Abstract

We introduce a local signal decomposition method for the analysis of three-dimensional (3D) diffraction fields involving curved surfaces. We decompose a given field on a two-dimensional curved surface into a sum of properly shifted and modulated Gaussian-shaped elementary signals. Then we write the 3D diffraction field as a sum of Gaussian beams, each of which corresponds to a modulated Gaussian window function on the curved surface. The Gaussian beams are propagated according to a derived approximate expression that is based on the Rayleigh–Sommerfeld diffraction model. We assume that the given curved surface is smooth enough that the Gaussian window functions on it can be treated as written on planar patches. For the surfaces that satisfy this assumption, the simulation results show that the proposed method produces quite accurate 3D field solutions.

© 2012 Optical Society of America

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  1. L. Onural, F. Yaraş, and H. Kang, “Digital holographic three-dimensional video displays,” Proc. IEEE 99, 576–589 (2011).
    [CrossRef]
  2. N. Sato, H. Aritake, M. Kato, M. Ishimoto, and M. Nakashima, “Stereoscopic display apparatus,” U.S. patent 5,594,559(14January1997).
  3. J. Hahn, H. Kim, Y. Lim, G. Park, and B. Lee, “Wide viewing angle dynamic holographic stereogram with a curved array of spatial light modulators,” Opt. Express 16, 12372–12386 (2008).
    [CrossRef]
  4. W. J. Dallas, “Computer-generated holograms,” in Digital Holography and Three-Dimensional Display, T. C. Poon, ed. (Springer, 2006), pp. 1–49.
  5. C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” Computer 38, 46–53 (2005).
    [CrossRef]
  6. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996), 2nd ed.
  7. J. D. Gaskill, “The propagation and diffraction of optical wave fields,” in Linear Systems, Fourier Transforms, and Optics (Wiley, 1978), Chap. 10.
  8. J. Waters, “Holographic image synthesis utilizing theoretical methods,” Appl. Phys. Lett. 9, 405–407 (1966).
    [CrossRef]
  9. T. Yatagai, “Stereoscopic approach to 3-D display using computer-generated holograms,” Appl. Opt. 15, 2722–2729 (1976).
    [CrossRef]
  10. K. Matsushima and M. Takai, “Recurrence formulas for fast creation of synthetic three-dimensional holograms,” Appl. Opt. 39, 6587–6594 (2000).
    [CrossRef]
  11. M. Lucente, “Optimization of hologram computation for real-time display,” Proc. SPIE 1667, 32–43 (1992).
    [CrossRef]
  12. K. Matsushima, “Computer-generated holograms for three-dimensional surface objects with shade and texture,” Appl. Opt. 44, 4607–4614 (2005).
    [CrossRef]
  13. M. Janda, I. Hanák, and L. Onural, “Hologram synthesis for photorealistic reconstruction,” J. Opt. Soc. Am. A 25, 3083–3096 (2008).
    [CrossRef]
  14. L. Ahrenberg, “Methods for transform, analysis and rendering of complete light representations,” Ph.D. thesis (Max-Planck-Institut für Informatik, 2010).
  15. G. C. Sherman, “Application of the convolution theorem to Rayleigh’s integral formulas,” J. Opt. Soc. Am. 57, 546–547 (1967).
    [CrossRef]
  16. G. B. Esmer, “Calculation of scalar optical diffraction field from its distributed samples over the space,” Ph.D. thesis (Bilkent University, 2010).
  17. G. B. Esmer, L. Onural, and H. M. Ozaktas, “Exact diffraction calculation from fields specified over arbitrary curved surfaces,” Opt. Commun. 284, 5537–5548 (2011).
    [CrossRef]
  18. D. Gabor, “Theory of communication,” J. IEE 93, 429–457 (1946).
  19. M. J. Bastiaans, “Gabor’s signal expansion and the Zak transform,” Appl. Opt. 33, 5241–5255 (1994).
    [CrossRef]
  20. M. J. Bastiaans, “Expansion of an optical signal into a discrete set of Gaussian beams,” Optik 57, 95–102 (1980).
  21. L. Onural, “Exact solution for scalar diffraction between tilted and translated planes using impulse functions over a surface,” J. Opt. Soc. Am. A 28, 290–295 (2011).
    [CrossRef]
  22. P. Flandrin, Time-Frequency/Time-Scale Analysis (Academic, 1999).
  23. M. J. Bastiaans, “Oversampling in Gabor’s signal expansion by an integer factor,” in Proceedings of the IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis, 1994 (IEEE, 1994), pp. 280–283.
  24. A. Janssen, “Gabor representation of generalized functions,” J. Math. Anal. Appl. 83, 377–394 (1981).
    [CrossRef]
  25. K. Tang, Mathematical Methods for Engineers and Scientists: Fourier Analysis, Partial Differential Equations and Variational Models (Springer, 2007).
  26. M. J. Bastiaans and A. J. van Leest, “Gabor’s signal expansion and the Gabor transform based on a non-orthogonal sampling geometry,” in Sixth International Symposium on Signal Processing and Its Applications, 2001 (IEEE, 2001), pp. 162–163.

2011 (3)

L. Onural, F. Yaraş, and H. Kang, “Digital holographic three-dimensional video displays,” Proc. IEEE 99, 576–589 (2011).
[CrossRef]

G. B. Esmer, L. Onural, and H. M. Ozaktas, “Exact diffraction calculation from fields specified over arbitrary curved surfaces,” Opt. Commun. 284, 5537–5548 (2011).
[CrossRef]

L. Onural, “Exact solution for scalar diffraction between tilted and translated planes using impulse functions over a surface,” J. Opt. Soc. Am. A 28, 290–295 (2011).
[CrossRef]

2008 (2)

2005 (2)

C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” Computer 38, 46–53 (2005).
[CrossRef]

K. Matsushima, “Computer-generated holograms for three-dimensional surface objects with shade and texture,” Appl. Opt. 44, 4607–4614 (2005).
[CrossRef]

2000 (1)

1994 (1)

1992 (1)

M. Lucente, “Optimization of hologram computation for real-time display,” Proc. SPIE 1667, 32–43 (1992).
[CrossRef]

1981 (1)

A. Janssen, “Gabor representation of generalized functions,” J. Math. Anal. Appl. 83, 377–394 (1981).
[CrossRef]

1980 (1)

M. J. Bastiaans, “Expansion of an optical signal into a discrete set of Gaussian beams,” Optik 57, 95–102 (1980).

1976 (1)

1967 (1)

1966 (1)

J. Waters, “Holographic image synthesis utilizing theoretical methods,” Appl. Phys. Lett. 9, 405–407 (1966).
[CrossRef]

1946 (1)

D. Gabor, “Theory of communication,” J. IEE 93, 429–457 (1946).

Ahrenberg, L.

L. Ahrenberg, “Methods for transform, analysis and rendering of complete light representations,” Ph.D. thesis (Max-Planck-Institut für Informatik, 2010).

Aritake, H.

N. Sato, H. Aritake, M. Kato, M. Ishimoto, and M. Nakashima, “Stereoscopic display apparatus,” U.S. patent 5,594,559(14January1997).

Bastiaans, M. J.

M. J. Bastiaans, “Gabor’s signal expansion and the Zak transform,” Appl. Opt. 33, 5241–5255 (1994).
[CrossRef]

M. J. Bastiaans, “Expansion of an optical signal into a discrete set of Gaussian beams,” Optik 57, 95–102 (1980).

M. J. Bastiaans, “Oversampling in Gabor’s signal expansion by an integer factor,” in Proceedings of the IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis, 1994 (IEEE, 1994), pp. 280–283.

M. J. Bastiaans and A. J. van Leest, “Gabor’s signal expansion and the Gabor transform based on a non-orthogonal sampling geometry,” in Sixth International Symposium on Signal Processing and Its Applications, 2001 (IEEE, 2001), pp. 162–163.

Cameron, C.

C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” Computer 38, 46–53 (2005).
[CrossRef]

Dallas, W. J.

W. J. Dallas, “Computer-generated holograms,” in Digital Holography and Three-Dimensional Display, T. C. Poon, ed. (Springer, 2006), pp. 1–49.

Esmer, G. B.

G. B. Esmer, L. Onural, and H. M. Ozaktas, “Exact diffraction calculation from fields specified over arbitrary curved surfaces,” Opt. Commun. 284, 5537–5548 (2011).
[CrossRef]

G. B. Esmer, “Calculation of scalar optical diffraction field from its distributed samples over the space,” Ph.D. thesis (Bilkent University, 2010).

Flandrin, P.

P. Flandrin, Time-Frequency/Time-Scale Analysis (Academic, 1999).

Gabor, D.

D. Gabor, “Theory of communication,” J. IEE 93, 429–457 (1946).

Gaskill, J. D.

J. D. Gaskill, “The propagation and diffraction of optical wave fields,” in Linear Systems, Fourier Transforms, and Optics (Wiley, 1978), Chap. 10.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996), 2nd ed.

Hahn, J.

Hanák, I.

Ishimoto, M.

N. Sato, H. Aritake, M. Kato, M. Ishimoto, and M. Nakashima, “Stereoscopic display apparatus,” U.S. patent 5,594,559(14January1997).

Janda, M.

Janssen, A.

A. Janssen, “Gabor representation of generalized functions,” J. Math. Anal. Appl. 83, 377–394 (1981).
[CrossRef]

Kang, H.

L. Onural, F. Yaraş, and H. Kang, “Digital holographic three-dimensional video displays,” Proc. IEEE 99, 576–589 (2011).
[CrossRef]

Kato, M.

N. Sato, H. Aritake, M. Kato, M. Ishimoto, and M. Nakashima, “Stereoscopic display apparatus,” U.S. patent 5,594,559(14January1997).

Kim, H.

Lee, B.

Lim, Y.

Lucente, M.

M. Lucente, “Optimization of hologram computation for real-time display,” Proc. SPIE 1667, 32–43 (1992).
[CrossRef]

Matsushima, K.

Nakashima, M.

N. Sato, H. Aritake, M. Kato, M. Ishimoto, and M. Nakashima, “Stereoscopic display apparatus,” U.S. patent 5,594,559(14January1997).

Onural, L.

L. Onural, F. Yaraş, and H. Kang, “Digital holographic three-dimensional video displays,” Proc. IEEE 99, 576–589 (2011).
[CrossRef]

G. B. Esmer, L. Onural, and H. M. Ozaktas, “Exact diffraction calculation from fields specified over arbitrary curved surfaces,” Opt. Commun. 284, 5537–5548 (2011).
[CrossRef]

L. Onural, “Exact solution for scalar diffraction between tilted and translated planes using impulse functions over a surface,” J. Opt. Soc. Am. A 28, 290–295 (2011).
[CrossRef]

M. Janda, I. Hanák, and L. Onural, “Hologram synthesis for photorealistic reconstruction,” J. Opt. Soc. Am. A 25, 3083–3096 (2008).
[CrossRef]

Ozaktas, H. M.

G. B. Esmer, L. Onural, and H. M. Ozaktas, “Exact diffraction calculation from fields specified over arbitrary curved surfaces,” Opt. Commun. 284, 5537–5548 (2011).
[CrossRef]

Park, G.

Sato, N.

N. Sato, H. Aritake, M. Kato, M. Ishimoto, and M. Nakashima, “Stereoscopic display apparatus,” U.S. patent 5,594,559(14January1997).

Sherman, G. C.

Slinger, C.

C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” Computer 38, 46–53 (2005).
[CrossRef]

Stanley, M.

C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” Computer 38, 46–53 (2005).
[CrossRef]

Takai, M.

van Leest, A. J.

M. J. Bastiaans and A. J. van Leest, “Gabor’s signal expansion and the Gabor transform based on a non-orthogonal sampling geometry,” in Sixth International Symposium on Signal Processing and Its Applications, 2001 (IEEE, 2001), pp. 162–163.

Waters, J.

J. Waters, “Holographic image synthesis utilizing theoretical methods,” Appl. Phys. Lett. 9, 405–407 (1966).
[CrossRef]

Yaras, F.

L. Onural, F. Yaraş, and H. Kang, “Digital holographic three-dimensional video displays,” Proc. IEEE 99, 576–589 (2011).
[CrossRef]

Yatagai, T.

Appl. Opt. (4)

Appl. Phys. Lett. (1)

J. Waters, “Holographic image synthesis utilizing theoretical methods,” Appl. Phys. Lett. 9, 405–407 (1966).
[CrossRef]

Computer (1)

C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” Computer 38, 46–53 (2005).
[CrossRef]

J. IEE (1)

D. Gabor, “Theory of communication,” J. IEE 93, 429–457 (1946).

J. Math. Anal. Appl. (1)

A. Janssen, “Gabor representation of generalized functions,” J. Math. Anal. Appl. 83, 377–394 (1981).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

G. B. Esmer, L. Onural, and H. M. Ozaktas, “Exact diffraction calculation from fields specified over arbitrary curved surfaces,” Opt. Commun. 284, 5537–5548 (2011).
[CrossRef]

Opt. Express (1)

Optik (1)

M. J. Bastiaans, “Expansion of an optical signal into a discrete set of Gaussian beams,” Optik 57, 95–102 (1980).

Proc. IEEE (1)

L. Onural, F. Yaraş, and H. Kang, “Digital holographic three-dimensional video displays,” Proc. IEEE 99, 576–589 (2011).
[CrossRef]

Proc. SPIE (1)

M. Lucente, “Optimization of hologram computation for real-time display,” Proc. SPIE 1667, 32–43 (1992).
[CrossRef]

Other (10)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996), 2nd ed.

J. D. Gaskill, “The propagation and diffraction of optical wave fields,” in Linear Systems, Fourier Transforms, and Optics (Wiley, 1978), Chap. 10.

N. Sato, H. Aritake, M. Kato, M. Ishimoto, and M. Nakashima, “Stereoscopic display apparatus,” U.S. patent 5,594,559(14January1997).

W. J. Dallas, “Computer-generated holograms,” in Digital Holography and Three-Dimensional Display, T. C. Poon, ed. (Springer, 2006), pp. 1–49.

L. Ahrenberg, “Methods for transform, analysis and rendering of complete light representations,” Ph.D. thesis (Max-Planck-Institut für Informatik, 2010).

G. B. Esmer, “Calculation of scalar optical diffraction field from its distributed samples over the space,” Ph.D. thesis (Bilkent University, 2010).

P. Flandrin, Time-Frequency/Time-Scale Analysis (Academic, 1999).

M. J. Bastiaans, “Oversampling in Gabor’s signal expansion by an integer factor,” in Proceedings of the IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis, 1994 (IEEE, 1994), pp. 280–283.

K. Tang, Mathematical Methods for Engineers and Scientists: Fourier Analysis, Partial Differential Equations and Variational Models (Springer, 2007).

M. J. Bastiaans and A. J. van Leest, “Gabor’s signal expansion and the Gabor transform based on a non-orthogonal sampling geometry,” in Sixth International Symposium on Signal Processing and Its Applications, 2001 (IEEE, 2001), pp. 162–163.

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Figures (10)

Fig. 1.
Fig. 1.

Local coordinate system on the input curved surface.

Fig. 2.
Fig. 2.

Mutual couplings between patches on a curved surface (2D cross sections of the surface and beams are shown for the sake of simplicity). (a) Mutual coupling between S 1 and S 3 patches, (b) mutual coupling between S 1 and S 2 patches (in addition to the mutual coupling between S 1 and S 3 patches).

Fig. 3.
Fig. 3.

2D simulation setup, which is periodic along the x axis (the fields used in the simulations repeat themselves between each dashed horizontal line shown in the figure).

Fig. 4.
Fig. 4.

Discrete Gaussian synthesis window function g d [ n ] = exp { L s 2 n 2 / σ 2 } for σ = 2.20 μm and L s = 0.26 μm .

Fig. 5.
Fig. 5.

Discrete analysis window function corresponding to the Gaussian synthesis window function g d [ n ] = exp { L s 2 n 2 / σ 2 } , for σ = 2.20 μm and L s = 0.26 μm , for the case that the shift parameters in space and frequency are M = 16 and K = 32 , respectively.

Fig. 6.
Fig. 6.

Periodic Gaussian beams induced by the periodic pattern over the periodic curved line.

Fig. 7.
Fig. 7.

Real part of the discrete input signal, u d [ n ] , on the circular arc with a measure of 30° (imaginary part is zero).

Fig. 8.
Fig. 8.

Magnitude of the 2D diffraction field due to the input signal u d [ n ] , one period of which is shown in Fig. 7, on the circular arc with a measure of 30°.

Fig. 9.
Fig. 9.

Magnitude of the 2D diffraction field due to a single shifted and modulated Gaussian window function, with parameters m = 16 and k = 1 , on the circular arc with a measure of 30°.

Fig. 10.
Fig. 10.

Real parts of the original, u d [ n ] , and reconstructed, u d r [ n ] , signals on the circular arc with a measure of 30°.

Equations (30)

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u ( x , y , z ) = B A ( f x , f y ) exp { j 2 π ( f x x + f y y + f z z ) } d f x d f y ,
A ( f x , f y ) = u ( x , y , 0 ) exp { j 2 π ( f x x + f y y ) } d x d y .
A ( f x , f y ) = a ( x , y ) exp { j 2 π ( f x x + f y y ) } d x d y .
Ψ = Φ a .
u ( x , y , 0 ) = a ( ξ , η , f x , f y ) g ( x ξ , y η ) exp { j 2 π ( f x x + f y y ) } d ξ d η d f x d f y ,
a ( ξ , η , f x , f y ) = u ( x , y , 0 ) g * ( x ξ , y η ) exp { j 2 π ( f x x + f y y ) } d x d y .
u ( x , y , 0 ) = m n k l a m n k l g ( x m X , y n Y ) exp { j 2 π ( k F x x + l F y y ) } ,
a m n k l = u ( x , y , 0 ) w * ( x m X , y n Y ) exp { j 2 π ( k F x x + l F y y ) } d x d y .
u ( x ) = a ( ξ , η , f x , f y ) g h ( x r , f x , f y ) d ξ d η d f x d f y .
u z ( x , y ) h z ( x , y ) U 0 ( x λ x 2 + y 2 + z 2 , y λ x 2 + y 2 + z 2 ) .
h z ( x , y ) = z j λ exp { j 2 π λ x 2 + y 2 + z 2 } x 2 + y 2 + z 2
1 | f ( x , y ) , f ˜ ( x , y ) | 2 f ( x , y ) , f ( x , y ) f ˜ ( x , y ) , f ˜ ( x , y ) ,
F { g ( x ξ , y η ) exp ( j 2 π μ x ) exp ( j 2 π ν y ) } = c σ 2 π exp { π 2 σ 2 [ ( f x μ ) 2 + ( f y ν ) 2 ] } exp { j 2 π [ ( f x μ ) ξ + ( f y ν ) η ] } .
g h ( x r , f x , f y ) h z ( x ξ , y η ) c σ 2 π exp { π 2 σ 2 [ ( x ξ λ ( x ξ ) 2 + ( y η ) 2 + z 2 f x ) 2 + ( y η λ ( x ξ ) 2 + ( y η ) 2 + z 2 f y ) 2 ] } exp { j 2 π [ ( x ξ λ ( x ξ ) 2 + ( y η ) 2 + z 2 f x ) ξ + ( y η λ ( x ξ ) 2 + ( y η ) 2 + z 2 f y ) η ] } .
u ( x ) a ( ξ , η , f x , f y ) h z ( x ξ , y η ) c σ 2 π exp { π 2 σ 2 [ ( x ξ λ ( x ξ ) 2 + ( y η ) 2 + z 2 f x ) 2 + ( y η λ ( x ξ ) 2 + ( y η ) 2 + z 2 f y ) 2 ] } exp { j 2 π [ ( x ξ λ ( x ξ ) 2 + ( y η ) 2 + z 2 f x ) ξ + ( y η λ ( x ξ ) 2 + ( y η ) 2 + z 2 f y ) η ] } d ξ d η d f x d f y .
u ( x ) m n k l a m n k l h z ( x m X , y n Y ) c σ 2 π exp { π 2 σ 2 [ ( x m X λ ( x m X ) 2 + ( y n Y ) 2 + z 2 k F x ) 2 + ( y n Y λ ( x m X ) 2 + ( y n Y ) 2 + z 2 l F y ) 2 ] } exp { j 2 π [ ( x m X λ ( x m X ) 2 + ( y n Y ) 2 + z 2 k F x ) m X + ( y n Y λ ( x m X ) 2 + ( y n Y ) 2 + z 2 l F y ) n Y ] } ,
g r ( x r , y r ) = c exp { ( x r 2 + y r 2 ) σ 2 } .
g r ( x r , y r ) exp { j 2 π ( f x r x r + f y r y r ) } = c exp { ( x r 2 + y r 2 ) σ 2 } exp { j 2 π ( f x r x r + f y r y r ) } .
a ( r , f x r , f y r ) = u r ( x r , y r ) g r * ( x r , y r ) exp { j 2 π ( f x r x r + f y r y r ) } d x r d y r V T r u r ( x r , y r ) g r * ( x r , y r ) exp { j 2 π ( f x r x r + f y r y r ) } d x r d y r .
x r = R r x r .
g h ( x , f x , f y ) h z ( x , y ) c σ 2 π exp { π 2 σ 2 [ ( x λ | x | f x ) 2 + ( y λ | x | f y ) 2 ] } .
u ( x ) S a ( r , f x r , f y r ) g h ( x r , f x r , f y r ) d f x r d f y r d S S a ( r , f x r , f y r ) h z r ( x r , y r ) c σ 2 π exp { π 2 σ 2 [ ( x r λ | x r | f x r ) 2 + ( y r λ | x r | f y r ) 2 ] } d f x r d f y r d S ,
u ( x ) i = 1 n k l a r i k l g h ( x r i , k F x , l F y ) i = 1 n k l a r i k l h z r i ( x r i , y r i ) c σ 2 π exp { π 2 σ 2 [ ( x r i λ | x r i | k F x ) 2 + ( y r i λ | x r i | l F y ) 2 ] } ,
a r i k l = u r i ( x r i , y r i ) w r i * ( x r i , y r i ) exp { j 2 π ( k F x x r i + l F y y r i ) } d x r i d y r i V T r i u r i ( x r i , y r i ) w r i * ( x r i , y r i ) exp { j 2 π ( k F x x r i + l F y y r i ) } d x r i d y r i .
a m k = n = N 2 N 2 1 u ˜ d [ n ] w ˜ d * [ n m M ] exp ( j 2 π k K n ) , m { N 2 M , N 2 M + 1 , , N 2 M 1 } , k { K 2 , K 2 + 1 , , K 2 1 } ,
a m k = n 1 = K 2 K 2 1 n 2 = N 2 K N 2 K 1 u ˜ d [ n 1 + n 2 K ] w ˜ d * [ n 1 + n 2 K m M ] exp ( j 2 π k K n 1 ) , m { N 2 M , N 2 M + 1 , , N 2 M 1 } , k { K 2 , K 2 + 1 , , K 2 1 } .
u ˜ w d [ n 1 ] = n 2 = N 2 K N 2 K 1 u ˜ d [ n 1 + n 2 K ] w ˜ d * [ n 1 + n 2 K m M ]
u ( x , z ) m = N 2 M N 2 M 1 k = K 2 K 2 1 b S w a m k g h ( R r ( m M L s ) [ x b X p , z ] T r ( m M L s ) , k K L s ) , S w = { b m k , b m k + 1 , , b m k + c } ,
u ( x , z ) = 1 N k = N 2 N 2 1 A k exp { j 2 π ( k X s N x + 1 λ 2 k 2 X s 2 N 2 ( z z o ) ) } ,
A k = n = N 2 N 2 1 u ( n X s , z o ) exp ( j 2 π k n N ) , k { N 2 , N 2 + 1 , , N 2 1 } ,

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