Zernike Circle Polynomials Research Articles

Phase estimation for Zernike circle polynomials using Phase Locked Loops for investigations of camera aberrations

This paper proposes a method to investigate aberrations of cameras from phase recovered fringe patterns using first-order Phase Locked Loop (PLL). Images of fringe patterns of different spatial periods are captured using different cameras. First-order PLL is applied to recover phase information from the captured images. Zernike circle polynomials are fitted to the phase recovered and Zernike coefficients are computed using the Gaussian elimination method. Comparison of cameras in terms of aberrations is done using these Zernike coefficients. Phase is also recovered and fitted for fringe patterns captured using the same camera with different lens combinations.

Wavefront fitting and comparison of camera aberrations using Zernike circle polynomials

Zernike polynomials are in widespread use for wavefront analysis because of their representation of classical aberrations. This paper presents a method to investigate wavefront aberrations which does not require complicated and expensive wavefront sensors. Images of checkerboard pattern are captured using mobile phone cameras, CCTV camera, Firewire camera, Handy cam and Stereo camera and wavefront is constructed using displacement in the grid corners of checkerboard patterns. This wavefront is fitted into Zernike circle polynomials and Zernike coefficients are computed using Gaussian elimination method. The computed Zernike polynomials and Zernike coefficients are further utilized to compare the aberrations of different cameras. The experiment and simulations confirm that displacement in the grid corners of the checkerboard pattern can be treated as a wavefront and fitted to Zernike polynomials. Stereo camera is also investigated for both raw and rectified set of images.

Sound Radiation from a Loudspeaker, from a Spherical Pole Cap, and from a Piston in an Infinite Baffle

Loudspeakers are often modelled as a rigid piston in an infinite baffle. As a model for real loudspeakers, this approach is limited in two ways. One issue is that a loudspeaker cone is not rigid, and a second issue is that a loudspeaker is mostly used in a cabinet. Both issues are addressed here by developing the velocity of the radiator in terms of orthogonal polynomials known from optical diffraction theory as Zernike circle polynomials. Using these polynomials we develop semi-analytic expressions for the sound pressure from the radiator in two different cases: as a flexible flat radiator mounted in an infinite baffle, and as the cap of a rigid sphere. In the latter case the comparison is done not only for the pressure but also for other quantities viz. the baffle-step response, sound power and directivity, and the acoustic centre of the radiator. These quantities are compared with those from a real loudspeaker. Finally, in the case of the baffled-piston radiation the spatial impulse response is presented.

New analytic results for the Zernike circle polynomials from a basic result in the Nijboer-Zernike diffraction theory

Several quantities related to the Zernike circle polynomials admit an expression, via the basic identity in the diffraction theory of Nijboer and Zernike, as an infinite integral involving the product of two or three Bessel functions. In this paper these integrals are identified and evaluated explicitly for the cases of (a) the expansion coefficients of scaled-and-shifted circle polynomials, (b) the expansion coefficients of the correlation of two circle polynomials, (c) the Fourier coefficients occurring in the cosine representation of the radial part of the circle polynomials.

Computation of Hopkins’ 3-circle integrals using Zernike expansions

The integrals occurring in optical diffraction theory under conditions of partial coherence have the form of an incomplete autocorrelation integral of the pupil function of the optical system. The incompleteness is embodied by a spatial coherence function of limited extent. In the case of circular optical systems and coherence functions supported by a disk, this gives rise to Hopkins’ 3-circle integrals. In this paper, a computation scheme for these integrals (initially with coherence functions that are constant on their disks) is proposed where the required integral is expressed semi-analytically in the Zernike expansion coefficients of the pupil function. To this end, the Zernike expansion coefficients of a shifted pupil function restricted to the coherence disk are expressed in terms of the pupil function’s Zernike expansion coefficients. Next, the required integral is expressed as an infinite series involving two sets of Zernike coefficients of restricted pupils using Parseval’s theorem for orthogonal series. Due to a convenient separation of the radial parameters and the spatial variables, the method avoids a cumbersome administration involving separate consideration of various overlap situations. The computation method is extended to the case of coherence functions that are not necessarily constant on their supporting disks by using a result on linearization of the product of two Zernike circle polynomials involving Wigner coefficients.

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.

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.

Orthonormal polynomials in wavefront analysis: analytical solution

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. In recent papers, we derived closed-form polynomials that are orthonormal over a hexagonal pupil, such as the hexagonal segments of a large mirror. We extend our work to elliptical, rectangular, and square pupils. Using the circle polynomials as the basis functions for their orthogonalization over such pupils, we derive closed-form polynomials that are orthonormal over them. These polynomials are unique in that they are not only orthogonal across such pupils, but also represent balanced classical aberrations, just as the Zernike circle polynomials are unique in these respects for circular pupils. The polynomials are given in terms of the circle polynomials as well as in polar and Cartesian coordinates. Relationships between the orthonormal coefficients and the corresponding Zernike coefficients for a given pupil are also obtained. The orthonormal polynomials for a one-dimensional slit pupil are obtained as a limiting case of a rectangular 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.

Orthonormal polynomials for hexagonal pupils

The problem of determining the orthonormal polynomials for hexagonal pupils by the Gram-Schmidt orthogonalization of Zernike circle polynomials is revisited, and closed-form expressions for the hexagonal polynomials are given. We show how the orthonormal coefficients are related to the corresponding Zernike coefficients for a hexagonal pupil and emphasize that it is the former that should be used for any quantitative wavefront analysis for such a pupil.

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.

Testing of refractive silicon microlenses by use of a lateral shearing interferometer in transmission

Wave aberrations of refractive photoresist microlenses and silicon microlenses were measured with a lateral shearing interferometer. Because of the silicon elements, a near-infrared working wavelength (lambda = 1.32 mum) was used. The wave front was evaluated by a phase step technique with four steps. Integration of the phase distributions was performed with a computationally efficient Fourier transformation algorithm. The influence of the detector vidicon nonlinearity on the measured wave front was calculated. The defocusing behavior of the interferometer was investigated by fitting the measured wave fronts with the help of Zernike circle polynomials. It is shown that the reproducibility can be kept below lambda/100 rms. Examples for the measured wave fronts of plano-convex silicon microlenses are discussed in detail.

Optical consequences of asymmetries in normal corneas

A survey of videokeratographs of normal corneas shows many with substantial peripheral asymmetries. For sufficiently large pupils (5.5 mm in this study) these asymmetries lead to coma-like axial aberrations large enough to produce measurable losses in vision in a number of cases. Starting from the output of a videokeratographic instrument, a method of estimating the optical effects of corneal asymmetries using Zernike circle polynomials is outlined. It is further shown that in a first approximation corneal asymmetries can be identified with the primary aberration coma and that this aberration is approximately due to a uniform gradient of refractive power across the cornea. Calculations for a representative case predict that a significant improvement in modulation transfer would follow from correction of this aberration.

Optical consequences of asymmetries in normal corneas

SummaryA survey of videokeratographs of normal corneas shows many with substantial peripheral asymmetries. For sufficiently large pupils (5.5 mm in this study) these asymmetries lead to coma‐like axial aberrations large enough to produce measurable losses in vision in a number of cases. Starting from the output of a videokeratographic instrument, a method of estimating the optical effects of corneal asymmetries using Zernike circle polynomials is outlined. It is further shown that in a first approximation corneal asymmetries can be identified with the primary aberration coma and that this aberration is approximately due to a uniform gradient of refractive power across the cornea. Calculations for a representative case predict that a significant improvement in modulation transfer would follow from correction of this aberration.

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 Circle Polynomials and Optical Aberrations of Systems with Circular Pupils

Zernike circle polynomials, their numbering scheme, and relationship to balanced optical aberrations of systems with circular pupils are discussed.

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.

Development of higher-order Zernike polynomials employed in the analysis of acoustic sensing antennas

Zernike circle polynomials have been found to be invaluable analytical tools for investigating the characteristics of acoustic sensing antennas [S. A. Adekola, Acustica 47, 114–131 (1981)]. New analytical results obtained from an investigation of the generalized Rnm(r) Zernike circle polynomials, summarized in the above-mentioned reference, are of higher degrees of accuracy than those presently available in the literature. Indeed, these polynomials are generated to 30° and 29° of freedom describing the even- and odd-numbered modes, respectively; and the field radiated by the acoustic sensing antenna, which is then expressed in terms of the polynomials, is evaluated through direct routine integration. Some of the new analytical results evaluated and numerically computed are here displayed in graphical forms for completeness. Also, illustrative examples given show not only how the Zernike polynomials can be employed in the interpretations of the radiation characteristics of the acoustic echosounding systems, but also how use can be made of the polynomials to describe significantly tapered source distributions essential in successful acoustic remote sensing of the lower atmosphere. [Work supported by Lagos State University, Badagry Expressway, Ojo, Lagos, through a Visiting Professorial Appointment.]

The Structure of Fraunhofer Diffraction Patterns of Apertures with N-fold Rotational Symmetry

The moment theorem is used to show that the innermost part of the Fraunhofer diffraction pattern of any real aperture with higher than two-fold rotational symmetry is rotationally invariant. Then a formalism is presented in which aperture transmission-functions are represented by series of Zernike circle polynomials and diffracted field-amplitudes by series of Bessel functions, from which it is easily shown that the diffraction patterns of such apertures consist of regions, contained between well-defined values of the radius, whose rotational symmetries are integral multiples of that of the aperture. The central region, extending from = 0 to , N ( measures the diffraction angle, and N is the degree of rotational symmetry of the aperture) is rotationally invariant, and successive circumjacent regions have progressively higher rotational symmetries. The diffraction patterns of sectoral apertures and of rings of pinholes are derived and shown to exemplify these general conclusions. Finally it is shown how the diffraction patterns of some apertures (‘chiral apertures’) with rotational symmetries but no mirror symmetry can be deduced from the diffraction pattern of a related aperture with mirror symmetries, to which a chiral perturbation is applied.

Zernike annular polynomials for imaging systems with annular pupils

The aberrations of imaging systems with uniformly illuminated annular pupils are discussed in terms of a complete set of polynomials that are orthogonal over an annular region. These polynomials, which we call Zernike annular polynomials, are similar to the Zernike circle polynomials and reduce to them as the annulus approaches the full circle. The Zernike-polynomial expansion of an aberration function is compared with its power-series expansion. The orthogonal aberrations given by Zernike annular polynomials describe how a higher-order classical aberration of a power-series expansion is balanced with one or more lower-order classical aberrations to minimize its variance. It is shown that as the obscuration ratio increases, the standard deviation of an orthogonal as well as a classical primary aberration decreases in the case of spherical aberration and field curvature and increases in the case of coma, astigmatism, and distortion. The only exception is the case of orthogonal coma, for which the standard deviation first increases and then decreases. The orthogonal aberrations for nonuniformly illuminated annular pupils are also considered, and, as an example, Gaussian illumination is discussed. It is shown that the standard deviation of an orthogonal primary aberration for a given amount of the corresponding classical aberration is somewhat smaller for a Gaussian pupil than that for a uniformly illuminated pupil.