Wavefront analysis is used extensively to measure and characterize the wavefront of light beams in different applications, including optical testing, microscopy, and high-resolution imaging. This article discusses wavefront analysis using wavefront sensors, advances in wavefront analysis, and the use of wavefront analysis for high-resolution imaging systems.
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Wavefront Analysis Using Wavefront Sensors
In the propagation of light, a wavefront is a crucial parameter that can be utilized to characterize optical surfaces, improve the optical system performance, and align optical assemblies. Light is primarily an electromagnetic radiation of any wavelength that propagates as waves. The set of all points where the wave possesses a similar phase of the sinusoid is referred to as the wavefront.
Wavefront sensors can perform direct wavefront measurements without using interferences between beams for wavefront reconstruction and are used to measure optical wavefront aberrations. Wavefront sensors provide a direct measure of the intensity and phase of a wavefront.
A wavefront sensor can deliver several parameters, including tip and tilt, curvature, refractive power, focal point positions, wavefront peak to valley (PV) and root mean square (RMS), intensity, spot diagram, Zernike coefficients, encircled energy, modulation transfer function (MTF), point spread function (PSF), and MSquare.
Wavefront PV and RMS and intensity are the measures of the wavefront aberrations magnitude and brightness of the light, respectively. Zernike coefficients are utilized to mathematically represent the wavefront aberrations, while encircled energy is the measure of the light contained in a given circle.
MTF and PSF indicate the ability of the optical system to reproduce high-frequency patterns and to spread out a point of light, respectively, while MSquare is the measure of overall wavefront quality.
The Shack–Hartmann wavefront sensor (SHWFS) is used extensively for wavefront analysis in laser diagnostics and optical metrology. This device combines a two-dimensional (2D) detector with a lenslet array and has been developed for adaptive optics.
The best factory-calibrated SHWFSs can provide nanometric accuracy for thousands of dynamic range waves with 99.9% linearity. This performance, coupled with the intrinsic properties of the devices, such as insensitivity to achromaticity, speed, and vibrations, make SHWFSs a critical tool for wavefront analysis in various applications in research and industry.
A SHWFS can be used to measure both intensity and phase in the same plane, which allows the calculation of several parameters that describe the propagation of light, such as the MTF and PSF, with less than one % accuracy. Mathematical techniques, such as curvature sensing or phase imaging, can also be used for wavefront analysis.
These techniques can provide wavefront estimations and are only limited by the digital image resolution used for wavefront measurement computation, while Shack-Hartmann lenslet arrays are limited in lateral resolution to the lenslet array size. However, curvature sensing/phase imaging has linearity issues and is significantly less efficient compared to the original SHWFS sensor.
Wavefront Analysis for High-resolution Imaging
Conventional wavefront sensors, such as SHWFS, can only attain a resolution of a few thousand pixels as they have to maintain a balance between phase estimation and spatial resolution.
In a paper published in the journal Light: Science and Applications, researchers proposed a novel computational-imaging-based technique, designated as the Wavefront Imaging Sensor with High resolution (WISH), which can offer multi-megapixel resolution, high frame rate, and robustness to vibrations to address this issue.
WISH contained a processor, a complementary metal oxide semiconductor (CMOS) sensor, and a spatial light modulator (SLM). Researchers replaced the microlens array in SHWFS with an SLM and utilized a computational phase-retrieval algorithm for recovering the incident wavefront.
In WISH imaging, the optical field was initially modulated using several random SLM patterns, and the corresponding intensity-only measurements were captured using a CMOS sensor. Subsequently, a computational phase-retrieval algorithm was used to process the acquired data. The algorithm estimated the complex optical field incident on the SLM.
The spatial resolution of the recovered field was higher than 10 megapixels, a 1000 × improvement compared to the conventional SHFWS as the computational-imaging-based method shifted the complexity from hardware to algorithm.
Thus, the high-resolution, noninterferometric wavefront sensor could effectively measure highly variant optical fields at over multi-megapixel resolution with fine phase estimation. This resolution was also an order of magnitude higher compared to the existing non interferometric wavefront sensors.
Researchers also presented three applications that cover an extensive range of spatial scales to demonstrate the capability of WISH. Specifically, they demonstrated the recovery of high-resolution images of objects obscured by scattering and the feasibility of using WISH as a microscope without an objective lens/for lensless microscopy.
The findings displayed that WISH, which is based on the designing principle combining computational algorithms and optical modulators, can significantly improve the performance of several existing high-resolution imaging systems such as microscopy and medical imaging.
Advances in Wavefront Analysis
Zernike circle polynomials are used extensively for wavefront analysis in optical testing and design as they are orthogonal over a unit circle/display orthogonality over a circular pupil and represent balanced aberrations of systems with circular pupils.
However, the Zernike circle polynomials are not orthogonal or represent balanced aberrations for systems with noncircular pupils. Thus, circle polynomials are not suitable for wavefront analysis of noncircular pupils.
In a paper published in the Journal of the Optical Society of America A, researchers used circle polynomials as the basis functions for their orthogonalization over elliptical, rectangular, and square pupils to derive closed-form polynomials that are orthonormal over such non-circular pupils.
An annular pupil, which is common for a centered mirror telescope, is an example of noncircular pupil. The Zernike annular polynomials are the appropriate polynomials for annular pupils, which can be obtained from the circle polynomials by their orthogonalization over an annulus using the nonrecursive matrix approach or the recursive Gram–Schmidt process.
Researchers extended this approach to rectangular, square, and elliptical pupils and determined the polynomials that represent balanced aberrations for and orthonormal over such pupils to retain the advantages of orthogonal polynomials.
These polynomials were both orthogonal across noncircular pupils and represented balanced classical aberrations, similar to the Zernike circle polynomials. Additionally, the polynomials were obtained in polar and Cartesian coordinates and circle polynomials.
Researchers also obtained the relationships between the orthonormal coefficients and the corresponding Zernike coefficients for a specific pupil and the orthonormal polynomials/Legendre polynomials representing balanced aberrations of a one-dimensional (1D) slit pupil as a limiting case of a rectangular pupil.
References and Further Reading
Mahajan, V. N., Dai, G. -M. (2007). Orthonormal polynomials in wavefront analysis: analytical solution. Journal of the Optical Society of America A, 24, 9, 2994-3016. https://doi.org/10.1364/JOSAA.24.002994
Wu, Y., Sharma, M. K., Veeraraghavan, A. (2019). WISH: Wavefront imaging sensor with high resolution. Light: Science & Applications, 8(1), 1-10. https://doi.org/10.1038/s41377-019-0154-x
Wavefront Sensor Applications [Online] (Accessed on 01 October 2023)
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