Principles for the design of modern interferometers

Principles for the development of modern interferometers are discussed.     

Real-time digital control of optical interferometers by the mechanical-modulation technique

We discuss the application of digital systems to the automatic control of dual-wave optical interferometers. We show that, if the mechanical-modulation technique is used for error-signal extraction, digital techniques can be used both for error-signal extraction and for control-signal generation. Therefore, apart from two front/end amplifiers that are necessary to match the dynamics of the detectors and actuators to the dynamics of the analog-to-digital converters and digital-to-analog converters, no other analog devices are required. In particular, the mechanical-modulation technique requires the synchronous demodulation of the photodiode output signal. Hence we need to implement a digital lock-in amplifier whose algorithm is described here. Finally, we describe one of the possible applications of this digital control procedure, such as the control of a classic Mach-Zehnder interferometer in air.     

Phase-shifting technique for improving the imaging capacity of sparse-aperture optical interferometers

We describe the principle of a multiaperture interferometer that uses a phase-shifting technique and is suitable for quick snapshot imaging of astrophysical objects at extreme angular resolution through Fourier inversion. A few advantages of the proposed design are highlighted, among which are radiometric efficiency, field of view equivalent to those of Fizeau interferometers, and a preliminary calibration procedure allowing characterization of instrumental errors. For large telescope numbers, the proposed design also results in considerable simplification of the optical and mechanical design. Numerical simulations suggest that it should be possible to couple hundreds of telescopes on a single 4K × 4K detector array, using only conventional optical components or emerging technologies.     

Wedge simulation method for calculating the reliability of linear dynamical systems

In this paper the problem of calculating the probability of failure of linear dynamical systems subjected to random excitations is considered. The failure probability can be described as a union of failure events each of which is described by a linear limit state function. While the failure probability due to a union of non-interacting limit state functions can be evaluated without difficulty, the interaction among the limit state functions makes the calculation of the failure probability a difficult and challenging task. A novel robust reliability methodology, referred to as Wedge-Simulation-Method, is proposed to calculate the probability that the response of a linear system subjected to Gaussian random excitation exceeds specified target thresholds. A numerical example is given to demonstrate the efficiency of the proposed method which is found to be enormously more efficient than Monte Carlo Simulations.     

Computerization of white-light contact interferometers based on optical image processing

A technique of computerization of white-light contact interferometers intended for calibration of reference end gauges and other objects in the range 0.1–1000 mm with resolution 0.001 μm is considered. Results are presented from studies of contact interferometers based on digital processing of interference images with the use of corresponding mathematical software. __________ Translated from Izmeritel’naya Tekhnika, No. 7, pp. 35–38, July, 2006.     

Optimization of displacement-measuring quadrature interferometers considering the real properties of optical components

We present the influence of alignment and the real properties of optical components on the performance of a two-detector homodyne displacement-measuring quadrature laser interferometer. An experimental method, based on the optimization of visibility and sensitivity, was established and theoretically described to assess the performance and stability of the interferometer. We show that the optimal performance of such interferometers is achieved with the iterative alignment procedure described.     

Experimental demonstration of an automatic alignment system for optical interferometers

An automatic alignment system, based on a differential phase-sensing technique described in a companion paper [Appl. Opt.33, 0000, (1994)], has been experimentally demonstrated on the 10-m prototype laser interferometric gravitational wave detector in Glasgow. The alignment system developed was used to control the orientations of two mirrors in a 10-m-long suspended Fabry-Perot cavity with respect to the direction defined by the input laser beam. The results of the test and a discussion of the performance of the system are given.     

Propagation of elliptical Gaussian beams in apertured and misaligned optical systems

On the basis of the fact that a hard-edged elliptical aperture can be expanded approximately as a finite sum of complex Gaussian functions in tensor form, an analytical propagation expression for an elliptical Gaussian beam (EGB) through a misaligned optical system with an elliptical aperture is derived by use of vector integration. The approximate analytical results provide more convenience for studying the propagation and transformation of EGBs than the usual way by using a diffraction integral directly, and the efficiency of numerical calculation is improved. Some numerical simulations are illustrated for the propagation properties of EGBs through apertured optical transforming systems with misaligned thin lenses.     

Propagation of twisted gaussian Schell-model beams through a misaligned first-order optical system

Abstract

The propagation of twisted Gaussian Schell-model (TGSM) beams passing through a misaligned first-order optical system is studied. The explicit expressions for the cross-spectral density function and Wigner distribution function of the output beam are derived. As a result, generalized partially coherent beams called the decentred twisted Gaussian Schell-model (DTGSM) beams are introduced and their properties are discussed.     

Fractional fourier transform as a tool for analysing the diffraction of misaligned optical systems

Abstract

It is shown that fractional Fourier transform could be viewed as a tool for analysing the diffraction of misaligned optical systems, just as the fractional Fourier transforms are adapted to the mathematical expression of Fresnel diffraction. Some special cases are also discussed.     

Digital error-signal extraction technique for real-time automatic control of optical interferometers

We describe an efficient and robust method for the extraction of the longitudinal error signal for the automatic control of optical interferometers, which can also be applied when the uncontrolled optical system spans hundreds of fringes. The method is based on classic modulation techniques (phase modulation, mechanical modulation, etc.), but extends their performances by the use of the information available only at the output photodiode. We digitally implemented such a method by following modular hardware and software architectures. We then tested the whole procedure in the automatic control of a suspended Michelson interferometer, showing its feasibility and the good performances.     

Modified refraction-interference method for calculating the geometric and optical parameters of optical waveguides

We describe a ray method for analysis of the geometric and optical characteristics of two-layer fiber optical waveguides. We have analyzed the effect of the polarization of the illuminating beam on the overall pattern of the intensity distribution. We consider the characteristic types of rays for a single-layer and two-layer fiber optical waveguide. We have obtained definitive dependences of the beam intensity for each ray individually. We have developed a program for calculation of the scattering pattern in fiber optical waveguides illuminated by a broad beam. Translated from Izmeritel'naya Tekhnika, No. 3, pp. 39–42, March, 1996.     

高维非线性动力系统最简规范形的计算

运用可逆线性变换和近恒同变换,研究了不经计算传统规范形,直接计算高维非线性动力系统的最简规范形。引进可逆线性变换,将非线性动力系统的线性矩阵拓扑等价于符合实际研究需求的分块对角线矩阵：相伴矩阵分布在对角线上,其余元素均为0。利用低阶项来化简高阶项,得到了高维非线性动力系统的最简规范形。在该最简规范形中,对应于每一个相伴矩阵的非线性系数矩阵,只有最后一行含有非0元素,其余各行元素均为0。借助Mathematica语言,编制了计算任意高维非线性动力系统的最简规范形的通用程序。运行该程序,分别计算了4维、6维和7维非线性动力系统的直到4阶的最简规范形。     

Design of lenses to project the image of a pupil in optical testing interferometers

When an optical surface or lens in an interferometer (Twyman-Green or Fizeau interferometer) is tested, the wave front at the pupil of the element being tested does not have the same shape as at the observation plane, because this shape changes along its propagation trajectory if the wave front is not flat or spherical. An imaging lens must then be used, as reported many times in the literature, to project the image of the pupil of the system being tested over the observation plane. This lens is especially necessary if the deviation of the wave front from sphericity is large, as in the case of testing paraboloidal or hyperboloidal surfaces. We show that the wave front at both positions does not need to have the same shape. The only condition is that the interferograms at both places be identical, which is a different condition. This leads to some considerations that should be taken into account in the optical design of such lenses.     

Propagation of a random electromagnetic beam through a misaligned optical system in turbulent atmosphere

On the basis of the generalized diffraction integral formula for misaligned optical systems in the spatial domain, an analytical propagation expression for the elements of the cross-spectral density matrix of a random electromagnetic beam passing through a misaligned optical system in turbulent atmosphere is derived. Some analyses are illustrated by numerical examples relating to changes in the state of polarization of an electromagnetic Gaussian Schell-model beam propagating through such an optical system. It is shown that the misalignment has a significant influence on the intensity profile and the state of polarization of the beam, but the influence becomes smaller for the beam propagating in strong turbulent atmosphere. The method in this paper can be applied for sources that are either isotropic or anisotropic. It is shown that the isotropic sources and the anisotropic sources have different polarization properties on beam propagation.