The influence of cross-phase modulation on third-harmonic generation is theoretically studied. Generalized phase-matching conditions for third-harmonic generation including pump-intensity-dependent phase shifts related to self- and cross-phase modulation effects are discussed. The phase mismatch between the pump and third-harmonic pulses is shown to vary from the leading edge of the pump pulse to its trailing edge, resulting in an asymmetric spectral broadening of the third harmonic.
© 2002 Optical Society of America
Cross-phase modulation (XPM) [1, 2] is a result of nonlinear-optical interaction of at least two physically different light pulses (i.e., pulses with different frequencies, polarizations, modes, etc.) that puts a chirp on one of the pulses (the probe pulse) due to the intensity-dependent addition to the refractive index of the medium induced by the other pulse (the pump pulse). Similar to self-phase modulation (SPM), XPM can be employed to produce ultrashort light pulses. The sensitivity of the XPM-induced chirp of the probe pulse to the intensity of the pump pulse then provides an opportunity to control the phase and the duration of the probe pulse by varying the intensity of the pump pulse [3, 4]. XPM was shown also to be an efficient and convenient tool for studying the dynamics of fast nonlinear processes, including multiphoton ionization and plasma build-up dynamics, as well as for characterizing ultrashort light pulses through phase measurements on an ultrashort probe pulse [5, 6].
Cross-phase modulation in various pump-probe four-wave mixing processes has been intensely investigated in earlier studies. An illuminating discussion of XPM in optical fibers and a comprehensive overview of the relevant experimental and theoretical results are provided in the book by Agrawal . SPM- and XPM-induced detuning in second harmonic generation has been previously studied by Wang et al. [7 – 9] and Chien et al. .To put our time-dependent analysis of SPM and XPM effects in THG in context, we should mention also earlier work on the steady-state analysis of similar problems, reviewed by Cao et al. . The main purpose of this paper is to theoretically analyze XPM-related features in spectral and temporal evolution of a third-harmonic pulse produced in the field of an ultrashort pump pulse. In contrast to a pump-probe scheme, the probe pulse in the case of third-harmonic generation (THG) arises as a result of nonlinear-optical frequency tripling of the pump pulse. Thus, the pump pulse in our case simultaneously serves as a source of the probe (third-harmonic) pulse and puts a chirp on the third-harmonic pulse it generates. This leads, as will be shown below, to several effects of general physical interest and opens the way for efficient compression of frequency-upconverted light pulses.
Waveguide nonlinear-optical interactions can now be radically enhanced by using fibers of new type - microstructure [12–18] (also called holey and photonic-crystal) and tapered [19, 20] fibers. Such fibers provide a high degree of light confinement in the fiber core, allowing high efficiencies of nonlinear-optical processes to be achieved starting with rather low energies of femtosecond pulses at the input of the fiber [15, 16, 19, 20]. Although this theoretical work was motivated by the results of recent experiments on THG in microstructure fibers , we believe that the general features of XPM-induced spectral broadening of the third harmonic examined in this paper may be also observed in optical fibers of other types. Our idea is, therefore, to discuss the role of XPM in THG in a broader context, keeping in mind also other fiber configurations. With such an attitude, we will considerably extend simple qualitative explanations given in  and supplement them with a more detailed and accurate analysis of phase-matching and group-delay aspects of XPM in THG.
2. Basic relations
2.1. The amplitude and the phase of the third-harmonic pulse
We will consider in this paper a generic model of THG by a pump pulse with a central frequency ω in a fiber without specifying explicitly the field distribution and dispersion of waveguide modes. Our idea here is to gain some general understanding of phase matching, spectral broadening, and pulse evolution in THG including SPM- and XPM-induced phase shifts, which can be adapted later to a specific type of an optical fiber. We will assume, therefore, that the THG process involves a pair of waveguide modes and introduce formally propagation constants and field distributions of pump and third-harmonic radiation corresponding to these waveguide mode. Then, in accordance with the results of the slowly varying envelope analysis of THG in a fiber with a first-order dispersion [3,4], the amplitudes of the pump and third-harmonic pulses, A(ηp ,z) and B(ηh , z), can be represented as
where ηl = (t - z/νl )/τ is the time in the frame of reference running along the propagation coordinate z with the pump or the third-harmonic pulse (l = p, h, with subscripts p and h corresponding to the parameters of the pump and third-harmonic pulses, respectively) normalized to the duration τ of the incident pump pulse; νp and νh are the group velocities of the pump and third-harmonic pulses, respectively; ζ = (1/νh - 1/νp )/τ; ΔK = Kh - 3Kp is the phase mismatch; Kp and Kh are the propagation constants of the pump and third-harmonic pulses corresponding to the relevant eigenmodes of the fiber; A 0(ηp ) is the envelope of the pump pulse at the input of the fiber;
is the nonlinear phase of the pump pulse due to self-phase modulation; and
is the nonlinear phase of the third-harmonic pulse due to the modulation of the refractive index induced in the medium by the pump pulse at the frequency of the third harmonic (XPM effect). The nonlinear coefficients γ 1, γ 1, and β, appearing in Eqs. (2) – (4) can be expressed in terms of the nonlinear-optical cubic susceptibilities with the relevant frequency arguments responsible for SPM, XPM, and THG, respectively . Since we restrict ourselves to the first-order approximation of dispersion theory, the pump pulse in our model propagates through the fiber with no changes in its envelope, |A(ηp , z)| = |A 0(ηp )|.
We will consider pump pulses with a hyperbolic secant envelope:
where φ 0(ηp ) = arg[A 0(ηp )] is the initial chirp of the pump pulse. The nonlinear phase (φxpm (ηh , z) given by Eq. (4) can then be calculated analytically:
In Section 3, we will use Eqs. (1) – (6) to explore the spectral broadening and pulse evolution of the third harmonic including SPM- and XPM-induced phase shifts.
2.2. The phase mismatch
Now, we proceed with the analysis of phase matching for THG under conditions when SPM and XPM effects have to be taken into consideration. As it follows from Eq. (2), the phase shift between the third-harmonic field and the nonlinear polarization induced in the medium at the frequency of the third harmonic can be written as
Expression (7) shows that the total phase shift between the third-harmonic field and the nonlinear polarization induced in the medium at the frequency of the third harmonic is determined by the linear wave-vector mismatch related to the material dispersion, the initial phase of the pump pulse and intensity-dependent phase shifts induced by SPM and XPM. Generalized phase matching for THG under conditions when SPM and XPM effects, as well as the group delay of pump and third-harmonic pulses, have to be taken into consideration can be formulated in terms of the effective wave-vector mismatch
To clarify the physical meaning of generalized phase matching, it is instructive to represent Eq. (8) as
are the wave-vector mismatch components due to XPM and the group delay of the pump and third-harmonic pulses, respectively, and
is the dimensionless quantity representing the deviation of the instantaneous frequency of the pump pulse from its central frequency ω in units of 1/τ.
Perfect phase matching is achieved when the wave-vector mismatch (8) is exactly equal to zero. When the pump field is weak, leading to negligible SPM and XPM effects, and the group delay is small, the effective wave-vector mismatch defined by Eq. (8) is reduced to the conventional, linear-regime wave-vector mismatch. However, in a more general situation, SPM and XPM lead to the spectral broadening of the pump and third-harmonic pulses, giving rise to the dependence of the phase shift (7) and the effective wave-vector mismatch (8) on the propagation coordinate and the running time. Since the wave-vector mismatch under these conditions is no longer the same for the entire harmonic pulse, this mismatch can be minimized only for a certain part of the pulse. This leads, as will be shown in the next section, to the asymmetry of the spectral broadening of the third-harmonic pulse.
3. Results and discussion
To investigate the influence of group delay, as well as SPM and XPM effects on the THG process we perform numerical simulations using Eqs. (1), (3), (5), (6) – (8). We chose the linear wave-vector mismatch and the group-delay parameter, Δk = 2 cm-1, ζ= -0.2 cm-1, pump pulse τ= 30 fs, energy 0.3 nJ, and initial phase φ 0(ηp ) = , α = 0.13; nonlinear refractive index n 2 = 3.2·10-16 cm2/W, in such a way as to achieve the best agreement with the results of experiments on third-harmonic generation in microstructure fibers described in .
Figure 1 shows the effective wave-vector mismatch (curves 1, 2; the left-hand axis) and the instantaneous frequency of the pump pulse (curves 3, 4; the right-hand axis) related to the trailing edge of the third-harmonic pulse (ηh = 1.6) as functions of the propagation coordinate z in the regimes of low (curves 2, 4) and high (curves 1, 3) intensities. For low intensities of the pump pulse, when the nonlinear phase shifts [Eq. (10)] are negligible, the wave-vector mismatch (10) is determined by the linear dispersion of waveguide modes and remains virtually constant as the third-harmonic pulse propagates through the medium (curve 2 in Fig. 1). Slight variations in the phase mismatch in this case are due to the initial chirp of the pump pulse. The situation changes for higher intensities of the pump pulse, when SPM- and XPM- induced phase shifts come into play. The wave-vector mismatch for this regime is shown by line 1 in Fig. 1. The phase-matching condition Δk eff(ηh , z) = 0 can be then satisfied only for certain values of instantaneous frequencies of the pump pulse (line 3 in Fig. 1), which leads to an asymmetry of the third-harmonic spectrum.
The animation of Fig. 2 illustrates the evolution of temporal profiles and spectra of the pump and third-harmonic pulses and the effective wave-vector mismatch Δkeff for THG in a fiber with the nonlinear refractive index n 2 = 3.2-10-16 cm2/W. We present the behavior of the quantity CNL = , rather than the pump power itself, in this animation since simultaneously gives an idea of the pump pulse profile and the profile of the third-harmonic pulse in the regime when the group-delay, as well as self- and cross-phase modulation effects are absent. All the quantities given along the ordinate axes in Figs. 2a, 2b are normalized to their maximum values. The real values of these parameters can be reconstructed from the following relations: max[ω¯p]= 3, max [Δkeff ] = 4 cm-1 , max [Pp ] = 104 W . The amplitude of the third harmonic, of course, remains much less than the amplitude of the pump pulse, with the conversion efficiency being determined by the cubic nonlinearity of the medium, phase mismatch, group delay, and the overlapping of fiber modes involved in the nonlinear-optical process (we do no specify this mode configuration here).
As the pump pulse propagates through the fiber, it becomes chirped (Fig. 2a), its spectrum broadens (Fig. 2b), and it generates the third harmonic. Owing to the group-velocity mismatch, the leading edge of the third harmonic walks off from the leading edge of the pump pulse. The joint action of group-delay effects and changing phase matching, which improves toward the trailing edge of the pump pulse, leads to a continuous reshaping of the third harmonic (see Fig. 2a), with the maximum of the third harmonic being locked to the trailing edge of the pump pulse. The carrier frequency of the pump pulse in our case is red-shifted on the leading edge and blue-shifted on the trailing edge of the pulse. Therefore, the wave-vector mismatch for the THG process on the trailing edge of the pump pulse, as can be seen from the animation of Fig. 2, is less than Δk eff on the leading edge of this pulse, which gives rise to a predominantly blue shift of the third-harmonic spectrum (see Fig. 2). This prediction concerning the asymmetry of spectral broadening of the third-harmonic pulse qualitatively agrees with the results of THG experiments in a microstructure fiber reported by Fedotov et al. .
The asymmetry of spectral broadening of a probe pulse is generally typical of XPM in the standard pump-probe scheme of four-wave mixing in the case when the group delay of pump and probe pulses is nonnegligible. If the group velocity of the probe pulse in this scheme is, for example, lower than the group velocity of the pump pulse, then the chirp of the probe pulse is mainly determined by the trailing edge of the pump pulse, which leads to asymmetric spectral broadening (see  for details). We have shown in this paper that the dependence of the effective wave-vector mismatch on the internal time within the pulse adds more aspects to this asymmetric spectral broadening scenario in the case when the third harmonic generated in the field of a short pump pulse plays the role of the probe pulse. The phase mismatch between the pump and third-harmonic pulses then changes from the leading edge of the pump pulse to its trailing edge, resulting in the asymmetry in the spectrum of the third harmonic.
The sensitivity of the XPM-induced chirp and frequency shift of third-harmonic pulses to the intensity of pump pulses suggests the way to control the phase, the frequency, and (with an appropriate pulse compressor) the duration of third-harmonic pulses by simply varying the intensity of pump pulses at the input of the fiber. In practical terms, there are several ways to make this concept work. Gas-filled hollow fibers allow efficient nonlinear-optical interactions, including SPM, XPM, and high-order harmonic generation, to be implemented with the use of high-intensity laser pulses. As shown by Durfee III et al. , XPM in the pump-probe scheme of four-wave mixing allows pulses at the frequency of the third harmonic of Ti: sapphire laser radiation to be compressed down to 8 fs (with the initial fundamental-frequency pump-pulse duration equal to 35 fs). Another possibility is associated with the use of microstructure fibers. XPM effects have been recently observed in THG in microstructure fibers with subnanojoule femtosecond pulses of a Cr: forsterite laser . Hollow-core microstructure fibers  seem to be ideally suited for nonlinear-optical experiments with high-intensity laser pulses. A combination of optical harmonic generation with XPM in these new fibers suggests new ways of producing very short pulses of short-wavelength radiation.
This study was supported in part by the President of Russian Federation Grant no. 00-15-99304, the Russian Foundation for Basic Research project no. 00-02-17567, the Volkswagen Foundation (project I/76 869), and CRDF Awards nos. RP2-2266 and RP2-2275.
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