Propagation of a femtosecond laser pulse through a dispersion‐free spatially‐chirped medium

Abstract

INTRODUCTIONPropagation of a femtosecond laser pulse in a periodic structure has proved to be an important problem in modern optics, with applications ranging from acousto-optics and holography to waveguides and integrated optics. Such media are used in many optical devices, such as holograms, lasers with distributed feedback, lasers with distributed Bragg reflection, Bragg reflectors with high reflectivity, compressors, etc. [1]. In many cases, approximate methods in the form of the Bloch-Floquet and coupled-mode theories are used to solve this problem, but not for femtosecond pulses [2][3][4].In this paper, the analytical solution of the wave equation that describes the propagation of a femtosecond laser pulse in a dispersion-free finite-length spatially chirped medium with 1D periodic inhomogeneity is presented. The analytical solution of the wave equation is valid for weakly modulated media with the assumption that the number of acoustic wavelengths spanning the medium length is much less than unity. We show that it is possible to modify the pulse duration by selecting the chirp rate and depth of modulation. Our analytical solution is corroborated by our numerical calculations of Maxwell's equations based on the finite-difference time-domain (FDTD) scheme. Reference [5] presents the shape of a 10-fs pulse with a 1-m wavelength, passed through a dispersion-free linear medium with a spatial periodic inhomogeneity length of 20 periods (1 period ϭ 0.33 m). This result was obtained by numerical integration of the corresponding Maxwell's equations for a spatial inhomogeneity with constant period. Obtaining an analytical solution for the femtosecond pulse propagation problem through an inhomogeneous media is therefore a useful result. In this section, we find the analytical description of femtosecond laser pulse propagation in a dispersion-free finitelength spatially chirped medium with 1D periodic inhomogeneity. ANALYTICAL SOLUTION OF THE WAVE EQUATIONIf the spectrum of the femtosecond laser pulse falls in the zero-dispersion region of the medium, then the medium can be considered to be dispersion-free. This condition holds, for example, for fused quartz when the wavelength of the femtosecond laser pulse is equal to 1.3 m. We consider the normal plane wave, incident upon a periodic medium with thickness z ϭ L, in which the speed of propagation of the wave is a quasi-periodic function of the spatial coordinate z, given bywhere c 0 is the speed of light in a vacuum, n 0 is the medium reflection index, K ϭ 2/⌳, ⌳/ 2 is the grating's spatial quasiperiod, is the depth of modulation of the medium refraction index, and ␣ is the coefficient that describes the rate of change of the lattice's spatial period (chirp). For a chosen x polarization of the incoming electric field, we can write the wave equation that describes the propagation of a femtosecond laser pulse in a dispersion-free medium along the z axis as follows [6]:where E ϭ E x is the x component of the electric field. For the spectral component amplitudes of the w...