Abstract

We derive the phase error of a hologram that is due to the imperfection of a wave plate, the azimuth angle error of a wave plate, and the azimuth angle error of a linear polarizer and analyze the effect of the phase error with a modified triangular interferometer of the three-dimensional image reconstruction of a hologram.

© 2009 Optical Society of America

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References

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  1. L. Mertz and N. O. Young, “Fresnel transformations of optics,” in Proceedings of the Conference on Optical Instruments and Techniques, K. J. Habell, ed. (Chapman & Hall, 1962), p. 305.
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  7. S.-G. Kim, B. Lee, and E.-S. Kim, “Removal of bias and the conjugate image in incoherent on-axis triangular holography and real-time reconstruction of the complex hologram,” Appl. Opt. 36, 4784-4791 (1997).
    [CrossRef] [PubMed]
  8. W. Steinchen and L. Yang, Digital Shearography: Theory and Application of Digital Speckle Pattern Shearing Interferometry, Press Monograph Vol. PM100 (SPIE Press, 2003), Chap. 4.
    [PubMed]
  9. D. Kim and Y. J. Cho, “3-D surface profile measurement using an acousto-optic tunable filter based spectral phase shifting technique,” J. Opt. Soc. Korea 12, 281-287(2008).
    [CrossRef]
  10. J. Hahn, H. Kim, S.-W. Cho, and B. Lee, “Phase-shifting interferometry with genetic algorithm-based twin image noise elimination,” Appl. Opt. 47, 4068-4076 (2008).
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    [CrossRef]
  12. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 2003) Chap. 5.

2008 (2)

2001 (1)

2000 (1)

1997 (1)

1992 (1)

1969 (1)

1966 (1)

1965 (1)

Cho, S.-W.

Cho, Y. J.

Cochran, G.

Hahn, J.

Indebetouw, G.

Kim, D.

Kim, E.-S.

Kim, H.

Kim, S.-G.

Kim, T.

Kozma, A.

Lee, B.

Lohmann, A. W.

Massey, N.

Mertz, L.

L. Mertz and N. O. Young, “Fresnel transformations of optics,” in Proceedings of the Conference on Optical Instruments and Techniques, K. J. Habell, ed. (Chapman & Hall, 1962), p. 305.

Mugnier, L. M.

Poon, T.-C.

Schilling, B. W.

Shinoda, K.

Sirat, G. Y.

Steinchen, W.

W. Steinchen and L. Yang, Digital Shearography: Theory and Application of Digital Speckle Pattern Shearing Interferometry, Press Monograph Vol. PM100 (SPIE Press, 2003), Chap. 4.
[PubMed]

Suzuki, Y.

Wu, M. H.

Yang, L.

W. Steinchen and L. Yang, Digital Shearography: Theory and Application of Digital Speckle Pattern Shearing Interferometry, Press Monograph Vol. PM100 (SPIE Press, 2003), Chap. 4.
[PubMed]

Yariv, A.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 2003) Chap. 5.

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 2003) Chap. 5.

Yi, C.-W.

Young, N. O.

L. Mertz and N. O. Young, “Fresnel transformations of optics,” in Proceedings of the Conference on Optical Instruments and Techniques, K. J. Habell, ed. (Chapman & Hall, 1962), p. 305.

Appl. Opt. (4)

J. Opt. Soc. Am. (2)

J. Opt. Soc. Korea (1)

Opt. Lett. (2)

Other (3)

L. Mertz and N. O. Young, “Fresnel transformations of optics,” in Proceedings of the Conference on Optical Instruments and Techniques, K. J. Habell, ed. (Chapman & Hall, 1962), p. 305.

W. Steinchen and L. Yang, Digital Shearography: Theory and Application of Digital Speckle Pattern Shearing Interferometry, Press Monograph Vol. PM100 (SPIE Press, 2003), Chap. 4.
[PubMed]

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 2003) Chap. 5.

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Figures (4)

Fig. 1
Fig. 1

Modified triangular interferometer.

Fig. 2
Fig. 2

Phase error from the imperfection of a wave plate and the azimuth angle error of polarization elements as a function of phase difference ϕ.

Fig. 3
Fig. 3

Change rate of the phase error from the imperfection of a wave plate and the azimuth angle error of polarization elements as a function of phase difference ϕ.

Fig. 4
Fig. 4

Intensity of the reconstructed image of a point-source hologram as a function of coordinate z for four error sums.

Tables (1)

Tables Icon

Table 1 Intensity Patterns by Combination of a Wave Plate and a Linear Polarizer

Equations (41)

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p c w ( x , y ) = i k 2 2 π z 0 exp [ i k 2 z 0 { ( α x x 0 ) 2 + ( α y y 0 ) 2 } ] = a exp { i θ cw ( x , y ) } ,
p ccw ( x , y ) = i k 2 2 π z 0 exp [ i k 2 z 0 { ( β x x 0 ) 2 + ( β y y 0 ) 2 } ] = b exp { i θ ccw ( x , y ) } ,
ϕ ( x , y ) = θ cw ( x , y ) θ ccw ( x , y ) = ( k / 2 z 1 ) { ( x x 1 ) 2 + ( y y 1 ) 2 ( x 1 2 + y 1 2 ) } , x 1 = x 0 / ( α + β ) , y 1 = y 0 / ( α + β ) , z 1 = z 0 / ( α 2 β 2 ) .
H ( x , y ) = exp { ± i ϕ ( x , y ) } .
E out = A ( φ 2 ) W ( φ 1 ) E in ,
E in = ( p ccw p cw ) = ( b exp ( i θ ccw ) a exp ( i θ cw ) ) ,
A ( φ 2 ) = ( cos 2 φ 2 1 / 2 sin 2 φ 2 1 / 2 sin 2 φ 2 sin 2 φ 2 ) ,
W ( φ 1 ) = ( 2 i sin 2 φ 1 sin Γ 2 + exp ( i Γ 2 ) i sin 2 φ 1 sin Γ 2 i sin 2 φ 1 sin Γ 2 2 i sin 2 φ 1 sin Γ 2 + exp ( i Γ 2 ) ) ,
I = | E out | 2 .
I 1 = 1 2 a 2 ( 1 cos Γ ) + 1 2 b 2 ( 1 + cos Γ ) a b sin ϕ sin Γ .
I 2 = 1 2 a 2 ( 1 cos Γ ) + 1 2 b 2 ( 1 + cos Γ ) + a b sin ϕ sin Γ .
I 3 = 1 2 a 2 + 1 2 b 2 + a b cos ϕ .
I 4 = 1 2 a 2 + 1 2 b 2 a b cos ϕ .
tan ϕ = I 2 I 1 I 3 I 4 = tan ϕ sin Γ ,
Γ = π 2 + ω ,
tan ϕ = tan ϕ cos ω .
tan ϕ = tan ( ϕ + Δ ϕ ) tan ϕ + Δ ϕ sec 2 ϕ ,
Δ ϕ W P = 1 4 ω 2 sin ( 2 ϕ ) .
I 1 = 1 2 a 2 + b 2 ( 1 2 + 2 β 1 2 ) a b ( 2 β 1 cos ϕ + sin ϕ ) .
I 2 = 1 2 a 2 + b 2 ( 1 2 + 2 β 2 2 ) a b ( 2 β 2 cos ϕ sin ϕ ) .
I 3 = 1 2 a 2 ( 1 + 2 β 1 + 2 β 1 2 ) + 1 2 b 2 ( 1 2 β 1 + 2 β 1 2 ) + a b ( cos ϕ 2 β 1 sin ϕ ) .
I 4 = 1 2 a 2 ( 1 2 β 1 + 2 β 1 2 ) + 1 2 b 2 ( 1 + 2 β 1 + 2 β 1 2 ) a b ( cos ϕ 2 β 1 sin ϕ ) .
tan ϕ = I 2 I 1 I 3 I 4 = b a ( β 2 2 β 1 2 ) ( β 2 β 1 ) cos ϕ + sin ϕ 2 β 1 cot 2 β + cos ϕ 2 β 1 sin ϕ tan ϕ + 1 + sin 2 ϕ 2 cot 2 β sin ϕ cos 2 ϕ β 1 β 2 ,
Δ ϕ a W P = ( 1 + sin 2 ϕ 2 cot 2 β sin ϕ ) β 1 β 2 cos 2 ϕ .
I 1 = 1 2 ( a 2 + b 2 ) + a b ( 2 γ 1 cos ϕ sin ϕ ) .
I 2 = 1 2 ( a 2 + b 2 ) + a b ( 2 γ 1 cos ϕ + sin ϕ ) .
I 3 = 1 2 ( a 2 + b 2 ) + a b ( cos ϕ + 2 γ 2 sin ϕ ) .
I 4 = 1 2 ( a 2 + b 2 ) a b ( cos ϕ + 2 γ 3 sin ϕ ) .
tan ϕ ( x , y ) = I 2 I 1 I 3 I 4 = sin ϕ [ cos ϕ + ( γ 2 + γ 3 ) sin ϕ ] tan ϕ tan 2 ϕ ( γ 2 + γ 3 ) .
Δ ϕ a L P = ( γ 2 + γ 3 ) sin 2 ϕ .
( Δ ϕ W P ) = 1 2 ω 2 cos ( 2 ϕ ) ,
( Δ ϕ a W P ) = ( β 1 + β 2 ) sin ( 2 ϕ ) ,
( Δ ϕ a L P ) = ( γ 2 + γ 3 ) sin ( 2 ϕ ) .
Δ ϕ = 1 4 ω 2 sin ( 2 ϕ ) + ( 1 + sin 2 ϕ 2 cot 2 β sin ϕ ) β 1 β 2 cos 2 ϕ ( γ 2 + γ 3 ) sin 2 ϕ .
Δ ϕ = 1 4 ω 2 sin ( 2 ϕ ) + ( 1 + sin 2 ϕ ) β 1 β 2 cos 2 ϕ ( γ 2 + γ 3 ) sin 2 ϕ .
H r ( x , y ) = exp ( i ϕ tot ) ,
ϕ tot = ϕ 1 4 ω 2 sin 2 ϕ + β 1 β 2 cos 2 ϕ ( γ 2 + γ 3 β 1 ) sin 2 ϕ .
H r ( x , y ) = exp i { ( k / 2 z 1 ) ( x 2 + y 2 ) 1 4 ω 2 sin [ ( k / z 1 ) ( x 2 + y 2 ) ] + β 1 β 2 cos 2 [ ( k / 2 z 1 ) ( x 2 + y 2 ) ] ( γ 2 + γ 3 β 1 ) sin 2 [ ( k / 2 z 1 ) ( x 2 + y 2 ) ] } .
U ( x , y , z ) = i λ z exp ( i k z ) ξ 2 + η 2 R 2 H r ( ξ , η ) exp { i k 2 z [ ( x ξ ) 2 + ( y η ) 2 ] } d ξ d η ,
U ( 0 , 0 , z ) = i λ z exp ( i k z ) ξ 2 + η 2 R 2 H r ( ξ , η ) exp [ i k 2 z ( ξ 2 + η 2 ) ] d ξ d η .
I ( 0 , 0 , z ) = k 2 R 4 4 z 2 m = J m 2 ( 1 4 ω 2 ) q = J q 2 { 1 2 ( β 1 + β 2 γ 2 γ 3 ) } sinc 2 { k 4 ( 1 z 1 1 z 2 m z 1 2 q z 1 ) R 2 } ,

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