Shack-Hartmann sensor for fast infrared wave-front testing

Abstract

A Shack-Hartmann sensor has been designed for testing the wave front of CO₂ lasers. Fabrication of a lens array and a detector array with tight tolerances on position accuracy are essential steps. Parallel electronics allow for high-speed wave-front measurements with 1 kHz sampling frequency. The device has been used to investigate the behaviour of a high-power CO₂ laser. Besides the expected thermal drifts of beam direction at the beginning of laser action, periodic changes of beam direction, have been detected. The Shack-Hartmann sensor seems the appropriate device for controlling adaptive optics in high-power laser applications.     

Interferometer for use in wave-front testing with a cross slit

A simple and novel interferometric setup for wave-front testing that uses a cross slit is described. In this method, the test beam illuminates a cross slit placed at the front focal plane of a Fourier lens. It selects two orthogonal slices from the test beam, and the interference of these two beams is observed at a slightly defocused plane near the back focal plane. Fringes of different conical forms (circular, elliptical, or hyperbolic) so obtained can be used for testing a coherent wave front in general. The theory, supported by some experimental results, is presented. An application of the method to the study of the nature of asymmetry in the beam profile of a semiconductor diode laser beam is demonstrated.     

Iterative Fourier transform algorithm with regularization for the optimal design of diffractive optical elements

There is a trade-off between uniformity and diffraction efficiency in the design of diffractive optical elements. It is caused by the inherent ill-posedness of the design problem itself. For the optimal design, the optimum trade-off needs to be obtained. The trade-off between uniformity and diffraction efficiency in the design of diffractive optical elements is theoretically investigated based on the Tikhonov regularization theory. A novel scheme of an iterative Fourier transform algorithm with regularization to obtain the optimum trade-off is proposed.     

Iterative algorithm for the design of free-space diffractive optical elements for fiber coupling

We present a design method based on the Gerchberg-Saxton algorithm for the design of high-performance diffractive optical elements. Results from this algorithm are compared with results from simulated annealing and the iterative Fourier-transform algorithm. The element performance is comparable with those designed by simulated annealing, whereas the design time is similar to the iterative Fourier-transform method. Finally, we present results for a demanding beam-shaping task that was beyond the capabilities of either of the traditional algorithms. The element performances demonstrate greater than 85% efficiency and less than 2% uniformity error.     

Iterative algorithm for magnetostatic problems with saturable media

This paper presents an iterative algorithm for computing a static magnetic field in an unbounded region in which the medium is saturable. An analysis is presented that shows the algorithm converges to the correct answer for all starting conditions.     

Iterative reverse optimization procedure for calibration of aspheric wave-front measurements on a nonnull interferometer

The accuracy of interferometric measurements made with a nonnull configuration is degraded by test-dependent aberrations. Calibration of the data can be done with reverse optimization methods. An iterative reverse optimization process that improves efficiency and eliminates sensitive merit function weighting issues is described. The process is shown to calibrate a nonnull interferometric measurement of a wave front with more than 200 waves of departure to an accuracy of 0.16 waves peak to valley and 0.02 waves rms.     

Method for measuring small optical absorption coefficients with use of a Shack-Hartmann wave-front detector

We present a method for measuring absorption at the 1 x 10(-5) cm(-1) level in high-quality optical materials. Using a Shack-Hartmann wave-front detector, thermal lensing in these materials may be measured. Then, the absorption coefficient may be estimated by fitting the observed deformation to a thermal lensing model based on the temperature dependences of the refractive index and the thermal expansion coefficient. For a particular sample of fused silica, the absorption coefficient was determined to be 1.8 +/- 0.4 x 10(-5) cm(-1). Obtaining this result requires a resolution in the optical path length better than +/- 0.1 nm.     

Orthonormal aberration polynomials for anamorphic optical imaging systems with rectangular pupils

The classical aberrations of an anamorphic optical imaging system, representing the terms of a power-series expansion of its aberration function, are separable in the Cartesian coordinates of a point on its pupil. We discuss the balancing of a classical aberration of a certain order with one or more such aberrations of lower order to minimize its variance across a rectangular pupil of such a system. We show that the balanced aberrations are the products of two Legendre polynomials, one for each of the two Cartesian coordinates of the pupil point. The compound Legendre polynomials are orthogonal across a rectangular pupil and, like the classical aberrations, are inherently separable in the Cartesian coordinates of the pupil point. They are different from the balanced aberrations and the corresponding orthogonal polynomials for a system with rotational symmetry but a rectangular pupil.     

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.     

Iterative algorithm for subaperture stitching test with spherical interferometers

Recently we have proposed an iterative algorithm for subaperture stitching interferometry. It was referred to as the subaperture stitching and localization (SASL) algorithm. The limitation of the algorithm is that three-dimensional Cartesian coordinates are required, whereas the standard spherical interferometer can read out only the phase differences on the pixels. On the basis of the SASL algorithm, we propose an iterative algorithm for a spherical subaperture stitching test. It deals with data directly from the spherical interferometer. Unknown radii of best-fit spheres for a null test of subapertures are included in the optimization variables. The developed algorithm inherits the advantages of the SASL algorithm.     

Iterative estimation of eigenmodes for acoustic cavities

In this paper, a stochastic estimation method of the number of eigenvalues of nonlinear eigenproblem (initially proposed by Maeda et al., JSIAM Letters 3, 61-64, (2011)) is employed iteratively in order to identify eigenvalues of acoustic cavities. Applied to several discretization formulations of the Helmholtz equation, the proposed approach handles complex acoustic cavities. Specific developments are carried out concerning the approximation of the stochastic estimator for the solved kernels. The method's accuracy is illustrated with academic nonlinear eigenproblems with various boundary conditions. In particular, a sphere problem with constant surface impedance is solved and validated by comparison with results issued from a finite element method software.     

Iterative boundary method for diffuse optical tomography

The recent application of tomographic methods to three-dimensional imaging through tissue by use of light often requires modeling of geometrically complex diffuse-nondiffuse boundaries at the tissue-air interface. We have recently investigated analytical methods to model complex boundaries by means of the Kirchhoff approximation. We generalize this approach using an analytical approximation, the N-order diffuse-reflection boundary method, which considers higher orders of interaction between surface elements in an iterative manner. We present the general performance of the method and demonstrate that it can improve the accuracy in modeling complex boundaries compared with the Kirchhoff approximation in the cases of small diffuse volumes or low absorption. Our observations are also contrasted with exact solutions. We furthermore investigate optimal implementation parameters and show that a second-order approximation is appropriate for most in vivo investigations.     

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.     

超声层析成像的迭代重建

利用变形Born迭代方法,建立了超声衍射重建算法。在迭代过程中,为了解决超声逆散射问题中的非线性性,需要反复地求解前向散射方程和逆散射方程,以达到全场和未知函数的近似,较好地重建物体内部的断层图象。由于逆散射方程是一个不适定性的方程组,要用正则化方法处理方程的不适定性问题,使迭代方法收敛于问题的真实解,才能成功地应用于较高对比度物体的图象重建问题。用Picard准则对不适定问题进行了分析,给出了通过简单图形．确定模型受噪声污染情况以及正则化方法适用范围的方法。在重建实验中。对建立的图像重建算法进行了实验仿真。达到了较好的效果。     

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.     

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.     

Cyclic interferometers for optical testing

In this paper, we describe two compact cyclic interferometers which can be used for the testing of right-angle prisms; with some modifications they can also be used for testing convergent wave fronts.     

Iterative algorithm for subaperture stitching interferometry for general surfaces

A novel iterative algorithm for subaperture stitching interferometry for general surfaces is presented. It is based on the alternating optimization technique and the successive linearization method. The computer-aided-design model of the tested surface is used to determine the overlapping region precisely. Subapertures are simultaneously stitched by minimizing deviations among them as well as deviations from the nominal surface. Precise prior knowledge of the six degrees-of-freedom nulling and alignment motion is no longer required.     

Large-scale wave-front reconstruction for adaptive optics systems by use of a recursive filtering algorithm

We propose a new recursive filtering algorithm for wave-front reconstruction in a large-scale adaptive optics system. An embedding step is used in this recursive filtering algorithm to permit fast methods to be used for wave-front reconstruction on an annular aperture. This embedding step can be used alone with a direct residual error updating procedure or used with the preconditioned conjugate-gradient method as a preconditioning step. We derive the Hudgin and Fried filters for spectral-domain filtering, using the eigenvalue decomposition method. Using Monte Carlo simulations, we compare the performance of discrete Fourier transform domain filtering, discrete cosine transform domain filtering, multigrid, and alternative-direction-implicit methods in the embedding step of the recursive filtering algorithm. We also simulate the performance of this recursive filtering in a closed-loop adaptive optics system.     

Characterization and control of a multielement dual-frequency liquid-crystal device for high-speed adaptive optical wave-front correction

Multielement nematic liquid-crystal devices have been used by others and ourselves for closed-loop adaptive control of optical wave-front distortions. Until recently the phase retardance of available devices could be controlled rapidly in only one direction. The phase retardance of the dual-frequency device can be controlled rapidly in both directions. Understanding the dynamics of the phase retardance change is critical to the development of a high-speed control algorithm. We describe measurements and experiments leading to the closed-loop control of a multielement dual-frequency liquid-crystal adaptive optic.