Hans J. Frankena

Linear approximation for measurement errors in phase shifting interferometry

This paper shows how measurement errors in phase shifting interferometry (PSI) can be described to a high degree of accuracy in a linear approximation. System error sources considered here are light source instability, imperfect reference phase shifting, mechanical vibrations, nonlinearity of the detector, and quantization of the detector signal. The measurement inaccuracies resulting from these errors are calculated in linear approximation for several formulas commonly used for PSI. The results are presented in tables for easy calculation of the measurement error magnitudes for known system errors. In addition, this paper discusses the measurement error reduction which can be achieved by choosing an appropriate phase calculation formula.

Real-time displacement measurement using a multicamera phase-stepping speckle interferometer

Surface point displacements can be measured by using standard phase-stepping speckle interferometry, but the measured data are vulnerable to disturbances during the interferogram recordings. To overcome this an interferometer with a computational system has been developed to record three phase-stepped interferograms simultaneously and to calculate displacements. This system is evaluated by measurements of the out-of-plane rotation of a flat surface. The surface displacement is calculated at a rate of 25 times/s. If one reduces noise by filtering, hardware limitations decrease the speed to 12.5 displacement calculations/s. With this system displacements can be measured with an accuracy exceeding λ/55 if filtering is applied.

Nonpolarizing beam splitter design.

A three-step design procedure is developed for dielectric stacks which are required to be nonpolarizing for a given wavelength lambdar and angle of incidence theta 0,r, at which the reflectance Rr is prescribed. The method leads to solutions in which only three layer materials occur and can be applied for a wide range of values of theta0,r and Rr. The media can be chosen from the available coating materials. Furthermore, the procedure offers the possibility of optimizing with respect to the behavior of the reflectance in the neighborhood of lambdar and theta0,r. An example is elaborated, and its results are compared with an actually produced coating.

Integrated optical sensors using micromechanical bridges and cantilevers

Micromechanical bridges and cantilevers sensitive to external forces have been fabricated upon Si substrates. They are used as optical waveguides and part of sensor circuits. The waveguides consist of sandwiched layers of an SiO2 buffer, an Al2O3 waveguide and an SiO2 cover. The bridges and cantilevers with very small dimensions such as 100 micrometers in length, 5 micrometers in width and 2.5 micrometers in thickness have been successfully produced. Such bridge- or cantilever-shaped waveguide structures have been applied in acoustic signal detection and noise monitoring. In this paper, the bridge and cantilever structures will be analyzed and experimental results on sound measurement will be presented.

Polarization phase stepping with a Savart element

The imaging properties of a real-time shearing interferometer are presented. The use of Savart elements, both as a beam displacer and an analyzer in a polarization phase-stepping scheme, is demonstrated in a real-time, two-camera, four-bucket shearing interferometer. A simple calculation scheme for ray propagation through uniaxial, birefringent elements is presented, and the effects on the image formation through 6-cm-long Savart elements is discussed.

Numerical evaluation of diffraction integrals for apertures of complicated shape

An algorithm for fast numerical integration of near-field scalar diffraction formulas is presented, based on the local approximation of the integrand of the diffraction equation by a variant of the Fresnel kernel. The two-dimensional local propagation integral is solved analytically for an integration domain enclosed between two mutually perpendicular line segments and a parabolic arc. We show that, by combining rectangular and arched elements, one can achieve accurate computation of the field diffracted at complicated aperture shapes without having to resort to time-consuming numerical quadrature techniques. The numerical accuracy and the computational speed of the algorithm are assessed and compared with the performance of the linear-phase approximation method developed by Hopkins and Yzuel [ Opt. Acta17, 157 ( 1970)].

Field analysis of two-dimensional integrated optical gratings

A method to determine the field scattered by a two-dimensional rectangular grating of finite length is developed. Both periodic or aperiodic and lossy or lossless structures can be treated. The grating is built up as a finite sequence of two alternating types of waveguide sections connected by means of step discontinuities. The mode-matching method is employed to determine the field scattered by a single step. In combining the scattering properties of the separate steps and waveguide sections to describe the entire grating, the scattering matrix formalism was employed. With the proposed method we have examined the 10- and the 20-period configurations described by Shigesawa Tsuji [ IEEE Trans. Microwave Theory Tech.37, 3– 14 ( 1989)] and by Liu Chew [ IEEE Trans. Microwave Theory Tech.39, 422– 430 ( 1991)], which show good agreement. Furthermore, we have successfully extended these configurations to gratings with 100 periods.

Modeling and characterization of an electrooptic polarization controller on LiNbO/sub 3/

A model for describing the operation of an integrated electrooptical polarization controller on LiNbO/sub 3/ as a function of the applied voltages is presented. This model contains several parameters; for their determination, a special measurement procedure has been developed. This technique is used to determine the parameters of a developed polarization controller. With these parameters, the model describes the operation of the controller accurately. Moreover, the model can be used to calculate the voltages needed to reach a specific polarization conversion. >

Production and measurement of superpolished surfaces

The influence of polishing time on the roughness of ultrasmooth bowl-feed-polished surfaces is studied. A large improvement of the surface quality is obtained within the first 10 minutes, but increasing the polishing time from 10 to 60 minutes did not yield a significant difference. A Linnik interference miscroscope, adapted for phase-shifting interferometry, was used for roughness measurements. Preliminary measurements have been performed with a setup determining the scattered intensity within a small solid angle. This relatively simple setup, which is also suitable for uncoated glass surfaces, clearly showed the improvement of surface quality by bowl-feed polishing.

Nonparaxial theory of curved holograms

Nonparaxial imaging by holograms on a spherical surface, with a limited circular cross section, is formulated mathematically. Starting from a point-shaped object, the image point (called the point of reference) is redefined such that the deviation in the positions of the actual image-forming wave front and a spherical wave front centered around this point contains no terms of the first order in the hologram coordinates. Furthermore, their second-order contribution, averaged over the hologram, vanishes. The remaining terms up to the fourth order describe the primary aberrations. Compared with earlier theories, this formulation yields a more accurate position for the holographic image. In addition, expressions are obtained for the primary aberrations, which not only consist of astigmatism, coma, and spherical aberration but also contain three additional contributions, which occurred as higher-order aberrations in previous literature. Experiments support our conclusions.