Laser dye stability. Part 2

The Zernike polynomials are orthogonal functions defined on the unit circle, which have been used primarily in the diffraction theory of optical aberrations. A summary of their principal properties is given. It is shown that the polynomials, which are closely related to the general spherical harmonics, are especially useful in numerical calculations. In particular, by using the polynomials as a basis to represent the commonly encountered functions of optical theory, it is often possible to avoid numerical quadrature and computations are reduced to the simple manipulation of expansion coefficients.     

Scaling pseudo-Zernike expansion coefficients to different pupil sizes

Orthogonal polynomials are routinely used to represent complex surfaces over a specified domain. In optics, Zernike polynomials have found wide application in optical testing, wavefront sensing, and aberration theory. This set is orthogonal over the continuous unit circle matching the typical shape of optical components and pupils. A variety of techniques has been developed to scale Zernike expansion coefficients to concentric circular subregions to mimic, for example, stopping down the aperture size of an optical system. Here, similar techniques are used to rescale the expansion coefficients to new pupil sizes for a related orthogonal set: the pseudo-Zernike polynomials.     

Zernike-basis expansion of the fractional and radial Hilbert phase masks

The linear Hilbert phase mask or transform has found applications in image processing and spectroscopy. An optical version of the fractional Hilbert mask is considered here, comprising an imaging system with a circular, unobscured pupil in which a variable phase delay is introduced into one half of the pupil, split bilaterally.The radial Hilbert phase mask is also used in image processing and to produce optical vortices which have applications in optical tweezers and the detection of exoplanets.We subjected the fractional and radial Hilbert phase masks to Zernike function expansion in order to compute the image plane electromagnetic field distribution using Nijboer-Zernike theory. The Zernike functions form an orthogonal basis on the unit circle. The complex-valued Zernike expansion coefficients for these two phase masks were derived for use in the context of the Extended Nijboer-Zernike (ENZ) theory of image formation. The ENZ approach is of interest in that it allows a greater range of defocus to be dealt with, provides a simple means of taking a finite source size into account and has been adapted to high Numerical Aperture (NA) imaging applications.Our image plane results for the fractional Hilbert mask were verified against a numerical model implemented in the commercial optical design and analysis code, Zemax^®. It was found that the Nijboer-Zernike result converged to the Zemax^® result from below as the number of Zernike terms in the expansion was increased.     

Orthogonal Aberration Functions for Microlithographic Optics

Newly developed orthogonal aberration functions are reviewed. The new functions can be utilized to express aberrations of a high NA and wide field optical system like a microlithographic projection lens. The new functions are orthogonal to each other and expressed by a simple combination of Zernike function(s) of pupil coordinates and Zernike function(s) of field coordinates.     

波面重构中非圆域Zernike正交基底构造方法

针对非圆域波面拟合中Zernike多项式失去正交特性、拟合系数交叉耦合的问题,提出非圆域Zernike正交基底函数构造方法。以圆Zernike为基底,采用Gram-Schimdt正交组构造方法,线性表出单位正交基底。通过构造不同遮光比环形光阑下的正交基底与环Zernike多项式进行比较,验证了此方法的正确性。然后采用圆Zernike多项式和构造的新基底对矩形光阑下的波面进行了拟合,从拟合残余误差、各项基底系数的稳定性、传递矩阵的条件数等分析,结果表明针对特定的非圆域构造的新基底可靠性和抗扰动能力优于圆Zernike多项式。此方法不需要具体求出基底的解析表达式,不同非圆域仅是正交化系数矩阵发生改变,为非圆域正交基底构造提供了一种新途径。     

Focal field computation of an arbitrarily polarized beam using fast Fourier transforms

In this work the vectorial diffraction theory of Richards and Wolf is extended to compute the focal field components of an arbitrarily polarized beam using fast Fourier transform (FFT) operations. Here the arbitrarily polarized pupil function is written as the vector sum of two mutually perpendicularly polarized pupil functions. The FFT based focal field expressions are particularly useful to compute the focal field components of pupil functions without a simple analytical form. We have then applied these expressions to simulate the effects of Zernike mode aberrations on the point spread functions of a number of important cylindrical-vector beam profiles such as radially and azimuthally polarized and helical light beams.     

Model independent control of lightly damped noise/vibration systems

Feedforward control is a popular strategy of active noise/vibration control. In well-damped noise/vibration systems, path transfer functions from actuators to sensors can be modeled by finite impulse response (FIR) filters with negligible errors. It is possible to implement noninvasive model independent feedforward control by a recently proposed method called orthogonal adaptation. In lightly damped noise/vibration systems, however, path transfer functions have infinite impulse responses (IIRs) that cause difficulties in design and implementation of broadband feedforward controllers. A major source of difficulties is model error if IIR path transfer functions are approximated by FIR filters. In general, active control performance deteriorates as model error increases. In this study, a new method is proposed to design and implement model independent feedforward controllers for broadband in lightly damped noise/vibration systems. It is shown analytically that the proposed method is able to drive the convergence of a noninvasive model independent feedforward controller to improve broadband control in lightly damped noise/vibration systems. The controller is optimized in the minimum H2 norm sense. Experiment results are presented to verify the analytical results.     

自适应光学系统的模式法数值模拟

建立了利用模式法笃自适应光学系统进行数值模拟的理论模型，编制了计算程序，并与激光大气传输计算程序衔接起来，进行了大量数值模拟计算。首次发现：存在泽尼特多项式展开的最佳项数。大于一定项数的展开式的效果迅速变坏，竖排和斜排经特面式展开有类似的结果。文献中认为可以采用的１５项经特多项式展开的效果不好，最佳项数随着横向风速的增加而减小，在风速较大时最佳项数下的模范地结果稍好于直接斜率控制法的结果。     

The influence of the sampling dots on the analysis of the wave front aberration by using the covariance matrix method

The covariance matrix method is a simple method for solving the Zernike polynomial with the higher fitting precision. In this paper, it was used to analyze the several optical wave fronts of the fine polished aluminum disk surface captured by a Twyman-Green interferometer system. We had found that the PV (peak-to-valley) and rms (root-mean-square) values of the wave front aberration changes with changing the Zernike term and the expressions for the several optical wave fronts with the different sampling dots were wrong. By analyzing the relations among the condition number of the coefficients matrix, the Zernike term, and the number of the sampling dots, it was indicated that the number of the sampling dots had only reduced the fluctuation the PV and the rms value while the Zernike term increases, but did not change the case that the expressions for the wave front aberration were wrong when the Zernike term is larger than 14, especially when the number of the sampling dots is less. Such an analysis will be valuable in solving the Zernike polynomial for the wave front aberration analysis by using the covariance matrix method in optical testing.     

Kinetic theory of field effects in nonuniform molecular gases

Effects of magnetic and electric fields on transport phenomena in dilute polyatomic gases are reviewed within the framework of first order Enskog theory. The established technique of approximate operator inversion is used to give first order approximations of the transport coefficients. Instead of the customary expansion of polarization into orthogonal polynomials a more general treatment is chosen here so as to accomodate recent experimental observations. The polarizations produced by macroscopic fluxes are assumed to be eigenfunctions of the collision operator within the subspace of functions anisotropic in angular momentum. The formalism is extended to mixtures in a way to let the final expressions assume the same form as for pure gases. The obtained transport coefficients obey several symmetry relations and inequalities. Additional inequalities are now also derived for the matrix describing the saturated field effects.     

Adaptive optics for airborne platforms—Part 1: modeling

Adaptive optics systems mitigate the atmospheric turbulence-induced distortion of a propagating light wavefront. The use of adaptive optics entails the design of a feedback controller, which requires the development of a model of the plant to be controlled. In adaptive optics, the plant consists of the atmosphere through which light is traveling. Moreover, a distinct feature of the adaptive optics control application is the presence of random signals in the plant. In optics, Zernike orthonormal polynomials are commonly used as a basis set for the expansion of wavefront phase distortions. Due to the atmospheric turbulence-induced random nature of the underlying physical process, the spatial-temporal correlation functions of the Zernike polynomial phase distortion expansion coefficients must be evaluated if a proper stochastic model of the plant is to be developed and adaptive optics is to be employed. In Part 1 of this paper, these correlation functions are developed using a layered atmospheric model and calculations for the first few low-order Zernike modes are performed. Using these correlation functions, an underlying stochastic linear dynamical system, which is adequate for control design, is synthesized. This system models the plant and, in turn, provides the basis for the employment of advanced model-based control and estimation concepts in an adaptive optics system for an airborne platform application.     

The radiation impedance of a rectangular piston

The radiation impedance of a rectangular piston is expressed as the Fourier transform of its impulse response, which is obtained from the recent work of Lindermann [1]. The analytical evaluation of the transform is performed and new integral expressions are presented for both the radiation resistance and reactance. The integrals are readily evaluated in terms of elementary functions at both the low and high frequency limits. The integrals are also expressed as series of Bessel functions which are valid for all frequencies and aspect ratios. Numerical results are presented to illustrate the behavior of the radiation resistance and reactance as a function of the aspect ratio of the piston and a normalized frequency parameter. Additional numerical results are then presented to illustrate the accuracy of the analytical expressions for the radiation resistance and reactance at low and high frequencies. Finally, numerical results are presented to illustrate the application and accuracy of using standard FFT algorithms to evaluate the radiation resistance and reactance directly from the impulse responses.     

离轴抛物面反射镜模拟空间环境镜面变形分析

在卫星红外多光谱扫描仪模拟空间环境辐射定标试验中,环境模拟器舱内的稳、瞬态温度场会给光学系统成像质量带来影响.以试验中的实测温度变化作为温度载荷,借助有限元软件,分析了定标系统中通光孔径为φ400mm的离轴抛物面主镜的温度场分布和热弹性变形.选用在单位圆内正交的Zernike多项式为拟合基函数,通过坐标变换,将抛物镜表...     

Karhunen–Loeve functions for non-Kolmogorov turbulence

Karhunen–Loeve functions for the case of non-Kolmogorov turbulence are calculated. The calculations are performed using the covariance matrix of Zernike coefficients which is derived in analytical form. It is shown that the Karhunen–Loeve expansion is more efficient than the Zernike one for simulations of high-order, turbulence-induced phase distortions.     

Diagnostics of the scale of turbulence using a divergent laser beam

A method for estimating the parameters of turbulence that was originally developed for collimated laser beams is applied to the case of a divergent beam. The estimate is based on an analysis of the correlation functions of the Zernike expansion coefficients of the phase. The results suggest that the method can be used for the estimation of the inner scale of turbulence in an experiment with a divergent laser beam. In the case of isotropic turbulence, it is also possible to estimate the outer scale of turbulence from the correlation functions of lower-order modes.     

An improved algorithm for balanced POD through an analytic treatment of impulse response tails

We present a modification of the balanced proper orthogonal decomposition (balanced POD) algorithm for systems with simple impulse response tails. In this new method, we use dynamic mode decomposition (DMD) to estimate the slowly decaying eigenvectors that dominate the long-time behavior of the direct and adjoint impulse responses. This is done using a new, low-memory variant of the DMD algorithm, appropriate for large datasets. We then formulate analytic expressions for the contribution of these eigenvectors to the controllability and observability Gramians. These contributions can be accounted for in the balanced POD algorithm by simply appending the impulse response snapshot matrices (direct and adjoint, respectively) with particular linear combinations of the slow eigenvectors. Aside from these additions to the snapshot matrices, the algorithm remains unchanged. By treating the tails analytically, we eliminate the need to run long impulse response simulations, lowering storage requirements and speeding up ensuing computations. To demonstrate its effectiveness, we apply this method to two examples: the linearized, complex Ginzburg–Landau equation, and the two-dimensional fluid flow past a cylinder. As expected, reduced-order models computed using an analytic tail match or exceed the accuracy of those computed using the standard balanced POD procedure, at a fraction of the cost.     

Analysis and demonstration of multiplexed phase computer-generated hologram for modal wavefront sensing

We propose to implement a modal wavefront sensor (MWS) using a multiplexed phase computer-generated hologram (MPCGH). Based on general orthogonal aberration modes, the theoretical treatments of the MWS employing a MPCGH are presented with scalar diffraction approximations and Fourier analysis. Under the small aberration approximations, we give the analytical formula for characterizing the relationship between the sensor signal and the amplitude of the aberration mode. We design several MPCGHs with an effective method of modified off-axis reference beam holograms superposition, and code some common orthogonal Zernike aberration modes into the MPCGH. The numerical simulation is carried out to investigate the performance of MWS to detect particular aberration mode(s). The results exhibit the expected responses of the corresponding symmetric spot pair, and indicate that the wavefront distorted by a special Zernike aberration mode, after modulated by the MPCGH, can be transformed into beams with an intensity-normalized differential signal, which can reflect the change trend of the aberration coefficients in the test wavefront. The experimental demonstration with designed MPCGHs in conjunction with two phase-only spatial light modulators was carried out to test the performance of the MWS.     

Comparing seven spectral methods for interpolation and for solving the Poisson equation in a disk: Zernike polynomials,Logan–Shepp ridge polynomials,Chebyshev–Fourier Series,cylindrical Robert functions,Bessel–Fourier expansions,square-to-disk conformal mapping and radial basis functions

We compare seven different strategies for computing spectrally-accurate approximations or differential equation solutions in a disk. Separation of variables for the Laplace operator yields an analytic solution as a Fourier–Bessel series, but this usually converges at an algebraic (sub-spectral) rate. The cylindrical Robert functions converge geometrically but are horribly ill-conditioned. The Zernike and Logan–Shepp polynomials span the same space, that of Cartesian polynomials of a given total degree, but the former allows partial factorization whereas the latter basis facilitates an efficient algorithm for solving the Poisson equation. The Zernike polynomials were independently rediscovered several times as the product of one-sided Jacobi polynomials in radius with a Fourier series in θ. Generically, the Zernike basis requires only half as many degrees of freedom to represent a complicated function on the disk as does a Chebyshev–Fourier basis, but the latter has the great advantage of being summed and interpolated entirely by the Fast Fourier Transform instead of the slower matrix multiplication transforms needed in radius by the Zernike basis. Conformally mapping a square to the disk and employing a bivariate Chebyshev expansion on the square is spectrally accurate, but clustering of grid points near the four singularities of the mapping makes this method less efficient than the rest, meritorious only as a quick-and-dirty way to adapt a solver-for-the-square to the disk. Radial basis functions can match the best other spectral methods in accuracy, but require slow non-tensor interpolation and summation methods. There is no single “best” basis for the disk, but we have laid out the merits and flaws of each spectral option.     

Dynamic measurement of room impulse responses using a moving microphone

A technique for the recording of large sets of room impulse responses or head-related transfer functions is presented. The technique uses a microphone moving with constant speed. Given a setup (e.g., length of the room impulse response), a careful choice of the recording parameters (excitation signal, speed of movement) leads to the reconstruction of all impulse responses along the trajectory. In the case of a moving microphone along a circle, the maximal angular speed is given as a function of the length of the impulse response, its maximal temporal frequency, the speed of sound propagation, and the radius of the circle. As a result of the presented algorithm, head-related transfer functions sampled at 44.1 kHz can be measured at all angular positions along the horizontal plane in less than 1 s. The presented theory is compared with a real system implementation using a precision moving microphone holder. The practical setup is discussed together with its limitations.     

Anharmonic vibrational wave functions, infrared band intensities, and dipole moments of water

A new and simple method to expand the anharmonic vibrational wave functions with respect to the harmonic oscillator wave functions is proposed. The coefficients of the expansion are given as matrix elements of the S function of the contact transformation in the perturbation theory and the explicit expressions of these coefficients are given within the approximation to the second order in λ. As an example of the expansion, the wave functions of water molecules were calculated and applied to the calculation of infrared band intensities and average values of dipole moments in several states.