Abstract

The complex amplitude distributions on two spherical reference surfaces of given curvature and spacing are simply related by a fractional Fourier transform. The order of the fractional Fourier transform is proportional to the Gouy phase shift between the two surfaces. This result provides new insight into wave propagation and spherical mirror resonators as well as the possibility of exploiting the fractional Fourier transform as a mathematical tool in analyzing such systems.

© 1994 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
    [CrossRef]
  2. H. M. Ozaktas, D. Mendlovic, Opt. Commun. 101, 163 (1993).
    [CrossRef]
  3. D. Mendlovic, H. M. Ozaktas, J. Opt. Soc. Am. A 10, 1875 (1993).
    [CrossRef]
  4. H. M. Ozaktas, D. Medlovic, J. Opt. Soc. Am. A 10, 2522 (1993).
    [CrossRef]
  5. D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, Appl. Opt. 33, 6188 (1994).
    [CrossRef] [PubMed]
  6. A. C. McBride, F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
    [CrossRef]
  7. A. W. Lohmann, J. Opt. Soc. Am. A 10, 2181 (1993).
    [CrossRef]
  8. H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, J. Opt. Soc. Am. A 11,547 (1994).
    [CrossRef]
  9. S. G. Lipson, H. Lipson, J. Opt. Soc. Am. A 10, 2088 (1993); Optical Physics, 2nd ed. (Cambridge U. Press, Cambridge, 1981), p. 191.
    [CrossRef]
  10. D. Mendlovic, H. M. Ozaktas, A. W. Lohmann, Opt. Commun. 105, 36 (1994).
    [CrossRef]

1994 (3)

1993 (5)

1987 (1)

A. C. McBride, F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
[CrossRef]

Barshan, B.

Kerr, F. H.

A. C. McBride, F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
[CrossRef]

Lipson, H.

Lipson, S. G.

Lohmann, A. W.

McBride, A. C.

A. C. McBride, F. H. Kerr, IMA J. Appl. Math. 39, 159 (1987).
[CrossRef]

Medlovic, D.

Mendlovic, D.

Onural, L.

Ozaktas, H. M.

Saleh, B. E. A.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
[CrossRef]

Teich, M. C.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
[CrossRef]

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 (1)

Fig. 1
Fig. 1

Spherical surfaces of given radii and separation: the complex amplitude distribution on the second surface is the fractional Fourier transform of that on the first surface. ĝ(u1, v1) and ĥ(u2, v2) denote the complex amplitude distributions on the scaled coordinate systems on surfaces 1 and 2, respectively. In the figure z1 < 0 and z2 > 0, but the results remain valid if both surfaces are on the same side of the z = 0 plane.

Equations (13)

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

ψ l ( u ) = 2 1 / 4 2 l l ! H l ( 2 π u ) exp ( - π u 2 ) ,
f 0 ( x , y ) = l = 0 m = 0 A l m 1 s 0 ψ l ( x s 0 ) ψ m ( y s 0 ) ,
A l m = - - f 0 ( x , y ) 1 s 0 ψ l ( x s 0 ) ψ m ( y s 0 ) d x d y ,
f n ( x , y ) = l = 0 m = 0 A l m 1 s n ψ l ( x s n ) ψ m ( y s n ) × exp [ i k z n + i k ( x 2 + y 2 ) 2 R n - i ( l + m + 1 ) ζ n ] .
f n ( x , y ) = s 0 s n ( F a n f ^ 0 ) ( x s n , y s n ) exp ( i k z n - i ζ n ) × exp [ i k ( x 2 + y 2 ) 2 R n ] ,
a n = ζ n π / 2 .
h ^ ( u 2 , v 2 ) = s 1 s 2 ( F a g ^ ) ( u 2 , v 2 ) ,
a = a 2 - a 1 = ζ 2 - ζ 1 π / 2 .
a π 2 = arctan [ ( 2 R / d - 1 ) 1 / 2 R / d - 1 ] .
0 [ cos ( a π / 2 ) ] 2 1.
( F a f ^ ) ( u , v ) = - - B a ( u , v ; u , v ) f ^ ( u , v ) d u d v ,
B a ( u , v ; u , v ) = l = 0 m = 0 exp ( - i a l π / 2 ) exp ( - i a m π / 2 ) × ψ l ( u ) ψ m ( v ) ψ l ( u ) ψ m ( v ) .
F a [ ψ l ( u ) ψ m ( v ) ] = exp ( - i a l π / 2 ) exp ( - i a m π / 2 ) × [ ψ l ( u ) ψ m ( v ) ] .

Metrics