Phase retrieval by means of a spatial light modulator in the Fourier domain of an imaging system

Simple phase retrieval method based on two intensity measurements on a single plane

In this work, a simple phase retrieval method is proposed by observing two intensity patterns on a single plane, which are generated with and without a lens. Rigorous theoretical derivations show that the two fields constitute the Fourier transform pairs, and a modified Gerchberg-Saxton algorithm is proposed to recover the phase patterns from the Fourier pairs. The proposed method does not require the intensity patterns to be measured on two different planes along the propagation distance, and this is quite beneficial in a system with a phase tuning element like a spatial light modulator, which can form a virtual lens by creating a parabola-like phase distribution. Experiments are conducted to demonstrate the effectiveness of the proposed phase retrieval method.

Fast 3D form measurement using a tunable lens profiler based on imaging with LED illumination

We present a fast shape measurement of micro-parts based on depth discrimination in imaging with LED illumination. It is based on a 4f-setup with an electrically adjusted tunable lens at the common Fourier plane. Using such a configuration, the opportunity to implement a fast depth scan by means of a tunable lens without the requirement of mechanically moving parts and depth discrimination using the limited spatial coherence of LED illumination is investigated. The technique allows the use of limited spatially partially coherent illumination which can be easily adapted to the test object by selecting the geometrical parameters of the system accordingly. Using this approach, we demonstrate the approach by measuring the 3D form of a tilted optically rough surface and a cold-formed micro-cup. The approach is robust, fast since required images are captured in less than a second, and eye-safe and offers an extended depth of focus in the range of few millimetres. Using a step height standard, we determine a height error of ±1.75 μm (1σ). This value may be further decreased by lowering the spatial coherence length of the illumination or by increasing the numerical aperture of the imaging system.

Fast form measurements using a digital micro-mirror device in imaging with partially coherent illumination

We present a new technique for fast form measurement based on imaging with partially coherent illumination. It consists of a 4f-imaging system with a digital micro-mirror device (DMD) located in the Fourier plane of its two lenses. The setup benefits from spatially partially coherent illumination that allows for depth discrimination and a DMD that enables a fast depth scan. Evaluating the intensity contrast, the 3D form of an object is reconstructed. We show that the technique additionally offers extended depth of focus imaging in microscopy and short measurement times of less than a second.

Phase defect detection of large-aperture optics with static multiplanar coherent diffraction imaging

Phase defect detection with micrometer scale on large aperture optical elements is one of the challenges in precision optical systems. An efficient scheme is proposed to detect phase defects. First, the defects are positioned in a large aperture by dark-field imaging based on large aperture photon sieves to improve the detection efficiency with a relatively low cost. Second, static multiplanar coherent diffraction imaging is used to retrieve the phase of the positioned defects in a small field of view. Here, a spatial light modulator is used as a multifocal negative lens to eliminate the mechanical errors in multiplanar imaging. The use of a negative lens instead of a positive lens has the advantage of a larger imaging space for the system configuration. Compared to the traditional interferometry system, this diffraction detection system has a simpler optical path and doesn't require sparse distribution of the defects. Experiment results demonstrate the success of the proposed scheme with a detection resolution better than 50 µm. We believe this work provides an effective method to rapidly detect phase defects on large aperture optics with high accuracy and high resolution.

Spatial multiplexing and autofocus in holographic contouring for inspection of micro-parts

We present a method for fast geometrical inspection of micro deep drawing parts. It is based on single-shot two-wavelength contouring digital holographic microscopy (DHM). Within the capturing process, spatial multiplexing is utilized in order to record the two required holograms in a single-shot. For fast evaluation, determining the locations where the object is in focus and stitching all focus object's areas together is achieved digitally without the need for any external intervention using an autofocus algorithm. Thus, the limited depth of focus of the microscope objective is improved. The autofocus algorithm is based on minimizing the total variation (TV) of phase difference residuals of the two-wavelength measurements. In contrast to standard DHM, an object side telecentric microscope objective is used for overcoming the image scaling distortions caused by a conventional microscope objective. The method is used to reconstruct the 3D geometrical shape of a cold drawing micro cup. Experimental results verify the improvement of DHM's depth of focus.

Fourier phase retrieval with a single mask by Douglas-Rachford algorithms

The Fourier-domain Douglas-Rachford (FDR) algorithm is analyzed for phase retrieval with a single random mask. Since the uniqueness of phase retrieval solution requires more than a single oversampled coded diffraction pattern, the extra information is imposed in either of the following forms: 1) the sector condition on the object; 2) another oversampled diffraction pattern, coded or uncoded. For both settings, the uniqueness of projected fixed point is proved and for setting 2) the local, geometric convergence is derived with a rate given by a spectral gap condition. Numerical experiments demonstrate global, power-law convergence of FDR from arbitrary initialization for both settings as well as for 3 or more coded diffraction patterns without oversampling. In practice, the geometric convergence can be recovered from the power-law regime by a simple projection trick, resulting in highly accurate reconstruction from generic initialization.

High-resolution transport-of-intensity quantitative phase microscopy with annular illumination

For quantitative phase imaging (QPI) based on transport-of-intensity equation (TIE), partially coherent illumination provides speckle-free imaging, compatibility with brightfield microscopy, and transverse resolution beyond coherent diffraction limit. Unfortunately, in a conventional microscope with circular illumination aperture, partial coherence tends to diminish the phase contrast, exacerbating the inherent noise-to-resolution tradeoff in TIE imaging, resulting in strong low-frequency artifacts and compromised imaging resolution. Here, we demonstrate how these issues can be effectively addressed by replacing the conventional circular illumination aperture with an annular one. The matched annular illumination not only strongly boosts the phase contrast for low spatial frequencies, but significantly improves the practical imaging resolution to near the incoherent diffraction limit. By incorporating high-numerical aperture (NA) illumination as well as high-NA objective, it is shown, for the first time, that TIE phase imaging can achieve a transverse resolution up to 208 nm, corresponding to an effective NA of 2.66. Time-lapse imaging of in vitro Hela cells revealing cellular morphology and subcellular dynamics during cells mitosis and apoptosis is exemplified. Given its capability for high-resolution QPI as well as the compatibility with widely available brightfield microscopy hardware, the proposed approach is expected to be adopted by the wider biology and medicine community.

Characterizing a spatial light modulator using ptychography

Ptychography is used to characterize the phase response of a spatial light modulator (SLM). We use the technique to measure and correct the optical curvature and the gamma curve of the device. Ptychography's unique ability to extend field of view is then employed to test performance by mapping the phase profile generated by a test image to subpixel resolution over the entire active region of the SLM.

Wavefront sensing based on a spatial light modulator and incremental binary random sampling

A wavefront sensing method based on a spatial light modulator (SLM) and an incremental binary random sampling (IBRS) algorithm is proposed. In this method, the recording setup is built just by a transmittance SLM and an image sensor. The tested wavefront incident to the SLM plane can be quantitatively retrieved from the diffraction intensities of the wavefront passed through the SLM displaying a IBRS pattern. Because only two modulation states (opaque and transparent) of the SLM are used, the method does not need to know the concrete modulation function of the SLM in advance. In addition by introducing the concept of the incremental random sampling into wavefront sensing, the adaptability of phase retrieving based on the diffraction intensities is significantly improved. To the best of our knowledge, no previous study has used this concept for the same purpose. Some experimental results are given for demonstrating the feasibility of our method.

Holographic display system for dynamic synthesis of 3D light fields with increased space bandwidth product

We present a new method for the generation of a dynamic wave field with high space bandwidth product (SBP). The dynamic wave field is generated from several wave fields diffracted by a display which comprises multiple spatial light modulators (SLMs) each having a comparably low SBP. In contrast to similar approaches in stereoscopy, we describe how the independently generated wave fields can be coherently superposed. A major benefit of the scheme is that the display system may be extended to provide an even larger display. A compact experimental configuration which is composed of four phase-only SLMs to realize the coherent combination of independent wave fields is presented. Effects of important technical parameters of the display system on the wave field generated across the observation plane are investigated. These effects include, e.g., the tilt of the individual SLM and the gap between the active areas of multiple SLMs. As an example of application, holographic reconstruction of a 3D object with parallax effects is demonstrated.

Enhanced intensity variation for multiple-plane phase retrieval using a spatial light modulator as a convenient tunable diffuser

In the multiple-plane phase retrieval method, a tedious-to-fabricate phase diffuser plate is used to increase the axial intensity variation for a nonstagnating iterative reconstruction of a smooth object wavefront. Here we show that a spatial light modulator (SLM) can be used as an easily controllable diffuser for phase retrieval. The polarization modulation at the SLM facilitates independent formation of orthogonally polarized scattered and specularly reflected beams. Through an analyzer, the polarization states are filtered enabling beam interference, thereby efficiently encoding the phase information in the axially diverse speckle intensity measurements. The technique is described using wave propagation and Jones calculus, and demonstrated experimentally on technical and biological samples.

Phase retrieval and diffractive imaging based on Babinet's principle and complementary random sampling

We proposed an iterative method for phase retrieval and diffractive imaging based on Babinet's principle and complementary random sampling (CRS). We demonstrated that the whole complex amplitude (not sieved) of an object wave can be accurately retrieved from the diffraction intensities of the object wave sampled by a group of binary CRS masks and the diffractive imaging for the object can be realized through a single digital inverse diffraction. Some experimental results are given for the demonstration. Our experimental results reveal that, using CRS, the influence of a binary random sampling mask on the retrieved field can be well eliminated, and the accuracy and efficiency of the phase retrieval can be greatly improved.

Phase unwrapping with a virtual Hartmann-Shack wavefront sensor

The use of a spatial light modulator for implementing a digital phase-shifting (PS) point diffraction interferometer (PDI) allows tunability in fringe spacing and in achieving PS without the need for mechanically moving parts. However, a small amount of detector or scatter noise could affect the accuracy of wavefront sensing. Here, a novel method of wavefront reconstruction incorporating a virtual Hartmann-Shack (HS) wavefront sensor is proposed that allows easy tuning of several wavefront sensor parameters. The proposed method was tested and compared with a Fourier unwrapping method implemented on a digital PS PDI. The rewrapping of the Fourier reconstructed wavefronts resulted in phase maps that matched well the original wrapped phase and the performance was found to be more stable and accurate than conventional methods. Through simulation studies, the superiority of the proposed virtual HS phase unwrapping method is shown in comparison with the Fourier unwrapping method in the presence of noise. Further, combining the two methods could improve accuracy when the signal-to-noise ratio is sufficiently high.

Quantitative phase retrieval with arbitrary pupil and illumination

We present a general algorithm for combining measurements taken under various illumination and imaging conditions to quantitatively extract the amplitude and phase of an object wave. The algorithm uses the weak object transfer function, which incorporates arbitrary pupil functions and partially coherent illumination. The approach is extended beyond the weak object regime using an iterative algorithm. We demonstrate the method on measurements of Extreme Ultraviolet Lithography (EUV) multilayer mask defects taken in an EUV zone plate microscope with both a standard zone plate lens and a zone plate implementing Zernike phase contrast.

Single-shot phase imaging with a coded aperture

We present a method of quantitatively acquiring a large complex field, containing not only amplitude information but also phase information, based on single-shot phase imaging with a coded aperture (SPICA). In SPICA, the propagating field from an object illuminated by partially coherent visible light is sieved by a coded mask, and the sieved field propagates to an image sensor, where it is captured. The sieved field is recovered from the single captured intensity image via a phase retrieval algorithm with an amplitude support constraint using the mask pattern, and then the object's complex field is reconstructed from the recovered sieved field by an algorithm employing a sparsity constraint based on compressive sensing. The system model and the theoretical bounds of SPICA are derived. We also verified the concept with numerical demonstrations.

Coded aperture pair for quantitative phase imaging

This Letter proposes a novel quantitative phase-imaging approach by optically encoding light fields into a complementary image pair followed by computational reconstruction. We demonstrate that the axial intensity derivative for phase recovery can be well estimated by a coded-aperture image pair without z axial scanning. The experimental results demonstrate that our approach can achieve higher accuracy and robustness compared with conventional transport-of-intensity equation (TIE) based approaches under partial coherence illumination.

A new optical surface measurement method with iterative sparsity-constrained threshold phase retrieval algorithm

Due to its low complexity and acceptable accuracy, phase retrieval technique has been proposed as an alternative to solve the classic optical surface measurement task. However, to capture the overall wave field, phase retrieval based optical surface measurement (PROSM) system has to moderate the CCD position during the multiple-sampling procedure. The mechanical modules of CCD movement may bring about unexpectable deviation to the final results. To overcome this drawback, we propose a new PROSM method based on spatial light modulator (SLM). The mechanical CCD movement can be replaced by an electrical moderation of SLM patterns; thus the deviation can be significantly suppressed in the new PROSM method. In addition, to further improve the performance, we propose a new iterative threshold phase retrieval algorithm with sparsity-constraint to effectively reconstruct the phase of wave field. Experimental results show that the new method provides a more simple and robust solution for the optical surface measurement than the traditional techniques and achieves higher accuracy.

High-speed transport-of-intensity phase microscopy with an electrically tunable lens

We present a high-speed transport-of-intensity equation (TIE) quantitative phase microscopy technique, named TL-TIE, by combining an electrically tunable lens with a conventional transmission microscope. This permits the specimen at different focus position to be imaged in rapid succession, with constant magnification and no physically moving parts. The simplified image stack collection significantly reduces the acquisition time, allows for the diffraction-limited through-focus intensity stack collection at 15 frames per second, making dynamic TIE phase imaging possible. The technique is demonstrated by profiling of microlens array using optimal frequency selection scheme, and time-lapse imaging of live breast cancer cells by inversion the defocused phase optical transfer function to correct the phase blurring in traditional TIE. Experimental results illustrate its outstanding capability of the technique for quantitative phase imaging, through a simple, non-interferometric, high-speed, high-resolution, and unwrapping-free approach with prosperous applications in micro-optics, life sciences and bio-photonics.

Noninterferometric single-shot quantitative phase microscopy

We present a noninterferometric single-shot quantitative phase microscopy technique with the use of the transport of intensity equation (TIE). The optical configuration is based on a Michelson-like architecture attached to a nonmodified inverted transmission bright field microscope. Two laterally separated images from different focal planes can be obtained simultaneously by a single camera exposure, enabling the TIE phase recovery to be performed at frame rates that are only camera limited. Precise measurement of a microlens array validates the principle and demonstrates the accuracy of the method. Investigations of chemical-induced apoptosis and the phagocytosis process of macrophages are then presented, suggesting that the method developed can provide promising applications in the dynamic study of cellular processes.

Investigation of composite materials using SLM-based phase retrieval

We present a robust method to inspect a typical composite material constructed of carbon fiber reinforced plastic (CFRP). It is based on optical surface contouring using the spatial light modulator (SLM)-based phase retrieval technique. The method utilizes multiple intensity observations of the wave field, diffracted by the investigated object, captured at different planes along the optical axis to recover the phase information across the object plane. The SLM-based system allows for the recording of the required consecutive intensity measurements in various propagation states across a common recording plane. This overcomes the mechanical shifting of a camera sensor required within the capturing process. In contrast to existing phase retrieval approaches, the measuring time is considerably reduced, since the switching time of the SLM is less than 50 ms. This enables nondestructive testing under thermal load. Experimental results are presented that demonstrate the approach can be used to assess structural properties of technical components made from CFRP.

Phase retrieval in 4f optical system: background compensation and sparse regularization of object with binary amplitude

Generally, wave field reconstructions obtained by phase-retrieval algorithms are noisy, blurred, and corrupted by various artifacts such as irregular waves, spots, etc. These distortions, arising due to many factors, such as nonidealities of the optical system (misalignment, focusing errors), dust on optical elements, reflections, and vibration, are hard to localize and specify. It is assumed that there is a cumulative disturbance called "background," which describes mentioned distortions in the coherent imaging system manifested at the sensor plane. Here we propose a novel iterative phase-retrieval algorithm compensating for these distortions in the optical system. An estimate of this background is obtained via special calibration experiments, and then it is used for the object reconstruction. The algorithm is based on the maximum likelihood approach targeting on the optimal object reconstruction from noisy data and imaging enhancement using a priori information on the object amplitude. In this work we demonstrate the compensation of the distortions of the optical trace for a complex-valued object with a binary amplitude. The developed algorithm results in state-of-the-art filtering, and sharp reconstruction imaging of the object amplitude can be achieved.

Enhanced deterministic phase retrieval using a partially developed speckle field

A technique for enhanced deterministic phase retrieval using a partially developed speckle field (PDSF) and a spatial light modulator (SLM) is demonstrated experimentally. A smooth test wavefront impinges on a phase diffuser, forming a PDSF that is directed to a 4f setup. Two defocused speckle intensity measurements are recorded at the output plane corresponding to axially-propagated representations of the PDSF in the input plane. The speckle intensity measurements are then used in a conventional transport of intensity equation (TIE) to reconstruct directly the test wavefront. The PDSF in our technique increases the dynamic range of the axial intensity derivative for smooth phase objects, resulting in a more robust solution to the TIE. The SLM setup enables a fast and accurate recording of speckle intensity. Experimental results are in good agreement with those obtained using the iterative phase retrieval and digital holographic methods of wavefront reconstruction.

Phase retrieval via spatial light modulator phase modulation in 4f optical setup: numerical inverse imaging with sparse regularization for phase and amplitude

The 4f optical setup is considered with a wave field modulation by a spatial light modulator located in the focal plane of the first lens. Phase as well as amplitude of the wave field are reconstructed from noisy multiple-intensity observations. The reconstruction is optimal due to a constrained maximum likelihood formulation of the problem. The proposed algorithm is iterative with decoupling of the inverse of the forward propagation of the wave field and the filtering of phase and amplitude. The sparse modeling of phase and amplitude enables the advanced high-accuracy filtering and sharp imaging of the complex-valued wave field. Artifacts typical for the conventional algorithms (wiggles, ringing, waves, etc.) and attributed to optical diffraction can be suppressed by the proposed algorithm.

Binary encoded computer generated holograms for temporal phase shifting

The trend towards real-time optical applications predicates the need for real-time interferometry. For real-time interferometric applications, rapid processing of computer generated holograms is crucial as the intractability of rapid phase changes may compromise the input to the system. This paper introduces the design of a set of binary encoded computer generated holograms (CGHs) for real-time five-frame temporal phase shifting interferometry using a binary amplitude spatial light modulator. It is suitable for portable devices with constraints in computational power. The new set of binary encoded CGHs is used for measuring the phase of the generated electric field for a real-time selective launch in multimode fiber. The processing time for the new set of CGHs was reduced by up to 65% relative to the original encoding scheme. The results obtained from the new interferometric technique are in good agreement with the results obtained by phase shifting by means of a piezo-driven flat mirror.

Automated compensation of misalignment in phase retrieval based on a spatial light modulator

In this paper, the issue of misalignment in phase retrieval by means of optical linear filtering is discussed. The filtering setup is based on a 4f configuration with a spatial light modulator (SLM) as an active element, located in the Fourier domain. From the analysis, crucial parameters for the alignment procedure of the setup's optical axes and the center of the SLM are identified. Furthermore, a method to automatically as well as electronically compensate such effects by modifying the phase pattern displayed on the SLM is introduced. Experimental results are presented that validate the compensation approach.

Quantitative phase retrieval of complex-valued specimens based on noninterferometric imaging

In recent years, the interferometric imaging method has been applied to analyze the structure of various specimens, such as crystals and biological tissues. However, the interferometric imaging method may require a relatively complex optical recording system, such as a reference wave and temporal coherence. In this paper, we propose a method based on noninterferometric imaging for quantitative phase retrieval of complex-valued specimens. A strategy using different focal lengths in the lens function is developed, and a series of diffraction intensity maps is recorded. Numerical simulation results are presented to demonstrate the feasibility and effectiveness of the proposed method.

Quantization analysis of speckle intensity measurements for phase retrieval

Speckle intensity measurements utilized for phase retrieval (PR) are sequentially taken with a digital camera, which introduces quantization error that diminishes the signal quality. Influences of quantization on the speckle intensity distribution and PR are investigated numerically and experimentally in the static wavefront sensing setup. Results show that 3 to 4 bits are adequate to represent the speckle intensities and yield acceptable reconstructions at relatively fast convergence rates. Computer memory requirements may be eased down by 2.4 times if a 4 bit instead of an 8 bit camera is used. This may facilitate rapid speckle data acquisition for dynamic wavefront sensing.

Single-plane multiple speckle pattern phase retrieval using a deformable mirror

A design for a single-plane multiple speckle pattern phase retrieval technique using a deformable mirror (DM) is analyzed within the formalism of complex ABCD-matrices, facilitating its use in conjunction with dynamic wavefronts. The variable focal length DM positioned at a Fourier plane of a lens comprises the adaptive optical (AO) system that replaces the time-consuming axial displacements in the conventional free-space multiple plane setup. Compared with a spatial light modulator, a DM has a smooth continuous surface which avoids pixelation, pixel cross-talk and non-planarity issues. The calculated distances for the proposed AO-system are evaluated experimentally using the conventional free-space phase retrieval setup. Two distance ranges are investigated depending on whether the measurement planes satisfy the Nyquist detector sampling condition or not. It is shown numerically and experimentally that speckle patterns measured at the non-Nyquist range still yield good reconstructions. A DM with a surface height of 25 microns and an aperture diameter of 5.2 mm may be used to reconstruct spherical phase patterns with 50-micron fringe spacing.