Spectral measurement of birefringence using particle swarm optimization analysis

Maciej Kraszewski; Marcin Strąkowski; Jerzy Pluciński; Bogdan B. Kosmowski

doi:10.1364/AO.54.000076

Applied Optics
Vol. 54,
Issue 1,
pp. 76-84
(2015)
•https://doi.org/10.1364/AO.54.000076

Spectral measurement of birefringence using particle swarm optimization analysis

Maciej Kraszewski, Marcin Strąkowski, Jerzy Pluciński, and Bogdan B. Kosmowski

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Maciej Kraszewski, Marcin Strąkowski, Jerzy Pluciński, and Bogdan B. Kosmowski, "Spectral measurement of birefringence using particle swarm optimization analysis," Appl. Opt. 54, 76-84 (2015)

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Table of Contents Category
- Instrumentation, Measurement, and Metrology
Optics & Photonics Topics
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About this Article
History
- Original Manuscript: July 14, 2014
- Revised Manuscript: November 14, 2014
- Manuscript Accepted: November 20, 2014
- Published: December 23, 2014

Abstract

Measurement of birefringence is useful for the examination of technical and biological objects. One of the main problems, however, is that the polarization state of light in birefringent media changes periodically. Without knowledge of the period number, the birefringence of a given medium cannot be reliably determined. We propose to analyze the spectrum of light in order to determine the birefringence. We use a particle swarm optimization algorithm for an automatic processing spectra of light transmitted through birefringent material for two orthogonal states of polarization. We have tested the described algorithm on a liquid crystal cell with varying effective birefringence. The proposed method can be used for the measurement of uniaxial positive birefringence without knowing the number of retardation periods or an approximate value of the measurement result. This fact makes the proposed method useful for automatic measurements, when hundreds or thousands of spectra need to be analyzed.

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Parameter	Symbol
Number of particles	$N$
Measured spectrum	$\tilde{Γ} (k)$
Initial birefringence range	$Δ n_{\min}$ , $Δ n_{\max}$
Iterations limit	$K_{\max}$
Mean square error threshold	${MSE}_{\max}$
Velocity weight limits	$A_{1}$ , $A_{2}$ , $A_{3}$
Birefringence of $i$ th particle	$Δ n_{i}$
Velocity of $i$ th particles	$v_{i}$
Best birefringences found by $i$ th particle	$Δ n_{l, i}$
Smallest error obtained by $i$ th particle	${MSE}_{l, i}$
Best birefringence found by all particles	$Δ n_{g}$
Smallest error obtained by all particles	${MSE}_{g}$

$a_{n}$	$r_{a}$	$b_{n}$	$r_{b}$	$N_{p}$	$K_{\max}$	$b$ (%)
0	0	0	0	20	20	4.2
$\leq 0.015$	0.06	0	0	50	20	$- 4.1$
$\leq 0.015$	0.06	$\leq 0.008$	$= 0.03$	50	50	$- 6.2$

$a_{n}$	$r_{a}$	$b_{n}$	$r_{b}$	$N_{p}$	$K_{\max}$	$b$ (%)
0	0	0	0	20	20	7.3
$\leq 0.015$	0.06	0	0	50	20	$- 3.1$
$\leq 0.015$	0.06	$\leq 0.008$	$= 0.03$	50	50	$- 6.7$

Parameter	Symbol
Number of particles	$N$
Measured spectrum	$\tilde{Γ} (k)$
Initial birefringence range	$Δ n_{\min}$ , $Δ n_{\max}$
Iterations limit	$K_{\max}$
Mean square error threshold	${MSE}_{\max}$
Velocity weight limits	$A_{1}$ , $A_{2}$ , $A_{3}$
Birefringence of $i$ th particle	$Δ n_{i}$
Velocity of $i$ th particles	$v_{i}$
Best birefringences found by $i$ th particle	$Δ n_{l, i}$
Smallest error obtained by $i$ th particle	${MSE}_{l, i}$
Best birefringence found by all particles	$Δ n_{g}$
Smallest error obtained by all particles	${MSE}_{g}$

$a_{n}$	$r_{a}$	$b_{n}$	$r_{b}$	$N_{p}$	$K_{\max}$	$b$ (%)
0	0	0	0	20	20	4.2
$\leq 0.015$	0.06	0	0	50	20	$- 4.1$
$\leq 0.015$	0.06	$\leq 0.008$	$= 0.03$	50	50	$- 6.2$

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Applied Optics