Abstract

We discuss exact solutions of Maxwell’s equations that describe the evolution of single-cycle electromagnetic pulses. The solutions are applied to recent observations of the diffraction transformation of terahertz pulses. In particular, we elucidate the role of the Gouy shift in the temporal reshaping and polarity reversals of single-cycle terahertz pulses.

© 1998 Optical Society of America

Full Article  |  PDF Article

Corrections

Simin Feng, Herbert G. Winful, and Robert W. Hellwarth, "Gouy shift and temporal reshaping of focused single-cycle electromagnetic pulses:?errata," Opt. Lett. 23, 1141-1141 (1998)
https://www.osapublishing.org/ol/abstract.cfm?uri=ol-23-14-1141

References

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  1. X.-C. Zhang, B. B. Hu, J. T. Darrow, and D. H. Auston, Appl. Phys. Lett. 56, 1011 (1990); B. I. Greene, J. F. Federici, D. R. Dykaar, R. R. Jones, and P. H. Bucksbaum, Appl. Phys. Lett. 59, 893 (1991); Ch. Fattinger and D. Grischkowsky, Appl. Phys. Lett. 54, 490 (1989); P. R. Smith, D. H. Auston, and M. C. Nuss, IEEE J. Quantum Electron. 24, 255 (1988).
    [CrossRef]
  2. A. Nahata and T. F. Heinz, IEEE J. Sel. Topics Quantum Electron. 2, 701 (1996).
    [CrossRef]
  3. J. Bromage, S. Radic, G. P. Agrawal, C. R. Stroud, P. M. Fauchet, and R. Sobolewski, Opt. Lett. 22, 627 (1997).
    [CrossRef] [PubMed]
  4. J. N. Brittingham, J. Appl. Phys. 54, 1179 (1983).
    [CrossRef]
  5. R. W. Ziolkowski, J. Math. Phys. 26, 861 (1985).
    [CrossRef]
  6. R. W. Ziolkowski, Phys. Rev. A 39, 2005 (1989).
    [CrossRef] [PubMed]
  7. R. W. Hellwarth and P. Nouchi, Phys. Rev. E 54, 889 (1996).
    [CrossRef]
  8. M. Van Exter, Ch. Fattinger, and D. Grischkowsky, Appl. Phys. Lett. 55, 337 (1989); D. Grischkowsky, S. Keiding, M. Van Exter, and Ch. Fattinger, J. Opt. Soc. Am. B 7, 2006 (1990).
    [CrossRef]
  9. P. U. Jepsen and S. R. Keiding, Opt. Lett. 20, 807 (1994).
    [CrossRef]
  10. C. R. Gouy, C. R. Acad. Sci. Paris 110, 1251 (1890); see also Lasers, A. E. Siegman, ed. (University Science, Singapore, 1986), pp. 682–685. Note that Gouy’s name is misspelled as “Guoy” in the latter.
  11. D. You and P. Bucksbaum, J. Opt. Soc. Am. B 14, 1651 (1997).
    [CrossRef]
  12. S. Feng, H. G. Winful, and R. W. Hellwarth, in Conference on Lasers and Electro-Optics, Vol. 11 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), paper CThO3.

1997 (2)

1996 (2)

R. W. Hellwarth and P. Nouchi, Phys. Rev. E 54, 889 (1996).
[CrossRef]

A. Nahata and T. F. Heinz, IEEE J. Sel. Topics Quantum Electron. 2, 701 (1996).
[CrossRef]

1994 (1)

1990 (1)

X.-C. Zhang, B. B. Hu, J. T. Darrow, and D. H. Auston, Appl. Phys. Lett. 56, 1011 (1990); B. I. Greene, J. F. Federici, D. R. Dykaar, R. R. Jones, and P. H. Bucksbaum, Appl. Phys. Lett. 59, 893 (1991); Ch. Fattinger and D. Grischkowsky, Appl. Phys. Lett. 54, 490 (1989); P. R. Smith, D. H. Auston, and M. C. Nuss, IEEE J. Quantum Electron. 24, 255 (1988).
[CrossRef]

1989 (2)

R. W. Ziolkowski, Phys. Rev. A 39, 2005 (1989).
[CrossRef] [PubMed]

M. Van Exter, Ch. Fattinger, and D. Grischkowsky, Appl. Phys. Lett. 55, 337 (1989); D. Grischkowsky, S. Keiding, M. Van Exter, and Ch. Fattinger, J. Opt. Soc. Am. B 7, 2006 (1990).
[CrossRef]

1985 (1)

R. W. Ziolkowski, J. Math. Phys. 26, 861 (1985).
[CrossRef]

1983 (1)

J. N. Brittingham, J. Appl. Phys. 54, 1179 (1983).
[CrossRef]

1890 (1)

C. R. Gouy, C. R. Acad. Sci. Paris 110, 1251 (1890); see also Lasers, A. E. Siegman, ed. (University Science, Singapore, 1986), pp. 682–685. Note that Gouy’s name is misspelled as “Guoy” in the latter.

Agrawal, G. P.

Auston, D. H.

X.-C. Zhang, B. B. Hu, J. T. Darrow, and D. H. Auston, Appl. Phys. Lett. 56, 1011 (1990); B. I. Greene, J. F. Federici, D. R. Dykaar, R. R. Jones, and P. H. Bucksbaum, Appl. Phys. Lett. 59, 893 (1991); Ch. Fattinger and D. Grischkowsky, Appl. Phys. Lett. 54, 490 (1989); P. R. Smith, D. H. Auston, and M. C. Nuss, IEEE J. Quantum Electron. 24, 255 (1988).
[CrossRef]

Brittingham, J. N.

J. N. Brittingham, J. Appl. Phys. 54, 1179 (1983).
[CrossRef]

Bromage, J.

Bucksbaum, P.

Darrow, J. T.

X.-C. Zhang, B. B. Hu, J. T. Darrow, and D. H. Auston, Appl. Phys. Lett. 56, 1011 (1990); B. I. Greene, J. F. Federici, D. R. Dykaar, R. R. Jones, and P. H. Bucksbaum, Appl. Phys. Lett. 59, 893 (1991); Ch. Fattinger and D. Grischkowsky, Appl. Phys. Lett. 54, 490 (1989); P. R. Smith, D. H. Auston, and M. C. Nuss, IEEE J. Quantum Electron. 24, 255 (1988).
[CrossRef]

Fattinger, Ch.

M. Van Exter, Ch. Fattinger, and D. Grischkowsky, Appl. Phys. Lett. 55, 337 (1989); D. Grischkowsky, S. Keiding, M. Van Exter, and Ch. Fattinger, J. Opt. Soc. Am. B 7, 2006 (1990).
[CrossRef]

Fauchet, P. M.

Feng, S.

S. Feng, H. G. Winful, and R. W. Hellwarth, in Conference on Lasers and Electro-Optics, Vol. 11 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), paper CThO3.

Gouy, C. R.

C. R. Gouy, C. R. Acad. Sci. Paris 110, 1251 (1890); see also Lasers, A. E. Siegman, ed. (University Science, Singapore, 1986), pp. 682–685. Note that Gouy’s name is misspelled as “Guoy” in the latter.

Grischkowsky, D.

M. Van Exter, Ch. Fattinger, and D. Grischkowsky, Appl. Phys. Lett. 55, 337 (1989); D. Grischkowsky, S. Keiding, M. Van Exter, and Ch. Fattinger, J. Opt. Soc. Am. B 7, 2006 (1990).
[CrossRef]

Heinz, T. F.

A. Nahata and T. F. Heinz, IEEE J. Sel. Topics Quantum Electron. 2, 701 (1996).
[CrossRef]

Hellwarth, R. W.

R. W. Hellwarth and P. Nouchi, Phys. Rev. E 54, 889 (1996).
[CrossRef]

S. Feng, H. G. Winful, and R. W. Hellwarth, in Conference on Lasers and Electro-Optics, Vol. 11 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), paper CThO3.

Hu, B. B.

X.-C. Zhang, B. B. Hu, J. T. Darrow, and D. H. Auston, Appl. Phys. Lett. 56, 1011 (1990); B. I. Greene, J. F. Federici, D. R. Dykaar, R. R. Jones, and P. H. Bucksbaum, Appl. Phys. Lett. 59, 893 (1991); Ch. Fattinger and D. Grischkowsky, Appl. Phys. Lett. 54, 490 (1989); P. R. Smith, D. H. Auston, and M. C. Nuss, IEEE J. Quantum Electron. 24, 255 (1988).
[CrossRef]

Jepsen, P. U.

Keiding, S. R.

Nahata, A.

A. Nahata and T. F. Heinz, IEEE J. Sel. Topics Quantum Electron. 2, 701 (1996).
[CrossRef]

Nouchi, P.

R. W. Hellwarth and P. Nouchi, Phys. Rev. E 54, 889 (1996).
[CrossRef]

Radic, S.

Sobolewski, R.

Stroud, C. R.

Van Exter, M.

M. Van Exter, Ch. Fattinger, and D. Grischkowsky, Appl. Phys. Lett. 55, 337 (1989); D. Grischkowsky, S. Keiding, M. Van Exter, and Ch. Fattinger, J. Opt. Soc. Am. B 7, 2006 (1990).
[CrossRef]

Winful, H. G.

S. Feng, H. G. Winful, and R. W. Hellwarth, in Conference on Lasers and Electro-Optics, Vol. 11 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), paper CThO3.

You, D.

Zhang, X.-C.

X.-C. Zhang, B. B. Hu, J. T. Darrow, and D. H. Auston, Appl. Phys. Lett. 56, 1011 (1990); B. I. Greene, J. F. Federici, D. R. Dykaar, R. R. Jones, and P. H. Bucksbaum, Appl. Phys. Lett. 59, 893 (1991); Ch. Fattinger and D. Grischkowsky, Appl. Phys. Lett. 54, 490 (1989); P. R. Smith, D. H. Auston, and M. C. Nuss, IEEE J. Quantum Electron. 24, 255 (1988).
[CrossRef]

Ziolkowski, R. W.

R. W. Ziolkowski, Phys. Rev. A 39, 2005 (1989).
[CrossRef] [PubMed]

R. W. Ziolkowski, J. Math. Phys. 26, 861 (1985).
[CrossRef]

Appl. Phys. Lett. (2)

M. Van Exter, Ch. Fattinger, and D. Grischkowsky, Appl. Phys. Lett. 55, 337 (1989); D. Grischkowsky, S. Keiding, M. Van Exter, and Ch. Fattinger, J. Opt. Soc. Am. B 7, 2006 (1990).
[CrossRef]

X.-C. Zhang, B. B. Hu, J. T. Darrow, and D. H. Auston, Appl. Phys. Lett. 56, 1011 (1990); B. I. Greene, J. F. Federici, D. R. Dykaar, R. R. Jones, and P. H. Bucksbaum, Appl. Phys. Lett. 59, 893 (1991); Ch. Fattinger and D. Grischkowsky, Appl. Phys. Lett. 54, 490 (1989); P. R. Smith, D. H. Auston, and M. C. Nuss, IEEE J. Quantum Electron. 24, 255 (1988).
[CrossRef]

C. R. Acad. Sci. Paris (1)

C. R. Gouy, C. R. Acad. Sci. Paris 110, 1251 (1890); see also Lasers, A. E. Siegman, ed. (University Science, Singapore, 1986), pp. 682–685. Note that Gouy’s name is misspelled as “Guoy” in the latter.

IEEE J. Sel. Topics Quantum Electron. (1)

A. Nahata and T. F. Heinz, IEEE J. Sel. Topics Quantum Electron. 2, 701 (1996).
[CrossRef]

J. Appl. Phys. (1)

J. N. Brittingham, J. Appl. Phys. 54, 1179 (1983).
[CrossRef]

J. Math. Phys. (1)

R. W. Ziolkowski, J. Math. Phys. 26, 861 (1985).
[CrossRef]

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

Opt. Lett. (2)

Phys. Rev. A (1)

R. W. Ziolkowski, Phys. Rev. A 39, 2005 (1989).
[CrossRef] [PubMed]

Phys. Rev. E (1)

R. W. Hellwarth and P. Nouchi, Phys. Rev. E 54, 889 (1996).
[CrossRef]

Other (1)

S. Feng, H. G. Winful, and R. W. Hellwarth, in Conference on Lasers and Electro-Optics, Vol. 11 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), paper CThO3.

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Figures (4)

Fig. 1
Fig. 1

Temporal profiles Eyr0, ρ, t (left) and Eyi0, ρ, t (right) at ρ=0 (solid curves), ρ=we (dashed curves), and ρ=2we (dotted–dashed curves) at the focal plane z=0. Here we=q1q2/21/2 is an effective beam radius at the focus. The pulse width increases off axis. Here and in Figs.  24 q1=0.025 mm and q2=1000 mm.

Fig. 2
Fig. 2

Spatiotemporal evolution of the imaginary solution Eyir, t from z=-3 m before the focus (left), through the focus (center), and then to the far field z=3 m (right). Note the polarity reversal and temporal reshaping.

Fig. 3
Fig. 3

Spatial evolution of the pulse shape on the axis ρ=0 at five propagation distances Z. Clearly, the real and the imaginary pulses evolve into each other during propagation.

Fig. 4
Fig. 4

Amplitude spectra at z=0 for various radial positions. Note the red shift as one moves from the center of the pulse to the edge. The effective beam radius is we=3.5 mm. The spectra are normalized by the peak value at ρ=0.

Equations (6)

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

2-1c22t2fr, t=0.
fr, t=f0/q1+iτs+q2
Eyr, t=2f0μ00q1+iτ2-q2-iσ2ρ2+q1+iτq2-iσ3,
Hxr, t=2f02ρ2 cos2φ+q1+iτ2+q2-iσ2ρ2+q1+iτq2-iσ3,
Eyrρ=0, z, T=ATq22+4z21/2cosαT+ϕz, Eyiρ=0, z, T=ATq22+4z21/2sinαT+ϕz,
E˜yrr, ω=-f0μ0πq2ωc3/2exp-ωcq1×exp-ikz+iωωϕwexp-ikρ22R-ρ2w2,

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