基于复关联系数测量的无参考光单像素全息

Reference-Free Single-Pixel Holography Based on Complex Correlation Coefficient Measurement

摘要: 针对现有干涉型单像素全息的系统复杂且噪声敏感，以及迭代型单像素全息的相位恢复计算量大且收敛慢等问题，本文提出一种基于复关联系数测量的无参考光单像素全息方法。该方法将物光看作为一组完备正交基的线性组合，通过在空间光调制器件上加载预先设计的调制图案，利用单像素测量并解析求解每个基函数的复关联系数，最后线性叠加重建物光的复振幅分布。依据衍射理论和复系数测量原理建立了模型，在0~50 dB高斯随机噪声条件下，以透镜、复合实物和复杂光场为目标物体，使用Hadamard、傅里叶、DCT和随机正交四种基函数调制图案开展了仿真实验。通过引入数字微镜阵列(DMD)和空间光调制器(SLM)的硬件误差，验证了该方法在非理想调制条件下的实用性，并与经典的Gerchberg-Saxton (GS)迭代算法进行了性能对比。当信噪比达到20 dB时，振幅归一化均方误差(NMSE)低于5%，相位均方根误差(RMSE)低于0.2 rad；在无环境噪声理想条件下，各类目标物体的重建振幅与相位误差均达到10⁻¹⁵量级；随机正交基调制图案在低信噪比下的噪声鲁棒性最优，而所有基函数调制图案在无环境噪声的理想条件下均能实现高精度无偏重建。此外，初步的压缩感知应用表明，该方法能够在欠采样下重构出原始物体。该方法无需干涉参考光，无需迭代计算，具有系统简洁、计算高效、抗噪性能良好的优点，在生物细胞相位成像、透明元件检测、涡旋光场表征等领域具有广阔的应用前景。

Abstract: To address the complex optical setup and environmental sensitivity in interferometric single-pixel holography, as well as the heavy computational load and slow convergence of iterative phase retrieval algorithms in single-pixel holography, we propose a reference-free single-pixel holography based on complex correlation coefficient measurement. In this method, the object field is considered as a linear combination of a complete set of orthogonal bases. By loading specifically designed modulation patterns onto spatial light modulators, the square of the modulus of the complex correlation coefficient for each modulation pattern is measured using a single-pixel detector. The complex correlation coefficients are analytically solved using their single-pixel measurements, and the complex amplitude of the object field is finally reconstructed through linear superposition. A theoretical model is established based on diffraction theory and the principle of complex coefficient measurement. Simulation experiments are carried out under Gauss random noise from 0 dB to 50 dB, using a lens, a compound real object, and a complex optical field as target objects, with four types of basis modulation patterns, Hadamard, Fourier, DCT, and random orthogonal bases. Furthermore, by introducing the hardware errors of DMD and SLM, the practicability of this method under non-ideal modulation conditions was verified. And a comprehensive performance comparison was conducted with the classic Gerchberg-Saxton (GS) iterative algorithm. When the signal to noise ratio (SNR) reaches 20 dB, the normalized mean square error (NMSE) of amplitude is below 5% and the root mean square error (RMSE) of phase is below 0.2 rad. Under noiseless conditions, the reconstruction errors of both amplitude and phase for all types of target objects reach the order of 10⁻¹⁵. The random orthogonal basis modulation pattern exhibits the best noise robustness at low SNR, while all basis modulation patterns achieve high-precision unbiased reconstruction under ideal noise-free conditions. Preliminary applications of compressive sensing show that this method can reconstruct the original object under under-sampling. The proposed method requires neither an interferometric reference beam nor iterative computation, featuring a simple system configuration, high computational efficiency, and good noise immunity. It holds broad application prospects in areas such as phase imaging of biological cells, inspection of transparent optical components, and characterization of vortex optical fields.