Numerical Analysis of Interference Patterns

This page is currently a work in progress.

Phase Shifting Technique

• requires three phase shifted fringe patterns
• the phase shift must be known
• carefully controlled conditions must be maintained

Fourier Analysis Method

A "Mexican Hat" function. (courtesy of [1])

The Fourier Analysis method of interferograms was created in 1982 by M. Takeda, H. Ina and S. Kobayashi and originally intended as an alternative to Moire Topography and the phase-shifting technique [1,2]. However, this method was ineffective at analyzing closed fringe patterns. A revision to the method, solved this problem by utilizing a Cartesian to polar coordinate transform [3]. The result of which could then be analyzed using the original method proposed.

The revised Fourier Analysis method does have several limitations. The first requirement is that the "measurement wave front be a monotonic function in the direction of the carrier frequency" [3]. For instance, if the surface to be analyzed resemble the image to the right were analyzed by the above method, it would look no different than a surface that decreased or increased from top to bottom. In order to analyze such a fringe pattern generated by such a surface, an additional fringe pattern giving the carrier frequency must be used.

• requires carrier frequency, narrow frequency, low noise and open fringes
• estimates the phase wrapped (via arctan)

[1] takeda et al 1982 [2] cuevas et al 2002 [3] ge et al 2001

Regularized Phase Tracking Algorithms

A shifted fringe Pattern.

The regularization technique for fringe pattern analysis involves using a cost function to find the difference between an estimated function and the observed interferogram and

Phase-Locked Loop Algorithm

• computer simulated oscillator (VCO) needed
• phase error b/w the fringe pattern and the VCO vanishes

Artificial Neural Network Method

• requires carrier phase
• non-algorithmic (i.e. must have learning phase)
• types of learning include: supervised, unsupervised and reinforcement
• multi-layer: input, output, hidden neurons present

The artificial neural network approach utilizes the ability fo

Genetic Algorithm

• compares generated surfaces to goal via a cost function
• mutations alter generated solutions which are then evaluated
• this process loops until cost function conditions (set by the user) are met

Simulated Annealing

ParSA

Here [2] is the link the the ParSA documentation.

The ParSA (Parallel Simulated Annealing) library is a set of classes written in C++ that can be used to solve optimization problems via a process know as simulated annealing.

The ParSA library contains many different types of

The Equation for convergence speed is:

$P\left(X_{n}\notin Cost_{min}\right)\approx \left({\frac {K}{n}}\right)^{\alpha }$

(1)

Where K and $\alpha$ are problem specific constants and $X_{n}$ is a solution of length n. Using equation (1) and test runs on smaller problems of lower order, K and $\alpha$ can be determined. Along with some suggestions provided in the ParSA documentation, progress can be made towards finding higher quality solutions at a much faster rate.

$\ln P=\alpha \left(\ln K-\ln n\right)$

(2)

The equation for warming temperature in the Aarts scheduler:

$T={\bar {\Delta C^{(|)}}}\left(\ln {\frac {m_{2}}{m_{2}\chi _{0}-(1-\chi _{0})m_{1}}}\right)^{-1}$

(3)

Table of Proposed Runs

Clustering

SA Clustering Solver

Clustering Scheduler

• SA Aarts
• SA Easy Scheduler

Multiple Independent Runs (M.I.R.)

MIR_Solver

MIR_Scheduler

Use Combinations of the Different Solver/Scheduler Classes