Abstract

The principles for measuring the extinction ratio and transmittance of a polarizer are formulated by use of the principal Mueller matrix, which includes both polarization and depolarization. The extinction ratio is about half of the depolarization, and the contrast is the inverse of the extinction ratio. Errors in the extinction ratio caused by partially polarized incident light and the misalignment of polarizers can be corrected by the devised zone average method and the null method. Used with a laser source, the null method can measure contrasts for very good polarizers. Correct algorithms are established to deduce the depolarization for three comparable polarizers calibrated mutually. These methods are tested with wire-grid polarizers used in the 3–5-μm wavelength region with a laser source and also a lamp source. The contrasts obtained from both methods agree.

© 1998 Optical Society of America

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References

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  1. J. M. Bennett, H. E. Bennett, “Polarization,” in Handbook of Optics (McGraw-Hill, New York, 1978), pp. 10.13–10.14; J. M. Bennett, “Polarization,” in Handbook of Optics, M. Bass, E. W. Van Stryland, eds. (McGraw-Hill, New York, 1995), Vol. 1, pp. 5.12–5.13.
  2. T. A. Leonard, “Infrared polarizer selection,” in Los Alamos Conference on Optics, D. H. Liebenberg, ed., Proc. SPIE288, 129–135 (1981).
    [CrossRef]
  3. S.-M. F. Nee, “Polarization of specular reflection and near-specular scattering by a rough surface,” Appl. Opt. 35, 3570–3582 (1996).
    [CrossRef]
  4. S.-M. F. Nee, “Effects of near-specular scattering on polarimetry,” in Polarization Analysis and Measurement II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE2265, 304–313 (1994).
    [CrossRef]
  5. S.-M. F. Nee, “The effects of incoherent scattering on ellipsometry,” in Polarization Analysis and Measurement, R. A. Chipman, D. H. Goldstein, eds., Proc. SPIE1746, 119–127 (1992).
    [CrossRef]
  6. S.-M. F. Nee, “Birefringence characterization using transmission ellipsometry,” in Polarization Analysis and Measurement, R. A. Chipman, D. H. Goldstein, eds., Proc. SPIE1746, 269–280 (1992).
    [CrossRef]
  7. A. Röseler, “Problem of polarization degree in spectroscopic photometric ellipsometry (polarimetry),” J. Opt. Soc. Am. A 9, 1124–1131 (1992).
    [CrossRef]
  8. A. B. Kostinski, “Depolarization criterion for incoherent scattering,” Appl. Opt. 31, 3506–3508 (1992).
    [CrossRef] [PubMed]
  9. E. S. Fry, G. W. Kattawar, “Relationships between elements of the Stokes matrix,” Appl. Opt. 20, 2811–2814 (1981).
    [CrossRef] [PubMed]
  10. J. J. Gil, E. Bernabeu, “A depolarization criterion in Müller matrices,” Opt. Acta 32, 259–261 (1985).
    [CrossRef]
  11. R. Simon, “Mueller matrices and depolarization criteria,” J. Mod. Opt. 34, 569–575 (1987).
    [CrossRef]
  12. K. Kim, L. Mandel, E. Wolf, “Relationship between Jones and Mueller matrices for random media,” J. Opt. Soc. Am. A 4, 433–437 (1987).
    [CrossRef]
  13. C. R. Givens, A. B. Kostinski, “A simple necessary and sufficient condition for the physical realizability of Mueller matrices,” J. Mod. Opt. 40, 471–481 (1993).
    [CrossRef]
  14. S.-M. F. Nee, H. E. Bennett, “Characterization of optical blacks by infrared ellipsometry and reflectometry,” in Stray Radiation in Optical System, R. P. Breault, ed., Proc. SPIE1331, 249–260 (1990).
    [CrossRef]
  15. S.-M. F. Nee, “Ellipsometric view on reflection and scattering from optical blacks,” Appl. Opt. 31, 1549–1556 (1992).
    [CrossRef] [PubMed]
  16. S.-M. F. Nee, J. M. Bennett, P. C. Archibald, “Reflection, scattering and polarization from very rough surfaces,” Optical Scattering: Applications, Measurement, and Theory II, J. C. Stover, ed., Proc. SPIE1995, 202–212 (1993).
    [CrossRef]
  17. S.-M. F. Nee, “Polarization characterization for target surfaces,” in Targets and Backgrounds: Characterization and Representation, W. R. Watkins, D. Clement, eds., Proc. SPIE2469, 220–230 (1995).
    [CrossRef]
  18. S.-M. F. Nee, P. C. Archibald, J. M. Bennett, D. K. Burge, T. W. Nee, C Yoo, “Polarization by rough painted surfaces,” presented at the Workshop on Infrared and Millimeter Wave Polarimetry, Dec. 5–7, 1995. Redstone Arsenal, Alabama; Proceedings published as Special Report RD-MG-96-8 (U.S. Army Missile Command, Redstone Arsenal, Alabama, 1996), pp. 527–542.
  19. R. A. Chipman, “Polarimetry,” in Handbook of Optics, (McGraw Hill, New York, 1995), Vol. II, Chap. 22.
  20. E. L. Church, P. Z. Takacs, “Subsurface and volume scattering from smooth surfaces,” in Scatter from Optical Components, J. C. Stover, ed., Proc. SPIE1165, 31–41 (1989).
    [CrossRef]

1996

1993

C. R. Givens, A. B. Kostinski, “A simple necessary and sufficient condition for the physical realizability of Mueller matrices,” J. Mod. Opt. 40, 471–481 (1993).
[CrossRef]

1992

1987

1985

J. J. Gil, E. Bernabeu, “A depolarization criterion in Müller matrices,” Opt. Acta 32, 259–261 (1985).
[CrossRef]

1981

Archibald, P. C.

S.-M. F. Nee, J. M. Bennett, P. C. Archibald, “Reflection, scattering and polarization from very rough surfaces,” Optical Scattering: Applications, Measurement, and Theory II, J. C. Stover, ed., Proc. SPIE1995, 202–212 (1993).
[CrossRef]

Bennett, H. E.

J. M. Bennett, H. E. Bennett, “Polarization,” in Handbook of Optics (McGraw-Hill, New York, 1978), pp. 10.13–10.14; J. M. Bennett, “Polarization,” in Handbook of Optics, M. Bass, E. W. Van Stryland, eds. (McGraw-Hill, New York, 1995), Vol. 1, pp. 5.12–5.13.

S.-M. F. Nee, H. E. Bennett, “Characterization of optical blacks by infrared ellipsometry and reflectometry,” in Stray Radiation in Optical System, R. P. Breault, ed., Proc. SPIE1331, 249–260 (1990).
[CrossRef]

Bennett, J. M.

J. M. Bennett, H. E. Bennett, “Polarization,” in Handbook of Optics (McGraw-Hill, New York, 1978), pp. 10.13–10.14; J. M. Bennett, “Polarization,” in Handbook of Optics, M. Bass, E. W. Van Stryland, eds. (McGraw-Hill, New York, 1995), Vol. 1, pp. 5.12–5.13.

S.-M. F. Nee, J. M. Bennett, P. C. Archibald, “Reflection, scattering and polarization from very rough surfaces,” Optical Scattering: Applications, Measurement, and Theory II, J. C. Stover, ed., Proc. SPIE1995, 202–212 (1993).
[CrossRef]

Bernabeu, E.

J. J. Gil, E. Bernabeu, “A depolarization criterion in Müller matrices,” Opt. Acta 32, 259–261 (1985).
[CrossRef]

Chipman, R. A.

R. A. Chipman, “Polarimetry,” in Handbook of Optics, (McGraw Hill, New York, 1995), Vol. II, Chap. 22.

Church, E. L.

E. L. Church, P. Z. Takacs, “Subsurface and volume scattering from smooth surfaces,” in Scatter from Optical Components, J. C. Stover, ed., Proc. SPIE1165, 31–41 (1989).
[CrossRef]

Fry, E. S.

Gil, J. J.

J. J. Gil, E. Bernabeu, “A depolarization criterion in Müller matrices,” Opt. Acta 32, 259–261 (1985).
[CrossRef]

Givens, C. R.

C. R. Givens, A. B. Kostinski, “A simple necessary and sufficient condition for the physical realizability of Mueller matrices,” J. Mod. Opt. 40, 471–481 (1993).
[CrossRef]

Kattawar, G. W.

Kim, K.

Kostinski, A. B.

C. R. Givens, A. B. Kostinski, “A simple necessary and sufficient condition for the physical realizability of Mueller matrices,” J. Mod. Opt. 40, 471–481 (1993).
[CrossRef]

A. B. Kostinski, “Depolarization criterion for incoherent scattering,” Appl. Opt. 31, 3506–3508 (1992).
[CrossRef] [PubMed]

Leonard, T. A.

T. A. Leonard, “Infrared polarizer selection,” in Los Alamos Conference on Optics, D. H. Liebenberg, ed., Proc. SPIE288, 129–135 (1981).
[CrossRef]

Mandel, L.

Nee, S.-M. F.

S.-M. F. Nee, “Polarization of specular reflection and near-specular scattering by a rough surface,” Appl. Opt. 35, 3570–3582 (1996).
[CrossRef]

S.-M. F. Nee, “Ellipsometric view on reflection and scattering from optical blacks,” Appl. Opt. 31, 1549–1556 (1992).
[CrossRef] [PubMed]

S.-M. F. Nee, J. M. Bennett, P. C. Archibald, “Reflection, scattering and polarization from very rough surfaces,” Optical Scattering: Applications, Measurement, and Theory II, J. C. Stover, ed., Proc. SPIE1995, 202–212 (1993).
[CrossRef]

S.-M. F. Nee, H. E. Bennett, “Characterization of optical blacks by infrared ellipsometry and reflectometry,” in Stray Radiation in Optical System, R. P. Breault, ed., Proc. SPIE1331, 249–260 (1990).
[CrossRef]

S.-M. F. Nee, “Effects of near-specular scattering on polarimetry,” in Polarization Analysis and Measurement II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE2265, 304–313 (1994).
[CrossRef]

S.-M. F. Nee, “The effects of incoherent scattering on ellipsometry,” in Polarization Analysis and Measurement, R. A. Chipman, D. H. Goldstein, eds., Proc. SPIE1746, 119–127 (1992).
[CrossRef]

S.-M. F. Nee, “Birefringence characterization using transmission ellipsometry,” in Polarization Analysis and Measurement, R. A. Chipman, D. H. Goldstein, eds., Proc. SPIE1746, 269–280 (1992).
[CrossRef]

S.-M. F. Nee, “Polarization characterization for target surfaces,” in Targets and Backgrounds: Characterization and Representation, W. R. Watkins, D. Clement, eds., Proc. SPIE2469, 220–230 (1995).
[CrossRef]

Röseler, A.

Simon, R.

R. Simon, “Mueller matrices and depolarization criteria,” J. Mod. Opt. 34, 569–575 (1987).
[CrossRef]

Takacs, P. Z.

E. L. Church, P. Z. Takacs, “Subsurface and volume scattering from smooth surfaces,” in Scatter from Optical Components, J. C. Stover, ed., Proc. SPIE1165, 31–41 (1989).
[CrossRef]

Wolf, E.

Appl. Opt.

J. Mod. Opt.

R. Simon, “Mueller matrices and depolarization criteria,” J. Mod. Opt. 34, 569–575 (1987).
[CrossRef]

C. R. Givens, A. B. Kostinski, “A simple necessary and sufficient condition for the physical realizability of Mueller matrices,” J. Mod. Opt. 40, 471–481 (1993).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Acta

J. J. Gil, E. Bernabeu, “A depolarization criterion in Müller matrices,” Opt. Acta 32, 259–261 (1985).
[CrossRef]

Other

J. M. Bennett, H. E. Bennett, “Polarization,” in Handbook of Optics (McGraw-Hill, New York, 1978), pp. 10.13–10.14; J. M. Bennett, “Polarization,” in Handbook of Optics, M. Bass, E. W. Van Stryland, eds. (McGraw-Hill, New York, 1995), Vol. 1, pp. 5.12–5.13.

T. A. Leonard, “Infrared polarizer selection,” in Los Alamos Conference on Optics, D. H. Liebenberg, ed., Proc. SPIE288, 129–135 (1981).
[CrossRef]

S.-M. F. Nee, “Effects of near-specular scattering on polarimetry,” in Polarization Analysis and Measurement II, D. H. Goldstein, D. B. Chenault, eds., Proc. SPIE2265, 304–313 (1994).
[CrossRef]

S.-M. F. Nee, “The effects of incoherent scattering on ellipsometry,” in Polarization Analysis and Measurement, R. A. Chipman, D. H. Goldstein, eds., Proc. SPIE1746, 119–127 (1992).
[CrossRef]

S.-M. F. Nee, “Birefringence characterization using transmission ellipsometry,” in Polarization Analysis and Measurement, R. A. Chipman, D. H. Goldstein, eds., Proc. SPIE1746, 269–280 (1992).
[CrossRef]

S.-M. F. Nee, H. E. Bennett, “Characterization of optical blacks by infrared ellipsometry and reflectometry,” in Stray Radiation in Optical System, R. P. Breault, ed., Proc. SPIE1331, 249–260 (1990).
[CrossRef]

S.-M. F. Nee, J. M. Bennett, P. C. Archibald, “Reflection, scattering and polarization from very rough surfaces,” Optical Scattering: Applications, Measurement, and Theory II, J. C. Stover, ed., Proc. SPIE1995, 202–212 (1993).
[CrossRef]

S.-M. F. Nee, “Polarization characterization for target surfaces,” in Targets and Backgrounds: Characterization and Representation, W. R. Watkins, D. Clement, eds., Proc. SPIE2469, 220–230 (1995).
[CrossRef]

S.-M. F. Nee, P. C. Archibald, J. M. Bennett, D. K. Burge, T. W. Nee, C Yoo, “Polarization by rough painted surfaces,” presented at the Workshop on Infrared and Millimeter Wave Polarimetry, Dec. 5–7, 1995. Redstone Arsenal, Alabama; Proceedings published as Special Report RD-MG-96-8 (U.S. Army Missile Command, Redstone Arsenal, Alabama, 1996), pp. 527–542.

R. A. Chipman, “Polarimetry,” in Handbook of Optics, (McGraw Hill, New York, 1995), Vol. II, Chap. 22.

E. L. Church, P. Z. Takacs, “Subsurface and volume scattering from smooth surfaces,” in Scatter from Optical Components, J. C. Stover, ed., Proc. SPIE1165, 31–41 (1989).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Simulated contrast C for α between -0.75 and 0.75. C becomes highly inaccurate as α deviates greatly from 0.

Fig. 2
Fig. 2

Contrasts simulated by the zone average method for samples A with slight misalignments of (a) A and B and (b) B and C. In each plot the misalignment of the third polarizer was assumed to be zero. When two polarizers are misaligned, the errors are enhanced nonlinearly.

Fig. 3
Fig. 3

Contrasts simulated by the null method for samples A with slight misalignments of (a) A and B and (b) B and C. The null method can eliminate most of the systematic errors caused by partially polarized incident light and imperfect polarizers, as well as misalignments of polarizers.

Fig. 4
Fig. 4

Apparent transmittance I/ I 0 of a polarizer at a different polarizer angle for partially polarized incident light. The average transmittance for pairs of points at 90° apart is 0.3922 and is shown as the dotted horizontal line.

Fig. 5
Fig. 5

Spectra of α and β for light emitted from a Nernst glower and passing through a grating monochromator for wavelengths between 2.5 and 5.0 μm.

Fig. 6
Fig. 6

Apparent contrast I max/I min for partially polarized light passing through polarizers P and M when the front polarizer is at various different angles. The contrast of P is 50 times better than that of M. Circles, P is in front and M is rotating; squares, M is in front and P is rotating.

Tables (7)

Tables Icon

Table 1 Measured Transmittance T, Depolarization D, Copolarized and Cross-Polarized NSS u and v for Two Wire-grid Polarizersa

Tables Icon

Table 2 Measured Intensities for Light Passing through a Wire-Grid Polarizer at λ = 3.39 μma

Tables Icon

Table 3 Intensities for Light Passing through a Pair of Wire-Grid Polarizers Measured at λ = 3.39 μm for the Zone Average Method with a Lamp Source

Tables Icon

Table 4 T, α, ℰ, and C for Wire-Grid Polarizers at λ = 3.39 μm Determined from Data of Table 3 by the Zone Average Method

Tables Icon

Table 5 Intensities for Light Passing through Three Wire-Grid Polarizers at Different Orientations, Measured by the Null Method with a He–Ne Laser Source at λ 3.39 μm

Tables Icon

Table 6 Intensities for Light Passing Through a Pair of Wire-Grid Polarizers Measured Using the Zone Average Methoda

Tables Icon

Table 7 Extinction Ratios and Contrasts of Wire-Grid Polarizers for the Null and the Zone Average Methods with a Laser Source at λ = 3.39 μm

Equations (26)

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T=T 1+u+v-cos 2ψ00-cos 2ψ1+u-v0000sin 2ψ cos Δ-sin 2ψ sin Δ00sin 2ψ sin Δsin 2ψ cos Δ,
T=T* 1-P cos 2ψ00-P cos 2ψ1-2 v0000P sin 2ψ cos Δ-P sin 2ψ sin Δ00P sin 2ψ sin ΔP sin 2ψ cos Δ,
P=1/1+u+v,D=1-P=u+v/1+u+v=Pu+v,T*=T1+u+v=T/P,v=Pv.
T0=T 1+u+v10011+u-v0000000000.
Tθ=T1+u+vcos 2θsin 2θ0cos 2θ1+u-vcos2 2θ1+u-vcos 2θ sin 2θ0sin 2θ1+u-vcos 2θ sin 2θ1+u-vsin2 2θ00000.
T*=I/I0=T1+u+v.
Iθ=ITθS0=TI01+u+v+cos 2θ.
=1C=IminImax=u+v2+u+v=1-P1+P=D2-D.
vp=2IA90°P90°S0/IA0°P0°S0,up=2IA0°P90°S0/IA0°P0°S0.
S0=I01, α, β, γ;
Iθ=TI01+u+v+α cos 2θ+β sin 2θ.
T*=I0°+I90°/2I0=T1+u+v.
α=I0°-I90°/I0°+I90°,β=I45°-I-45°/I45°+I-45°.
IP, A=TaTpI01+ua+va+up+vp+cos 2A-P+α cos 2P+β sin 2P1+ua+va+1+up-vpcos 2A-P.
I0, A=TaTI01+α1+ua+va+u+v+1+α1+u-vcos 2A,=1+αu+v+ua+va-2αu1+α2+u+v+ua+va-2αv.
I0, θ=TpTI01+α1+u+v+up+vp+1+α1+up-vpcos 2θ,=1+αu+v+up+vp-2αup1+α2+u+v+up+vp-2αvp.
Tp*=I0, 0+I90, 90+I0, 90+I90, 02Ia0+Ia90=Tp1+up+vp,α=I0, 0-I90, 90I0, 0+I90, 90.
=I0, 90+I90, 0I0, 0+I90, 90=up+vp+ua+va2+up+vp+ua+va.
up+vp+ua+va=2I0, 90+I90, 0I0, 0+I90, 90-I0, 90-I90, 0.
δI  -sin2A-PδA-P.
IP, A=TpTaTI01+u+v+cos 2P+cos 2A+1+u-vcos 2P cos 2A.
I±45°, A=TaTpTI01+u+v+cos 2A=TaTpTI0u+v+2 sin2A-90°.
u+v=4 sin2 ΔA I±45°, A0I±45°, A0+ΔA+I±45°, A0-ΔA-2I±45°, A0.
IP, A=TaTpTI01+Da+D+Dp+1+Dacos×2P+1+u-vcos 2A cos 2P+1+Dpcos 2A+α cos 2P+β sin 2P×1+D+Da+1+Da+up-vpcos 2P+cos 2A+1+u-v+up-vpcos 2P× cos2A.
I±45°, A=TaTpTI01+Dp±β×1+Da+D+cos 2A.
D+Da=4sin2 ΔAI±45°, A0I±45°, A0+ΔA+I±45°, A0-ΔA-2I±45°, A0.

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