Abstract

All 16 elements of the Mueller matrix of an optical system (sample) can be encoded onto, hence can be retrieved from, a single detected signal using a class of photopolarimeters with modulated polarizing and analyzing optics. The general theory of operation of such polarimeters is presented. We also propose a specific new photopolarimeter whose polarizing and analyzing optics are modulated by synchronously rotating two quarter-wave retarders at angular speeds ω and 5ω. When the light flux leaving such polarimeter is linearly detected, a periodic signal J=a0+n=112(ancosnωft+bnsinnωft) is generated, with fundamental frequency ωf = 2ω. From the Fourier amplitudes a0, an, bn, to be measured by performing a discrete Fourier transform (DFT) of the signal , the 16 elements of the Mueller matrix are simply determined.

© 1978 Optical Society of America

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References

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  1. Also called the phase matrix when considering light scattering by an inhomogeneous medium or a system of particles.
  2. R. M. A. Azzam, Surf. Sci. 56, 6 (1976).
    [CrossRef]
  3. P. S. Hauge, Proc. SPIE 112, 2 (1977).
  4. P. S. Hauge, Opt. Commun. 17, 74 (1976).
    [CrossRef]
  5. In this case, some, but not all, of the elements of the Mueller matrix are encoded on one signal. See, e.g., A. J. Hunt, D. R. Huffman, Rev. Sci. Instrum. 44, 1753 (1973).
    [CrossRef]
  6. R. Thompson, E. Fry, J. Bottiger, Proc. SPIE 112, 152 (1977).
  7. Q is obviously the matrix of the coefficients μkijc,s of the Mueller-matrix elements mij in Eq. (8).
  8. R. M. A. Azzam, Opt. Commun., in press.
  9. R. M. A. Azzam, J. Opt. Soc. Am.68 (1978), in press.

1977

P. S. Hauge, Proc. SPIE 112, 2 (1977).

R. Thompson, E. Fry, J. Bottiger, Proc. SPIE 112, 152 (1977).

1976

R. M. A. Azzam, Surf. Sci. 56, 6 (1976).
[CrossRef]

P. S. Hauge, Opt. Commun. 17, 74 (1976).
[CrossRef]

1973

In this case, some, but not all, of the elements of the Mueller matrix are encoded on one signal. See, e.g., A. J. Hunt, D. R. Huffman, Rev. Sci. Instrum. 44, 1753 (1973).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam, Surf. Sci. 56, 6 (1976).
[CrossRef]

R. M. A. Azzam, Opt. Commun., in press.

R. M. A. Azzam, J. Opt. Soc. Am.68 (1978), in press.

Bottiger, J.

R. Thompson, E. Fry, J. Bottiger, Proc. SPIE 112, 152 (1977).

Fry, E.

R. Thompson, E. Fry, J. Bottiger, Proc. SPIE 112, 152 (1977).

Hauge, P. S.

P. S. Hauge, Proc. SPIE 112, 2 (1977).

P. S. Hauge, Opt. Commun. 17, 74 (1976).
[CrossRef]

Huffman, D. R.

In this case, some, but not all, of the elements of the Mueller matrix are encoded on one signal. See, e.g., A. J. Hunt, D. R. Huffman, Rev. Sci. Instrum. 44, 1753 (1973).
[CrossRef]

Hunt, A. J.

In this case, some, but not all, of the elements of the Mueller matrix are encoded on one signal. See, e.g., A. J. Hunt, D. R. Huffman, Rev. Sci. Instrum. 44, 1753 (1973).
[CrossRef]

Thompson, R.

R. Thompson, E. Fry, J. Bottiger, Proc. SPIE 112, 152 (1977).

Opt. Commun.

P. S. Hauge, Opt. Commun. 17, 74 (1976).
[CrossRef]

Proc. SPIE

R. Thompson, E. Fry, J. Bottiger, Proc. SPIE 112, 152 (1977).

P. S. Hauge, Proc. SPIE 112, 2 (1977).

Rev. Sci. Instrum.

In this case, some, but not all, of the elements of the Mueller matrix are encoded on one signal. See, e.g., A. J. Hunt, D. R. Huffman, Rev. Sci. Instrum. 44, 1753 (1973).
[CrossRef]

Surf. Sci.

R. M. A. Azzam, Surf. Sci. 56, 6 (1976).
[CrossRef]

Other

Also called the phase matrix when considering light scattering by an inhomogeneous medium or a system of particles.

Q is obviously the matrix of the coefficients μkijc,s of the Mueller-matrix elements mij in Eq. (8).

R. M. A. Azzam, Opt. Commun., in press.

R. M. A. Azzam, J. Opt. Soc. Am.68 (1978), in press.

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Figures (2)

Fig. 1
Fig. 1

General polarimeter with arbitrary polarizing and analyzing optics.

Fig. 2
Fig. 2

A Fourier photopolarimeter capable of measuring all 16 elements of the Mueller matrix of an optical system.

Tables (1)

Tables Icon

Table 1 Relations between the Signal Fourier Amplitudes and the Elements of the Scaled Mueller Matrix M′

Equations (16)

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S f = MS i ,
J = c AMP ,
J = c i , j = 1 4 a i p j m i j ,
J = c i , j = 1 4 μ i j m i j ,
μ i j = a i p j .
μ = k ( μ k c cos ω k t + μ k s sin ω k t ) .
J = k ( J k c cos ω k t + J k s sin ω k t ) ,
J k c , s = c i , j = 1 4 μ k i j c , s m i j .
μ = I τ { 1 cos 2 2 C sin 2 C cos 2 C sin 2 C cos 2 2 C cos 2 2 C cos 2 2 C sin 2 C cos 2 C cos 2 2 C sin 2 C cos 2 2 C sin 2 C cos 2 C cos 2 2 C sin 2 C cos 2 C sin 2 C cos 2 C sin 2 C cos 2 C sin 2 C sin 2 C cos 2 C - sin 2 C - cos 2 2 C sin 2 C - sin 2 C cos 2 C sin 2 C - sin 2 C sin 2 C } ,
C = ω t ,             C = 5 ω t .
J = a 0 + n = 1 12 ( a n cos n ω f t + b n sin n ω f t ) ,
M = i 0 M ,
i 0 = c I τ .
m 24 = - 4 b 9 = 4 b 11 = 2 ( b 11 - b 9 ) , m 34 = 4 a 9 = - 4 a 11 = 2 ( a 9 - a 11 ) , m 44 = - 2 a 4 = 2 a 6 = ( a 6 - a 4 ) , m 43 = - 4 a 3 = 4 a 7 = 2 ( a 7 - a 3 ) , m 42 = - 4 b 3 = - 4 b 7 = - 2 ( b 3 + b 7 ) .
m 22 = 4 ( a 8 + a 12 ) ,             m 23 = 4 ( - b 8 + b 12 ) , m 32 = 4 ( b 8 + b 12 ) ,             m 33 = 4 ( a 8 - a 12 ) .
m 12 = 2 a 2 - 1 2 m 22 ,             m 13 = 2 b 2 - 1 2 m 23 , m 21 = 2 a 10 - 1 2 m 22 ,             m 31 = 2 b 10 - 1 2 m 32 , m 11 = a 0 - 1 2 m 12 - 1 2 m 21 - 1 4 m 22 .

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