Abstract

The linear canonical transform (LCT) is the name of a parameterized continuum of transforms that include, as particular cases, many widely used transforms in optics such as the Fourier transform, fractional Fourier transform, and Fresnel transform. It provides a generalized mathematical tool for representing the response of any first-order optical system in a simple and insightful way. In this work we present four uncertainty relations between LCT pairs and discuss their implications in some common optical systems.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |

  1. K. R. Wolf, Integral Transforms in Science and Engineering (Plenum, 1979), Chaps. 9, 10.
  2. H. M. Ozaktas, M. A. Kutay, and Z. Zalevsky, The Fractional Fourier Transform With Applications in Optics and Signal Processing (Wiley, 2000).
  3. S. C. Pei and J. J. Ding, "Relations between fractional operations and time-frequency distributions, and their applications," IEEE Trans. Signal Process. 49, 1638-1655 (2001).
    [CrossRef]
  4. S. C. Pei and J. J. Ding, "Eigenfunctions of the offset Fourier, fractional Fourier, and linear canonical transforms," J. Opt. Soc. Am. A 20, 522-532 (2003).
    [CrossRef]
  5. S. C. Pei and J. J. Ding, "Generalized prolate spheroidal wave functions for optical finite fractional Fourier and linear canonical transforms," J. Opt. Soc. Am. A 22, 460-474 (2005).
    [CrossRef]
  6. B. M. Hennelly and J. T. Sheridan, "Generalizing, optimizing, and inventing numerical algorithms for the, fractional Fourier, Fresnel, and linear canonical transforms," J. Opt. Soc. Am. A 22, 917-927 (2005).
    [CrossRef]
  7. B. M. Hennelly and J. T. Sheridan, "Fast numerical algorithm for the linear canonical transform," J. Opt. Soc. Am. A 22, 998-937 (2005).
    [CrossRef]
  8. A. Stern, "Why is the linear canonical transform so little known?," AIP Conf. Proc. 860, 225-234 (2006).
    [CrossRef]
  9. A. Stern, "Sampling of linear canonical transformed signals," Signal Process. 86, 1421-1425 (2006).
    [CrossRef]
  10. D. F. V. James and G. S. Agarwal, "The generalized Fresnel transform and its application to optics," Opt. Commun. 126, 207-212 (1996).
    [CrossRef]
  11. L. M. Bernardo, "ABCD matrix formalism of fractional Fourier optics," Opt. Eng. (Bellingham) 35, 732-740 (1996).
    [CrossRef]
  12. S. A. Collins, "Lens-system diffraction integral written in terms of matrix optics," J. Opt. Soc. Am. 60, 1168-1174 (1970).
    [CrossRef]
  13. M. Nazarathy and J. Shamir, "First-order optics--a canonical operator representation: lossless systems," J. Opt. Soc. Am. 72, 356-364 (1982).
    [CrossRef]
  14. M. Moshinsky and C. Quesne, "Linear canonical transformations and their unitary representations," J. Math. Phys. 12, 1772-1783 (1971).
    [CrossRef]
  15. S. C. Pei and J. J. Ding, "Eigenfunctions of the offset Fourier, fractional Fourier, and linear canonical transforms," J. Opt. Soc. Am. A 20, 522-532 (2003).
    [CrossRef]
  16. A. Stern, "Sampling of compact signals in offset linear canonical transform domains," Signal Image Video Process. 1, 359-367 (2007).
    [CrossRef]
  17. S. Abe and 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]
  18. A. Papoulis, "Pulse compression, fiber communication, and diffraction: a unified approach," J. Opt. Soc. Am. A 11, 3-13 (1994).
    [CrossRef]
  19. D. Mendlovic, H. M. Ozaktas, and A. W. Lohmann, "Graded index fibers, Wigner distribution, and the fractional Fourier transform," Appl. Opt. 33, 6188-6193 (1994).
    [CrossRef] [PubMed]
  20. H. M. Ozaktas and O. Aytur,"Fractional Fourier domains," Signal Process. 46, 119-124 (1995).
    [CrossRef]
  21. S. Shinde and V. M. Gadre, "An uncertainty principle for real signals in the fractional Fourier transform domain," IEEE Trans. Signal Process. 49, 2545-2548 (2001).
    [CrossRef]
  22. G. P. Agrawal, Fiber-Optic Communication Systems, 3rd ed. (Wiley, 2002), pp. 47-49.
  23. S. Mallat, A Wavelet Tour of Signal Processing, 2nd ed. (Academic, 1999), pp. 31-32.
  24. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, 1968), Chap. 11.

2007 (1)

A. Stern, "Sampling of compact signals in offset linear canonical transform domains," Signal Image Video Process. 1, 359-367 (2007).
[CrossRef]

2006 (2)

A. Stern, "Why is the linear canonical transform so little known?," AIP Conf. Proc. 860, 225-234 (2006).
[CrossRef]

A. Stern, "Sampling of linear canonical transformed signals," Signal Process. 86, 1421-1425 (2006).
[CrossRef]

2005 (3)

2003 (2)

2001 (2)

S. C. Pei and J. J. Ding, "Relations between fractional operations and time-frequency distributions, and their applications," IEEE Trans. Signal Process. 49, 1638-1655 (2001).
[CrossRef]

S. Shinde and V. M. Gadre, "An uncertainty principle for real signals in the fractional Fourier transform domain," IEEE Trans. Signal Process. 49, 2545-2548 (2001).
[CrossRef]

1996 (2)

D. F. V. James and G. S. Agarwal, "The generalized Fresnel transform and its application to optics," Opt. Commun. 126, 207-212 (1996).
[CrossRef]

L. M. Bernardo, "ABCD matrix formalism of fractional Fourier optics," Opt. Eng. (Bellingham) 35, 732-740 (1996).
[CrossRef]

1995 (1)

H. M. Ozaktas and O. Aytur,"Fractional Fourier domains," Signal Process. 46, 119-124 (1995).
[CrossRef]

1994 (3)

1982 (1)

1971 (1)

M. Moshinsky and C. Quesne, "Linear canonical transformations and their unitary representations," J. Math. Phys. 12, 1772-1783 (1971).
[CrossRef]

1970 (1)

AIP Conf. Proc. (1)

A. Stern, "Why is the linear canonical transform so little known?," AIP Conf. Proc. 860, 225-234 (2006).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Signal Process. (2)

S. Shinde and V. M. Gadre, "An uncertainty principle for real signals in the fractional Fourier transform domain," IEEE Trans. Signal Process. 49, 2545-2548 (2001).
[CrossRef]

S. C. Pei and J. J. Ding, "Relations between fractional operations and time-frequency distributions, and their applications," IEEE Trans. Signal Process. 49, 1638-1655 (2001).
[CrossRef]

J. Math. Phys. (1)

M. Moshinsky and C. Quesne, "Linear canonical transformations and their unitary representations," J. Math. Phys. 12, 1772-1783 (1971).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (6)

Opt. Commun. (1)

D. F. V. James and G. S. Agarwal, "The generalized Fresnel transform and its application to optics," Opt. Commun. 126, 207-212 (1996).
[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)

Signal Image Video Process. (1)

A. Stern, "Sampling of compact signals in offset linear canonical transform domains," Signal Image Video Process. 1, 359-367 (2007).
[CrossRef]

Signal Process. (2)

A. Stern, "Sampling of linear canonical transformed signals," Signal Process. 86, 1421-1425 (2006).
[CrossRef]

H. M. Ozaktas and O. Aytur,"Fractional Fourier domains," Signal Process. 46, 119-124 (1995).
[CrossRef]

Other (5)

G. P. Agrawal, Fiber-Optic Communication Systems, 3rd ed. (Wiley, 2002), pp. 47-49.

S. Mallat, A Wavelet Tour of Signal Processing, 2nd ed. (Academic, 1999), pp. 31-32.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, 1968), Chap. 11.

K. R. Wolf, Integral Transforms in Science and Engineering (Plenum, 1979), Chaps. 9, 10.

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

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.


Tables (1)

Tables Icon

Table 1 Properties of LCT a

Equations (37)

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

f ̃ M ( y ) = O M { f ( x ) } ( y ) = f ( x ) , C M * ( y , x ) f ( x ) C M ( y , x ) d x ,
C M ( y , x ) = 1 j 2 π b e j 2 b ( a x 2 2 x y + d y 2 ) , b 0 ,
f ̃ M ( y ) = O M { f ( x ) } ( y ) = d e j 2 c d y 2 f ( d y ) , b = 0 .
O M 1 O M 2 = O M 1 M 2 .
f ̃ M ( y ) g ̃ M ( y ) * d y = f ( x ) g ( x ) * d x ,
f ̃ M ( y ) = f ( x ) ,
O M 1 { y f ̃ A ( y ) } ( x ) = a x f ( x ) j b d d x f ( x ) ,
σ f ( x ) 2 = 1 E ( x u ) 2 f ( x ) 2 d x ,
σ f ̃ ( y ) 2 = 1 E ( y ξ ) 2 f ̃ M ( y ) 2 d y ,
u = 1 E x f ( x ) 2 d x ,
ξ = 1 E y f ̃ M ( y ) 2 d y .
σ f ( x ) 2 σ f ̂ M ( y ) 2 b 2 4 .
σ f ̂ M ( y ) = b 2 σ .
σ ω y 2 = 1 E ( ω y η ) 2 F { f ̃ M } ( ω y ) 2 d ω y ,
η = 1 E ω y F { f ̃ M } ( ω y ) 2 d ω y .
σ x 2 σ ω y 2 d 2 4 ,
F { f ̃ M } = O M F O M { f ( x ) } = O M eq { f ( x ) } ,
M eq = ( 0 1 1 0 ) ( a b c d ) .
σ x 2 σ y 2 ( a σ x 2 ) 2 + b 2 4 ,
σ y σ x 2 + ( λ z ) 2 16 π 2 σ x 2 ,
w z 2 = w 0 2 + ( z λ ) 2 ( 2 π ) 2 w 0 2 .
f z ( t ) = F { F { f 0 ( t ) } e j 2 β 2 ω t 2 } ( t ) = 1 j 2 π β 2 z f 0 ( t ) e j 2 β 2 z ( t t ) 2 d t ,
σ z σ 0 = 1 + ( β 2 L 2 σ 0 2 ) 2 ,
σ x 2 σ ω y 2 ( c σ x 2 ) 2 + d 2 4 .
G ( y ) f ( x ) e j 2 b a x 2 e j b y x d x = j 2 π b e j 2 b d y 2 f ̃ M ( y ) .
σ f ̃ ( y ) 2 = 1 E ( y ξ ) 2 f ̃ M ( y ) 2 d y = 1 E ( y ξ ) 2 G ( y ) 2 d y σ G ( y ) 2 ,
σ G ( y b ) 2 = 1 b 2 σ G ( y ) 2 = 1 b 2 σ f ̃ ( y ) 2 .
σ g ( x ) 2 σ G ( r ) 2 1 4 ,
σ g ( x ) 2 1 b 2 σ f ̃ ( y ) 2 1 4 ,
σ x 2 σ y 2 = 1 E 2 x f ( x ) 2 d x y f ̃ M ( y ) 2 d y = 1 E 2 x f ( x ) 2 d x a x f ( x ) j b d d x f ( x ) 2 d x .
a x f ( x ) j b d d x f ( x ) 2 d x = a x f ( x ) 2 d x + b d d x f ( x ) 2 d x ,
σ x 2 σ y 2 = a 2 E 2 [ x f ( x ) 2 d x ] 2 + b 2 E 2 x f ( x ) 2 d x d d x f ( x ) 2 d x .
x f ( x ) 2 d x d d x f ( x ) 2 d x { x f * ( x ) d d x f ( x ) d x } 2 ,
x f * ( x ) d d x f ( x ) d x 1 2 f * ( x ) d d x f ( x ) + f ( x ) d d x f * ( x ) d x = 1 2 x d d x f ( x ) 2 d x .
1 2 x d d x f ( x ) 2 d x = 1 2 [ x f ( x ) 2 f ( x ) 2 d x ] = E 2 .
ξ = 1 E y f ̃ M ( y ) 2 d y = 1 E [ y f ̃ M ( y ) ] [ f ̃ M ( y ) ] * d y = 1 E [ a x f ( x ) j b d d x f c ( x ) ] f ̃ * ( y ) d y = a E x f ( x ) 2 d x j b x f * ( x ) d d x f ( x ) d x = a u .
σ f ̃ c ( y ) 2 = 1 E ( y ξ c ) 2 f ̃ c M ( y ) 2 d y = 1 E y 2 f ̃ c M ( y ) 2 d y = 1 E y 2 e j c u y e j 2 a c u 2 f ̃ M ( y a u ) 2 d y = y = y a u 1 E ( y a u ) 2 f ̃ M ( y ) 2 d y a u = ξ σ f ̃ ( y ) 2 .

Metrics