Orthonormal aberration polynomials for anamorphic optical imaging systems with circular pupils

In a recent paper, we considered the classical aberrations of an anamorphic optical imaging system with a rectangular pupil, representing the terms of a power series expansion of its aberration function. These aberrations are inherently separable in the Cartesian coordinates (x,y) of a point on the pupil. Accordingly, there is x-defocus and x-coma, y-defocus and y-coma, and so on. We showed that the aberration polynomials orthonormal over the pupil and representing balanced aberrations for such a system are represented by the products of two Legendre polynomials, one for each of the two Cartesian coordinates of the pupil point; for example, L(l)(x)L(m)(y), where l and m are positive integers (including zero) and L(l)(x), for example, represents an orthonormal Legendre polynomial of degree l in x. The compound two-dimensional (2D) Legendre polynomials, like the classical aberrations, are thus also inherently separable in the Cartesian coordinates of the pupil point. Moreover, for every orthonormal polynomial L(l)(x)L(m)(y), there is a corresponding orthonormal polynomial L(l)(y)L(m)(x) obtained by interchanging x and y. These polynomials are different from the corresponding orthogonal polynomials for a system with rotational symmetry but a rectangular pupil. In this paper, we show that the orthonormal aberration polynomials for an anamorphic system with a circular pupil, obtained by the Gram-Schmidt orthogonalization of the 2D Legendre polynomials, are not separable in the two coordinates. Moreover, for a given polynomial in x and y, there is no corresponding polynomial obtained by interchanging x and y. For example, there are polynomials representing x-defocus, balanced x-coma, and balanced x-spherical aberration, but no corresponding y-aberration polynomials. The missing y-aberration terms are contained in other polynomials. We emphasize that the Zernike circle polynomials, although orthogonal over a circular pupil, are not suitable for an anamorphic system as they do not represent balanced aberrations for such a system.     

Aberrations of anamorphic optical systems III: the primary aberration theory for toroidal anamorphic systems

We apply a new method for optical aberration derivation to anamorphic systems made from toroidal surfaces and obtain a complete set of primary aberration coefficients. This set is written in a form similar to the well-known Seidel aberrations for rotationally symmetrical optical systems and includes first-order quantities only, thus it can be easily applied to anamorphic lens design practice. By tracing four nonskew paraxial marginal and chief rays, the 16 anamorphic primary aberration coefficients can be easily calculated.     

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.     

Diffraction analysis of rotationally symmetric optical systems using computer-generated aberration polynomials

A ray trace scheme for the automatic generation of optical aberration polynomials to arbitrary orders pioneered by T. B. Andersen is successfully applied to the diffraction analysis of rotationally symmetric optical systems on a desk-top computer. The diffraction-based optical transfer functions at various field positions are computed using the relatively new Winograd Fourier transform algorithm. The coding includes aspherical and reflective surfaces.     

Orthogonal aberration functions for high-aperture optical systems

Aberration functions that are a complete, orthogonal, and normalized set over a weighted spherical pupil are developed. A general weighting is considered, for which special cases are applicable to systems satisfying the Abbe sine condition and the Herschel condition. Paraboloidal mirrors are also considered. This weighting can also be used to account empirically for Fresnel reflection losses in the optical system. The functions can be expressed in an analytic form. Expressions are given for 24 low-order aberrations.     

Multichannel sampling schemes for optical imaging systems

We introduce a framework of focal-plane coding schemes for multichannel sampling in optical systems. A particular objective is to develop an ultrathin imager without compromising image resolution. We present a complete f/2.1 optical system with a thickness of 2.2 mm. The resolution is maintained in the thin optical system by an integrated design of the encoding scheme, the process of making the coding elements, and the decoding algorithms.     

Lensless anamorphic Fourier transform hologram recorded with prism systems

A novel configuration for recording a lensless anamorphic Fourier transform hologram of a given object's light distribution is proposed. The method is based on the use of prism anamorphic optical systems coupled with phase cancellation at the hologram plane. Anamorphic systems with cylindrical lenses and prisms are critically evaluated through computer simulations for their suitability in anamorphic Fourier transform holographic configurations. A complete theoretical analysis and experimental demonstration of the recording and reconstruction of a lensless anamorphic Fourier transform hologram are presented.     

Generalized confocal imaging systems for free-space optical interconnections

A generalized confocal imaging system, which is composed of two confocal lenses and one field lens, is proposed for free-space optical interconnections. Unlike in a conventional 4-f system, both the object distance and the image distance can be almost arbitrarily chosen. This advantage is especially important for practical setups in which the object distance and the image distance cannot be designed to be the same. As a concrete example, we have designed and experimentally tested a planar-integrated micro-optical imaging system. The result is in good agreement with the theoretical prediction. Similarly to the conventional 4-f imaging system and the light-pipe imaging system, the system proposed here can also be used as one important part of a hybrid imaging setup.     

Determining the aberration characteristics of optical systems containing acousto-optical diffraction elements

Technical Physics Letters - We describe a method and present an example of the calculation of aberration images that arise in optical systems (OSs) containing acousto-optical (AO) diffraction...     

Two-dimensional orthogonal polynomials for vibration of rectangular composite plates

The two-dimensional orthogonal plate function generated by the Gram-Schmidt orthogonalization process in the Rayleigh-Ritz procedure has been used to study the vibration of composite plates. The effect of fibre orientation on the vibration behaviour of rectangular single-layer, parallel-fibre plates with various combinations of boundary conditions is studied. Some numerical results are presented and compared with other numerical methods in the literature. The present study will show how boundary conditions, aspect ratio and fibre orientation affect the natural frequencies of free-vibration plates using the computationally efficient and accurate orthogonalization process.     

Compositional synthesis of optical imaging systems

This paper introduces a new systematic and complete compositional technique for conceptual design of optical imaging systems from behavioral specifications. There are two key ideas: (1) modeling the structure and behavior of optical components using affine transformations, and (2) using a systematic search algorithm integrated with a powerful algebraic constraint solver to compute three-dimensional layouts of optical components in the design. By embedding our tool in a constraint programming environment, we are able to automate the conceptual design of optical imaging systems – a step performed manually by human designers until now. We demonstrate the power of the method by recreating several designs for imaging systems of copiers. Electronic Publication     

Iterative zonal wave-front estimation algorithm for optical testing with general-shaped pupils

An iterative zonal wave-front estimation algorithm for slope or gradient-type data in optical testing acquired with regular or irregular pupil shapes is presented. In the mathematical model proposed, the optical surface, or wave-front shape estimation, which may have any pupil shape or size, shares a predefined wave-front estimation matrix that we establish. Owing to the finite pupil of the instrument, the challenge of wave front shape estimation in optical testing lies in large part in how to properly handle boundary conditions. The solution we propose is an efficient iterative process based on Gerchberg-type iterations. The proposed method is validated with data collected from a 15 x 15-grid Shack-Hartmann sensor built at the Nanjing Astronomical Instruments Research Center in China. Results show that the rms deviation error of the estimated wave front from the original wave front is less than lambda/130-lambda/150 after approximately 12 iterations and less than lambda/100 (both for lambda = 632.8 nm) after as few as four iterations. Also, a theoretical analysis of algorithm complexity and error propagation is presented.     

Extended ABCD matrix formalism for the description of femtosecond diffraction patterns; application to femtosecond digital in-line holography with anamorphic optical systems

We present a new model to predict diffraction patterns of femtosecond pulses through complex optical systems. The model is based on the extension of an ABCD matrix formalism combined with generalized Huygens-Fresnel transforms (already used in the CW regime) to the femtosecond regime. The model is tested to describe femtosecond digital in-line holography experiments realized in situ through a cylindrical Plexiglas pipe. The model allows us to establish analytical relations that link the holographic reconstruction process to the experimental parameters of the pipe and of the incident beam itself. Simulations and experimental results are in good concordance. Femtosecond digital in-line holography is shown to allow significant coherent noise reduction, and this model will be particularly efficient to describe a wide range of optical geometries. More generally, the model developed can be easily used in any experiment where the knowledge of the precise evolution of femtosecond transverse patterns is required.     

Eigenmode super-resolution imaging in arbitrary optical systems

We investigate optical super-resolution by means of eigenmode decomposition in arbitrary imaging systems. This technique is applicable for arbitrary objects but requires a knowledge of the eigenmodes of the imaging system. We outline a reconstruction technique that can be applied even to systems in which the eigenmodes are not orthogonal, and we present numerical simulations of eigenmode super-resolution in systems with resolution limited both by diffraction and by aberrations. Our results indicate that optical super-resolution by direct eigenmode decomposition provides a versatile method of sub-diffraction and distortion-free imaging in arbitrary optical systems.     

A phase aberration correction method for ultrasound imaging

A computationally efficient method for phase aberration correction in ultrasound imaging is presented. The method is based on time delay estimation via minimization of the sum of absolute differences between radio frequency samples of adjacent array elements. Effects of averaging estimated aberration patterns over scan angle and truncation to a single bit wordlength are examined. Phase distortions due to near-field inhomogeneities are simulated using silicone rubber aberrators. Performance of the method is tested using experimental data. Simulation studies addressing different factors affecting efficiency of the method, such as the number of iterations, window length, and the number of scan angles used for averaging, are presented. Images of a standard resolution phantom are reconstructed and used for qualitative testing.