## Fast and optimal broad-band Stokes/Mueller polarimeter design by the use of a genetic algorithm |

Optics Express, Vol. 18, Issue 22, pp. 23095-23103 (2010)

http://dx.doi.org/10.1364/OE.18.023095

Acrobat PDF (871 KB)

### Abstract

A fast multichannel Stokes/Mueller polarimeter with no mechanically moving parts has been designed to have close to optimal performance from 430 – 2000 nm by applying a genetic algorithm. Stokes (Mueller) polarimeters are characterized by their ability to analyze the full Stokes (Mueller) vector (matrix) of the incident light (sample). This ability is characterized by the condition number, *κ*, which directly influences the measurement noise in polarimetric measurements. Due to the spectral dependence of the retardance in birefringent materials, it is not trivial to design a polarimeter using dispersive components. We present here both a method to do this optimization using a genetic algorithm, as well as simulation results. Our results include fast, broad-band polarimeter designs for spectrographic use, based on 2 and 3 Ferroelectric Liquid Crystals, whose material properties are taken from measured values. The results promise to reduce the measurement noise significantly over previous designs, up to a factor of 4.5 for a Mueller polarimeter, in addition to extending the spectral range.

© 2010 Optical Society of America

## 1. Introduction

1. A. M. Gandorfer, “Ferroelectric retarders as an alternative to piezoelastic modulators for use in solar Stokes vector polarimetry,” Opt. Eng. **38**, 1402–1408 (1999). [CrossRef]

4. J. D. Howe, M. A. Miller, R. V. Blumer, T. E. Petty, M. A. Stevens, D. M. Teale, and M. H. Smith, “Polarization sensing for target acquisition and mine detection,” in *Polarization Analysis, Measurement, and Remote Sensing III*, D. B. Chenault, M. J. Duggin, W. G. Egan, and D. H. Goldstein, eds., Proc. SPIE **4133**, 202–213 (2000).

6. R. N. Weinreb, S. Shakiba, and L. Zangwill, “Scanning laser polarimetry to measure the nerve fiber layer of normal and glaucomatous eyes,” Am. J. Ophthalmol. **119**, 627–636 (1995). [PubMed]

7. M. Foldyna, A. D. Martino, R. Ossikovski, E. Garcia-Caurel, and C. Licitra, “Characterization of grating structures by Mueller polarimetry in presence of strong depolarization due to finite spot size,” Opt. Commun. **282**, 735–741 (2009). [CrossRef]

8. I. S. Nerbø, S. Le Roy, M. Foldyna, M. Kildemo, and E. Søndergård, “Characterization of inclined GaSb nanopillars by Mueller matrix ellipsometry,” J. Appl. Phys. **108**, 014307 (2010). [CrossRef]

9. L. Jin, M. Kasahara, B. Gelloz, and K. Takizawa, “Polarization properties of scattered light from macrorough surfaces,” Opt. Lett. **35**, 595–597 (2010). [CrossRef] [PubMed]

11. T. Germer, “Measurement of roughness of two interfaces of a dielectric film by scattering ellipsometry,” Phys. Rev. Lett. **85**, 349–352 (2000). [CrossRef] [PubMed]

*κ*) of these system matrices is minimized [12

12. F. Stabo-Eeg, M. Kildemo, I. Nerbø, and M. Lindgren, “Well-conditioned multiple laser Mueller matrix ellipsometer,” Opt. Eng. **47**, 073604 (2008). [CrossRef]

13. J. S. Tyo, “Noise equalization in Stokes parameter images obtained by use of variable-retardance polarimeters,” Opt. Lett. **25**, 1198–1200 (2000). [CrossRef]

*e.g.*rotating retarders [14

14. D. S. Sabatke, M. R. Descour, E. L. Dereniak, W. C. Sweatt, S. A. Kemme, and G. S. Phipps, “Optimization of retardance for a complete Stokes polarimeter,” Opt. Lett. **25**, 802–804 (2000). [CrossRef]

15. R. M. A. Azzam and A. De, “Optimal beam splitters for the division-of-amplitude photopolarimeter,” J. Opt. Soc. Am. A **20**, 955–958 (2003). [CrossRef]

16. R. M. A. Azzam, “Photopolarimetric measurement of the Mueller matrix by Fourier analysis of a single detected signal,” Opt. Lett. **2**, 148 (1978). [CrossRef] [PubMed]

17. J. M. Bueno, “Polarimetry using liquid-crystal variable retarders: theory and calibration,” J. Opt. A: Pure Appl. Opt.2, 216–222 (2000). [CrossRef]

*e.g.*by using a Charge-Coupled Device (CCD) based spectrograph) [18

18. E. Garcia-Caurel, A. D. Martino, and B. Drévillon, “Spectroscopic Mueller polarimeter based on liquid crystal devices,” Thin Solid Films **455–456**, 120–123 (2004). [CrossRef]

18. E. Garcia-Caurel, A. D. Martino, and B. Drévillon, “Spectroscopic Mueller polarimeter based on liquid crystal devices,” Thin Solid Films **455–456**, 120–123 (2004). [CrossRef]

20. L. M. S. Aas, P. G. Ellingsen, M. Kildemo, and M. Lindgren, “Dynamic Response of a fast near infra-red Mueller matrix ellipsometer,” J. Mod. Opt. (accepted) (2010). [CrossRef]

*e.g.*Ref. [23]. GAs have previously been applied in ellipsometry to solve the inversion problem for the thickness and dielectric function of multiple thin layers, see

*e.g.*Ref. [24

24. A. Kudla, “Application of the genetic algorithms in spectroscopic ellipsometry,” Thin Solid Films455–456, 804–808 (2004). [CrossRef]

26. V. R. Fernandes, C. M. S. Vicente, N. Wada, P. S. André, and R. A. S. Ferreira, “Multi-objective genetic algorithm applied to spectroscopic ellipsometry of organic-inorganic hybrid planar waveguides,” Opt. Express **18**, 16580–16586 (2010). [CrossRef] [PubMed]

## 2. Overdetermined polarimetry

**S**can then be expressed as

**S**=

**A**

^{−1}

**b**, where

**A**is a system matrix describing the PSA and

**b**is a vector containing the intensity measurements.

**A**

^{−1}denotes the matrix inverse of

**A**, which in the case of overdetermined polarimetry with more than 4 projection states will denote the Moore–Penrose

*pseudoinverse*. The analyzing matrix

**A**is constructed from the first rows of the Mueller matrices of the PSA for the different states. The noise in the measurements of

**b**will be amplified by the condition number of

**A**,

*κ*

**, in the inversion to find**

_{A}**S**. Therefore

*κ*

**should be as small as possible, which correspond to do as independent measurements as possible (**

_{A}*i.e.*to use projection states that are as orthogonal as possible).

**M**describes how an interaction changes the polarization state of light, by transforming an incoming Stokes vector

**S**

_{in}to the outgoing Stokes vector

**S**

_{out}=

**MS**

_{in}. To measure the Mueller matrix of a sample it is necessary to generate at least 4 different polarization states by a polarization state generator (PSG) and measure the outgoing Stokes vector by at least 4 measurements for each generated state. The measured intensities can then be arranged in a matrix

**B**=

**AMW**, where the system matrix

**W**of the PSG contains the generated Stokes vectors as its columns. These generated Stokes vectors are found simply as the first column of the Mueller matrix of the PSG in the respective states.

**M**can then be found by inversion as

**M**=

**A**

^{−1}

**BW**

^{−1}. The error Δ

**M**in

**M**is then bounded by the condition numbers according to [27

27. F. Stabo-Eeg, M. Kildemo, E. Garcia-Caurel, and M. Lindgren, “Design and characterization of achromatic 132° retarders in CaF_{2} and fused silica,” J. Mod. Opt. **55**, 2203–2214 (2008). [CrossRef]

*κ*

**= ||**

_{A}**A**||||

**A**

^{−1}||, which for the the 2-norm can be calculated from the ratio of the largest to the smallest singular value [28]. Δ

**A**and Δ

**W**are calibration errors, which increase with

*κ*when calibration methods using matrix inversion are applied. The PSG can be constructed from the same optical elements as the PSA, placed in the reverse order, which would give

*κ*

**=**

_{A}*κ*

**≡**

_{W}*κ*. As the error in Mueller matrix measurements is proportional to

*κ*

^{2}, it is very important to keep this value as low as possible.

14. D. S. Sabatke, M. R. Descour, E. L. Dereniak, W. C. Sweatt, S. A. Kemme, and G. S. Phipps, “Optimization of retardance for a complete Stokes polarimeter,” Opt. Lett. **25**, 802–804 (2000). [CrossRef]

^{3}= 8 different states can be analyzed (generated) by the PSA (PSG). To accurately measure the Stokes vector, the system matrix

**A**needs to be well known. For a Mueller polarimeter generating and analyzing 4 states in the PSG and PSA, the eigenvalue calibration method (ECM) [29

29. E. Compain, S. Poirier, and B. Drevillon, “General and self-consistent method for the calibration of polarization modulators, polarimeters, and Mueller-matrix ellipsometers,” Appl. Opt. **38**, 3490–3502 (1999). [CrossRef]

**A**and

**W**), without relying on exact knowledge or modeling of the optical components. However, the ECM is based on the inversion of a product of measured intensity matrices

**B**for measurements on a set of calibration samples. This product becomes singular for a system analyzing and generating more than four states. A workaround of this problem is to choose the subset of 4 out of 8 states which gives the lowest

*κ*value, and build a

**B**matrix of those states to find 4 of the 8 rows (columns) of

**A**(

**W**). More rows (columns) of

**A**(

**W**) can then be found by calibrating on a different subset of the 8 states, giving the second lowest

*κ*value, and so on. By repeating the calibration on different subsets of states, all the 8 rows (columns) of

**A**(

**W**) can be found with low relative error ||Δ

**A**||/||

**A**|| (||Δ

**W**||/||

**W**||).

## 3. Genetic optimization

*κ*(

*λ*), one can conceivably employ a variety of optimization algorithms, from simple brute-force exhaustive search to more advanced algorithms, such as

*e.g.*Levenberg–Marquardt, simulated annealing, and particle swarm optimization. Our group has previously performed optimization of a polarimeter design based on fixed components, namely, two FLCs and two waveplates. In this case, the optimization problem reduces to searching the space of 4 orientation angles. With a resolution of 1° per angle, this gives a search space consisting of 180

^{4}≈ 10

^{9}states to evaluate; on modern computer hardware, this direct search can be performed. In order to optimize the retardances of the components as well, the total number of states increases to about (10

^{9})

^{2}= 10

^{18}. Obviously, brute force exhaustive search is unfeasible for such large search spaces.

*i.e.*0 → 1 or vice versa. Sexual reproduction is simulated by using multi-point crossover,

*i.e.*simply cutting and pasting two genomes together, as described by Holland [22].

*i.e.*for each orientation angle and each retardance, we select

*m*bits in the genome (typically,

*m*= 8) and interpret this number as an integer in the range from 1 to 2

*. The integer is subsequently interpreted as a real number in a predefined range,*

^{m}*e.g.*,

*θ*∈ [0°, 180°]. In order to avoid excessively large jumps in the search space due to mutations, we chose to implement the interpretation of bits into integers by using the Gray code, also known as the reflected binary code. The most important parameter values in our GA are shown in Table 2. Making good choices for each of these parameters is often essential in order to ensure good convergence.

*κ*(

*λ*). As discussed,

*κ*

^{−1}(

*λ*) maximally takes on the value

*e*, as

*λ*=

_{n}*λ*

_{min}+ (

*n*– 1)Δ

*λ*, with

*n*= 1, 2,...,

*N*and Δ

_{λ}*λ*= 5 nm.

*λ*

_{min}and

*N*are determined by the wavelength range we are interested in. The choice of taking the difference between

_{λ}*κ*

^{−1}(

*λ*) and the optimal value to power 4 is done in order to “punish” peaks in the condition number more severely. As GAs conventionally seek to maximize the fitness function, we define an individual’s fitness as

*f*takes on real and positive values where higher values represents more optimal polarimeter designs.

## 4. Results

*κ*(

*λ*) by varying the orientation angle,

*θ*, and the retardance,

*δ*, of all the elements. This yields a 12-dimensional search space,

*i.e.*, 6 retardances and 6 orientation angles.

*θ*is the angle between the fast axis of the retarder (WP or FLC) and the transmission axis of the polarizer (see Fig. 2), taken to be in the range

*θ*∈ [0°,180°]. The retardance,

*δ*, is modeled using a modified Sellmeier equation, where

*A*,

_{UV}*A*,

_{IR}*λ*, and

_{UV}*λ*are experimentally determined parameters for an FLC (

_{IR}*λ*/2@510 nm, Displaytech Inc.) and a Quartz zero order waveplate (

*λ/*4@465 nm) taken directly from Refs. [19

19. J. Ladstein, M. Kildemo, G. Svendsen, I. Nerbø, and F. Stabo-Eeg, “Characterisation of liquid crystals for broadband optimal design of Mueller matrix ellipsometers,” in *Liquid Crystals and Applications in Optics*, M. Glogarova, P. Palffy-Muhoray, and M. Copic, eds. Proc. SPIE **6587**, 65870D (2007).

*A*= 0).

_{IR}*L*is a normalized thickness, with

*L*= 1 corresponding to a retardance of

*λ/*2@510 nm for the FLCs and

*λ*/4@465 nm for the waveplates. Each

*L*and

*θ*are represented by 8 bits each in the genome. We use experimental values to ensure that our design is based on as realistic components as possible.

*κ*

^{−1}(

*λ*) in Fig. 4. The inverse condition number,

*κ*

^{−1}, is larger than 0.5 over most parts of the spectrum, which is close to the optimal inverse condition number (

*κ*

^{−1}≈ 0.33. The new design promise a decrease in noise amplification by up to a factor of 21 for a Stokes polarimeter, and up to factor of 4.5 for a Mueller polarimeter. In addition the upper spectral limit is extended from 1700 nm to 2000 nm. Shorter wavelengths than 430 nm were not considered as the FLC material will be degraded by exposure to UV light. Previous designs often suffer from

*κ*

^{−1}(

*λ*) oscillating as a function of wavelength, whereas our solution is more uniform over the wavelength range we are interested in. This uniformity in

*κ*(

*λ*) will, according to Eq. (1), give a more uniform noise distribution over the spectrum.

*f*[see Eq. (3)] as a function of the generation number is shown in Fig. 5. The mean population fitness (

*μ*) and standard deviation (

*σ*) is also shown. As so often happens with genetic algorithms, we see that the maximal and average fitness increases dramatically in the first few generations. Following this fast initial progress, evolution slows down considerably, before it finally converges after 600 generations. The parameters used in our GA to obtain these results are shown in Table 2.

## 5. Conclusion

## Acknowledgments

## References and links

1. | A. M. Gandorfer, “Ferroelectric retarders as an alternative to piezoelastic modulators for use in solar Stokes vector polarimetry,” Opt. Eng. |

2. | P. Collins, R. Redfern, and B. Sheehan, “Design, construction and calibration of the Galway astronomical Stokes polarimeter (GASP),” in |

3. | A. Alvarez-Herrero, V. Martínez-Pillet, J. del Toro Iniesta, and V. Domingo, “The IMaX polarimeter for the solar telescope SUNRISE of the NASA long duration balloon program,” in Proceedings of API’09, E. Garcia-Caurel, ed. (EPJ Web of Conferences, 2010), vol. 5, p. 05002. |

4. | J. D. Howe, M. A. Miller, R. V. Blumer, T. E. Petty, M. A. Stevens, D. M. Teale, and M. H. Smith, “Polarization sensing for target acquisition and mine detection,” in |

5. | M. H. Smith, P. D. Burke, A. Lompado, E. A. Tanner, and L. W. Hillman, “Mueller matrix imaging polarimetry in dermatology,” in |

6. | R. N. Weinreb, S. Shakiba, and L. Zangwill, “Scanning laser polarimetry to measure the nerve fiber layer of normal and glaucomatous eyes,” Am. J. Ophthalmol. |

7. | M. Foldyna, A. D. Martino, R. Ossikovski, E. Garcia-Caurel, and C. Licitra, “Characterization of grating structures by Mueller polarimetry in presence of strong depolarization due to finite spot size,” Opt. Commun. |

8. | I. S. Nerbø, S. Le Roy, M. Foldyna, M. Kildemo, and E. Søndergård, “Characterization of inclined GaSb nanopillars by Mueller matrix ellipsometry,” J. Appl. Phys. |

9. | L. Jin, M. Kasahara, B. Gelloz, and K. Takizawa, “Polarization properties of scattered light from macrorough surfaces,” Opt. Lett. |

10. | T. A. Germer, “Polarized light scattering by microroughness and small defects in dielectric layers.” J. Opt. Soc. Am. A |

11. | T. Germer, “Measurement of roughness of two interfaces of a dielectric film by scattering ellipsometry,” Phys. Rev. Lett. |

12. | F. Stabo-Eeg, M. Kildemo, I. Nerbø, and M. Lindgren, “Well-conditioned multiple laser Mueller matrix ellipsometer,” Opt. Eng. |

13. | J. S. Tyo, “Noise equalization in Stokes parameter images obtained by use of variable-retardance polarimeters,” Opt. Lett. |

14. | D. S. Sabatke, M. R. Descour, E. L. Dereniak, W. C. Sweatt, S. A. Kemme, and G. S. Phipps, “Optimization of retardance for a complete Stokes polarimeter,” Opt. Lett. |

15. | R. M. A. Azzam and A. De, “Optimal beam splitters for the division-of-amplitude photopolarimeter,” J. Opt. Soc. Am. A |

16. | R. M. A. Azzam, “Photopolarimetric measurement of the Mueller matrix by Fourier analysis of a single detected signal,” Opt. Lett. |

17. | J. M. Bueno, “Polarimetry using liquid-crystal variable retarders: theory and calibration,” J. Opt. A: Pure Appl. Opt.2, 216–222 (2000). [CrossRef] |

18. | E. Garcia-Caurel, A. D. Martino, and B. Drévillon, “Spectroscopic Mueller polarimeter based on liquid crystal devices,” Thin Solid Films |

19. | J. Ladstein, M. Kildemo, G. Svendsen, I. Nerbø, and F. Stabo-Eeg, “Characterisation of liquid crystals for broadband optimal design of Mueller matrix ellipsometers,” in |

20. | L. M. S. Aas, P. G. Ellingsen, M. Kildemo, and M. Lindgren, “Dynamic Response of a fast near infra-red Mueller matrix ellipsometer,” J. Mod. Opt. (accepted) (2010). [CrossRef] |

21. | D. Cattelan, E. Garcia-Caurel, A. De Martino, and B. Drevillon, “Device and method for taking spectroscopic polarimetric measurements in the visible and near-infrared ranges,” Patent application 2937732, France (2010). |

22. | J. H. Holland, “Genetic algorithms,” Scientific American |

23. | D. Floreano and C. Mattiussi, |

24. | A. Kudla, “Application of the genetic algorithms in spectroscopic ellipsometry,” Thin Solid Films455–456, 804–808 (2004). [CrossRef] |

25. | G. Cormier and R. Boudreau, “Genetic algorithm for ellipsometric data inversion of absorbing layers,” J. Opt. Soc. Am. A |

26. | V. R. Fernandes, C. M. S. Vicente, N. Wada, P. S. André, and R. A. S. Ferreira, “Multi-objective genetic algorithm applied to spectroscopic ellipsometry of organic-inorganic hybrid planar waveguides,” Opt. Express |

27. | F. Stabo-Eeg, M. Kildemo, E. Garcia-Caurel, and M. Lindgren, “Design and characterization of achromatic 132° retarders in CaF |

28. | W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, |

29. | E. Compain, S. Poirier, and B. Drevillon, “General and self-consistent method for the calibration of polarization modulators, polarimeters, and Mueller-matrix ellipsometers,” Appl. Opt. |

30. | J. Ladstein, F. Stabo-Eeg, E. Garcia-Caurel, and M. Kildemo, “Fast near-infra-red spectroscopic Mueller matrix ellipsometer based on ferroelectric liquid crystal retarders,” Phys. Status Solidi C |

**OCIS Codes**

(120.2130) Instrumentation, measurement, and metrology : Ellipsometry and polarimetry

(120.4570) Instrumentation, measurement, and metrology : Optical design of instruments

(300.0300) Spectroscopy : Spectroscopy

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: September 9, 2010

Revised Manuscript: October 5, 2010

Manuscript Accepted: October 5, 2010

Published: October 18, 2010

**Citation**

Paul Anton Letnes, Ingar Stian Nerbø, Lars Martin S. Aas, Pål Gunnar Ellingsen, and Morten Kildemo, "Fast and optimal broad-band Stokes/Mueller polarimeter design by the use of a genetic algorithm," Opt. Express **18**, 23095-23103 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-22-23095

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### References

- A. M. Gandorfer, "Ferroelectric retarders as an alternative to piezoelastic modulators for use in solar Stokes vector polarimetry," Opt. Eng. 38, 1402-1408 (1999). [CrossRef]
- P. Collins, R. Redfern, and B. Sheehan, "Design, construction and calibration of the Galway astronomical Stokes polarimeter (GASP)," in AIP Conference Proceedings, D. Phelan, O. Ryan, and A. Shearer, eds. (AIP, Edinburgh (Scotland), 2008), vol. 984, p. 241. [CrossRef]
- A. Alvarez-Herrero, V. Martínez-Pillet, J. del Toro Iniesta, and V. Domingo, "The IMaX polarimeter for the solar telescope SUNRISE of the NASA long duration balloon program," in Proceedings of API’09, E. Garcia-Caurel, ed. (EPJ Web of Conferences, 2010), vol. 5, p. 05002.
- J. D. Howe, M. A. Miller, R. V. Blumer, T. E. Petty, M. A. Stevens, D. M. Teale, and M. H. Smith, "Polarization sensing for target acquisition and mine detection," in Polarization Analysis, Measurement, and Remote Sensing III, D. B. Chenault, M. J. Duggin, W. G. Egan, and D. H. Goldstein, eds., Proc. SPIE 4133, 202-213 (2000).
- M. H. Smith, P. D. Burke, A. Lompado, E. A. Tanner, and L. W. Hillman, "Mueller matrix imaging polarimetry in dermatology," in Biomedical Diagnostic, Guidance, and Surgical-Assist Systems II, T. Vo-Dinh, W. S. Grundfest, and D. A. Benaron, eds., Proc. SPIE 3911, 210-216 (2000).
- R. N. Weinreb, S. Shakiba, and L. Zangwill, "Scanning laser polarimetry to measure the nerve fiber layer of normal and glaucomatous eyes," Am. J. Ophthalmol. 119, 627-636 (1995). [PubMed]
- M. Foldyna, A. D. Martino, R. Ossikovski, E. Garcia-Caurel, and C. Licitra, "Characterization of grating structures by Mueller polarimetry in presence of strong depolarization due to finite spot size," Opt. Commun. 282, 735-741 (2009). [CrossRef]
- I. S. Nerbø, S. Le Roy, M. Foldyna, M. Kildemo, and E. Søndergård, "Characterization of inclined GaSb nanopillars by Mueller matrix ellipsometry," J. Appl. Phys. 108, 014307 (2010). [CrossRef]
- L. Jin, M. Kasahara, B. Gelloz, and K. Takizawa, "Polarization properties of scattered light from macrorough surfaces," Opt. Lett. 35, 595-597 (2010). [CrossRef] [PubMed]
- T. A. Germer, "Polarized light scattering by microroughness and small defects in dielectric layers," J. Opt. Soc. Am. A 18, 1279-1288 (2001). [CrossRef]
- T. Germer, "Measurement of roughness of two interfaces of a dielectric film by scattering ellipsometry," Phys. Rev. Lett. 85, 349-352 (2000). [CrossRef] [PubMed]
- F. Stabo-Eeg, M. Kildemo, I. Nerbø, and M. Lindgren, "Well-conditioned multiple laser Mueller matrix ellipsometer," Opt. Eng. 47, 073604 (2008). [CrossRef]
- J. S. Tyo, "Noise equalization in Stokes parameter images obtained by use of variable-retardance polarimeters," Opt. Lett. 25, 1198-1200 (2000). [CrossRef]
- D. S. Sabatke, M. R. Descour, E. L. Dereniak, W. C. Sweatt, S. A. Kemme, and G. S. Phipps, "Optimization of retardance for a complete Stokes polarimeter," Opt. Lett. 25, 802-804 (2000). [CrossRef]
- R. M. A. Azzam, and A. De, "Optimal beam splitters for the division-of-amplitude photopolarimeter," J. Opt. Soc. Am. A 20, 955-958 (2003). [CrossRef]
- R. M. A. Azzam, "Photopolarimetric measurement of the Mueller matrix by Fourier analysis of a single detected signal," Opt. Lett. 2, 148 (1978). [CrossRef] [PubMed]
- J. M. Bueno, "Polarimetry using liquid-crystal variable retarders: theory and calibration," J. Opt. A, Pure Appl. Opt. 2, 216-222 (2000). [CrossRef]
- E. Garcia-Caurel, A. D. Martino, and B. Drévillon, "Spectroscopic Mueller polarimeter based on liquid crystal devices," Thin Solid Films 455-456, 120-123 (2004). [CrossRef]
- J. Ladstein, M. Kildemo, G. Svendsen, I. Nerbø, and F. Stabo-Eeg, "Characterisation of liquid crystals for broadband optimal design of Mueller matrix ellipsometers," in Liquid Crystals and Applications in Optics, M. Glogarova, P. Palffy-Muhoray, and M. Copic, eds. Proc. SPIE 6587, 65870D (2007).
- L. M. S. Aas, P. G. Ellingsen, M. Kildemo, and M. Lindgren, "Dynamic Response of a fast near infra-red Mueller matrix ellipsometer," J. Mod. Opt. (accepted). [CrossRef]
- D. Cattelan, E. Garcia-Caurel, A. De Martino, and B. Drevillon, "Device and method for taking spectroscopic polarimetric measurements in the visible and near-infrared ranges," Patent application 2937732, France (2010).
- J. H. Holland, "Genetic algorithms," Sci. Am. 267, 44-50 (1992).
- D. Floreano, and C. Mattiussi, Bio-Inspired Artificial Intelligence: Theories, Methods, and Technologies (The MIT Press, 2008).
- A. Kudla, "Application of the genetic algorithms in spectroscopic ellipsometry," Thin Solid Films 455-456, 804-808 (2004). [CrossRef]
- G. Cormier, and R. Boudreau, "Genetic algorithm for ellipsometric data inversion of absorbing layers," J. Opt. Soc. Am. A 17, 129-134 (2000). [CrossRef]
- V. R. Fernandes, C. M. S. Vicente, N. Wada, P. S. André, and R. A. S. Ferreira, "Multi-objective genetic algorithm applied to spectroscopic ellipsometry of organic-inorganic hybrid planar waveguides," Opt. Express 18, 16580-16586 (2010). [CrossRef] [PubMed]
- F. Stabo-Eeg, M. Kildemo, E. Garcia-Caurel, and M. Lindgren, "Design and characterization of achromatic 132◦ retarders in CaF2 and fused silica," J. Mod. Opt. 55, 2203-2214 (2008). [CrossRef]
- W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, 2007).
- E. Compain, S. Poirier, and B. Drevillon, "General and self-consistent method for the calibration of polarization modulators, polarimeters, and Mueller-matrix ellipsometers," Appl. Opt. 38, 3490-3502 (1999). [CrossRef]
- J. Ladstein, F. Stabo-Eeg, E. Garcia-Caurel, and M. Kildemo, "Fast near-infra-red spectroscopic Mueller matrix ellipsometer based on ferroelectric liquid crystal retarders," Phys. Status Solidi C 5, 1097-1100 (2008). [CrossRef]

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