# Enhanced Phase Retrieval Method Based on Random Phase Modulation
## 1 Introduction  
The profile and inner structure of an object lie in the complex field scattered from the object. However, current detectors are only sensitive to amplitude, leaving the phase information unrecorded. The problem that remains to be solved is how to get the phase information of a wavefront, because of the lack of apparatus with a response frequency faster than the wave frequency. Interferometric imaging approaches use a known reference wave to hide the phase information in the interference pattern produced by the reference and target wavefront [1, 2]. However, the existence of extra reference light increases the complexity and disturbance of the system [3]. Beam-propagation-based approaches are another powerful tool to reconstruct the complex amplitude of sample. The simple experimental set-up makes this technique capable of robustness to external influence and suitability for various wavelengths [4]. The beam-propagation-based methods utilize variants of iterative phase retrieval techniques which originate from the Gerchberg-Saxton (GS) algorithm to recover the complete wavefront [5]. The iterative phase retrieval works via numerical propagation of the light field back and forth between the recording plane and the object plane. The iterative methods which require only a single measurement always suffer from low convergence speed and much sensitivity to the initial value [6, 7, 8]. Although successful reconstruction could be obtained with these methods, the difficulties encountered in obtaining a tight support constraint limit the application scope. To overcome these restrictions, multi-image phase retrieval techniques that use different kinds of diversities in data collection were proposed [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]. Methods to generate multiple-diversityintensity measurements include, among others, illumination area overlap [9; 10], multiple recording distance [11; 12; 13; 14], multiple wavelength [15], and illumination beam tilting [16; 17]. Except for the multiple wavelength method, the other three types of methods require employing mechanical platforms, which inevitably lead to low acquisition speed and cause potential mechanical error. The method which uses varied wavelength to offer diversities may lead to stagnation and ambiguity in the iterative process due to the lack of intensity variation when the wavelength changes are relatively small. In fact, random phase modulation can provide effective diversity to realize quality imaging results in multi-image phase retrieval methods. Since the \"uniqueness\" of the random phase can break any symmetry that may exist in the object field, it allows for wiping out any ambiguity and stagnation problem [22; 23]. Furthermore, variable modulation can be implemented using a spatial light modulator (SLM); thus, data acquisition speed is improved and mechanical error is avoided. The spread-spectrum phase retrieval (SSPR) method can realize rapid convergence rate by using random phase modulation and a serial reconstructed framework [20; 21]. However, the introduction of random phase modulation and a coherent light source brings noise, and the SSPR accumulates noise as the iteration progresses since the serial reconstruction algorithm has no noise suppression operation. A parallel reconstructed framework proposed in Reference [19] can be used in the random phase modulation method to enhance the noise immunity and robustness, due to the existence of an average operation in every iteration. Nevertheless, the parallel reconstructed algorithm suffers from slow convergence speed. In this study, we propose a beam-propagation-based phase retrieval method based on random phase modulation and a serial-parallel cascaded reconstruction framework, which can suppress the noise generated during the imaging process while realizing rapid convergence. In the proposed method, we use the serial algorithm from SSPR to generate an initial guess of the parallel reconstruction algorithm.