Theory of self-phase modulation and spectral broadening

Abstract

Self-phase modulation refers to the phenomenon in which a laser beam propagating in a medium interacts with the medium and imposes a phase modulation on itself. It is one of those very fascinating effects discovered in the early days of nonlinear optics (Bloembergen and Lallemand, 1966; Brewer, 1967; Cheung et al., 1968; Lallemand, 1966; Jones and Stoicheff, 1964; Shimizu, 1967; Stoicheff, 1963). The physical origin of the phenomenon lies in the fact that the strong field of a laser beam is capable of inducing an appreciable intensity-dependent refractive index change in the medium. The medium then reacts back and inflicts a phase change on the incoming wave, resulting in self-phase modulation (SPM). Since a laser beam has a finite cross section, and hence a transverse intensity profile, SPM on the beam should have a transverse spatial dependence, equivalent to a distortion of the wave front. Consequently, the beam will appear to have self-diffracted. Such a selfdiffraction action, resulting from SPM in space, is responsible for the wellknown nonlinear optical phenomena of self-focusing and self-defocusing (Marburger, 1975; Shen, 1975). It can give rise to a multiple ring structure in the diffracted beam if the SPM is sufficiently strong (Durbin et al., 1981; Santamato and Shen, 1984). In the case of a pulsed laser input, the temporal variation of the laser intensity leads to an SPM in time. Since the time derivative of the phase of a wave is simply the angular frequency of the wave, SPM also appears as a frequency modulation. Thus, the output beam appears with a self-induced spectral broadening (Cheung et al., 1968; Gustafson et al., 1969; Shimizu, 1967). In this chapter we are concerned mainly with SPM that leads to spectral broadening (Bloembergen and Lallemand, 1966; Brewer, 1967; Cheung et al., 1968; Lallemand, 1966; Jones and Stoicheff, 1964; Shimizu, 1967; Stoicheff, 1963). For large spectral broadening, we need a strong SPM in time (i.e., a large time derivative in the phase change). This obviously favors the use of short laser pulses. Consider, for example, a phase change of 6π occurring in 10-12 s. Such a phase modulation would yield a spectral broadening of ~100cm -1. In practice, with sufficiently intense femtosecond laser pulses, a spectral broadening of 20,000cm-1 is readily achievable by SPM in a condensed medium, which is essentially a white continuum (Alfano and Shapiro, 1970). The pulse duration of any frequency component (uncertainty limited) in the continuum is not very different from that of the input pulse (Topp and Rentzepis, 1971). This spectrally superbroadened output from SPM therefore provides a much needed light source in ultrafast spectroscopic studies - tunable femtosecond light pulses (Busch et al., 1973; Alfano and Shapiro, 1971). If the SPM and hence the frequency sweep in time on a laser pulse are known, then it is possible to send the pulse through a properly designed dispersive delay system to compensate the phase modulation and generate a compressed pulse with little phase modulation (Treacy, 1968, 1969). Such a scheme has been employed to produce the shortest light pulses ever known (Fork et al., 1987; Ippen and Shank, 1975; Nakatsuka and Grischkowsky, 1981; Nakatsuka et al., 1981; Nikolaus and Grischkowsky, 1983a, 1983b). Self-phase modulation was first proposed by Shimizu (1967) to explain the observed spectrally broadened output from self-focusing of a Q-switched laser pulse in liquids with large optical Kerr constants (Bloembergen and Lallemand, 1966; Brewer, 1967; Cheung et al., 1968; Jones and Stoicheff, 1964; Lallemand, 1966; Shimizu, 1967; Stoicheff, 1963). In this case, the spectral broadening is generally of the order of a hundred reciprocal centimeters. Alfano and Shapiro (1970) showed that with picosecond laser pulses, it is possible to generate by SPM a spectrally broadened output extending over 10,000cm-1 in almost any transparent condensed medium. Self-focusing is believed to have played an important role in the SPM process in the latter case. In order to study the pure SPM process, one would like to keep the beam cross section constant over the entire propagation distance in the medium. This can be achieved in an optical fiber since the beam cross section of a guided wave should be constant and the self-focusing effect is often negligible. Stolin and Lin (1978) found that indeed the observed spectral broadening of a laser pulse propagating through a long fiber can be well explained by the simple SPM theory. Utilizing a well-defined SPM from an optical fiber, Grischkowsky and co-workers were then able to design a pulse compression system that could compress a laser pulse to a few hundredths of its original width (Nakatsuka and Grischkowsky, 1981; Nakatsuka et al., 1981; Nikolaus and Grischkowsky, 1983a, 1983b). With femtosecond laser pulses, a strong SPM on the pulses could be generated by simply passing the pulses through a thin film. In this case, the beam cross section is practically unchanged throughout the film, and one could again expect a pure SPM process. Fork et al. (1983) observed the generation of a white continuum by focusing an 80-fs pulse to an intensity of ~1014W/cm2 on a 500-μm ethylene glycol film. Their results can be understood by SPM along with the self-steepening effect (Manassah et al., 1985, 1986; Yang and Shen, 1984). Among other experiments, Corkum et al. (1985) demonstrated that SPM and spectral broadening can also occur in a medium with infrared laser pulses. More recently, Corkum et al. (1986) and Glownia et al. (1986) have independently shown that with femtosecond pulses it is even possible to generate a white continuum in gas media. The phase modulation induced by one laser pulse can also be transferred to another pulse at a different wavelength via the induced refractive index change in a medium. A number of such experiments have been carried out by Alfano and co-workers (1986, 1987). Quantitative experiments on spectral superbroadening are generally difficult. Self-focusing often complicates the observation. Even without self-focusing, quantitative measurements of a spectrum that is generated via a nonlinear effect by a high-power laser pulse and extends from infrared to ultraviolet are not easy. Laser fluctuations could lead to large variations in the output. The simple theory of SPM considering only the lower-order effect is quite straightforward (Gustafson et al., 1969; Shimizu, 1967). Even the more rigorous theory including the higher-order contribution is not difficult to grasp as long as the dispersive effect can be neglected (Manassah et al., 1985, 1986; Yang and Shen, 1984). Dispersion in the material response, however, could be important in SPM, and resonances in the medium would introduce pronounced resonant structure in the broadened spectrum. The SPM theory with dispersion is generally very complex; one often needs to resort to a numerical solution (Fischen and Bischel, 1975; Fisher et al., 1983). It is possible to describe the spectral broadening phenomenon as resulting from a parametric wave mixing process (in the pump depletion limit) (Bloembergen and Lallemand, 1966; Lallemand, 1966; Penzkofer, 1974; Penzkofer et al., 1973, 1975). In fact, in the studies of spectral broadening with femtosecond pulses, four-wave parametric generation of new frequency components in the phasematched directions away from the main beam can be observed together with the spectrally broadened main beam. Unfortunately, a quantitative estimate of spectral broadening due to the parametric process is not easy. In the presence of self-focusing, more complication arises. Intermixing of SPM in space and SPM in time makes even numerical solution very difficult to manage, especially since a complete quantitative description of self-focusing is not yet available. No such attempt has ever been reported. Therefore, at present, we can only be satisfied with a qualitative, or at most a semiquantitative, description of the phenomenon (Marburger, 1975; Shen, 1975). This chapter reviews the theory of SPM and associated spectral broadening. In the following section, we first discuss briefly the various physical mechanisms that can give rise to laser-induced refractive index changes responsible for SPM. Then in Section 3 we present the simple physical picture and theory of SPM and the associated spectral broadening. SPM in space is considered only briefly. Section 4 deals with a more rigorous theory of SPM that takes into account the higher-order effects of the induced refractive index change. Finally, in Section 5, a qualitative picture of how self-focusing can influence and enhance SPM and spectral broadening is presented. Some semiquantitative estimates of the spectral broadening are given and compared with experiments, including the recent observations of supercontinuum generation in gases. © 2006, 1989 Springer Science+Business Media, Inc.