Abstract

The focal shift for a lens of finite value of Fresnel number can be defined in terms of the second moment of the intensity distribution in transverse planes. The connection with the optical transfer function is described. The specification of the focused amplitude in terms of the fractional Fourier transform is discussed, and the connections among the fractional Fourier transform, the Wigner distribution, and the ambiguity function are described, leading to a model for effects of Fresnel number in terms of a rotation in phase space. The uncertainty principle is discussed, including the significance of the beam propagation factor M2 and the width of optical fiber beam modes. Calculation of the moments in terms of the modulus and the phase of the illuminating wave is presented, and the use of the Kaiser–Teager energy operator is also described.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. De Nicola, D. Anderson, M. Lisak, “Focal shift effects in diffracted focused beams,” Pure Appl. Opt. 7, 1249–1259 (1998).
    [CrossRef]
  2. Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
    [CrossRef]
  3. A. Papoulis, “Ambiguity function in Fourier optics,” J. Opt. Soc. Am. 64, 779–788 (1974).
    [CrossRef]
  4. Y. Li, E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,” J. Opt. Soc. Am. A 1, 801–808 (1984).
    [CrossRef]
  5. C. J. R. Sheppard, “Imaging in optical systems of finite Fresnel number,” J. Opt. Soc. Am. A 3, 1428–1432 (1986).
    [CrossRef]
  6. R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).
  7. N. Bareket, “Second moment of the diffraction point spread function as an image quality criterion,” J. Opt. Soc. Am. 69, 1311–1312 (1979).
    [CrossRef]
  8. M. R. Teague, “Irradiance moments: their propagation and use for unique retrieval of phase,” J. Opt. Soc. Am. 72, 1199–1209 (1982).
    [CrossRef]
  9. R. Martinez-Herrero, P. M. Meijas, “Second-order spatial characterization of hard-edge diffracted beams,” Opt. Lett. 18, 1669–1671 (1993).
    [CrossRef] [PubMed]
  10. E. U. Condon, “Immersion of the Fourier transform in a continuous group of functional transformations,” Proc. Natl. Acad. Sci. USA 23, 158–164 (1937).
    [CrossRef] [PubMed]
  11. T. Alieva, V. Lopez, F. Aguillo-Lopez, L. B. Almeida, “The fractional Fourier transform in optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
    [CrossRef]
  12. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier transform as a tool for analysing beam propagation and spherical mirror resonators,” Opt. Lett. 19, 1678–1680 (1994).
    [CrossRef] [PubMed]
  13. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
    [CrossRef]
  14. P. Pellat-Finet, “Fresnel diffraction and the fractional-order Fourier transform,” Opt. Lett. 19, 1388–1390 (1994).
    [CrossRef] [PubMed]
  15. C. J. R. Sheppard, “Free-space diffraction and the fractional Fourier transform,” J. Mod. Opt. 45, 2097–2104 (1998).
    [CrossRef]
  16. D. Dragoman, “The relation between light diffraction and the fractional Fourier transform,” J. Mod. Opt. 45, 2117–2124 (1998).
    [CrossRef]
  17. W. H. Carter, “Focal shift and concept of effective Fresnel number for a Gaussian laser beam,” Appl. Opt. 21, 1989–1994 (1982).
    [CrossRef] [PubMed]
  18. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  19. A. W. Lohmann, B. H. Soffer, “Relationships between the Radon–Wigner and fractional Fourier transforms,” J. Opt. Soc. Am. A 11, 1798–1801 (1994).
    [CrossRef]
  20. L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
    [CrossRef]
  21. D. Mustard, “The fractional Fourier transform and the Wigner distribution,” J. Aust. Math. Soc. Ser. B 38, 209–219 (1996).
    [CrossRef]
  22. M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
    [CrossRef]
  23. K.-H. Brenner, A. W. Lohmann, J. Ojeda-Castañada, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
    [CrossRef]
  24. J. Bertrand, P. Bertrand, “Tomographic procedures for constructing phase space representations,” in The Physics of Phase Space, Y. S. Kim, W. W. Zachary, eds. (Springer, New York, 1987).
  25. W. D. MacMillan, Dynamics of Rigid Bodies (Dover, New York, 1960).
  26. H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1969).
  27. M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 82, 173–181 (1989).
  28. D. Mustard, “Uncertainty principles invariant under the fractional Fourier transform,” J. Aust. Math. Soc. Ser. B, 33, 180–191 (1991).
    [CrossRef]
  29. A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
    [CrossRef]
  30. H. Kogelnik, T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966).
    [CrossRef] [PubMed]
  31. S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970).
    [CrossRef]
  32. P. A. Bélanger, “Beam propagation and the ABCD ray matrices,” Opt. Lett. 16, 196–198 (1991).
    [CrossRef] [PubMed]
  33. 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]
  34. K. G. Larkin, “Efficient nonlinear algorithm for envelope detection in white light interferometry,” J. Opt. Soc. Am. A 13, 832–843 (1996).
    [CrossRef]
  35. S. C. Pohlig, “Signal duration and the Fourier transform,” Proc. IEEE 68, 629–630 (1980).
    [CrossRef]
  36. K. Petermann, “Constraints for fundamental-mode spot size for broadband dispersion-compensated single-mode fibres,” Electron. Lett. 19, 712–714 (1983).
    [CrossRef]
  37. C. Pask, “Physical interpretation of Petermann’s strange spot size for single-mode fibres,” Electron. Lett. 20, 144–145 (1984).
    [CrossRef]
  38. A. Liang, C.-C. Fan, “Mode-field radius of noncircular field single-mode fibre: new definition and application to calculation of splice loss and waveguide dispersion,” Electron. Lett. 24, 646–647 (1988).
    [CrossRef]
  39. P. Grivet, Electron Optics (Pergamon, Oxford, UK, 1965), p. 419.
  40. S. C. Fleming, “Measurement of mode field radius distribution and evaluation of mode field radius in single mode optical waveguides,” in Tests, Measurements, and Characterization of Electro-Optic Devices and Systems, S. G. Wadekar, ed., Proc. SPIE1180, 95–106 (1989).
    [CrossRef]
  41. A. Papoulis, “Apodization for optimum imaging of smooth objects,” J. Opt. Soc. Am. 62, 1423–1429 (1972).
    [CrossRef]
  42. P. Maragos, J. F. Kaiser, T. F. Quaterieri, “On amplitude and frequency demodulation using energy operators,” IEEE Trans. Signal Process. 41, 3024–3051 (1993).
    [CrossRef]
  43. P. Maragos, A. C. Bovik, “Image demodulation using multidimensional energy separation,” J. Opt. Soc. Am. A 12, 1867–1876 (1995).
    [CrossRef]
  44. R. Hamila, J. Astola, M. Alaya Cheikh, M. Gabbouj, M. Renfors, “Teager energy and the ambiguity function,” IEEE Trans. Signal Process. 47, 260–262 (1999).
    [CrossRef]
  45. C. J. R. Sheppard, Z. S. Hegedus, “Axial behaviour of pupil plane filters,” J. Opt. Soc. Am. A 5, 643–647 (1988).
    [CrossRef]

1999

R. Hamila, J. Astola, M. Alaya Cheikh, M. Gabbouj, M. Renfors, “Teager energy and the ambiguity function,” IEEE Trans. Signal Process. 47, 260–262 (1999).
[CrossRef]

1998

C. J. R. Sheppard, “Free-space diffraction and the fractional Fourier transform,” J. Mod. Opt. 45, 2097–2104 (1998).
[CrossRef]

D. Dragoman, “The relation between light diffraction and the fractional Fourier transform,” J. Mod. Opt. 45, 2117–2124 (1998).
[CrossRef]

S. De Nicola, D. Anderson, M. Lisak, “Focal shift effects in diffracted focused beams,” Pure Appl. Opt. 7, 1249–1259 (1998).
[CrossRef]

1996

D. Mustard, “The fractional Fourier transform and the Wigner distribution,” J. Aust. Math. Soc. Ser. B 38, 209–219 (1996).
[CrossRef]

K. G. Larkin, “Efficient nonlinear algorithm for envelope detection in white light interferometry,” J. Opt. Soc. Am. A 13, 832–843 (1996).
[CrossRef]

1995

1994

1993

1991

P. A. Bélanger, “Beam propagation and the ABCD ray matrices,” Opt. Lett. 16, 196–198 (1991).
[CrossRef] [PubMed]

D. Mustard, “Uncertainty principles invariant under the fractional Fourier transform,” J. Aust. Math. Soc. Ser. B, 33, 180–191 (1991).
[CrossRef]

1989

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

1988

A. Liang, C.-C. Fan, “Mode-field radius of noncircular field single-mode fibre: new definition and application to calculation of splice loss and waveguide dispersion,” Electron. Lett. 24, 646–647 (1988).
[CrossRef]

C. J. R. Sheppard, Z. S. Hegedus, “Axial behaviour of pupil plane filters,” J. Opt. Soc. Am. A 5, 643–647 (1988).
[CrossRef]

1986

1984

C. Pask, “Physical interpretation of Petermann’s strange spot size for single-mode fibres,” Electron. Lett. 20, 144–145 (1984).
[CrossRef]

Y. Li, E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,” J. Opt. Soc. Am. A 1, 801–808 (1984).
[CrossRef]

1983

K. Petermann, “Constraints for fundamental-mode spot size for broadband dispersion-compensated single-mode fibres,” Electron. Lett. 19, 712–714 (1983).
[CrossRef]

K.-H. Brenner, A. W. Lohmann, J. Ojeda-Castañada, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

1982

1981

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

1980

S. C. Pohlig, “Signal duration and the Fourier transform,” Proc. IEEE 68, 629–630 (1980).
[CrossRef]

1979

1978

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[CrossRef]

1974

1972

1970

1966

1937

E. U. Condon, “Immersion of the Fourier transform in a continuous group of functional transformations,” Proc. Natl. Acad. Sci. USA 23, 158–164 (1937).
[CrossRef] [PubMed]

Abe, S.

Aguillo-Lopez, F.

T. Alieva, V. Lopez, F. Aguillo-Lopez, L. B. Almeida, “The fractional Fourier transform in optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
[CrossRef]

Alieva, T.

T. Alieva, V. Lopez, F. Aguillo-Lopez, L. B. Almeida, “The fractional Fourier transform in optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
[CrossRef]

Almeida, L. B.

T. Alieva, V. Lopez, F. Aguillo-Lopez, L. B. Almeida, “The fractional Fourier transform in optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
[CrossRef]

L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
[CrossRef]

Anderson, D.

S. De Nicola, D. Anderson, M. Lisak, “Focal shift effects in diffracted focused beams,” Pure Appl. Opt. 7, 1249–1259 (1998).
[CrossRef]

Astola, J.

R. Hamila, J. Astola, M. Alaya Cheikh, M. Gabbouj, M. Renfors, “Teager energy and the ambiguity function,” IEEE Trans. Signal Process. 47, 260–262 (1999).
[CrossRef]

Bareket, N.

Bastiaans, M. J.

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

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[CrossRef]

Bélanger, P. A.

Bertrand, J.

J. Bertrand, P. Bertrand, “Tomographic procedures for constructing phase space representations,” in The Physics of Phase Space, Y. S. Kim, W. W. Zachary, eds. (Springer, New York, 1987).

Bertrand, P.

J. Bertrand, P. Bertrand, “Tomographic procedures for constructing phase space representations,” in The Physics of Phase Space, Y. S. Kim, W. W. Zachary, eds. (Springer, New York, 1987).

Bovik, A. C.

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).

Brenner, K.-H.

K.-H. Brenner, A. W. Lohmann, J. Ojeda-Castañada, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

Carter, W. H.

Cheikh, M. Alaya

R. Hamila, J. Astola, M. Alaya Cheikh, M. Gabbouj, M. Renfors, “Teager energy and the ambiguity function,” IEEE Trans. Signal Process. 47, 260–262 (1999).
[CrossRef]

Collins, S. A.

Condon, E. U.

E. U. Condon, “Immersion of the Fourier transform in a continuous group of functional transformations,” Proc. Natl. Acad. Sci. USA 23, 158–164 (1937).
[CrossRef] [PubMed]

De Nicola, S.

S. De Nicola, D. Anderson, M. Lisak, “Focal shift effects in diffracted focused beams,” Pure Appl. Opt. 7, 1249–1259 (1998).
[CrossRef]

Dragoman, D.

D. Dragoman, “The relation between light diffraction and the fractional Fourier transform,” J. Mod. Opt. 45, 2117–2124 (1998).
[CrossRef]

Fan, C.-C.

A. Liang, C.-C. Fan, “Mode-field radius of noncircular field single-mode fibre: new definition and application to calculation of splice loss and waveguide dispersion,” Electron. Lett. 24, 646–647 (1988).
[CrossRef]

Fleming, S. C.

S. C. Fleming, “Measurement of mode field radius distribution and evaluation of mode field radius in single mode optical waveguides,” in Tests, Measurements, and Characterization of Electro-Optic Devices and Systems, S. G. Wadekar, ed., Proc. SPIE1180, 95–106 (1989).
[CrossRef]

Gabbouj, M.

R. Hamila, J. Astola, M. Alaya Cheikh, M. Gabbouj, M. Renfors, “Teager energy and the ambiguity function,” IEEE Trans. Signal Process. 47, 260–262 (1999).
[CrossRef]

Goldstein, H.

H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1969).

Grivet, P.

P. Grivet, Electron Optics (Pergamon, Oxford, UK, 1965), p. 419.

Hamila, R.

R. Hamila, J. Astola, M. Alaya Cheikh, M. Gabbouj, M. Renfors, “Teager energy and the ambiguity function,” IEEE Trans. Signal Process. 47, 260–262 (1999).
[CrossRef]

Hegedus, Z. S.

Kaiser, J. F.

P. Maragos, J. F. Kaiser, T. F. Quaterieri, “On amplitude and frequency demodulation using energy operators,” IEEE Trans. Signal Process. 41, 3024–3051 (1993).
[CrossRef]

Kogelnik, H.

Larkin, K. G.

Li, T.

Li, Y.

Liang, A.

A. Liang, C.-C. Fan, “Mode-field radius of noncircular field single-mode fibre: new definition and application to calculation of splice loss and waveguide dispersion,” Electron. Lett. 24, 646–647 (1988).
[CrossRef]

Lisak, M.

S. De Nicola, D. Anderson, M. Lisak, “Focal shift effects in diffracted focused beams,” Pure Appl. Opt. 7, 1249–1259 (1998).
[CrossRef]

Lohmann, A. W.

Lopez, V.

T. Alieva, V. Lopez, F. Aguillo-Lopez, L. B. Almeida, “The fractional Fourier transform in optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
[CrossRef]

MacMillan, W. D.

W. D. MacMillan, Dynamics of Rigid Bodies (Dover, New York, 1960).

Maragos, P.

P. Maragos, A. C. Bovik, “Image demodulation using multidimensional energy separation,” J. Opt. Soc. Am. A 12, 1867–1876 (1995).
[CrossRef]

P. Maragos, J. F. Kaiser, T. F. Quaterieri, “On amplitude and frequency demodulation using energy operators,” IEEE Trans. Signal Process. 41, 3024–3051 (1993).
[CrossRef]

Martinez-Herrero, R.

Meijas, P. M.

Mendlovic, D.

Mustard, D.

D. Mustard, “The fractional Fourier transform and the Wigner distribution,” J. Aust. Math. Soc. Ser. B 38, 209–219 (1996).
[CrossRef]

D. Mustard, “Uncertainty principles invariant under the fractional Fourier transform,” J. Aust. Math. Soc. Ser. B, 33, 180–191 (1991).
[CrossRef]

Ojeda-Castañada, J.

K.-H. Brenner, A. W. Lohmann, J. Ojeda-Castañada, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

Ozaktas, H. M.

Papoulis, A.

Pask, C.

C. Pask, “Physical interpretation of Petermann’s strange spot size for single-mode fibres,” Electron. Lett. 20, 144–145 (1984).
[CrossRef]

Pellat-Finet, P.

Petermann, K.

K. Petermann, “Constraints for fundamental-mode spot size for broadband dispersion-compensated single-mode fibres,” Electron. Lett. 19, 712–714 (1983).
[CrossRef]

Pohlig, S. C.

S. C. Pohlig, “Signal duration and the Fourier transform,” Proc. IEEE 68, 629–630 (1980).
[CrossRef]

Quaterieri, T. F.

P. Maragos, J. F. Kaiser, T. F. Quaterieri, “On amplitude and frequency demodulation using energy operators,” IEEE Trans. Signal Process. 41, 3024–3051 (1993).
[CrossRef]

Renfors, M.

R. Hamila, J. Astola, M. Alaya Cheikh, M. Gabbouj, M. Renfors, “Teager energy and the ambiguity function,” IEEE Trans. Signal Process. 47, 260–262 (1999).
[CrossRef]

Sheppard, C. J. R.

Sheridan, J. T.

Siegman, A. E.

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
[CrossRef]

Soffer, B. H.

Teague, M. R.

Wolf, E.

Appl. Opt.

Electron. Lett.

K. Petermann, “Constraints for fundamental-mode spot size for broadband dispersion-compensated single-mode fibres,” Electron. Lett. 19, 712–714 (1983).
[CrossRef]

C. Pask, “Physical interpretation of Petermann’s strange spot size for single-mode fibres,” Electron. Lett. 20, 144–145 (1984).
[CrossRef]

A. Liang, C.-C. Fan, “Mode-field radius of noncircular field single-mode fibre: new definition and application to calculation of splice loss and waveguide dispersion,” Electron. Lett. 24, 646–647 (1988).
[CrossRef]

IEEE Trans. Signal Process.

L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42, 3084–3091 (1994).
[CrossRef]

P. Maragos, J. F. Kaiser, T. F. Quaterieri, “On amplitude and frequency demodulation using energy operators,” IEEE Trans. Signal Process. 41, 3024–3051 (1993).
[CrossRef]

R. Hamila, J. Astola, M. Alaya Cheikh, M. Gabbouj, M. Renfors, “Teager energy and the ambiguity function,” IEEE Trans. Signal Process. 47, 260–262 (1999).
[CrossRef]

J. Aust. Math. Soc. Ser. B

D. Mustard, “The fractional Fourier transform and the Wigner distribution,” J. Aust. Math. Soc. Ser. B 38, 209–219 (1996).
[CrossRef]

D. Mustard, “Uncertainty principles invariant under the fractional Fourier transform,” J. Aust. Math. Soc. Ser. B, 33, 180–191 (1991).
[CrossRef]

J. Mod. Opt.

T. Alieva, V. Lopez, F. Aguillo-Lopez, L. B. Almeida, “The fractional Fourier transform in optical propagation problems,” J. Mod. Opt. 41, 1037–1044 (1994).
[CrossRef]

C. J. R. Sheppard, “Free-space diffraction and the fractional Fourier transform,” J. Mod. Opt. 45, 2097–2104 (1998).
[CrossRef]

D. Dragoman, “The relation between light diffraction and the fractional Fourier transform,” J. Mod. Opt. 45, 2117–2124 (1998).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

M. J. Bastiaans, “The Wigner distribution function applied to optical signals and systems,” Opt. Commun. 25, 26–30 (1978).
[CrossRef]

K.-H. Brenner, A. W. Lohmann, J. Ojeda-Castañada, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

Opt. Lett.

Optik (Stuttgart)

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

Proc. IEEE

S. C. Pohlig, “Signal duration and the Fourier transform,” Proc. IEEE 68, 629–630 (1980).
[CrossRef]

Proc. Natl. Acad. Sci. USA

E. U. Condon, “Immersion of the Fourier transform in a continuous group of functional transformations,” Proc. Natl. Acad. Sci. USA 23, 158–164 (1937).
[CrossRef] [PubMed]

Pure Appl. Opt.

S. De Nicola, D. Anderson, M. Lisak, “Focal shift effects in diffracted focused beams,” Pure Appl. Opt. 7, 1249–1259 (1998).
[CrossRef]

Other

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).

J. Bertrand, P. Bertrand, “Tomographic procedures for constructing phase space representations,” in The Physics of Phase Space, Y. S. Kim, W. W. Zachary, eds. (Springer, New York, 1987).

W. D. MacMillan, Dynamics of Rigid Bodies (Dover, New York, 1960).

H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1969).

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
[CrossRef]

P. Grivet, Electron Optics (Pergamon, Oxford, UK, 1965), p. 419.

S. C. Fleming, “Measurement of mode field radius distribution and evaluation of mode field radius in single mode optical waveguides,” in Tests, Measurements, and Characterization of Electro-Optic Devices and Systems, S. G. Wadekar, ed., Proc. SPIE1180, 95–106 (1989).
[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 (2)

Fig. 1
Fig. 1

Order of fractional Fourier transformation for a slit aperture illuminated by a converging wave for different values of the Fresnel number N0.

Fig. 2
Fig. 2

Locus of the radius of gyration of the Wigner distribution function in phase space.

Equations (78)

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

U(x, z)=1λ(f+z)expikf+z+x22(f+z)-iπ4×-a+aU(x,-f)
×exp-ikx22f+ikx22(f+z)-ikxxf+zdx,
ξ=xa,
v=kx af+z,u=kz a2f(f+z).
U(v, u)=aλsin[α(z)]1/2 exp[ik(f+z)]exp-iπ4×expiv22ka sin[α(z)]×-11U(ξ,-f)exp(-ivξ)×exp-iuξ22dξ.
vkx sin[α(z)],ukz sin α0 sin[α(z)],
U(v, u)=iaλsin[α(z)]exp[ik(f+z)]expiv22ka sin[α(z)]×01U(ρ,-f)J0(vρ)expiuρ22ρ dρ,
N(z)=a2λ(f+z),
U(v, u)=N(z) exp[ik(f+z)]exp-iπ4expiv24πN(z)×-11U(ξ,-f)exp(-ivξ)×exp-iuξ22dξ.
FβU(p)=1(2π sin β)1/2×exp-iπ4+iβ2expip2 cot β2×-+U(p)expip2 cot β2-ippsinβdp.
p2 cot β=-uξ2,ppsin β=vξ.
cot β=-u,
sin β=11+u2,
p=v1+u2.
Uu(v)=2πN(z)(1+u2)1/4×exp[ik(f+z)]expi2cot-1 u×expiv2212πN(z)+u1+u2×F-cot-1 uUv1+u2.
Uu(v)=2πN0-u(1+u2)1/4×exp[ik(f+z)]expi2cot-1 u×expiv2(1+2πN0u)2(2πN0-u)(1+u2)×F-cot-1 uUv1+u2.
zf=-11+4π2N02
Iu(v)=(2πN0-u)(1+u2)1/2×F-cot-1 uUv1+u22.
Uu(v)=2πN0(1+u2)1/4×exp[ik(f+z)]expi2cot-1 u×expiv2u2(1+u2)F-cot-1 uUv1+u2.
U˜(p)=12π-U(ξ)exp(-ipξ)dξ.
W(ξ, p)=12π-U(ξ+ξ/2)U*(ξ-ξ/2)×exp(-ipξ)dξ=12π-U˜(p+p/2)U˜*(p-p/2)×exp(ipξ)dp.
12π-W(ξ, p)dp=|U(ξ)|2,
12π-W(ξ, p)dξ=|U˜(p)|2,
12π- W(ξ, p)dpdξ=E,
mξ=1E|U(ξ)|2ξ dξ,
mp=1E|U˜(p)|2p dp,
mξξ=1E|U(ξ)|2(ξ-mξ)2 dξ,
mpp=1E|U˜(p)|2(p-mp)2 dp.
mξp=mpξ=12πEW(ξ, p)(ξ-mξ)(p-mp)dpdξ.
mξξ(β)=mξξ cos2 β+mpp sin2 β-2mpξ sin β cos β.
A(ξ, p)=12π-U(ξ+ξ/2)U*(ξ-ξ/2)×exp(-ipξ)dξ=12π-U˜(p+p/2)U˜*(p-p/2)×exp(ipξ)dp.
A(ξ, 0)=12π|U˜(p)|2 exp(ipξ)dp,
A(0, p)=12π|U(ξ)|2 exp(-ipξ)dξ,
A(0, 0)=E2π.
mξ=-2πEA(0, p)pp=0,
mp=-2πEA(ξ, 0)ξξ=0
mξξ=-2πE2A(0, p)p2p=0-mξ2,
mpp=-2πE2A(ξ, 0)ξ2ξ=0-mp2,
mξp=-2πE2A(ξ, p)ξpξ=0, p=0-mξmp.
zf=-11+2πN0 tan β
v=xa(cos β+2πN0 sin β)sin β,
Iβ(x)=cos β+2πN0 sin βsin2 β×FβUxa(cos β+2πN0 sin β)2.
w2=a2 mξξ cos2 β+mpp sin2 β-2mξp sin β cos β(cos β+2πN0 sin β)2.
cot β=mpp+2πN0mξp2πN0mξξ+mξp,
zf=-mpp+2πN0mξp4π2N02mξξ+mpp+4πN0mξp.
w2=a2 mξξmpp-mξp24π2N02mξξ+mpp+4πN0mξp.
θ2=4π2N02mξξ+mpp+4πN0mξp4π2N02f2.
4mξξmpp1,
M2=2(mξξmpp-mξp2)1/2,
M4=4 det(M),
M=mξξmξpmξpmpp.
M=RTMR,
R=cos βsin β-sin βcos β.
mξ=mξ cos β+mp sin β,
mp=mp cos β-mξ sin β,
mpp=mξξ sin2 β+mpp cos2 β+2mξp sin β cos β,
mξp=(mξξ-mpp)sin β cos β+mξp cos 2β.
x0=a mξ cos β+mp sin βcos β+2πN0 sin β.
mW=mξξ+mpp,
m1,2=12(mW±mW2-M4).
m1m2=M4/4.
mp=i2EdU(ξ)dξU*(ξ)-U(ξ) dU*(ξ)dξdξ.
U(ξ)=a(ξ)exp[iθ(ξ)],
mp=-1Ea2(ξ) dθ(ξ)dξdξ,
mpp=1Eddξ[U(ξ)exp(impξ)]2dξ.
mpp=1Eda(ξ)dξ2+a2(ξ)dθ(ξ)dξ+mp2dξ.
mpp=1Ea(ξ) da(ξ)dξ-11--11a(ξ) d2a(ξ)dξ2dξ+-11a2(ξ)dθ(ξ)dξ2 dξ-mp2.
mppπ24,
U(ξ)=cos πξ2,
mξξ=π2-63π2,
M2=π2-631/2=1.136.
mpp=14E2dU(ξ)dξ2-d2U(ξ)dξ2U*(ξ)+U(ξ) d2U*(ξ)dξ2dξ-mp2,
mpp=12EΨ(U(ξ))dξ-mp2,
Ψ(U(ξ))=dU(ξ)dξ2-12d2U(ξ)dξ2U*(ξ)+U(ξ) d2U*(ξ)dξ2.
mpp=12Eda(ξ)dξ2-a(ξ) d2a(ξ)dξ2+2a2(ξ)dθ(ξ)dξ2dξ-mp2.
-a(ξ) d2a(ξ)dξ2+da(ξ)dξ2dξ=a(ξ) da(ξ)dξ-.
mξp=-i2EU(ξ) dU*(ξ)dξ-U*(ξ) dU(ξ)dξξ dξ-mpmξ,
mξp=-1Ea2(ξ) dθ(ξ)dξξ dξ-mpmξ,

Metrics