Abstract

We present a polarimetric instrument suitable for the simultaneous measurement of angle resolved normalized Mueller matrices for polar angles ranging from 0° to 60° and all azimuths. The polarimetric modulation and analysis are performed by means of an optimized polarization state generator and analyzer based on nematic liquid crystals. A high numerical aperture (0.95) microscope objective is used in double pass to illuminate the sample, with its rear focal plane imaged on a low noise CCD. This polarimeter can be used either in reflection, with the sample set in the objective front focal plane, or in transmission, for thin transparent samples. This latter configuration, which involves an additional spherical mirror with its center of curvature at the objective front focus, is described in detail. This instrument was used to accurately determine the directions of the optic axes and the angular dependence of the retardation of a biaxial polyethylene terephthalate (PET) plastic substrate in spite of the strong depolarization essentially due to the source 10nm spectral width or the limitation in angular resolution due to the pixels distribution of the CCD combined with the sample large retardation.

© 2009 Optical Society of America

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References

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  1. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, 2nd ed., (North-Holland, 1986).
  2. R. A. Chipman, “Polarimetry,” in Handbook of Optics, 2nd ed., M. Bass, ed. (McGraw-Hill, 1995), Chap. 22.
  3. M. Schubert and W. Dollase, “Generalized ellipsometry for biaxial absorbing materials: determination of crystal orientation and optical constants of Sb2S3,” Opt. Lett. 27, 2073-2075 (2002).
    [CrossRef]
  4. T. Novikova, A. De Martino, S. Ben Hatit, and B. Drévillon, “Application of Mueller polarimetry in conical diffraction for CD measurements in microelectronics,” Appl. Opt. 45, 3688-3697 (2006).
    [CrossRef] [PubMed]
  5. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 2002).
  6. A. Beaudry, Y. Zhao, and R. Chipman, “Dielectric tensor measurement from a single Mueller matrix image,” J. Opt. Soc. Am. A 24, 814-824 (2007).
    [CrossRef]
  7. A. De Martino, S. Ben Hatit, and M. Foldyna, “Mueller polarimetry in the back focal plane,” Proc. SPIE 6518, 65180X (2007).
    [CrossRef]
  8. A. De Martino, Y.-K. Kim, E. Garcia-Caurel, B. Laude, and B. Drévillon, “Optimized Mueller polarimeter with liquid crystals,” Opt. Lett. 28, 616-618 (2003).
    [CrossRef] [PubMed]
  9. S. Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13, 1106-1113 (1996).
    [CrossRef]
  10. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley-Interscience, 2003).
  11. P. Yeh, Optical Waves in Layered Media (Wiley-Interscience, 2005).
  12. T. Scharf, Polarized Light in Liquid Crystals and Polymers (Wiley-Interscience, 2007).

2007 (2)

A. Beaudry, Y. Zhao, and R. Chipman, “Dielectric tensor measurement from a single Mueller matrix image,” J. Opt. Soc. Am. A 24, 814-824 (2007).
[CrossRef]

A. De Martino, S. Ben Hatit, and M. Foldyna, “Mueller polarimetry in the back focal plane,” Proc. SPIE 6518, 65180X (2007).
[CrossRef]

2006 (1)

2003 (1)

2002 (1)

1996 (1)

Azzam, R. M. A.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, 2nd ed., (North-Holland, 1986).

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, 2nd ed., (North-Holland, 1986).

Beaudry, A.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 2002).

Chipman, R.

Chipman, R. A.

S. Y. Lu and R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13, 1106-1113 (1996).
[CrossRef]

R. A. Chipman, “Polarimetry,” in Handbook of Optics, 2nd ed., M. Bass, ed. (McGraw-Hill, 1995), Chap. 22.

De Martino, A.

Dollase, W.

Drévillon, B.

Foldyna, M.

A. De Martino, S. Ben Hatit, and M. Foldyna, “Mueller polarimetry in the back focal plane,” Proc. SPIE 6518, 65180X (2007).
[CrossRef]

Garcia-Caurel, E.

Hatit, S. Ben

Kim, Y.-K.

Laude, B.

Lu, S. Y.

Novikova, T.

Scharf, T.

T. Scharf, Polarized Light in Liquid Crystals and Polymers (Wiley-Interscience, 2007).

Schubert, M.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 2002).

Yariv, A.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley-Interscience, 2003).

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley-Interscience, 2003).

P. Yeh, Optical Waves in Layered Media (Wiley-Interscience, 2005).

Zhao, Y.

Appl. Opt. (1)

J. Opt. Soc. Am. A (2)

Opt. Lett. (2)

Proc. SPIE (1)

A. De Martino, S. Ben Hatit, and M. Foldyna, “Mueller polarimetry in the back focal plane,” Proc. SPIE 6518, 65180X (2007).
[CrossRef]

Other (6)

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 2002).

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, 2nd ed., (North-Holland, 1986).

R. A. Chipman, “Polarimetry,” in Handbook of Optics, 2nd ed., M. Bass, ed. (McGraw-Hill, 1995), Chap. 22.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley-Interscience, 2003).

P. Yeh, Optical Waves in Layered Media (Wiley-Interscience, 2005).

T. Scharf, Polarized Light in Liquid Crystals and Polymers (Wiley-Interscience, 2007).

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Figures (14)

Fig. 1
Fig. 1

Architecture of the polarimeter with its different elements.

Fig. 2
Fig. 2

Alignment of the objective in (a) reflection mode, (b) transmission mode.

Fig. 3
Fig. 3

Images in the Fourier plane (a) nonaligned, (b) aligned objective with the spherical mirror.

Fig. 4
Fig. 4

(a) Scalar retardance Δ ( ° ) , (b) fast axis orientation (°), (c) Scalar diattenuation angle Ψ ( ° ) and (d) high transmission axis orientation (°) of the microscope objective measured in double transmission.

Fig. 5
Fig. 5

(a) Calculated scalar retardance Δ ( ° ) and (b) diattenuation Ψ ( ° ) of the microscope objective in single transmission.

Fig. 6
Fig. 6

Calculated Mueller matrix of the objective.

Fig. 7
Fig. 7

Normal surface of a biaxial medium represented in the principal planes.

Fig. 8
Fig. 8

Refraction of the two main polarizations.

Fig. 9
Fig. 9

Normalized Mueller matrix in double transmission of PET with objective correction.

Fig. 10
Fig. 10

(a) Retardance and (b) depolarization of a biaxial PET plate in double transmission.

Fig. 11
Fig. 11

Measured retardance as function of the angle of incidence in the principal planes.

Fig. 12
Fig. 12

Retardance reconstructed in x z and y z planes: the points represent measurement, and the lines represent simulation.

Fig. 13
Fig. 13

Depolarization versus angle of incidence (in degrees) measured (dots) and simulated (continuous line) in the two principal planes.

Fig. 14
Fig. 14

Additional depolarization around the optical axes as function of the maximal retardance measured on the same pixel.

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

M = M ρ M R M D .
ρ = 1 | M ρ 22 | + | M ρ 33 | + | M ρ 44 | 3 ,
tan θ 0 = n z n x ( n y 2 n x 2 n z 2 n y 2 ) 1 / 2 .
δ = 2 × 2 π h λ ( n 2 sin 2 θ n 2 sin 2 θ ) ,
δ = 2 × 2 π h λ ( n y 2 sin 2 θ n x 2 + n z 2 sin 2 θ ) ,
M δ = [ 1 0 0 0 0 1 0 0 0 0 cos δ sin δ 0 0 sin δ cos δ ] ,
M = 1 Δ δ δ Δ δ / 2 δ + Δ δ / 2 , M δ d δ = [ 1 0 0 0 0 1 0 0 0 0 a 0 0 0 0 a ] M δ ,
ρ = 2 3 [ 1 | a | ] ,
Δ δ 1 = δ · Δ λ λ .
Δ δ 2 = δ θ Δ θ ,

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