Abstract

We apply digital in-line holography to image opaque objects through a thick plano–concave pipe. Opaque fibers and opaque particles are considered. Analytical expression of the intensity distribution in the CCD sensor plane is derived using a generalized Fresnel transform. The proposed model has the ability to deal with various pipe shapes and thicknesses and compensates for the lack of versatility of classical digital in-line holography models. Holograms obtained with a 12mm thick plano–concave pipe are then reconstructed using a fractional Fourier transform. This method allows us to get rid of astigmatism. Numerical and experimental results are presented.

© 2008 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
  3. E. Malkiel, J. Sheng, J. Katz, and J. Strickler, “The three-dimensional flow field generated by a feeding calanoid copepod measured using digital holography,” J. Exp. Biol. 206, 3657-3666 (2003).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  8. S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattini, “Wave front reconstruction of Fresnel off-axis holograms with compensation of aberrations by means of phase-shifting digital holography,” Opt. Lasers Eng. 37, 331-340 (2002).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2008 (1)

2007 (1)

2006 (5)

2005 (2)

2004 (1)

M. Malek, D. Allano, S. Coëtmellec, C. Özkul, and D. Lebrun, “Digital in-line holography for three-dimensional-two-components particle tracking velocimetry,” Meas. Sci. Technol. 15, 699-705 (2004).
[CrossRef]

2003 (3)

E. Malkiel, J. Sheng, J. Katz, and J. Strickler, “The three-dimensional flow field generated by a feeding calanoid copepod measured using digital holography,” J. Exp. Biol. 206, 3657-3666 (2003).
[CrossRef]

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245-258 (2003).
[CrossRef]

G. Pan and H. Meng, “Digital holography of particle fields: reconstruction by use of complex amplitude,” Appl. Opt. 42, 827-833 (2003).
[CrossRef]

2002 (1)

S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattini, “Wave front reconstruction of Fresnel off-axis holograms with compensation of aberrations by means of phase-shifting digital holography,” Opt. Lasers Eng. 37, 331-340 (2002).
[CrossRef]

2001 (3)

1999 (1)

S.-C. Pei, M.-H. Yeh, and C.-C. Tseng, “Discrete fractional Fourier transform based on orthogonal projections,” IEEE Trans. Signal Process. 47, 1335-1348 (1999).
[CrossRef]

1998 (1)

1997 (1)

1993 (2)

1992 (1)

C. S. Vikram, “Particle field holography,” in Cambridge Studies in Modern Optics (Cambridge U. Press, 1992).

1988 (1)

J. J. Wen and M. Breazeale, “A diffraction beam expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[CrossRef]

1987 (4)

L. Onural and P. Scott, “Digital decoding of in-line holograms,” Opt. Eng. 26, 1124-1132 (1987).

J. J. Wen and M. Breazeale, “Gaussian beam functions as a base function set for acoustical field calculations,” Proc. IEEE Ultrason. Symp. 1137-1140 (1987).

H. T. Yura and S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am. A 4, 1931-1948 (1987).

A. C. McBride and F. H. Kerr, “On Namias's fractional Fourier transforms,” IMA J. Appl. Math. 39, 159-175(1987).
[CrossRef]

1986 (1)

A. E. Siegman, Lasers (University Science Books, 1986).

1984 (1)

1982 (1)

1980 (1)

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241-265 (1980).
[CrossRef]

1949 (1)

D. Gabor, “Microscopy by reconstructed wave-fronts,” Proc. R. Soc. London Ser. A 197, 454-487 (1949).

Allano, D.

Bagini, V.

Breazeale, M.

J. J. Wen and M. Breazeale, “A diffraction beam expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[CrossRef]

J. J. Wen and M. Breazeale, “Gaussian beam functions as a base function set for acoustical field calculations,” Proc. IEEE Ultrason. Symp. 1137-1140 (1987).

Brunel, M.

Coëtmellec, S.

N. Verrier, S. Coëtmellec, M. Brunel, D. Lebrun, and A. J. E. M. Janssen, “Digital in-line holography with an elliptical, astigmatic Gaussian beam: wide-angle reconstruction,” J. Opt. Soc. Am. A 25, 1459-1466 (2008).
[CrossRef]

F. Nicolas, S. Coëtmellec, M. Brunel, and D. Lebrun, “Digital in-line holography with a sub-picosecond laser beam,” Opt. Commun. 268, 27-33 (2006).
[CrossRef]

M. Malek, D. Allano, S. Coëtmellec, C. Özkul, and D. Lebrun, “Digital in-line holography for three-dimensional-two-components particle tracking velocimetry,” Meas. Sci. Technol. 15, 699-705 (2004).
[CrossRef]

Crane, J. S.

De Nicola, S.

Du, X.

Dunn, P.

Ferraro, P.

Finizio, A.

Fraser, D.

Gabor, D.

D. Gabor, “Microscopy by reconstructed wave-fronts,” Proc. R. Soc. London Ser. A 197, 454-487 (1949).

Garcia-Sucerquia, J.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts and Company, 2005).

Gouesbet, G.

Grehan, G.

Grilli, S.

Hanson, S. G.

Hernández, C.

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245-258 (2003).
[CrossRef]

Illueca, C.

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245-258 (2003).
[CrossRef]

Janssen, A. J.

Janssen, A. J. E. M.

Jericho, M.

Jericho, S.

Katz, J.

E. Malkiel, J. Sheng, J. Katz, and J. Strickler, “The three-dimensional flow field generated by a feeding calanoid copepod measured using digital holography,” J. Exp. Biol. 206, 3657-3666 (2003).
[CrossRef]

Kerr, F. H.

A. C. McBride and F. H. Kerr, “On Namias's fractional Fourier transforms,” IMA J. Appl. Math. 39, 159-175(1987).
[CrossRef]

Klages, P.

Knapp, J. Z.

Kreuzer, H.

Kutay, M. A.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform: with Applications in Optics and Signal Processing (Wiley, 2001).

Lambert, A. J.

Lebrun, D.

N. Verrier, S. Coëtmellec, M. Brunel, D. Lebrun, and A. J. E. M. Janssen, “Digital in-line holography with an elliptical, astigmatic Gaussian beam: wide-angle reconstruction,” J. Opt. Soc. Am. A 25, 1459-1466 (2008).
[CrossRef]

F. Nicolas, S. Coëtmellec, M. Brunel, and D. Lebrun, “Digital in-line holography with a sub-picosecond laser beam,” Opt. Commun. 268, 27-33 (2006).
[CrossRef]

F. Nicolas, S. Coëtmellec, M. Brunel, D. Allano, D. Lebrun, and A. J. Janssen, “Application of the fractional Fourier transformation to digital holography recorded by an elliptical, astigmatic Gaussian beam,” J. Opt. Soc. Am. A 22, 2569-2577(2005).
[CrossRef]

M. Malek, D. Allano, S. Coëtmellec, C. Özkul, and D. Lebrun, “Digital in-line holography for three-dimensional-two-components particle tracking velocimetry,” Meas. Sci. Technol. 15, 699-705 (2004).
[CrossRef]

Lohmann, A. W.

Malek, M.

M. Malek, D. Allano, S. Coëtmellec, C. Özkul, and D. Lebrun, “Digital in-line holography for three-dimensional-two-components particle tracking velocimetry,” Meas. Sci. Technol. 15, 699-705 (2004).
[CrossRef]

Malkiel, E.

E. Malkiel, J. Sheng, J. Katz, and J. Strickler, “The three-dimensional flow field generated by a feeding calanoid copepod measured using digital holography,” J. Exp. Biol. 206, 3657-3666 (2003).
[CrossRef]

Mas, D.

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245-258 (2003).
[CrossRef]

McBride, A. C.

A. C. McBride and F. H. Kerr, “On Namias's fractional Fourier transforms,” IMA J. Appl. Math. 39, 159-175(1987).
[CrossRef]

Meng, H.

Meucci, R.

Miret, J. J.

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245-258 (2003).
[CrossRef]

Namias, V.

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241-265 (1980).
[CrossRef]

Nicolas, F.

Onural, L.

L. Onural, “Diffraction from a wavelet point of view,” Opt. Lett. 18, 846-848 (1993).

L. Onural and P. Scott, “Digital decoding of in-line holograms,” Opt. Eng. 26, 1124-1132 (1987).

Ozaktas, H. M.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform: with Applications in Optics and Signal Processing (Wiley, 2001).

Özkul, C.

M. Malek, D. Allano, S. Coëtmellec, C. Özkul, and D. Lebrun, “Digital in-line holography for three-dimensional-two-components particle tracking velocimetry,” Meas. Sci. Technol. 15, 699-705 (2004).
[CrossRef]

Palma, C.

Pan, G.

Pei, S.-C.

S.-C. Pei, M.-H. Yeh, and C.-C. Tseng, “Discrete fractional Fourier transform based on orthogonal projections,” IEEE Trans. Signal Process. 47, 1335-1348 (1999).
[CrossRef]

Pérez, J.

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245-258 (2003).
[CrossRef]

Pierattini, and G.

Pierattini, G.

S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattini, “Wave front reconstruction of Fresnel off-axis holograms with compensation of aberrations by means of phase-shifting digital holography,” Opt. Lasers Eng. 37, 331-340 (2002).
[CrossRef]

S. Grilli, P. Ferraro, S. De Nicola, A. Finizio, G. Pierattini, and R. Meucci, “Whole optical wavefields reconstruction by digital holography,” Opt. Express 9, 294-302 (2001).

Scott, P.

L. Onural and P. Scott, “Digital decoding of in-line holograms,” Opt. Eng. 26, 1124-1132 (1987).

Sheng, J.

E. Malkiel, J. Sheng, J. Katz, and J. Strickler, “The three-dimensional flow field generated by a feeding calanoid copepod measured using digital holography,” J. Exp. Biol. 206, 3657-3666 (2003).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986).

Slimani, F.

Strickler, J.

E. Malkiel, J. Sheng, J. Katz, and J. Strickler, “The three-dimensional flow field generated by a feeding calanoid copepod measured using digital holography,” J. Exp. Biol. 206, 3657-3666 (2003).
[CrossRef]

Thompson, B. J.

Tseng, C.-C.

S.-C. Pei, M.-H. Yeh, and C.-C. Tseng, “Discrete fractional Fourier transform based on orthogonal projections,” IEEE Trans. Signal Process. 47, 1335-1348 (1999).
[CrossRef]

Vázquez, C.

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245-258 (2003).
[CrossRef]

Verrier, N.

Vikram, C. S.

C. S. Vikram, “Particle field holography,” in Cambridge Studies in Modern Optics (Cambridge U. Press, 1992).

Wen, J. J.

J. J. Wen and M. Breazeale, “A diffraction beam expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[CrossRef]

J. J. Wen and M. Breazeale, “Gaussian beam functions as a base function set for acoustical field calculations,” Proc. IEEE Ultrason. Symp. 1137-1140 (1987).

Xu, W.

Yeh, M.-H.

S.-C. Pei, M.-H. Yeh, and C.-C. Tseng, “Discrete fractional Fourier transform based on orthogonal projections,” IEEE Trans. Signal Process. 47, 1335-1348 (1999).
[CrossRef]

Yura, H. T.

Zalevsky, Z.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform: with Applications in Optics and Signal Processing (Wiley, 2001).

Zeiss, J.

Zhao, D.

Zheng, C.

C. Zheng, D. Zhao, and X. Du, “Analytical expression of elliptical Gaussian beams through nonsymmetric systems with an elliptical aperture,” Optik (Jena) 117, 296-298 (2006).
[CrossRef]

Appl. Opt. (5)

IEEE Trans. Signal Process. (1)

S.-C. Pei, M.-H. Yeh, and C.-C. Tseng, “Discrete fractional Fourier transform based on orthogonal projections,” IEEE Trans. Signal Process. 47, 1335-1348 (1999).
[CrossRef]

IMA J. Appl. Math. (1)

A. C. McBride and F. H. Kerr, “On Namias's fractional Fourier transforms,” IMA J. Appl. Math. 39, 159-175(1987).
[CrossRef]

J. Acoust. Soc. Am. (1)

J. J. Wen and M. Breazeale, “A diffraction beam expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[CrossRef]

J. Exp. Biol. (1)

E. Malkiel, J. Sheng, J. Katz, and J. Strickler, “The three-dimensional flow field generated by a feeding calanoid copepod measured using digital holography,” J. Exp. Biol. 206, 3657-3666 (2003).
[CrossRef]

J. Inst. Math. Appl. (1)

V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241-265 (1980).
[CrossRef]

J. Opt. Soc. Am. A (8)

Meas. Sci. Technol. (1)

M. Malek, D. Allano, S. Coëtmellec, C. Özkul, and D. Lebrun, “Digital in-line holography for three-dimensional-two-components particle tracking velocimetry,” Meas. Sci. Technol. 15, 699-705 (2004).
[CrossRef]

Opt. Commun. (2)

D. Mas, J. Pérez, C. Hernández, C. Vázquez, J. J. Miret, and C. Illueca, “Fast numerical calculation of Fresnel patterns in convergent systems,” Opt. Commun. 227, 245-258 (2003).
[CrossRef]

F. Nicolas, S. Coëtmellec, M. Brunel, and D. Lebrun, “Digital in-line holography with a sub-picosecond laser beam,” Opt. Commun. 268, 27-33 (2006).
[CrossRef]

Opt. Eng. (1)

L. Onural and P. Scott, “Digital decoding of in-line holograms,” Opt. Eng. 26, 1124-1132 (1987).

Opt. Express (1)

Opt. Lasers Eng. (1)

S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattini, “Wave front reconstruction of Fresnel off-axis holograms with compensation of aberrations by means of phase-shifting digital holography,” Opt. Lasers Eng. 37, 331-340 (2002).
[CrossRef]

Opt. Lett. (2)

Optik (Jena) (1)

C. Zheng, D. Zhao, and X. Du, “Analytical expression of elliptical Gaussian beams through nonsymmetric systems with an elliptical aperture,” Optik (Jena) 117, 296-298 (2006).
[CrossRef]

Proc. IEEE Ultrason. Symp. (1)

J. J. Wen and M. Breazeale, “Gaussian beam functions as a base function set for acoustical field calculations,” Proc. IEEE Ultrason. Symp. 1137-1140 (1987).

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

D. Gabor, “Microscopy by reconstructed wave-fronts,” Proc. R. Soc. London Ser. A 197, 454-487 (1949).

Other (4)

C. S. Vikram, “Particle field holography,” in Cambridge Studies in Modern Optics (Cambridge U. Press, 1992).

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform: with Applications in Optics and Signal Processing (Wiley, 2001).

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts and Company, 2005).

A. E. Siegman, Lasers (University Science Books, 1986).

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Figures (11)

Fig. 1
Fig. 1

(a) Schematic representation of the optical setup (not to scale). Definition of the numerical and experimental parameters. (b) Close-up of the pipe used in the simulations and experiments.

Fig. 2
Fig. 2

Schematic representation of the object.

Fig. 3
Fig. 3

Simulation of the diffraction pattern of a 51.8 μm opaque fiber parallel to the axis of a glass pipe, with n 2 = 1.33 , λ = 632.8 nm , z = 23 mm , and δ = 18 mm .

Fig. 4
Fig. 4

Experimental diffraction pattern of a 51.8 μm opaque fiber parallel to the axis of a glass pipe, with n 2 = 1.33 , λ = 632.8 nm , z = 23 mm , and δ = 18 mm .

Fig. 5
Fig. 5

Comparison of the simulated and the experimental intensity distributions.

Fig. 6
Fig. 6

Normalized intensity distribution recorded by the CCD: comparison of the thin lens and thick lens models with e = 0 mm . Simulation parameters are 2 b = 51.8 µm , a , z p = 325 mm , n 2 = 1 , δ = 18 mm , and z = 23 mm .

Fig. 7
Fig. 7

Reconstruction of the fiber image from the diffraction pattern of Fig. 3 by FRFT with a x opt = 0.337 and a y opt = 0.273 .

Fig. 8
Fig. 8

Reconstruction of the fiber image from the diffraction pattern of Fig. 4 by FRFT with a x opt = 0.33 and a y opt = 0.27 .

Fig. 9
Fig. 9

Comparison between theoretical fractional orders and optimal fractional orders estimated from the experimental holograms.

Fig. 10
Fig. 10

Experimental diffraction pattern of 100 µm latex beads. z p = 325 mm , z = 23 mm .

Fig. 11
Fig. 11

Reconstruction of the image of the latex beads (a)  with thin lens parameters and (b) using the thick lens approach.

Equations (52)

Equations on this page are rendered with MathJax. Learn more.

G ( μ , ν ) = exp ( μ 2 + ν 2 w 2 ) ,
G 1 ( ξ , η ) = exp ( i 2 π λ E 1 ) i λ B 1 x B 1 y R 2 G ( μ , v ) exp [ i π λ B 1 x ( A 1 x μ 2 2 ξ μ + D 1 x ξ 2 ) ] × exp [ i π λ B 1 x ( A 1 x v 2 2 η v + D 1 x η 2 ) ] d μ d v ,
G 2 ( x , y ) = exp ( i 2 π λ E 2 ) i λ B 2 x B 2 y R 2 G 1 ( ξ , η ) [ 1 T ( ξ , η ) ] exp [ i π λ B 2 x ( A 2 x ξ 2 2 x ξ + D 2 x x 2 ) ] × exp [ i π λ B 2 y ( A 2 y η 2 2 y η + D 2 y η 2 ) ] d ξ d η ,
T ( r ) = k = 1 N A k exp ( r T R T P k R r ) ,
R = ( cos θ sin θ sin θ cos θ ) ,
P k = ( B k a 2 0 0 B k b 2 ) .
T ( ξ , η ) = k = 1 N A k exp [ B k b 2 ( ξ 2 + R ell 2 η 2 ) ] .
G 2 ( x , y ) = exp ( i 2 π λ E 2 ) i λ B 2 x B 2 y [ R ( x , y ) O ( x , y ) ] ,
R ( x , y ) = R 2 G 1 ( ξ , η ) exp [ i π λ B 2 x ( A 2 x ξ 2 2 x ξ + D 2 x x 2 ) ] × exp [ i π λ B 2 x ( A 2 x η 2 2 y η + D 2 y y 2 ) ] d ξ d η ,
O ( x , y ) = R 2 G 1 ( ξ , η ) T ( ξ , η ) exp [ i π λ B 2 x ( A 2 x ξ 2 2 x ξ + D 2 x x 2 ) ] × exp [ i π λ B 2 x ( A 2 x η 2 2 y η + D 2 y y 2 ) ] d ξ d η .
R ( x , y ) = exp ( i 2 π λ E 1 ) i λ B 1 x B 1 y K 1 x K 1 y K 2 x K 2 y × exp [ π λ ( N x B 2 x x 2 + N y B 2 y y 2 ) ] exp [ i π λ ( M x B 2 x x 2 + M y B 2 y y 2 ) ] ,
O ( x , y ) = exp ( i 2 π λ E 1 ) i λ B 1 x B 1 y K 1 x K 1 y exp [ i π λ ( D 2 x B 2 x x 2 + D 2 y B 2 y y 2 ) ] k = 1 N A k K 2 x eq K 2 y eq × exp [ π λ ( N x eq B 2 x x 2 + N y eq B 2 y y 2 ) ] exp [ i π λ ( M x eq B 2 x x 2 + M y eq B 2 y y 2 ) ] .
I ( x , y ) = G 2 ( x , y ) G 2 ( x , y ) ¯ = 1 λ 2 B 2 x B 2 y ( | R | 2 2 { R O ¯ } + | O | 2 ) ,
Φ ( x l , y l ) = exp [ i π λ ( x l 2 f x + y l 2 f y ) ] .
I thin ( x , y ) = | exp ( i 2 π λ E 2 ) i λ E 2 [ R thin ( x , y ) O thin ( x , y ) ] | 2 .
lim e 0 I ( x , y ) = I thin ( x , y ) ,
α x , α y [ I ( x , y ) ] ( x a , y a ) = R 2 N α x ( x , x a ) N α y ( y , y a ) I ( x , y ) d x d y ,
N α p ( x , x a ) = C ( α p ) exp ( i π x 2 + x a 2 s p 2 tan α p ) exp ( i 2 π x a x s p 2 sin α p ) ,
C ( α p ) = exp ( i ( π 4 sign ( sin α p ) α p 2 ) ) | s p 2 sin α p | 1 / 2 .
φ = π λ [ ( M x D 2 x B 2 x ) x 2 + ( M y D 2 y B 2 y ) y 2 ] ,
{ R O ¯ } = | R O ¯ | cos ( i φ ) .
φ a = π ( cot α x s x 2 x 2 + cot α y s y 2 y 2 ) .
α x , α y [ I ( x , y ) ] α x , α y [ | R | 2 + | O | 2 ] 2 F α x , α y [ | R O ¯ | cos φ ] .
α x , α y [ I ( x , y ) ] α x , α y [ | R | 2 + | O | 2 ] C ( α x ) C ( α y ) 2 | R O ¯ | exp [ i ( ϕ a ϕ ) ] exp [ i 2 π ( x a x s x 2 sin α x + y a y s y 2 sin α x ) ] d x d y C ( α x ) C ( α y ) 2 | R O ¯ | exp [ i ( ϕ a + ϕ ) ] exp [ i 2 π ( x a x s x 2 sin α x + y a y s y 2 sin α x ) ] d x d y .
φ a ± φ = 0.
α x opt = arctan [ B 2 x λ s x 2 ( M x D 2 x ) ] , α y opt = arctan [ B 2 y λ s y 2 ( M y D 2 y ) ] .
a x opt = 0.337 , a y opt = 0.273 .
M z p = ( 1 z p 0 1 ) .
M R x 1 = ( 1 0 n 0 n 1 R x 1 1 ) , M R 1 y = ( 1 0 n 0 n 1 R y 1 1 ) .
M e = ( 1 e n 1 0 1 ) .
M R x 2 = ( 1 0 n 1 n 2 R x 2 1 ) , M R y 2 = ( 1 0 n 1 n 2 R y 2 1 ) .
M δ = ( 1 δ n 2 0 1 ) .
M z i = ( 1 z i n 2 0 1 ) .
M R x 3 = ( 1 0 n 2 n 1 R x 3 1 ) , M R y 3 = ( 1 0 n 2 n 1 R y 3 1 ) ,
M e = ( 1 e n 1 0 1 ) ,
M R x 4 = ( 1 0 n 1 n 0 R x 4 1 ) , M R y 4 = ( 1 0 n 1 n 0 R y 4 1 ) ,
M z = ( 1 z 0 1 ) ,
M 1 x , y = M δ × M L 1 x , y × M z p = ( A 1 x , y B 1 x , y C 1 x , y D 1 x , y ) .
M 2 x , y = M z × M L 2 x , y × M z i = ( A 2 x , y B 2 x , y C 2 x , y D 2 x , y ) .
M L 1 x = M R x 2 × M e × M R x 1 , M L 1 y = M R y 2 × M e × M R y 1 .
M L 2 x = M R x 4 × M e × M R x 3 , M L 2 y = M R y 4 × M e × M R y 3 .
G 1 ( ξ , η ) = exp ( i 2 π λ E 1 ) i λ B 1 x B 1 y K 1 x K 1 y exp [ ( ξ 2 ω 1 x 2 + η 2 ω 1 y 2 ) ] exp [ i π λ ( ξ 2 R 1 x + η 2 R 1 y ) ] ,
K 1 x , y = ( π ω 2 1 i A 1 x , y π ω 2 λ B 1 x , y ) 1 / 2 ,
ω 1 x , y = ( λ B 1 x , y π ω ) [ 1 + ( A 1 x , y π ω 2 λ B 1 x , y ) 2 ] 1 / 2 , R 1 x , y = B 1 x , y D 1 x , y A 1 x , y ( π ω 2 λ B 1 x , y ) 2 1 + ( A 1 x , y π ω 2 λ B 1 x , y ) 2 .
R ( x , y ) = exp ( i 2 π λ E 1 ) i λ B 1 x B 1 y K 1 x K 1 y K 2 x K 2 y × exp [ π λ ( N x B 2 x x 2 + N y B 2 y y 2 ) ] exp [ i π λ ( M x B 2 x x 2 + M y B 2 y y 2 ) ] ,
M x , y = D 2 x , y + ( π ω 1 x , y 2 λ B 2 x , y ) 2 ( B 2 x , y R 1 x , y A 2 x , y ) 1 + ( π ω 1 x , y 2 λ B 2 x , y ) 2 ( B 2 x , y R 1 x , y A 2 x , y ) 2 , N x , y = π ω 1 x , y 2 λ B 2 x , y 1 + ( π ω 1 x , y 2 λ B 2 x , y ) 2 ( B 2 x , y R 1 x , y A 2 x , y ) 2 ,
K 2 x , y = [ π ω 1 x , y 2 1 + i π ω 1 x , y 2 λ B 2 x , y ( B 2 x , y R 1 x , y - A 2 x , y ) ] 1 / 2 .
1 ω 1 x eq 2 = 1 ω 1 x 2 + { B k } b 2 , 1 ω 1 y eq 2 = 1 ω 1 y 2 + R ell 2 { B k } b 2 ,
1 R 1 x eq = 1 R 1 x + { B k } λ π b 2 , 1 R 1 y eq = 1 R 1 y + R ell 2 { B k } λ π b 2
O ( x , y ) = exp ( i 2 π λ E 1 ) i λ B 1 x B 1 y K 1 x K 1 y exp [ i π λ ( D 2 x B 2 x x 2 + D 2 y B 2 y y 2 ) ] k = 1 N A k K 2 x eq K 2 y eq × exp [ π λ ( N x eq B 2 x x 2 + N y eq B 2 y y 2 ) ] exp [ i π λ ( M x eq B 2 x x 2 + M y eq B 2 y y 2 ) ] ,
M x , y eq = ( π ω 1 x , y eq 2 λ B 2 x , y ) 2 ( B 2 x , y R 1 x , y eq A 2 x , y ) 1 + ( π ω 1 x , y eq 2 λ B 2 x , y ) 2 ( B 2 x , y R 1 x , y eq A 2 x , y ) 2 , N x , y eq = π ω 1 x , y eq 2 λ B 2 x , y 1 + ( π ω 1 x , y eq 2 λ B 2 x , y ) 2 ( B 2 x , y R 1 x , y eq A 2 x , y ) 2 ,
K 2 x , y eq = [ π ω 1 x , y eq 2 1 + i π ω 1 x , y eq 2 λ B 2 x , y ( B 2 x , y R 1 x , y eq - A 2 x , y ) ] 1 / 2 .

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