁵, while the overwhelming ambient shot noise determines that conventional imaging schemes have a low signal-to-noise ratio (SNR)⁴⁶⁻⁴⁸. Consequently, how to realize quantitative phase reconstruction with a high SNR under low signal light conditions is still a critical problem.

In this paper, based on multi-channel metasurface and quantum entanglement source, a simple and integrated quantum analog operation system is proposed to realize quantitative phase reconstruction with a high SNR under a low signal photon level. In the experiment, four differential operations are implemented simultaneously on the same metasurface, which are required for obtaining the phase gradient. Then, the quantitative reconstruction can be realized by performing Fourier integral operation and solving optimization problem for the phase gradient. Combined with the quantum entanglement source, sister photons of the imaging photon can be used as the external trigger signal of the detector to control the shutter switch, which can filter out most of the ambient noise and obtain the images with an improved SNR. Besides, the multi-channel and non-local switching of mode selection is realized through the polarization correlation of entangled photon pairs. The proposed method of phase reconstruction may pave the way for the applications of metasurfaces in optical chip, wave function reconstruction, and label-free biology imaging.

Methods

Based on a multifunctional metasurface, the concept of metasurface-integrated analog operation system is proposed, which has four polarization-dependent channels to perform different modes. On this basis, combined with the quantum entanglement source system, the remote switching of the working mode can be realized. As shown in Fig. 1(a), the entangled photon pair is in the first Bell state:

Figure 1.  Phase reconstruction via metasurface-integrated quantum analog operation. (a) Schematic of non-local mode selection by metasurface-integrated quantum analog operation. ${\hat{F}}_{1}$, ${\hat{F}}_{2}$, ${\hat{F}}_{3}$, and ${\hat F_4}$ are the differential operators. The green area has been designed to construct differential operators, and the blue area can be extended to construct other operators. (b) Three modes of metasurface-integrated quantum analog operation system and corresponding differential output results through non-local mode selection. (c) Theoretical phase gradient in x-direction. (d) Theoretical phase gradient in y-direction. (e) Theoretical two-dimensional (2D) phase gradient. (f) Theoretical phase distribution calculated by the 2D phase gradient.

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where the subscripts $\mathrm{i}$ and $\mathrm{\mathrm{ }t}$ represent imaging photon and trigger photon, used to extract the sample information and control the detector switch, respectively. $|H\rangle$ and $|V\rangle$ are horizontal and vertical polarization states. Equation (1) indicates that when the polarization state of trigger photon is set as $|{H_{\mathrm{t}}}\rangle$ $(|{V_{\mathrm{t}}}\rangle )$, the imaging photon will be collapsed to $|{H_{\mathrm{t}}}\rangle$ $(|{V_{\mathrm{t}}}\rangle )$ simultaneously. The optical axes of the four working regions on the metasurface all change periodically in one dimensional (1D) direction with the spatial period of $\Lambda$. After passing through arbitrary working region of the metasurface, the imaging photon will produce a spin-dependent shift $\delta = - {\sigma _ \pm }(\lambda z/\Lambda )$, where $\sigma = \pm 1$ indicate the left-circularly polarized (LCP) and right-circularly polarized light (RCP), respectively, and $z$ is the transmission distance of the imaging photon⁴⁹.

Assuming that the imaging photon may pass through the center of four regions, four differential operators ${\hat{F}}_{1}=(\alpha +{\mathrm{i}}\delta (\partial /\partial x))$, ${\hat{F}}_{2}=(\alpha -{\mathrm{i}}\delta { }(\partial /\partial x))$, ${\hat{F}}_{3}=(\beta + {\mathrm{i}}\delta { }(\partial /\partial y))$, and ${ }{\hat{F}}_{4}=(\beta -{\mathrm{i}}\delta { }(\partial /\partial y))$ can be constructed. Here, $\alpha$ and $\beta$ are the tiny angles which the optical axes deviating from the $H$- and $V$-directions, respectively. The initial spatial wave function of the imaging photon can be expressed as $|{\Psi _{{\mathrm{sp}}}}{\rangle _{{\mathrm{in}}}}$. After being affected by ${\hat F_i}{\mathrm{ }}(i = 1,2,3,4),$ the wave function will evolve to $|{\Psi _{{\mathrm{sp}}}}{\rangle _{{\mathrm{out}}}} = {\hat F_{{i}}}|{\Psi _{{\mathrm{sp}}}}{\rangle _{{\mathrm{in}}}}$. Eventually, the intensity distribution of photons on the detector can be given as (see Section 1 in the Supplementary information):

where $\gamma$ is the polarization angle, and ${P_{{\mathrm{in}}}}(x,y) = \langle {\Psi _{{\mathrm{sp}}}}|{\Psi _{{\mathrm{sp}}}}\rangle$ represents the initial intensity distribution. Notably, the four-channel metasurface helps to simultaneously capture the four results shown in Eq. (2), which are necessary to achieve phase reconstruction based on differential operations. In conventional schemes, multiple adjustments of the optical path are involved to capture the above results, inevitably increasing complexity and decreasing accuracy. After linearly combining the above Eq. (2), quantitative phase gradient can be obtained by

where ${\mathrm{\Delta}} P_{{\mathrm{out}}}^x$ and ${\mathrm{\Delta}} P_{{\mathrm{out}}}^y$ can be expressed as

Specially, since the imaging and the trigger photons have polarization entanglement, when the trigger photon is polarized at $\gamma = {0^\circ},{90^\circ},{45^\circ}$, the imaging photon will collapse into the same polarization state simultaneously. Under this basic principle, the non-local control of the differential operator can be achieved [Fig. 1(b)]. In particular, the quantum analog operation system can still function properly when the light source does not match the designed working wavelength⁵⁰ (see Section 2 in the Supplementary information).

Considering the case in which the light is emitted to phase objects, such as transparent biological cells⁵⁰⁻⁵³, the amplitude of the input light field is hardly affected due to the weak absorption feature of light, and only phase information $\varphi (x,y)$ of the object will be introduced. Therefore, after the imaging photon passes through the phase object, the spatial wave function can be expressed as $\left| {{\Psi _{{\mathrm{sp}}}}(x,y)} \right\rangle = {\mathrm{exp}} [{\mathrm{i}}\varphi (x,y)]\left| {\psi (x,y)} \right\rangle$, where $\left| {\psi (x,y)} \right\rangle$ represents the initial wave function of the imaging photon. If the polarization state of trigger photon is set as $\left| D \right\rangle$, the wave function input to the metasurface can be expressed as

Subsequently, according to the Fourier integral theorem, normalized phase distribution can be calculated as⁴⁵

where $\mathcal{F}$ and ${\mathcal{F}^{ - 1}}$ represent the Fourier and inverse Fourier transforms, ${k_x}$ and ${k_y}$ represent wave vectors in the $x$- and $y$-directions, respectively. However, compared with the true phase distribution $\varphi (x,y)$, the normalized reconstructed phase distribution $\phi (x,y)$ differs by a normalized factor $K$, which is necessary for quantitative phase reconstruction. Since differential operations are linear operations, the following relation holds:

where $\nabla \phi (x,y)$ is calculated numerically by $\phi (x,y)$. According to Eq. (7), the residual difference, which is between the measured phase gradient $\nabla \varphi (x,y)$ and the reconstructed phase gradient $K\nabla \phi (x,y)$, can be defined as

By calculating the minimum value of the residual difference $S(K)$, the optimal normalization coefficient $K$ is obtained. Consequently, quantitative phase reconstruction results can be described as

Results and discussion

Conclusions

In the proposed metasurface-integrated quantum analog operation system, the non-local correlation of polarized entangled photon pairs is a prerequisite for the remote switching of mode selection. Through the polarization selection of the trigger photon, the imaging photon can be induced to collapse to the desired state simultaneously, realizing the analog operation in different modes. As a demonstration, four differential operations in the metasurface-integrated quantum analog operation system are designed for realizing quantitative phase reconstruction. Theoretically, more diverse operational modes are expected to be achieved by designing multiple functional structures on the metasurface, which can be freely switched by trigger photons. Therefore, the proposed metasurface-integrated quantum analog operation system is extremely malleable. Furthermore, the quantum analog operation provides clearer imaging results than classical analog operations. When the signal is equivalent (even smaller than) to the ambient noise, the signal in the classical method will be drowned by ambient noise inevitably. However, in the quantum method, the sister photon of the signal photon can be used as an external trigger signal to control the switch of the detector, which suppresses the ambient noise effectively.

In conclusion, based on multi-channel metasurface and quantum entanglement source, a simple and integrated quantum analog operation system is proposed to realize quantitative phase reconstruction with a high SNR under a low signal photon level. Specifically, due to the non-local correlation of entangled photons, the mode selection of the metasurface-integrated quantum analog operation system can be switched remotely without changing the imaging arm. Notably, the four differential operators required for phase reconstruction can be obtained simultaneously by an integrated metasurface. As a result, the relative error of quantitative phase reconstruction is 2.71 %, which indicate the accuracy of the proposed method. Besides, the SNR of differential results is improved by $2.69$ times compared with the classical approach. The results demonstrate our proposed scheme may potentially empower the metasurface in optical chip, wave function reconstruction, and label-free biology imaging.