Zonal Wavefront Research Articles

A frequency-response-optimized Shack-Hartmann zonal wavefront reconstructor based on Fan's model.

This paper introduces an optimized method for zonal wavefront reconstruction utilizing Fan's model, specifically tailored to enhance the frequency response. Analysis of the system frequency response demonstrates a 27% increase in bandwidth compared to the Southwell model. Examination of reconstruction errors at various frequency points reveals consistently smaller values when compared to the Southwell model. Validation through numerical simulations and real experiments underscores the superior performance of the proposed reconstructor, particularly noticeable at higher response levels within the mid- and high-frequency domains.

Fast and Highly Accurate Zonal Wavefront Reconstruction from Multi-Directional Slope and Curvature Information Using Subregion Cancelation

The wavefront reconstruction is a crucial step in determining the performance of wavefront detection instruments. The wavefront reconstruction algorithm is primarily evaluated in three dimensions: accuracy, speed, and noise immunity. In this paper, we propose a hybrid zonal reconstruction algorithm that introduces slope and curvature information in the diagonal, anti-diagonal, horizontal, and vertical directions by dividing the neighbor sampling points into subregions in groups of four. By canceling the same parameters in integration equations, an algorithm using multi-directional slope–curvature information is achieved with only two sets of integration equations in each subregion, reducing the processing time. Simulation experiments show that the relative root-mean-square reconstruction error of this algorithm is improved by about 4 orders of magnitude compared with existing algorithms that use multi-directional slope information or slope–curvature information alone. Compared with the hybrid multi-directional slope–curvature algorithm, the proposed algorithm can reduce computation time by about 50% as well as provide better noise immunity and reconstruction accuracy. Finally, the validity of the proposed algorithm is verified by the null test experiment.

Comparison of modal and zonal wavefront measurements of refractive extended depth of focus intraocular lenses.

Extended depth-of-focus (EDoF) intraocular lenses (IOLs) are typically evaluated using commercially available aberrometers. Given the intricate optical design of these IOLs, employing an appropriate wavefront reconstruction method with a sufficient sampling resolution of the aberrometer is crucial. A high-resolution Shack-Hartmann wavefront sensor was developed by magnifying the pupil aperture by a factor of five onto a lenslet array (pitch: 133 µm) and utilizing a full-frame CMOS sensor (24 by 36 mm), resulting in a 26.6 µm sampling resolution. Zonal wavefront reconstruction was used and compared with Zernike-based modal wavefront reconstruction to retain detailed local slope irregularities. Four refractive EDoF IOLs with a power of 20D were examined, and the wavefront difference between the zonal and modal methods, expressed as the root mean squared error (RMSE), remained significant for two of the IOLs up to the 16th-order Zernike spherical aberrations (SAs). Conversely, a negligibly small RMSE was observed for the other two IOLs, as long as the Zernike SAs were higher than the 6th order. The raytracing simulation results from the zonal wavefronts exhibited a stronger correlation with the results of recent optical bench studies than those from the modal wavefronts. The study suggests that certain recent refractive EDoF IOLs possess a complex optical profile that cannot be adequately characterized by limited orders of SAs.

Experimental demonstration of in situ surface and thickness profile measurements of thin film during deposition using a grating array based wavefront sensor.

Here we introduce an in situ and non-intrusive surface and thickness profile monitoring scheme of thin-film growth during deposition. The scheme is implemented using a programmable grating array based zonal wavefront sensor integrated with a thin-film deposition unit. It provides both 2D surface and thickness profiles of any reflecting thin film during deposition without requiring the properties of the thin-film material. The proposed scheme comprises a mechanism to nullify the effect of vibrations which is normally built in with the vacuum pumps of thin-film deposition systems and is largely immune to the fluctuations in the probe beam intensity. The final thickness profile obtained is compared with independent off-line measurement and the two results are observed to be in agreement.

Inverse Hartmann test for radius of curvature measurement in a corneal topography calibration sphere

In this article, the use of a square Hartmann screen test to measure the radius of curvature of a corneal topography calibration test sphere is presented. The proposed technique is based on the image formation principle by specular reflection on convex reflective surfaces. Applying an inverse Hartmann test, a de-magnified virtual image (Hartmanngram) is obtained; considering their own scaled reference screen plate, a zonal wavefront retrieval approach is used and the radius of curvature obtained. Experimental setup along the obtained results is presented. A simulated spherical wavefront is used as a method to evaluate the error in the wavefront reconstruction. Since the measurements of radius of curvature fits in to ISO 10343, through suitable modifications the proposed method is potentially applicable in small F/# convex specular surfaces, as is the case in keratometry and corneal topography measurements.

Hybrid wavefront reconstruction from multi-directional slope and full curvature measurements using integral equations with higher-order truncation errors for wavefront sensors

Wavefront reconstruction is one of the most important steps in zonal wavefront sensors and has great influence on measurement accuracy. An improved hybrid wavefront reconstruction algorithm that adds two diagonal slope and curvature values at each grid point is proposed to improve reconstruction accuracy and noise immunity. Hybrid reconstruction equations with smaller truncation errors in diagonal and anti-diagonal directions are deduced. Furthermore, multi-directional hybrid reconstruction equations incorporating vertical, horizontal, diagonal and anti-diagonal directions are also formulated. Numerical experiments show that the relative reconstruction errors of the proposed algorithms are reduced by about four orders of magnitude compared with the existing hybrid reconstruction algorithm with noise ignored. Considering the measurement noise, the proposed algorithms have better noise immunity and reconstruction accuracy, especially for the reconstruction of high-order aberrations.

Enhancement of wavefront measurement sensitivity in a zonal wavefront sensor without curtailing the sensing speed

In this paper we propose a zonal wavefront sensing scheme that facilitates wavefront measurement with enhanced sensitivity at the standard video rate. We achieve this enhanced sensitivity by implementing a sequential display of binary holograms described over each zone sampling the incident wavefront with the help of a ferroelectric liquid crystal spatial light modulator. By keeping the number of active zones as 24 and using a camera with an imaging frame rate equal to the binary hologram display rate of the spatial light modulator, we are able to reach a sensing frame rate of 60 Hz. In addition to enhanced sensitivity, the proposed scheme facilitates zone-wise tuning of binary holograms and eliminates the possibility of any crosstalk between adjacent zones. We perform a proof-of-principle experiment that validates the proposed zonal wavefront sensing scheme and demonstrates its advantages.

Improved wavefront estimation accuracy in a programmable grating array based wavefront sensor

Shack–Hartmann wavefront sensor (SHWS) is one of the popular zonal wavefront sensor that typically consists of a two-dimensional (2D) array of lenslets and a camera. Another variant of a zonal wavefront sensor is the programmable grating array based wavefront sensor (PGAWS) that replaces the lenslets array of the SHWS with a 2D array of programmable diffraction gratings and a single focusing lens to result in a 2D array of +1 order spots. The ability of a PGAWS to correctly measure the incident wavefront depends on the accuracy with which each +1 order spot centroid position in the camera plane is calculated. However, in practical applications the intensities of the +1 order spots are usually non-uniform and there is presence of higher order spots that lead to erroneous calculation of the centroid position. In the present work, an improved PGAWS is proposed and by exploiting its programmable facility the transmittance function of the grating array is defined to generate an array of uniform intensity +1 order spots with negligible contribution from unwanted higher order spots by breaking them into noise. We present proof-of-concept simulation results that illustrate the generation of uniform intensity +1 order spots and also quantify its uniformity numerically by defining a quality metric. Furthermore, the working of the proposed PGAWS is substantiated through simulation results which demonstrate its accurate wavefront estimation capability in comparison to the conventional PGAWS, in the presence of non-uniform intensity +1 order spots.

3D reconstruction of specular surface by combined binocular vision and zonal wavefront reconstruction.

This paper proposes a framework for the three-dimensional (3D) reconstruction of specular surfaces. The framework begins with a modified reconstruction method that uses gradient descent to accurately search for the correspondence matches in the binocular vision system. An analysis of the binocular vision system shows that the reconstruction result is not necessarily unique and the reconstruction accuracy is significantly compromised by a bilinear interpolation error. An error-based selection algorithm is proposed to select the correct result. To improve the reconstruction accuracy, binocular vision and zonal wavefront reconstruction are incorporated. The reconstruction result from the binocular vision system is taken as a reference surface for slope estimation. The measured slope is then integrated into a height map by the zonal wavefront reconstruction. The overall reconstruction process provides an accurate reconstruction result without the need for prior surface information. Simulations and experiments were conducted to verify the feasibility of the proposed algorithms. A relative error of 0.59% or less is experimentally recorded in the step-size measurement of a stair-like object.

Zonal Reconstruction of Photonic Wavefunction via Momentum Weak Measurement

AbstractDirect measurement of wavefunctions has attracted great interest and many different methods have been developed. However, the precision of current techniques is limited by the use of Fourier transform lenses. These measurements require to shear cut the part of particles with momentum , which greatly restricts the efficiency and application of the approaches. Here, a method to directly measure 2D photonic wavefunctions is proposed and experimentally demonstrated by combining the momentum weak measurement technology and the zonal wavefront restoration algorithm. Both the Gaussian and Laguerre–Gaussian wavefunctions are experimentally well reconstructed. This method avoids using the Fourier lens and post selection on the momentum . This is further applied to measure wavefronts with ultra‐high spatial frequency and achieve a pixel resolution, which is difficult for traditional Shack–Hartmann wavefront sensing technologies. This work extends the ability of quantum measurement techniques and will be useful for high‐resolution wavefront sensing.

Improved linear response in a modal wavefront sensor.

The modal wavefront sensor provides a direct and less computationally intensive way, relative to a zonal wavefront sensor, of estimating the various aberration modes present in a beam of light. Such wavefront sensors are particularly useful when the optical system concerned is affected by a limited number of aberration modes. Unfortunately, the basic design of the modal wavefront sensor suffers from reduced linear response and intermodal crosstalk. As a result of the reduced linear response, the magnitude of an aberration mode estimated by a modal wavefront sensor may differ from the actual strength of the aberration mode. This difference is even more if other aberration modes, in addition to the aberration mode to be detected, also referred to as the sensor mode, are present in the beam. In this paper we propose a modal wavefront sensing scheme where the sensor output has a significantly enhanced linear response. The proposed scheme also provides superior intermodal crosstalk immunity, especially for a few selected aberration modes. We first show theoretically how by incorporating a variable magnitude of the sensor mode into the beam the precise strength of the same mode can be estimated. Results from numerical simulation and an experiment using a proof of principle setup demonstrate the improved performance by the proposed scheme.

Estimation of inter-modal cross talk in a modal wavefront sensor

A modal wavefront sensor provides a direct way of measuring the presence of a given type of aberration in the incident beam and is less computationally intensive compared to a zonal wavefront sensor. Such a sensor is particularly useful when the incident beam is aberrated with a limited number of lower order aberrations. The modal wavefront sensor has found applications in diverse areas and research is on to further improve the performance of the sensor. However, one major issue of the modal wavefront sensor is the inter-modal cross talk, that is the effect on the sensor output due to the presence of other aberration modes present in the beam. Although inter-modal cross talk is an inherent phenomenon in a modal wavefront sensor, so far there has not been any comprehensive study on the same in general. Besides there does not exist a model for quick numerical evaluation of the inter-modal cross talk in a modal wavefront sensor. In this paper we develop the theoretical expressions to define an inter-modal cross talk co-efficient and further expand it to a polynomial form. The polynomial form provides a quick way to calculate the effect of inter-modal cross talk. The correctness of the polynomial form thus developed is demonstrated by comparing the results with those obtained using the basic theory and the experimental implementation of a modal wavefront sensor. The polynomial form of the cross talk co-efficient is further used to investigate the effect of inter-modal cross talk in the sensing of a few important low order aberrations.

Zonal wavefront reconstruction of Shack–Hartmann and Hartmann patterns with hexagonal cells

In this article we will develop a method to integrate Shack–Hartmann and Hartmann pattern with hexagonal cells, using a polynomial representation (modal integration) over each hexagonal cell. Since each hexagonal has six sampling points, one at each vertex, instead of the typical four sampling points in square cells, it is possible to have a different representation of the wavefront in each cell, each with different aberration terms. The local curvatures and low order aberrations in each cell are calculated more accurately than for square cells. All the analytical functions over each hexagonal cell have a different unknown piston term, that is calculated with a method to be described here. As a result, wavefront retrieval and representation of freeform optical surfaces for some optical systems can be made, due to the calculation of aberrations in each hexagonal cell.

A zonal wavefront sensor with multiple detector planes

A conventional zonal wavefront sensor estimates the wavefront from the data captured in a single detector plane using a single camera. In this paper, we introduce a zonal wavefront sensor which comprises multiple detector planes instead of a single detector plane. The proposed sensor is based on an array of custom designed plane diffraction gratings followed by a single focusing lens. The laser beam whose wavefront is to be estimated is incident on the grating array and one of the diffracted orders from each grating is focused on the detector plane. The setup, by employing a beam splitter arrangement, facilitates focusing of the diffracted beams on multiple detector planes where multiple cameras can be placed. The use of multiple cameras in the sensor can offer several advantages in the wavefront estimation. For instance, the proposed sensor can provide superior inherent centroid detection accuracy that can not be achieved by the conventional system. It can also provide enhanced dynamic range and reduced crosstalk performance. We present here the results from a proof of principle experimental arrangement that demonstrate the advantages of the proposed wavefront sensing scheme.

A Holography-Based Modal Wavefront Sensor for the Precise Positioning of a Light Emitter Using a High-Resolution Computer-Generated Hologram

In certain applications, modal wavefront sensors (MWFSs) can outperform zonal wavefront sensors, which are widely used due to their high flexibility. In this paper, a holography-based MWFS as described is developed for the fast position control of a light emitter in a deep parabolic mirror. The light source is located in the vicinity of the focal point. Instead of Zernike polynomials, more complex phase functions, which are related to certain dislocations of the light source are used as detector modes. The performance of the sensor is verified with a test setup, where the test wavefront is generated by a spatial light modulator instead of a real parabolic mirror. The design and fabrication of the required high-resolution holographic element is described and an easy way of multiplexing several single mode sensors is demonstrated.

Improvement in error propagation in the Shack-Hartmann-type zonal wavefront sensors.

Estimation of the wavefront from measured slope values is an essential step in a Shack-Hartmann-type wavefront sensor. Using an appropriate estimation algorithm, these measured slopes are converted into wavefront phase values. Hence, accuracy in wavefront estimation lies in proper interpretation of these measured slope values using the chosen estimation algorithm. There are two important sources of errors associated with the wavefront estimation process, namely, the slope measurement error and the algorithm discretization error. The former type is due to the noise in the slope measurements or to the detector centroiding error, and the latter is a consequence of solving equations of a basic estimation algorithm adopted onto a discrete geometry. These errors deserve particular attention, because they decide the preference of a specific estimation algorithm for wavefront estimation. In this paper, we investigate these two important sources of errors associated with the wavefront estimation algorithms of Shack-Hartmann-type wavefront sensors. We consider the widely used Southwell algorithm and the recently proposed Pathak-Boruah algorithm [J. Opt.16, 055403 (2014)JOOPDB0150-536X10.1088/2040-8978/16/5/055403] and perform a comparative study between the two. We find that the latter algorithm is inherently superior to the Southwell algorithm in terms of the error propagation performance. We also conduct experiments that further establish the correctness of the comparative study between the said two estimation algorithms.

Zonal wavefront reconstruction in quadrilateral geometry for phase measuring deflectometry.

There are wide applications for zonal reconstruction methods in slope-based metrology due to its good capability of reconstructing the local details on surface profile. It was noticed in the literature that large reconstruction errors occur when using zonal reconstruction methods designed for rectangular geometry to process slopes in a quadrilateral geometry, which is a more general geometry with phase measuring deflectometry. In this work, we present a new idea for the zonal methods for quadrilateral geometry. Instead of employing the intermediate slopes to set up height-slope equations, we consider the height increment as a more general connector to establish the height-slope relations for least-squares regression. The classical zonal methods and interpolation-assisted zonal methods are compared with our proposal. Results of both simulation and experiment demonstrate the effectiveness of the proposed idea. In implementation, the modification on the classical zonal methods is addressed. The new methods preserve many good aspects of the classical ones, such as the ability to handle a large incomplete slope dataset in an arbitrary aperture, and the low computational complexity comparable with the classical zonal method. Of course, the accuracy of the new methods is much higher when integrating the slopes in quadrilateral geometry.

Local Slope and Curvature Tests via Wave Front Modulations in the Shack–Hartmann Sensor

To provide slope and full curvature measurements with a single image for each subaperture in zonal wave front sensing, we propose to introduce a wave front modulation to each subaperture of the Shack–Hartmann sensor. This idea is preliminarily realized on a new sensor, which uses a four-hole mask array to modulate the test wave front incident on the lenslet array. Centroid estimation for the local slope measurements can be performed with reduced variance, thanks to the reduced size of the central lobe offered by the four-hole mask modulation. Meanwhile, the side lobes in the focal image give three intensity-normalized differential signals for the local curvature measurements. Each of the three signals shows approximately linear response to the corresponding second order term of the local wave front. In this way, the full curvature information for each subaperture can be directly determined without any complicated post processing algorithms required in other focal plane sensing techniques. Accuracy of the sensor in a single subaperture is examined by simulation.

基于子孔径波前振幅调制的斜率和曲率信号提取算法

The slope and curvature hybrid wavefront sensing technique can realize the precision measurement of high-order aberration by using all the first and second derivative information of the wavefront in the subaperture. And it is an important development direction of zonal wavefront sensing technology. Based on the slope and curvature hybrid wavefront sensor used subaperture amplitude modulation, a slope and curvature signal extraction algorithm was proposed. The centroid detection algorithm was utilized to realize the direct measurement of wavefront slope, Laplacian curvature and twist curvature in each subaperture. The simulation results show that the detection accuracy of the slope and curvature signals are all within 0.02(=632.8 nm) under the ideal condition. In the case where the noise level is less than 10%, the detection accuracy of the slope and curvature signals are all within 0.08. The algorithm has high detection precision and good noise immunity.

Quasi-absolute surface figure test with two orthogonal transverse spatial shifts

A new zonal wavefront reconstruction algorithm with pixel-level spatial resolution and high accuracy is proposed, which is able to reconstruct the original wavefront of general aperture shape from only two difference wavefronts measured at two orthogonal shear directions with shear amounts equaling arbitrary moderate integral multiples of the sample interval. Based on this algorithm, a quasi-absolute surface figure test method is presented, which requires only two additional translational measurements with shifts of arbitrary moderate integral multiples of sample interval along x and y directions besides the original position measurement. Optical schemes of the proposed method for testing flat, spherical and cylindrical surfaces are investigated, and special considerations and challenges for calibrating spheres and cylinders are also briefly formulated theoretically. Thorough errors analysis is formulated for obtaining high accuracy test result. Simulations and experiments on a flat surface are conducted to validate the proposed algorithm and method. Compared with existing absolute test methods with Pseudo-Shear Interferometry (PSI) technique, the presented method has advantages, like, less number of measurements, arbitrary moderate shear amounts and the high signal-to-noise ratio it can reach.