Abstract

The general theories of single-and double-exposure polarization holography in Jones calculus, Mueller calculus, coherence calculus, Poincare sphere, and the geometric meaning method can be introduced into the problems of polarization holography. From the results obtained, we offer an interpretation of the reference light in polarization holography.

© 1986 Optical Society of America

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References

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  1. R. C. Jones, “A New Calculus for the Treatment of Optical System. 1: Description and Discussion of the Calculus,” J. Opt. Soc. Am. 31, 488 (1941).
    [Crossref]
  2. R. C. Jones, “Transmittance of a Train of Three Polarizers,” J. Opt. Soc. Am. 46, 528 (1956).
    [Crossref]
  3. H. Mueller, “The Foundation of Optics,” J. Opt. Soc. Am. 38, 361 (1948).
  4. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Chap. 10.
  5. C. K. Lee, C. P. Hu, “The Coherence Calculus,” in Procetdings, Seventh International Conference on Experimental Stress Analysis, Haifa, Israel (23–27 Aug. 1982.), 429.
  6. H. Poincare, Theorie Mathematique de la Lumiere, Vol. 2 (Gauthiers-Villars, Paris, 1892), Chap. 12.
  7. C. K. Lee, C. P. Hu, “Further Development of the Analytic Methods for Polarization Optics (2): The Geometric Meaning Method,” in Proceedings, Seventh International Conference on Experimental Stress Analysis, Haifa, Israel (23–27 Aug. 1982), p.431.
  8. M. E. Fourney, “Application of Holography to Photoelasticity,” Exp. Mech. 8, 33 (1968).
    [Crossref]
  9. R. J. Sanford, “A General Theory of Polarization and Its Applications to Holography,” Proc. Soc. Photo-Opt. Instrum. Eng.331 (1972).
  10. D. Gabor, “Microscopy by Reconstructed Wave-Front,” Proc. R. Soc. London Ser. A 197, 454 (1949).
    [Crossref]
  11. E. N. Leith, J. Upatnieks, “Wavefront Reconstruction with Continuous-Tone Objects,” J. Opt. Soc. Am. 53, 1377 (1963).
    [Crossref]
  12. D. G. Falconer, “Role of the Photographic Process in Holography,” Photogr. Sci. Eng. 10, 133 (1966).

1972 (1)

R. J. Sanford, “A General Theory of Polarization and Its Applications to Holography,” Proc. Soc. Photo-Opt. Instrum. Eng.331 (1972).

1968 (1)

M. E. Fourney, “Application of Holography to Photoelasticity,” Exp. Mech. 8, 33 (1968).
[Crossref]

1966 (1)

D. G. Falconer, “Role of the Photographic Process in Holography,” Photogr. Sci. Eng. 10, 133 (1966).

1963 (1)

1956 (1)

1949 (1)

D. Gabor, “Microscopy by Reconstructed Wave-Front,” Proc. R. Soc. London Ser. A 197, 454 (1949).
[Crossref]

1948 (1)

H. Mueller, “The Foundation of Optics,” J. Opt. Soc. Am. 38, 361 (1948).

1941 (1)

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Chap. 10.

Falconer, D. G.

D. G. Falconer, “Role of the Photographic Process in Holography,” Photogr. Sci. Eng. 10, 133 (1966).

Fourney, M. E.

M. E. Fourney, “Application of Holography to Photoelasticity,” Exp. Mech. 8, 33 (1968).
[Crossref]

Gabor, D.

D. Gabor, “Microscopy by Reconstructed Wave-Front,” Proc. R. Soc. London Ser. A 197, 454 (1949).
[Crossref]

Hu, C. P.

C. K. Lee, C. P. Hu, “The Coherence Calculus,” in Procetdings, Seventh International Conference on Experimental Stress Analysis, Haifa, Israel (23–27 Aug. 1982.), 429.

C. K. Lee, C. P. Hu, “Further Development of the Analytic Methods for Polarization Optics (2): The Geometric Meaning Method,” in Proceedings, Seventh International Conference on Experimental Stress Analysis, Haifa, Israel (23–27 Aug. 1982), p.431.

Jones, R. C.

Lee, C. K.

C. K. Lee, C. P. Hu, “Further Development of the Analytic Methods for Polarization Optics (2): The Geometric Meaning Method,” in Proceedings, Seventh International Conference on Experimental Stress Analysis, Haifa, Israel (23–27 Aug. 1982), p.431.

C. K. Lee, C. P. Hu, “The Coherence Calculus,” in Procetdings, Seventh International Conference on Experimental Stress Analysis, Haifa, Israel (23–27 Aug. 1982.), 429.

Leith, E. N.

Mueller, H.

H. Mueller, “The Foundation of Optics,” J. Opt. Soc. Am. 38, 361 (1948).

Poincare, H.

H. Poincare, Theorie Mathematique de la Lumiere, Vol. 2 (Gauthiers-Villars, Paris, 1892), Chap. 12.

Sanford, R. J.

R. J. Sanford, “A General Theory of Polarization and Its Applications to Holography,” Proc. Soc. Photo-Opt. Instrum. Eng.331 (1972).

Upatnieks, J.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Chap. 10.

Exp. Mech. (1)

M. E. Fourney, “Application of Holography to Photoelasticity,” Exp. Mech. 8, 33 (1968).
[Crossref]

J. Opt. Soc. Am. (4)

Photogr. Sci. Eng. (1)

D. G. Falconer, “Role of the Photographic Process in Holography,” Photogr. Sci. Eng. 10, 133 (1966).

Proc. R. Soc. London Ser. A (1)

D. Gabor, “Microscopy by Reconstructed Wave-Front,” Proc. R. Soc. London Ser. A 197, 454 (1949).
[Crossref]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

R. J. Sanford, “A General Theory of Polarization and Its Applications to Holography,” Proc. Soc. Photo-Opt. Instrum. Eng.331 (1972).

Other (4)

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), Chap. 10.

C. K. Lee, C. P. Hu, “The Coherence Calculus,” in Procetdings, Seventh International Conference on Experimental Stress Analysis, Haifa, Israel (23–27 Aug. 1982.), 429.

H. Poincare, Theorie Mathematique de la Lumiere, Vol. 2 (Gauthiers-Villars, Paris, 1892), Chap. 12.

C. K. Lee, C. P. Hu, “Further Development of the Analytic Methods for Polarization Optics (2): The Geometric Meaning Method,” in Proceedings, Seventh International Conference on Experimental Stress Analysis, Haifa, Israel (23–27 Aug. 1982), p.431.

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Figures (2)

Fig. 1
Fig. 1

Typical holographic optical system for a light field.

Fig. 2
Fig. 2

Representation of the operation on the Poincare sphere.

Equations (32)

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a c sum = a 1 + a 2 .
a i c sum = s 1 + s 2 = J ˜ 1 + J ˜ 2 ,
a E = α a I ,
I E / I I = α α ,
α = ( F τ I ) - γ / 2 ,
α = F τ I .
( a )             α u = F τ ( I o + I R )
( b )             α v = F τ ( a R a o )
( c )             α Re = F τ ( a R a o ) *
I v / I R C = F 2 τ 2 a o J ˜ R F a o .
I Re / I R C = F 2 τ 2 a R J ˜ o a R ,
I α a o + J ˜ R a o .
a o J ˜ R a o = a o ( ½ [ S R 0 I ˜ + S R 1 σ ˜ z + S R 2 σ ˜ x + S R 3 σ ˜ y ] ) a o = ½ [ S R 0 ( a o I ˜ a o ) + S R 1 ( a o σ ˜ z a o ) + S R 2 ( a o σ ˜ x a o ) + S R 3 ( a o σ ˜ y a o ) = ½ ( S R 0 S 0 + S R 1 S 01 + S R 2 S 02 + S R 3 S 03 ) = ½ ( S R · S 0 )
I ˜ = [ 1 0 0 1 ] ,             σ ˜ x = [ 0 1 1 0 ] ,             σ ˜ y = [ 0 - i i 0 ] ,             σ ˜ z = [ 0 1 1 0 ]
a o J ˜ R a o = ( a x o * a y o * ) [ ( J x x ) R ( J x y ) R ( J y x ) R ( J y y ) R ] [ a x o a y o ] = ( J x x ) R ( J x x ) o + ( J x y ) R ( J y x ) o + ( J y x ) R ( J x y ) o + ( J y y ) R ( J y y ) o = trace ( J ˜ R J ˜ o ) = trace ( J ˜ o J ˜ R ) .
a o J ˜ R a o = trace ( J ˜ R J ˜ o = trace ( J ˜ R J ˜ o J ˜ r ) .
½ ( S R · S o ) = ½ [ 1 + ( S R 1 , S R 2 , S R 3 ) · ( S 01 , S 02 , S 03 ) ] = ½ [ 1 + cos O R ] = cos 2 O R 2 ,
I α 1 + S ( Pr _ - 1 ) 2 = 1 + S ( rP _ - 1 ) 2 ,
R β ( δ ) = exp ( - i π / λ ) ( n 1 + n 2 - 2 n ) t × [ cos θ - sin θ sin θ cos θ ] [ exp ( + i δ / 2 ) 0 0 exp ( - i δ / 2 ) ] × [ cos θ sin θ - sin θ cos θ ] ,
δ = - 2 π t λ ( n 1 - n 2 ) ,
a o = 1 2 exp ( - i π / λ ) ( n 1 + n 2 - 2 n ) t [ cos 2 θ exp ( i δ / 2 ) + sin 2 θ exp ( - i δ / 2 ) 2 i sin θ cos θ sin δ / 2 2 i sin θ cos θ sin δ / 2 cos 2 θ exp ( - i δ / 2 ) + sin 2 θ exp ( i δ / 2 ) ] [ 1 i i 1 ] [ 0 0 0 1 ] [ A x exp ( i δ x ) A y exp ( i δ y ) ] .
a = 1 2 A y exp ( i δ y ) exp ( - i π / λ ) ( n 1 + n 2 - 2 n ) t [ i [ cos 2 r exp ( i δ / 2 ) + sin 2 θ exp ( - i δ / 2 ) + 2 cos θ sin θ sin δ / 2 ] cos 2 θ exp ( - i δ / 2 ) + sin 2 θ exp ( i δ / 2 ) - 2 cos θ sin θ sin δ / 2 ] .
a R = 1 / 2 [ 1 i i 1 ] [ 0 0 0 1 ] [ A x exp ( i δ x ) A y exp ( i δ y ) ] ,
a R = 1 / 2 A y exp ( i δ y ) [ i 1 ] .
I V / I R α cos 2 δ / 2.
J R = ½ ( [ 1 i i 1 ] [ 0 0 0 1 ] ) [ A x 2 A x A y exp [ - i ( δ y - δ x ) ] A x A y exp [ i ( δ y - δ x ) ] A y 2 ] ( [ 1 i i 1 ] [ 0 0 0 1 ] ) ,
J O = ½ ( T [ 1 i i 1 ] [ 0 0 0 1 ] ) [ A x 2 A x A y exp [ - i ( δ y - δ x ) ] A x A y exp [ t ( δ y - δ x ) ] A y 2 ] ( T [ 1 i i 1 ] [ 0 0 0 1 ] ) ,
T = [ cos 2 θ exp ( i δ / 2 ) + sin 2 θ exp ( - i δ / 2 ) 2 i sin θ cos θ sin δ / 2 2 i sin θ cos θ sin δ / 2 cos 2 θ exp ( - i δ / 2 ) + sin 2 θ exp ( i δ / 2 ) ] .
S o = ½ [ 1 0 0 0 0 cos 2 2 θ + sin 2 2 θ cos δ ( 1 - cos δ ) sin 2 θ cos 2 θ - sin 2 θ sin δ 0 ( 1 - cos δ ) sin 2 θ cos 2 θ sin 2 2 θ + cos 2 2 θ cos δ cos 2 θ cos δ 0 sin 2 sin δ - cos 2 θ sin δ cos δ ] ,
S R = ½ [ 1 0 0 0 0 0 0 - 1 0 0 1 0 0 1 0 0 ] [ 1 - 1 0 0 - 1 1 0 0 0 0 0 0 0 0 0 0 ] [ S 0 S 1 S 2 S 3 ] [ 1 0 0 0 0 0 0 - 1 0 0 1 0 0 1 0 0 ] [ 1 - 1 0 0 - 1 1 0 0 0 0 0 0 0 0 0 0 ] [ S 0 S 1 S 2 S 3 ] .
P _ o = ( q _ β q _ 45 ] ) ( - i _ ) ( q _ β q _ 45 ° ) - 1 ,
P _ R = ( q _ 45 ° ) ( - i _ ) ( q _ 45 ° ) - 1 ,

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