Abstract

Based on two theorems, the importance of the root-mean-square (rms) width for the characterization of ultrashort optical pulses is demonstrated. First, it is shown that one can directly determine the rms width from the autocorrelation without making any assumptions about the specific form of the pulse envelope. Second, it is shown that a bandwidth-limited (unchirped) wave packet has the smallest possible rms time–bandwidth product. This reveals a natural definition for a rms chirp that is easily accessible to experimental measurement and that presents a useful measure for the quality of pulse compression techniques.

© 2000 Optical Society of America

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References

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  1. K. L. Sala, G. A. Kenney-Wallace, and G. E. Hall, IEEE J. Quantum Electron. 16, 990–996 (1980).
    [CrossRef]
  2. For a review, see the feature issue on ultrashort-laser-pulse intensity and phase measurement and applications, IEEE J. Quantum Electron. 35, 418–523 (1999), and references therein.
  3. U. Morgner, F. X. Kärtner, S. H. Cho, Y. Chen, H. A. Haus, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, and T. Tschudi, Opt. Lett. 24, 411–413 (1999); D. H. Sutter, G. Steinmeyer, L. Gallmann, N. Matuschek, F. Morier-Genoud, U. Keller, V. Scheuer, G. Angelow, and T. Tschudi, Opt. Lett. 24, 631–633 (1999).
    [CrossRef]
  4. A. Baltuska, Z. Wei, M. S. Pshenichnikov, and D. A. Wiersma, Opt. Lett. 22, 102–104 (1997).
    [CrossRef] [PubMed]
  5. S. Sartania, Z. Cheng, M. Lenzner, G. Tempea, Ch. Spielmann, F. Krausz, and K. Ferencz, Opt. Lett. 22, 1562–1564 (1997).
    [CrossRef]
  6. T. Brabec and F. Krausz, Phys. Rev. Lett. 78, 3282 (1997).
    [CrossRef]
  7. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1995).
  8. K. Naganuma, K. Mogi, and H. Yamada, IEEE J. Quantum Electron. 25, 1225 (1989).
    [CrossRef]
  9. For the derivation, we need to make the following assumptions. We require that the complex envelopes U and tU are absolutely integrable; that is, ∫−∞|tnU(t)|dt for n=0, 1 is finite. The absolute integrability ensures that the Fourier transform of U and its first derivative, Ũ(ω), dŨ/dω→ 0 for ω→±∞.
  10. G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, San Diego, Calif., 1995).
  11. A. H. Liang and H. K. Tsang, IEEE J. Quantum Electron. 32, 2064 (1996); A. H. Liang, H. K. Tsang, and L. Y. Chan, J. Opt. Soc. Am. B 13, 2464 (1996).
    [CrossRef]
  12. D. Yelin, D. Meshulach, and Y. Silberberg, Opt. Lett. 22, 1793–1795 (1997); T. Baumert, T. Brixner, V. Setfried, M. Strehle, and G. Gerber, Appl. Phys. B 65, 779 (1997); A. Efimov, M. D. Moores, N. M. Beach, J. L. Krause, and D. H. Reitze, Opt. Lett. OPLEDP 23, 1915–1917 (1998).
    [CrossRef]
  13. R. N. Bracewell, Fourier Transform and Its Applications, 3rd ed. (McGraw-Hill, New York, 1986), p. 172.
  14. If the operator L does not contain explicit functions of time and the boundary conditions are defined at t=±∞, then the Green’s function G(t, t) has the form G(t−t). Most, if not all, physical processes of interest satisfy these conditions.

1997 (3)

1989 (1)

K. Naganuma, K. Mogi, and H. Yamada, IEEE J. Quantum Electron. 25, 1225 (1989).
[CrossRef]

1980 (1)

K. L. Sala, G. A. Kenney-Wallace, and G. E. Hall, IEEE J. Quantum Electron. 16, 990–996 (1980).
[CrossRef]

Baltuska, A.

Brabec, T.

T. Brabec and F. Krausz, Phys. Rev. Lett. 78, 3282 (1997).
[CrossRef]

Cheng, Z.

Ferencz, K.

Hall, G. E.

K. L. Sala, G. A. Kenney-Wallace, and G. E. Hall, IEEE J. Quantum Electron. 16, 990–996 (1980).
[CrossRef]

Kenney-Wallace, G. A.

K. L. Sala, G. A. Kenney-Wallace, and G. E. Hall, IEEE J. Quantum Electron. 16, 990–996 (1980).
[CrossRef]

Krausz, F.

Lenzner, M.

Mogi, K.

K. Naganuma, K. Mogi, and H. Yamada, IEEE J. Quantum Electron. 25, 1225 (1989).
[CrossRef]

Naganuma, K.

K. Naganuma, K. Mogi, and H. Yamada, IEEE J. Quantum Electron. 25, 1225 (1989).
[CrossRef]

Pshenichnikov, M. S.

Sala, K. L.

K. L. Sala, G. A. Kenney-Wallace, and G. E. Hall, IEEE J. Quantum Electron. 16, 990–996 (1980).
[CrossRef]

Sartania, S.

Spielmann, Ch.

Tempea, G.

Wei, Z.

Wiersma, D. A.

Yamada, H.

K. Naganuma, K. Mogi, and H. Yamada, IEEE J. Quantum Electron. 25, 1225 (1989).
[CrossRef]

IEEE J. Quantum Electron. (2)

K. Naganuma, K. Mogi, and H. Yamada, IEEE J. Quantum Electron. 25, 1225 (1989).
[CrossRef]

K. L. Sala, G. A. Kenney-Wallace, and G. E. Hall, IEEE J. Quantum Electron. 16, 990–996 (1980).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. Lett. (1)

T. Brabec and F. Krausz, Phys. Rev. Lett. 78, 3282 (1997).
[CrossRef]

Other (9)

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1995).

For the derivation, we need to make the following assumptions. We require that the complex envelopes U and tU are absolutely integrable; that is, ∫−∞|tnU(t)|dt for n=0, 1 is finite. The absolute integrability ensures that the Fourier transform of U and its first derivative, Ũ(ω), dŨ/dω→ 0 for ω→±∞.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, San Diego, Calif., 1995).

A. H. Liang and H. K. Tsang, IEEE J. Quantum Electron. 32, 2064 (1996); A. H. Liang, H. K. Tsang, and L. Y. Chan, J. Opt. Soc. Am. B 13, 2464 (1996).
[CrossRef]

D. Yelin, D. Meshulach, and Y. Silberberg, Opt. Lett. 22, 1793–1795 (1997); T. Baumert, T. Brixner, V. Setfried, M. Strehle, and G. Gerber, Appl. Phys. B 65, 779 (1997); A. Efimov, M. D. Moores, N. M. Beach, J. L. Krause, and D. H. Reitze, Opt. Lett. OPLEDP 23, 1915–1917 (1998).
[CrossRef]

R. N. Bracewell, Fourier Transform and Its Applications, 3rd ed. (McGraw-Hill, New York, 1986), p. 172.

If the operator L does not contain explicit functions of time and the boundary conditions are defined at t=±∞, then the Green’s function G(t, t) has the form G(t−t). Most, if not all, physical processes of interest satisfy these conditions.

For a review, see the feature issue on ultrashort-laser-pulse intensity and phase measurement and applications, IEEE J. Quantum Electron. 35, 418–523 (1999), and references therein.

U. Morgner, F. X. Kärtner, S. H. Cho, Y. Chen, H. A. Haus, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, and T. Tschudi, Opt. Lett. 24, 411–413 (1999); D. H. Sutter, G. Steinmeyer, L. Gallmann, N. Matuschek, F. Morier-Genoud, U. Keller, V. Scheuer, G. Angelow, and T. Tschudi, Opt. Lett. 24, 631–633 (1999).
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Relative deviation 1-σ/σ as a function of rms width σ for various pulse shapes. The rms width σ is determined from the IAC, as described in the text.

Fig. 2
Fig. 2

(a) Intensity profile of an optical pulse after propagation in a monomode fiber (dotted curve) and after a dispersive delay line (solid curve). For the parameters, see the text. (b) Intensity spectrum and the actual phase of the spectrum after the fiber (dotted curve). The dashed curve denotes the least-squares fit of a quadratic phase (which is compensated in a dispersive delay line) to the actual phase. As the actual phase is nearly quadratic, the dotted and dashed curves overlap. (c) AC functions after the fiber (dotted curve) and after the delay line (solid curve).

Tables (1)

Tables Icon

Table 1 Comparison of the Pulse Duration and of the Time–Bandwidth Product as Determined by the rms Width and by the FWHM Width for Some Characteristic Pulse Shapesa

Equations (25)

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σ=t2-t2,
tn=1N-tnI(t)d t,N=-I(t)d t,
σ2=12σ2(g) 12N2-τ2g(τ)dτ,
tn=inN-U˜*(ω) dndωnU˜(ω)dω,
σp2-σ02=1N-dωA˜2dφdω2-1N-dω A˜2 dφdω2.
1N-dωA˜2 dφdω21N2-dωA˜2dφdω2-dωA˜2=1N-dωA˜2dφdω2.
σp2-σ020.
C=1σ0dφdω2-dφdω21/2=σpσ02-11/2
σ2(f·g)=σ2(f)+σ2(g),
L f(t)=Ipump(t),
S(τ)-f(t)Iprobe(t+τ)d t.
f(t)=-G(t-t)Ipump(t)d t.
σ2(S)=σ2(G)+σ2(Ipump)+σ2(Iprobe).
exp[-(t/τ)2]
2 ln 2τ
2 ln 2/π
sech(t/τ)
(π/3)τ
2 ln(2+1)τ
tanh(π/23)
11+(t/τ)2
22-1τ
2/π
2-1 ln 2/π
exp[-(t/τ)4]

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