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