Abstract

Prolate spheroidal wave functions (PSWFs) are known to be useful for analyzing the properties of the finite-extension Fourier transform (fi-FT). We extend the theory of PSWFs for the finite-extension fractional Fourier transform, the finite-extension linear canonical transform, and the finite-extension offset linear canonical transform. These finite transforms are more flexible than the fi-FT and can model much more generalized optical systems. We also illustrate how to use the generalized prolate spheroidal functions we derive to analyze the energy-preservation ratio, the self-imaging phenomenon, and the resonance phenomenon of the finite-sized one-stage or multiple-stage optical systems.

© 2005 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–63 (1961).
    [Crossref]
  2. H. J. Landau, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—II,” Bell Syst. Tech. J. 40, 65–84 (1961).
    [Crossref]
  3. H. J. Landau, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—III,” Bell Syst. Tech. J. 41, 1295–1336 (1962).
    [Crossref]
  4. D. Slepian, “Some asymptotic expansions for prolate spheroidal wave functions,” J. Math. Phys. 44, 99–140 (1965).
  5. D. Slepian, “Prolate spheroidal wave functions, Fourier analysis, and uncertainty—V: the discrete case,” Bell Syst. Tech. J. 57, 1371–1430 (1977).
    [Crossref]
  6. J. H. Wilkinson, The Algebraic Eigenvalue Problem (Oxford U. Press, London, 1988).
  7. B. R. Frieden, “Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions,” in Progress in Optics, Vol. IX, E. Wolf, ed. (North-Holland, Amsterdam, 1971), pp. 311–407.
  8. A. Papoulis, M. S. Bertran, “Digital filtering and prolate functions,” IEEE Trans. Circuit Theory 19, 674–681 (1972).
    [Crossref]
  9. H. M. Ozaktas, M. A. Kutay, Z. Zalevsky, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2000).
  10. V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. 25, 241–265 (1980).
    [Crossref]
  11. K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979), Chap. 9, pp. 381–416.
  12. L. M. Bernardo, “ABCD matrix formalism of fractional Fourier optics,” Opt. Eng. (Bellingham) 35, 732–740 (1996).
    [Crossref]
  13. M. J. Bastiaans, “Propagation laws for the secondorder moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 82, 173–181 (1989).
  14. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
    [Crossref]
  15. P. Pellat-Finet, G. Bonnet, “Fractional order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
    [Crossref]
  16. S. Abe, J. T. Sheridan, “Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation,” Opt. Lett. 19, 1801–1803 (1994).
    [Crossref] [PubMed]
  17. S. C. Pei, J. J. Ding, “Eigenfunctions of the offset Fourier, fractional Fourier, and linear canonical transforms,” J. Opt. Soc. Am. A 20, 522–532 (2003).
    [Crossref]
  18. K. Khare, N. George, “Fractional finite Fourier transform,” J. Opt. Soc. Am. A 21, 1179–1185 (2004).
    [Crossref]
  19. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).

2004 (1)

2003 (1)

1996 (1)

L. M. Bernardo, “ABCD matrix formalism of fractional Fourier optics,” Opt. Eng. (Bellingham) 35, 732–740 (1996).
[Crossref]

1995 (1)

1994 (2)

1989 (1)

M. J. Bastiaans, “Propagation laws for the secondorder moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 82, 173–181 (1989).

1980 (1)

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

1977 (1)

D. Slepian, “Prolate spheroidal wave functions, Fourier analysis, and uncertainty—V: the discrete case,” Bell Syst. Tech. J. 57, 1371–1430 (1977).
[Crossref]

1972 (1)

A. Papoulis, M. S. Bertran, “Digital filtering and prolate functions,” IEEE Trans. Circuit Theory 19, 674–681 (1972).
[Crossref]

1965 (1)

D. Slepian, “Some asymptotic expansions for prolate spheroidal wave functions,” J. Math. Phys. 44, 99–140 (1965).

1962 (1)

H. J. Landau, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—III,” Bell Syst. Tech. J. 41, 1295–1336 (1962).
[Crossref]

1961 (2)

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–63 (1961).
[Crossref]

H. J. Landau, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—II,” Bell Syst. Tech. J. 40, 65–84 (1961).
[Crossref]

Abe, S.

Bastiaans, M. J.

M. J. Bastiaans, “Propagation laws for the secondorder moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 82, 173–181 (1989).

Bernardo, L. M.

L. M. Bernardo, “ABCD matrix formalism of fractional Fourier optics,” Opt. Eng. (Bellingham) 35, 732–740 (1996).
[Crossref]

Bertran, M. S.

A. Papoulis, M. S. Bertran, “Digital filtering and prolate functions,” IEEE Trans. Circuit Theory 19, 674–681 (1972).
[Crossref]

Bonnet, G.

P. Pellat-Finet, G. Bonnet, “Fractional order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
[Crossref]

Ding, J. J.

Frieden, B. R.

B. R. Frieden, “Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions,” in Progress in Optics, Vol. IX, E. Wolf, ed. (North-Holland, Amsterdam, 1971), pp. 311–407.

George, N.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).

Khare, K.

Kutay, M. A.

H. M. Ozaktas, M. A. Kutay, Z. Zalevsky, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2000).

Landau, H. J.

H. J. Landau, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—III,” Bell Syst. Tech. J. 41, 1295–1336 (1962).
[Crossref]

H. J. Landau, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—II,” Bell Syst. Tech. J. 40, 65–84 (1961).
[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]

Ozaktas, H. M.

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
[Crossref]

H. M. Ozaktas, M. A. Kutay, Z. Zalevsky, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2000).

Papoulis, A.

A. Papoulis, M. S. Bertran, “Digital filtering and prolate functions,” IEEE Trans. Circuit Theory 19, 674–681 (1972).
[Crossref]

Pei, S. C.

Pellat-Finet, P.

P. Pellat-Finet, G. Bonnet, “Fractional order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
[Crossref]

Pollak, H. O.

H. J. Landau, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—III,” Bell Syst. Tech. J. 41, 1295–1336 (1962).
[Crossref]

H. J. Landau, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—II,” Bell Syst. Tech. J. 40, 65–84 (1961).
[Crossref]

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–63 (1961).
[Crossref]

Sheridan, J. T.

Slepian, D.

D. Slepian, “Prolate spheroidal wave functions, Fourier analysis, and uncertainty—V: the discrete case,” Bell Syst. Tech. J. 57, 1371–1430 (1977).
[Crossref]

D. Slepian, “Some asymptotic expansions for prolate spheroidal wave functions,” J. Math. Phys. 44, 99–140 (1965).

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–63 (1961).
[Crossref]

Wilkinson, J. H.

J. H. Wilkinson, The Algebraic Eigenvalue Problem (Oxford U. Press, London, 1988).

Wolf, K. B.

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979), Chap. 9, pp. 381–416.

Zalevsky, Z.

H. M. Ozaktas, M. A. Kutay, Z. Zalevsky, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2000).

Bell Syst. Tech. J. (4)

D. Slepian, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell Syst. Tech. J. 40, 43–63 (1961).
[Crossref]

H. J. Landau, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—II,” Bell Syst. Tech. J. 40, 65–84 (1961).
[Crossref]

H. J. Landau, H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—III,” Bell Syst. Tech. J. 41, 1295–1336 (1962).
[Crossref]

D. Slepian, “Prolate spheroidal wave functions, Fourier analysis, and uncertainty—V: the discrete case,” Bell Syst. Tech. J. 57, 1371–1430 (1977).
[Crossref]

IEEE Trans. Circuit Theory (1)

A. Papoulis, M. S. Bertran, “Digital filtering and prolate functions,” IEEE Trans. Circuit Theory 19, 674–681 (1972).
[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. Math. Phys. (1)

D. Slepian, “Some asymptotic expansions for prolate spheroidal wave functions,” J. Math. Phys. 44, 99–140 (1965).

J. Opt. Soc. Am. A (3)

Opt. Commun. (1)

P. Pellat-Finet, G. Bonnet, “Fractional order Fourier transform and Fourier optics,” Opt. Commun. 111, 141–154 (1994).
[Crossref]

Opt. Eng. (Bellingham) (1)

L. M. Bernardo, “ABCD matrix formalism of fractional Fourier optics,” Opt. Eng. (Bellingham) 35, 732–740 (1996).
[Crossref]

Opt. Lett. (1)

Optik (Stuttgart) (1)

M. J. Bastiaans, “Propagation laws for the secondorder moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 82, 173–181 (1989).

Other (5)

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979), Chap. 9, pp. 381–416.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996).

H. M. Ozaktas, M. A. Kutay, Z. Zalevsky, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2000).

J. H. Wilkinson, The Algebraic Eigenvalue Problem (Oxford U. Press, London, 1988).

B. R. Frieden, “Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions,” in Progress in Optics, Vol. IX, E. Wolf, ed. (North-Holland, Amsterdam, 1971), pp. 311–407.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

Finite-sized optical system consisting of two media and one free space.

Fig. 2
Fig. 2

Thickness and the extension of a medium.  

Fig. 3
Fig. 3

Two special orders of the generalized prolate spheroidal functions of the optical system in Fig. 1. (a) ϕ 0 ( x ) ϕ 0 ( y ) , (b) output corresponding to ϕ 0 ( x ) ϕ 0 ( y ) , (c) ϕ 5 ( x ) ϕ 5 ( y ) , (d) output corresponding to ϕ 5 ( x ) ϕ 5 ( y ) .

Fig. 4
Fig. 4

Finite-sized optical system consisting of multiple stages.

Fig. 5
Fig. 5

Spherical-mirror-pair system.

Equations (234)

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

FT : F ( ω ) = ( 2 π ) - 1 / 2 - exp ( - j ω x ) f ( x ) d x ,
x , ω ( - ,   ) .
fi - FT : F ˜ ( ω ) = ( 2 π ) - 1 / 2 - T T exp ( - j ω x ) f ( x ) d x .
- | F ( ω ) | 2 d ω = - | f ( x ) | 2 d x .
0 < energy preservation ratio = - Ω Ω | F ˜ ( ω ) | 2 d ω - T T | f ( x ) | 2 d x < 1 .
- T T K F , Ω ( x ˜ ,   x ) ψ n , T , Ω ( x ) d x = λ n , T , Ω ψ n , T , Ω ( x ˜ ) ,
K F , Ω ( x ˜ ,   x ) = sin [ Ω ( x ˜ - x ) ] π ( x ˜ - x ) .
1 > λ 0 , T , Ω > λ 1 , T , Ω > λ 2 , T , Ω > > 0 ( all of λ n , T , Ω   real ) .
- T T ψ m ( x ) ψ n ( x ) d x = λ n δ m , n ,
- ψ m ( x ) ψ n ( x ) d x = δ m , n .
ψ n ( x ) C n exp ( - x 2 / 2 ) H n ( x ) , T × Ω ,
H n ( x ) : Hermite polynomials .
R F , Ω [ k ,   h ] = K F , Ω ( k Δ ,   h Δ ) × Δ = sin [ Ω Δ ( k - h ) ] π ( k - h )
( m ,   n = - N 0 ,   - N 0 + 1   , ,   N 0 ) ,
T [ p ,   p - 1 ] = T [ p - 1 ,   p ] = β p , T [ p ,   p ] = α p ,
T [ p ,   q ] = 0 otherwise ,
β p = 2 ( p - 1 ) ( N - p + 1 ) ,
α p = [ N + 1 - 2 p ] 2 cos ( Ω Δ ) ,
p , q = 1 ,   2 ,   3 , ,   N .
p r ( μ ) = ( α r - μ ) p r - 1 ( μ ) - β r 2 p r - 2 ( μ )
[ p 1 ( μ ) = α 1 - μ and defining p 0 ( μ ) = 1 ] .
v n [ 1 ] = 1 , v n [ r ] = ( - 1 ) r - 1 p r - 1 ( μ n ) ( β 2 β 3 β r ) - 1 .
λ n τ n ,
ψ n { [ k - 2 - 1 ( N + 1 ) ] Δ } A n v n [ k ] ,
A n = λ n 1 / 2 n v n 2 [ n ] Δ - 1 / 2 .
f ( x ) = n = 0 a n ψ n ( x ) , x [ - T ,   T ] .
- Ω Ω | F ˜ ( ω ) | 2 d ω = - Ω Ω F ˜ ( ω ) F ˜ * ( ω ) d ω = 1 2 π   - Ω Ω - T T exp ( - j ω x ) f ( x ) d x × - T T exp ( j ω x ˜ ) f * ( x ˜ ) d x d ω = 1 2 π   - T T - T T - Ω Ω exp [ j ω ( x ˜ - x ) ] d ω f ( x ) f * ( x ˜ ) d x d x ˜ = - T T - T T K F , Ω ( x ,   x ˜ ) f ( x ) f * ( x ˜ ) d x d x ˜ = - T T n = 0 a n - T T K F , Ω ( x ,   x ˜ ) ψ n ( x ) d x × m = 0 a m * ψ m ( x ˜ ) d x ˜ = - T T n = 0 λ n a n ψ n ( x ˜ ) m = 0 a m * ψ m ( x ˜ ) d x ˜ = n = 0 m = 0 λ n a n a m * - T T ψ n ( x ˜ ) ψ m ( x ˜ ) d x ˜ = n = 0 m = 0 λ n a n a m * λ n δ m , n = n = 0 | a n | 2 λ n 2 ,
- T T | f ( x ) | 2 d x = - T T f ( x ) f * ( x ) d ω = - T T n = 0 a n ψ n ( x ) m = 0 a m * ψ m ( x ) d x = n = 0 m = 0 a n a m * - T T ψ n ( x ) ψ m ( x ) d x = n = 0 m = 0 a n a m * λ n δ m , n = n = 0 | a n | 2 λ n
energy - preservation ratio = - Ω Ω | F ˜ ( ω ) | 2 d ω - T T | f ( x ) | 2 d x = n = 0 | a n | 2 λ n 2 n = 0 | a n | 2 λ n .
f ( x ) = a 0 ψ 0 ( x ) .
G a ( u ) = O F α [ g ( x ) ] = 1 - j cot α 2 π 1 / 2 - exp j 2   u 2 cot α - jux csc α + j 2   x 2 cot α g ( x ) d x .
G ( a , b , c , d ) ( u ) = O F ( a , b , c , d ) [ g ( x ) ] = 1 j 2 π b 1 / 2 - exp j 2   d b   u 2 - j   u b   x + j 2   a b   x 2 g ( x ) d x .
O F a [ g ( x ) ] = [ exp ( j α ) ] 1 / 2 O F ( cos α , sin α , - sin α , cos α ) [ g ( x ) ] .
O F ( a , b , c , d , τ , ρ ) [ g ( x ) ]
= ( j 2 π b ) - 1 / 2 exp ( j ρ u ) - exp j 2   d b   ( u - τ ) 2 - j   u - τ b   x + j 2   a b   x 2 g ( x ) d x .
G ˜ a ( u ) = O F ˜ a [ g ( x ) ] = 1 - j cot α 2 π 1 / 2 × T 1 T 2 exp j 2   u 2 cot α - jux csc α + j 2   x 2 cot α g ( x ) d x ,
x [ T 1 ,   T 2 ] , u [ Ω 1 ,   Ω 2 ] .
G ˜ α ( u ) = T 1 / 2 Ω - 1 / 2 T 1 / 2 Ω - 1 / 2 K ˆ α ( u ,   x ) g ( T 1 / 2 Ω - 1 / 2 x ) d x ,
K ˆ α ( u ,   x ) = n = 0 i - β ( T Ω - 1 λ n ) - β ϕ n ( T 1 / 2 Ω - 1 / 2 u ) × ϕ n ( T 1 / 2 Ω - 1 / 2 x ) , β = 2 α π - 1 .
G ˜ ( a , b , c , d ) ( u ) = O F ˜ ( a , b , c , d ) [ g ( x ) ] = 1 j 2 π b 1 / 2 × T 1 T 2 exp j 2   d b   u 2 - j   u b   x + j 2   a b   x 2 g ( x ) d x ,
ad - bc = 1 , x [ T 1 ,   T 2 ] , u [ Ω 1 ,   Ω 2 ] .
G ˜ ( a , b , c , d , τ , ρ ) ( u ) = O F ˜ ( a , b , c , d , τ , ρ ) [ g ( x ) ] = exp ( j ρ u ) ( j 2 π b ) 1 / 2   T 1 T 2 exp jd 2 b   ( u - τ ) 2 - j   u - τ b   x + ja 2 b   x 2 g ( x ) d x ,
ad - bc = 1 , x [ T 1 ,   T 2 ] , u [ Ω 1 ,   Ω 2 ] .
[ T 1 ,   T 2 ] = [ - T ,   T ] , [ Ω 1 ,   Ω 2 ] = [ - Ω ,   Ω ] .
R ( energy - pres . ratio ) = - Ω Ω | G ˜ ( a , b , c , d ) ( u ) | 2 d u - T T | g ( x ) | 2 d x .
- Ω Ω | G ˜ ( a , b , c , d ) ( u ) | 2 d u
= - Ω Ω G ˜ ( a , b , c , d ) ( u ) G ˜ ( a , b , c , d ) * ( u ) d u = 1 2 π | b |   - Ω Ω - T T - T T exp j 2   d b   u 2 - j   u b   x + j 2   a b   x 2 × exp - j 2   d b   u 2 + j   u b   x ˜ - j 2   a b   x ˜ 2 g ( x ) g * ( x ˜ ) d x d x ˜ d u = 1 2 π | b |   - T T - T T - Ω Ω exp j   u b   ( x ˜ - x ) d u × exp j 2   a b   ( x 2 - x ˜ 2 ) g ( x ) g * ( x ˜ ) d x d x ˜
= - T T - T T   sin [ Ω ( x ˜ - x ) | b | - 1 ] π ( x ˜ - x ) × exp j 2   a b   ( x 2 - x ˜ 2 ) g ( x ) g * ( x ˜ ) d x d x ˜ .
η n , T , Ω ϕ n , T , Ω ( x ˜ ) = - T T K ( a , b , c , d ) , Ω ( x ˜ ,   x ) ϕ n , T , Ω ( x ) d x ,
K ( a , b , c , d ) , Ω ( x ˜ ,   x ) = sin [ Ω ( x ˜ - x ) | b | - 1 ] π ( x ˜ - x ) × exp j 2   a b   ( x 2 - x ˜ 2 ) ,
x , x ˜ [ - T ,   T ] .
K ( a , b , c , d ) , Ω ( x ˜ ,   x ) = | b | - 1 K F , Ω ( | b | - 1 x ˜ ,   | b | - 1 x ) exp ja 2 b   ( x 2 - x ˜ 2 ) ,
λ n , T | b | - 1 , Ω ψ n , T | b | - 1 , Ω ( x ˜ )
= - T | b | - 1 T | b | - 1 K F , Ω ( x ˜ ,   x ) ψ n , T | b | - 1 , Ω ( x ) d x ,
ϕ n , T , Ω ( x ) = | b | - 1 / 2 exp - j 2   a b   x 2 ψ n , T | b | - 1 , Ω ( | b | - 1 x ) ,
η n , T , Ω = λ n , T | b | - 1 , Ω ,
- T T K ( a , b , c , d ) , Ω ( x ˜ ,   x ) ϕ n , T , Ω ( x ) d x = | b | - 3 / 2 exp [ - ja x ˜ 2 ( 2 b ) - 1 ] × - T T K F , Ω ( | b | - 1 x ˜ ,   | b | - 1 x ) ψ n , T | b | - 1 , Ω ( | b | - 1 x ) d x = | b | - 1 / 2 exp [ - ja x ˜ 2 ( 2 b ) - 1 ] × - T | b | - 1 T | b | - 1 K F , Ω ( | b | - 1 x ˜ ,   x 1 ) ψ n , T | b | - 1 , Ω ( x 1 ) d x 1 = λ n , T | b | - 1 , Ω | b | - 1 / 2 exp - j 2   a b   x ˜ 2 ψ n , T | b | - 1 , Ω ( | b | - 1 x ˜ ) = η n , T , Ω ϕ n , T , Ω ( x ˜ ) .
1 > η 0 , T , Ω > η 1 , T , Ω > η 2 , T , Ω > η 3 , T , Ω > > 0 .
- T T ϕ n , T , Ω ( x ) ϕ m , T , Ω * ( x ) d x
= | b | - 1 - T T ψ n , T | b | - 1 , Ω ( | b | - 1 x ) ψ m , T | b | - 1 , Ω * ( | b | - 1 x ) d x = - T | b | - 1 T | b | - 1 ψ n , T | b | - 1 , Ω ( x 1 ) ψ m , T | b | - 1 , Ω * ( x 1 ) d x 1 = λ n , T | b | - 1 , Ω δ m , n = η n , T , Ω δ m , n ,
- ϕ n , T , Ω ( x ) ϕ m , T , Ω * ( x ) d x
= | b | - 1 - ψ n , T | b | - 1 , Ω ( | b | - 1 x ) ψ m , T | b | - 1 , Ω * ( | b | - 1 x ) d x = - ψ n , T | b | - 1 , Ω ( x 1 ) ψ m , T | b | - 1 , Ω * ( x 1 ) d x 1 = δ m , n .
g ( x ) = n = 0 σ n ϕ n ( x ) , x [ - T ,   T ] ,
σ n = η n - 1 - T T g ( x ) ϕ n * ( x ) d x .
- T T | g ( x ) | 2 d x = n = 0 m = 0 σ n σ m * - T T ϕ n ( x ) ϕ m * ( x ) d x = n = 0 | σ n | 2 η n .
- Ω Ω | G ˜ ( a , b , c , d ) ( u ) | 2 d u
= - T T - T T K ( a , b , c , d ) , Ω ( x ˜ ,   x ) g ( x ) g * ( x ˜ ) d x d x ˜ = - T T - T T K ( a , b , c , d ) , Ω ( x ˜ ,   x ) n = 0 σ n ϕ n ( x ) d x × m = 0 σ m * ϕ m * ( x ˜ ) d x ˜ = - T T n = 0 σ n η n ϕ n ( x ˜ ) m = 0 σ m * ϕ m * ( x ˜ ) d x ˜ = n = 0 m = 0 σ n σ m * η n - T T ϕ n ( x ˜ ) ϕ m * ( x ˜ ) d x ˜ = n = 0 | σ n | 2 η n 2 ;
R = - Ω Ω | G ˜ ( a , b , c , d ) ( u ) | 2 d u - T T | g ( x ) | 2 d x = n = 0 | σ n | 2 η n 2 n = 0 | σ n | 2 η n .
Max ( R ) = η 0 1 , g ( x ) = σ 0 ϕ 0 ( x ) ,
min ( R ) = η n 0 , g ( x ) = σ n ϕ n ( x ) , n .
ϕ n , T , Ω ( x ) = | csc α | 1 / 2 × exp - j 2   x 2 cot α ψ n , T | csc α | , Ω ( | csc α | x ) ,
η n , T , Ω = λ n , T | csc α | , Ω ,
λ n , T | csc α | , Ω ψ n , T | csc α | , Ω ( x ˜ )
= - T | csc α | T | csc α |   sin [ Ω ( x ˜ - x ) ] π ( x ˜ - x )   ψ n , T | csc α | , Ω ( x ) d x .
R = - Ω Ω | G ˜ α ( u ) | 2 d u - T T | g ( x ) | 2 d x = n = 0 | σ n | 2 η n 2 n = 0 | σ n | 2 η n ,
σ n = η n - 1 - T T g ( x ) ϕ n * ( x ) d x .
η n , T , Ω ϕ n , T , Ω ( x ˜ ) = - T T K ( a , b , c , d , τ , ρ ) , Ω ( x ˜ ,   x ) ϕ n , T , Ω ( x ) d x ,
K ( a , b , c , d , τ , ρ ) , Ω ( x ˜ ,   x )
= sin [ Ω ( x ˜ - x ) | b | - 1 ] π ( x ˜ - x ) × exp j τ b   ( x - x ˜ ) + a 2 b   ( x 2 - x ˜ 2 ) ,
x , x ˜ [ - T ,   T ] .
K ( a , b , c , d , τ , ρ ) , Ω ( x ˜ ,   x ) = exp j τ b   ( x - x ˜ ) K ( a , b , c , d ) , Ω ( x ˜ ,   x ) .
ϕ n , T , Ω ( x ) = exp ( - j τ x / b ) ϕ n , T , Ω ( fi - LCT ) ( x ) ,
ϕ n , T , Ω ( x ) = | b | - 1 / 2 exp - j a 2 b   x 2 + τ b   x × ψ n , T | b | - 1 , Ω ( | b | - 1 x ) ,
η n , T , Ω = λ n , T | b | - 1 , Ω ,
G ˆ ( u ) = T 1 T 2 Γ ( a , b , c , d , τ , ρ ) ( u ,   x ) g ( x ) d x ,
Γ ( a , b , c , d , τ , ρ ) ( u ,   x )
= exp ( j ρ u ) ( j 2 π b ) 1 / 2 exp jd 2 b   ( u - τ ) 2 - j   u - τ b   x + ja 2 b   x 2 ,
x [ T 1 ,   T 2 ] , u [ Ω 1 ,   Ω 2 ] , T 1 - T 2 ,
Ω 1 - Ω 2 .
h ( x ) = g ( x + T 0 ) for h ( x ) ,   x [ - T 3 ,   T 3 ] ,
H ˆ ( u ) = G ˆ ( u + Ω 0 ) for H ˆ ( u ) , u [ - Ω 3 ,   Ω 3 ] ,
T 0 = 2 - 1 ( T 1 + T 2 ) , Ω 0 = 2 - 1 ( Ω 1 + Ω 2 ) ,
T 3 = 2 - 1 ( T 2 - T 1 ) , Ω 3 = 2 - 1 ( Ω 2 - Ω 1 ) .
H ˆ ( u ) = - T 3 T 3 Γ ( a , b , c , d , τ , ρ ) ( u + Ω 0 ,   x + T 0 ) h ( x ) d x .
Γ ( a , b , c , d , τ , ρ ) ( u + Ω 0 ,   x + T 0 )
= exp ( j φ ) Γ ( a , b , c , d , τ 1 , ρ 1 ) ( u ,   x ) ,
τ 1 = τ + aT 0 - Ω 0 ,
ρ 1 = ρ + b - 1 ( ad - 1 ) T 0 = ρ + cT 0 .
H ˆ ( u ) = exp ( j φ ) - T 3 T 3 Γ ( a , b , c , d , τ 1 , ρ 1 ) ( u ,   x ) h ( x ) d x ,
x [ - T 3 ,   T 3 ] , u [ - Ω 3 ,   Ω 3 ] .
ϕ n , T 3 , Ω 3 ( x ) = | b | - 1 / 2 exp - j a 2 b   x 2 + τ + aT 0 - Ω 0 b   x × ψ n , T 3 | b | - 1 , Ω 3 ( | b | - 1 x ) .
ϕ n , T 1 , T 2 , Ω 1 , Ω 2 ( x ) = ϕ n , T 3 , Ω 3 ( x - T 0 ) = | b | - 1 / 2 exp - ja 2 b   x 2 + j   Ω 0 - τ b   x × ψ n , T 3 | b | - 1 , Ω 3 x - T 0 | b | ,
η n , T 1 , T 2 , Ω 1 , Ω 2 = λ n , T 3 | b | - 1 , Ω 3 ,
T 1 T 2 ϕ n ( x ) ϕ m * ( x ) d x = η n δ m , n ,
- ϕ n ( x ) ϕ m * ( x ) d x = δ m , n .
R = Ω 1 Ω 2 | G ˜ ( a , b , c , d ) ( u ) | 2 d u T 1 T 2 | g ( x ) | 2 d x = n = 0 | σ n | 2 η n 2 n = 0 | σ n | 2 η n ,
σ n = T 1 T 2 g ( x ) ϕ n * ( x ) d x η n .
ϕ n , T 1 , T 2 , Ω 1 , Ω 2 ( T 0 + x ) = ( - 1 ) n ϕ n , T 1 , T 2 , Ω 1 , Ω 2 ( T 0 - x ) ,
T 0 = 2 - 1 ( T 1 + T 2 ) .
ψ n , σ T , Ω σ - 1 ( x ) = σ - 1 / 2 ψ n , T , Ω ( σ - 1 x ) .
x [ T 4 ,   T 5 ] = [ T 0 - σ T 3 ,   T 0 + σ T 3 ] ,
ω [ Ω 4 ,   Ω 5 ] = [ Ω 0 - σ - 1 Ω 3 ,   Ω 0 + σ - 1 Ω 3 ] ,
ϕ n , T 4 , T 5 , Ω 4 , Ω 5 ( x ) = | b | - 1 / 2 exp - ja 2 b   x 2 + j   Ω 0 - τ b   x × ψ n , σ T 3 | b | - 1 , Ω 3 σ - 1 x - T 0 | b | ,
[ applying Eq . ( 65 ) ]
= | σ b | - 1 / 2 exp - ja 2 b   x 2 + j   Ω 0 - τ b   x ψ n , T 3 | b | - 1 , Ω 3 x - T 0 σ | b | ,
| ϕ n , T 4 , T 5 , Ω 4 , Ω 5 ( x + T 0 ) |
= σ - 1 / 2 ϕ n , T 1 , T 2 , Ω 1 , Ω 2 x + T 0 σ .
P = T 3 Ω 3 | b | - 1 = ( T 2 - T 1 ) ( Ω 2 - Ω 1 ) ( 4 | b | ) - 1 .
j - n λ n , T , T 1 / 2 ψ n , T , T ( ω ) = ( 2 π ) - 1 / 2 - T T exp ( - j ω x ) ψ n , T , T ( x ) d x .
j - n ( λ n , T , Ω T Ω - 1 ) 1 / 2 ψ n , T , Ω ( T Ω - 1 ω )
= ( 2 π ) - 1 / 2 - T T exp ( - j ω x ) ψ n , T , Ω ( x ) d x .
O F ˜ ( a , b , c , d , τ , ρ ) [ ϕ n , T 1 , T 2 , Ω 1 , Ω 2 ( x ) ]
= exp ( j ρ u ) ( j 2 π b ) 1 / 2   T 1 T 2 exp jd 2 b   ( u - τ ) 2 - j   u - τ b   x + ja 2 b   x 2 ϕ n , T 1 , T 2 , Ω 1 , Ω 2 ( x ) d x .
O F ˜ ( a , b , c , d , τ , ρ ) [ ϕ n , T 1 , T 2 , Ω 1 , Ω 2 ( x ) ] = exp ( j ρ u ) ( j 2 π b | b | ) 1 / 2   T 1 T 2 exp jd 2 b   ( u - τ ) 2 - j   u - Ω 0 b   x ψ n , T 3 / | b | , Ω 3 x - T 0 | b | d x = exp [ j ρ u - jb - 1 ( u - Ω 0 ) T 0 ] [ j 2 π   sgn ( b ) ] 1 / 2 × exp jd 2 b   ( u - τ ) 2 - T 3 | b | - 1 T 3 | b | - 1 exp - j   u - Ω 0 sgn ( b )   x 1 × ψ n , T 3 | b | - 1 , Ω 3 ( x 1 ) d x 1 = j - n - 1 / 2 λ n , T 3 | b | - 1 , Ω 3 T 3 b Ω 3 1 / 2 exp j   Ω 0 - u b   T 0 + j ρ u + jd 2 b   ( u - τ ) 2 ψ n , T 3 | b | - 1 , Ω 3 T 3 ( u - Ω 0 ) b Ω 3 = j ( - n - 1 / 2 ) sgn ( b ) λ n , T 3 | b | - 1 , Ω 3 T 3 | b | Ω 3 1 / 2 exp j   Ω 0 - u b   T 0 + j ρ u + jd 2 b   ( u - τ ) 2 ψ n , T 3 | b | - 1 , Ω 3 T 3 ( u - Ω 0 ) | b | Ω 3 .
O F ˜ ( a , b , c , d , τ , ρ ) [ ϕ n , T 1 , T 2 , Ω 1 , Ω 2 ( x ) ]
= μ n k 1 - 1 / 2 exp ( jk 3 u 2 + jk 4 u ) ϕ n , T 1 , T 2 , Ω 1 , Ω 2
× u - k 2 k 1 ,
k 1 = Ω 3 T 3 - 1 , k 2 = Ω 0 - Ω 3 T 0 T 3 - 1 ,
k 3 = d 2 b + aT 3 2 2 b Ω 3 2 , k 4 = ρ - d b   τ - T 0 b - T 3 b Ω 3   Ω 0 aT 3 Ω 3 + 1 - τ - aT 0 ,
μ n = j ( - n - 1 / 2 ) sgn ( b ) η n , T 1 , T 2 , Ω 1 , Ω 2 1 / 2 exp j b   d 2   τ 2 + Ω 0 T 0 + a 2   k 2 2 k 1 2 + k 2 k 1   ( Ω 0 - τ ) ,
T 1 = Ω 1 , T 2 = Ω 2 ( i . e . , T 0 = Ω 0 , T 3 = Ω 3 ) ,
d = - a , ρ = b - 1 [ ( d - 1 ) τ + 2 T 0 ] ,
Medium 1 . Thickness : h 0 + h 1 x + h 2 y + h 3 x 2
+ h 4 y 2 ,
extension : x [ - B 1 ,   B 1 ] ,
y [ - P 1 ,   P 1 ] ,
Medium 2 . Thickness : s 0 + s 1 u + s 2 v + s 3 u 2
+ s 4 v 2 ,
extension : u [ - B 2 ,   B 2 ] ,
v [ - P 2 ,   P 2 ] ,
Free space . Length : z .
g o ( u ,   v ) = exp ( j ) O F y ( a 1 , b 1 , c 1 , d 1 , τ 1 , ρ 1 ) { O F x ( a , b , c , d , τ , ρ ) [ g i ( x ,   y ) ] } ,
a b c d = 1 + 2 z ( n - 1 ) h 3 z / k 2 k ( n - 1 ) ( h 3 + s 3 ) + ζ 1 + 2 z ( n - 1 ) s 3 ,
τ ρ = z ( n - 1 ) h 1 k ( n - 1 ) ( h 1 + s 1 ) + κ ,
a 1 b 1 c 1 d 1
= 1 + 2 z ( n - 1 ) h 4 z / k 2 k ( n - 1 ) ( h 4 + s 4 ) + ζ 1 1 + 2 z ( n - 1 ) s 4 ,
τ 1 ρ 1 = z ( n - 1 ) h 2 k ( n - 1 ) ( h 2 + s 2 ) + κ 1 ,
ς = 4 zk ( n - 1 ) 2 h 3 s 3 , ς 1 = 4 zk ( n - 1 ) 2 h 4 s 4 ,
κ = 2 kz ( n - 1 ) 2 h 1 s 3 , κ 1 = 2 kz ( n - 1 ) 2 h 2 s 4 ,
k = 2 π / λ , λ , wavelength ;
n , refractive index of the two media .
ϕ m ( x ) ϕ n ( y ) ,
ϕ m ( x ) = k z 1 / 2 exp - jk 1 + 2 z ( n - 1 ) h 3 2 z   x 2 + ( n - 1 ) h 1 x ψ m , kB 1 / z , B 2 k z   x ,
ϕ n ( y ) = k z 1 / 2 exp - jk 1 + 2 z ( n - 1 ) h 4 2 z   x 2 + ( n - 1 ) h 2 x ψ n , kP 1 / z , P 2 k z   y ,
- kB 1 z - 1 kB 1 z - 1   sin [ B 2 ( x ˜ - x ) ] π ( x ˜ - x )   ψ n , kB 1 z - 1 , B 2 ( x ) d x
= λ n ψ n , kB 1 z - 1 , B 2 ( x ˜ ) .
R = - P 2 P 2 - B 2 B 2 | g o ( x ,   y ) | 2 d x d y - P 1 P 1 - B 1 B 1 | g i ( x ,   y ) | 2 d x d y = n = 0 m = 0 | σ m , n | 2 λ m 2 λ n 2 n = 0 m = 0 | σ m , n | 2 λ m λ n ,
σ m , n = λ m - 1 λ n - 1 - P 1 P 1 - B 1 B 1 g i ( x ,   y ) ϕ m * ( x ) ϕ n * ( y ) d x d y .
Medium 1 . Thickness : - 2.5 ( x 2 + y 2 ) + 10 - 2 ( x + y ) + 8 × 10 - 3 ;
extension : x ,   y [ - 0.036 m ,   0.036 m ] ,
Medium 2 . Thickness : - 3.5 ( x 2 + y 2 ) + 1.2 × 10 - 2 ( x + y ) + 7 × 10 - 3 ;
extension : x , y [ - 0.03 m ,   0.03 m ] ,
z = 1.8 m , λ = 5 × 10 - 4 ,
n = 1.5 .
ϕ 0 ( x ) ϕ 0 ( y ) , ϕ 5 ( x ) ϕ 5 ( y ) ,
λ 0 = 1 - 5.16 × 10 - 6 , λ 5 = 0.2111 ,
for ϕ 0 ( x ) ϕ 0 ( y ) : R = ( 1 - 5.16 × 10 - 6 ) 2 = 1 - 1.0312 × 10 - 5 ,
for ϕ 5 ( x ) ϕ 5 ( y ) : R = 0.2111 2 = 0.0446 .
g q - 1 ( x ) = n = 0 σ q , n ϕ q , n ( x ) ,
σ q , n = η q , n - 1 T q - 1 , 1 T q - 1 , 2 g q - 1 ( x ) ϕ q , n * ( x ) d x .
O F q [ ϕ q , n ( x ) ] = μ q , n ζ q , n ( u ) .
ϕ q , n ( x ) = ϕ n , T 1 , T 2 , Ω 1 , Ω 2 ( x ) ,
ζ q , n ( u ) = μ n k 1 - 1 / 2 ( k 1 ) - 1 / 2 exp ( jk 3 u 2 + jk 4 u ) ϕ n , T 1 , T 2 , Ω 1 , Ω 2 [ k 1 - 1 ( u - k 2 ) ] ,
μ q , n = exp ( j φ ) μ n ,
g q ( u ) = O F q [ g q - 1 ( x ) ] = n = 0 σ q , n μ q , n ζ q , n ( u ) .
η q , n < when n > N q .
g q ( u ) g ˆ q ( u ) = n = 0 N q σ q , n μ q , n ζ q , n ( u ) .
g ˆ q - 1 ( x ) = n = 0 N q σ q , n ϕ q , n ( x ) ,
σ q , n = η q , n - 1 T q - 1 , 1 T q - 1 , 2 g q - 1 ( x ) ϕ q , n * ( x ) d x .
g ˆ q - 1 ( x ) = p q Φ q , g ˆ q ( x ) = p q Λ q Z q ,
p q = [ σ q , 0 ,   σ q , 1 ,   σ q , 2 , ,   σ q , N q ] ,
Φ q = [ ϕ q , 0 ( x ) ,   ϕ q , 1 ( x ) ,   ϕ q , 2 ( x ) , ,   ϕ q , N q ( x ) ] T ,
Z q = [ ζ q , 0 ( x ) ,   ζ q , 1 ( x ) ,   ζ q , 2 ( x ) , ,   ζ q , N q ( x ) ] T ,
Λ q = μ q , 0 0 0 0 0 μ q , 1 0 0 0 0 μ q , N q - 1 0 0 0 0 μ q , N q ;
Z q = R q & Fgr ; q + 1 ,
R q = r q , 0 , 0 r q , 0 , 1 r q , 0 , N q + 1 r q , 1 , 0 r q , 1 , 1 r q , 1 , N q + 1 r q , N q , 0 r q , N q , 1 r q , N q , N q + 1 ,
r q , m , n = η q + 1 , n - 1 T q , 1 T q , 2 ζ q , m ( x ) ϕ q + 1 , n * ( x ) d x .
g ˆ q ( x ) = p q Λ q Z q = p q Λ q R q & Fgr ; q + 1 .
p q + 1 = p q Λ q R q , p Q = p 1 Λ 1 R 1 Λ 2 R 2 Λ Q - 1 R Q - 1 .
input : g 0 ( x ) = n = 0 σ 1 , n ϕ 1 , n ( x ) ,
output : g Q ( x ) n = 0 N Q β n ζ Q , n ( x ) ,
p 1 = [ σ q , 0 ,   σ q , 1 ,   σ q , 2 , ,   σ q , N 1 ] ,
p o = [ β 0 ,   β 1 ,   β 2 , ,   β N Q ] ,
p o = p 1 A ,
A = Λ 1 R 1 Λ 2 R 2 Λ Q - 1 R Q - 1 Λ Q .
R = T Q , 1 T Q , 2 | g Q ( u ) | 2 d u T 0 , 1 T 0 , 2 | g 0 ( x ) | 2 d x n = 0 T Q | μ n | 2 η Q , n n = 0 | σ 1 , n | 2 η 1 , n = C   n = 0 T Q | μ n | 2 η Q , n n = 0 T 1 | σ 1 , n | 2 η 1 , n = C   p o | Λ Q | 2 p o H p 1 | Λ 1 | 2 p 1 H = C   p 1 A | Λ Q | 2 A H p 1 H p 1 | Λ 1 | 2 p 1 H = C   p Δ | Λ 1 | - 1 A | Λ Q | 2 A H | Λ 1 | - 1 p Δ H p Δ p Δ H ,
C = n = 0 T 1 | σ 1 , n | 2 η 1 , n n = 0 | σ 1 , n | 2 η 1 , n - 1 ,
σ 1 , n = η 1 , n - 1 T 0 , 1 T 0 , 2 g 0 ( x ) ϕ 1 , n * ( x ) d x ,
p Δ = p 1 | Λ 1 | .
B = | Λ 1 | - 1 A | Λ Q | 2 A H | Λ 1 | - 1 , f n B = κ n f n .
1 > κ 0 > κ 1 > κ 2 > > κ N 1 > 0 ,
p Δ = n = 0 N 1 α n f n , α n = p Δ f n H = p 1 | Λ 1 | f n H ,
R = C   p Δ Bp Δ H p Δ p Δ H = C   n = 0 N 1 m = 0 N 1 α n α m * f n Bf m H n = 0 N 1 m = 0 N 1 α n α m * f n f m H = C   n = 0 N 1 m = 0 N 1 α n α m * κ n f n f m H n = 0 N 1 m = 0 N 1 α n α m * f n f m H = C   n = 0 N 1 | α n | 2 κ n n = 0 N 1 | α n | 2 ,
p Δ = α 0 f 0 , i . e . , p 1 = α 0 f 0 | Λ 1 | - 1 ,
R = C κ 0 .
O F ˜ ( a , b , c , d , τ , ρ ) [ ϕ n ( x ) ] = μ n ζ n ( u ) ,
ζ n ( u ) = ( k 1 ) - 1 / 2 exp ( jk 3 u 2 + jk 4 u ) ϕ n u - k 2 k 1 .
Optical system : f o ( x ) = exp ( j φ ) O F ˜ ( a , b , c , d , τ , ρ ) [ f i ( x ) ] ,
f i ( x ) : input , f o ( x ) : output .
f o ( x ) = exp ( j φ ) C μ n ζ n ( x ) .
| f o ( x ) | = | C μ n ζ n ( x ) | = ( k 1 - 1 η n ) 1 / 2 C ϕ n x - k 2 k 1 = ( k 1 - 1 η n ) 1 / 2 f i x - k 2 k 1 .
1   >   | μ n | > 1 - Δ ;
0 n N ˜ , Δ very small .
μ n μ m j - ( n + 1 / 2 ) sgn ( b ) exp ( j θ ) , m , n N ˜ ,
m - n = 4 M , where M is an integer ,
θ = b - 1 [ 2 - 1 d τ 2 + Ω 0 T 0 + 2 - 1 ak 1 - 2 k 2 2 + k 1 - 1 k 2 ( Ω 0 - τ ) ] .
f i ( x ) = s = 0 N s α s ϕ r + 4 s ( x ) , r = 0 ,   1 ,   2 , or 3 ,
r + 4 N s N ˜ < r + 4 ( N s + 1 ) ,
f o ( u ) = exp ( j φ ) O F ˜ ( a , b , c , d , τ , ρ ) s = 0 N s α s ϕ r + 4 s ( x ) = exp ( j φ ) s = 0 N s α s μ r + 4 s ζ r + 4 s ( x ) exp [ j ( φ + θ ) ] j - ( n + 1 / 2 ) sgn ( b ) s = 0 N s α s ζ r + 4 s ( u ) = k 1 - 1 / 2 s = 0 N s α s ϕ r + 4 s [ k 1 - 1 ( u - k 2 ) ] ,
| f o ( u ) | k 1 - 1 / 2 f i u - k 2 k 1 ,
mirror A : x ,   y [ - B 1 ,   B 1 ] ;
mirror B : x ,   y [ - B 2 ,   B 2 ] .
f o , 0 ( x ,   y ) = exp ( j φ 1 ) O F ˜ y ( a 1 , b 1 , c 1 , d 1 , τ ˜ 1 , ρ ˜ 1 ) { O F ˜ x ( a 1 , b 1 , c 1 , d 1 , τ 1 , ρ 1 ) × [ f i , 0 ( x ,   y ) ] } ,
a 1 b 1 c 1 d 1 = 1 - R 1 - 1 D - D / k k ( R 1 - 1 + R 2 - 1 - R 1 - 1 R 2 - 1 D ) 1 - R 2 - 1 D ,
k = 2 π / λ , τ 1 = τ ˜ 1 = 0 ,
ρ 1 = - kx 0 / R 2 , ρ ˜ 1 = - ky 0 / R 2 ,
f i , 1 ( x ,   y ) = exp ( j φ 2 ) O F ˜ y ( a 2 , b 2 , c 2 , d 2 , τ ˜ 2 , ρ ˜ 2 ) { O F ˜ x ( a 2 , b 2 , c 2 , d 2 , τ 2 , ρ 2 ) [ f o , 0 ( x ,   y ) ] } ,
a 2 b 2 c 2 d 2 = 1 - R 2 - 1 D - D / k k ( R 1 - 1 + R 2 - 1 - R 1 - 1 R 2 - 1 D ) 1 - R 1 - 1 D ,
τ 2 ρ 2 = x 0 DR 2 - 1 kx 0 R 2 - 1 ( R 1 - 1 D - 1 ) ,
τ ˜ 2 ρ ˜ 2 = y 0 DR 2 - 1 ky 0 R 2 - 1 ( R 1 - 1 D - 1 ) .
f i , 1 ( x ,   y ) = ρ f i , 0 ( x ,   y ) , ρ is some constant ,
f i , 0 ( x ,   y ) = n = 0 N 1 σ 1 , n ϕ 1 , n ( x )   n = 0 N 1 σ 1 , n ϕ 1 , n ( y ) ,
H = h 0 , 0 h 0 , 1 h 0 , N 2 + 1 h 1 , 0 h 1 , 1 h 1 , N 2 + 1 h N 2 , 0 h N 2 , 1 h N 2 , N 2 + 1 ,
h m , n = η 1 , n - 1 - B 1 B 1 ζ 2 , m ( x ) ϕ 1 , n * ( x ) d x ,

Metrics