Abstract

The Sommerfeld diffraction theory is extended to the case of extremely short pulses. We show that in the far field the energy-density distribution of the diffractional pattern is transformed into a Gaussian distribution, when the plane wave with uniform radial amplitude distribution and one oscillation period falls upon the circular aperture. When one focuses the Gaussian beam with one oscillation period, the energy-density distribution in the focal plane differs from the Gaussian distribution.

© 1999 Optical Society of America

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References

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  1. L. Xu, Ch. Spielman, F. Krausz, R. Szipocs, “Ultrabroadband ring oscillator for sub-10-fs pulse generation,” Opt. Lett. 21, 1259–1261 (1996).
    [CrossRef] [PubMed]
  2. E. M. Belenov, A. V. Nazarkin, “Nonstationary diffraction effects in the process of propagation of electromagnetic field clots in vacuum,” JETP Lett. 53, 188–191 (1991).
  3. Z. Wang, Z. Zhang, Z. Xu, Q. Lin, “Space–time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33, 566–573 (1997).
    [CrossRef]
  4. E. M. Belenov, A. V. Nazarkin, V. A. Ushchapovskii, “The propagation and interaction dynamics of electromagnetic field clots in two-level media,” Zh. Eksp. Teor. Fiz. 100, 762–775 (1991) [Sov. Phys. JETP 73, 422–435 (1991)].

1997 (1)

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

1996 (1)

1991 (2)

E. M. Belenov, A. V. Nazarkin, “Nonstationary diffraction effects in the process of propagation of electromagnetic field clots in vacuum,” JETP Lett. 53, 188–191 (1991).

E. M. Belenov, A. V. Nazarkin, V. A. Ushchapovskii, “The propagation and interaction dynamics of electromagnetic field clots in two-level media,” Zh. Eksp. Teor. Fiz. 100, 762–775 (1991) [Sov. Phys. JETP 73, 422–435 (1991)].

Belenov, E. M.

E. M. Belenov, A. V. Nazarkin, V. A. Ushchapovskii, “The propagation and interaction dynamics of electromagnetic field clots in two-level media,” Zh. Eksp. Teor. Fiz. 100, 762–775 (1991) [Sov. Phys. JETP 73, 422–435 (1991)].

E. M. Belenov, A. V. Nazarkin, “Nonstationary diffraction effects in the process of propagation of electromagnetic field clots in vacuum,” JETP Lett. 53, 188–191 (1991).

Krausz, F.

Lin, Q.

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

Nazarkin, A. V.

E. M. Belenov, A. V. Nazarkin, “Nonstationary diffraction effects in the process of propagation of electromagnetic field clots in vacuum,” JETP Lett. 53, 188–191 (1991).

E. M. Belenov, A. V. Nazarkin, V. A. Ushchapovskii, “The propagation and interaction dynamics of electromagnetic field clots in two-level media,” Zh. Eksp. Teor. Fiz. 100, 762–775 (1991) [Sov. Phys. JETP 73, 422–435 (1991)].

Spielman, Ch.

Szipocs, R.

Ushchapovskii, V. A.

E. M. Belenov, A. V. Nazarkin, V. A. Ushchapovskii, “The propagation and interaction dynamics of electromagnetic field clots in two-level media,” Zh. Eksp. Teor. Fiz. 100, 762–775 (1991) [Sov. Phys. JETP 73, 422–435 (1991)].

Wang, Z.

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

Xu, L.

Xu, Z.

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

IEEE J. Quantum Electron. (1)

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

JETP Lett. (1)

E. M. Belenov, A. V. Nazarkin, “Nonstationary diffraction effects in the process of propagation of electromagnetic field clots in vacuum,” JETP Lett. 53, 188–191 (1991).

Opt. Lett. (1)

Zh. Eksp. Teor. Fiz. (1)

E. M. Belenov, A. V. Nazarkin, V. A. Ushchapovskii, “The propagation and interaction dynamics of electromagnetic field clots in two-level media,” Zh. Eksp. Teor. Fiz. 100, 762–775 (1991) [Sov. Phys. JETP 73, 422–435 (1991)].

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Figures (4)

Fig. 1
Fig. 1

Variation in energy density with distance z in the case of the diffraction of a pulse consisting of one electromagnetic-oscillation period.

Fig. 2
Fig. 2

Energy density versus transverse coordinate ρ when z/L d is, 1, 0.3, 2, 0.6, 3, 1.2, 4, 6.0, 5, 12.

Fig. 3
Fig. 3

Temporal behavior of a diffracted short pulse with the Gaussian transverse distribution of the field amplitude at distance z = 2L d . The distance from the z axis, ρ/ρ0, is 1, 0, 2, 1.0, 3, 2.0, 4, 3.0, 5, 5.0.

Fig. 4
Fig. 4

Energy density in the focal plane as a function of the transverse coordinate ρ for, 1, the monochromatic wave and, 2, the one-period optical pulse.

Equations (31)

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Ex, y, z=2  Ex, y, 0Gzz=0dxdy,
Gzz=0ik RzexpikR4πR=-ik zRexpikR4πR.
Ex, y, z, t=t  Ex, y, 0, ω×exp-iωt2zcRexpiωR/c4πRdxdydω.
Ex, y, z, t=t  Ex, y, 0, t×z2πcR2 δt-tdtdxdy.
Ex, y, z, t=12πc  t Ex, y, 0, t-R/c×cos ϑRdxdy.
E0, 0, z, t=12πczz2+ρ021/2t×E0t-R/ccos ϑR 2πRdR.
E0, 0, z, t=-zz2+ρ021/2R E0t-R/cdR=E0t-z/c-E0t-z2+ρ021/2/c.
W= E20, 0, z, tdt=2  E02tdt,
W0= E02tdt.
E0, 0, z, t=E0t-z/c-E0t-z2+ρ021/2/cE0t-z/ctρ022zc,
Ex, y, z, t=12πc  t Ex, y, 0, t-Rc+ρ22fc×zR2dxdy.
E0, 0, f, t=fc  t Eρ, 0, t-fcρdρf2+ρ2.
E0, 0, f, t=fct ETt-fc  ESρρdρf2+ρ2.
ESρ=exp-ρ2/ρ02,
E0, 0, f, t=-f2ct ETt-fc×expf2ρ02--f2/ρ02expααdα.
E0, 0, f, t=f2ct ETt-fclnf2+ρ02f2.
E0, 0, f, t=ρ022fct ETt-fc.
2Ex2+2Ey2+2Ez2 =1c22Et2.
Ex, y, z, t=12π3/2  Fkx, ky, z, ω×expikxx+kyy+ωtdkxdkydω,
2Fz2+ω2c2-kx2-ky2F=0.
Fkx, ky, z, ω=F0kx, ky, ω×exp-izω2c2-kx2-ky21/2,
F0kx, ky, ω=12π  Fωexp-x2+y2/ρ02×exp-ikxx+kyydxdy
Ex, y, z, t=12π1/2  Fωexpiωt-z/c×ωρ02ωρ02-2izc exp-ωx2+y2ωρ02-2izcdω.
Eρ, z, t=12π1/2  Fωexpiωt-z/c×2ρ02fω2ρ02fω-2ρ02zω-4ifzc×exp-2ρ2fω-iρ2ω2ρ02/c2ρ02fω-2ρ02zω-4ifzcdω,
ETt=t/τ0exp-t2/τ02.
Eρ, f, t=ρ02τ0216fc p-3/21+q22pexpq24p,
qρ, t=iρ22fc-t-fc,  pρ=ρ2ρ024f2c2+τ024.
ωq=- F2ωω2dω- F2ωdω1/21.73τ0-1
ETt=sin2πt/τ0  when |t|τ0/2,  ETt=0  when |t|>τ0/2,
ρpulse=0 Wρρdρ0 Wρdρ=4.6×10-5ρ0,
WzW0=1-zf2+z2LG2-1,

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