Lateral Shearing Interferometry Research Articles

Wavefront reconstruction based on multi-directional orthogonal lateral shearing interferometry

Wavefront reconstruction based on multi-directional orthogonal lateral shearing interferometry

Linear and nonlinear optical properties of femtosecond laser inscribed waveguides into GLS glass

Gallium lanthanum sulfide (GLS) glass is a promising material for mid-infrared photonics due to its wide transmission window and its high nonlinear refractive index that is almost three orders of magnitude higher than that of fused silica. In this paper, we present the results of a detailed study into the linear and nonlinear optical properties of waveguides fabricated in GLS glass via ultrafast laser direct-inscription using three different techniques: cumulative heating in the thermal regime as well as multi-scan and half-scan in the athermal regime. Using quadriwave lateral shearing interferometry, we fully characterized the refractive index profiles of such inscribed waveguides and found no difference between half-scan and multi-scan writing which indicates the absence of laser-induced stress in this soft glass in stark contrast to fused silica. In terms of nonlinearity, we utilized self-phase modulation (SPM)-induced spectral broadening experiments at mid-IR wavelengths to demonstrate that waveguides fabricated in the athermal regime preserve the high intrinsic nonlinearity of the GLS bulk material, outperforming those written in the thermal regime based. These findings pave the way for the fabrication of fiber-coupled optical waveguide chips for nonlinear mid-infrared photonics.

High-definition quadriwave lateral shearing interferometry

We present an evolution of quadriwave lateral shearing interferometry that improves the definition of quantitative phase images. It is now possible to produce images with as many pixels as the camera that records the interferogram. This is done by moving a diffraction grating in front of the camera and linearly combining at least nine acquisitions. In this paper, we present the principle of this technique and illustrate it by several examples acquired on a microscope with both calibrated and biological samples. We demonstrate the possibility of producing quantitative phase images with 5.5 million pixels, which are to our knowledge the largest images ever produced by a wavefront sensor.

Phase recovery from Fresnel incoherent correlation holography using differential Zernike fitting.

Fresnel incoherent correlation holography (FINCH) was created to improve imaging resolution and 3D imaging capabilities using spatially incoherent illumination. The optical setup of a FINCH-based interferometer is closely related to a radial shearing interferometer, which measures the radial phase difference of an input wavefront. By using phase retrieval methodologies from lateral shearing interferometry, namely, differential Zernike fitting (DZF), we show that FINCH-based and radial shearing interferometry can be used for phase retrieval and adaptive optics (AO). In this paper, we describe the phase retrieval algorithm using least squares-based DZF and demonstrate a simple adaptive optics loop with an aberrated point spread function using wave optics simulation. We find that FINCH-based phase retrieval has the advantages of fast phase retrieval measurements, thanks to well-studied least squares-based phase reconstruction methods, improved resolution compared to the Shack-Hartmann-based wavefront sensing, and the simplified optical setup of radial shearing interferometry.

Fast and direct calibration for high numerical aperture quadriwave lateral shearing interferometry using geometry model with higher-order Taylor expansion

The current research on quadriwave lateral shearing interferometry (LSI) is primarily focused on the measurements of plane wave or quasiplane wave. When directly measuring spherical waves with high numerical apertures (NA>0.35), it will generate additional systematic errors, which affect measurement accuracy. To make the quadriwave LSI applicable to the measurements of a high-NA spherical wave, this paper proposes a fast and direct calibration method by establishing the geometry model of the quadriwave LSI with high-NA spherical wave incidence. The expression for the optical path difference represented systematic errors introduced by high-NA spherical waves in the shearing wavefronts are derived, which utilize a ninth-order Taylor expansion and Zernike polynomials fitting to achieve the high accuracy required by the high-NA spherical wave incidence. Then, the systematic errors are directly calibrated in the shearing wavefronts of four directions. This paper presents the theoretical analysis and verifies the feasibility and reliability of the proposed method through simulations and experiments, achieving a good measurement accuracy.

Error correction analysis of wavefront testing in quadriwave lateral shearing interferometry

Quadriwave lateral shearing interferometry (QWLSI) is based on double birefringent crystals of a beam displacer (DBCs-BD), which can generate the lateral shearing interference wavefront of four beams of overlapped replicas in the DBCs-BD orthogonal directions. When the replica waves are overlapped incident to the analyzer and the direction of the transmission axis is set as 45° or 135°, the QWLSI’s polarization interferogram can be obtained. This paper deduces the principle of QWLSI based on the DBCs-BD and presents the analysis of orthogonal error influence based on the DBCs-BD and the phase retrieval error of QWLSI when shear displacement by tilted incident on the DBCs-BD. In our investigation, we have established the correction range of the PBD’s orthogonal angle error is within −0.5∘−0.5∘, the maximum error in PV is 0.0012λ, and the maximum error in RMS is 1.3789×10−4λ in wavefront reconstruction. Moreover, when the testing light tilts to incident on the PBD in the range of −0.4∘−0.4∘, correction of the shear distance is used for wavefront reconstruction to achieve a high-precision wavefront testing result. Finally, the experiment shows that QWLSI based on the DBCs-BD exhibits feasibility and high precision.

Analysis of Stokes singularities using a lateral shear interferometer

Polarization and Poincaré singularities in the optical fields can be studied by analyzing the phase singularities of mathematically constructed Stokes vector fields. The wider applicability of the Stokes construction is found by exploring the generation and detection methods for various types of Stokes singularities and their analysis. Here, we detect and analyze all forms of the Stokes singularities through lateral shear interferometry. Specifically, the projections of a Stokes singularity on three pairs of orthogonal polarization basis states, defined by the eigen polarization states of Pauli’s matrices, are analyzed through unique fork patterns in the shearogram pairs. These interference patterns also provide the topological indices of the singularities. Such a self-referencing interferometric method also helps to remove the degeneracy in the Stokes index and polarization. Through both, simulations and experiments, we have analyzed specific beams represented by higher order Poincaré sphere and hybrid order Poincaré sphere topological constructs.

Polarization-multiplexed snapshot lateral shearing interferometric sensor for surface roughness measurements

A compact optical roughness measurement sensor based on lateral shearing interferometry is theoretically described and experimentally verified. By adopting the advanced surface roughness parameter extraction methods based on statistical machine learning technique and the partial integration approach, the sensor using a polarization grating and a polarization camera can instantaneously obtain a surface gradient map and provide reliable surface roughness parameters. To verify the operation and the performance of the sensor, a flat mirror and roughness standard specimens were measured, and the results were compared with those of a white light scanning interferometer as the reference values. As a result, the roughness parameters were very close to the reference values with <5 nm accuracy and <0.1 nm repeatability.

Lateral shearing interferometry method based on double-checkerboard grating by suppressing aliasing effect

Ronchi lateral shearing interferometry is a promising wavefront sensing technology with the advantages of simple structure and no reference light, which can realize a high-precision wavefront aberration measurement. To obtain shear information in both directions, the conventional double-Ronchi interferometer sequentially applies two orthogonal one-dimensional Ronchi gratings as the object-plane splitting element of the optics under test. Simultaneously, another Ronchi grating is positioned on the image plane in the same orientation to capture two sets of interferograms, thereby enabling two-dimensional wavefront reconstruction. Mechanical errors will inevitably be introduced during grating conversion, affecting reconstruction accuracy. Based on this, we propose a lateral shearing interferometry applying double-checkerboard grating. Only unidirectional phase shift is needed to obtain shear information in two directions while evading the grating conversion step, aiming to streamline operational processes and mitigate the potential for avoidable errors. We employ scalar diffraction theory to analyze the full optical path propagation process of the double-checkerboard shearing interferometry and introduce a new reconstruction algorithm to effectively extract the two-dimensional shear phase by changing the grating morphology, suppressing the aliasing effect of irrelevant diffraction orders. We reduce the fitting error through iterative optimization to realize high-precision wavefront reconstruction. Compared with conventional Ronchi lateral shearing interferometry, the proposed method exhibits better robustness and stability in noisy environments.

In situ calibration of shear ratio for quadriwave lateral shearing interferometry using double-slit interference.

A new method, to the best of our knowledge, based on double-slit (DS) interference is proposed to accurately estimate the shear ratio of the system, with plane wave or spherical wave incidence. Existing shear ratio calibration methods, designed primarily for lateral shearing interferometry (LSI) with plane wave incidence, are not applicable to LSIs directly testing divergent or convergent spherical waves. Equations for calculating the shear ratio using the fringe spacing of the DS interferogram and the NA of the incident spherical wave are derived in this paper. The simulation result shows that the relative error of the shear ratio value is about 0.3%, when the shear ratio is 0.1. In the experiment, the quadriwave LSI is designed with a plug-in feature. The shear ratio at integer multiples of 1/6 Talbot distance from the modified Hartmann mask was calibrated using a DS, and the results were in good agreement with theoretical values, confirming the accuracy of the method. Subsequently, with the assistance of an inductance micrometer, the shear ratio was calibrated at intervals of 0.5mm, and the results closely matched the theoretical variation of the shear ratio caused by displacement, confirming the high precision of the method.

High-Precision Correction for Spatial Light Modulator Based on Quadriwave Lateral Shearing Interferometry

High-Precision Correction for Spatial Light Modulator Based on Quadriwave Lateral Shearing Interferometry

Distortion measurement of a lithography projection lens based on multichannel grating lateral shearing interferometry.

The optical distortion of the lithographic projection lens can reduce imaging quality and cause overlay errors in lithography, thus preventing the miniaturization of printed patterns. In this paper, we propose a technique to measure the optical distortion of a lithographic projection lens by sensing the wavefront aberrations of the lens. A multichannel dual-grating lateral shearing interferometer is used to measure the wavefront aberrations at several field points in the pupil plane simultaneously. Then, the distortion at these field points is derived according to the proportional relationship between the Z 2 and Z 3 Zernike terms (the tilt terms) and the image position shifts. Without the need for additional devices, our approach can simultaneously retrieve both the wavefront aberrations and the image distortion information. Consequently, it improves not only measurement speed and accuracy but also enables accounting for displacement stage positioning error. Experiments were conducted on a lithographic projection lens with a numerical aperture of 0.57 to verify the feasibility of the proposed method.

Two-frame wavefront reconstruction method with nonlinear optimization for Ronchi lateral shearing interferometry

Ronchi lateral shearing interferometry is widely applied in wavefront measurement of advanced lithography tools. In Ronchi lateral shearing interferometry, higher-order diffraction beam interference is a common challenge, currently mitigated by introducing more phase shifts, resulting in extended measurement times and added errors. In this study, we propose a nonlinear optimization approach that combines a 2-frame phase-shifting algorithm to suppress the influence of high diffraction orders, reducing the required number of phase shifts by more than 75%. We formulate a numerical model that aligns with the interferometry, construct a highly robust error function, and enable high-precision wavefront reconstruction by optimizing Zernike coefficients. The accuracy and robustness of the proposed method are verified through simulations and experiments.

Accuracy analysis of pseudo lateral shearing interferometry measuring complex spatio-temporal couplings

Accuracy analysis of pseudo lateral shearing interferometry measuring complex spatio-temporal couplings

Dynamic imaging of three-dimensional temperature field via three-directional wavelength modulated lateral shearing interferometry

A three-directional wavelength modulated lateral shearing interference system was proposed for the dynamic three-dimensional visualization of a temperature field. The laser emitted from a distributed feedback (DFB) laser was divided into three beams by a fiber beam splitter. The beams were spatially expanded and passed through the flame in three directions of 0°, 120° and 240°, respectively. The three interferograms formed by reflections from the front and back surfaces of the parallel glass plates were spatially converged into the lens of a high-speed camera. Multiple plane mirrors were utilized to reflect the beams. The size of the region of interest (ROI) was −20 ∼ 20 mm in both x and z directions and 5 ∼ 30 mm in y direction, with a spatial resolution of 200 μm in all directions. A three-dimensional sensitivity matrix was modeled, and the simultaneous algebraic reconstruction technique (SART) was used for layer-by-layer imaging to achieve three-dimensional flame monitoring. Numerical simulations verified the effectiveness of the algorithm qualitatively and quantitatively. In experiments, the temperature distributions of four cases of flames, namely, steady diffused flame, partially blocked diffused flame, dynamic acoustically excited flame and the flame after ignition, were reconstructed at 1k frames per second (fps). The flame structures and temporal variations of temperature distributions were clearly visualized, demonstrating the ability of the system in fast monitoring of three-dimensional dynamic temperature fields.

Absolute measurement of focusing properties of a large-aperture diffractive lens.

Diffractive lenses are popular in large optical systems owing to their lightweight and multifunctional design. However, they are difficult to calibrate accurately due to the cross talk between the first-order diffraction and the background light. Here, a quadriwave lateral shearing interferometry (QWLSI) with spherical wave illumination was proposed to absolutely measure the focusing properties of diffractive lenses by means of the reference background light, in which the corresponding theoretical modeling was first derived, and then the single-shot experiment on a 210 mm-diameter beam was carried out. The results showed that the measurement error of the focal length was 0.59%, and the consistency error was 0.008%.

Fast measurement of wavefront aberration in double-grating Ronchi lateral shearing interferometry

The accuracy of the shear-phase retrieval method is critical for accurate wavefront aberration measurements in double-grating Ronchi lateral shearing interferometry. Currently, the suppression of the interferences of unwanted diffractions generated by the Ronchi grating at the image plane is eliminated by adding more phase shifts, which increases the measurement time. Here, a stepped phase-shifting algorithm is proposed to suppress the unwanted diffractions and retrieve the shear phase between ±1st orders accurately with fewer phase shifts, and the measurement efficiency can be increased by 25% at least. The minimum number of phase shifts, which depends on the high diffraction orders existing in the interferograms, is analyzed. The proposed method was verified via numerical simulations and experiments. The wavefront measurement was validated via a comparison with the results of point-diffracted interferometry, and the root-mean-square difference was within 2.0 nm.

Image-plane coherent diffractive imaging using variable-ratio lateral-shearing interferometry

Coherent diffractive imaging (CDI) is an imaging technique that directly recovers the amplitude and phase information of the test object from the recorded diffraction patterns with an iterative algorithm. Different from the traditional CDI, here an image-plane CDI is proposed based on the variable-ratio lateral-shearing interferometry. Variable-ratio operation can realize three optical functions, the first is the array imaging, the second is the image relay, and the third is variable-ratio shear. In experiment, a frame of array diffraction patterns is recorded in one single exposure. Then, each sub-diffraction pattern is extracted through the image features and is sequentially used to recover the test object by phase retrieval. The experimental results of phase-only and amplitude-only object verify the effectiveness and robustness of our proposed method. This method not only circumvents the calibration problem of gratings but also strengthens the convergence conditions required for phase retrieval, which has broad applications in fields such as optical imaging and wavefront sensing.

Investigations Towards the Measurement of Displacement using Lateral Shearing Interferometry and Fourier Fringe Analysis Technique

In the present communication we report the measurement of small displacement of reflecting surface (plane mirror) using wedge plate lateral shearing interferometry. With the help of simple mathematical analysis relation between specimen displacement and difference phase is undertaken. Fourier based fringe analysis technique is used for determination of difference phase. Theoretical and experimental investigations are carried out to check the sensitivity of the technique. The detailed discussion regarding sources of errors and uncertainty analysis is also incorporated and the expanded uncertainty is found to be ±0.00236mm. The technique is simple and can be used in industry environment.

Wavefront Microscopy Using Quadriwave Lateral Shearing Interferometry: From Bioimaging to Nanophotonics

Common cameras are only sensitive to the intensity of light, discarding an essential feature of a light wave: its phase profile or, equivalently, its wavefront profile. This Review focuses on a rising wavefront imaging technique called quadriwave lateral shearing interferometry (QLSI), based on the simple use of a 2-dimensional diffraction grating, aka a cross-grating, in front of a regular camera. We detail the working principle of QLSI and its implementation on an optical microscope. We highlight its microscopy applications in bioimaging and nanophotonics, in particular for the characterization of living cells, nanoparticles, 2D materials, metasurfaces, microscale temperature gradients, and surface topography. Finally, we draw a critical comparison of QLSI with current quantitative phase microscopy techniques, namely, digital holography microscopy (DHM), spatial light interference microscopy (SLIM), and diffraction phase microscopy (DPM).