Liquid-crystal point-diffraction interferometer for wave-front measurements

Point-diffraction interferometer wavefront sensor with birefringent crystal

A key technique in direct imaging of extrasolar planets with ground-based telescopes is extreme adaptive optics. It requires a wavefront sensor capable of achieving high accuracy with a small number of photons. Imada et al. [Appl. Opt.54, 7870 (2015)APOPAI0003-693510.1364/AO.54.007870] proposed a type of wavefront sensor that employs a point-diffraction interferometer (PDI). This type of sensor has problems concerning a low photon-usage efficiency and manufacturing feasibility. In addition, they did not give sufficient study on the optimum pinhole size. Here, we propose a novel PDI, with which these problems are overcome, and study the optimum pinhole size for it. The sensor is incorporated with birefringent crystal as the key component to achieve high efficiency and is feasible to manufacture realistically. We run numerical simulations to optimize the pinhole size, where the photon noise is evaluated.

Complex amplitude reconstruction for dynamic beam quality M2 factor measurement with self-referencing interferometer wavefront sensor

A real-time complex amplitude reconstruction method for determining the dynamic beam quality M² factor based on a Mach-Zehnder self-referencing interferometer wavefront sensor is developed. By using the proposed complex amplitude reconstruction method, full characterization of the laser beam, including amplitude (intensity profile) and phase information, can be reconstructed from a single interference pattern with the Fourier fringe pattern analysis method in a one-shot measurement. With the reconstructed complex amplitude, the beam fields at any position z along its propagation direction can be obtained by first utilizing the diffraction integral theory. Then the beam quality M² factor of the dynamic beam is calculated according to the specified method of the Standard ISO11146. The feasibility of the proposed method is demonstrated with the theoretical analysis and experiment, including the static and dynamic beam process. The experimental method is simple, fast, and operates without movable parts and is allowed in order to investigate the laser beam in inaccessible conditions using existing methods.

Measurement of M²-Curve for Asymmetric Beams by Self-Referencing Interferometer Wavefront Sensor

For asymmetric laser beams, the values of beam quality factor Mx2 and My2 are inconsistent if one selects a different coordinate system or measures beam quality with different experimental conditionals, even when analyzing the same beam. To overcome this non-uniqueness, a new beam quality characterization method named as M²-curve is developed. The M²-curve not only contains the beam quality factor Mx2 and My2 in the x-direction and y-direction, respectively; but also introduces a curve of Mxα2 versus rotation angle α of coordinate axis. Moreover, we also present a real-time measurement method to demonstrate beam propagation factor M²-curve with a modified self-referencing Mach-Zehnder interferometer based-wavefront sensor (henceforth SRI-WFS). The feasibility of the proposed method is demonstrated with the theoretical analysis and experiment in multimode beams. The experimental results showed that the proposed measurement method is simple, fast, and a single-shot measurement procedure without movable parts.

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.

Self-calibrating common-path interferometry

A quantitative phase measuring technique is presented that estimates the object phase from a series of phase shifted interferograms that are obtained in a common-path configuration with unknown phase shifts. The derived random phase shifting algorithm for common-path interferometers is based on the Generalized Phase Contrast theory [pl. Opt.40(2), 268 (2001)10.1063/1.1404846], which accounts for the particular image formation and includes effects that are not present in two-beam interferometry. It is shown experimentally that this technique can be used within common-path configurations employing nonlinear liquid crystal materials as self-induced phase filters for quantitative phase imaging without the need of phase shift calibrations. The advantages of such liquid crystal elements compared to spatial light modulator based solutions are given by the cost-effectiveness, self-alignment, and the generation of diminutive dimensions of the phase filter size, giving unique performance advantages.

Digital phase-shifting point diffraction interferometer

A digital phase-shifting (PS) point diffraction interferometer is demonstrated with a transmitting liquid crystal spatial light modulator. This novel wavefront sensor allows tunability in the choice of pinhole size and eliminates the need for mechanically moving parts to achieve PS. It is shown that this wavefront sensor is capable of sensing Zernike aberrations introduced with a deformable mirror. The results obtained are compared with those of a commercial Hartmann-Shack wavefront sensor.

Synthetic phase-shifting for optical testing: point-diffraction interferometry without null optics or phase shifters

An innovative iterative search method called the synthetic phase-shifting (SPS) algorithm is proposed. This search algorithm is used for maximum-likelihood (ML) estimation of a wavefront that is described by a finite set of Zernike Fringe polynomials. In this paper, we estimate the coefficient, or parameter, values of the wavefront using a single interferogram obtained from a point-diffraction interferometer (PDI). In order to find the estimates, we first calculate the squared-difference between the measured and simulated interferograms. Under certain assumptions, this squared-difference image can be treated as an interferogram showing the phase difference between the true wavefront deviation and simulated wavefront deviation. The wavefront deviation is the difference between the reference and the test wavefronts. We calculate the phase difference using a traditional phase-shifting technique without physical phase-shifters. We present a detailed forward model for the PDI interferogram, including the effect of the finite size of a detector pixel. The algorithm was validated with computational studies and its performance and constraints are discussed. A prototype PDI was built and the algorithm was also experimentally validated. A large wavefront deviation was successfully estimated without using null optics or physical phase-shifters. The experimental result shows that the proposed algorithm has great potential to provide an accurate tool for non-null testing.

Phase-shifting point-diffraction interferometer developed by using the electro-optic effect in ferroelectric crystals

A novel and simple phase-shifting point-diffraction interferometer using a z-cut lithium niobate wafer is proposed. The pinhole is realized by an optical lithography process, aluminum deposition, and subsequent lift-off on the surface of the wafer. The phase shifting is obtained by inducing the electro-optic effect along the z crystal axis. We demonstrate experimentally the possibility of retrieving an aberrated wavefront.

Polarization phase-shifting point-diffraction interferometer

A new instrument, the polarization phase-shifting point-diffraction interferometer, has been developed by use of a birefringent pinhole plate. The interferometer uses polarization to separate the test and reference beams, interfering what begin as orthogonal polarization states. The instrument is compact, simple to align, and vibration insensitive and can phase shift without moving parts or separate reference optics. The theory of the interferometer is presented, along with properties and fabrication techniques for the birefringent pinhole plate and a new model used to determine the quality of the reference wavefront from the pinhole as a function of pinhole size and test optic aberrations. The performance of the interferometer is also presented, along with a detailed error analysis and experimental results.

Wavefront reconstruction by spatial-phase-shift imaging interferometry

Common-path imaging interferometers offer some advantages over other interferometers, such as insensitivity to vibrations and the ability to be attached to any optical system to analyze an imaged wavefront. We introduce the spatial-phase-shift imaging interferometry technique for surface measurements and wavefront analysis in which different parts of the wavefront undergo certain manipulations in a certain plane along the optical axis. These manipulations replace the reference-beam phase shifting of existing interferometry methods. We present the mathematical algorithm for reconstructing the wavefront from the interference patterns and detail the optical considerations for implementing the optical system. We implemented the spatial phase shift into a working system and used it to measure a variety of objects. Measurement results and comparison with other measurement methods indicate that this approach improves measurement accuracy with respect to existing quantitative phase-measurement methods.

Experimental comparison of a liquid-crystal point-diffraction interferometer (LCPDI) and a commercial phase-shifting interferometer and methods to improve LCPDI accuracy

We compare the phase measurements of a fused-silica witness sample made with a liquid-crystal point-diffraction interferometer (LCPDI) with measurements made with a Zygo Mark IV xp phase-shifting interferometer and find close agreement. Two phase-shift-error sources in the LCPDI that contribute to measurement discrepancies are frame-to-frame intensity changes caused by the dichroism of the dye and alignment distortions of the host liquid crystal. An empirical model of the phase-shift error caused by the host alignment distortions is presented and used to investigate the performance of two different phase-detection algorithms. It is suggested that by proper choice of LCPDI fabrication parameters and phase-acquisition methods, the device's accuracy can be significantly improved.

Optimal phase contrast in common-path interferometry

We have developed an analytical model for the design and optimization of common-path interferometers (CPI's) based on spatial filtering. We describe the mathematical analysis in detail and show how its application to the optimization of a range of different CPI's results in the development of a graphical framework to characterize quantitatively CPI performance. A detailed analytical treatment of the effect of curvature in the synthetic reference wave is undertaken. We show that it is possible to improve the linearity and fringe accuracy of certain standard interferometers by a modification of the Fourier filter, and we propose and analyze a dual CPI system for the unambiguous mapping of phase to intensity over the complete input phase range.

Interferometric optical fourier-transform processor for calculation of selected spatial frequencies

A novel interferometric optical Fourier-transform processor is presented that calculates the complex-valued Fourier transform of an image at preselected points on the spatial-frequency plane. The Fourier spectrum of an arbitrary input image is interfered with that of a reference image in a common-path interferometer. Both the real and the imaginary parts of the complex-valued spectrum are determined. The source and the reference images are easily matched to guarantee good fringe visibility. At least six interferograms are postprocessed to extract the real and the imaginary parts of the Fourier spectrum at preselected points. The proposed hybrid optical-digital technique is computationally appropriate when the number of desired spatial frequencies is small compared with the number of pixels in the image. When the number of desired points is comparable with the number of image pixels, a conventional or pruned two-dimensional fast Fourier transform is more appropriate. The number of digital operations required by the hybrid optical-digital Fourier processor is proportional to the number of desired spatial frequencies rather than the number of pixels in the image. The points may be regularly distributed over the spatial-frequency plane or concentrated in one or several irregularly shaped regions of interest. The interferometric optical Fourier processor is demonstrated in a moving-object trajectory estimation system. The system successfully estimates the trajectory of multiple objects moving over both stationary and white-noise backgrounds. A comparison of performance was made with all-digital computation. With everything else equal, our hybrid optical-digital calculation was more than 3 orders of magnitude faster.

Phase-shift error as a result of molecular alignment distortions in a liquid-crystal point-diffraction interferometer

We report the observation of nematic director distortions around the diffracting element of a liquid-crystal point-diffraction interferometer. The observed director field distortions are similar to those reported in the literature for other liquid-crystal guest-host systems. We show how the alignment distortion changes as a function of the voltage applied to the liquid-crystal cell, leading to an observed phase-shift error. Tailoring of surface anchoring conditions and judicious choice of phase-shift algorithm can improve device accuracy.