Abstract

An analytical approach to the theory of diffraction transformation of pulses with superbroad spectra and arbitrary time dependence, in particular half-cycle (unipolar), single-cycle, and multicycle pulses, was developed. Closed-form solutions were found for on-axis propagation of half-cycle pulses with initially Gaussian spatial profiles that have either cosh-1-like or Gaussian time dependence, for single-cycle pulses based on higher modes of these functions, and for multicycle pulses. The far-field propagation demonstrates common patterns of time-derivative behavior regardless of the initial spatiotemporal profile. It is also shown that the time width of an off-axis pulse increases with the angle of observation. Owing to time–space reciprocity, the pulse transformation that is due to diffraction can be reversed, e.g., by reflection of a pulse from a spherical concave mirror.

© 1998 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).
  2. A. Siegman, Lasers (University Science, Mill Valley, Calif., 1986); A. Yariv, Quantum Electronics (Wiley, New York, 1989).
  3. P. R. Smith, D. H. Auston, and M. S. Nuss, IEEE J. Quantum Electron. 24, 255 (1988).
    [CrossRef]
  4. D. Grischkowsky, S. Keidin, M. van Exter, and Ch. Fattinger, J. Opt. Soc. Am. B 7, 2006 (1990); R. A. Cheville and D. Grischkowsky, Opt. Lett. 20, 1646 (1995).
    [CrossRef] [PubMed]
  5. J. H. Glownia, J. A. Misewich, and P. P. Sorokin, J. Chem. Phys. 92, 3335 (1990).
    [CrossRef]
  6. B. B. Hu and M. S. Nuss, Opt. Lett. 20, 1716 (1995).
    [CrossRef]
  7. R. R. Jones, D. You, and P. H. Bucksbaum, Phys. Rev. Lett. 70, 1236 (1993); C. O. Reinhold, M. Melles, H. Shao, and J. Burgdorfer, J. Phys. B 26, L659 (1993).
    [CrossRef] [PubMed]
  8. A. E. Kaplan, Phys. Rev. Lett. 73, 1243 (1994); A. E. Kaplan and P. L. Shkolnikov, J. Opt. Soc. Am. B 13, 412 (1996).
    [CrossRef] [PubMed]
  9. A. E. Kaplan and P. L. Shkolnikov, Phys. Rev. Lett. 75, 2316 (1995); Int. J. Nonlin. Opt. Phys. Mater. 4, 831 (1995); A. E. Kaplan, S. F. Straub, and P. L. Shkolnikov, Opt. Lett. OPLEDP 22, 405 (1997); J. Opt. Soc. Am. B JOBPDE 14, 3013 (1997).
    [CrossRef] [PubMed]
  10. M. van Exeter and D. R. Grischkowsky, IEEE Trans. Microwave Theory Tech. 38, 1684 (1990); J. Bromage, S. Radic, G. P. Agrawal, C. R. Stroud, Jr., P. M. Fauchet, and R. Sobolevski, Opt. Lett. 22, 627 (1997).
    [CrossRef] [PubMed]
  11. R. W. Ziolkowski and J. B. Judkins, J. Opt. Soc. Am. B 9, 2021 (1992).
    [CrossRef]
  12. I. S. Gradshtein and I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1980).

1995 (1)

1992 (1)

R. W. Ziolkowski and J. B. Judkins, J. Opt. Soc. Am. B 9, 2021 (1992).
[CrossRef]

1990 (1)

J. H. Glownia, J. A. Misewich, and P. P. Sorokin, J. Chem. Phys. 92, 3335 (1990).
[CrossRef]

1988 (1)

P. R. Smith, D. H. Auston, and M. S. Nuss, IEEE J. Quantum Electron. 24, 255 (1988).
[CrossRef]

Auston, D. H.

P. R. Smith, D. H. Auston, and M. S. Nuss, IEEE J. Quantum Electron. 24, 255 (1988).
[CrossRef]

Glownia, J. H.

J. H. Glownia, J. A. Misewich, and P. P. Sorokin, J. Chem. Phys. 92, 3335 (1990).
[CrossRef]

Hu, B. B.

Judkins, J. B.

R. W. Ziolkowski and J. B. Judkins, J. Opt. Soc. Am. B 9, 2021 (1992).
[CrossRef]

Misewich, J. A.

J. H. Glownia, J. A. Misewich, and P. P. Sorokin, J. Chem. Phys. 92, 3335 (1990).
[CrossRef]

Nuss, M. S.

B. B. Hu and M. S. Nuss, Opt. Lett. 20, 1716 (1995).
[CrossRef]

P. R. Smith, D. H. Auston, and M. S. Nuss, IEEE J. Quantum Electron. 24, 255 (1988).
[CrossRef]

Smith, P. R.

P. R. Smith, D. H. Auston, and M. S. Nuss, IEEE J. Quantum Electron. 24, 255 (1988).
[CrossRef]

Sorokin, P. P.

J. H. Glownia, J. A. Misewich, and P. P. Sorokin, J. Chem. Phys. 92, 3335 (1990).
[CrossRef]

Ziolkowski, R. W.

R. W. Ziolkowski and J. B. Judkins, J. Opt. Soc. Am. B 9, 2021 (1992).
[CrossRef]

IEEE J. Quantum Electron. (1)

P. R. Smith, D. H. Auston, and M. S. Nuss, IEEE J. Quantum Electron. 24, 255 (1988).
[CrossRef]

J. Chem. Phys. (1)

J. H. Glownia, J. A. Misewich, and P. P. Sorokin, J. Chem. Phys. 92, 3335 (1990).
[CrossRef]

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

R. W. Ziolkowski and J. B. Judkins, J. Opt. Soc. Am. B 9, 2021 (1992).
[CrossRef]

Opt. Lett. (1)

Other (8)

I. S. Gradshtein and I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1980).

R. R. Jones, D. You, and P. H. Bucksbaum, Phys. Rev. Lett. 70, 1236 (1993); C. O. Reinhold, M. Melles, H. Shao, and J. Burgdorfer, J. Phys. B 26, L659 (1993).
[CrossRef] [PubMed]

A. E. Kaplan, Phys. Rev. Lett. 73, 1243 (1994); A. E. Kaplan and P. L. Shkolnikov, J. Opt. Soc. Am. B 13, 412 (1996).
[CrossRef] [PubMed]

A. E. Kaplan and P. L. Shkolnikov, Phys. Rev. Lett. 75, 2316 (1995); Int. J. Nonlin. Opt. Phys. Mater. 4, 831 (1995); A. E. Kaplan, S. F. Straub, and P. L. Shkolnikov, Opt. Lett. OPLEDP 22, 405 (1997); J. Opt. Soc. Am. B JOBPDE 14, 3013 (1997).
[CrossRef] [PubMed]

M. van Exeter and D. R. Grischkowsky, IEEE Trans. Microwave Theory Tech. 38, 1684 (1990); J. Bromage, S. Radic, G. P. Agrawal, C. R. Stroud, Jr., P. M. Fauchet, and R. Sobolevski, Opt. Lett. 22, 627 (1997).
[CrossRef] [PubMed]

D. Grischkowsky, S. Keidin, M. van Exter, and Ch. Fattinger, J. Opt. Soc. Am. B 7, 2006 (1990); R. A. Cheville and D. Grischkowsky, Opt. Lett. 20, 1646 (1995).
[CrossRef] [PubMed]

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

A. Siegman, Lasers (University Science, Mill Valley, Calif., 1986); A. Yariv, Quantum Electronics (Wiley, New York, 1989).

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Figures (2)

Fig. 1
Fig. 1

Evolution of the on-axis temporal profile. The curves depict the normalized field Eon/E0 versus the normalized time τ=t/t0 and the normalized amplitude spectrum |S|/E0t0 versus the normalized frequency νωt0 (inset) of the initially Gaussian half-cycle pulse [Eq. (3.4)] as it propagates along the axis ζ=zct0/r02. Curves: 1, ζ=0; 2, ζ=0.25; 3, ζ=0.5; 4, ζ=1; 5, ζ=2; 6, ζ=4. For comparison, each curve in the larger part of the figure is scaled up by the factor w-1/2(ζ).

Fig. 2
Fig. 2

Same as Fig. 1, but for the initially Gaussian single-cycle pulse [Eq. (4.3)].

Equations (40)

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zd=r02/ct0,
r0ct0orzd/r01.
-2i(ω/c)Eω/z+ΔEω=0,
-2(2Et˜/zt˜)+cΔEt˜=0,
Ez=0=E0(t)exp(-r2/2r02),
E(t˜, r, z)=(2π)-1/2-S(ω, r, z)exp(iωt˜)dω,
S(ω, r, z)=D(ω, z)S0(ω)exp[-(r2/2r02)D(ω, z)],
D(ω, z)=(1-izc/ωr02)-1
Son(ω, z)=D(ω, z)S0(ω)=iν(ζ+iν)-1S0(ν),
Eon/τ+ζEon=E0(τ)/τ.
Eon(τ, ζ)=E0(τ)-ζ exp(-ζτ)-s(ζ)τexp(ζt)E0(t)dt,
Eon(τ, ζ)=-n=1n= 1(-ζ)n nE0(τ)τn,
Eon(τ, ζ)ζ-1E0(τ)/τ.
E0(τ)=E0(1+|τ|)exp(-|τ|)
Eon(τ, ζ)E0=exp(-|τ|)1-s(τ)ζ 11-s(τ)ζ+|τ|+2ζ exp(-ζτ)[s(ζ)+s(τ)](1-ζ2)2.
E0=E0/cosh(τ),
E0(t)=E0 exp(-t2/2t02)=E0 exp(-τ2/2)
Eon(τ, ζ)=E0(τ)(1-ζ(π/2)1/2 exp[(τ-ζ)2/2]×{s(ζ)+erf[(τ-ζ)/2]}),
Won(ζ)=-|Son(ω, ζ)|2dω=-E2(t, ζ)dt,
w(ζ)Won(ζ)/Won(0)=1-ζπ exp(ζ2)[s(ζ) -erf(ζ)],
E0(τ)=E0τ(1+|τ|)exp(-|τ|)
E0(τ)=E0τ exp(-|τ|).
Eon(τ, ζ)E0=exp(-|τ|)1-s(τ)ζ ζ1-s(τ)ζ+τ-2ζ2 exp(-ζτ)[s(ζ)+s(τ)](1-ζ2)2.
E0(t)=E0τ exp(-τ2/2),τ=t/t0,
Eon(τ, ζ)E0=exp(-τ2/2)ζ+τ-ζ2π/2 exp(τ-ζ)22×{s(ζ)+erf[(τ-ζ)/2]}.
w(ζ)=1-2ζ2+2ζ3π exp(ζ2)[s(ζ)-erf(ζ)].
[E0(τ)]n=E0m=0nAmτm exp(-τ2/2),
E0(τ)=E0 cos(qτ)exp(-τ2/2)
E0(τ)=E0 sin(qτ)exp(-τ2/2),
Eon(τ, ζ)E0=exp-τ22 cos(qτ)-ζπ/2 exp(τ-ζ)2-q22(cos[q(τ-ζ)]+Re{exp[-iq(τ-ζ)]×erf(τ-ζ-iq)/2}),
Eon(τ, ζ)E0=exp-τ22sin(qτ)-ζπ/2×exp(τ-ζ)2-q22(sin[q(τ-ζ)]+Im{exp[-iq(τ-ζ)]×erf(τ-ζ-iq)/2}).
E0(τ)/E0=osc(qτ)(1+|τ|)exp(-|τ|),
w(ζ)=|D(ω0, z)|2=[1+(zc/ω0r0)2]-1=[1+(ζ/q)2]-1.
Soff(ν, θ, ζ)iνζ-1S0(ν)exp[-(iν+ν2)(θ2/2θd2)].
Eoff(τ, θ, ζ)E0ζΘ2 {exp[-(τ-τsp)2/2Θ2]}(τ/Θ),
Eoff(τ, θ, ζ)E0ζΘ3 2{exp[-(τ-τsp)2/2Θ2]}(τ/Θ)2,
S(ω, ρ, θ)=iω cos θ2πρc -S0(ω, x, y)dxdy,
E(t˜, ρ, θ)=(eE/2πρc)[q0(t˜)/t˜],
q0(t)=-E0(t, x, y)dxdy,
f1-1+f2-1=2Rm-1.

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