Abstract

We demonstrate that the effect of object shift in an elliptical, astigmatic Gaussian beam does not affect the optimal fractional orders used to reconstruct the holographic image of a particle or another opaque object in the field. Simulations and experimental results are presented.

© 2008 Optical Society of America

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References

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  1. W. Xu, M. H. Jericho, H. J. Kreuzer, and I. A. Meinertzhagen, “Tracking particles in four dimensions with in-line holographic microscopy,” Opt. Lett. 28, 164-166 (2003).
    [CrossRef] [PubMed]
  2. F. Dubois, L. Joannes, and J.-C. Legros, “Improved three-dimensional imaging with a digital holography microscope using a partial spatial coherent source,” Appl. Opt. 38, 7085-7094 (1999).
    [CrossRef]
  3. B. Skarman, K. Wozniac, and J. Becker, “Simultaneous 3D-PIV and temperature measurement using a new CCD based holographic interferometer,” Flow Meas. Instrum. 7, 1-6 (1996).
    [CrossRef]
  4. M. Sebesta and M. Gustafsson, “Object characterization with refractometric digital Fourier holography,” Opt. Lett. 30, 471-473 (2005).
    [CrossRef] [PubMed]
  5. U. Schnars, “Direct phase determination in hologram interferometry with use of digitally recorded holograms,” J. Opt. Soc. Am. A 11, 2011-2015 (1994).
    [CrossRef]
  6. G. Shen and R. Wei, “Digital holography particle image velocimetry for the measurement of 3Dt-3c flows,” Opt. Lasers Eng. 43, 1039-1055 (2002).
    [CrossRef]
  7. L. Onural, “Diffraction from a wavelet point of view,” Opt. Lett. 18, 846-848 (1993).
    [CrossRef] [PubMed]
  8. L. Onural and P. D. Scott, “Digital decoding of in-line holograms,” Opt. Eng. (Bellingham) 26, 1124-1132 (1987).
  9. E. Fogret and P. Pellat-Finet, “Agreement of fractional Fourier optics with the Huygens-Fresnel principle,” Opt. Commun. 272, 281-288 (2007).
    [CrossRef]
  10. S. Coëtmellec, D. Lebrun, and C. Özkul, “Characterization of diffraction patterns directly from in-line holograms with the fractional Fourier transform,” Appl. Opt. 41, 312-319 (2002).
    [CrossRef] [PubMed]
  11. 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]
  12. 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]
  13. A. J. E. M. Janssen, J. J. M. Braat, and P. Dirksen, “On the computation of the Nijboer-Zernike aberration integrals at arbitrary defocus,” J. Mod. Opt. 51, 687-703 (2004).
    [CrossRef]
  14. J. J. M. Braat, P. Dirksen, and 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]
  15. 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]
  16. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).
  17. W. Mecklenbräuker and F. Hlawatsch, The Wigner Distribution. Theory and Applications in Signal Processing (Elsevier, 1997), pp. 59-83.
  18. A. C. McBride and F. H. Kerr, “On Namias's fractional Fourier transforms,” IMA J. Appl. Math. 39, 159-175 (1987).
    [CrossRef]
  19. V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241-265 (1980).
    [CrossRef]
  20. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181-2186 (1993).
    [CrossRef]
  21. 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]
  22. H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform: With Applications in Optics and Signal Processing (Wiley, 2001).

2007 (1)

E. Fogret and P. Pellat-Finet, “Agreement of fractional Fourier optics with the Huygens-Fresnel principle,” Opt. Commun. 272, 281-288 (2007).
[CrossRef]

2006 (1)

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]

2005 (2)

2004 (1)

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

2003 (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]

W. Xu, M. H. Jericho, H. J. Kreuzer, and I. A. Meinertzhagen, “Tracking particles in four dimensions with in-line holographic microscopy,” Opt. Lett. 28, 164-166 (2003).
[CrossRef] [PubMed]

2002 (4)

1999 (1)

1996 (1)

B. Skarman, K. Wozniac, and J. Becker, “Simultaneous 3D-PIV and temperature measurement using a new CCD based holographic interferometer,” Flow Meas. Instrum. 7, 1-6 (1996).
[CrossRef]

1994 (1)

1993 (2)

1987 (2)

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

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

1980 (1)

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

Abramowitz, M.

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

Allano, D.

Becker, J.

B. Skarman, K. Wozniac, and J. Becker, “Simultaneous 3D-PIV and temperature measurement using a new CCD based holographic interferometer,” Flow Meas. Instrum. 7, 1-6 (1996).
[CrossRef]

Braat, J. J. M.

A. J. E. M. Janssen, J. J. M. Braat, and 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, and 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]

Brunel, M.

Coëtmellec, S.

Dirksen, P.

A. J. E. M. Janssen, J. J. M. Braat, and 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, and 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]

Dubois, F.

Fogret, E.

E. Fogret and P. Pellat-Finet, “Agreement of fractional Fourier optics with the Huygens-Fresnel principle,” Opt. Commun. 272, 281-288 (2007).
[CrossRef]

Gustafsson, M.

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]

Hlawatsch, F.

W. Mecklenbräuker and F. Hlawatsch, The Wigner Distribution. Theory and Applications in Signal Processing (Elsevier, 1997), pp. 59-83.

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. H.

Joannes, L.

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]

Kreuzer, H. J.

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).

Lebrun, D.

Legros, J.-C.

Lohmann, A. W.

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]

Mecklenbräuker, W.

W. Mecklenbräuker and F. Hlawatsch, The Wigner Distribution. Theory and Applications in Signal Processing (Elsevier, 1997), pp. 59-83.

Meinertzhagen, I. A.

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).
[CrossRef] [PubMed]

L. Onural and P. D. Scott, “Digital decoding of in-line holograms,” Opt. Eng. (Bellingham) 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.

Pellat-Finet, P.

E. Fogret and P. Pellat-Finet, “Agreement of fractional Fourier optics with the Huygens-Fresnel principle,” Opt. Commun. 272, 281-288 (2007).
[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]

Schnars, U.

Scott, P. D.

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

Sebesta, M.

Shen, G.

G. Shen and R. Wei, “Digital holography particle image velocimetry for the measurement of 3Dt-3c flows,” Opt. Lasers Eng. 43, 1039-1055 (2002).
[CrossRef]

Skarman, B.

B. Skarman, K. Wozniac, and J. Becker, “Simultaneous 3D-PIV and temperature measurement using a new CCD based holographic interferometer,” Flow Meas. Instrum. 7, 1-6 (1996).
[CrossRef]

Stegun, I. A.

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

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]

Wei, R.

G. Shen and R. Wei, “Digital holography particle image velocimetry for the measurement of 3Dt-3c flows,” Opt. Lasers Eng. 43, 1039-1055 (2002).
[CrossRef]

Wozniac, K.

B. Skarman, K. Wozniac, and J. Becker, “Simultaneous 3D-PIV and temperature measurement using a new CCD based holographic interferometer,” Flow Meas. Instrum. 7, 1-6 (1996).
[CrossRef]

Xu, W.

Zalevsky, Z.

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

Appl. Opt. (2)

Flow Meas. Instrum. (1)

B. Skarman, K. Wozniac, and J. Becker, “Simultaneous 3D-PIV and temperature measurement using a new CCD based holographic interferometer,” Flow Meas. Instrum. 7, 1-6 (1996).
[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. 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. Mod. Opt. (1)

A. J. E. M. Janssen, J. J. M. Braat, and 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 (5)

Opt. Commun. (3)

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]

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]

E. Fogret and P. Pellat-Finet, “Agreement of fractional Fourier optics with the Huygens-Fresnel principle,” Opt. Commun. 272, 281-288 (2007).
[CrossRef]

Opt. Eng. (Bellingham) (1)

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

Opt. Lasers Eng. (1)

G. Shen and R. Wei, “Digital holography particle image velocimetry for the measurement of 3Dt-3c flows,” Opt. Lasers Eng. 43, 1039-1055 (2002).
[CrossRef]

Opt. Lett. (3)

Other (3)

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

W. Mecklenbräuker and F. Hlawatsch, The Wigner Distribution. Theory and Applications in Signal Processing (Elsevier, 1997), pp. 59-83.

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

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Figures (7)

Fig. 1
Fig. 1

Numerical and experimental optical set-up.

Fig. 2
Fig. 2

Diffraction pattern with ω ξ = 7 mm , ω η = 1.75 mm , R ξ = , R η = 50 mm , D = 150 μ m , λ = 632.8 nm , z = 120 mm , δ = 250 mm , ξ 0 = 0.5 mm , and η 0 = 0.2 mm .

Fig. 3
Fig. 3

Diffraction pattern with ω ξ = 7 mm , ω η = 1.75 mm , R ξ = , R η = 50 mm , D = 150 μ m , λ = 632.8 nm , z = 120 mm , δ = 150 mm , ξ 0 = 0.5 mm , and η 0 = 0.2 mm .

Fig. 4
Fig. 4

FRFT of the diffraction pattern with a x o p t = 0.564 and a y o p t = 0.850 .

Fig. 5
Fig. 5

FRFT the diffraction pattern with a x o p t = 0.564 and a y o p t = 0.664 .

Fig. 6
Fig. 6

Diffraction pattern of the word “ELECTRO.”

Fig. 7
Fig. 7

FRFT of the diffraction pattern with a x o p t = 0.505 and a y o p t = 0.785 .

Equations (58)

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A = exp ( i 2 π λ z ) i λ z R 2 E T ( ξ , η ) exp ( i π λ z [ ( ξ x ) 2 + ( η y ) 2 ] ) d ξ d η ,
c ξ = 1 ω ξ 2 i π λ R ξ , c η = 1 ω η 2 i π λ R η .
T ( ξ , η ) = { 1 , 0 < ξ 2 + η 2 < D 2 , 1 2 , 0 < ξ 2 + η 2 = D 2 , 0 , ξ 2 + η 2 > D 2 > 0 .
A ( x , y ) = exp ( i 2 π λ z ) i [ A 1 A 2 ] .
A 2 = π D 2 λ z exp [ Φ ( ξ 0 , η 0 ) ] exp ( i c z [ ( x ξ 0 ) 2 + ( y η 0 ) 2 ] ) k = 0 ( i ) k ε k T k ( r , γ ) cos ( 2 k θ ) ,
T k ( r , γ ) = p = 0 β 2 k + 2 p 2 k ( δ ) V 2 k + 2 p 2 k ( r , γ ) ,
V 2 k + 2 p 2 k ( r , γ ) = exp ( i γ 2 ) m = 0 ( 2 m + 1 ) i m j m ( γ 2 ) l = max ( 0 , m 2 k p , p m ) m + p ( 1 ) l ω m l J 2 k + 2 l + 1 ( r ) r .
j m ( γ 2 ) 1 ( 2 m + 1 ) 1 2 min ( 1 , ( π 2 ) 1 2 γ 4 m m ! ) .
J 2 k + 2 l + 1 ( r ) r min ( 1 , 1 2 r 2 k + 2 l exp ( r 2 ) ( 2 k + 2 l + 1 ) ! ) .
j m ( γ 2 ) J 2 k + 2 ( m + p ) + 1 ( r ) r .
V 0 0 ( r , γ ) exp ( i γ 2 ) j 0 ( γ 2 ) J 1 ( r ) r .
A 2 = π D 2 2 λ z β 0 0 ( δ ) exp [ Φ ( ξ 0 , η 0 ) ] exp ( i c z [ ( x ξ 0 ) 2 + ( y η 0 ) 2 ] ) V 0 0 ( r , γ ) .
I = A A ¯ = [ A 1 A 2 ] [ A 1 ¯ A 2 ¯ ] = [ A 1 2 + A 2 2 ] 2 R { A 1 A 2 ¯ } ,
f i ( x ) = 1 2 π φ ( x ) x = 1 2 π arg ( A 1 , 2 ) x = 0 .
A 1 A 2 ¯ = A 1 A 2 ¯ exp [ i arg ( A 1 A 2 ¯ ) ] ,
ϕ = c z ( x 2 ( M ξ 1 ) + y 2 ( M η 1 ) ) + 2 c z ( x ξ 0 + y η 0 ) arg ( J 1 ( r ) r ) ,
ϕ 0 = R ( γ ) 2 + I ( Φ ( ξ 0 , η 0 ) ) + arg ( j 0 ( γ 2 ) ) + arg ( β 0 0 ( δ ) ) arg ( K ξ K η ) ,
y 0 = Δ ± z Δ η 0 .
F 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 .
s p 2 = N p δ p 2 .
lim ε 0 1 i π ε exp ( x 2 i ε ) = δ ( x ) ,
F α [ g ( x ) ] = δ ( x a ) .
F α x , α y [ I ] = F α x , α y [ A 1 2 ] F α x , α y [ 2 A 1 A 2 ¯ cos ( ϕ ϕ 0 ) ] + F α x , α y [ A 2 2 ] .
F α x , α y [ 2 A 1 A 2 ¯ cos ( ϕ ϕ 0 ) ] = exp ( i π x a 2 s 2 tan α x ) exp ( i π y a 2 s 2 tan α y ) { I + I + } ,
I ± = C ( α x ) C ( α y ) R 2 A 1 A 2 ¯ exp [ i ( ϕ a ± ( ϕ ϕ 0 ) ) ] exp [ 2 i π s 2 ( x a x sin α x + y a y sin α y ) ] d x d y .
π cot α x o p t s 2 = c z ( M x 1 ) , π cot α y o p t s 2 = c z ( M y 1 ) ,
I = χ F [ J 1 ( r ) r exp ( π λ z ρ T N ρ ) ] ( u , v ) ,
u = x a s 2 sin ( α x o p t ) + c z ξ 0 π , v = y a s 2 sin ( α y o p t ) + c z η 0 π .
I = χ F [ J 1 ( r ) r ] * F [ exp ( π λ z ρ T N ρ ) ] .
F [ J 1 ( r ) r ] = 2 π ( λ z π D ) 2 exp [ i 2 π ( u X 0 + v Y 0 ) ] × { 1 , 0 < u 2 + v 2 < D 2 λ z , 1 2 , 0 < u 2 + v 2 = D 2 λ z , 0 , u 2 + v 2 > D 2 λ z > 0 ,
( s 2 ( u c z ξ 0 π ) tan α x o p t , s 2 ( v c z η 0 π ) tan α y o p t ) .
F α [ f ( x b ) ] ( u ) = exp ( i π b 2 sin α cos α ) exp ( i 2 π b u sin α ) F α [ f ( x ) ] ( u b cos α ) .
A 1 = K ξ K η exp ( π λ z ρ T N ρ ) exp ( i π λ z ρ T M ρ ) ,
K q = [ π ω q 2 λ z 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 ω q 2 λ z 1 + π 2 ω q 4 ( λ z ) 2 ( z R q 1 ) 2 ,
M q = 1 + π 2 ω q 4 ( λ z ) 2 ( z R q 1 ) 1 + π 2 ω q 4 ( λ z ) 2 ( z R q 1 ) 2 .
A 2 = exp [ i π λ z ( x 2 + y 2 ) ] λ z R 2 E ( ξ , η ) T ( ξ ξ 0 , η η 0 ) exp [ i π λ z ( ξ 2 + η 2 ) ] exp [ i 2 π λ z ( x ξ + y η ) ] d ξ d η ,
A 2 = exp [ i π λ z ( x 2 + y 2 ) ] λ z D E ( ξ + ξ 0 , η + η 0 ) exp [ i π λ z ( ( ξ + ξ 0 ) 2 + ( η + η 0 ) 2 ) ] exp [ i 2 π λ z ( x ( ξ + ξ 0 ) + y ( η + η 0 ) ) ] d ξ d η .
A 2 = D 2 4 λ z exp [ c ξ ξ 0 2 + c η η 0 2 + i c z [ ( x ξ 0 ) 2 + ( y η 0 ) 2 ] ] 0 1 0 2 π exp [ i γ σ 2 ] exp [ i δ σ 2 cos ( 2 φ ) ] exp [ i a σ cos φ + i b σ sin φ ] σ d σ d φ ,
γ = D 2 4 c z i D 2 8 ( c ξ + c η ) , δ = i D 2 8 ( c η c ξ ) ,
a = D c z [ ξ 0 ( 1 i c ξ c z ) x ] , b = D c z [ η 0 ( 1 i c η c z ) y ] .
a cos φ + b sin φ = r cos ( φ θ ) ,
a = r cos θ , b = r sin θ
exp [ i δ σ 2 cos ( 2 φ + 2 θ ) ] = J 0 ( δ σ 2 ) + 2 k = 1 + i k J k ( δ σ 2 ) cos 2 k ( φ + θ ) ,
1 2 π 0 2 π exp ( i n θ ) exp [ i x cos θ ] d θ = i n J n ( x ) ,
A 2 = π D 2 λ z exp [ Φ ( ξ 0 , η 0 ) ] exp ( i c z [ ( x ξ 0 ) 2 + ( y η 0 ) 2 ] ) k = 0 ( i ) k ε k T k ( r , γ ) cos ( 2 k θ ) ,
T k ( r , γ ) = p = 0 β 2 k + 2 p 2 k ( δ ) V 2 k + 2 p 2 k ( r , γ ) ,
V 2 k + 2 p 2 k ( r , γ ) = exp ( i γ 2 ) m = 0 ( 2 m + 1 ) i m j m ( γ 2 ) l = max ( 0 , m 2 k p , p m ) m + p ( 1 ) l ω m l J 2 k + 2 l + 1 ( r ) r .
τ ω = u , τ ω = v .
τ = ( r s ) 1 2 exp ( i ( α + β ) 2 ) ,
ω = exp ( i θ ) = ( r s ) 1 2 exp ( i ( α β ) 2 )
F π 2 [ I ( x ) ] ( x a ) = C ( π 2 ) + I ( x ) exp ( i 2 π x x a s 2 ) d x .
F π 2 [ I ( m ) ] ( k ) = C ( π 2 ) m = N 2 N 2 1 I ( m ) exp ( i 2 π m δ x k δ x a s 2 ) δ x ,
F π 2 [ I ( m ) ] ( k ) = C ( π 2 ) m = N 2 N 2 1 I ( m ) exp ( i 2 π m k N ) δ x .
δ x δ x a s 2 = 1 N so s 2 = N δ x δ x a = N δ x 2 .

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