Annular Polynomials Research Articles

Method of wavefront phase retrieval from wavefront curvature sensing using membrane modes.

Wavefront phase retrieval is one of the most critical problems in adaptive optics. Here, phase retrieval by solving the transport of intensity equation using membrane vibration modes is proposed. Our study shows that the wavefront curvature sensing signal on the pupil can be expanded as a set of corresponding membrane vibration modes. The analytic expressions of the reconstructed phase are given. The coefficients of the functions are obtained by the integral over the pupil and boundary. Several representative Zernike circular and annular polynomials are respectively fitted by eigenfunctions and membrane modes in the absence of noise. In addition, wavefront recovery from noisy curvature data of the simulated atmospheric turbulence phase based on Zernike modes and Kolmogorov spectrum is demonstrated to verify the accuracy and robustness of the proposed method.

Vector functions for direct analysis of annular wavefront slope data

In the aberration analysis of a wavefront over a certain domain, the polynomials that are orthogonal over and represent balanced aberrations for this domain are used. For example, Zernike circle polynomials are used for analysis of a circular wavefront. When the data available for analysis are the slopes of a wavefront, the polynomials used are those that are orthogonal to the gradient of Zernike polynomials, and are irrotational. Similarly, the annular polynomials are used to analyze the annular wavefronts for systems with annular pupils, as in a rotationally symmetric two-mirror system. In this paper we derive the vector functions that are orthogonal to the gradients of annular polynomials and are irrotational so that they propagate minimum noise from the slope data to the annular aberration coefficients. These vector functions can be used directly to obtain independent annular aberration coefficients as their inner products with the slope data of annular wavefronts. The utility of the functions is demonstrated in a numerical simulation of noisy wavefront slope data to determine the annular coefficients.

A high-resolution wavefront sensing method to investigate the annular Zernike polynomials behaviour in the indoor convective air turbulence in the presence of a 2D temperature gradient

We report on applications of a two-channel moiré deflectometry wavefront sensor in measurements of wavefront aberrations caused by convective air turbulence generated by a 2D temperature gradient (TG). Data of the wavefront sensor are analysed and the aberrated wavefront is reconstructed and expanded versus Zernike annular polynomials. The results show that variances of the Zernike annular coefficients (except the piston aberration) increase with highest temperature relative to ambient temperature. In particular, we found that the variances of X- and Y -tilt aberration coefficients are not equal when the highest temperature is larger than C. This means that the produced turbulence is not isotropic. Plots of the Zernike annular coefficient variances could be divided into three distinct regions with different slopes.

Modal wavefront reconstruction to obtain Zernike coefficient with no cross coupling in lateral shearing measurement

Modal cross coupling frequently occurs in modal approaches from wavefront gradient data such as lateral shearing measurement through Zernike circle polynomials, since the gradients of Zernike circle polynomials are not orthogonal. We use a modal approaches incorporating the Gram matrix, using the orthogonality of angular derivative of m≠0 modes with respect to weight function w(ρ) = ρ (polar coordinates), and the orthogonality of radial derivative of m = 0 modes with respect to weight function w(ρ) = ρ(1-ρ2) (polar coordinates). The Gram matrix method needs no auxiliary vector functions. The Zernike coefficients can be obtained with no modal cross coupling. The simulation results are given, which indicate that the modal cross coupling is avoided by using Gram matrix method. This method can be easily extended to annulus, and the coefficients of Zernike annular polynomials with no modal cross coupling can be obtained.

Engineering a coaxial visible/infrared imaging system based on monolithic multisurface optics.

All-reflective coaxial visible/infrared imaging systems based on monolithic multisurface optics have been a hot point of research in recent years. Since multiple surfaces share a single substrate, their relative positions are fundamentally guaranteed as fabricated without any alignment. In this paper, the coaxial system is designed with multifolded ideas. Both the visible subsystem and the infrared subsystem are comprised of two monolithic optical modules, which are machined by single-point diamond turning (SPDT). A novel method based on a computer-generated hologram (CGH) is then proposed to simultaneously measure the shape and position of monolithic multisurface optics. The effects of surface shape and position error on the wavefront aberration of the system are also discussed with the help of Zernike annular polynomials. Then the wavefront aberration of the system is measured, from which we subtract the contribution of surface shape and position error. The aberration induced by misalignment of the two monolithic modules is then estimated. It indicates that the concentricity is about 3μm. Finally, two similar systems with different clear apertures are assembled as a coaxial visible/infrared imaging system. Coaxial visible and infrared images are captured and fused to show clearer details.

Jacobi circle and annular polynomials: modal wavefront reconstruction from wavefront gradient.

Jacobi circle polynomials, which are orthogonal on the unit circle with orthogonal radial derivatives, have been developed previously. As the classical Zernike mode can be represented as a linear combination of Jacobi modes, Zernike wavefront modes can be reconstructed using Jacobi modes. Comparison of the Jacobi and Zernike modes for the modal approach indicates that a modal approach incorporating the Gram matrix with the Jacobi modes has potential application in high-sampling wavefront gradient sensors. The Gram matrix method using the Jacobi modes can be extended to annular pupils.

Aberration influence and compensation on beam mode features for asymmetric folded laser resonators

We demonstrate the influence on output coupling mode with introducing typical intracavity disturbation and results of aberrated mode compensation in a folded-type optical resonator used in high-energy lasers. The eigenmode profiles and relevant Zernike aberration with intracavity reflector misalignment are achieved by numerical calculation. Experimental result about the function of the reflector maladjustment and the mode features are described in terms of model reconstruction combined with Zernike annular polynomials. Results show that phase disturbation has significant influence on out-coupling beam property, and the uniform and symmetry of the mode are rapidly disrupted by a slight misalignment of the resonator reflectors. The influence on PSF profile in the far-field will be obviously different even if equivalent disturbation is inputted into the same cavity. The device of real-time controlling of the cavity reflector for compensating intracavity phase distortion is successfully constructed. In addition, Zernike defocus aberration is effectively controlled by precisely adjusting the resonator length. A high beam quality in the output coupling mode is realized, which is in good agreement with the analytical predictions.

Further investigations of stitching model for annular subaperture interferometric testing based on Zernike annular polynomials

The subaperture stitching algorithm as a key step of annular subaperture interferometric testing (ASIT) is used to sew all subaperture data together to get the full aperture information. The core part of stitching algorithm is that using the stitching model to eliminate the relative misalignment errors between the adjacent subapertures to realize the stitching treatment. Obviously, the stitching accuracy is mainly determined by the performance of the stitching model. Different stitching model for existing annular stitching algorithms are briefly discussed in this paper. Based on Zernike annular polynomials (ZAP), a more accurate and stabile stitching model for ASIT is proposed. By combining standard deviation and correlation coefficient of overlapping area, we can comprehensively evaluate the stitching performance. In addition, the fitting performance of the ZAP is evaluated in some pertinent cases: different obscuration ratios and noises. By comparing the results of proposed method with that of the classical auto-collimation method, it demonstrates that the accuracy and practicability of the proposed stitching model.

Imaging characteristics of Zernike and annular polynomial aberrations

The general equations for the point-spread function (PSF) and optical transfer function (OTF) are given for any pupil shape, and they are applied to optical imaging systems with circular and annular pupils. The symmetry properties of the PSF, the real and imaginary parts of the OTF, and the modulation transfer function (MTF) of a system with a circular pupil aberrated by a Zernike circle polynomial aberration are derived. The interferograms and PSFs are illustrated for some typical polynomial aberrations with a sigma value of one wave, and 3D PSFs and MTFs are shown for 0.1 wave. The Strehl ratio is also calculated for polynomial aberrations with a sigma value of 0.1 wave, and shown to be well estimated from the sigma value. The numerical results are compared with the corresponding results in the literature. Because of the same angular dependence of the corresponding annular and circle polynomial aberrations, the symmetry properties of systems with annular pupils aberrated by an annular polynomial aberration are the same as those for a circular pupil aberrated by a corresponding circle polynomial aberration. They are also illustrated with numerical examples.

Systematic comparison of the use of annular and Zernike circle polynomials for annular wavefronts

The theory of wavefront analysis of a noncircular wavefront is given and applied for a systematic comparison of the use of annular and Zernike circle polynomials for the analysis of an annular wavefront. It is shown that, unlike the annular coefficients, the circle coefficients generally change as the number of polynomials used in the expansion changes. Although the wavefront fit with a certain number of circle polynomials is identically the same as that with the corresponding annular polynomials, the piston circle coefficient does not represent the mean value of the aberration function, and the sum of the squares of the other coefficients does not yield its variance. The interferometer setting errors of tip, tilt, and defocus from a four-circle-polynomial expansion are the same as those from the annular-polynomial expansion. However, if these errors are obtained from, say, an 11-circle-polynomial expansion, and are removed from the aberration function, wrong polishing will result by zeroing out the residual aberration function. If the common practice of defining the center of an interferogram and drawing a circle around it is followed, then the circle coefficients of a noncircular interferogram do not yield a correct representation of the aberration function. Moreover, in this case, some of the higher-order coefficients of aberrations that are nonexistent in the aberration function are also nonzero. Finally, the circle coefficients, however obtained, do not represent coefficients of the balanced aberrations for an annular pupil. The various results are illustrated analytically and numerically by considering an annular Seidel aberration function.

Experimental study on measurement of aspheric surface shape with complementary annular subaperture interferometric method

Based on our previously reported annular subaperture reconstruction algorithm with Zernike annular polynomials and matrix method, we provide an experimental demonstration by testing a parabolic mirror with the complementary annular subaperture interferometric method. By comparing the results of annular subaperture method with that of the classical auto-collimation method, it is shown that the reconstruction results are in good agreement with the auto-collimation measurement. In addition, we discuss some limitations of characterizing annular subaperture measurement data with finite Zernike coefficients in our algorithm, and also show the possibility of characterizing higher spatial frequency information with adequate Zernike coefficients. It is believable that the reported method can be extended to test the surface shape of some large concave aspheric mirrors with acceptable accuracy.

Zernike annular polynomials and atmospheric turbulence

Imaging through atmospheric turbulence by systems with annular pupils is discussed using the Zernike annular polynomials. Fourier transforms of these polynomials are derived analytically to facilitate the calculation of variance and covariance of the aberration coefficients. Zernike annular shape functions are derived and used to calculate the Strehl ratio and the residual phase structure and mutual coherence functions when a certain number of modes are corrected using, say, a deformable mirror. Special cases of long- and short-exposure images are also considered. The results for systems with a circular pupil are obtained as a special case of the annular pupil.

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.

Analysis of the Naval Observatory Flagstaff Station 1-m telescope using annular Zernike polynomials

The Naval Observatory Flagstaff Station 1-m telescope is evaluated for the addition of adaptive optics capabilities to its instrumentation suite. Zernike decomposition of the optical system based on phase diversity measurements shows that the static optical aberrations are small enough that they will not degrade the performance of the deformable optical element. The analysis makes use of annular pupil Zernike polynomial reconstruction of the wavefront to accommodate the large obscuration in this telescope and compares this with the results from using filled circular Zernike polynomials.

Zernike-Gauss Polynomials and Optical Aberrations of Systems with Gaussian Pupils

In the first two Notes of this series,(l,2) we discussed Zernike circle and annular polynomials that represent optimally balanced classical aberrations of systems with uniform circular or annular pupils, respectively. Here we discuss Zernike-Gauss polynomials which are the corresponding polynomials for systems with Gaussian circular or annular pupils.(3-5) Such pupils, called apodized pupils, are used in optical imaging to reduce the secondary rings of the pointspread functions of uniform pupils.(6) Propagation of Gaussian laser beams also involves such pupils.

Zernike Annular Polynomials and Optical Aberrations of Systems with Annular Pupils

Zernike annular polynomials that represent orthogonal andbalanced aberrations suitable for systems with annular pupilsare described. Their numbering scheme is the same asfor Zernike circle polynomials. Expressions for standard deviationof primary and balanced primary aberrations are given.

Line of sight of an aberrated optical system

The line of sight of an aberrated optical system is defined in terms of the centroid of its point-spread function (PSF). It is also expressed in terms of its optical transfer function as well as its pupil function. Whereas the PSF’s obtained according to wave-diffraction optics and ray geometrical optics are quite different from each other, they have the same centroid. Although different amplitude distributions across an aberration-free pupil give the same centroid location, in the case of an aberrated pupil not only the phase but also the amplitude distribution affects the centroid location. If the amplitude across a pupil is uniform, then the centroid may be obtained from the aberration only along the perimeter of the pupil, without regard for the aberration across its interior irrespective of its shape. Next, an optical system with aberrated but uniformly illuminated annular pupil is considered. The aberration function is expanded in terms of Zernike annular polynomials. It is shown that only those aberrations that vary with angle as cos θ or sin θ contribute to the line of sight. A simple expression is obtained for the line of sight in terms of the Zernike aberration coefficients. Similar results are obtained for annular pupils with radially symmetric illumination. Finally, as numerical examples, some specific results are discussed for annular pupils aberrated by classical primary and secondary coma. As an example of a radially symmetric illumination, we obtain numerical results for Gaussian illumination of aberrated annular pupils. It is emphasized that the centroid and the peak of the PSF’s aberrated by coma are not coincident. Moreover, as the amount of the aberration increases, the separation of the centroid and peak locations also increases.

Zernike annular polynomials for imaging systems with annular pupils

Based on our earlier work on Zernike annular polynomials,1 expressions for some higher-order radial polynomials are given, thus providing a complete set up to the sixth order.

Zernike annular polynomials for imaging systems with annular pupils: errata

Zernike annular polynomials for imaging systems with annular pupils: errata