Abstract

A family of analytical solutions of the time-dependent wave equation, the ultrashort pulsed sinc-Gaussian light beams (UPSGLB’s), are presented in the paraxial approximation. Each of them has the product form of the monochromatic Gaussian light beam with the central frequency νc times the sinc function of the complex temporal–spatial beam parameter Pn. The complex temporal–spatial beam parameter Pn, which corresponds to the order n, is directly related to the temporal–spatial coupling properties of the nth-order UPSGLB. The UPSGLB’s are used, for the first time to our knowledge, as an analytical expansion set for bandwidth-limited ultrashort light pulses emitted from mode-locked lasers with stable resonators (ULPEMLLSR’s). Two special examples of bandwidth-limited ULPEMLLSR’s, a single zeroth-order UPSGLB and a novel model of a nearly temporal–spatial Gaussian beam, are analytically investigated and compared with experimental results.

© 2000 Optical Society of America

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References

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  1. M. Kempe, U. Stamm, B. Wilhelmi, and W. Rudolph, “Spatial and temporal transformation of femtosecond laser pulses by lenses and lens systems,” J. Opt. Soc. Am. B 9, 1158–1165 (1992).
    [CrossRef]
  2. Z. Bor and Z. L. Horváth, “Distortion of femtosecond pulses in lenses. Wave optical description,” Opt. Commun. 94, 249–258 (1992).
    [CrossRef]
  3. M. Kempe and W. Rudolph, “Femtosecond pulses in the focal region of lenses,” Phys. Rev. A 48, 4721–4729 (1993).
    [CrossRef] [PubMed]
  4. Z. L. Horváth and Z. Bor, “Focusing of femtosecond pulses having Gaussian spatial distribution,” Opt. Commun. 100, 6–12 (1993).
    [CrossRef]
  5. R. W. Ziolkowski and D. B. Davidson, “Designer pulsed beams for enhanced space–time focusing,” Opt. Lett. 19, 284–286 (1994).
    [CrossRef] [PubMed]
  6. A. S. Marathay, “Propagation of optical pulses with spatial and temporal dependence,” Appl. Opt. 33, 3139–3145 (1994).
    [CrossRef] [PubMed]
  7. E. Ibragimov, “Focusing of ultrashort laser pulses by the combination of diffractive and refractive elements,” Appl. Opt. 34, 7280–7285 (1995).
    [CrossRef] [PubMed]
  8. J. Paye and A. Migus, “Space–time Wigner functions and their application to the analysis of a pulse shaper,” J. Opt. Soc. Am. B 12, 1480–1490 (1995).
    [CrossRef]
  9. M. Gu and X. S. Gan, “Fresnel diffraction by circular and serrated apertures illuminated with an ultrashort pulsed-laser beam,” J. Opt. Soc. Am. A 13, 771–778 (1996).
    [CrossRef]
  10. Z. Wang, Z. Xu, and Z. Zhang, “Diffraction integral formulas of the pulsed wave field in the temporal domain,” Opt. Lett. 22, 354–356 (1997).
    [CrossRef] [PubMed]
  11. J. M. Anderson and C. Roychoudhuri, “Diffraction of an extremely short optical pulse,” J. Opt. Soc. Am. A 15, 456–463 (1998).
    [CrossRef]
  12. I. P. Christov, “Propagation of femtosecond light pulses,” Opt. Commun. 53, 364–366 (1985).
    [CrossRef]
  13. R. W. Ziolkowski and J. B. Judkins, “Propagation characteristics of ultrawide-bandwidth pulsed Gaussian beams,” J. Opt. Soc. Am. A 9, 2021–2030 (1992).
    [CrossRef]
  14. Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space–time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33, 566–573 (1997).
    [CrossRef]
  15. M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58, 1086–1093 (1998).
    [CrossRef]
  16. E. Heyman and L. B. Felsen, “Complex-source pulsed-beam fields,” J. Opt. Soc. Am. A 6, 806–817 (1989).
    [CrossRef]
  17. T. Melamed and L. B. Felsen, “Pulsed-beam propagation in lossless dispersive media. I. Theory,” J. Opt. Soc. Am. A 15, 1268–1276 (1998).
    [CrossRef]
  18. M. A. Porras, “Nonsinusoidal few-cycle pulsed light beams in free space,” J. Opt. Soc. Am. B 16, 1468–1474 (1999).
    [CrossRef]
  19. Q. Cao, “Pulsed negative-power-function light beams,” J. Opt. Soc. Am. B 16, 1786–1789 (1999).
    [CrossRef]
  20. S. Feng and H. G. Winful, “Spatiotemporal transformation of isodiffracting ultrashort pulses by nondispersive quadratic phase media,” J. Opt. Soc. Am. A 16, 2500–2509 (1999).
    [CrossRef]
  21. R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2033 (1989).
    [CrossRef] [PubMed]
  22. R. W. Hellwarth and P. Nouchi, “Focused one-cycle electromagnetic pulses,” Phys. Rev. E 54, 889–895 (1996).
    [CrossRef]
  23. S. Feng, H. G. Winful, and R. W. Hellwarth, “Gouy shift and temporal reshaping of focused single-cycle electromagnetic pulses,” Opt. Lett. 23, 385–387 (1998).
    [CrossRef]
  24. L. Xu, G. Tempea, C. Spielmann, F. Krausz, A. Stingl, K. Ferencz, and S. Takano, “Continuous-wave mode-locked Ti: sapphire laser focusable to 5×1013 W/cm2,” Opt. Lett. 23, 789–791 (1998).
    [CrossRef]
  25. M. Nisoli, S. De Silvestri, O. Svelto, R. Szipöcs, K. Ferencz, C. Spielmann, S. Sartania, and F. Krausz, “Compression of high-energy laser pulses below 5 fs,” Opt. Lett. 22, 522–524 (1997).
    [CrossRef] [PubMed]
  26. I. P. Bilinsky, J. G. Fujimoto, J. N. Walpole, and L. J. Missaggia, “Semiconductor-doped-silica saturable-absorber films for solid-state laser mode locking,” Opt. Lett. 23, 1766–1768 (1998).
    [CrossRef]
  27. U. Morgner, F. X. Kärtner, S. H. Cho, Y. Chen, H. A. Haus, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, and T. Tschudi, “Sub-two-cycle pulses from a Kerr-lens mode-locked Ti:sapphire laser,” Opt. Lett. 24, 411–413 (1999).
    [CrossRef]
  28. D. H. Sutter, G. Steinmeyer, L. Gallmann, N. Matuschek, F. M. Genoud, U. Keller, V. Scheuer, G. Angelow, and T. Tschudi, “Semiconductor saturable-absorber mirror-assisted Kerr-lens mode-locked Ti:sapphire laser producing pulses in the two-cycle regime,” Opt. Lett. 24, 631–633 (1999).
    [CrossRef]
  29. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, UK, 1975), pp. 494–499.
  30. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1566 (1966).
    [CrossRef] [PubMed]
  31. W. Koechner, Solid-State Laser Engineering, 3rd ed. (Springer-Verlag, Berlin, 1992), pp. 482–485 and pp. 192–201.
  32. P. M. Mejias and R. Martinez-Herrero, “Time-resolved spatial parametric characterization of pulsed light beams,” Opt. Lett. 20, 660–662 (1995).
    [CrossRef] [PubMed]
  33. Q. Cao and X. Deng, “Spatial parametric characterization of general polychromatic light beams,” Opt. Commun. 142, 135–145 (1997).
    [CrossRef]
  34. H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. 28, 2086–2096 (1992).
    [CrossRef]
  35. K. H. Lin and W. F. Hsieh, “Analytical design of symmetrical Kerr-lens mode-locking laser cavities,” J. Opt. Soc. Am. B 11, 737–741 (1994).
    [CrossRef]
  36. K. H. Lin, Y. Lai, and W. F. Hsieh, “Simple analytical method of cavity design for astigmatism-compensated Kerr-lens mode-locked ring lasers and its applications,” J. Opt. Soc. Am. B 12, 468–475 (1995).
    [CrossRef]
  37. H. Kogelnik, “Imaging of optical modes-resonators with internal lenses,” Bell Syst. Tech. J. 43, 455–494 (1965).
    [CrossRef]
  38. G. Herziger and H. Weber, “Equivalent optical resonators,” Appl. Opt. 23, 1450–1452 (1984).
    [CrossRef]

1999 (5)

1998 (6)

1997 (4)

Q. Cao and X. Deng, “Spatial parametric characterization of general polychromatic light beams,” Opt. Commun. 142, 135–145 (1997).
[CrossRef]

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space–time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33, 566–573 (1997).
[CrossRef]

Z. Wang, Z. Xu, and Z. Zhang, “Diffraction integral formulas of the pulsed wave field in the temporal domain,” Opt. Lett. 22, 354–356 (1997).
[CrossRef] [PubMed]

M. Nisoli, S. De Silvestri, O. Svelto, R. Szipöcs, K. Ferencz, C. Spielmann, S. Sartania, and F. Krausz, “Compression of high-energy laser pulses below 5 fs,” Opt. Lett. 22, 522–524 (1997).
[CrossRef] [PubMed]

1996 (2)

1995 (4)

1994 (3)

1993 (2)

M. Kempe and W. Rudolph, “Femtosecond pulses in the focal region of lenses,” Phys. Rev. A 48, 4721–4729 (1993).
[CrossRef] [PubMed]

Z. L. Horváth and Z. Bor, “Focusing of femtosecond pulses having Gaussian spatial distribution,” Opt. Commun. 100, 6–12 (1993).
[CrossRef]

1992 (4)

Z. Bor and Z. L. Horváth, “Distortion of femtosecond pulses in lenses. Wave optical description,” Opt. Commun. 94, 249–258 (1992).
[CrossRef]

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. 28, 2086–2096 (1992).
[CrossRef]

M. Kempe, U. Stamm, B. Wilhelmi, and W. Rudolph, “Spatial and temporal transformation of femtosecond laser pulses by lenses and lens systems,” J. Opt. Soc. Am. B 9, 1158–1165 (1992).
[CrossRef]

R. W. Ziolkowski and J. B. Judkins, “Propagation characteristics of ultrawide-bandwidth pulsed Gaussian beams,” J. Opt. Soc. Am. A 9, 2021–2030 (1992).
[CrossRef]

1989 (2)

E. Heyman and L. B. Felsen, “Complex-source pulsed-beam fields,” J. Opt. Soc. Am. A 6, 806–817 (1989).
[CrossRef]

R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2033 (1989).
[CrossRef] [PubMed]

1985 (1)

I. P. Christov, “Propagation of femtosecond light pulses,” Opt. Commun. 53, 364–366 (1985).
[CrossRef]

1984 (1)

1966 (1)

1965 (1)

H. Kogelnik, “Imaging of optical modes-resonators with internal lenses,” Bell Syst. Tech. J. 43, 455–494 (1965).
[CrossRef]

Anderson, J. M.

Angelow, G.

Bilinsky, I. P.

Bor, Z.

Z. L. Horváth and Z. Bor, “Focusing of femtosecond pulses having Gaussian spatial distribution,” Opt. Commun. 100, 6–12 (1993).
[CrossRef]

Z. Bor and Z. L. Horváth, “Distortion of femtosecond pulses in lenses. Wave optical description,” Opt. Commun. 94, 249–258 (1992).
[CrossRef]

Cao, Q.

Q. Cao, “Pulsed negative-power-function light beams,” J. Opt. Soc. Am. B 16, 1786–1789 (1999).
[CrossRef]

Q. Cao and X. Deng, “Spatial parametric characterization of general polychromatic light beams,” Opt. Commun. 142, 135–145 (1997).
[CrossRef]

Chen, Y.

Cho, S. H.

Christov, I. P.

I. P. Christov, “Propagation of femtosecond light pulses,” Opt. Commun. 53, 364–366 (1985).
[CrossRef]

Davidson, D. B.

De Silvestri, S.

Deng, X.

Q. Cao and X. Deng, “Spatial parametric characterization of general polychromatic light beams,” Opt. Commun. 142, 135–145 (1997).
[CrossRef]

Felsen, L. B.

Feng, S.

Ferencz, K.

Fujimoto, J. G.

Gallmann, L.

Gan, X. S.

Genoud, F. M.

Gu, M.

Haus, H. A.

Hellwarth, R. W.

Herziger, G.

Heyman, E.

Horváth, Z. L.

Z. L. Horváth and Z. Bor, “Focusing of femtosecond pulses having Gaussian spatial distribution,” Opt. Commun. 100, 6–12 (1993).
[CrossRef]

Z. Bor and Z. L. Horváth, “Distortion of femtosecond pulses in lenses. Wave optical description,” Opt. Commun. 94, 249–258 (1992).
[CrossRef]

Hsieh, W. F.

Ibragimov, E.

Ippen, E. P.

Judkins, J. B.

Kärtner, F. X.

Keller, U.

Kempe, M.

Kogelnik, H.

H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1566 (1966).
[CrossRef] [PubMed]

H. Kogelnik, “Imaging of optical modes-resonators with internal lenses,” Bell Syst. Tech. J. 43, 455–494 (1965).
[CrossRef]

Krausz, F.

Lai, Y.

Li, T.

Lin, K. H.

Lin, Q.

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space–time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33, 566–573 (1997).
[CrossRef]

Marathay, A. S.

Martinez-Herrero, R.

Matuschek, N.

Mejias, P. M.

Melamed, T.

Migus, A.

Missaggia, L. J.

Morgner, U.

Nisoli, M.

Nouchi, P.

R. W. Hellwarth and P. Nouchi, “Focused one-cycle electromagnetic pulses,” Phys. Rev. E 54, 889–895 (1996).
[CrossRef]

Paye, J.

Porras, M. A.

M. A. Porras, “Nonsinusoidal few-cycle pulsed light beams in free space,” J. Opt. Soc. Am. B 16, 1468–1474 (1999).
[CrossRef]

M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58, 1086–1093 (1998).
[CrossRef]

Roychoudhuri, C.

Rudolph, W.

Sartania, S.

Scheuer, V.

Spielmann, C.

Stamm, U.

Steinmeyer, G.

Stingl, A.

Sutter, D. H.

Svelto, O.

Szipöcs, R.

Takano, S.

Tempea, G.

Tschudi, T.

Walpole, J. N.

Wang, Z.

Z. Wang, Z. Xu, and Z. Zhang, “Diffraction integral formulas of the pulsed wave field in the temporal domain,” Opt. Lett. 22, 354–356 (1997).
[CrossRef] [PubMed]

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space–time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33, 566–573 (1997).
[CrossRef]

Weber, H.

Wilhelmi, B.

Winful, H. G.

Xu, L.

Xu, Z.

Z. Wang, Z. Xu, and Z. Zhang, “Diffraction integral formulas of the pulsed wave field in the temporal domain,” Opt. Lett. 22, 354–356 (1997).
[CrossRef] [PubMed]

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space–time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33, 566–573 (1997).
[CrossRef]

Zhang, Z.

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space–time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33, 566–573 (1997).
[CrossRef]

Z. Wang, Z. Xu, and Z. Zhang, “Diffraction integral formulas of the pulsed wave field in the temporal domain,” Opt. Lett. 22, 354–356 (1997).
[CrossRef] [PubMed]

Ziolkowski, R. W.

Appl. Opt. (4)

Bell Syst. Tech. J. (1)

H. Kogelnik, “Imaging of optical modes-resonators with internal lenses,” Bell Syst. Tech. J. 43, 455–494 (1965).
[CrossRef]

IEEE J. Quantum Electron. (2)

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space–time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33, 566–573 (1997).
[CrossRef]

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. 28, 2086–2096 (1992).
[CrossRef]

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

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

Opt. Commun. (4)

Q. Cao and X. Deng, “Spatial parametric characterization of general polychromatic light beams,” Opt. Commun. 142, 135–145 (1997).
[CrossRef]

Z. Bor and Z. L. Horváth, “Distortion of femtosecond pulses in lenses. Wave optical description,” Opt. Commun. 94, 249–258 (1992).
[CrossRef]

Z. L. Horváth and Z. Bor, “Focusing of femtosecond pulses having Gaussian spatial distribution,” Opt. Commun. 100, 6–12 (1993).
[CrossRef]

I. P. Christov, “Propagation of femtosecond light pulses,” Opt. Commun. 53, 364–366 (1985).
[CrossRef]

Opt. Lett. (9)

R. W. Ziolkowski and D. B. Davidson, “Designer pulsed beams for enhanced space–time focusing,” Opt. Lett. 19, 284–286 (1994).
[CrossRef] [PubMed]

P. M. Mejias and R. Martinez-Herrero, “Time-resolved spatial parametric characterization of pulsed light beams,” Opt. Lett. 20, 660–662 (1995).
[CrossRef] [PubMed]

Z. Wang, Z. Xu, and Z. Zhang, “Diffraction integral formulas of the pulsed wave field in the temporal domain,” Opt. Lett. 22, 354–356 (1997).
[CrossRef] [PubMed]

M. Nisoli, S. De Silvestri, O. Svelto, R. Szipöcs, K. Ferencz, C. Spielmann, S. Sartania, and F. Krausz, “Compression of high-energy laser pulses below 5 fs,” Opt. Lett. 22, 522–524 (1997).
[CrossRef] [PubMed]

S. Feng, H. G. Winful, and R. W. Hellwarth, “Gouy shift and temporal reshaping of focused single-cycle electromagnetic pulses,” Opt. Lett. 23, 385–387 (1998).
[CrossRef]

L. Xu, G. Tempea, C. Spielmann, F. Krausz, A. Stingl, K. Ferencz, and S. Takano, “Continuous-wave mode-locked Ti: sapphire laser focusable to 5×1013 W/cm2,” Opt. Lett. 23, 789–791 (1998).
[CrossRef]

I. P. Bilinsky, J. G. Fujimoto, J. N. Walpole, and L. J. Missaggia, “Semiconductor-doped-silica saturable-absorber films for solid-state laser mode locking,” Opt. Lett. 23, 1766–1768 (1998).
[CrossRef]

U. Morgner, F. X. Kärtner, S. H. Cho, Y. Chen, H. A. Haus, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, and T. Tschudi, “Sub-two-cycle pulses from a Kerr-lens mode-locked Ti:sapphire laser,” Opt. Lett. 24, 411–413 (1999).
[CrossRef]

D. H. Sutter, G. Steinmeyer, L. Gallmann, N. Matuschek, F. M. Genoud, U. Keller, V. Scheuer, G. Angelow, and T. Tschudi, “Semiconductor saturable-absorber mirror-assisted Kerr-lens mode-locked Ti:sapphire laser producing pulses in the two-cycle regime,” Opt. Lett. 24, 631–633 (1999).
[CrossRef]

Phys. Rev. A (2)

M. Kempe and W. Rudolph, “Femtosecond pulses in the focal region of lenses,” Phys. Rev. A 48, 4721–4729 (1993).
[CrossRef] [PubMed]

R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2033 (1989).
[CrossRef] [PubMed]

Phys. Rev. E (2)

R. W. Hellwarth and P. Nouchi, “Focused one-cycle electromagnetic pulses,” Phys. Rev. E 54, 889–895 (1996).
[CrossRef]

M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58, 1086–1093 (1998).
[CrossRef]

Other (2)

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, UK, 1975), pp. 494–499.

W. Koechner, Solid-State Laser Engineering, 3rd ed. (Springer-Verlag, Berlin, 1992), pp. 482–485 and pp. 192–201.

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Figures (6)

Fig. 1
Fig. 1

Stable spherical resonator.

Fig. 2
Fig. 2

Time delay with the increase of the order n at the point (r=0, z=0). (A) n=-1, (B) n=0, and (C) n=1. The parameters are chosen such that νc=400 THz, δ=100 THz, and b=3π2×108 m2 s-1.

Fig. 3
Fig. 3

Time delay with the increase of the radial coordinate r of the zeroth-order UPSGLB at the z=3.14 m plane. (A) r=0.0 mm, (B) r=1.0 mm, and (C) r=2.0 mm. The parameters are the same as in Fig. 2.

Fig. 4
Fig. 4

Time-integral intensity distribution In(r, 0) of a single UPSGLB at the z=0 plane. The parameters are chosen such that νc=400 THz, δ=300 THZ, z=0, and b=3π2×108 m2 s-1.

Fig. 5
Fig. 5

Pulse intensity V2(0, 0, t, 0) of a single zeroth-order UPSGLB at the point (r=0, z=0). The parameters are chosen such that νc=396 THz and δ=80 THz.

Fig. 6
Fig. 6

Novel model of a nearly temporal–spatial Gaussian beam. (A) The weighting function g(ν), (B) the pulse intensity V2(0, 0, t, 0) at the point (r=0, z=0), and (C) the transverse intensity distribution V2(r, 0, 0) at the z=0 plane at the t=0 moment. The parameters are chosen such that νc=358 THz, δ=17 THz, and b=3.75π2×108 m2 s-1.

Equations (33)

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

2x2+2y2V-1c22Vt2+2Vz2=0,
ϕ(x, y, t, z)=20φ(x, y, ν, z)exp(-i2πνt)dν,
φ(x, y, ν, z)=-V(x, y, t, z)exp(i2πνt)dt.
V(x, y, t, z)=Re[ϕ(x, y, t, z)].
2φ+k2φ=0,
i2kUz+2x2+2y2U=0.
b=πcL(R1-L)(R2-L)(R1+R2-L)(R1+R2-2L)21/2.
U(x, y, νm, 0)=gmπbexp-π2r2νmb,
φ(x, y, νm, z)=gmπb+iπzcexpi2πzcνm×exp-π2r2νmb+iπzc.
ϕ(x, y, t, z)=2m=m1m2φ(x, y, νm, z)exp(-i2πνmt),
ϕ(x, y, t, z)=2νlνhφ(x, y, ν, z)exp(-i2πνt)dν,
φ(x, y, ν, z)=g(ν)πb+iπzcexpi2πzcν×exp-π2r2νb+iπzc,
gn(ν)=An expinπ(ν-νc)δ
Φn(x, y, t, z)=-i2πδAn(b+iπzc)Pnexp-inπνcδ×expiPnνhδ-expiPnνlδ,
Pn=nπ-2πδτ+iδπ2r2b+iπzc,
ϕn(x, y, t, z)=4πδAnb+iπzcexp(-i2πνcτ)×exp-π2r2νcb+iπzcsinc(Pn),
ϕn(0, 0, t, z)=4πδAnb+iπzcexp(-i2πνcτ)×sinc(nπ-2πδτ).
ϕn(0, 0, t, 0)=4πδAnb-1 exp(-i2πνcτ)×sinc(nπ-2πδτ),
limz ϕn(0, 0, t, z)=4δAnzcexp(-iπ/2)exp(-i2πνcτ)×sinc(nπ-2πδτ).
1q=iπcb+iπzc.
Pn=-2πδτ-n2δ-r22cR+iδπr2cIm1q,
D(ν)=2-|φ(x, y, ν, z)|2dxdy.
Dn(ν)=πAn2b-1ν-1forνlννh0,elsewhere.
An=bπ[ln(νh)-ln(νl)]1/2.
In(x, y, z)=2An2π2b2+π2z2c2νlνh exp-2π2br2νb2+π2z2c2dν.
In(x, y, z)=2An2br2exp-2π2bνcr2b2+π2z2c2×sinh2π2bδr2b2+π2z2c2,
I(x, y, νc, z)=4An2π2δb2+π2z2c2exp-2π2bνcr2b2+π2z2c2,
I(x, y, νl, z)=4An2π2δb2+π2z2c2exp-2π2bνlr2b2+π2z2c2,
g(ν)=n=-cnAn expinπ(ν-νc)δ=-cngn(ν),
cn=12Anδνlνhg(ν)exp-inπ(ν-νc)δdν.
ϕ(x, y, t, z)=n=-cnϕn(x, y, t, z),
g(ν)=0.5An+0.5An cos[π(ν-νc)/δ]
ϕ(x, y, t, z)=0.5ϕ0(x, y, t, z)+0.25ϕ1(x, y, t, z)+0.25ϕ-1(x, y, t, z).

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