Abstract

The authors have studied the diffraction pattern produced by a particle field illuminated by an elliptic and astigmatic Gaussian beam. They demonstrate that the bidimensional fractional Fourier transformation is a mathematically suitable tool to analyse the diffraction pattern generated not only by a collimated plane wave [J. Opt. Soc. Am A 19, 1537 (2002) ], but also by an elliptic and astigmatic Gaussian beam when two different fractional orders are considered. Simulations and experimental results are presented.

© 2005 Optical Society of America

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  1. S. Coëtmellec, D. Lebrun, C. Özkul, “Application of the two-dimensional fractional-order Fourier transformation to particle field digital holography,” J. Opt. Soc. Am. A 19, 1537–1546 (2002).
    [CrossRef]
  2. M. Gustafsson, M. Sebesta, B. Bengtsson, S. G. Pettersson, P. Egelberg, T. Lenart, “High-resolution digital transmission microscopy—a Fourier holography approach,” Opt. Lasers Eng. 41, 553–563 (2004).
    [CrossRef]
  3. W. L. Anderson, H. Diao, “Two dimensional wavelet transform and application to holographic particle velocimetry,” Appl. Opt. 34, 249–255 (1995).
    [CrossRef] [PubMed]
  4. C. Buraga, S. Coëtmellec, D. Lebrun, C. Özkul, “Application of wavelet transform to hologram analysis: three-dimensional location of particles,” Opt. Lasers Eng. 33, 409–421 (2000).
    [CrossRef]
  5. L. Onural, “Diffraction from a wavelet point of view,” Opt. Lett. 18, 846–848 (1993).
    [CrossRef] [PubMed]
  6. T. M. Kreis, W. P.O. Jüptner “Suppression of the dc term in digital holography,” Opt. Eng. (Bellingham) 36, 2357–2360 (1997).
    [CrossRef]
  7. P. Pellat-Finet, “Diffraction entre un émetteur et un récepteur localement toriques. Application à l’étude des systèmes astigmates,” C. R. Acad. Sci., Ser. IIb: Mec., Phys., Chim., Astron. 327, 1269–1274 (1999).
  8. Yangjian Cai, Qiang Lin, “Propagation of elliptical Gaussian beam through misaligned optical systems in spatial domain and frequency domain,” Opt. Laser Technol. 34, 415–421 (2002).
    [CrossRef]
  9. H. Ozaktas, D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
    [CrossRef]
  10. Yangjian Cai, Qiang Lin, “Fractional Fourier transform for elliptical Gaussian beams,” Opt. Commun. 217, 7–13, (2003).
    [CrossRef]
  11. F. Slimani, G. Grehan, G. Gouesbet, D. Allano, “Near-field Lorenz–Mie theory and its application to microholography,” Appl. Opt. 23, 4140–4148 (1984).
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  12. G. A. Tyler, B. J. Thompson, “Fraunhofer holography applied to particle size analysis: a reassessment,” Opt. Acta 23, 261–304 (1976).
    [CrossRef]
  13. M. Born, E. Wolf, Principles of Optics, 7th. ed. (Cambridge U. Press, Cambridge, 1999).
    [CrossRef]
  14. A. C. McBride, F. H. Kerr, “On Namias’s Fractional Fourier Transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
    [CrossRef]
  15. V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
    [CrossRef]
  16. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  17. T. Alieva, M. L. Calvo, “Importance of the phase amplitude in the fractional Fourier domain,” J. Opt. Soc. Am. A 20, 533–541 (2003).
    [CrossRef]
  18. J. J.M. Braat, P. Dirksen, A. J.E.M. Janssen, “Assessment of an extended Nijboer–Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A 19, 858–870 (2002).
    [CrossRef]
  19. A. J.E.M. Janssen, “Extended Nijboer–Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A 19, 849–857 (2002).
    [CrossRef]
  20. A. J.E.M. Janssen, J. J.M. Braat, P. Dirksen, “On the computation of the Nijboer–Zernike aberration integrals at arbitrary defocus,” J. Mod. Opt. 51, 687–703 (2004).
    [CrossRef]
  21. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).
  22. G. E. Andrews, R. Askey, R. Roy, Special Functions (Cambridge U. Press, Cambridge, 1999).
    [CrossRef]

2004

M. Gustafsson, M. Sebesta, B. Bengtsson, S. G. Pettersson, P. Egelberg, T. Lenart, “High-resolution digital transmission microscopy—a Fourier holography approach,” Opt. Lasers Eng. 41, 553–563 (2004).
[CrossRef]

A. J.E.M. Janssen, J. J.M. Braat, P. Dirksen, “On the computation of the Nijboer–Zernike aberration integrals at arbitrary defocus,” J. Mod. Opt. 51, 687–703 (2004).
[CrossRef]

2003

Yangjian Cai, Qiang Lin, “Fractional Fourier transform for elliptical Gaussian beams,” Opt. Commun. 217, 7–13, (2003).
[CrossRef]

T. Alieva, M. L. Calvo, “Importance of the phase amplitude in the fractional Fourier domain,” J. Opt. Soc. Am. A 20, 533–541 (2003).
[CrossRef]

2002

2000

C. Buraga, S. Coëtmellec, D. Lebrun, C. Özkul, “Application of wavelet transform to hologram analysis: three-dimensional location of particles,” Opt. Lasers Eng. 33, 409–421 (2000).
[CrossRef]

1999

P. Pellat-Finet, “Diffraction entre un émetteur et un récepteur localement toriques. Application à l’étude des systèmes astigmates,” C. R. Acad. Sci., Ser. IIb: Mec., Phys., Chim., Astron. 327, 1269–1274 (1999).

1997

T. M. Kreis, W. P.O. Jüptner “Suppression of the dc term in digital holography,” Opt. Eng. (Bellingham) 36, 2357–2360 (1997).
[CrossRef]

1995

1993

1987

A. C. McBride, F. H. Kerr, “On Namias’s Fractional Fourier Transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

1984

1980

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

1976

G. A. Tyler, B. J. Thompson, “Fraunhofer holography applied to particle size analysis: a reassessment,” Opt. Acta 23, 261–304 (1976).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).

Alieva, T.

Allano, D.

Anderson, W. L.

Andrews, G. E.

G. E. Andrews, R. Askey, R. Roy, Special Functions (Cambridge U. Press, Cambridge, 1999).
[CrossRef]

Askey, R.

G. E. Andrews, R. Askey, R. Roy, Special Functions (Cambridge U. Press, Cambridge, 1999).
[CrossRef]

Bengtsson, B.

M. Gustafsson, M. Sebesta, B. Bengtsson, S. G. Pettersson, P. Egelberg, T. Lenart, “High-resolution digital transmission microscopy—a Fourier holography approach,” Opt. Lasers Eng. 41, 553–563 (2004).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th. ed. (Cambridge U. Press, Cambridge, 1999).
[CrossRef]

Braat, J. J.M.

A. J.E.M. Janssen, J. J.M. Braat, P. Dirksen, “On the computation of the Nijboer–Zernike aberration integrals at arbitrary defocus,” J. Mod. Opt. 51, 687–703 (2004).
[CrossRef]

J. J.M. Braat, P. Dirksen, A. J.E.M. Janssen, “Assessment of an extended Nijboer–Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A 19, 858–870 (2002).
[CrossRef]

Buraga, C.

C. Buraga, S. Coëtmellec, D. Lebrun, C. Özkul, “Application of wavelet transform to hologram analysis: three-dimensional location of particles,” Opt. Lasers Eng. 33, 409–421 (2000).
[CrossRef]

Cai, Yangjian

Yangjian Cai, Qiang Lin, “Fractional Fourier transform for elliptical Gaussian beams,” Opt. Commun. 217, 7–13, (2003).
[CrossRef]

Yangjian Cai, Qiang Lin, “Propagation of elliptical Gaussian beam through misaligned optical systems in spatial domain and frequency domain,” Opt. Laser Technol. 34, 415–421 (2002).
[CrossRef]

Calvo, M. L.

Coëtmellec, S.

S. Coëtmellec, D. Lebrun, C. Özkul, “Application of the two-dimensional fractional-order Fourier transformation to particle field digital holography,” J. Opt. Soc. Am. A 19, 1537–1546 (2002).
[CrossRef]

C. Buraga, S. Coëtmellec, D. Lebrun, C. Özkul, “Application of wavelet transform to hologram analysis: three-dimensional location of particles,” Opt. Lasers Eng. 33, 409–421 (2000).
[CrossRef]

Diao, H.

Dirksen, P.

A. J.E.M. Janssen, J. J.M. Braat, P. Dirksen, “On the computation of the Nijboer–Zernike aberration integrals at arbitrary defocus,” J. Mod. Opt. 51, 687–703 (2004).
[CrossRef]

J. J.M. Braat, P. Dirksen, A. J.E.M. Janssen, “Assessment of an extended Nijboer–Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A 19, 858–870 (2002).
[CrossRef]

Egelberg, P.

M. Gustafsson, M. Sebesta, B. Bengtsson, S. G. Pettersson, P. Egelberg, T. Lenart, “High-resolution digital transmission microscopy—a Fourier holography approach,” Opt. Lasers Eng. 41, 553–563 (2004).
[CrossRef]

Gouesbet, G.

Grehan, G.

Gustafsson, M.

M. Gustafsson, M. Sebesta, B. Bengtsson, S. G. Pettersson, P. Egelberg, T. Lenart, “High-resolution digital transmission microscopy—a Fourier holography approach,” Opt. Lasers Eng. 41, 553–563 (2004).
[CrossRef]

Janssen, A. J.E.M.

Jüptner, W. P.O.

T. M. Kreis, W. P.O. Jüptner “Suppression of the dc term in digital holography,” Opt. Eng. (Bellingham) 36, 2357–2360 (1997).
[CrossRef]

Kerr, F. H.

A. C. McBride, F. H. Kerr, “On Namias’s Fractional Fourier Transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

Kreis, T. M.

T. M. Kreis, W. P.O. Jüptner “Suppression of the dc term in digital holography,” Opt. Eng. (Bellingham) 36, 2357–2360 (1997).
[CrossRef]

Lebrun, D.

S. Coëtmellec, D. Lebrun, C. Özkul, “Application of the two-dimensional fractional-order Fourier transformation to particle field digital holography,” J. Opt. Soc. Am. A 19, 1537–1546 (2002).
[CrossRef]

C. Buraga, S. Coëtmellec, D. Lebrun, C. Özkul, “Application of wavelet transform to hologram analysis: three-dimensional location of particles,” Opt. Lasers Eng. 33, 409–421 (2000).
[CrossRef]

Lenart, T.

M. Gustafsson, M. Sebesta, B. Bengtsson, S. G. Pettersson, P. Egelberg, T. Lenart, “High-resolution digital transmission microscopy—a Fourier holography approach,” Opt. Lasers Eng. 41, 553–563 (2004).
[CrossRef]

Lin, Qiang

Yangjian Cai, Qiang Lin, “Fractional Fourier transform for elliptical Gaussian beams,” Opt. Commun. 217, 7–13, (2003).
[CrossRef]

Yangjian Cai, Qiang Lin, “Propagation of elliptical Gaussian beam through misaligned optical systems in spatial domain and frequency domain,” Opt. Laser Technol. 34, 415–421 (2002).
[CrossRef]

Lohmann, A. W.

McBride, A. C.

A. C. McBride, F. H. Kerr, “On Namias’s Fractional Fourier Transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

Mendlovic, D.

Namias, V.

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

Onural, L.

Ozaktas, H.

Özkul, C.

S. Coëtmellec, D. Lebrun, C. Özkul, “Application of the two-dimensional fractional-order Fourier transformation to particle field digital holography,” J. Opt. Soc. Am. A 19, 1537–1546 (2002).
[CrossRef]

C. Buraga, S. Coëtmellec, D. Lebrun, C. Özkul, “Application of wavelet transform to hologram analysis: three-dimensional location of particles,” Opt. Lasers Eng. 33, 409–421 (2000).
[CrossRef]

Pellat-Finet, P.

P. Pellat-Finet, “Diffraction entre un émetteur et un récepteur localement toriques. Application à l’étude des systèmes astigmates,” C. R. Acad. Sci., Ser. IIb: Mec., Phys., Chim., Astron. 327, 1269–1274 (1999).

Pettersson, S. G.

M. Gustafsson, M. Sebesta, B. Bengtsson, S. G. Pettersson, P. Egelberg, T. Lenart, “High-resolution digital transmission microscopy—a Fourier holography approach,” Opt. Lasers Eng. 41, 553–563 (2004).
[CrossRef]

Roy, R.

G. E. Andrews, R. Askey, R. Roy, Special Functions (Cambridge U. Press, Cambridge, 1999).
[CrossRef]

Sebesta, M.

M. Gustafsson, M. Sebesta, B. Bengtsson, S. G. Pettersson, P. Egelberg, T. Lenart, “High-resolution digital transmission microscopy—a Fourier holography approach,” Opt. Lasers Eng. 41, 553–563 (2004).
[CrossRef]

Slimani, F.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).

Thompson, B. J.

G. A. Tyler, B. J. Thompson, “Fraunhofer holography applied to particle size analysis: a reassessment,” Opt. Acta 23, 261–304 (1976).
[CrossRef]

Tyler, G. A.

G. A. Tyler, B. J. Thompson, “Fraunhofer holography applied to particle size analysis: a reassessment,” Opt. Acta 23, 261–304 (1976).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 7th. ed. (Cambridge U. Press, Cambridge, 1999).
[CrossRef]

Appl. Opt.

C. R. Acad. Sci., Ser. IIb: Mec., Phys., Chim., Astron.

P. Pellat-Finet, “Diffraction entre un émetteur et un récepteur localement toriques. Application à l’étude des systèmes astigmates,” C. R. Acad. Sci., Ser. IIb: Mec., Phys., Chim., Astron. 327, 1269–1274 (1999).

IMA J. Appl. Math.

A. C. McBride, F. H. Kerr, “On Namias’s Fractional Fourier Transforms,” IMA J. Appl. Math. 39, 159–175 (1987).
[CrossRef]

J. Inst. Math. Appl.

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

J. Mod. Opt.

A. J.E.M. Janssen, J. J.M. Braat, P. Dirksen, “On the computation of the Nijboer–Zernike aberration integrals at arbitrary defocus,” J. Mod. Opt. 51, 687–703 (2004).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Acta

G. A. Tyler, B. J. Thompson, “Fraunhofer holography applied to particle size analysis: a reassessment,” Opt. Acta 23, 261–304 (1976).
[CrossRef]

Opt. Commun.

Yangjian Cai, Qiang Lin, “Fractional Fourier transform for elliptical Gaussian beams,” Opt. Commun. 217, 7–13, (2003).
[CrossRef]

Opt. Eng. (Bellingham)

T. M. Kreis, W. P.O. Jüptner “Suppression of the dc term in digital holography,” Opt. Eng. (Bellingham) 36, 2357–2360 (1997).
[CrossRef]

Opt. Laser Technol.

Yangjian Cai, Qiang Lin, “Propagation of elliptical Gaussian beam through misaligned optical systems in spatial domain and frequency domain,” Opt. Laser Technol. 34, 415–421 (2002).
[CrossRef]

Opt. Lasers Eng.

M. Gustafsson, M. Sebesta, B. Bengtsson, S. G. Pettersson, P. Egelberg, T. Lenart, “High-resolution digital transmission microscopy—a Fourier holography approach,” Opt. Lasers Eng. 41, 553–563 (2004).
[CrossRef]

C. Buraga, S. Coëtmellec, D. Lebrun, C. Özkul, “Application of wavelet transform to hologram analysis: three-dimensional location of particles,” Opt. Lasers Eng. 33, 409–421 (2000).
[CrossRef]

Opt. Lett.

Other

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).

G. E. Andrews, R. Askey, R. Roy, Special Functions (Cambridge U. Press, Cambridge, 1999).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 7th. ed. (Cambridge U. Press, Cambridge, 1999).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Experimental optical setup.

Fig. 2
Fig. 2

Diffraction pattern with λ = 632.8 nm , ω ξ = 7 mm , ω η = 1.75 mm , R ξ = 2.36 10 5 m , R η = 50 mm , D = 150 μ m and z = 107 mm .

Fig. 3
Fig. 3

Image reconstructed by the fractional Fourier transformation with a x opt = 0.521 and a y opt = 0.816 .

Fig. 4
Fig. 4

Diffraction pattern with λ = 632.8 nm , ω ξ = 7 mm , ω η = 2.27 mm , R ξ = 4.38 10 5 m , R η = 65 mm , D = 150 μ m and z = 154 mm .

Fig. 5
Fig. 5

Image reconstructed by the fractional Fourier transformation with a x opt = 0.633 and a y opt = 0.718 .

Fig. 6
Fig. 6

Diffraction pattern with λ = 632.8 nm , ω ξ = 7 mm , ω η = 1.75 mm , R ξ = 2.36 10 5 m , R η = 50 mm , D = 150 μ m and z = 106 mm .

Fig. 7
Fig. 7

Image reconstructed by the fractional Fourier transformation with a x opt = 0.518 and a y opt = 0.813 .

Fig. 8
Fig. 8

Diffraction pattern with λ = 632.8 nm , ω ξ = 7 mm , ω η = 2.27 mm , R ξ = 4.38 10 5 m , R η = 65 mm , D = 150 μ m and z = 154 mm .

Fig. 9
Fig. 9

Image reconstructed by the fractional Fourier transformation with a x opt = 0.633 and a y opt = 0.718 .

Equations (58)

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

A = exp ( i 2 π λ z ) i λ z + + E ( ξ , η ) [ 1 T ( ξ , η ) ] exp ( i π λ z [ ( ξ x ) 2 + ( η y ) 2 ] ) d ξ d η ,
A 1 = + + E ( ξ , η ) exp ( i π λ z [ ( ξ x ) 2 + ( η y ) 2 ] ) d ξ d η ,
A 2 = + + E ( ξ , η ) T ( ξ , η ) exp ( i π λ z [ ( ξ x ) 2 + ( η y ) 2 ] ) d ξ d η ,
A = 1 i λ z [ A 1 A 2 ] .
T ( ξ , η ) = { 1 if ξ 2 + η 2 < D 2 1 2 if ξ 2 + η 2 = D 2 0 otherwise } .
E ( ξ , η ) = exp ( [ ξ 2 ω ξ 2 + η 2 ω η 2 ] ) exp ( i π λ [ ξ 2 R ξ + η 2 R η ] ) .
A 1 = K ( ω ξ , R ξ ) K ( ω η , R η ) exp ( π λ z r T N r ) exp ( i π λ z r T M r ) ,
K ( ω q , R q ) = ( π ω q 2 1 + i π ω q 2 λ z ( z R q 1 ) ) 1 2 .
N = [ N x 0 0 N y ] , M = [ M x 0 0 M y ] ,
N q = π λ z ω q 2 1 + π 2 λ 2 ω q 4 ( 1 R q 1 z ) 2 ,
M q = 1 + π λ z ω q 4 π λ ( 1 R q 1 z ) 1 + π 2 λ 2 ω q 4 ( 1 R q 1 z ) 2 ,
A 2 = D 2 4 exp ( i π r 2 λ z ) 0 1 0 2 π exp ( a σ 2 + b σ 2 cos 2 φ ) exp ( i π D λ z r σ cos ( φ θ ) ) σ d σ d φ ,
a = D 2 4 a 1 + i π D 2 4 ( b 1 1 λ z ) , b = D 2 4 a 2 + i π D 2 4 b 2 ,
a 1 = 1 2 ( 1 ω ξ 2 + 1 ω η 2 ) , a 2 = 1 2 ( 1 ω ξ 2 1 ω η 2 ) ,
b 1 = 1 2 λ ( 1 R ξ + 1 R η ) , b 2 = 1 2 λ ( 1 R ξ 1 R η ) .
A 2 = π D 2 exp ( i π r 2 λ z ) k = 0 ( i ) k ϵ k T k ( r ) cos ( 2 k θ ) ,
T k ( r ) = 0 1 exp ( i 1 2 u σ 2 ) J k ( γ 2 σ 2 ) J 2 k ( π D λ z r σ ) σ d σ ,
u = π D 2 2 λ z ( 1 + λ z b 1 ) i a 1 D 2 2 ,
γ 2 = D 2 4 ( π b 2 + i a 2 ) .
T k ( r ) = p = 0 β 2 k + 2 p 2 k ( γ 2 ) V 2 k + 2 p 2 k ( r , u ) ,
I = A A * = κ [ A 1 A 2 ] [ A 1 * A 2 * ] = κ [ A 1 2 + A 2 2 ] 2 κ R { A 1 A 2 * } ,
lim ω R κ A 1 2 = 1 .
lim ω R κ A 2 2 = U 1 2 ( π D 2 2 λ z , π D λ z r ) + U 2 2 ( π D 2 2 λ z , π D λ z r ) 1 ( λ z ) 2 ( π D 2 2 ) 2 J 1 2 ( π D λ z r ) ( π D λ z r ) 2 ,
lim ω R 2 κ R { A 1 A 2 * } π D 2 λ z sin ( π r 2 λ z ) J 1 ( π D λ z r ) ( π D λ z r ) .
F α x , α y [ f ] ( x a , y a ) = R 2 N α x ( x , x a ) N α y ( y , y a ) f d x d y ,
N α p ( x , x a ) = C ( α p ) exp ( i π x 2 + x a 2 s 2 tan α p ) exp ( i 2 π x a x s 2 sin α p ) ,
C ( α p ) = exp { i [ π 4 sign ( sin α p ) α p 2 ] } s 2 sin α p 1 2 .
h z ( x , y ) = 1 i 2 π exp [ i π λ z ( x 2 + y 2 ) ] .
A 1 A 2 * = A 1 A 2 * exp ( i φ ) ,
φ = π λ z ( x 2 ( M x 1 ) + y 2 ( M y 1 ) ) .
φ a = π s 2 ( x 2 cot α x + y 2 cot α y ) .
F α x , α y [ I ] ( x a , y a ) = κ F α x , α y [ A 1 2 + A 2 2 ] ( x a , y a ) 2 κ F α x , α y [ A 1 A 2 * cos ( φ ) ] ( x a , y a ) .
F α x , α y [ I ] ( x a , y a ) = κ F α x , α y [ A 1 2 + A 2 2 ] ( x a , y a ) κ C ( α x ) C ( α y ) R 2 A 1 A 2 * exp [ i ( φ a φ ) ] exp [ i 2 π s 2 ( x a x sin α x + y a y sin α y ) ] d x d y κ C ( α x ) C ( α y ) R 2 A 1 A 2 * exp [ i ( φ a + φ ) ] exp [ i 2 π s 2 ( x a x sin α x + y a y sin α y ) ] d x d y .
φ a ± φ = 0 .
tan α x opt = λ z s 2 ( M x 1 ) , tan α y opt = λ z s 2 ( M y 1 ) .
ω q = λ δ π ω [ 1 + ( π ω 2 δ λ ) 2 ( δ f q 1 ) 2 ] 1 2 , R q = δ [ 1 + ( π ω 2 δ λ ) 2 ( δ f q 1 ) 1 + ( π ω 2 δ λ ) 2 ( δ f q 1 ) 2 ] .
a x opt = 0.521 , a y opt = 0.816 .
a x opt = 0.633 , a y opt = 0.718 .
J k ( γ 2 σ 2 ) = p = 0 β 2 k + 2 p 2 k ( γ 2 ) R 2 k + 2 p 2 k ( σ ) .
β 2 k + 2 p 2 k ( γ 2 ) = 2 ( 2 k + 2 p + 1 ) 0 1 J k ( γ 2 σ 2 ) R 2 k + 2 p 2 k ( σ ) σ d σ .
T k ( r ) = p = 0 β 2 k + 2 p 2 k ( γ 2 ) V 2 k + 2 p 2 k ( r , u ) ,
V 2 k + 2 p 2 k ( r , u ) = 0 1 exp ( i 1 2 u σ 2 ) R 2 k + 2 p 2 k ( σ ) J 2 k ( π D λ z r σ ) σ d σ .
V 2 k + 2 p 2 k ( r , u ) = exp ( i u 4 ) m = 0 ( 2 m + 1 ) i m j m ( u 4 ) l = max ( 0 , m 2 k p , p m ) m + p ( 1 ) ω m l l J 2 k + 2 l + 1 ( π D λ z r ) ( π D λ z r ) .
J k ( γ 2 σ 2 ) = ( 1 2 γ 2 σ 2 ) k s = 0 ( 1 2 i γ 2 σ 2 ) 2 s s ! ( k + s ) ! .
β 2 k + 2 p 2 k ( γ 2 ) = 2 ( 2 k + 2 p + 1 ) ( 1 2 γ 2 ) k s = 0 ( 1 2 i γ 2 ) 2 s s ! ( k + s ) ! 0 1 σ 2 k + 4 s R 2 k + 2 p 2 k ( σ ) σ d σ .
0 1 σ 2 k + 4 s R 2 k + 2 p 2 k ( σ ) σ d σ = 1 2 ( 1 ) p ( 2 s ) p ( 2 k + 2 s + 1 ) p + 1 ,
β 2 k + 2 p 2 k ( γ 2 ) = 2 ( 2 k + 2 p + 1 ) ( 1 2 γ 2 ) k s = r ( 1 2 i γ 2 ) 2 s s ! ( k + s ) ! 1 2 ( 1 ) p ( 2 s ) p ( 2 k + 2 s + 1 ) p + 1 .
β 2 k + 2 p 2 k ( γ 2 ) = ( 2 k + 2 p + 1 ) ( 1 ) r ( 1 2 γ 2 ) k + 2 r j = 0 κ j ( 1 4 γ 2 2 ) j j !
κ j = j ! ( j + r ) ! ( k + j + r ) ! ( 1 ) p ( 2 j 2 r ) p ( 2 k + 2 j + 2 r + 1 ) p + 1 .
β 2 k + 2 p 2 k ( γ 2 ) = d 0 0 ( 1 ) r ( 2 k + 4 r + 1 ) ( 1 2 γ 2 ) k + 2 r × F 3 2 ( r + 1 2 k + r + 1 2 ; 1 4 γ 2 2 1 2 k + 2 r + 3 2 k + 2 r + 1 ) .
β 2 k + 2 p 2 k ( γ 2 ) = d 0 1 ( 1 ) r ( 2 k + 4 r 1 ) ( 1 2 γ 2 ) k + 2 r × F 3 2 ( r + 1 2 k + r + 1 2 ; 1 4 γ 2 2 3 2 k + 2 r + 1 k + 2 r + 1 2 ) .
d 0 0 = ( 2 r ) ! ( 2 k + 2 r ) ! r ! ( k + r ) ! ( 2 k + 4 r + 1 ) ! , d 0 1 = ( 2 r ) ! ( 2 k + 2 r ) ! r ! ( k + r ) ! ( 2 k + 4 r ) ! .
ω m l = s = 0 p t = 0 min ( m , s ) f p s 2 k b m s t g m + s 2 t , l 2 k ,
f p s 2 k = ( 1 ) p s 2 s + 1 p + s + 1 ( 2 k + p s 1 2 k 1 ) ( 2 k + p + s s ) ( p + s s ) , s = 0 , , p ,
g u l 2 k = 2 k + 2 l + 1 2 k + u + l + 1 ( 2 k u l ) ( u + l l ) ( 2 k + l + u 2 k + l ) , u = l , , l + 2 k , s 1 , s 2 = 0 , 1 , ,
b s 1 s 2 t = 2 s 1 + 2 s 2 4 t + 1 2 s 1 + 2 s 2 2 t + 1 A s 1 t A t A s 2 t A s 1 + s 2 t , t = 0 , , min ( s 1 , s 2 ) ,
A m = ( 2 m m ) .
j r ( z ) = π 2 z J r + 1 2 ( z ) .

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