Abstract

A method for active polarization compensation for a scanning goniometer is presented. This is part of instrumentation constructed for measurements of light scattering by particulates in a high voltage spark. Individual optical element characterization is made using ellipsometry. Mueller matrix calculations are used to model the optical system. The inverse of the optical system is used to calculate the necessary input polarization state. A polarized source with angularly controlled halfwave and quarterwave retarders is used to introduce the necessary polarization state into the goniometer.

© 1988 Optical Society of America

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References

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  1. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  2. A. K. Arora, “A New Setup for Measuring Angle-Resolved Polarized Light Scattering,” J. Phys E 17, 1119 (1984).
    [CrossRef]
  3. W. S. Bickel, “Optical System for Light Scattering Experiments,” Appl. Opt. 18, 1707 (1979).
    [CrossRef] [PubMed]
  4. D. Clarke, J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, Oxford, 1971).
  5. A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975).
  6. C.-R. Hu, G. W. Kattawar, M. E. Parkin, P. Herb, “Symmetry Theorems on the Forward and Backward Scattering Mueller Matrices for Light Scattering from a Nonspherical Dielectric Scatterer,” Appl. Opt. 26, 4159 (1987).
    [CrossRef] [PubMed]
  7. G. W. Kattawar, C.-R. Hu, M. E. Parkin, P. Herb, “Mueller Matrix Calculations for Dielectric Cubes: Comparison With Experiments,” Appl. Opt. 26, 4174 (1987).
    [CrossRef] [PubMed]
  8. J. Brazas, “The Effects of Highly Reflecting Materials on Nearby Fluorescent Probes,” Ph.D. Thesis, U. Illinois at Urbana-Champaign (1983).
  9. F. McCrackin, E. Passaglia, R. Stromberg, H. Steinberg, “Measurement of the Thickness and Refractive Index of Very Thin Films and the Optical Properties of Surfaces by Ellipsometry,” J. Res. Natl. Bur. Stand. Sect. A 67, 363 (1963).
    [CrossRef]
  10. R. H. Muller, “Principles of Ellipsometry,” Adv. Electrochem. Eng. 9, 167 (1973).

1987 (2)

1984 (1)

A. K. Arora, “A New Setup for Measuring Angle-Resolved Polarized Light Scattering,” J. Phys E 17, 1119 (1984).
[CrossRef]

1979 (1)

1973 (1)

R. H. Muller, “Principles of Ellipsometry,” Adv. Electrochem. Eng. 9, 167 (1973).

1963 (1)

F. McCrackin, E. Passaglia, R. Stromberg, H. Steinberg, “Measurement of the Thickness and Refractive Index of Very Thin Films and the Optical Properties of Surfaces by Ellipsometry,” J. Res. Natl. Bur. Stand. Sect. A 67, 363 (1963).
[CrossRef]

Arora, A. K.

A. K. Arora, “A New Setup for Measuring Angle-Resolved Polarized Light Scattering,” J. Phys E 17, 1119 (1984).
[CrossRef]

Bickel, W. S.

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Brazas, J.

J. Brazas, “The Effects of Highly Reflecting Materials on Nearby Fluorescent Probes,” Ph.D. Thesis, U. Illinois at Urbana-Champaign (1983).

Burch, J. M.

A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975).

Clarke, D.

D. Clarke, J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, Oxford, 1971).

Gerrard, A.

A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975).

Grainger, J. F.

D. Clarke, J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, Oxford, 1971).

Herb, P.

Hu, C.-R.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Kattawar, G. W.

McCrackin, F.

F. McCrackin, E. Passaglia, R. Stromberg, H. Steinberg, “Measurement of the Thickness and Refractive Index of Very Thin Films and the Optical Properties of Surfaces by Ellipsometry,” J. Res. Natl. Bur. Stand. Sect. A 67, 363 (1963).
[CrossRef]

Muller, R. H.

R. H. Muller, “Principles of Ellipsometry,” Adv. Electrochem. Eng. 9, 167 (1973).

Parkin, M. E.

Passaglia, E.

F. McCrackin, E. Passaglia, R. Stromberg, H. Steinberg, “Measurement of the Thickness and Refractive Index of Very Thin Films and the Optical Properties of Surfaces by Ellipsometry,” J. Res. Natl. Bur. Stand. Sect. A 67, 363 (1963).
[CrossRef]

Steinberg, H.

F. McCrackin, E. Passaglia, R. Stromberg, H. Steinberg, “Measurement of the Thickness and Refractive Index of Very Thin Films and the Optical Properties of Surfaces by Ellipsometry,” J. Res. Natl. Bur. Stand. Sect. A 67, 363 (1963).
[CrossRef]

Stromberg, R.

F. McCrackin, E. Passaglia, R. Stromberg, H. Steinberg, “Measurement of the Thickness and Refractive Index of Very Thin Films and the Optical Properties of Surfaces by Ellipsometry,” J. Res. Natl. Bur. Stand. Sect. A 67, 363 (1963).
[CrossRef]

Adv. Electrochem. Eng. (1)

R. H. Muller, “Principles of Ellipsometry,” Adv. Electrochem. Eng. 9, 167 (1973).

Appl. Opt. (3)

J. Phys E (1)

A. K. Arora, “A New Setup for Measuring Angle-Resolved Polarized Light Scattering,” J. Phys E 17, 1119 (1984).
[CrossRef]

J. Res. Natl. Bur. Stand. Sect. A (1)

F. McCrackin, E. Passaglia, R. Stromberg, H. Steinberg, “Measurement of the Thickness and Refractive Index of Very Thin Films and the Optical Properties of Surfaces by Ellipsometry,” J. Res. Natl. Bur. Stand. Sect. A 67, 363 (1963).
[CrossRef]

Other (4)

J. Brazas, “The Effects of Highly Reflecting Materials on Nearby Fluorescent Probes,” Ph.D. Thesis, U. Illinois at Urbana-Champaign (1983).

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

D. Clarke, J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, Oxford, 1971).

A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975).

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Figures (10)

Fig. 1
Fig. 1

Goniometer mirror arrangement.

Fig. 2
Fig. 2

Goniometer optical layout: A, B, fixed mirror; C, D, F, rotating goniometer mirrors; E, translation stage; G, spark gap.

Fig. 3
Fig. 3

Uncompensated goniometer response. The Stokes output parameters are plotted as a function of goniometer angle: Ψ = 44.47; Δ = 173.04; Sin = [1,1,0,0].

Fig. 4
Fig. 4

Retarder correction angles for [1,1,0,0] goniometer output. (A) Ideal retarder correction angles: λ/2, halfwave angle; λ/4 quarterwave angle. (B) 10% retarder phase error correction angles: λ/2, halfwave retarder (162°); λ/4, quarterwave retarder (99°).

Fig. 5
Fig. 5

Correction angles for other polarization states: λ/2, halfwave retarder angle; λ/4, quarterwave retarder angle. (A) Correction angles for 45° polarized light: output polarization state, 45° linear; input polarization state, x-axis linear. (B) Correction angles for elliptically polarized light: output polarization state, [1,0.5774,0.5774,0.5774];input polarization state, [1,1,0,0].

Fig. 6
Fig. 6

Experimental compensation angle errors: (A) x-axis linearly polarized light; (B) 45° linearly polarized light.

Fig. 7
Fig. 7

Retarder angle sensitivity at analyzer maximum: goniometer angle 0.0°; unity amplifier gain; nominal output polarization [1,1,0,0].

Fig. 8
Fig. 8

Retarder angle sensitivity at analyzer minimum: goniometer angle 0.0°; amplifier gain 1000; nominal output polarization [1,1,0,0].

Fig. 9
Fig. 9

Experimental polarization corrected output measurements: crossed polarizaer amplifier gain 1000; parallel polarizer amplifier gain 1. (A) Output polarization measurement for x-axis linearly polarized light; (B) output polarization measurement for 45° linearly polarized light.

Fig. 10
Fig. 10

Alternate optical system. Matched orthogonal mirror pairs are used for all reflections.

Tables (2)

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Table I Correction Table for Retarder Angle Calculations

Tables Icon

Table II Polarization Correction Optical Parameters

Equations (52)

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[ I Q U V ]
I = E 2 ,
Q = E x 2 - E y 2 = I cos ( 2 ψ m )
U = 2 E x E y cos Δ = I sin ( 2 ψ m ) cos Δ m ,
V = 2 E x E y sin Δ = I sin ( 2 ψ m ) sin Δ m ,
ψ m = a t n ( E y / E x ) ,
Δ m = Δ y - Δ x .
[ S o ] = [ M pol ] [ S i ] .
[ S o ] = [ M 3 ] [ M 2 ] [ M 1 ] [ S i ] .
[ S i ] = [ M sys ] - 1 [ S o ] .
[ M θ ] = [ R - θ ] [ M std ] [ R θ ] .
tan Ψ e = R p / R s ,
Δ e = Δ p - Δ s ,
[ M Ψ , Δ ] = [ 1 K 1 0 0 K 1 1 0 0 0 0 K 2 cos Δ e K 2 sin Δ e 0 0 - K 2 sin Δ e K 2 cos Δ e ] ,
K 1 = 1 - tan Ψ e 1 + tan Ψ e ,
K 2 = 2 ( tan Ψ e ) 1 / 2 1 + tan Ψ e .
[ M g , θ ] = [ M 4 ] [ M 3 ] [ M 2 ] [ R θ ] [ M 1 ] [ R 90 ] [ M 0 ] [ R 90 ] ;
[ S o ] = [ M g , θ ] [ S i ] .
[ S i ] = [ M g , θ ] - 1 [ S o ] .
θ λ / 4 = tan - 1 ( U i / Q i ) 2 ,
θ λ / 2 = θ off + θ λ / 4 2 ,
tan θ off = [ I i - ( Q i 2 + U i 2 ) 1 / 2 I i + ( Q i 2 + U i 2 ) 1 / 2 ] 1 / 2 .
[ I 1 Q 1 U 1 V 1 ] = [ M 11 M 12 M 13 M 14 M 21 M 22 M 23 M 24 M 31 M 32 M 33 M 34 M 41 M 42 M 43 M 44 ] [ I 0 Q 0 U 0 V 0 ] .
I 1 = R x I x + R y I y ,
I 0 = I x + I y ,
Q 1 = R x I x + R y I y ,
Q 0 = I x - I y .
I x = I 0 + Q 0 2 ,
I y = I 0 - Q 0 2 .
I 1 = ½ [ ( R x + R y ) I 0 + ( R x - R y ) Q 0 ] ,
Q 1 = ½ [ ( R y - R y ) I 0 - ( R x + R y ) Q 0 ] .
tan Ψ = R y / R x .
x = R x + R y .
R x = x 1 + tan Ψ ,
R y = x tan Ψ 1 + tan Ψ .
I 1 = x 2 [ I 0 + Q 0 ( 1 - tan Ψ ) ( 1 + tan Ψ ) ] ,
Q 1 = x 2 [ Q 0 + I 0 ( 1 - tan Ψ ) ( 1 + tan Ψ ) ] .
M 11 = x / 2 M 13 = 0 , M 12 = x 2 ( 1 - tan Ψ ) / ( 1 + tan Ψ ) M 14 = 0 , M 21 = x 2 ( 1 - tan Ψ ) / ( 1 + tan Ψ ) M 23 = 0 , M 22 = x / 2 M 24 = 0.
U 1 = 2 ( I x R x I y R y ) 1 / 2 cos ( Δ 0 - Δ ) ,
U 0 = 2 ( I x I y ) 1 / 2 cos Δ 0 ,
V 1 = 2 ( I x R x I y R y ) 1 / 2 sin ( Δ 0 - Δ ) ,
V 0 = 2 ( I x I y ) 1 / 2 sin Δ 0 .
cos ( A ± B ) = cosA cosB sinA sinB ,
sin ( A ± B ) = sinA cosB ± cosA sinB .
U 1 = ( R x R y ) 1 / 2 [ 2 ( I x I y ) 1 / 2 cos Δ 0 cos Δ + 2 ( I x I y ) 1 / 2 sin Δ 0 sin Δ ] ,
V 1 = ( R x R y ) 1 / 2 [ 2 ( I x I y ) 1 / 2 sin Δ 0 cos Δ + 2 ( I x I y ) 1 / 2 cos Δ 0 sin Δ
U 1 = ( R x R y ) 1 / 2 ( U 0 cos Δ + V 0 sin Δ ) ,
V 1 = ( R x R y ) 1 / 2 ( V 0 cos Δ + U 0 sin Δ ) .
( R x R y ) 1 / 2 = x ( tan Ψ ) 1 / 2 1 + tan Ψ .
U 1 = x ( tan Ψ ) 1 / 2 1 + tan Ψ ( U 0 cos Δ + V 0 sin Δ ) ,
V 1 = x ( tan Ψ ) 1 / 2 1 + tan Ψ ( V 0 cos Δ + U 0 sin Δ ) .
M 31 = 0             M 33 = x ( tan Ψ ) 1 / 2 cos Δ 1 + tan Ψ , M 32 = 0 M 34 = x ( tan Ψ ) 1 / 2 sin Δ 1 + tan Ψ , M 41 = 0 M 43 = - x ( tan Ψ ) 1 / 2 sin Δ 1 + tan Ψ , M 42 = 0 M 44 = x ( tan Ψ ) 1 / 2 cos Δ 1 + tan Ψ .

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