Genetic Invention of Fast and Optimal Broad-bandStokes/Mueller Polarimeter Designs

Paul Anton Letnespaul.anton.letnes@gmail.com

Ingar Stian Nerbønerbo@ntnu.no

Lars Martin Sandvik Aaslars.martin.aas@gmail.com

Pål Gunnar Ellingsenpaalge@gmail.com

Department of PhysicsNorwegian University ofScience and Technology

N-7491 Trondheim, Norway

Morten Kildemomorten.kildemo@phys.ntnu.no

ABSTRACTWe have applied a genetic algorithm to generate optimalpolarimeter designs for a selected wavelength interval, as-suming known dispersion relations of the components. Ourresults are improvements on previous patented designs basedon ferroelectric liquid crystals.

Categories and Subject DescriptorsJ.2 [Physical Sciences and Engineering]: Physics; G.1.6[Optimization]: Global Optimization

General TermsDesign

KeywordsPolarimetry, Ellipsometry, Optical design, Genetic algorithms

1. INTRODUCTIONVision is perhaps the most important sense we have, and

light is widely used for measurement and detection in sci-ence and technology. Techniques based on light usually havethe advantages of being non-destructive and non-invasive,and have the possibility of performing measurements at highspeed and large (even astronomical) distances. Unfortu-nately, the human eye and many measurement techniquesare only able to detect the intensity of light. As light con-sists of oscillating electric and magnetic fields, it is a vec-torial quantity. The orientation properties of these fields isreferred to as the polarization of the light. A polarimeteris an instrument that measures the polarization of light togain additional information. Such instruments are appliedin a wide range of fields: from astronomy to characterizationof semiconductor components, remote sensing, and medicaldiagnostics.

Any polarization state of light can be described by a 4element Stokes vector, S [2]. A Stokes polarimeter mea-sure these elements by using optical components such as alinear polarizer and e.g. ferroelectric liquid crystals (FLC)to measure the intensity of light projected along different

Copyright is held by the author/owner(s).GECCO’11, July 12–16, 2011, Dublin, Ireland.ACM 978-1-4503-0690-4/11/07.

polarization states. For dispersive optical components, op-timal projection states can only be achieved for a singlewavelength. Finding the optimal polarimeter operating overa broad spectrum of wavelengths turns out to be a chal-lenging optimization problem, well suited for genetic algo-rithms (GA).

Figure 1: An example 2-FLC polarimeter where thepolarization state of incident light is analyzed bythe Polarization State Analyzer and a light intensitydetector.

The change of a polarization state can be described by a4 × 4 transformation matrix called a Mueller matrix, con-necting an incoming Stokes vector to an outgoing Stokesvector, Sout = MSin. The Mueller matrix can describe theeffect of any linear interaction of light with a sample or anoptical element. Polarization effects contained in a Muellermatrix could be diattenuation, retardance, and depolariza-tion. Due to dispersion the Mueller matrix will in generalbe wavelength dependent.

The measured intensities of a polarimetric measurementare arranged into a vector b, and is related to the unknownStokes vector S through b = AS, where A is a systemmatrix, which can be found from the Mueller matrices ofthe components. The unknown Stokes vector can then befound by inverting A, S = A−1b. The noise in the in-tensity measurements b will be amplified by the conditionnumber (κ) of A in the inversion. Therefore κ should be assmall as possible [4]. The condition number of A is givenas κ = ‖A‖‖A−1‖, which for the 2-norm equals the ra-tio of the largest to the smallest singular value. The bestcondition number that can be achieved for a polarimeter isκ =√

3 [4]. If more than 4 measurements are performed, Awill not be square, and the Moore-Penrose pseudoinverse isused to invert A.

To evaluate the performance of a polarimeter design, wecompare the inverse condition number to the theoretically

optimal value (1/√

3). We define an fitness function (f) as

1

f=

1

Nλ

Nλ∑n=1

(κ−1(λn)− 1/

√3)4.

We take(κ−1 − 1/

√3)

to the 4th power to punish unwantedpeaks in κ more severely.

While the GA is able to handle any kind of optical com-ponents, we focus on systems based on FLCs as polariza-tion modulators, with fixed waveplates “sandwiched” be-tween them to increase the condition number. The designwe set out to find was one based on three FLC retarders andthree fixed waveplates. Each FLC has two variables, namelyits thickness and its orientation angle. The same is true forthe fixed waveplates. This yields a 12-dimensional searchspace: six components with two variables each.

400 800 1200 1600 2000

Wavelength (λ) [nm]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Inv.

condit

ion

num

ber

(κ−

1)

GA design

Patent

1/√

3

Figure 2: Inverse condition number (κ−1(λ)) for aGA-generated and a previously patented design [1].

In our GA we represent polarimeter designs using a tradi-tional binary genome. Each component is assigned a numberof bits for the orientation angle and a number of bits for itsthickness. Given these parameters the condition number κof the system matrix A can be determined [3].

The genetic operator that was used for mutation was thesimple bit-flip operator. The mutation rate per individualwas typically set to 0.2 per generation. Crossover was per-formed by standard multi-point crossover. Experience indi-cates that two crossover points combined with a crossoverrate of 0.7 gives the best convergence performance. Theselection protocol we used was tournament selection withK = 4 individuals in the tournament pool and ε = 0.3 prob-ability of an“underdog”selection. The elitism rate was set to1 individual per generation. A population of 500 individualsevolving over 600 generations was used. Several equivalentsimulation runs were performed with different initializationsof the random number generator.

2. RESULTSThe best polarimeter design found by GA is shown in

Figure 2. For comparison with previous designs, we alsoshow a recently patented design. Both designs are basedon three FLCs. The new design is useful over a broaderspectral range and has lower noise amplification due to alower condition number. The condition number is improved

Figure 3: Fitness landscape around the optimalvalue for the GA-generated design. θ1 and θ2 arethe orientation angles of two of the components.

by up to a factor of 2, which means the noise amplificationcan be reduced by a factor of 2.

One can get an impression of how complex the fitnesslandscape is from Figure 3, where a plot of f depending on2 of the 12 parameters are shown.

In conclusion, the use of GA allows for quick generationof good polarimeter designs. The generated designs outper-form previously published and patented designs.

3. ACKNOWLEDGEMENTSWe thank prof. Keith Downing at the Dept. of Computer

and Information Science at NTNU for helpful discussions.

4. REFERENCES[1] D. Cattelan, E. Garcia-Caurel, A. de Martino, and

B. Drevillon. Device and method for takingspectroscopic polarimetric measurements in the visibleand near-infrared ranges. Patent application,2937732(France), 2010.

[2] E. Collett. Polarized light. Fundamentals andapplications. Marcel Dekker, Inc., New York, 1993.

[3] P. A. Letnes, I. S. Nerbø, L. M. S. Aas, P. G. Ellingsen,and M. Kildemo. Fast and optimal broad-bandStokes/Mueller polarimeter design by the use of agenetic algorithm. Opt. Express, 18(22):23095–23103,2010.

[4] J. S. Tyo. Noise equalization in Stokes parameterimages obtained by use of variable-retardancepolarimeters. Opt. Lett., 25:1198–1200, 2000.