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  • Editors: Andrew Dunn and Anthony Durkin
  • Vol. 8, Iss. 5 — Jun. 6, 2013
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Overdetermined broadband spectroscopic Mueller matrix polarimeter designed by genetic algorithms

Lars Martin Sandvik Aas, Pål Gunnar Ellingsen, Bent Even Fladmark, Paul Anton Letnes, and Morten Kildemo  »View Author Affiliations


Optics Express, Vol. 21, Issue 7, pp. 8753-8762 (2013)
http://dx.doi.org/10.1364/OE.21.008753


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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

Polarimeters measure the polarization state of electromagnetic waves. Methods based on polarimetry are thus non-invasive and have the possibility for remote sensing applications, which makes them attractive in many fields of science. In the range of optical frequencies, polarimetry has proven to be useful and promising in e.g. biomedical diagnostics [1

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

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] .

], remote sensing [4

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] .

] and astronomy [5

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.

]. The sample measuring polarimeter (ellipsometer) is a key characterization technique for thin films [6

6. R. Azzam and N. Bashara, Ellipsometry and Polarized light (North-Holland, 1977).

8

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

], with recent applications to e.g. gratings [9

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] .

], nanostructures [10

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

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] .

], plasmonics [13

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] .

], metamaterials [14

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] .

] and scattering from rough surfaces [15

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

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] .

].

A Mueller matrix ellipsometer/polarimeter consists of a complete polarization state generator (PSG) and polarization state analyzer (PSA), which determines all the polarization altering properties of a sample both in reflection and in transmission. A Stokes polarimeter consist only of a PSA and is used to determine the complete polarization state of partially polarized light.

A PSA/PSG generally consists of a diattenuating polarizer and an active birefringent optical component either modulated by azimuthal rotation or by an externally applied electric field [18

18. 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.

]. Typical examples are rotating (wave-plate/bi-prism) retarders [6

6. R. Azzam and N. Bashara, Ellipsometry and Polarized light (North-Holland, 1977).

, 19

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

], electro-optical modulation [20

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] .

], photoelastic modulators [21

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

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

], and liquid crystal retarders [23

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

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] .

]. Dispersion in the optical components is usually limiting the wavelengths range of the polarimeters, but novel system designs may overcome this problem for at least a limited spectral range [23

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] .

].

Certain wide band achromatic polarimeters (from the ultraviolet to the infrared) may be constructed using near non-dispersive retarders, by exploiting the total internal reflection from Fresnel prisms [19

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

]. These retarders do commonly have a small aperture, are sensitive to alignment and require mechanical azimuth rotation for operation, and are thus not really suitable for imaging and space applications. On the other hand, liquid crystal retarders have no moving parts and can easily be made with large apertures, but they are strongly dispersive and a liquid crystal based wide band polarimeter requires a more advanced design.

Advantages of overdetermined polarimeters and the use of genetic algorithms to design them, was proven theoretically for a broadband system based on ferroelectric liquid crystal components in [25

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] .

]. The genetic algorithm was generically implemented in order to create designs using any polarization modulating component with known dispersive properties. We here report for the first time an experimental implementation and testing of a genetic algorithm designed wide-band liquid crystal variable retarder Mueller matrix polarimeter.

2. Theory

Let us first briefly review the theory and notation used to describe the measurement of Stokes vectors and Mueller matrices using a PSA and a PSG, both for determined and overdetermined systems. The calibration of overdetermined Mueller matrix polarimeters is thereafter explained using a generalization of the eigenvalue calibration method (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] .

].

The polarization state of light, can generally be represented in vector form by the four element Stokes vector defined by
S=[s1s2s3s4]=[Ex(t)2+Ey(t)2Ex(t)2Ey(t)22Ex(t)Ey(t)cosδ(t)2Ex(t)Ey(t)sinδ(t)],
where Ex(t) and Ey(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.

The Mueller matrix is a 4 × 4 transfer matrix transforming an initial Stokes vector to the final by Sfinal = MsampleSinitial. 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

18. 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.

].

The ECM is explained in the original paper [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] .

] for systems using four Stokes vectors in the PSG and PSA. Here we present the generalization needed to calibrate a system with 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}. Bi is of size n × m and is given by
Bi=AMiW.
For convenience, reference sample M0 is chosen to be air, such that B0 = AW. Next, two sets of matrices, {C} and {C′}, are constructed using
Ci=B0Bi=WMiWandCi=BiB0=AMiA,
(1)
where denotes the Moore-Penrose pseudoinverse, which is the common way of defining the inverse of a non-square matrix with noise [30

30. A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications (Springer-Verlag, 2003).

]. In the case where Bi is of size four by four, the sets {C} and {C′} have the same eigenvalues as the set of Mueller matrices {M}. As Ci is independent of A, and Ci is independent of W, A and W may be found independently, and {M} may be found both from {C} and {C′} independently.

A robust solution to finding the correct eigenvalues of {M} from {C} and {C′}, is to (at each wavelength) reduce B0 and Bi in Eq. (1) to the 4 × 4 subset of B0 and Bi resulting in the lowest condition number for the reduced B0. When inverting B0, this ensures minimal noise propagation into Ci and to the eigenvalues of Mi. After finding the eigenvalues of {M}, the remainder of the calibration procedure follows the original paper by Compain et al. [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] .

], using the non reduced {B}.

Also worth noting is that the noise equation, Eq. (2), is the basis for the genetic optimisation, which tries to maximise the inverse condition number.

3. Experimental

The essential optical components in the Mueller matrix polarimeter presented here, are polarizers and liquid crystal variable retarders (LCVR). In the calibration, a polarizer and a waveplate was used. We used a high extinction ratio near infra-red polarizer (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.

LCVRs are wave retarders having the retardance as a function of applied voltage. Compared to ferroelectric liquid crystal retarders (previously proposed overdetermined polarimeter design [25

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] .

]) which have only one fixed retardance, but with two stable azimuth orientations, they typically have much longer transition times between two states [32

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] .

], but have the advantage of allowing the selection of all retardation values between a maximum and minimum value.

In order to design the optimal polarimeter, the retardance as a function of voltage and wavelength needs to be known with reasonable precision. Although the calibration routine handles small deviations in dispersive optical properties, high accuracy of the Mueller matrix elements is only insured as long as the condition number is not strongly degraded with respect to the design. In the instrument reported here, it was also found that there were, due to manufacturing uncertainties, differences in thickness between the individual crystals. The crystals were therefore characterized individually in the range of 450 – 1680 nm using a commercial available Mueller matrix polarimeter 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.

Fig. 1 (a) The measured retardance of a LCVR as a function of wavelength and the voltage applied. (b) The retardance of the LCVR measured at 0 V with the temperature stabilized at 26°C, 28°C and 30°C. The figure shows the relative difference to the LCVR retardance at 24°C.

It was found that the retardance was reduced significantly with the increased ambient temperature. Figure 1(b) shows the deviation in the wavelength dependent retardance for 26°C, 28°C and 30°C, relative to the retardance at 24°C at 0 V. Thus, for reproducible and accurate measurements, the LCVRs must be operated in an environment with a stable temperature.

Figure 2 shows a schematic drawing of a typical LCVR Mueller matrix polarimeter system design, based on a broadband white light source, a spectrograph and four temperature controlled LCVRs. The polarizer and the two crystals on the left side of the sample makes up the PSG, while the components in the opposite order on the right side of the sample makes up the PSA (i.e. a Stokes polarimeter).

Fig. 2 Schematic drawing of a typical spectroscopic Mueller matrix polarimeter using liquid crystal variable retarders (LCVR), a broad band light source and spectrometer.

The genetic optimization was performed using the settings given in Table 1. 8 bits was considered sufficient to represent the voltage and the rotation angle, i.e. a step size of 0.04 V and 0.7°. The wavelength range was selected to be 900 – 1700 nm, the range of a typical indium gallium arsenide (InGaAs) near infrared detector. In particular, we used the 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.

Table 1. General settings for the genetic optimization

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4. Results and discussion

Several Mueller matrix polarimeters were optimized in order to cover the near infrared spectral range. By using two LCVRs in both the PSG and the PSA, one may in principle generate a large number of states. However, a system that approaches the theoretically optimal inverse condition number 1/3 will need 16 states for every measured wavelength, as long as a retardance of 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 (42) 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 (62) 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 (82) 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.

Fig. 3 The inverse of the generalized condition number of W as a function of wavelength. Figure (a) shows the comparison between the best designs for a 2 × 2, 2 × 3 and 2 × 4 states system, where the systems are presented in terms of the number of retardance (voltage) states for each of the two LCVRs making up the PSG or the PSA. Figure (b) shows both the simulated, measured and calibrated inverse condition number of the experimentally realized polarimeter with 2 × 3 states.

Table 2. Configuration of the optimal polarimeters

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In comparison to previously reported ferroelectric liquid crystal (FLC) designs [25

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] .

] obtained using a similar genetic algorithm, the 2 × 3 LCVR design has a much narrower wavelength range and a slightly lower inverse condition number. This is as expected, since the design using three FLCs has three compensating wave-plates in addition to the liquid crystals, giving extra degrees of freedom. Specifically, these degrees of freedom arise from the fact that both the thickness (birefringence) and the azimuthal orientation of each component can be selected, and as a result one would expect an overall higher inverse generalized condition number. However, more optical components reduce the transmitted intensity, which in some cases results in a greatly reduced signal to noise ratio. A high signal to noise ratio is particularly important for applications with a limited flux, for example a large field of view imaging or space applications.

The 2 × 3 design was mounted in custom made temperature controlled holders, one for the PSG and one for the PSA. These were then separately mounted in the beam-path of the RC2, and by switching through all the 2 × 3 states of the LCVRs, the Stokes vectors of the 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.

An important measure of the Mueller matrix polarimeter accuracy is the measurement of air, whose Mueller matrix is simply the 4 × 4 identity matrix. Figure 4 shows the spectroscopic Mueller matrix measurement of air (normalized to the m11 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%.

Fig. 4 The measured spectroscopic Mueller matrix of air normalized to the m11 element.

It is recalled that the wide band LVCR design uses few components and thereby has small reflection losses from the optical interfaces. Hence, the designed Mueller matrix polarimeter should also be well suited for a fast imaging setup with low loss of light, compared to a FLC based setup, while insuring small measurement errors across the spectral range of operation, even suitable for hyperspectral imaging. The reduction in number of components also enables a more compact design. Another advantage of the setup will be the possibility to redesign the spectral characteristics without having to rotate the components, since the voltage of the LCVRs is software controlled. By using the genetic algorithm, it is possible to fix the rotation angles and only optimize on the LVCR voltage in order to fulfill other system specifications. Finally, one may envisage a system that can improve itself by self-characterization and intelligent design by implementing an in-line version of the genetic algorithm.

5. Conclusion

Genetic algorithms have been used to design multichannel Mueller matrix polarimeters based on liquid crystal variable retarders for the near infrared with 2 × 2, 2 × 3 or 2 × 4 voltage states for the polarization state generator and analyzer. The design using 2 × 3 states was experimentally realized and calibrated, based on its advantageous trade-off between total measurement time and overall performance with respect to error propagation (optimized inverse condition number). The resulting Mueller matrix polarimeter demonstrated here, shows good performance in the design wavelength range (900–1700 nm) with less than 0.1% error on the Mueller matrix of air, making it suitable in for example hyperspectral or multispectral imaging applications.

Acknowledgment

L.M.S.A acknowledges support from The Norwegian Research Center for Solar Cell Technology (project num. 193829).

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. 16, 116002 (2011) [CrossRef] [PubMed] .

2.

M. H. Smith, P. D. Burke, A. Lompado, E. A. Tanner, and L. W. Hillman, “Mueller matrix imaging polarimetry in dermatology,” Proc. SPIE 3911, 210–216 (2000) [CrossRef] .

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] .

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, Ellipsometry and Polarized light (North-Holland, 1977).

7.

H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications (John Wiley & Sons, Chichester, England; Hoboken, NJ, 2007) [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] .

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. 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] .

16.

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

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] .

18.

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.

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] .

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] .

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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] .

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] .

30.

A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications (Springer-Verlag, 2003).

31.

R. Hagen, S. Roch, and B. Silbermann, C* Algebras Numerical Analysis (Marcel Dekker, 2001).

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] .

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

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