Radial Shearing Interferometer Research Articles

Recording of incoherent vector holograms using elements of the spatial cross-spectral density matrix

We present an approach to record incoherent vector holograms using a highly stable field-based interferometer employing radial shearing. The vector holograms are recorded as the elements of the 2 × 2 cross-spectral density (CSD) matrix and reconstruction of the object encoded into these holograms is demonstrated by digitally propagating the elements of the CSD matrix. The CSD matrix contains its elements in the form of a complex spatial coherence function and is connected with an incoherent vector source by the vectorial van Cittert-Zernike theorem. The experimental measurement of the two-dimensional distributions of the complex elements of the CSD matrix is realized by a Sagnac radial shearing interferometer with a phase-shifting approach. The five-step phase-shifting technique is used to measure the fringe visibility and corresponding phase which give the complex elements of the CSD matrix. Recording of incoherent vector holograms is demonstrated from the experimentally recovered elements of the CSD matrix and results are presented for two different incoherent sources, namely polarized and unpolarized. The complex nature of the CSD matrix elements helps in the digital reconstruction and focusing of the incoherent source.

Surface figure measurement tool based on a radial shearing interferometer using a geometric phase lens with various spherical wavefronts.

We describe a compact radial shearing interferometer based on a geometric phase lens as a surface figure measurement tool. Based on the polarization and diffraction characteristics of a geometric phase lens, two radially sheared wavefronts are simply generated, and the surface figure of a specimen can be immediately reconstructed from the radial wavefront slope calculated with four phase-shifted interferograms obtained from a polarization pixelated complementary metal-oxide semiconductor camera. In order to increase the field of view, moreover, the incident wavefront is adjusted according to the shape of the target, which allows the reflected wavefront to become planar. By the combination of the incident wavefront formula and the measurement result by the proposed system, the whole surface figure of the target can be instantaneously reconstructed. As the experimental result, the surface figures of various optical components were reconstructed at the extended measurement area with less than 0.78µm deviations, and it was confirmed that the radial shearing ratio was fixed independent of the surface figures.

Spherical wavefront measurement on modified cyclic radial shearing interferometry.

We propose a radial shearing interferometric approach to measure spherical wavefronts as both of the reflective and transmissive optical configurations. The modified cyclic radial shearing interferometer uses a single lens in the optical layout, which can conveniently adjust the radial shearing ratio between two shearing spherical wavefronts, and the use of a polarization camera enables to reconstruct the wavefront by a single image. The wavefront mapped onto the camera plane can be identified and quantified throughout an optimized wavefront reconstruction algorithm. In the experiments, plano-convex lenses and concave mirrors were used to generate spherical wavefronts, and the proposed system was able to reconstruct the surface figures after system characterization and calibration. Further investigations were performed to evaluate the system measurement accuracy by the radius of curvature comparison with design value and a commercial Shack-Hartmann wavefront sensor.

Estimation of global visibility for low-quality fringe pattern using Fourier-polar transform

When the interferometers with adjustable fringe visibility is used for high precision wavefront measurement, it needs to estimate the visibility of experimental patterns so as to obtain the interference patterns with the highest visibility. A new method combining Fourier-polar transform with directional projection is developed to estimate the global visibility of carrier fringe pattern. This method can also be applied to low-quality fringe patterns such as low contrast and low signal-to-noise ratio (SNR) that often appear in the interferometer testing. Theoretical analysis shows that this method is obviously superior to the direct calculation method. In addition, an illustrative experiment based on the radial shearing interferometer is given. Results generated from this method are compared with the derived values from theoretical model of experimental system, and exemplary agreement between both is demonstrated.

Measurement of three-dimensional displacements by radial shearing interferometer

<sec> Moiré patterns formed by overlapping two circular gratings of slightly different pitches have been extensively used for measuring the two-dimensional (2D) and three-dimensional (3D) displacements. However, in the existing applications, Moiré patterns are analyzed based on geometric superposition, by which the 3D displacements cannot be instantaneously or simultaneously measured with a high accuracy. In this paper, radial shearing interferometry with double circular gratings of slightly different pitches is presented to realize the simultaneous measurement of 3D displacements. </sec><sec> The measurement is based on the principle that Moiré patterns produced by radial shearing interferometry are determined not only by the 2D in-plane displacements, but also by the out-of-plane displacement that brings about a phase shift between Moiré patterns of +1 and –1 diffraction orders. First, the production mechanism of Moiré patterns by radial shearing interferometry is studied based on the scalar diffraction theory and the intensity distribution of Moiré fringes of +1 and –1 orders is derived to establish the exact analytic relations between Moiré patterns and 3D displacements. Second, on the basis of spectrum characteristics of circular grating, a semicircular ring filter is proposed for spatial filtering to realize the simultaneous imaging of Moiré fringes of +1 and –1 orders. Then, the algorithm to quantitatively extract 3D displacements from Moiré patterns is proposed and demonstrated by numerical simulation. In the algorithm, Moiré patterns in the rectangular coordinate system are transformed into the polar coordinate system and skeletons are extracted to determine the feature points of the bright fringes. The in-plane displacements can be solved by feature points of +1 or –1 diffraction order, and the out-of-plane displacement can be computed by the feature points of +1 and –1 diffraction orders in the same bright fringe. Finally, experimental results prove that the maximum absolute error and mean error for in-plane displacements are 4.8 × 10<sup>–3</sup> mm and 2.0 × 10<sup>–4</sup> mm respectively, and 0.25 mm and 8.6 × 10<sup>–3</sup> mm for out-of-plane displacement. </sec><sec> In conclusion, by using the Moiré patterns of +1 and –1 diffraction orders imaged by radial shearing interferometer with double circular gratings of slightly different pitches, the 3D displacement can be simultaneously measured. The method has the advantages of simple device, high measurement accuracy, non-contact and instantaneous measurement, which provides an important guidance for practically measuring the 3D displacements. </sec>

Diverging cyclic radial shearing interferometry for single-shot wavefront sensing.

In this investigation, we describe a simple cyclic radial shearing interferometer for single-shot wavefront sensing. Instead of using the telescope lens system used in typical radial shearing interferometry, a single lens is used to generate two diverging radial shearing beams. This simple modification leads to the advantages of conveniently adjusting the radial shearing ratio, compactness of the system, and practical ease of alignment. With the aid of a polarization pixelated CMOS camera, the spatial phase-shifting technique is used to extract the phase with a single image. The most important feature is the fringe contrast enhancement by reducing the aberrations caused by the complicated optical system even though an incoherent light is used. The experimental results show the fringe contrast enhancement is at least 0.1 better than that of the conventional method, and the wavefronts are properly reconstructed with less than 0.071λ root-mean-squared wavefront error regardless of the coherence of the light.

Radial-shearing interferometric imaging with Theon-Kepler bifocal telescope.

A method of radial-shearing interferometric imaging based on the Theon-Kepler bifocal telescope is proposed. The Theon-Kepler bifocal telescope system consists of two identical Theon photon sieves with bifocal spots. The short focal length of the first photon sieve coincides with the long focal length of the second photon sieve. At the same time, the first focal length coincides with the second short focal length. This setup naturally constitutes two sets of 4f systems. When a plane wave is incident on the bifocal telescope, two plane waves of different sizes are emitted. These two beams undergo radial-shearing interference. The common-path setup of this new type of radial-shearing interferometer is simple, with low requirements for the experimental environment and strong anti-interference ability.

Measurement of flow density field by cyclic radial shearing interferometer

Transient measurements of high-speed airflow field are needed in the measurement of boundary layer. Digital interferometry can measure flow field quantitatively to obtain density information, which is very necessary in flow field measurement. In this paper, a common-path shearing interferometry method is introduced. It is insensitive to vibration and does not need a reference plane. It is suitable for flow field measurement. A fast algorithm based on spatial phase modulation, coupled with a pulse laser and a synchronous control system, is used to measure the disturbance density field quantitatively in real time. The acquisition resolution of the system is 200 pixels × 200 pixels, and the acquisition frequency can reach more than 1000 frames per second. The wavefront reconstruction method of the system has been simulated by computer, and the detection result is better than 1/20λ. The experimental results in a 0.6 m wind tunnel show that the system can restrain the vibration interference and distinguish the disturbance signal and the vibration noise remarkably. It has good application prospects.

Radial polarizing phase-shifting interferometry with applications to single-shot n interferogram measurements and potential usage for white light interferogram analysis.

In this research, we present an interferometric system to analyze transparent samples using interferograms generated by a phase-shifting radial shear grating interferometer for two cases: the first obtaining n simultaneous phase-shifting interferograms using a coherent light source and the second one using sequential phase steps with a white light source. For the first case, the simultaneous interferograms are generated using two optical systems: the first one generates the polarized pattern while the second one consists of a 4f system creating replicas of the output interferograms. By using a 2D sinusoidal phase grating, we have the advantage of obtaining up to nine replicated interferograms, all of them with comparable intensities and having amplitudes modulated by the 2D sinusoidal phase grating diffraction orders as zero-order Bessel's functions. To obtain the optical phase map, several phase shifts are generated by placing a polarizing filter covering each replicated interferogram. We highlight the advantage of using n simultaneous interferograms by comparing resulting optical phases processed by a conventional four-step algorithm against those obtained by an implemented n=N+1 method, reducing errors with noisy interferograms. Results for n=7 and n=9 cases are presented. In addition, we have tested the setup with white light interference techniques by employing the polarizer radial shearing interferometer; for this case, the optical phase is calculated with the four-step and the three-step algorithms. Results of testing the developed system to examine static and dynamic phase objects are also included.

Single-shot phase calibration of a spatial light modulator using geometric phase interferometry.

A vibration-insensitive, single-shot phase-calibration method for phase-only spatial light modulators (SLM) is reported. The proposed technique uses a geometric phase lens to form a phase-shifting radial shearing interferometer to enable common-path measurements. This configuration has several advantages: (a) unlike diffraction-based SLM calibration techniques, this technique is robust against intensity errors due to misalignment; (b) unlike two-beam interferometers, this technique offers a high environmental stability; and (c) unlike intensity-based methods, the phase-shifting capability provides a phase uncertainty routinely in the order of ${2}\pi /100$2π/100. The experimental results show a significantly higher accuracy when compared to the diffraction-based approaches.

Compact radial shearing interferometer with a randomly encoded cosinusoidal zone plate for wavefront measurements.

A radial shearing interferometer (RSI) using a randomly encoded cosinusoidal zone plate (RECZP) to measure the wavefront is proposed. The RECZP has two foci, i.e., a virtual focus and a real focus, so its Fresnel diffractions contain only two beams. These two beams can be regarded as the extended beam and the contracted beam in the RSI, respectively. This RSI is composed of a RECZP and a charge-coupled device (CCD). The radial shearing rate is continuously adjustable by changing the distance between the CCD and RECZP, which is good for measurement sensitivity and dynamic range for different situation requirements. In the simulation experiment, we analyzed the influence of beam tilt error, distance error of zone plate and CCD, CCD camera nonlinearity, and noise on wavefront reconstruction results. We also analyzed the effects of different fabrication errors (randomly encoded principle error, sidewall angle error, depth error, and alignment error of amplitude zone plate and phase zone plate) on the diffraction intensity distributions, which determine the fabrication tolerance of the RECZP. Experimentally compared with a ZYGO interferometer, the RECZP-RSI optical system can get good results.

Parallel phase shifting radial shear interferometry with complex fringes and unknown phase shift.

We present an interferometric method to analyze transparent samples using complex fringes generated by a parallel phase shifting radial shear interferometer using two coupled interferometers. Parallel interferograms are generated using two interferometers: the first one generates the polarized base pattern, and the second system is used to generate parallel interferograms allowing the adjustment of the x-y positions of the parallel interferograms. To obtain the optical phase map, parallel phase shift is generated by collocating polarizing filters at the output of the system; the polarizers are placed at arbitrary angles since they do not require adjustment because of the phase-recovery algorithm. The optical phase was processed using a two-step algorithm based on a modified Gram-Schmidt orthogonalization method. Such an algorithm has the advantage of not being iterative and is robust to amplitude modulation. The proposed method reduces the number of captures needed in phase-shifting interferometry. We applied the developed system to examine static and dynamics phase objects.

Radial Shearing Interferometer Based on a Cosinusoidal Zone Plate

A simple and compact radial shearing interferometer (RSI) based on a cosinusoidal zone plate (CZP) is proposed. The CZP can modulate both phase and amplitude, it has two foci, so as the nominal plane beam under test passes through the CZP, it is diffracted into two parts, i.e., a converging beam and a diverging beam. Then, the two beams can be considered as the compressed beam and the expanded beam in the RSI, respectively. A charge-coupled device detector located in the vicinity of the CZP records the interference pattern. This RSI needs only a single zone plate and therefore can be made very compact. The experiment validates the correctness of the proposed method. With the compactness and stability, the presented interferometer is expected to be used in the detection of moving objects and dynamic processes.

Common-path off-axis incoherent Fourier holography with a maximum overlapping interference area.

In this Letter, we present a new method for recording spatially incoherent common-path off-axis Fourier holograms. This method records the three-dimensional (3D) information of an object into a Fourier hologram without the need of any mechanical scanning with incoherent illumination. The proposed setup consists of two gratings to form a common-path configuration, and two customized cells to create a rotational and radial shearing interferometer. While the first grating is placed on the first image plane, the second grating shifts axially from the second image plane to build off-axis geometry. A lens is used to combine two beams to generate the maximum overlapping area at the hologram plane. Proof-of-concept experiments confirm the ability of such a system to achieve the maximum overlapping interference area, stability of the system against the vibration of surrounding environment, numerical reconstruction using only one fast Fourier transform, and 3D capability to capture a 3D object illuminated by an LED light.

Three-wave radial shearing interferometer.

We propose a simple and compact three-wave radial shearing interferometer (TWRSI). This interferometer is constructed by a Gabor zone plate (GZP) and a charge-coupled device (CCD). The nominal plane wavefront under test is diffracted by the GZP into three orders (i.e., a parallel beam, a converging spherical beam, and a diverging spherical beam) leading to an interference pattern. In the imaging plane, three beams can be considered as being the copies of the incident beam, the compressed beam, and the expanded beam, respectively. A CCD detector located in the vicinity of the GZP records this interference pattern. Wavefront can be reconstructed by a single interferogram. The experiment result shows that the proposed method is correct and accurate. This TWRSI is compact, vibration insensitive, suitable for measurement of continuous and transient wavefront.

Multiple-wave radial shearing interferometer based on a Fresnel zone plate.

A compact multiple-wave radial shearing interferometer (MWRSI) with strong adaptability, which is based on a Fresnel zone plate (FZP), is proposed. The nominally plane beam under test passes through the FZP and diffracts into multiple orders, including nominally plane beam, converging spherical beams, and diverging spherical beams. After propagating a distance, different apertures of diffraction orders cause multiple beams to meet and interfere in the imaging plane and form radial shear. The single interference pattern contains multiple wavefront differences in the overlapping region of beams. Then, a method that is employed in the wavefront difference retrieval from the MWRSI's interferogram has been given. This MWRSI needs only an FZP and a CCD; therefore, it can be made very compact. It is suitable to measure the continuous and transient wavefront. The numerical simulations and experiments validate the proposed method.

Direct emission of chirality controllable femtosecond LG01 vortex beam

Direct emission of a chirality controllable ultrafast LG01 mode vortex optical beam from a conventional z-type cavity design SESAM (SEmiconductor Saturable Absorber Mirror) mode locked LD pumped Yb:Phosphate laser has been demonstrated. A clean 360 fs vortex beam of ∼45.7 mW output power has been achieved. A radial shear interferometer has been built to determine the phase singularity and the wavefront helicity of the ultrafast output laser. Theoretically, it is found that the LG01 vortex beam is obtained via the combination effect of diagonal HG10 mode generation by off-axis pumping and the controllable Gouy phase difference between HG10 and HG01 modes in the sagittal and tangential planes. The chirality of the LG01 mode can be manipulated by the pump position to the original point of the laser cavity optical axis.

Iterative algorithm for beam complex amplitude reconstruction by radial shearing interferometry

An iterative algorithm for precisely reconstructing the complex amplitude of a laser beam based on a radial shearing interferometer is proposed. First, the complex amplitude of the modulation function that includes the test beam’s information is obtained from a cyclic radial shearing interferogram by the Fourier filtering method. Then the complex amplitude of the modulation function is used to estimate the complex amplitude of the laser output with a successive iterative process. The simulations and real experimental results show the performance of the proposed iteration method of beam complex amplitude reconstruction.

Compact single-shot radial shearing interferometer with random phase shift.

We propose a compact phase-shifting radial shearing interferometer based on a pixelated micro-retarder array (MRA). The MRA is made of a thin birefringence plate, and is composed of identical units that have pixels of four different thicknesses. We demonstrate that pixelated phase delay between two shearing beams can be introduced by applying the fast axis of birefringence plate in the horizontal or vertical orientation. We also present an approach to extract the wavefront under test with random phase shift, which dramatically reduces the machining difficulty of the MRA. The feasibility and accuracy of the proposed method are further validated through our numerical analysis. With the advantages of vibration immunity, simultaneous phase stepping, and broad spectral range, the presented interferometer is expected to be of potential use in a variety of applications, such as the detection of moving objects and dynamic processes.

A novel wavefront reconstruction algorithm based on interpolation coefficient matrix for radial shearing interferometry

A novel wavefront reconstruction algorithm for radial shearing interferometer (RSI) is proposed in this paper. Based on the shearing relationship of RSI, an interpolation coefficient matrix is established by the radial shearing ratio and the number of discrete points of test wavefront. Accordingly, the expanded wavefront is characterized by the interpolation coefficient matrix and the test wavefront. Consequently the test wavefront can be calculated from the phase difference wavefront. The numerical simulation is conducted to confirm the correctness of the proposed algorithm. Compared with the previous wavefront reconstruction methods, the proposed algorithm is more accurate and stable.