Abstract

A simple but exact treatment of spatiotemporal behavior of ultrawideband pulses under an arbitrarily tight focusing is developed. The model makes use of the oblate spheroidal coordinate system to represent free scalar field as if generated by a point-like source-and-sink pair placed at a complex location. The results, illustrated by animated 3D plots, demonstrate characteristic temporal reshaping of the pulses in the course of propagation through the focus, which is a spectacular manifestation of the Gouy phase shift. It is shown that the salient features of the reshaping, which were recently established for the paraxial limit, remain valid beyond it. The treatment is particularly applicable to an ultrawideband isodiffracting ultrashort terahertz-domain or light pulses in high-aperture resonators, such as microcavities, and it is usable in femto- and attosecond optics in general.

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References

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  1. C. J. R. Sheppard and H. J. Matthews, "Imaging in high-aperture optical systems," J. Opt. Soc. Am. A 4, 1354-1360 (1987).
    [CrossRef]
  2. C. J. R. Sheppard and S. Saghafi, "Beam modes beyond the paraxial approximation: A scalar treatment," Phys. Rev. A 57, 2971-2979 (1998).
    [CrossRef]
  3. Z. Ulanowski and I. K. Ludlow, "Scalar field of nonparaxial Gaussian beams," Opt. Lett. 25, 1792-1794 (2000).
    [CrossRef]
  4. G. A. Deschamps, "Gaussian beam as a bundle of complex rays," Electron. Lett. 7, 684-685 (1971).
    [CrossRef]
  5. R. W. Hellwarth and P. Nouchi, "Focused one-cycle electromagnetic pulses," Phys. Rev. E 54, 889-896 (1996).
    [CrossRef]
  6. A. E. Kaplan, "Diffraction-induced transformation of near-cycle and subcycle pulses," J. Opt. Soc. Am. B 15, 951-956 (1998).
    [CrossRef]
  7. M. A. Porras, "Ultrashort pulsed Gaussian light beams," Phys. Rev. E 58, 1086-1093 (1998).
    [CrossRef]
  8. S. M. Feng, H. G. Winful, and R. W. Hellwarth "Spatiotemporal evolution of focused single-cycle electromagnetic pulses," Phys. Rev. E 59, 4630-4649 (1999).
    [CrossRef]
  9. Z. L. Horv�th and Zs. Bor, "Reshaping of femtosecond pulses by the Gouy phase shift," Phys. Rev. E 60, 2337-2345 (1999).
    [CrossRef]
  10. S. M. Feng and H. G. Winful, "Spatiotemporal structure of isodiffracting ultrashort electromagnetic pulses," Phys. Rev. E 61, 862-873 (2000).
    [CrossRef]
  11. B. T. Landesman and H. H. Barret, "Gaussian amplitude functions that are exact solutions to the scalar Helmholtz equation," J. Opt. Soc. Am. A 5, 1610-1619 (1988).
    [CrossRef]
  12. P. Saari, "Superluminal localized waves of electromagnetic field in vacuo," in Proc. Intern. Conf. "Time's Arrows, Quantum Measurements and Superluminal Behaviour," Naples, Oct.2-5, 2000 (to be published by the Italian NCR), Los Alamos preprint archive, http://xxx.lanl.gov/abs/physics/0103054
  13. E. Heyman, B. Z. Steinberg, and L. B. Felsen, "Spectral analysis of focus wave modes," J. Opt. Soc. Am. A 4, 2081-2091 (1987).
    [CrossRef]
  14. R. W. Ziolkowski, I. M. Besieris, and A. M. Shaarawi, "Aperture realizations of exact solutions to homogeneous wave equations," J. Opt. Soc. Am. A 10, 75-87 (1993).
    [CrossRef]
  15. P. Saari and K. Reivelt, "Evidence of X-shaped propagation-invariant localized light waves," Phys. Rev. Lett. 79, 4135-4138 (1997).
    [CrossRef]
  16. K. Reivelt and P. Saari, "Optical generation of focus wave modes,"J. Opt. Soc. Am. A 17, 1785-1790 (2000).
    [CrossRef]
  17. C. J. R. Sheppard and S. Saghafi, "Electromagnetic Gaussian beams beyond the paraxial approximation," J. Opt. Soc. Am. A 16, 1381-1386 (1999).
    [CrossRef]

Other (17)

C. J. R. Sheppard and H. J. Matthews, "Imaging in high-aperture optical systems," J. Opt. Soc. Am. A 4, 1354-1360 (1987).
[CrossRef]

C. J. R. Sheppard and S. Saghafi, "Beam modes beyond the paraxial approximation: A scalar treatment," Phys. Rev. A 57, 2971-2979 (1998).
[CrossRef]

Z. Ulanowski and I. K. Ludlow, "Scalar field of nonparaxial Gaussian beams," Opt. Lett. 25, 1792-1794 (2000).
[CrossRef]

G. A. Deschamps, "Gaussian beam as a bundle of complex rays," Electron. Lett. 7, 684-685 (1971).
[CrossRef]

R. W. Hellwarth and P. Nouchi, "Focused one-cycle electromagnetic pulses," Phys. Rev. E 54, 889-896 (1996).
[CrossRef]

A. E. Kaplan, "Diffraction-induced transformation of near-cycle and subcycle pulses," J. Opt. Soc. Am. B 15, 951-956 (1998).
[CrossRef]

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

S. M. Feng, H. G. Winful, and R. W. Hellwarth "Spatiotemporal evolution of focused single-cycle electromagnetic pulses," Phys. Rev. E 59, 4630-4649 (1999).
[CrossRef]

Z. L. Horv�th and Zs. Bor, "Reshaping of femtosecond pulses by the Gouy phase shift," Phys. Rev. E 60, 2337-2345 (1999).
[CrossRef]

S. M. Feng and H. G. Winful, "Spatiotemporal structure of isodiffracting ultrashort electromagnetic pulses," Phys. Rev. E 61, 862-873 (2000).
[CrossRef]

B. T. Landesman and H. H. Barret, "Gaussian amplitude functions that are exact solutions to the scalar Helmholtz equation," J. Opt. Soc. Am. A 5, 1610-1619 (1988).
[CrossRef]

P. Saari, "Superluminal localized waves of electromagnetic field in vacuo," in Proc. Intern. Conf. "Time's Arrows, Quantum Measurements and Superluminal Behaviour," Naples, Oct.2-5, 2000 (to be published by the Italian NCR), Los Alamos preprint archive, http://xxx.lanl.gov/abs/physics/0103054

E. Heyman, B. Z. Steinberg, and L. B. Felsen, "Spectral analysis of focus wave modes," J. Opt. Soc. Am. A 4, 2081-2091 (1987).
[CrossRef]

R. W. Ziolkowski, I. M. Besieris, and A. M. Shaarawi, "Aperture realizations of exact solutions to homogeneous wave equations," J. Opt. Soc. Am. A 10, 75-87 (1993).
[CrossRef]

P. Saari and K. Reivelt, "Evidence of X-shaped propagation-invariant localized light waves," Phys. Rev. Lett. 79, 4135-4138 (1997).
[CrossRef]

K. Reivelt and P. Saari, "Optical generation of focus wave modes,"J. Opt. Soc. Am. A 17, 1785-1790 (2000).
[CrossRef]

C. J. R. Sheppard and S. Saghafi, "Electromagnetic Gaussian beams beyond the paraxial approximation," J. Opt. Soc. Am. A 16, 1381-1386 (1999).
[CrossRef]

Supplementary Material (5)

» Media 1: MOV (687 KB)     
» Media 2: MOV (423 KB)     
» Media 3: MOV (2234 KB)     
» Media 4: MOV (2518 KB)     
» Media 5: MOV (974 KB)     

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Figures (6)

Fig. 1.
Fig. 1.

The elementary free field D(t,R). The first 0.7MB video clip shows its temporal evolution. The field is nonzero only on the transparent spherical surface, which has been made blue to mark the negative values and red – the positive values of the wave function.

Fig. 2.
Fig. 2.

Intersection of a surface ξ=const (red) and a surface θ=const (blue) with the coordinate plane φ=±π/2 in the case of the oblate spheroidal coordinate system with the parameter d=0.5, see also Fig.3. (0.4 MB video clip shows the tranformation from the spherical coordinate system).

Fig. 3.
Fig. 3.

The z-x plane (or two meridian planes φ=0 and φ=π) and lines of intersection with the coordinate surfaces ξ=0.5, 1, 1.5, 2,…. and θ=0, 0.1π, 0.2π,… Note that the pair of blue θ-hyperbolas, nearest to the axis z, together with the arcs of red ξ-ellipses between them, are reminiscent of a scheme of the (paraxial) Gaussian beam.

Fig. 4.
Fig. 4.

The frame t=0 of the animated plot (2.2 MB) of the pulse given by the real part of Eq. (5). The propagation axis z has been directed from left to right. For a better visualization a color “lighting” of the surface plot has been used and a colored contour plot of the same data is shown at the bottom. The Rayleigh range d=0.5, the pulsewidth a=0.05 (values given relative to the confocal parameter, the same unit of length has been used in the scales on the basal plane). The vertical scale of the amplitude has been normalized to the unit “charge” q=1. Fast start and end frame jumps have been caused by cutting the clip shorter in order to meet the file length limit. The “shakes” of the scales in the animation indicate the time moments of the penetration of the basal plane by the pulse peak.

Fig. 5.
Fig. 5.

The frame t=0 of the animated plot (2.5 MB) of the pulse given by the imaginary part of Eq. (5). Other parameters and units are the same as in Fig. 4. The pulse peak has been partially cut off, as its highest value (~19) falls outside the vertical scale.

Fig. 6.
Fig. 6.

A frame of the animated 2-D plot (1 MB) showing the on-axis behavior of the real part (blue line), the imaginary part (magenta), and the modulus (dotted) of Eq. (5). Other parameters and units are the same as in Fig. 4. The frame depicts the distribution of the field along the z axis at the converging stage (at instant 2d/c prior to the focus).

Equations (9)

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D ( t , R ) = c 4 πR [ δ ( R ct ) δ ( R + ct ) ]
ρ ( t , r ) = ( r ) t 0 it + Δ ,
ϕ ( t , r ) = c R dt qt 0 it + Δ { δ [ R c ( t t ) ] δ [ R + c ( t t ) ] } =
= q R [ ct 0 i ( R ct ) + c Δ ct 0 i ( R + ct ) + c Δ ] = q 2 ict 0 R 2 + c 2 ( it + Δ ) 2 .
x = d 1 + ξ 2 sin θ cos φ , y = d 1 + ξ 2 sin θ sin φ , z = d ξ cos θ ,
0 ξ , 0 θ π , 0 φ 2 π .
ϕ ( t , ξ , η ) = qd 2 i d 2 ( ξ ) 2 + ( ict + d + a ) 2 .
ϕ ( τ , ξ , η ) = 2 q d e i [ π 2 + tan 1 ( τ δ ) tan 1 ( 2 ξ τ δ + ) ] ( τ 2 + δ - 2 ) 1 2 [ ( 2 ξ τ ) 2 + δ + 2 ] 1 2 .
ϕ ( τ , ξ , η ) q d e i [ π 2 + tan 1 ( τ δ ) tan 1 ( ξ ) ] ( τ 2 + δ 2 ) 1 2 ( ξ 2 + 1 ) 1 2 .

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