Abstract

A Fourier-optical analysis of the transformation of ultrashort light pulses by lenses is given. Inserting the material dispersion up to second order, we find a coupling between the temporal, spectral, and spatial properties of the light pulse. In general, this coupling leads to a drastic increase in pulse duration and width of the spatial intensity distribution in the focal plane of the lens, which can be avoided with the use of achromatic lens doublets. The results are discussed for Gaussian-shaped input pulses.

© 1992 Optical Society of America

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References

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  1. W. Kaiser, ed., Ultrashort Laser Pulses and Applications (Springer, New York, 1988).
  2. T. Wilson and C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Pergamon, London, 1984).
  3. A. Andreoni, R. Cubeddu, S. De Silvestri, P. Laporta, and O. Svelto, in Lasers and Applications, W. O. U. Guimaraes, ed. (Springer, Berlin, 1981).
  4. F. Docchio, R. Ramponi, C. A. Sacchi, G. Bottiroli, and I. Freitas, J. Microsc. 134, 151 (1984).
    [Crossref] [PubMed]
  5. H. Schneckenburger, in Proceedings of the 16th International Congress on High Speed Photography and Photonics, M. L. Andre and M. Hugenschmidt, eds., Proc. Soc. Photo-Opt. Instrum. Eng.491, 363 (1984).
    [Crossref]
  6. T. Minami, M. Kawahigashi, Y. Sakai, K. Shimamoto, and S. Hirayama, J. Lumin. 35, 217 (1986).
    [Crossref]
  7. J. G. Fujimoto, S. De Silvestri, E. P. Ippen, C. A. Puliafito, and R. Margolis, Opt. Lett. 11, 150 (1986).
    [Crossref]
  8. S. Kinoshita, T. Seki, T. F. Liu, and T. Kushida, J. Photochem. Photobiol. B 2, 195 (1988).
    [Crossref] [PubMed]
  9. I. Bugiel, K. König, and H. Wabnitz, Laser Life Sci. 3, 47 (1989).
  10. A. Olsson, D. J. Erskine, E. Y. Xu, and C. L. Tang, Appl. Phys. Lett. 41, 659 (1982).
    [Crossref]
  11. M. C. Downer, R. I. Fork, and C. V. Shank, J. Opt. Soc. Am. B 2, 595 (1985).
    [Crossref]
  12. L. M. Smith, D. R. Wake, J. P. Wolfe, D. Levi, M. V. Klein, J. Klem, T. Henderson, and H. Morkoc, Phys. Rev. B 38, 5788 (1988).
    [Crossref]
  13. H. Bergner, T. Damm, U. Stamm, and K.-P. Stolberg, Int. J. Optoelectron. 4, 583 (1989).
  14. H. Bergner, T. Damm, U. Stamm, M. Müller, and K.-P. Stolberg, in Frontiers of Electro-Optics, presented at the Electro-Optics and Lasers 90 Conference, Birmingham, Ala., 1990.
  15. T. A. Louis, G. Ripamonti, and A. Lacaita, Rev. Sci. Instrum. 61, 11 (1990).
    [Crossref]
  16. H. Azechi, S. Oda, T. Sasaki, T. Yamanaka, and C. Yamanaka, in Proceedings of the 13th International Congress on High Speed Photography and Photonics, S. Hyodo, ed., Proc. Soc. Photo-Opt. Instrum. Eng.189, 536 (1978).
    [Crossref]
  17. J. A. Valdmanis and G. Mourou, IEEE J. Quantum Electron. QE-22, 69 (1986).
    [Crossref]
  18. Z. Bor, J. Mod. Opt. 35, 1907 (1988).
    [Crossref]
  19. M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1989).
  20. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1988).
  21. W. Rudolph and B. Wilhelmi, Light Pulse Compression (Harwood, Zurich, 1989).
  22. Z. Bor and B. Racz, Appl. Opt. 24, 3440 (1985).
    [Crossref] [PubMed]
  23. The authors acknowledge that this point was suggested during the discussion of the paper of Z. Bor at CLEO’90, Anaheim, Calif.
  24. T. Wilson, Oxford University, Oxford, UK (personal communication).

1990 (1)

T. A. Louis, G. Ripamonti, and A. Lacaita, Rev. Sci. Instrum. 61, 11 (1990).
[Crossref]

1989 (2)

H. Bergner, T. Damm, U. Stamm, and K.-P. Stolberg, Int. J. Optoelectron. 4, 583 (1989).

I. Bugiel, K. König, and H. Wabnitz, Laser Life Sci. 3, 47 (1989).

1988 (3)

L. M. Smith, D. R. Wake, J. P. Wolfe, D. Levi, M. V. Klein, J. Klem, T. Henderson, and H. Morkoc, Phys. Rev. B 38, 5788 (1988).
[Crossref]

S. Kinoshita, T. Seki, T. F. Liu, and T. Kushida, J. Photochem. Photobiol. B 2, 195 (1988).
[Crossref] [PubMed]

Z. Bor, J. Mod. Opt. 35, 1907 (1988).
[Crossref]

1986 (3)

J. A. Valdmanis and G. Mourou, IEEE J. Quantum Electron. QE-22, 69 (1986).
[Crossref]

T. Minami, M. Kawahigashi, Y. Sakai, K. Shimamoto, and S. Hirayama, J. Lumin. 35, 217 (1986).
[Crossref]

J. G. Fujimoto, S. De Silvestri, E. P. Ippen, C. A. Puliafito, and R. Margolis, Opt. Lett. 11, 150 (1986).
[Crossref]

1985 (2)

1984 (1)

F. Docchio, R. Ramponi, C. A. Sacchi, G. Bottiroli, and I. Freitas, J. Microsc. 134, 151 (1984).
[Crossref] [PubMed]

1982 (1)

A. Olsson, D. J. Erskine, E. Y. Xu, and C. L. Tang, Appl. Phys. Lett. 41, 659 (1982).
[Crossref]

Andreoni, A.

A. Andreoni, R. Cubeddu, S. De Silvestri, P. Laporta, and O. Svelto, in Lasers and Applications, W. O. U. Guimaraes, ed. (Springer, Berlin, 1981).

Azechi, H.

H. Azechi, S. Oda, T. Sasaki, T. Yamanaka, and C. Yamanaka, in Proceedings of the 13th International Congress on High Speed Photography and Photonics, S. Hyodo, ed., Proc. Soc. Photo-Opt. Instrum. Eng.189, 536 (1978).
[Crossref]

Bergner, H.

H. Bergner, T. Damm, U. Stamm, and K.-P. Stolberg, Int. J. Optoelectron. 4, 583 (1989).

H. Bergner, T. Damm, U. Stamm, M. Müller, and K.-P. Stolberg, in Frontiers of Electro-Optics, presented at the Electro-Optics and Lasers 90 Conference, Birmingham, Ala., 1990.

Bor, Z.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1989).

Bottiroli, G.

F. Docchio, R. Ramponi, C. A. Sacchi, G. Bottiroli, and I. Freitas, J. Microsc. 134, 151 (1984).
[Crossref] [PubMed]

Bugiel, I.

I. Bugiel, K. König, and H. Wabnitz, Laser Life Sci. 3, 47 (1989).

Cubeddu, R.

A. Andreoni, R. Cubeddu, S. De Silvestri, P. Laporta, and O. Svelto, in Lasers and Applications, W. O. U. Guimaraes, ed. (Springer, Berlin, 1981).

Damm, T.

H. Bergner, T. Damm, U. Stamm, and K.-P. Stolberg, Int. J. Optoelectron. 4, 583 (1989).

H. Bergner, T. Damm, U. Stamm, M. Müller, and K.-P. Stolberg, in Frontiers of Electro-Optics, presented at the Electro-Optics and Lasers 90 Conference, Birmingham, Ala., 1990.

De Silvestri, S.

J. G. Fujimoto, S. De Silvestri, E. P. Ippen, C. A. Puliafito, and R. Margolis, Opt. Lett. 11, 150 (1986).
[Crossref]

A. Andreoni, R. Cubeddu, S. De Silvestri, P. Laporta, and O. Svelto, in Lasers and Applications, W. O. U. Guimaraes, ed. (Springer, Berlin, 1981).

Docchio, F.

F. Docchio, R. Ramponi, C. A. Sacchi, G. Bottiroli, and I. Freitas, J. Microsc. 134, 151 (1984).
[Crossref] [PubMed]

Downer, M. C.

Erskine, D. J.

A. Olsson, D. J. Erskine, E. Y. Xu, and C. L. Tang, Appl. Phys. Lett. 41, 659 (1982).
[Crossref]

Fork, R. I.

Freitas, I.

F. Docchio, R. Ramponi, C. A. Sacchi, G. Bottiroli, and I. Freitas, J. Microsc. 134, 151 (1984).
[Crossref] [PubMed]

Fujimoto, J. G.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1988).

Henderson, T.

L. M. Smith, D. R. Wake, J. P. Wolfe, D. Levi, M. V. Klein, J. Klem, T. Henderson, and H. Morkoc, Phys. Rev. B 38, 5788 (1988).
[Crossref]

Hirayama, S.

T. Minami, M. Kawahigashi, Y. Sakai, K. Shimamoto, and S. Hirayama, J. Lumin. 35, 217 (1986).
[Crossref]

Ippen, E. P.

Kawahigashi, M.

T. Minami, M. Kawahigashi, Y. Sakai, K. Shimamoto, and S. Hirayama, J. Lumin. 35, 217 (1986).
[Crossref]

Kinoshita, S.

S. Kinoshita, T. Seki, T. F. Liu, and T. Kushida, J. Photochem. Photobiol. B 2, 195 (1988).
[Crossref] [PubMed]

Klein, M. V.

L. M. Smith, D. R. Wake, J. P. Wolfe, D. Levi, M. V. Klein, J. Klem, T. Henderson, and H. Morkoc, Phys. Rev. B 38, 5788 (1988).
[Crossref]

Klem, J.

L. M. Smith, D. R. Wake, J. P. Wolfe, D. Levi, M. V. Klein, J. Klem, T. Henderson, and H. Morkoc, Phys. Rev. B 38, 5788 (1988).
[Crossref]

König, K.

I. Bugiel, K. König, and H. Wabnitz, Laser Life Sci. 3, 47 (1989).

Kushida, T.

S. Kinoshita, T. Seki, T. F. Liu, and T. Kushida, J. Photochem. Photobiol. B 2, 195 (1988).
[Crossref] [PubMed]

Lacaita, A.

T. A. Louis, G. Ripamonti, and A. Lacaita, Rev. Sci. Instrum. 61, 11 (1990).
[Crossref]

Laporta, P.

A. Andreoni, R. Cubeddu, S. De Silvestri, P. Laporta, and O. Svelto, in Lasers and Applications, W. O. U. Guimaraes, ed. (Springer, Berlin, 1981).

Levi, D.

L. M. Smith, D. R. Wake, J. P. Wolfe, D. Levi, M. V. Klein, J. Klem, T. Henderson, and H. Morkoc, Phys. Rev. B 38, 5788 (1988).
[Crossref]

Liu, T. F.

S. Kinoshita, T. Seki, T. F. Liu, and T. Kushida, J. Photochem. Photobiol. B 2, 195 (1988).
[Crossref] [PubMed]

Louis, T. A.

T. A. Louis, G. Ripamonti, and A. Lacaita, Rev. Sci. Instrum. 61, 11 (1990).
[Crossref]

Margolis, R.

Minami, T.

T. Minami, M. Kawahigashi, Y. Sakai, K. Shimamoto, and S. Hirayama, J. Lumin. 35, 217 (1986).
[Crossref]

Morkoc, H.

L. M. Smith, D. R. Wake, J. P. Wolfe, D. Levi, M. V. Klein, J. Klem, T. Henderson, and H. Morkoc, Phys. Rev. B 38, 5788 (1988).
[Crossref]

Mourou, G.

J. A. Valdmanis and G. Mourou, IEEE J. Quantum Electron. QE-22, 69 (1986).
[Crossref]

Müller, M.

H. Bergner, T. Damm, U. Stamm, M. Müller, and K.-P. Stolberg, in Frontiers of Electro-Optics, presented at the Electro-Optics and Lasers 90 Conference, Birmingham, Ala., 1990.

Oda, S.

H. Azechi, S. Oda, T. Sasaki, T. Yamanaka, and C. Yamanaka, in Proceedings of the 13th International Congress on High Speed Photography and Photonics, S. Hyodo, ed., Proc. Soc. Photo-Opt. Instrum. Eng.189, 536 (1978).
[Crossref]

Olsson, A.

A. Olsson, D. J. Erskine, E. Y. Xu, and C. L. Tang, Appl. Phys. Lett. 41, 659 (1982).
[Crossref]

Puliafito, C. A.

Racz, B.

Ramponi, R.

F. Docchio, R. Ramponi, C. A. Sacchi, G. Bottiroli, and I. Freitas, J. Microsc. 134, 151 (1984).
[Crossref] [PubMed]

Ripamonti, G.

T. A. Louis, G. Ripamonti, and A. Lacaita, Rev. Sci. Instrum. 61, 11 (1990).
[Crossref]

Rudolph, W.

W. Rudolph and B. Wilhelmi, Light Pulse Compression (Harwood, Zurich, 1989).

Sacchi, C. A.

F. Docchio, R. Ramponi, C. A. Sacchi, G. Bottiroli, and I. Freitas, J. Microsc. 134, 151 (1984).
[Crossref] [PubMed]

Sakai, Y.

T. Minami, M. Kawahigashi, Y. Sakai, K. Shimamoto, and S. Hirayama, J. Lumin. 35, 217 (1986).
[Crossref]

Sasaki, T.

H. Azechi, S. Oda, T. Sasaki, T. Yamanaka, and C. Yamanaka, in Proceedings of the 13th International Congress on High Speed Photography and Photonics, S. Hyodo, ed., Proc. Soc. Photo-Opt. Instrum. Eng.189, 536 (1978).
[Crossref]

Schneckenburger, H.

H. Schneckenburger, in Proceedings of the 16th International Congress on High Speed Photography and Photonics, M. L. Andre and M. Hugenschmidt, eds., Proc. Soc. Photo-Opt. Instrum. Eng.491, 363 (1984).
[Crossref]

Seki, T.

S. Kinoshita, T. Seki, T. F. Liu, and T. Kushida, J. Photochem. Photobiol. B 2, 195 (1988).
[Crossref] [PubMed]

Shank, C. V.

Sheppard, C.

T. Wilson and C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Pergamon, London, 1984).

Shimamoto, K.

T. Minami, M. Kawahigashi, Y. Sakai, K. Shimamoto, and S. Hirayama, J. Lumin. 35, 217 (1986).
[Crossref]

Smith, L. M.

L. M. Smith, D. R. Wake, J. P. Wolfe, D. Levi, M. V. Klein, J. Klem, T. Henderson, and H. Morkoc, Phys. Rev. B 38, 5788 (1988).
[Crossref]

Stamm, U.

H. Bergner, T. Damm, U. Stamm, and K.-P. Stolberg, Int. J. Optoelectron. 4, 583 (1989).

H. Bergner, T. Damm, U. Stamm, M. Müller, and K.-P. Stolberg, in Frontiers of Electro-Optics, presented at the Electro-Optics and Lasers 90 Conference, Birmingham, Ala., 1990.

Stolberg, K.-P.

H. Bergner, T. Damm, U. Stamm, and K.-P. Stolberg, Int. J. Optoelectron. 4, 583 (1989).

H. Bergner, T. Damm, U. Stamm, M. Müller, and K.-P. Stolberg, in Frontiers of Electro-Optics, presented at the Electro-Optics and Lasers 90 Conference, Birmingham, Ala., 1990.

Svelto, O.

A. Andreoni, R. Cubeddu, S. De Silvestri, P. Laporta, and O. Svelto, in Lasers and Applications, W. O. U. Guimaraes, ed. (Springer, Berlin, 1981).

Tang, C. L.

A. Olsson, D. J. Erskine, E. Y. Xu, and C. L. Tang, Appl. Phys. Lett. 41, 659 (1982).
[Crossref]

Valdmanis, J. A.

J. A. Valdmanis and G. Mourou, IEEE J. Quantum Electron. QE-22, 69 (1986).
[Crossref]

Wabnitz, H.

I. Bugiel, K. König, and H. Wabnitz, Laser Life Sci. 3, 47 (1989).

Wake, D. R.

L. M. Smith, D. R. Wake, J. P. Wolfe, D. Levi, M. V. Klein, J. Klem, T. Henderson, and H. Morkoc, Phys. Rev. B 38, 5788 (1988).
[Crossref]

Wilhelmi, B.

W. Rudolph and B. Wilhelmi, Light Pulse Compression (Harwood, Zurich, 1989).

Wilson, T.

T. Wilson, Oxford University, Oxford, UK (personal communication).

T. Wilson and C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Pergamon, London, 1984).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1989).

Wolfe, J. P.

L. M. Smith, D. R. Wake, J. P. Wolfe, D. Levi, M. V. Klein, J. Klem, T. Henderson, and H. Morkoc, Phys. Rev. B 38, 5788 (1988).
[Crossref]

Xu, E. Y.

A. Olsson, D. J. Erskine, E. Y. Xu, and C. L. Tang, Appl. Phys. Lett. 41, 659 (1982).
[Crossref]

Yamanaka, C.

H. Azechi, S. Oda, T. Sasaki, T. Yamanaka, and C. Yamanaka, in Proceedings of the 13th International Congress on High Speed Photography and Photonics, S. Hyodo, ed., Proc. Soc. Photo-Opt. Instrum. Eng.189, 536 (1978).
[Crossref]

Yamanaka, T.

H. Azechi, S. Oda, T. Sasaki, T. Yamanaka, and C. Yamanaka, in Proceedings of the 13th International Congress on High Speed Photography and Photonics, S. Hyodo, ed., Proc. Soc. Photo-Opt. Instrum. Eng.189, 536 (1978).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

A. Olsson, D. J. Erskine, E. Y. Xu, and C. L. Tang, Appl. Phys. Lett. 41, 659 (1982).
[Crossref]

IEEE J. Quantum Electron. (1)

J. A. Valdmanis and G. Mourou, IEEE J. Quantum Electron. QE-22, 69 (1986).
[Crossref]

Int. J. Optoelectron. (1)

H. Bergner, T. Damm, U. Stamm, and K.-P. Stolberg, Int. J. Optoelectron. 4, 583 (1989).

J. Lumin. (1)

T. Minami, M. Kawahigashi, Y. Sakai, K. Shimamoto, and S. Hirayama, J. Lumin. 35, 217 (1986).
[Crossref]

J. Microsc. (1)

F. Docchio, R. Ramponi, C. A. Sacchi, G. Bottiroli, and I. Freitas, J. Microsc. 134, 151 (1984).
[Crossref] [PubMed]

J. Mod. Opt. (1)

Z. Bor, J. Mod. Opt. 35, 1907 (1988).
[Crossref]

J. Opt. Soc. Am. B (1)

J. Photochem. Photobiol. B (1)

S. Kinoshita, T. Seki, T. F. Liu, and T. Kushida, J. Photochem. Photobiol. B 2, 195 (1988).
[Crossref] [PubMed]

Laser Life Sci. (1)

I. Bugiel, K. König, and H. Wabnitz, Laser Life Sci. 3, 47 (1989).

Opt. Lett. (1)

Phys. Rev. B (1)

L. M. Smith, D. R. Wake, J. P. Wolfe, D. Levi, M. V. Klein, J. Klem, T. Henderson, and H. Morkoc, Phys. Rev. B 38, 5788 (1988).
[Crossref]

Rev. Sci. Instrum. (1)

T. A. Louis, G. Ripamonti, and A. Lacaita, Rev. Sci. Instrum. 61, 11 (1990).
[Crossref]

Other (11)

H. Azechi, S. Oda, T. Sasaki, T. Yamanaka, and C. Yamanaka, in Proceedings of the 13th International Congress on High Speed Photography and Photonics, S. Hyodo, ed., Proc. Soc. Photo-Opt. Instrum. Eng.189, 536 (1978).
[Crossref]

The authors acknowledge that this point was suggested during the discussion of the paper of Z. Bor at CLEO’90, Anaheim, Calif.

T. Wilson, Oxford University, Oxford, UK (personal communication).

H. Bergner, T. Damm, U. Stamm, M. Müller, and K.-P. Stolberg, in Frontiers of Electro-Optics, presented at the Electro-Optics and Lasers 90 Conference, Birmingham, Ala., 1990.

M. Born and E. Wolf, Principles of Optics (Pergamon, London, 1989).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1988).

W. Rudolph and B. Wilhelmi, Light Pulse Compression (Harwood, Zurich, 1989).

H. Schneckenburger, in Proceedings of the 16th International Congress on High Speed Photography and Photonics, M. L. Andre and M. Hugenschmidt, eds., Proc. Soc. Photo-Opt. Instrum. Eng.491, 363 (1984).
[Crossref]

W. Kaiser, ed., Ultrashort Laser Pulses and Applications (Springer, New York, 1988).

T. Wilson and C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Pergamon, London, 1984).

A. Andreoni, R. Cubeddu, S. De Silvestri, P. Laporta, and O. Svelto, in Lasers and Applications, W. O. U. Guimaraes, ed. (Springer, Berlin, 1981).

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Figures (8)

Fig. 1
Fig. 1

Geometry of (a) the Fourier optical treatment and (b) the lens parameters.

Fig. 2
Fig. 2

Intensity distribution I(υ; t) in the focal plane for (a) τ/T = 0, (b) τ/T = 5, and (c) τ/T = 20.

Fig. 3
Fig. 3

Spatially integrated intensity distribution versus normalized time for various parameters τ/T.

Fig. 4
Fig. 4

Spatial FWHM and temporal pulse width of the intensity distribution versus τ/T without GVD (solid curves), with GVD for bandwidth-limited incident pulses (lens 3 of Table 1) (dotted curves), and with GVD for optimum chirped pulses (for the same lens) (dashed curves).

Fig. 5
Fig. 5

Temporally integrated intensity distribution versus the optical coordinate υ for various parameters τ/T. (Note that this distribution is symmetrical with respect to υ = 0.)

Fig. 6
Fig. 6

Total energy contained in a circle about the optical axis with radius υ for (a) various parameters τ/T and (b) the optical coordinate υ50 versus τ/T without GVD (solid curve), with GVD for bandwidth-limited pulses (lens 3 of Table 1 (dotted curve), and with GVD for optimum chirped pulses (for the same lens) (dashed curve).

Fig. 7
Fig. 7

I(υ; t) for a bandwidth-limited incident pulse in the focal plane of lens 3 (Table 1) with τ/T = 20.

Fig. 8
Fig. 8

Schematic pulse transformation by a thin lens (a) without considering GVD and (b) with GVD.

Tables (1)

Tables Icon

Table 1 Dispersion Parameters of Various Lenses (BK7 Glass) at 620 nm

Equations (40)

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

[ 2 + k 2 ( ω ) ] U ( r , ω ) = 0 .
U ( x 2 , y 2 , z ; ω ) d x 1 d y 1 P ( x 1 , y 1 ) A ( ω ) Θ ( x 1 , y 1 ) × exp { j k a 2 z [ ( x 2 x 1 ) 2 + ( y 2 y 1 ) 2 ] } .
Θ ( x 1 , y 1 ) = exp ( j k l d ) × exp [ j ( k l k a ) x 1 2 + y 1 2 2 ( 1 R 1 1 R 2 ) ] ,
k l = ω c n ( ω ) k 0 n 0 [ 1 + a 1 Δ ω + a 2 ( Δ ω ) 2 ] ,
a 1 = 1 ω 0 + 1 n 0 d n d ω 0 | ω = ω 0 , a 2 = 1 ω 0 n 0 d n d ω 0 | ω = ω 0 + 1 2 n 0 d 2 n d ω 2 | ω = ω 0 , k a = ω c = k 0 ( 1 + Δ ω ω 0 ) , k l k a = ω c [ n ( ω ) 1 ] k 0 ( n 0 1 ) [ 1 + b 1 Δ ω + b 2 ( Δ ω ) 2 ] .
1 f 0 = ( n 0 1 ) ( 1 R 1 1 R 2 ) .
U ( x 2 , y 2 , z ; Δ ω ) A ( Δ ω ) exp [ j k 0 n 0 d Δ ω ( a 1 + a 2 Δ ω ) ] × exp [ j k 0 2 f 0 ( x 2 2 + y 2 2 ) ( 1 + Δ ω ω 0 ) ] × d x 1 d y 1 P ( x 1 , y 1 ) exp [ j k 0 2 ( x 1 2 + y 1 2 ) ( 1 f 0 1 z ) ] × exp [ j k 0 f 0 ( x 1 x 2 + y 1 y 2 ) ( 1 + Δ ω ω 0 ) ] × exp [ j k 0 2 f 0 ( x 1 2 + y 1 2 ) Δ ω ( b 1 f 0 z ω 0 + b 2 Δ ω ) ] .
P ( x 1 , y 1 ) = { 1 , if x 1 2 + y 1 2 = r 1 2 = ( r a ) 2 , r [ 0 , 1 ] 0 , else
U ( υ , u ; Δ ω ) A ( Δ ω ) exp [ j k 0 n 0 d Δ ω ( a 1 + a 2 Δ ω ) ] × exp [ j υ 2 4 N ( 1 + Δ ω ω 0 ) ] × 0 1 d r r J 0 [ r υ ( 1 + Δ ω ω 0 ) ] exp ( j u 2 r 2 ) × exp [ j k 0 2 f 0 ( a r ) 2 Δ ω ( b 1 f 0 z ω 0 + b 2 Δ ω ) ] ,
U ( υ , u ; t ) d ( Δ ω ) exp [ j ( Δ ω ) t ] U ( υ , u ; Δ ω ) ,
U ( υ , u ; t ) d ( Δ ω ) A ( Δ ω ) exp [ j k 0 n 0 d a 2 ( Δ ω ) 2 ] × 0 1 d r r J 0 [ r υ ( 1 + Δ ω ω 0 ) ] exp ( j u 2 r 2 ) × exp { j Δ ω [ t k 0 n 0 d a 1 + ( a r ) 2 k 0 2 f 0 ( b 1 f 0 z ω 0 + b 2 Δ ω ) ] } .
U ( υ , u ; t ) d ( Δ ω ) A ( Δ ω ) 0 1 d r r J 0 [ r υ ( 1 + Δ ω ω 0 ) ] × exp ( j u 2 r 2 ) exp { j ( Δ ω ) 2 [ δ δ r 2 ] } × exp { j Δ ω [ t τ + τ ( u ) r 2 ] } ,
δ = a 2 k 0 2 f 0 b 2 = a 2 k 0 2 f 0 ( n 0 1 ) ( 1 ω 0 d n d ω | ω = ω 0 + 1 2 d 2 n d ω 2 | ω = ω 0 ) , δ = k 0 n 0 d a 2 = k 2 d ( 1 ω 0 d n d ω | ω = ω 0 + 1 2 d 2 n d ω 2 | ω = ω 0 ) , τ ( u ) = a 2 k 0 2 f 0 ( b 1 f 0 z ω 0 ) = a 2 k 0 2 f 0 ( n 0 1 ) d n d ω | ω = ω 0 + u 2 ω 0 , τ = k 0 n 0 d a 1 = k 0 d ( n 0 ω 0 + d n d ω | ω = ω 0 ) .
Δ ω δ r 2 τ r 2 = Δ ω ω 0 + Δ ω 2 d 2 n d ω 2 | ω = ω 0 ( d n d ω | ω = ω 0 ) 1 ,
d 2 n d ω 2 | ω = ω 0 = C ( 2 ) 1 ω 0 d n d ω | ω = ω 0 ,
Δ ω δ r 2 τ r 2 < 2 Δ ω ω 0 .
A ( t ) = exp [ ( t / T ) 2 ] ,
m = | 0 1 d r r J 0 [ υ r ( 1 + Δ ω ω 0 ) ] 0 1 d r r J 0 ( υ r ) | υ 0 1 d r r J 0 ( υ r ) υ .
U ( υ , u ; t ) 0 1 d r r J 0 ( r υ ) exp ( j u 2 r 2 ) [ 1 + j δ ( r ; T ) 1 + j δ 2 ( r ; T ) ] 1 / 2 × exp { ( t + τ τ 2 ) 2 T 2 [ 1 + δ 2 ( r ; T ) ] [ 1 + j δ ( r ; T ) ] } ,
δ ( r ; T ) = 4 ( δ δ r 2 ) / T 2 .
U ( υ ; t ) 0 1 d r r J 0 ( r υ ) exp [ ( t T + r 2 τ T ) 2 ] .
I ( t ) 0 d υ υ | U ( υ ; t ) | 2 ,
I ( υ ) d t | U ( υ ; t ) | 2 .
L ( υ ) = 1 2 I 0 0 υ d υ υ I ( υ ) ,
δ ( r ; T ) T 2 = 4 ( δ δ r 2 ) = 4 k 0 { [ d a 2 r 2 2 f 0 ( n 0 1 ) ] × ( 1 ω 0 d n d ω | ω = ω 0 + 1 2 d 2 n d ω 2 | ω = ω 0 ) } , δ ( r ; T ) T 2 6.4 d c d n d ω | ω = ω 0 6.4 τ ω 0 r 2 .
A ( t ) = exp ( { t T [ 1 + ( δ ) 2 ] 1 / 2 } 2 ( 1 + j δ ) ) ,
δ ( r ; T ) = 4 ( δ δ r 2 ) / T 2 δ .
δ = 4 ( δ δ ) / T 2 ,
d d ω 1 f | ω = ω 0 = 0 .
Θ ( x 1 , y 1 ) = exp { j [ k l ( 1 ) d ( 1 ) + k l ( 2 ) d ( 2 ) ] } × exp { j ( x 1 2 + y 1 2 ) 2 [ k l ( 1 ) k a R 1 k l ( 2 ) k a R 2 ] } ,
1 f 0 = n 0 ( 1 ) 1 R 1 n 0 ( 2 ) 1 R 2 .
1 f ( ω ) = n ( 1 ) ( ω ) 1 R 1 n ( 2 ) ( ω ) 1 R 2
1 R 1 d n ( 1 ) d ω | ω = ω 0 = 1 R 2 d n ( 2 ) d ω | ω = ω 0 .
Θ ( x 1 , y 1 ) exp [ j k 0 ( x 1 2 + y 1 2 ) f 0 2 ( 1 + Δ ω ω 0 ) ] × exp { j k 0 ( x 1 2 + y 1 2 ) × 1 4 [ 1 R 1 d 2 n ( 1 ) d ω 2 | ω = ω 0 1 R 2 d 2 n ( 2 ) d ω 2 | ω = ω 0 ] ( Δ ω ) 2 } × exp ( j k 0 { n 0 ( 1 ) d ( 1 ) + n 0 ( 2 ) d ( 2 ) + [ n 0 ( 1 ) d ( 1 ) a 1 ( 1 ) + n 0 ( 2 ) d ( 2 ) a ( 2 ) ] Δ ω + [ n 0 ( 1 ) d ( 1 ) a 2 ( 1 ) + n 0 ( 2 ) d ( 2 ) a 2 ( 2 ) ] ( Δ ω ) 2 } ) .
Θ ( r 1 ) exp [ j r 2 δ ¯ ( Δ ω ) 2 ] exp ( j { a 2 r 2 k 0 2 f 0 ( 1 + Δ ω ω 0 ) [ τ / ( 1 ) + τ / ( 2 ) ] Δ ω [ δ / ( 1 ) + δ / ( 2 ) ] ( Δ ω ) 2 } ) ,
δ ¯ = a 2 k 0 4 [ 1 R 1 d 2 n ( 1 ) d ω 2 | ω = ω 0 1 R 2 d 2 n ( 2 ) d ω 2 | ω = ω 0 ] .
δ ¯ = C ( 2 ) a 2 4 c [ 1 R 1 d n ( 1 ) d ω | ω = ω 0 1 R 2 d n ( 2 ) d ω | ω = ω 0 ] ,
δ ¯ = 0
δ = δ / ( 1 ) + δ / ( 2 ) , τ = τ / ( 1 ) + τ / ( 2 ) ,
δ = 0 , τ = 0 .

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