Gram-Schmidt orthonormalization of Zernike polynomials for general aperture shapes

X-ray focusing below 3 nm with aberration-corrected multilayer Laue lenses

Multilayer Laue lenses are volume diffractive optical elements for hard X-rays with the potential to focus beams to sizes as small as 1 nm. This ability is limited by the precision of the manufacturing process, whereby systematic errors that arise during fabrication contribute to wavefront aberrations even after calibration of the deposition process based on wavefront metrology. Such aberrations can be compensated by using a phase plate. However, current high numerical aperture lenses for nanometer resolution exhibit errors that exceed those that can be corrected by a single phase plate. To address this, we accumulate a large wavefront correction by propagation through a linear array of 3D-printed phase correcting elements. With such a compound refractive corrector, we report on a point spread function with a full-width at half maximum area of 2.9 × 2.8 nm2 at a photon energy of 17.5 keV.

Rapid aberration correction for diffractive X-ray optics by additive manufacturing

Diffraction-limited hard X-ray optics are key components for high-resolution microscopy, in particular for upcoming synchrotron radiation sources with ultra-low emittance. Diffractive optics like multilayer Laue lenses (MLL) have the potential to reach unprecedented numerical apertures (NA) when used in a crossed geometry of two one-dimensionally focusing lenses. However, minuscule fluctuations in the manufacturing process and technical limitations for high NA X-ray lenses can prevent a diffraction-limited performance. We present a method to overcome these challenges with a tailor-made refractive phase plate. With at-wavelength metrology and a rapid prototyping approach we demonstrate aberration correction for a crossed pair of MLL, improving the Strehl ratio from 0.41(2) to 0.81(4) at a numerical aperture of 3.3 × 10^-3. This highly adaptable aberration-correction scheme provides an important tool for diffraction-limited hard X-ray focusing.

Impact of microvibration on the optical performance of an airborne camera

Aiming to investigate the connection between camera structure and optical systems, a comprehensive analysis needs to be performed for the airborne camera. An integrated analysis method was proposed to design and analyze optical and mechanical structures. Based on the designed small airborne camera, the impact of microvibration on the optical performance of the airborne camera was studied by integrated optomechanical analysis. In addition, the change of optical surface accuracy was analyzed. First, static and dynamic analysis of the designed airborne camera was performed to verify the stability of the camera structure and obtain the data for integrated optomechanical analysis. Then, a calculation method for rigid body displacement was proposed, and the impact of rigid body displacements on the optical system was analyzed. To evaluate the change of surface accuracy, the parameters root mean square (RMS) and peak to valley (PV) were calculated by fitting the surface distortion data. Based on the Zernike polynomial coefficients, the response of the optical system was calculated and analyzed utilizing ZEMAX to analyze the impact of microvibration on the optical performance of the airborne camera. The analysis results show that microvibration has no significant impact on optical performance of the designed small airborne camera. Finally, the analysis results were verified through experiments.

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.

Efficient representation of freeform gradient-index profiles for non-rotationally symmetric optical design

Conventional optical designs with gradient index (GRIN) use rotationally-invariant GRIN profiles described by polynomials with no orthogonality. These GRIN profiles have limited effectiveness at correcting aberrations from tilted/decentered or freeform systems. In this paper, a three-dimensional orthogonal polynomial basis set (the FGRIN basis) is proposed, which enables the design of GRIN profiles with both rotational and axial variations. The FGRIN basis is then demonstrated via the design of a 3D GRIN corrector plate targeted to correct the rotationally-variant aberrations induced from a tilted spherical mirror. A sample corrector is manufactured and tested, showing significant correction of astigmatism. The FGRIN basis opens a new design space of 3D rotational variant GRIN profiles, which has the potential of replacing multiple freeform surfaces and simplifying complex systems.

Numerical analysis of a novel two-stage enlargement and adaptive correction approach for the annular aberration compensation

The annular laser beam (ALB) is widely used in many fields, which could be affected by laser power and beam quality. To effectively and flexibly improve the beam quality of high-power large-aperture thin-wall ALB, a two-stage enlargement and adaptive correction configuration (TEACC) consisting of a novel outer-surface tubular deformable mirror (OTDM) and two extra prism groups (EPGs) is proposed in this paper. The correction principle and design principle of the TEACC are derived and analyzed. Based on the principle, a typical OTDM prototype and EPG structure are designed. Annular aberrations are compensated by applying the OTDM's influence functions and the least-square algorithm in simulation. The results show that the TEACC could perfectly compensate the wavefront distortions described by the 2^nd to 36^th order Zernike annular aberrations.

Theoretical research on the novel adaptive optics configuration based on the tubular deformable mirror for the aberration correction of the annular laser beam

The annular laser beam (ALB) has been widely used in many fields for its unique intensity distribution. Especially, in the materials processing, the power and the beam quality of the large-aperture thin-wall ALB are of vital. However, limited by the aperture, the actuators' spacing or the damage threshold, the existing deformable mirrors (DMs) are not suitable for the correction of the ALB. Considering the stretching effect of the oblique incidence, in this paper, by using the tubular DM (TDM), a novel adaptive optics (AO) configuration is promoted to increase the number of the effective actuators covered by the input ALB. The coordinate transformation equations and correction principle of the novel AO configuration are derived based on the ray tracing. A typical TDM prototype is designed based on the coordinate transformation equations. The influence function characteristics of the TDM is analyzed using the finite element method, and the correction ability of the novel AO configuration based on the TDM is verified. Simulation results show that the TDM could perfectly compensate the wavefront distortions described by the 2^th to 15^th order Zernike annular aberrations.

On-the-fly surface manufacturability constraints for freeform optical design enabled by orthogonal polynomials

When leveraging orthogonal polynomials for describing freeform optics, designers typically focus on the computational efficiency of convergence and the optical performance of the resulting designs. However, to physically realize these designs, the freeform surfaces need to be fabricated and tested. An optimization constraint is described that allows on-the-fly calculation and constraint of manufacturability estimates for freeform surfaces, namely peak-to-valley sag departure and maximum gradient normal departure. This constraint's construction is demonstrated in general for orthogonal polynomials, and in particular for both Zernike polynomials and Forbes 2D-Q polynomials. Lastly, this optimization constraint's impact during design is shown via two design studies: a redesign of a published unobscured three-mirror telescope in the ball geometry for use in LWIR imaging and a freeform prism combiner for use in AR/VR applications. It is shown that using the optimization penalty with a fixed number of coefficients enables an improvement in manufacturability in exchange for a tradeoff in optical performance. It is further shown that, when the number of coefficients is increased in conjunction with the optimization penalty, manufacturability estimates can be improved without sacrificing optical performance.

Degeneracy in freeform surfaces described with orthogonal polynomials

Orthogonal polynomials offer useful mathematical properties for describing freeform optical surfaces. Their advantages are best leveraged by understanding the interactions between variables such as tip and tilt, base sphere and conic variables, and packaging variables that define the problem of design for manufacture. These interactions can cause degeneracy, which can complicate the interpretation of design specifications in manufacturing and, consequently, negatively impact the cost of fabrication and assembly. Optimization constraints to break degeneracy during design are also discussed.

Strehl ratio and amplitude-weighted generalized orthonormal Zernike-based polynomials

The concept of orthonormal polynomials is revisited by developing a Zernike-based orthonormal set for a non-circular pupil that is transmitting an aberrated, non-uniform field. We refer to this pupil as a general pupil. The process is achieved by using the matrix form of the Gram-Schmidt procedure on Zernike circle polynomials and is interpreted as a process of balancing each Zernike circle polynomial by adding those of lower order in the general pupil, a procedure which was previously performed using classical aberrations. We numerically demonstrate this concept by comparing the representation of phase in a square-Gaussian pupil using the Zernike-Gauss square and Zernike circle polynomials. As expected, using the Strehl ratio, we show that only specific lower-order aberrations can be used to balance specific aberrations, for example, tilt cannot be used to balance spherical aberration. In the process, we present a possible definition of the Maréchal criterion for the analysis of the tolerance of systems with apodized pupils.

Modal wavefront reconstruction in radial shearing interferometry with general aperture shapes

Wavefront reconstruction in radial shearing interferometry with general aperture shapes is challenging because the problem may be ill-conditioned. Here we propose a Gram-Schmidt orthogonalization method to cope with off-axis wavefront reconstruction with any aperture type. The proposed method constructs a set of orthogonal basis functions and computes the corresponding expansion coefficients, which are converted into another set of expansion coefficients to reproduce the original wavefront. The method can effectively alleviate the ill-conditioning of the problem, and is numerically stable compared with the classic least-squares method, especially for non-circular apertures and in the presence of noise. Computer simulation and experimental results are presented to demonstrate the performance of the algorithm.

Zernike-like systems in polygons and polygonal facets

Zernike polynomials are commonly used to represent the wavefront phase on circular optical apertures, since they form a complete and orthonormal basis on the unit disk. In [Opt. Lett.32, 74 (2007)10.1364/OL.32.000074OPLEDP0146-9592] we introduced a new Zernike basis for elliptic and annular optical apertures based on an appropriate diffeomorphism between the unit disk and the ellipse and the annulus. Here, we present a generalization of this Zernike basis for a variety of important optical apertures, paying special attention to polygons and the polygonal facets present in segmented mirror telescopes. On the contrary to ad hoc solutions, most of them based on the Gram-Smith orthonormalization method, here we consider a piecewise diffeomorphism that transforms the unit disk into the polygon under consideration. We use this mapping to define a Zernike-like orthonormal system over the polygon. We also consider ensembles of polygonal facets that are essential in the design of segmented mirror telescopes. This generalization, based on in-plane warping of the basis functions, provides a unique solution, and what is more important, it guarantees a reasonable level of invariance of the mathematical properties and the physical meaning of the initial basis functions. Both the general form and the explicit expressions for a typical example of telescope optical aperture are provided.

Tunable liquid crystal cylindrical micro-optical array for aberration compensation

A tunable aberration compensation device for rectangular micro-optical systems is proposed and demonstrated. This device, which is based in nematic liquid crystal and a micro-electrode structure, forms gradients in the index of refraction as a function of voltage. We have developed a fringe skeletonizing application in order to extract the 3D wavefront from an interference pattern. This software tool obtains the optical aberrations using Chebyshev polynomials. By using phase shifted electrical signals the aberrations can be controlled independently. A complete independent control over the spherical and coma aberration has been demonstrated. Also, an independent control over the astigmatism aberration has been demonstrated in a broad range. This device has promising applications where aberration compensation is required. The independent compensation achieved for some coefficients, such as astigmatism for example, is more than 2.4 waves.

Orthonormal polynomials for annular pupil including a cross-shaped obstruction

By nonrecursive matrix method using the Zernike circle polynomials as the basis functions, we derived a set of polynomials up to fourth order which is approximately orthonormal for optical systems with an annular pupil having a cross-shaped obstruction. The performance of the polynomials is compared with the strictly orthonormal polynomials with some numerical examples.

Performance analysis of a mirror by numerical iterative method

Zernike polynomials are generally used to predict the optical performance of a mirror. However, it can also be done by a numerical iterative method. As piston, tip, tilt, and defocus (P.T.T.F) aberrations can be easily removed by optical alignment, we iteratively used a rotation transformation and a paraboloid graph subtraction for removal of the aberrations from a raw deformation of the optical surface through a Finite Element Method (FEM). The results of a 30 cm concave circular mirror corrected by the iterative method were almost the same as those yielded by Zernike polynomial fitting, and the computational time was fast. In addition, a concave square mirror whose surface area is π was analyzed in order to visualize the deformation maps of a general mirror aperture shape. The iterative method can be applicable efficiently because it does not depend on the mirror aperture shape.

Generalization of Zernike polynomials for regular portions of circles and ellipses

Zernike polynomials are commonly used to represent the wavefront phase on circular optical apertures, since they form a complete and orthonormal basis on the unit circle. Here, we present a generalization of this Zernike basis for a variety of important optical apertures. On the contrary to ad hoc solutions, most of them based on the Gram-Schmidt orthonormalization method, here we apply the diffeomorphism (mapping that has a differentiable inverse mapping) that transforms the unit circle into an angular sector of an elliptical annulus. In this way, other apertures, such as ellipses, rings, angular sectors, etc. are also included as particular cases. This generalization, based on in-plane warping of the basis functions, provides a unique solution and what is more important, it guarantees a reasonable level of invariance of the mathematical properties and the physical meaning of the initial basis functions. Both, the general form and the explicit expressions for most common, elliptical and annular apertures are provided.

Orthonormal polynomials for elliptical wavefronts with an arbitrary orientation

We generalize the analytical form of the orthonormal elliptical polynomials for any arbitrary aspect ratio to arbitrary orientation and give expression for them up to the 4th order. The utility of the polynomials is demonstrated by obtaining the expansion up to the 8th order in two examples of an off-axis wavefront exiting from an optical system with a vignetted pupil.

Orthonormal aberration polynomials for optical systems with circular and annular sector pupils

Using the Zernike circle polynomials as the basis functions, we obtain the orthonormal polynomials for optical systems with circular and annular sector pupils by the Gram-Schmidt orthogonalization process. These polynomials represent balanced aberrations yielding minimum variance of the classical aberrations of rotationally symmetric systems. Use of the polynomials obtained is illustrated with numerical examples.

Long-exposure filtering of turbulence-degraded wavefronts

The quasi-static aberrations of optical telescopes are often determined using light from a star as the reference wavefront. We calculate the exposure time necessary to determine the amplitude of the phase aberrations for a given telescope to a given accuracy in the presence of atmospheric seeing. We implement a computational simulation of the atmosphere and present the root mean square of the generated wavefront Zernike amplitudes for a given exposure time. We find the exposure time τ required to reach a desired precision is strongly dependent on telescope diameter (τ∝D(8/3)) and can be many tens of minutes in extreme cases. We present the results so τ can be calculated for a range of telescopes and atmospheric parameters.

Representation of wavefronts in free-form transmission pupils with Complex Zernike Polynomials

Purpose

To propose and evaluate Complex Zernike polynomials (CZPs) to represent general wavefronts with non uniform intensity (amplitude) in free-from transmission pupils.

Methods

They consist of three stages: (1) theoretical formulation; (2) numerical implementation; and (3) two studies of the fidelity of the reconstruction obtained as a function of the number of Zernike modes used (36 or 91). In the first study, we generated complex wavefronts merging wave aberration data from a group of 11 eyes, with a generic Gaussian model of the Stiles-Crawford effective pupil transmission. In the second study we simulated the wavefront passing through different pupil stop shapes (annular, semicircular, elliptical and triangular).

Results

The reconstructions of the wave aberration (phase of the generalized pupil function) were always good, the reconstruction RMS error was of the order of 10⁻⁴ wave lengths, no matter the number of modes used. However, the reconstruction of the amplitude (effective transmission) was highly dependent of the number of modes used. In particular, a high number of modes is necessary to reconstruct sharp edges, due to their high frequency content.

Conclusions

CZPs provide a complete orthogonal basis able to represent generalized pupil functions (or complex wavefronts). This provides a unified general framework in contrast to the previous variety of ad oc solutions. Our results suggest that complex wavefronts require a higher number of CZP, but they seem especially well-suited for inhomogeneous beams, pupil apodization, etc.

A comprehensive approach to deal with instrumental optical aberrations effects in high-accuracy photon's orbital angular momentum spectrum measurements

With the current and upcoming applications of beams carrying orbital angular momentum (OAM), there will be the need to generate beams and measure their OAM spectrum with high accuracy. The instrumental OAM spectrum distortion is connected to the effect of its optical aberrations on the OAM content of the beams that the instrument creates or measures. Until now, the effect of the well-known Zernike aberrations has been studied partially, assuming vortex beams with trivial radial phase components. However, the traditional Zernike polynomials are not best suitable when dealing with vortex beams, as their OAM spectrum is highly sensitive to some Zernike terms, and completely insensitive to others. We propose the use of a new basis, the OAM-Zernike basis, which consists of the radial aberrations as described by radial Zernike polynomials and of the azimuthal aberrations described in the OAM basis. The traditional tools for the characterization of aberrations of optical instruments can be used, and the results translated to the new basis. This permits the straightforward calculation of the effect of any optical system, such as an OAM detection stage, on the OAM spectrum of an incoming beam. This knowledge permits to correct, a posteriori, the effect of instrumental OAM spectrum distortion on the measured spectra. We also found that the knowledge of the radial aberrations is important, as they affect the efficiency of the detection, and in some cases its accuracy. In this new framework, we study the effect of aberrations in common OAM detection methods, and encourage the characterization of those systems using this approach.

Orthonormal polynomials in wavefront analysis: error analysis

Zernike circle polynomials are in widespread use for wavefront analysis because of their orthogonality over a circular pupil and their representation of balanced classical aberrations. However, they are not appropriate for noncircular pupils, such as annular, hexagonal, elliptical, rectangular, and square pupils, due to their lack of orthogonality over such pupils. We emphasize the use of orthonormal polynomials for such pupils, but we show how to obtain the Zernike coefficients correctly. We illustrate that the wavefront fitting with a set of orthonormal polynomials is identical to the fitting with a corresponding set of Zernike polynomials. This is a consequence of the fact that each orthonormal polynomial is a linear combination of the Zernike polynomials. However, since the Zernike polynomials do not represent balanced aberrations for a noncircular pupil, the Zernike coefficients lack the physical significance that the orthonormal coefficients provide. We also analyze the error that arises if Zernike polynomials are used for noncircular pupils by treating them as circular pupils and illustrate it with numerical examples.

Comparison of annular wavefront interpretation with Zernike circle polynomials and annular polynomials

A general wavefront fitting procedure with Zernike annular polynomials for circular and annular pupils is proposed. For interferometric data of typical annular wavefronts with smaller and larger obscuration ratios, the results fitted with Zernike annular polynomials are compared with those of Zernike circle polynomials. Data are provided demonstrating that the annular wavefront expressed with Zernike annular polynomials is more accurate and meaningful for the decomposition of aberrations, the calculation of Seidel aberrations, and the removal of misalignments in interferometry. The primary limitations of current interferogram reduction software with Zernike circle polynomials in analyzing wavefronts of annular pupils are further illustrated, and some reasonable explanations are provided. It is suggested that the use of orthogonal basis functions on the pupils of the wavefronts analyzed is more appropriate.

Full-aperture wavefront reconstruction from annular subaperture interferometric data by use of Zernike annular polynomials and a matrix method for testing large aspheric surfaces

We propose a more accurate and efficient reconstruction method used in testing large aspheric surfaces with annular subaperture interferometry. By the introduction of the Zernike annular polynomials that are orthogonal over the annular region, the method proposed here eliminates the coupling problem in the earlier reconstruction algorithm based on Zernike circle polynomials. Because of the complexity of recurrence definition of Zernike annular polynomials, a general symbol representation of that in a computing program is established. The program implementation for the method is provided in detail. The performance of the reconstruction algorithm is evaluated in some pertinent cases, such as different random noise levels, different subaperture configurations, and misalignments.

Gram-Schmidt orthogonalization of the Zernike polynomials on apertures of arbitrary shape

An orthonormal hexagonal Zernike basis set is generated from circular Zernike polynomials apodized by a hexagonal mask by use of the Gram-Schmidt orthogonalization technique. Results for the first 15 hexagonal Zernike polynomials are shown. The Gram-Schmidt orthogonalization technique presented can be extended to both apertures of arbitrary shape and other basis functions.

Zernike polynomials as a basis for wave-front fitting in lateral shearing interferometry

A new method for handling Zernike polynomials is presented. Owing to its efficiency, this method enables the use of Zernike polynomials as a basis for wave-front fitting in shearography systems. An excerpt of a C(++) class is presented to show how the polynomials are calculated and represented in computer memory.

Pupil exploration and wave-front-polynomial fitting of optical systems

Pupil exploration and wave-front-polynomial fitting algorithms are tools that are often employed in image-quality evaluation techniques, such as optical-transfer-function and point-spread-function calculations. These techniques require that aberration data be determined for a large number of points across the pupil. With optical systems increasing in complexity, it is necessary that these algorithms become more sophisticated to ensure that the proper pupil shapes and aberration maps are used to represent the wave fronts. Such algorithms are described. These algorithms can handle systems that not only lack the symmetry found with the more conventional lens systems but those that also have apertures with unusual shapes. As practical demonstrations the treatments employed in the pupil exploration and the wave-front-polynomial fitting have been applied to various lens arrangements and the results discussed.

Visualization of surface figure by the use of Zernike polynomials

Commercial software in modern interferometers used in optical testing frequently fit the wave-front or surface-figure error to Zernike polynomials; typically 37 coefficients are provided. We provide visual representations of these data in a form that may help optical fabricators decide how to improve their process or how to optimize system assembly.