Spline algorithms for continuum functions

For the surface shape measurement of a semiconductor with a highly reflective index, it is im portant to effectively suppress the harmonic signal s from multi ple reflection s. In application, the phase extraction algorithm should have a maximum value when there is no phase -shift miscalibration. In this presentation, a new 4N - 3 phase extraction algorithm , which has the ability to suppress harmonic signal s and exhibits a fringe contrast maximum value when there is no phase -shift error , was derived. This n ew 4N - 3 algorithm consists of a new polynomial window function and a discrete Fourier transform term and has the ability to compensate for 2 nd -order nonlineari ty in the phase shift. The suppression ability of the new polynomial window function is compared with other conventional window functions. The sampling functions of the new 4N - 3 algorithm have much smaller amplitudes in the vicinity of the detection freq uency than does synchronous detection or other phase extraction algorithm s with conventional window functions . Keywords: H ighly reflective index, p hase measurement, nonlinearity, window function

Eigensolutions of boundary value problems using inverse iteration

Discretisations of differential eigenvalue problems have a sensitivity to perturbations which is asymptotically least as h ? 0 when the differential equation is in first order system form. Both second and fourth order accurate discretisations of the first order system are straightforward to derive and lead to generalised eigenvalue problems of the form ?? where both A and B are narrow-banded, block bidiagonal (hence unsymmetric) matrices, and typically B is singular. Solutions of the differential equation associated with eigenvalues of small magnitude are best determined by the discretisations. Thus Krylov subspace methods (for example) require A to be invertible and seek large solutions of ?? This already requires rational methods in principle. It follows that rapidly convergent methods based on inverse iteration applied to the original formulation as a nonstandard generalised eigenvalue problem prove attractive for the narrow-banded systems considered here. Also they have the advantage that they are applicable under the weaker condition A ? ?B ? =? . We have had extensive experience with a method combining aspects of Newton's method and inverse iteration and having a convergence rate of 3.56 . Our implementation combines this basic algorithm with a limiting form of Weilandt deflation to find a sequence of eigenvalues. It has proved extremely satisfactory in a range of applications. This formulation has the further advantage that it is easy to insert the eigenvalue calculation inside an outer loop to satisfy a constraint on an auxiliary parameter. Examples to illustrate both the robustness of the deflation and the flexibility of the approach are provided.

In this paper the mechanisms causing phase drift in the dual active bridge topology are presented and two compensation methods are proposed to overcome their effects. The phase drift causes discrepancy between the predicted phase shift and the actual phase shift measured from the transformer primary and secondary voltages. The error in the phase shift causes the input power to be larger or smaller than predicted and it can also interfere with the operation of several modulation methods. By using the phase drift compensation methods presented in this paper the converter input power can be predicted accurately by using the classical power equation. The effects of dead-time on the phase drift are also discussed and a method to calculate a suitable dead-time with respect to the converter operating point is proposed. The value of the phase drift compensation is presented along with a variable frequency modulation method, which enables zero voltage switching at a wide load range with a minimal circulating current.

An algorithm is proposed for extraction of the unknown phase shift steps and correction of the phase shift error in partially coherent illuminated phase-shifting digital holography. The mean contrast of noise (MCN) of the reconstructed amplitude image is developed to evaluate the noise level resulting from phase shift error. Actual phase shift steps are first extracted by minimizing the MCN. Then the extracted actual phase shift steps are used to retrieve the original object wave without phase shift error. The validity of the proposed method is verified by simulations, and then implemented in partially coherent illuminated phase-shifting digital holography where a spatial light modulator with unknown phase modulation characteristic is used as the phase shifter. The results demonstrated that the actual phase shift steps can be extracted and the phase shift error is corrected effectively.

The phase shift interferometer can be used to detect the wavefront phase of optical system and the piezoelectric transducer is widely used as a phase shifter in phase shift interferometer. However, the displacement of piezoelectric transducer presents hysteresis nonlinearity. Hence, when the equal-step phase shift algorithm (e.g. the Schwider-Hariharan phase-shifting algorithm which obtains five images and sets the phase-shifting between images to 90 degrees) is used in the phase shift interferometer to calculate the wavefront phase, the nonlinearity of piezoelectric transducer will introduce error to the 90 degrees phase-shifting between images. Then the calculated wavefront phase will contain the error. Hence, it is necessary to propose a method to diminish the phase error. The method by choosing an optimized phase-shifting algorithm has been used to mitigate the phase error. However, the method does not directly focus on piezoelectric transducer itself. In the paper, we propose a method by compensating for the hysteresis nonlinearity displacement of piezoelectric transducer to mitigate the phase error. We analyze the relationship between the nonlinearity displacement of piezoelectric transducer and phase shift error and establish the error models to analyze the relationship between the phase shift error and phase error. Furtherly, we propose the Proportion Integration Differentiation (PID) closed-loop feedback control method to compensate for the hysteresis nonlinearity displacement of piezoelectric transducer. The results indicate that, with compensated, the maximum tracking error is reduced from 18.89 % to 5.5 %. And in the ascending stage, the PV error and RMS error with compensated are less 12.35nm and 4.34nm than the PV error and RMS error without compensated. In the descending stage, the PV error and RMS error with compensated are less 19.15nm and 6.76nm than the PV error and RMS error without compensated. Hence, the proposed compensation method can improve the detection accuracy of the phase shift interferometer.

A phase shift selection method is proposed to design algorithms immune against phase shift errors in two-step generalized phase-shifting interferometry. A general formula for wavefront reconstruction error is derived, and its specific expressions for two common errors are also given. Calculation results suggest that the proper range of phase shift for general application is about from 0.5 to 2.0 rad for both the fixed and linear phase shift errors. Computer simulations have demonstrated the effectiveness of this phase shift selection method by decreasing the wave reconstruction errors to one-fifth.

A common material for read-write sliders is a composite of alumina (Al(2)O(3)) and titanium carbide (TiC), with a grain size of the order of 1 mum. I derive the effective complex reflectivity of this material, using scalar diffraction theory and the known indices of refraction of Al(2)O(3) and TiC. The effective reflectivity is a function of the relative surface area of the exposed TiC grains as well as of the numerical aperture of the collection optics. The theory resolves several known discrepancies between ellipsometry and reflectometry of Al(2)O(3)-TiC. The theory also predicts a systematic error in the phase shift on reflection calculation. These results are of considerable interest for surface shape metrology of the slider as well as for optical flying-height testing and control of pole-tip recession.

Two dominant systematic errors in the absolute interferometric test of high numerical-aperture sphericals are discussed and measured experimentally in pursuit of an accuracy of λ=100. Gravitational sag deformation of a 4-inch spherical concave surface was measured in a vertical phase-shifting Fizeau interferometer. The surface shapes of two identical transmission spherical concaves were measured via the two-surface comparison method using three positional measurements. One of the surfaces was then rotated around the optical axis and the interference phases were averaged to extract a rotationally symmetric component of the aberration. The gravitational sag was then determined by the aberration component independent of the rotation. The geometrical error in the phase shift is also estimated theoretically and corrected in the experiment. Experimental results show that the both errors amount to 6 to 7 nm peak-to-valley lengths, the magnitudes of which are comparable to that of the total aberration of the spherical surface.

Optical absorption in asymmetric triple quantum well structures

Laser-diode interferometry

In this paper we consider a rational eigenvalue problem governing the vibrations of a tube bundle immersed in an inviscid compressible fluid. Taking advantage of eigensolutions of appropriate sparse linear eigenproblems the large nonlinear eigenvalue problem is projected to a much smaller one which is solved by inverse iteration.

A Padé Extrapolated Inverse Power Method for Coupled Schrödinger-like Equations Applied to the Two-Body Relativistic Bound State Problem in Quantum Electrodynamics

A Generalised Eigenvalue Problem and The Lanczos Algorithm

Linear stability theory was applied in the present paper to analyze the stability of the basic state solution of thermocapillary convection in a liquid bridge with liquid encapsulation. Discretizing the linearized disturbance equations by using finite-difference approximation, stability analysis is evolved to a complex generalized eigenvalue problem with a complicated band structure of matrix. The influence of dimensionless parameters on the stability of the system is revealed through solving the complex generalized eigenvalue problem by inverse iteration. The results provide a theoretical and numerical foundation for crystal growth by float-zone method and for other engineering applications.