Phase Retrieval Method Research Articles

Broad-band phase retrieval method for transient radial shearing interference using chirp Z transform technique* *Project supported by the National Natural Science Foundation of China (Grant No. 61705254) and the Key Research and Development Program of Shaanxi Province of China (Grant No. 2020GY-114).

The transient radial shearing interferometry technique based on fast Fourier transform (FFT) provides a means for the measurement of the wavefront phase of transient light field. However, which factors affect the spatial bandwidth of the wavefront phase measurement of this technology and how to achieve high-precision measurement of the broad-band transient wavefront phase are problems that need to be studied further. To this end, a theoretical model of phase-retrieved bandwidth of radial shearing interferometry is established in this paper. The influence of the spatial carrier frequency and the calculation window on phase-retrieved bandwidth is analyzed, and the optimal carrier frequency and calculation window are obtained. On this basis, a broad-band transient radial shearing interference phase-retrieval method based on chirp Z transform (CZT) is proposed, and the corresponding algorithm is given. Through theoretical simulation, a known phase is used to generate the interferogram and it is retrieved by the traditional method and the proposed method respectively. The residual wavefront RMS of the traditional method is 0.146λ, and it is 0.037λ for the proposed method, which manifests an improvement of accuracy by an order of magnitude. At the same time, different levels of signal-to-noise ratios (SNRs) from 50 dB to 10 dB of the interferogram are simulated, and the RMS of the residual wavefront is from 0.040λ to 0.066λ. In terms of experiments, an experimental verification device based on a phase-only spatial light modulator is built, and the known phase on the modulator is retrieved from the actual interferogram. The RMS of the residual wavefront retrieved through FFT is 0.112λ, and it decreases to 0.035λ through CZT. The experimental results verify the effectiveness of the method proposed in this paper. Furthermore, the method can be used in other types of spatial carrier frequency interference, such as lateral shearing interference, rotational shearing interference, flipping shearing interference, and four-wave shearing interference.

On inverse problem with phase retrieval for an inclined plane in the parabolic approximation

The inverse problem of amplitude reconstruction on an inclined plane based on the values of amplitude or its module as recorded on semi-infinite plane orthogonal to the beam propagation direction is considered within the framework of 2D parabolic equation. It is demonstrated that this inverse problem, in case when the complex image plane amplitude is known, can be reduced to a singular Cauchy-type integral equation. The existence of its solutions requires that certain conditions be met but if a solution exists it is necessary unique. The obtained integral equation is then approximated piece-wisely and the resulting linear algebraic system is solved numerically while applying necessary regularization procedures to enhance the stability of its solutions. Finally, an iterative method of phase retrieval is developed and a set of numerical experiments is performed.

Polarization-Encoded Fully-Phase Encryption Using Transport-of-Intensity Equation

In this study, we propose a novel method to encrypt fully-phase information combining the concepts of the transport of intensity equation and spatially variant polarization encoding. The transport of intensity equation is a non-iterative and non-interferometric phase-retrieval method which recovers the phase information from defocused intensities. Spatially variant polarization encoding employs defocused intensity measurements. The proposed cryptosystem uses a two-step optical experimentation process—primarily, a simple set-up for defocused intensities recording for phase retrieval and then a set-up for encoding. Strong security, convenient intensity-based measurements, and noise-free decryption are the main features of the proposed method. The simulation results have been presented in support of the proposed idea. However, the TIE section of the cryptosystem, as of now, has been experimentally demonstrated for micro-lens.

Optimized phase-coding method for absolute phase retrieval based on K-means algorithm

The phase-coding method for absolute phase measurement possesses a strong robustness because its codewords are encoded into phase domain rather than intensity. However, a misalignment problem arises between fringe order and wrapped phase in this method owing to random noises and image blurring, especially in the high-frequency fringes situation. Therefore, the measurement result is unavailable. To overcome this problem, an optimized phase-coding algorithm is proposed in the study. We use the sinusoidal patterns to generate two opposite binary masks, which can locate half of each codeword. Then, the K-means algorithm is used to correct the codewords errors. The accurate fringe order is obtained by merging two halves of codewords and the absolute phase is recovered afterward. The used method is experimentally verified by an established measurement system, and the results show the proposed method is robustness and efficiency.

Dynamic sampling iterative phase retrieval for holographic data storage.

A dynamic sampling iterative phase retrieval method, which dynamically samples the Fourier intensity distribution of the reconstruction beam captured by the detector, is proposed to shorten the iterative number and decrease the phase error rate of phase retrieval in the phase-modulated holographic data storage. By the dynamic sampling method, that keeping relatively low frequency component of Fourier intensity spectrum at the beginning of iteration and gradually releasing more high frequency component at the subsequent iterations, we shortened the iterative number by 2 times and decreased the phase error rate to some extent because our method provided a better convergent path to the phase retrieval. We also believe the thought of our method can be used in more image retrieval fields.

Enhancing Multi-Distance Phase Retrieval via Unequal Interval Measurements

In the conventional methods of multi-distance phase retrieval, the diffraction intensity patterns are recorded at equal intervals, which can induce slow convergence or stagnation in the subsequent reconstruction process. To solve this problem, a measurement method with unequal intervals is proposed in this paper. The interval spacings between adjacent measurement planes are decreased gradually. A large gap accelerates retrieval progress, and a short span helps to recover detailed information. The proposed approach makes full use of the available measured dataset and simultaneously generates variations in diversity amplitude, which is a crucial issue for the techniques of multi-image phase retrieval. Both computational simulations and experiments are performed. The results demonstrate that this method can improve the convergence speed by 2 to 3 times and enhance the quality of reconstruction results in comparison to that of the conventional methods.

High-resolution multi-planar coherent diffraction imaging with multimode fiber source

Coherent diffractive imaging (CDI) is a lensless, high-resolution imaging method. In order to overcome the problems of poor-quality image and slow convergence of iterative algorithm in traditional CDI, an innovative setup for the multimode-based phase retrieval method is proposed. Here the multimode fiber source was selected as the light source to replace the traditional plane-wave incidence. Due to the nature of the multimode fiber, a speckle light field can be generated on the output fiber facet and then is used to illuminate the object. Thus the diffraction field can be rapidly changed at a very short distance so as to realize much higher resolution and consume less iteration time. Then a phase-only spatial light modulator (SLM) is used as a negative lens instead of a positive lens in a 4f imaging system for the advantage of larger detection space. This proposed setup has many advantages such as a simpler structure and static recording. The comparison experiments with plane waves confirm that our proposed method has higher reconstruction resolution and smaller iteration errors.

Phase retrieval from the FTIR transmittance spectrum of thick materials with anchoring points in THz-TDS measurement

Extraction of the refractive index for optically thick and low-loss materials with a single Fourier transform infrared spectrometer (FTIR) transmittance spectrum becomes difficult with the conventional root-finding method at THz frequencies. A phase-retrieval method was proposed based on Kramers-Kronig (K-K) relations with anchoring points from independently measured transmission data in THz-TDS. Two approaches, i.e., multiply subtractive K-K relations and fast Fourier transform, were utilized to successfully obtain the phase spectrum for 3.52 mm TPX and 1.08 mm crystal quartz with two and three anchoring points, respectively. The results agree with the phase from TDS in the overlapped frequency range from 1.5 to 4.5 THz. An iterative optimization algorithm was finally employed to extract the refractive index of the two materials with amplitude and retrieved the phase spectrum of the transmitted signal within the FTIR measured frequency range.

Optical Single Sideband Signal Reconstruction Based on Time-Domain Iteration

Due to its low cost, simple architecture and robustness to fiber dispersion, single sideband (SSB) transmission with direct detection (DD) system is an attractive solution for 80-km inter data center interconnects (DCIs). However, it will suffer performance penalty caused by the signal-to-signal beating interference (SSBI). Kramers-Kronig (KK) receiver has been extensively investigated for SSBI elimination by reconstructing the SSB signal. The nonlinear operations in KK algorithm require up-sampling to cope with spectral broadening, which results in high complexity for practical application. Optical signal phase retrieval method based on the minimum phase signal has also been investigated for SSB signal recovery, in which the SSB and DC-Value properties are iteratively imposed on the amplitude signal in frequency domain. In this paper, we propose a low complexity iterative algorithm for minimum phase signal recovery without up-sampling in time domain. Finite impulse response (FIR) filter is applied to iteratively generate the SSB signal and update the phase component. Based on the proposed scheme, the transmission of 30 GHz SSB 16-QAM discrete multitone (DMT) signal over 80 km single mode fiber (SMF) is successfully demonstrated with the bit error rate (BER) below the hard-decision forward error correction (HD-FEC) threshold of 3.8 × 10−3. The experimental results show that, the BER performance of KK scheme with up-sampling factor of 2, frequency-domain iterative scheme and our proposed scheme is almost the same. However, compared with the KK scheme, the proposed method can save the numbers of adders and multipliers by the factors of 29 and 7, while the factors are 5.5 and 4 comparing to the frequency-domain iteration scheme.

Complex imaging of phase domains by deep neural networks.

The reconstruction of a single-particle image from the modulus of its Fourier transform, by phase-retrieval methods, has been extensively applied in X-ray structural science. Particularly for strong-phase objects, such as the phase domains found inside crystals by Bragg coherent diffraction imaging (BCDI), conventional iteration methods are time consuming and sensitive to their initial guess because of their iterative nature. Here, a deep-neural-network model is presented which gives a fast and accurate estimate of the complex single-particle image in the form of a universal approximator learned from synthetic data. A way to combine the deep-neural-network model with conventional iterative methods is then presented to refine the accuracy of the reconstructed results from the proposed deep-neural-network model. Improved convergence is also demonstrated with experimental BCDI data.

Ultrafast exciton dynamics reconstruction with a ptychographic approach

Excitons characterize the ultrafast response of many materials of technological interest. While the development of attosecond science has unlocked the possibility of performing experiments with a suitable timeresolution, the access to the exciton properties remains a non-trivial step. We propose therefore a novel approach to disclose the physical properties behind the ultrafast exciton dynamics based on a phase-retrieval method.

Suppression of artifacts in X-ray phase-contrast images retrieved by Fourier transform

Over the last two decades, the grating-based phase-contrast imaging has aroused the interest of a number of researchers. It could provide an access to three complementary signals simultaneously: the conventional absorption contrast, the differential phase contrast related to refraction of incident wave, and the dark-field contrast that relates to ultra small angle scattering in a sample. The grating-based phase-contrast signals have higher contrast sensitivity for some types of soft samples than the absorption signals. Dark-field signals have better diagnostic effects in the detection of lung tumors, pneumothorax and the identification of microcalcifications in breast. There are two main phase retrieval methods in grating-based X-ray phase-contrast imaging, i.e. phase stepping method and Fourier transform method. The phase signals retrieved by phase stepping is high precise and has low noise. But the sample suffers high dose due to at least three exposures. The phase signals retrieved by Fourier transform is low-dose due to the fact that only one image with sample is needed, but it is easily affected by artifacts when the size of the filtering window is too large. However, when the size of the filtering window is too small, the high-frequency information of the phase-contrast image will be lost, and the image will become blurred. A trade-off between definitions of the image and artifacts should be made. Since the phase-contrast signal and the dark-field signal of the sample are modulated by carrier fringes, the frequency spectrum of the detected image consists of many different harmonics. The artifacts in the retrieved signals originate from the spectrum aliasing between primary peak around zero spatial frequency and first-order harmonic peaks. Therefore, the subtraction between two images with phase difference can remove the primary peak, and the artifacts in the phase-contrast signals and dark-field signals will be suppressed. In order to further suppress the artifacts, we increase the frequency of carrier fringes, which results in a larger distance between first-order harmonic peaks in frequency domain. We finally attain artifact-free phase-contrast images and dark-field images while maintaining high definition of the images. The method proposed here is not only applicable to incoherent imaging system, but also to Talbot-Lau interferometer, and it would be useful in fast and low-dose X-ray phase-contrast and dark-field imaging.

A self‐tracking phase retrieval algorithm used in distributed disturbance sensing

AbstractThe phase‐retrieval method with delayed calibration is widely used in fiber‐optic sensing structures to extract the distributed vibration messages. Yet, the performance of this method is not satisfactory when sudden changes are introduced in sensing path, for the DC compensation is not a slow variable under this situation. To improve the adjustment of interferometric fiber‐optic structure, a self‐tracking phase retrieval method is proposed. In the novel algorithm, the static state is tracked by calibrating DC compensation value in each retrieval period. When tested in the real scene with a 20 km‐long sensing fiber, the performance of the novel retrieval algorithm has been proved. Experimental results show that, compared with the conventional delayed calibration method, self‐tracking algorithm brings a better accuracy and uniformity performance in intrusion location test and shows a better adaptability to the environment.

Fast measurement of primary reflector for the Tianma telescope based on two beams

Primary reflector of a large radio telescope deforms considerably due to gravitational, thermal, and other effects. An active surface system alleviates these large-scale deformations and improves the electrical performance of the telescope. The fast reflector measurement is the major challenge for active-systems to work efficiently. Currently, the measurement takes approximately 20 min, which is a substantial amount of time. A phase-retrieval holography method based on two beams is proposed. The method employs the principle of overlap and similarity of the two beams and reduces the scanning area to shorten the observation time. Moreover, this method maintains the signal-to-noise ratio of the measured data. The time is reduced by half when compared with the single beam observation. The maximum time required for each measurement is below 8 min, and the achieved accuracy is approximately 55 μm. We performed several measurements to ensure the accuracy of the proposed method at the Tianma telescope. The proposed method successfully corrected gravitational and thermal deformations. Furthermore, it improved the aperture efficiency. We recommend that the proposed method can be used in other large radio telescopes that have multibeam systems.

High-fidelity pixel-super-resolved complex field reconstruction via adaptive smoothing.

Pixel super-resolution (PSR) techniques have been developed to overcome the sampling limit in lensless digital holographic imaging. However, the inherent non-convexity of the PSR phase retrieval problem can potentially degrade reconstruction quality by causing the iterations to tend toward a false local minimum. Furthermore, the ill posedness of the up-sampling procedure renders PSR algorithms highly susceptible to noise. In this Letter, we propose a heuristic PSR algorithm with adaptive smoothing (AS-PSR) to achieve high-fidelity reconstruction. By automatically adjusting the intensity constraints on the estimated field, the algorithm can effectively locate the optimal solution and converge with high reconstruction quality, pushing the resolution toward the diffraction limit. The proposed method is verified experimentally within a coherent modulation phase retrieval framework, achieving a twofold improvement in resolution. The AS-PSR algorithm can be further applied to other phase retrieval methods based on alternating projection.

Quantitative phase retrieval with low photon counts using an energy resolving quantum detector.

X-ray phase contrast imaging (PCI) combined with phase retrieval has the potential to improve soft-material visibility and discrimination. This work examined the accuracy, image quality gains, and robustness of a spectral phase retrieval method proposed by our group. Spectroscopic PCI measurements of a physical phantom were obtained using state-of-the-art photon-counting detectors in combination with a polychromatic x-ray source. The phantom consisted of four poorly attenuating materials. Excellent accuracy was demonstrated in simultaneously retrieving the complete refractive properties (photoelectric absorption, attenuation, and phase) of these materials. Approximately 10 times higher SNR was achieved in retrieved images compared to the original PCI intensity image. These gains are also shown to be robust against increasing quantum noise, even for acquisition times as low as 1 s with a low-flux microfocus x-ray tube (average counts of 250 photons/pixels). We expect that this spectral phase retrieval method, adaptable to several PCI geometries, will allow significant dose reduction and improved material discrimination in clinical and industrial x-ray imaging applications.

Absolute phase retrieval for colored objects based on three phase-shifting amount codes.

We propose an absolute phase retrieval method based on three phase-shifting amount codes (3-PSA-codes) to measure the colored object with one additional pattern. 3-PSA-codes adopt the coding concept of 3-digit-codes, in which the code elements of three consecutive periods are treated as a unique code word for one period. However, to measure the colored object more effectively in the proposed method, each code element is embedded into the PSA domain and retrieved from the phase difference. Fringe patterns for the wrapped phase are artfully employed in the code element retrieval. Hence, for the first time, to the best of our knowledge, the code element related to the phase can be determined by one additional pattern. It breaks the constraint that temporal methods require multiple additional patterns to overcome the adverse effect of the surface color of objects on absolute phase retrieval. Experimental results demonstrate that the proposed 3-PSA-codes have strong robustness in the measurement of the colored object.

Four-pattern, phase-step non-sensitive phase shifting method based on Carré algorithm

Carré algorithm is a classic phase shifting measurement method due to its advantage in resisting the error caused by inaccurate phase step. The conventional Carré algorithm only calculates the wrapped phase, so multiple frequencies are used to unwrap the phase so as to obtain the absolute phase, which increases the number of patterns and decreases the measuring speed. We propose a high-accuracy and 4-pattern phase retrieval method based on Carré algorithm. The phase step in Carré algorithm is encoded as a linear variable. With the linear phase step, we obtain both the wrapped phase and absolute phase simultaneously from only 4 patterns. The proposed method is non-sensitive to the errors in phase step, the same as the conventional Carré algorithm. In addition, because only 4 patterns are required for obtaining the wrapped phase and absolute phase, the proposed method is practical in high-speed 3D measurement. Experimental results demonstrate its performance in high accuracy (absolute accuracy is higher than 0.02 mm and relative accuracy is higher than 0.03%), efficiency (requires only 4 patterns) and robustness.

Determination of nonlinear refractive index by an iterative phase retrieval method.

We present a simple and robust technique for measuring the nonlinear refractive index. The principle is based on an iterative phase retrieval algorithm with a pump-probe system. Different strong phase modulations are intentionally introduced into the probe beam, and corresponding diffraction intensity patterns are recorded. The recordings are used in the phase retrieval algorithm to reconstruct the pump-induced phase on the probe beam. The nonlinear refractive index is then extracted from the reconstructed phase. The reconstruction method offers a straightforward procedure and a simple lensless setup. Simulations validate the proposed method. The effects of different characteristics of the pump and probe beams on the quality of reconstructions are investigated. The obtained results demonstrate that the reconstructions are accurate even for the probe beams with complex-valued fields and non-Gaussian pump beams; it removes the requirement for smooth fields of the pump and probe beams. The validity of the method in noisy conditions is also shown.

Noninvasive THz spectroscopy for bunch current profile reconstructions at MHz repetition rates

X-ray free-electron lasers based on superconducting accelerator technology deliver ultrashort photon pulses with unprecedented peak brilliance at high repetition rates. Continuous and noninvasive monitoring of the current profile of the electron bunches is essential for the operation and control of the accelerator. Longitudinal diagnostics based on coherent radiation have already shown their potential at various free-electron laser facilities, and the multi-GeV electron beams of x-ray free-electron lasers are powerful sources for the generation of broadband coherent diffraction radiation. We present noninvasive current profile measurements with a few femtoseconds resolution based on spectroscopy of coherent diffraction radiation in the frequency range 0.7--58 THz. The current profiles, reconstructed from the spectroscopic data with an advanced phase retrieval method, are compared with measurement results obtained with a transverse deflecting structure. For the first time, bunch-resolved current profiles have been recorded simultaneously to user operation at European XFEL for all bunches in the bunch train at MHz repetition rates.