## Overdetermined broadband spectroscopic Mueller matrix polarimeter designed by genetic algorithms |

Optics Express, Vol. 21, Issue 7, pp. 8753-8762 (2013)

http://dx.doi.org/10.1364/OE.21.008753

Acrobat PDF (1062 KB)

### Abstract

This paper reports on the design and implementation of a liquid crystal variable retarder based overdetermined spectroscopic Mueller matrix polarimeter, with parallel processing of all wavelengths. The system was designed using a modified version of a recently developed genetic algorithm [Letnes et al. Opt. Express **18**, 22, 23095 (2010)]. A generalization of the eigenvalue calibration method is reported that allows the calibration of such overdetermined polarimetric systems. Out of several possible designs, one of the designs was experimentally implemented and calibrated. It is reported that the instrument demonstrated good performance, with a measurement accuracy in the range of 0.1% for the measurement of air.

© 2013 OSA

## 1. Introduction

1. P. G. Ellingsen, M. B. Lilledahl, L. M. S. Aas, C. d. L. Davies, and M. Kildemo, “Quantitative characterization of articular cartilage using Mueller matrix imaging and multiphoton microscopy,” J. Biomed. Opt. **16**, 116002 (2011) [CrossRef] [PubMed] .

3. 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] .

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,” Proc. SPIE **4133**, 202–213 (2000) [CrossRef] .

8. H. Tompkins and E. A. Irene, *Handbook of Ellipsometry* (William Andrew, 2005) [CrossRef] .

9. 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] .

10. D. Schmidt, A. C. Kjerstad, T. Hofmann, R. Skomski, E. Schubert, and M. Schubert, “Optical, structural, and magnetic properties of cobalt nanostructure thin films,” J. Appl. Phys. **105**, 113508 (2009) [CrossRef] .

12. 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] .

13. T. Oates, H. Wormeester, and H. Arwin, “Characterization of plasmonic effects in thin films and metamaterials using spectroscopic ellipsometry,” Prog. Surf. Sci. **86**, 328–376 (2011) [CrossRef] .

14. B. Gallas, K. Robbie, R. Abdeddaïm, G. Guida, J. Yang, J. Rivory, and a. Priou, “Silver square nanospirals mimic optical properties of U-shaped metamaterials.” Opt. Express **18**, 16335–16344 (2010) [CrossRef] [PubMed] .

15. T. A. Germer, “Polarized light scattering by microroughness and small defects in dielectric layers.” J. Opt. Soc. Am. A **18**, 1279–1288 (2001) [CrossRef] .

17. Ø. Svensen, M. Kildemo, J. Maria, J. J. Stamnes, and O. Frette, “Mueller matrix measurements and modeling pertaining to Spectralon white reflectance standards.” Opt. Express **20**, 15045–15053 (2012) [CrossRef] [PubMed] .

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

20. E. Compain and B. Drevillon, “Complete high-frequency measurement of Mueller matrices based on a new coupled-phase modulator,” Rev. Sci. Instrum. **68**, 2671 (1997) [CrossRef] .

21. O. Arteaga, J. Freudenthal, B. Wang, and B. Kahr, “Mueller matrix polarimetry with four photoelastic modulators: theory and calibration.” Appl. Optics **51**, 6805–6817 (2012) [CrossRef] .

22. G. E. Jellison and F. a. Modine, “Two-channel polarization modulation ellipsometer.” Appl. Optics **29**, 959–974 (1990) [CrossRef] .

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

24. L. Aas, P. Ellingsen, and M. Kildemo, “Near infra-red Mueller matrix imaging system and application to retardance imaging of strain,” Thin Solid Films **519**, 2737–2741 (2010) [CrossRef] .

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

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

25. P. Letnes, I. Nerbø, L. Aas, P. Ellingsen, and M. Kildemo, “Fast and optimal broad-band Stokes/Mueller polarimeter design by the use of a genetic algorithm,” Opt. Express **18**, 23095–23103 (2010) [CrossRef] [PubMed] .

25. P. Letnes, I. Nerbø, L. Aas, P. Ellingsen, and M. Kildemo, “Fast and optimal broad-band Stokes/Mueller polarimeter design by the use of a genetic algorithm,” Opt. Express **18**, 23095–23103 (2010) [CrossRef] [PubMed] .

26. J. H. Holland, “Genetic algorithms,” Sci. Am. **267**, 44–50 (1992) [CrossRef] .

25. P. Letnes, I. Nerbø, L. Aas, P. Ellingsen, and M. Kildemo, “Fast and optimal broad-band Stokes/Mueller polarimeter design by the use of a genetic algorithm,” Opt. Express **18**, 23095–23103 (2010) [CrossRef] [PubMed] .

## 2. Theory

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

*E*(

_{x}*t*) and

*E*(

_{y}*t*) are time dependent, electric field amplitudes of the

*x*− and

*y*−components, of an electric field propagating in the

*z*−direction. 〈·〉 denotes time averages and

*δ*(

*t*) is the time dependent phase difference between the

*x*− and

*y*−components of the electric field. Note that the averaging of time varying amplitudes and phases, results in a reduced degree of polarization.

**S**

_{final}=

**M**

_{sample}

**S**

_{initial}. A Mueller matrix can describe all changes in the polarization state of light upon the interaction with a sample, with quantifiable effects, like for instance polarizance, diattenuation, retardance and depolarization [18].

**M**, one needs at least four probing Stokes vectors [18]. Consequently these Stokes vectors need to be measured by a polarimeter/PSA. A polarimeter/PSA projects the incoming intensity to at least four carefully selected polarization states. These states are the Stokes vectors in the PSA, organized into the rows of the PSA matrix

**A**. The intensity vector,

**b**=

**AS**, for an incoming Stokes vector can then be measured, and the Stokes vector found by inversion;

**S**=

**A**

^{−1}

**b**. Similarly, for the PSG in the Mueller matrix polarimeter/ellipsometer, the generated Stokes vectors are organized as columns in the

**W**matrix. The product

**MW**gives the Stokes vectors for the PSA to analyze, yielding the total intensity measurement matrix

**B**=

**AMW**. As a result, the Mueller matrix can then, in principle, readily be calculated by inversion of

**A**and

**W**,

**M**=

**A**

^{−1}

**BW**

^{−1}. There are several ways of finding

**A**and

**W**for a system, but a common method is the robust and increasingly popular ECM [28

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

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

24. L. Aas, P. Ellingsen, and M. Kildemo, “Near infra-red Mueller matrix imaging system and application to retardance imaging of strain,” Thin Solid Films **519**, 2737–2741 (2010) [CrossRef] .

29. S. B. Hatit, M. Foldyna, A. De Martino, and B. Drévillon, “Angle-resolved Mueller polarimeter using a microscope objective,” Phys. Stat. Sol. (a) **205**, 743–747 (2008) [CrossRef] .

**A**and

**W**result from six states in the PSA and PSG, corresponding to 12 specific Stokes vectors

**S**

_{W}_{1−6}and

**S**

_{A}_{1−6}

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

*n*Stokes vectors in the PSA and

*m*Stokes vectors in the PSG. We start with a set of reference Mueller matrices, {

**M**}, corresponding to a set of intensity measurements, {

**B**}.

**B**

*is of size*

_{i}*n*×

*m*and is given by For convenience, reference sample

**M**

_{0}is chosen to be air, such that

**B**

_{0}=

**AW**. Next, two sets of matrices, {

**C**} and {

**C**′}, are constructed using where

^{†}denotes the Moore-Penrose pseudoinverse, which is the common way of defining the inverse of a non-square matrix with noise [30]. In the case where

**B**

*is of size four by four, the sets {*

_{i}**C**} and {

**C**′} have the same eigenvalues as the set of Mueller matrices {

**M**}. As

**C**

*is independent of*

_{i}**A**, and

**C**′

*is independent of*

_{i}**W**,

**A**and

**W**may be found independently, and {

**M**} may be found both from {

**C**} and {

**C**′} independently.

**M**} from {

**C**} and {

**C**′}, is to (at each wavelength) reduce

**B**

_{0}and

**B**

*in Eq. (1) to the 4 × 4 subset of*

_{i}**B**

_{0}and

**B**

*resulting in the lowest condition number for the reduced*

_{i}**B**

_{0}. When inverting

**B**

_{0}, this ensures minimal noise propagation into

**C**

*and to the eigenvalues of*

_{i}**M**

*. After finding the eigenvalues of {*

_{i}**M**}, the remainder of the calibration procedure follows the original paper by Compain et al. [28

**38**, 3490–3502 (1999) [CrossRef] .

**B**}.

## 3. Experimental

*LPNIR*) from

*Thorlabs*and a true zero-order quarter waveplate at 1310 nm from

*Casix*. The LCVRs were custom made for the near infrared from

*Meadowlark Optics*.

**18**, 23095–23103 (2010) [CrossRef] [PubMed] .

32. L. Aas, P. Ellingsen, M. Kildemo, and M. Lindgren, “Dynamic Response of a fast near infra-red Mueller matrix ellipsometer,” J. Mod. Opt. **57**, 1603–1610 (2010) [CrossRef] .

*RC2*from

*J.A. Woollam Co.*. Figure 1(a) shows as an example, a surface plot of the resulting retardance as a function of voltage and wavelength for one of the LCVRs. It is noted that for lower voltages the retardance reaches a threshold at the critical voltage 1.5 V, while it approaches a low residual retardance for high voltages. The large retardance in the visible insures the possibility for a reasonable retardance variation in the longer wavelengths of the NIR spectrum.

*Pyevolve*library [33

33. C. S. Perone, “Pyevolve,” http://pyevolve.sourceforge.net/.

*e*is the error function defined as Here

*N*is the total number of wavelengths and

_{λ}*κ*(

*λ*) is the generalized condition number of

_{n}**A**or

**W**for a given wavelength

*λ*. This fitness function is similar to the one previously defined in [25

_{n}**18**, 23095–23103 (2010) [CrossRef] [PubMed] .

**A**or

**W**. As for the previous fitness function, it punishes inverse condition numbers far away from the theoretical maximum inverse condition number (

*NirQuest512*spectrograph from

*Ocean Optics*. The number of generations, population size and mutation rate were found by trial and error, by encouraging diversity and avoiding formation of large groups of individuals focused around one minimum. Elitism (copying of the best individual from one generation to the next), two point crossover and tournament were also used.

## 4. Results and discussion

*π*/2 is available. For a large number of wavelengths (typically > 500), such a system results in an unreasonable high total measurement time. Therefore, several Mueller matrix polarimeters were designed and evaluated with only a limited number of states in the PSG and the PSA, in order to keep the measurement time low. Three Mueller matrix polarimeter designs were optimized and evaluated. All designs used an equal number of states in the PSG and the PSA. The first design used two states in each of the LCVRs, totaling 4 (2 × 2) states for the PSA or the PSG, and 16 (4

^{2}) states for the complete system. The second design had two states in the first LCVR and three in the second LCVR, totaling 6 (2 × 3) states for the PSA/PSG and 36 (6

^{2}) in the complete system. Similarly the last design had two states in the first and four in the second LCVR, resulting in a total of 8 (2 × 4) states for the PSA/PSG, and 64 (8

^{2}) for the complete polarimeter. For simplicity, these three designs will from now on be denoted as 2 × 2, 2 × 3 and 2 × 4. The resulting designs are summarized in Table 2, whereas the resulting inverse condition numbers are shown in Fig. 3. It is evident that by going from a 2 × 2 to a 2 × 3 design the condition number is increased on the two edges of the spectrum. By moving to 2 × 4 states, the condition number increases over the whole spectrum. It is clear that there will be a trade-off between the measurement time and the gain in the increased condition number, and hence the noise reduction at the edges of the spectrum. As a compromise, we found it practical to use the 2 × 3 design for the experimental realization of the Mueller matrix polarimeter.

**18**, 23095–23103 (2010) [CrossRef] [PubMed] .

**W**and

**A**matrix was determined in the range 350 – 1680 nm, by selecting the first column of the measured Mueller matrix. The resulting inverse generalized condition number is shown in Fig. 3(b) (in green stippled lines), together with the simulated inverse generalized condition number resulting from the design obtained by the genetic algorithm (solid line). As seen, the correspondence between the experimental and simulated inverse generalized condition numbers is excellent. This demonstrates the power of such a system design and implementation, utilizing genetic algorithms and re-characterization of the optical components after arrival. The PSG and the PSA were then mounted in the transmission geometry shown in Fig. 2, making up the complete LCVR based Mueller matrix polarimeter.

**A**and

**W**were found using the ECM, as discussed in section 2. The generalized condition number of the calibrated

**W**matrix is plotted in Fig. 3(b). It is observed from the figure that the general spectral features in the optimized condition number are reproduced in the experimental version, although it suffers from small offsets in some parts of the spectrum. The latter offsets are possible due to that the components of the PSA/PSG were slightly realigned between the measurement using the commercial Mueller matrix ellipsometer compared to the final implementation of the LCVR Mueller matrix polarimeter. On the other hand, it shows that the modified ECM automatically compensates for alignment errors during assembly, and the final system calibrate correctly with only minor changes in the propagation of noise.

*m*

_{11}element). The deviation from the identity matrix is small, since the error is less than 0.1% over most of the spectrum, and is never more than 0.17%.

## 5. Conclusion

## Acknowledgment

## References and links

1. | P. G. Ellingsen, M. B. Lilledahl, L. M. S. Aas, C. d. L. Davies, and M. Kildemo, “Quantitative characterization of articular cartilage using Mueller matrix imaging and multiphoton microscopy,” J. Biomed. Opt. |

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

3. | 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. |

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,” Proc. SPIE |

5. | 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 API’09, (2010), pp. 05002. |

6. | R. Azzam and N. Bashara, |

7. | H. Fujiwara, |

8. | H. Tompkins and E. A. Irene, |

9. | 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. |

10. | D. Schmidt, A. C. Kjerstad, T. Hofmann, R. Skomski, E. Schubert, and M. Schubert, “Optical, structural, and magnetic properties of cobalt nanostructure thin films,” J. Appl. Phys. |

11. | L. M. S. Aas, M. Kildemo, Y. Cohin, and E. Søndergård, “Determination of small tilt angles of short gasb nanopillars using uv-visible mueller matrix ellipsometry,” Thin Solid Films (2012) [CrossRef] . |

12. | 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. |

13. | T. Oates, H. Wormeester, and H. Arwin, “Characterization of plasmonic effects in thin films and metamaterials using spectroscopic ellipsometry,” Prog. Surf. Sci. |

14. | B. Gallas, K. Robbie, R. Abdeddaïm, G. Guida, J. Yang, J. Rivory, and a. Priou, “Silver square nanospirals mimic optical properties of U-shaped metamaterials.” Opt. Express |

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

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

17. | Ø. Svensen, M. Kildemo, J. Maria, J. J. Stamnes, and O. Frette, “Mueller matrix measurements and modeling pertaining to Spectralon white reflectance standards.” Opt. Express |

18. | J. M. Bennet, R. Chipman, and R. M. A. Azzam, “Polarized light,” in |

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

20. | E. Compain and B. Drevillon, “Complete high-frequency measurement of Mueller matrices based on a new coupled-phase modulator,” Rev. Sci. Instrum. |

21. | O. Arteaga, J. Freudenthal, B. Wang, and B. Kahr, “Mueller matrix polarimetry with four photoelastic modulators: theory and calibration.” Appl. Optics |

22. | G. E. Jellison and F. a. Modine, “Two-channel polarization modulation ellipsometer.” Appl. Optics |

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

24. | L. Aas, P. Ellingsen, and M. Kildemo, “Near infra-red Mueller matrix imaging system and application to retardance imaging of strain,” Thin Solid Films |

25. | P. Letnes, I. Nerbø, L. Aas, P. Ellingsen, and M. Kildemo, “Fast and optimal broad-band Stokes/Mueller polarimeter design by the use of a genetic algorithm,” Opt. Express |

26. | J. H. Holland, “Genetic algorithms,” Sci. Am. |

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

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

29. | S. B. Hatit, M. Foldyna, A. De Martino, and B. Drévillon, “Angle-resolved Mueller polarimeter using a microscope objective,” Phys. Stat. Sol. (a) |

30. | A. Ben-Israel and T. N. E. Greville, |

31. | R. Hagen, S. Roch, and B. Silbermann, |

32. | L. Aas, P. Ellingsen, M. Kildemo, and M. Lindgren, “Dynamic Response of a fast near infra-red Mueller matrix ellipsometer,” J. Mod. Opt. |

33. | C. S. Perone, “Pyevolve,” http://pyevolve.sourceforge.net/. |

**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: February 8, 2013

Revised Manuscript: March 15, 2013

Manuscript Accepted: March 19, 2013

Published: April 2, 2013

**Virtual Issues**

Vol. 8, Iss. 5 *Virtual Journal for Biomedical Optics*

**Citation**

Lars Martin Sandvik Aas, Pål Gunnar Ellingsen, Bent Even Fladmark, Paul Anton Letnes, and Morten Kildemo, "Overdetermined broadband spectroscopic Mueller matrix polarimeter designed by genetic algorithms," Opt. Express **21**, 8753-8762 (2013)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-21-7-8753

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### References

- P. G. Ellingsen, M. B. Lilledahl, L. M. S. Aas, C. d. L. Davies, and M. Kildemo, “Quantitative characterization of articular cartilage using Mueller matrix imaging and multiphoton microscopy,” J. Biomed. Opt.16, 116002 (2011). [CrossRef] [PubMed]
- M. H. Smith, P. D. Burke, A. Lompado, E. A. Tanner, and L. W. Hillman, “Mueller matrix imaging polarimetry in dermatology,” Proc. SPIE3911, 210–216 (2000). [CrossRef]
- 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]
- 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,” Proc. SPIE4133, 202–213 (2000). [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 API’09, (2010), pp. 05002.
- R. Azzam and N. Bashara, Ellipsometry and Polarized light (North-Holland, 1977).
- H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications (John Wiley & Sons, Chichester, England; Hoboken, NJ, 2007). [CrossRef]
- H. Tompkins and E. A. Irene, Handbook of Ellipsometry (William Andrew, 2005). [CrossRef]
- 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]
- D. Schmidt, A. C. Kjerstad, T. Hofmann, R. Skomski, E. Schubert, and M. Schubert, “Optical, structural, and magnetic properties of cobalt nanostructure thin films,” J. Appl. Phys.105, 113508 (2009). [CrossRef]
- L. M. S. Aas, M. Kildemo, Y. Cohin, and E. Søndergård, “Determination of small tilt angles of short gasb nanopillars using uv-visible mueller matrix ellipsometry,” Thin Solid Films (2012). [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]
- T. Oates, H. Wormeester, and H. Arwin, “Characterization of plasmonic effects in thin films and metamaterials using spectroscopic ellipsometry,” Prog. Surf. Sci.86, 328–376 (2011). [CrossRef]
- B. Gallas, K. Robbie, R. Abdeddaïm, G. Guida, J. Yang, J. Rivory, and a. Priou, “Silver square nanospirals mimic optical properties of U-shaped metamaterials.” Opt. Express18, 16335–16344 (2010). [CrossRef] [PubMed]
- T. A. Germer, “Polarized light scattering by microroughness and small defects in dielectric layers.” J. Opt. Soc. Am. A18, 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]
- Ø. Svensen, M. Kildemo, J. Maria, J. J. Stamnes, and O. Frette, “Mueller matrix measurements and modeling pertaining to Spectralon white reflectance standards.” Opt. Express20, 15045–15053 (2012). [CrossRef] [PubMed]
- J. M. Bennet, R. Chipman, and R. M. A. Azzam, “Polarized light,” in Handbook of Optics, M. Bass and V. Mahajan, eds. (McGraw-Hill, Inc., 2010), pp. 12.3–16.21.
- F. Stabo-Eeg, M. Kildemo, I. Nerbø, and M. Lindgren, “Well-conditioned multiple laser Mueller matrix ellipsometer,” Opt. Eng.47, 073604 (2008). [CrossRef]
- E. Compain and B. Drevillon, “Complete high-frequency measurement of Mueller matrices based on a new coupled-phase modulator,” Rev. Sci. Instrum.68, 2671 (1997). [CrossRef]
- O. Arteaga, J. Freudenthal, B. Wang, and B. Kahr, “Mueller matrix polarimetry with four photoelastic modulators: theory and calibration.” Appl. Optics51, 6805–6817 (2012). [CrossRef]
- G. E. Jellison and F. a. Modine, “Two-channel polarization modulation ellipsometer.” Appl. Optics29, 959–974 (1990). [CrossRef]
- E. Garcia-Caurel, A. D. Martino, and B. Drevillon, “Spectroscopic Mueller polarimeter based on liquid crystal devices,” Thin Solid Films455, 120–123 (2004). [CrossRef]
- L. Aas, P. Ellingsen, and M. Kildemo, “Near infra-red Mueller matrix imaging system and application to retardance imaging of strain,” Thin Solid Films519, 2737–2741 (2010). [CrossRef]
- P. Letnes, I. Nerbø, L. Aas, P. Ellingsen, and M. Kildemo, “Fast and optimal broad-band Stokes/Mueller polarimeter design by the use of a genetic algorithm,” Opt. Express18, 23095–23103 (2010). [CrossRef] [PubMed]
- J. H. Holland, “Genetic algorithms,” Sci. Am.267, 44–50 (1992). [CrossRef]
- D. Floreano and C. Mattiussi, Bio-Inspired Artificial Intelligence: Theories, Methods, and Technologies (The MIT Press, 2008).
- E. Compain, S. Poirier, and B. Drevillon, “General and self-consistent method for the calibration of polarization modulators, polarimeters, and mueller-matrix ellipsometers.” Appl. Optics38, 3490–3502 (1999). [CrossRef]
- S. B. Hatit, M. Foldyna, A. De Martino, and B. Drévillon, “Angle-resolved Mueller polarimeter using a microscope objective,” Phys. Stat. Sol. (a)205, 743–747 (2008). [CrossRef]
- A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications (Springer-Verlag, 2003).
- R. Hagen, S. Roch, and B. Silbermann, C* Algebras Numerical Analysis (Marcel Dekker, 2001).
- L. Aas, P. Ellingsen, M. Kildemo, and M. Lindgren, “Dynamic Response of a fast near infra-red Mueller matrix ellipsometer,” J. Mod. Opt.57, 1603–1610 (2010). [CrossRef]
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