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

1992

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

1990

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

1988

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.

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

J. Chem. Phys.

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

J. Opt. Soc. Am. B

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

Opt. Lett.

Other

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).

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

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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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