Abstract

We show how nondispersive quadratic phase media (e.g., lenses, curved mirrors, and gradient-index rods) transform the spatial and temporal profiles of isodiffracting single-cycle pulses. Because all the frequency components of an isodiffracting pulsed beam have the same Rayleigh range, the entire pulse can be transformed directly in the time domain by use of ABCD matrices. The accumulated phase shift plays an important role in changing the spatial and temporal profiles. These pulses form natural spatiotemporal modes of a stable cavity resonator.

© 1999 Optical Society of America

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References

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  1. For a collection of papers on terahertz generation see J. Opt. Soc. Am. B 11, 2454–2580 (1994).
  2. A. Nahata, T. F. Heinz, “Reshaping of freely propagating terahertz pulses by diffraction,” IEEE J. Sel. Top. Quantum Electron. 2, 701–798 (1996).
    [CrossRef]
  3. A. E. Kaplan, “Diffraction-induced transformation of near-cycle and subcycle pulses,” J. Opt. Soc. Am. B 15, 951–956 (1998).
    [CrossRef]
  4. D. You, P. H. Bucksbaum, “Propagation of half-cycle far infrared pulses,” J. Opt. Soc. Am. B 14, 1651–1655 (1997).
    [CrossRef]
  5. R. W. Ziolkowski, J. B. Judkins, “Propagation characteristics of ultrawide-bandwidth pulsed Gaussian beams,” J. Opt. Soc. Am. A 9, 2021–2030 (1992).
    [CrossRef]
  6. S. Feng, H. G. Winful, R. W. Hellwarth, “Gouy shift and temporal reshaping of focused single-cycle electromagnetic pulses,” Opt. Lett. 23, 385–387 (1998); errata 23, 1141 (1998).
    [CrossRef]
  7. S. Feng, H. G. Winful, R. W. Hellwarth, “Spatiotemporal evolution of focused single-cycle electromagnetic pulses,” Phys. Rev. E 59, 4630–4649 (1999).
    [CrossRef]
  8. E. Heyman, “Pulsed beam propagation in inhomogeneous medium,” IEEE Trans. Antennas Propag. 42, 311–319 (1994).
    [CrossRef]
  9. T. Melamed, L. B. Felsen, “Pulsed-beam propagation in lossless dispersive media. I. Theory,” J. Opt. Soc. Am. A 15, 1268–1276 (1998).
    [CrossRef]
  10. A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).
  11. M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58, 1086–1092 (1998);S. Feng, H. G. Winful, “Spatiotemporal structure of isodiffracting ultrashort electromagnetic pulses,” Phys. Rev. E (to be published).
    [CrossRef]
  12. D. Grischkowsky, S. Keiding, M. van Exter, Ch. Fattinger, “Far-infrared time-domain spectroscopy with terahertz beams of dielectrics and semiconductors,” J. Opt. Soc. Am. B 7, 2006–2015 (1990).
    [CrossRef]

1999

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

1998

1997

1996

A. Nahata, T. F. Heinz, “Reshaping of freely propagating terahertz pulses by diffraction,” IEEE J. Sel. Top. Quantum Electron. 2, 701–798 (1996).
[CrossRef]

1994

For a collection of papers on terahertz generation see J. Opt. Soc. Am. B 11, 2454–2580 (1994).

E. Heyman, “Pulsed beam propagation in inhomogeneous medium,” IEEE Trans. Antennas Propag. 42, 311–319 (1994).
[CrossRef]

1992

1990

Bucksbaum, P. H.

Fattinger, Ch.

Felsen, L. B.

Feng, S.

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

S. Feng, H. G. Winful, R. W. Hellwarth, “Gouy shift and temporal reshaping of focused single-cycle electromagnetic pulses,” Opt. Lett. 23, 385–387 (1998); errata 23, 1141 (1998).
[CrossRef]

Grischkowsky, D.

Heinz, T. F.

A. Nahata, T. F. Heinz, “Reshaping of freely propagating terahertz pulses by diffraction,” IEEE J. Sel. Top. Quantum Electron. 2, 701–798 (1996).
[CrossRef]

Hellwarth, R. W.

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

S. Feng, H. G. Winful, R. W. Hellwarth, “Gouy shift and temporal reshaping of focused single-cycle electromagnetic pulses,” Opt. Lett. 23, 385–387 (1998); errata 23, 1141 (1998).
[CrossRef]

Heyman, E.

E. Heyman, “Pulsed beam propagation in inhomogeneous medium,” IEEE Trans. Antennas Propag. 42, 311–319 (1994).
[CrossRef]

Judkins, J. B.

Kaplan, A. E.

Keiding, S.

Melamed, T.

Nahata, A.

A. Nahata, T. F. Heinz, “Reshaping of freely propagating terahertz pulses by diffraction,” IEEE J. Sel. Top. Quantum Electron. 2, 701–798 (1996).
[CrossRef]

Porras, M. A.

M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58, 1086–1092 (1998);S. Feng, H. G. Winful, “Spatiotemporal structure of isodiffracting ultrashort electromagnetic pulses,” Phys. Rev. E (to be published).
[CrossRef]

van Exter, M.

Winful, H. G.

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

S. Feng, H. G. Winful, R. W. Hellwarth, “Gouy shift and temporal reshaping of focused single-cycle electromagnetic pulses,” Opt. Lett. 23, 385–387 (1998); errata 23, 1141 (1998).
[CrossRef]

Yariv, A.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

Yeh, P.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

You, D.

Ziolkowski, R. W.

IEEE J. Sel. Top. Quantum Electron.

A. Nahata, T. F. Heinz, “Reshaping of freely propagating terahertz pulses by diffraction,” IEEE J. Sel. Top. Quantum Electron. 2, 701–798 (1996).
[CrossRef]

IEEE Trans. Antennas Propag.

E. Heyman, “Pulsed beam propagation in inhomogeneous medium,” IEEE Trans. Antennas Propag. 42, 311–319 (1994).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Lett.

Phys. Rev. E

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

M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58, 1086–1092 (1998);S. Feng, H. G. Winful, “Spatiotemporal structure of isodiffracting ultrashort electromagnetic pulses,” Phys. Rev. E (to be published).
[CrossRef]

Other

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

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Figures (10)

Fig. 1
Fig. 1

Schematic of lens transformation of single-cycle pulses. Both z1 and z2 are measured on axis. z1 is the distance from the beam waist of the input pulse to the curved lens interface. z2 is the distance from the curved interface to the output wave front.

Fig. 2
Fig. 2

(a) Spatiotemporal wave forms of a single-cycle pulse at the image location (d2) for different object locations (d1). All waveforms are plotted at the beam waists. With the same object pulse (zR=10 cm) and focal length (f=20 cm), the image pulses are different in both space (beam size) and time (pulse width and phase) for different input pulse locations (d1). The phase shift changes the temporal profiles of the pulse. Here τ0=0.3 ps. The values of the fields are normalized by the peak value of the input pulse. (b) Off-axis temporal profiles corresponding to (a). The space–time coupling results in variations of off-axis time and frequency quantities with distance z.

Fig. 3
Fig. 3

Effect of the refractive index on the spatiotemporal waveforms of single-cycle pulses. All pulses are plotted at their beam waists. For the same object pulse (zR=10 cm) and the same input position (d1=25 cm) the image pulses are different in both space and time for the lens and the spherical dielectric interface (n=2.5) of the same focal length (f=25 cm). The pulse width on axis is invariant, as shown in the bottom plots. However, the beam size, phase shift, and hence the temporal profile and symmetry are affected by the index of refraction. The values of the fields are normalized by the peak value of the input pulse.

Fig. 4
Fig. 4

Successive transformations of a single-cycle pulse (zR=5 cm) by two thin lenses of the same focal length separated by 2 f, with d1=f=20 cm. All the pulses are plotted at the beam waists. A π/2 phase shift at the image beam waist by the first lens transforms a symmetric pulse into an antisymmetric pulse of larger spot size. Another π/2 phase shift, by the second lens, results in an inverted symmetric pulse at the final focal plane. After these two lenses, the pulse reproduces itself in space and time but with opposite polarity. All the pulses are normalized by themselves. τ0=0.3 ps.

Fig. 5
Fig. 5

Spatiotemporal evolution of a single-cycle pulse (zR=10 cm) in a stable lens waveguide of the same focal length f=8 cm and the separation L=30 cm. The position of the input beam waist is at d1=12 cm from the first lens. The fields are plotted in successive planes halfway between each pair of lenses. The beam size, radius of curvature, temporal waveform, symmetry, and polarity change quasi-periodically with propagation. Here the values of the fields are normalized by the peak value of the input pulse. τ0=0.3 ps.

Fig. 6
Fig. 6

Off-axis temporal waveforms of a single-cycle pulse in a lens waveguide corresponding to Fig. 5. The off-axis pulse width, waveform, symmetry, and polarity change quasi-periodically during propagation.

Fig. 7
Fig. 7

Oscillation of beam size and Rayleigh range. These variables have a repetition period that is one half of that of the electric field, as shown in Fig. 6.

Fig. 8
Fig. 8

Comparison of isodiffracting and nonisodiffracting pulses transformed by a four-lens waveguide. The focal lengths are 73.83, 27.42, 32.54, and 107.44 mm. The distances between the lenses are 86.34, 55.20, and 242.40 mm. The input distance of the beam waist is 107.84 mm. Those numbers are randomly generated by computer. The pulses are plotted at the beam waist after each lens element.

Fig. 9
Fig. 9

Comparison of the isodiffracting and nonisodiffracting pulses that propagate a random distance (here 191.94 mm) after passing through the four-lens waveguide in Fig. 8.

Fig. 10
Fig. 10

Spatiotemporal evolution of a single-cycle pulse in a quadratic index medium (l=1 mm). The pulse is coupled into the medium at the beam waist (z0=0). The Rayleigh range of the input pulse is zR=2 mm (top) and zR=1 mm (bottom). For zRl the beam size, radius of curvature, pulse width, peak frequency, temporal profile, symmetry, and polarity oscillate periodically during propagation. For zR=l the pulse temporal profile, symmetry, and polarity change periodically but the beam size, radius of curvature (R), pulse width, and peak frequency stay constant during propagation. The single-cycle pulse resembles a spatiotemporal soliton in the case zR=l. τ0=0.1 fs, n0=2.5.

Equations (41)

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E(r, t)=A0 G(z)exp[-i3 tan-1(T)](1+T2)3/21+ρ2a2(z)3,
T(r, t)t-1cn z+ρ22R(z)τ01+ρ2a2(z),
1q=1R(z)+i 2cnτ0a2(z).
q2=Aq1+BCq1+DQH,
R=|Q|2Re(HQ*),a2=-2cτ0Im(q1) |Q|2.
G=-i/Q.
T(r, t)=t-nc z1n+z2+ρ22Rτ01+ρ2a2,
10F1/n,
1z201.
1+Fz2z2/nF1/n.
q2=(1+Fz2)q1+z2/nFq1+1/nQH,
d2=-Fn2(d12+zR2)+nd1(1+nFd1)2+n2F2zR2,
zRi=zR(1+nFd1)2+n2F2zR2,
a2(d2)=2cτ0zR(1+nFd1)2+n2F2zR2.
ϕ(d2)=tan-1(y, x)=tan-1(-nFzR, 1+nFd1).
ϕ(d2)=tan-1(zR, f-d1).
tan-1(d1/zR)+tan-1(d2/z0)=tan-1(zR, f-d1),
n(ρ)=n01-ρ22l2,ρl,
q(z)=q0 cos(z/l)+l sin(z/l)-q0/l sin(z/l)+cos(z/l),
ϕ(Z)=tan-1Z0+tan ZZR,
η(r)=1+ρ2a2(z),
a2(z)=2cnτ0zR cos2 Z1+Z0+tan ZZR2.
τp(r)=τpoη(r),ΔωFWHM(r)=ΔωFWHMoη(r),
fp(r)=fpoη(r),
R(Z)=l tan Z1-ZR2 1+ZRtan Z2.
a2(Z)=2cnτ0zR cos2 Z1+tan ZZR2.
G(Z)=exp[-iϕ(Z)]l cos Z(tan2 Z+ZR2)1/2,
ϕ(Z)=tan-1tan ZZR.
b=2zR2πw02λ=L(2R-L).
w0=λL(2R-L)4π21/4.
F˜p(ω)=(ω-ω0)p exp[-(ω-ω0)τ0](ω>ω0>0),
2+2ik zΨ(r, ω)=0,
Ψ(r, ω)=1z-izR expikρ22(z-izR),
E˜p(r, ω)=F˜p(ω) exp(ikz)z-izR expikρ22(z-izR)
(ω>ω0>0).
Ep(r, t)=Ap G(z)exp[-i(p+1)tan-1(T)](1+T2)(p+1)/2η(r)p+1×exp[-iω0τ-ρ2/w2(z)]
τ(r, t)=t-1cn z+ρ22R(z)
η(r)=1+ρ2a2(z).
exp-ωρ2zR2cn(z2+zR2)
=exp-(ω-ω0)ρ2zR2cn(z2+zR2)exp-ω0ρ2zR2cn(z2+zR2).
1q=1R(z)+i λnπw2(z).

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