Spectral properties of the Kerr-lens mode-locked Cr 4 :YAG laser

Transcription

1 2084 J. Opt. Soc. Am. B/ Vol. 20, No. 10/ October 2003 Kalashnikov et al. Spectral properties of the Kerr-lens mode-locked Cr 4 :YAG laser Vladimir L. Kalashnikov, Evgeni Sorokin, Sergei Naumov, and Irina T. Sorokina Photonics Institute, Technical University of Vienna, Gusshausstrasse 27/387, A-1040, Vienna, Austria Received January 27, 2003; revised manuscript received June 11, 2003 The spectral characteristics of the Kerr-lens mode-locked Cr 4 :YAG laser have been investigated both experimentally and theoretically. It was demonstrated that third-order dispersion provides a frequency downshift of the pulse spectrum (by as much as 70 nm) in combination with extra broadening (as much as 400 nm), in analogy with the generation of a spectral continuum in the fibers in the vicinity of zero-dispersion wavelength. The large redshift of the few optical cycle pulses (greater than 100 nm) and the dependence of the spectrum s position on dispersion can be explained only if stimulated Raman scattering in the active medium is taken into account. It was shown that third-order dispersion can act as either a stabilizing or a destabilizing factor, depending on the amount of group-delay dispersion. Stabilization is found to be due to the negative feedback produced by the spectral sidebands Optical Society of America OCIS codes: , , INTRODUCTION The Cr 4 YAG laser, lasing in the m wavelength range, 1,2 is of interest for numerous applications near 1.5 m, e.g., for optical communications, optical coherent tomography, and remote sensing. The combination of good material properties with a wide gain band and its suitability for direct diode pumping 3,4 make this laser especially attractive. The crystal also facilitates high-power few-optical-cycle pulse generation. Since the first report of mode locking in a Cr 4 :YAG laser, 5 generation of the few-optical-cycle pulse was achieved with a Kerr-lens mode-locking (KLM) technique with dispersion compensation by chirped mirrors. 6 An observed feature of the mode-locked Cr 4 :YAG laser is a redshift of the ultrashort-pulse spectrum with respect to the gain maximum and continuous-wave (cw) oscillation, which occurs in a wide range of the pulse durations from hundreds of femtoseconds to fewer than 30 fs A similar redshift exists in femtosecond Cr:LiSGaF, Cr:LiSAF, 13,14 and Cr:forsterite, lasers. 15 Obviously, this phenomenon prevents achieving the minimum pulse duration and reduces the maximum pulse energy, because imperfect overlap of the pulse spectrum and the gain band reduces the usable gain. Control of the selffrequency shift can thus broaden the achievable spectral region of femtosecond oscillation. Despite the importance of the problem, at this moment there is no clear and unambiguous explanation of the femtosecond redshift in near-ir lasers. Numerous hypotheses have been proposed. First, it was proposed that the ultrashort pulses in the Cr 4 :YAG laser can be affected by excited-stated absorption. 5 As was demonstrated in Ref. 16, the reabsorption of the generation field at the blue side of its spectrum in combination with gain saturation can result in some self-frequency redshifting (a few tens of nanometers). However, this effect is not sufficient to explain the real-world maximum shifts ( 100 nm). Moreover, excited-state absorption is not known in Cr:YAG. 9,17 Second, a similar effect can result from the absorption caused by water vapor. 8 However, purging the laser cavity with dry nitrogen does not significantly affect the ultrashort-pulse characteristics of a Cr 4 :YAG laser. 7 A third explanation takes into account the large value of third-order dispersion (TOD) in comparison with the low value of group-delay dispersion (GDD) for Cr 4 :YAG. 11 In this case, for a comparatively small amount of higher-order dispersion the generation of ultrashort pulses is affected by TOD in the vicinity of zero GDD. This situation reminds us that in optical fibers the soliton part of the radiation is pushed into the red part of the spectrum It should be noted that the requirement of a negligible contribution from higher-order dispersion is crucial because the presence of this dipersion in TOD-compensation techniques can conceal a selffrequency shift, 16 especially when the pulse spectrum s width is close to the gain bandwidth 23 (see Section 5 below). The complementary explanation suggests a contribution by stimulated Raman scattering (SRS) to the selffrequency redshift. 16,24,25 This explanation was best applicable for Cr:LiSGaF and Cr:LiSAF lasers (the resultant shifts were as large as 100 nm and strongly depended on the pulse s energy and duration). Thus there is an ambiguity concerning the nature of transformations of the pulse spectrum in the Cr 4 :YAG lasers. In this paper we analyze the various mechanisms of spectrum transformation in Kerr-lens mode-locked lasers: (i) the effect of the frequency-dependent dissipation produced by the output coupler and the water vapor; (ii) high-order dispersion, in particular, TOD, and (iii) SRS within the active medium. It is shown that the redshift of the pulse spectrum as well as its extrabroadening can be explained by a process that is an analog of the generation of spectral supercontinua from fibers in the vicinity of the zero-dispersion wavelength (ZDW). The basic requirement here is that the ZDW be located in the vicinity of the maximum gain band. As a result, TOD causes a /2003/ $ Optical Society of America

2 Kalashnikov et al. Vol. 20, No. 10/October 2003/J. Opt. Soc. Am. B 2085 frequency downshift of the ultrashort-pulse spectrum and, simultaneously, extra broadening of the spectrum as a result of sideband excitation. Also, we consider the stability of ultrashort pulses affected by the spectrum transformations. As will be seen, TOD causes a passive negative feedback, which affects the ultrashort pulse s stability. As a result, the behavior of the pulse characteristics differs essentially from that in the absence of TOD. However, as will be demonstrated, one can obtain a full explanation of the properties of the ultrashort-pulse spectrum only by taking into account SRS within the active medium. 2. EXPERIMENTAL DATA In the experiment we used the setup described in detail in Ref. 7 that consisted of an X-shaped linear cavity with a 20-mm Cr:YAG crystal (Fig. 1). Mode locking was achieved by soft-aperture KLM. 26 We obtained dispersion compensation by using a fused-silica prism pair, dispersion-compensating mirrors (DCMs), and a combination of prisms and DCMs. In a purely DCM setup the overall intracavity dispersion contains strong oscillating terms, which means the presence of strong higher-order dispersion. We therefore modeled prism-compensated and prism-dcm setups, in which TOD is the dominant high-order term. As pump sources, Yb:fiber lasers as well as high-power laser diodes have been used. For modeling, however, we used a setup with Yb:fiber laser pumping. This is so because in this case the pump mode s shape and divergence are known with high accuracy, whereas pump mode parameters in the diodepumped setup (a combination of as many as of four diodes 3 ) are strongly alignment dependent. The output coupling was varied from 0.2% to 0.5% and was spectrally centered at the gain maximum. The high reflectors and the DCMs were also centered about the gain peak of the Cr:YAG. During prism-pair adjustment it was verified that in the free-running cw regime the laser was operating near its gain maximum (1450 nm), i.e., that the prisms did not introduce any spectral filtering. During laser experiments the following parameters were changed: dispersion; pump power; the position of the laser crystal with respect to the beam focus, and the distance between the curved mirrors, which create the intracavity focus. The latter two parameters affected the strength of the KLM mechanism and were optimized to Fig. 1. Schematic diagram of the Cr:YAG laser. Mirrors M1, M2 (radius of curvature, 100 mm) and M3 are all highly reflecting at 1500 nm and transmitting at 1070 nm. OC, output coupler; FS, fused-silica; F, focusing lens. Fig. 2. Dependence of central wavelength on pulse duration. The open circles correspond to different pulse energies and intracavity dispersion. produce stable single-pulse self-starting operation. Tuning the pump power and exchange of the output couplers influenced the intracavity pulse energy. We controlled the dispersion by changing the prism s apex-to-apex distance from 50 to 3.5 cm. The pulse durations varied from 100 to 35 fs and could be further shortened only by use of a chirped mirror for TOD compensation. 7 Figure 2 shows the dependence of the pulse s central wavelength on the pulse duration. The data include pulses obtained at different intracavity dispersion and power levels, and this explains the relatively large scattering of the data in the vertical direction. Yet the trend (marked by the dashed line) is obvious and demonstrates clearly the shift of the pulse s central wavelength from gain maximum (1450 nm) and loss minimum (1470 nm). 3. MODEL We use the distributed model of KLM that was described in Refs. 16 and 27 and is formulated as a generalized Ginzburg Landau equation: a z, t z t g 2 2 t a z, t 2 1 a z, t 2 N i i m 2 m m m i a z, t 2 m! t m a z, t P SRS t. (1) Here a(z, t) is the slowly varying intracavity field [so a(z, t) 2 has a dimension of intensity], which depends on local time t and on longitudinal coordinate z normalized to the cavity length. The non-hamiltonian (laser) part of Eq. (1) is enclosed in the first set of brackets and describes (i) the action of the saturated gain with coefficient ; (ii) fieldindependent (linear) intracavity and output losses ; (iii) spectral filtering caused by the gain spectral profile with bandwidth 2 /t g (t g 5.4 fs for the Cr 4 :YAG laser), and (iv) a fast saturable absorber produced by self-focusing in the active medium. and are the fast saturable absorber s modulation depth and the inverse saturation intensity, respectively. To take into account the pulse spec-

3 2086 J. Opt. Soc. Am. B/ Vol. 20, No. 10/ October 2003 Kalashnikov et al. trum s transformation that is due to the frequencydependent output coupler s reflectivity and to the field reabsorption caused by the active medium and the water vapor, we consider linear loss coefficient 6 m m 0 m t, m where coefficients m are found from the polynomial approximation of the loss s spectral profile within the region of laser generation. The gain coefficient obeys the following equation: z, t t I p a max z, t p Table 1. Parameter Active Medium and Laser Parameters Value a (cm 2 ) g (cm 2 ) l g (cm) 2 m 0.28 n n 2 (esu) (nm) 1448 g ( s) 4 T cav (ns) 10.2 Table 2. Parameter Parameters of Raman Scattering a Lines E g E g A 1g E g A 1g A 1g g SRS (cm/gw) (cm 1 ) (cm 1 ) a Only lines of Eg and A 1g symmetry are Raman active in this scattering geometry. a z, t 2 z, t g z, t, (2) 0 g where a and g are the absorption and emission cross sections, respectively, of an active medium, g is the gain relaxation time, I p is the absorbed pump intensity, p is the pump frequency, max 2 g n g l g is the maximum gain coefficient, n g is the concentration of active centers, and l g is the length of the gain medium. The assumption that t p T cav (t p is the pulse duration; T cav is the cavity period) permits the integration of Eq. (2). Then the steadystate gain coefficient is expressed as m a I p T cav p ai p T cav E T 1 cav p E g g, (3) where E g 0 / g is the gain saturation energy flux T and E cav /2 Tcav a(t) 2 dt is the field energy stored within /2 the cavity period. The Hamiltonian part of Eq. (1) is the generalized higher-order nonlinear Schrödinger equation with dispersion terms m up to Nth order (the choice of sign corresponds to that Ref. 28). 2l g n 2 0 /c is the self-phase modulation coefficient (n 2 is the nonlinear refractive index). P SRS is the part of the nonlinear polarization caused by SRS within the active medium. The SRS-induced polarization for the pump component ( p) is P SRS l t i g SRS l 2 l t a s t G t t a p t a s * t exp i l t t dt. (4) Here s labels the Stokes component, l is the frequency of the optical phonon wave (l labels the corresponding Raman line), and g l SRS is the Raman gain coefficient. G(t t ) exp (t t )/ l is the so-called Raman response function Inasmuch as we consider the situation in which the pulse spectrum acts on itself (this is valid for the few-optical-cycle pulses in Cr 4 :YAG), the distinction between pump and Stokes components can be dropped (see Ref. 24). The simulations were performed by use of the fast- Fourier-transform split-step method. The local time grid, consisting of 2 12 points, had 3.24-fs steps. Convergence was provided by steps on z. Only stable single-pulse solutions were considered. A change of the pulse s peak intensity by less than 1% during the last 1000 transits was considered to be the steady-state pulse criterion. The basic simulation parameters are listed in Tables 1 and SPECTRAL TRANSFORMATION CAUSED BY OUTPUT LOSS AND INTRACAVITY REABSORPTION The most obvious source of the ultrashort-pulse spectral shift in solid-state lasers is frequency-dependent losses caused by the output coupler and the intracavity elements. 5,8,16 Figure 3 shows the spectral profiles of the loss coefficients for Cr 4 :YAG crystal, intracavity water vapor, and a 0.5% output mirror. The experiments showed that spectral dissipation cannot be considered the sole or the main mechanism for the Stokes shift of the pulse spectrum. This statement is based on the following facts: (i) the locations of the cw and the pulsed spectral maxima are practically unaffected by the interchanging output couplers that provide the shift of the reflection maximum within a m region and (ii) purging of the laser cavity by dry nitrogen, which eliminates water vapor, does not affect the pulse spectrum. To confirm this conclusion theoretically we performed numerical simulations on the basis of the model described above. Equation (1) was considered without the higherorder dispersion terms and the SRS that allowed the pure contribution of the frequency-dependent dissipation to be isolated and introduced into the term as a polynomial approximation of the spectral dependencies on loss presented in Fig. 3. The simulation demonstrates that the frequency downshift of the ultrashort-pulse spectrum induced by the

4 Kalashnikov et al. Vol. 20, No. 10/October 2003/J. Opt. Soc. Am. B 2087 Fig. 3. Spectra of measured intracavity loss owing to water vapor (solid curve), the output coupler (dashed curve), and reabsorption inside the active medium (dotted curve). losses does not exceed 20 nm. This shift is weakly dependent on the pulse duration and is too small to explain the experimental data for sub-100-fs pulses. As was demonstrated in Ref. 16, the source of this shift is the change in wavelength corresponding to the maximum net gain that is due jointly to intracavity and output losses and the gain saturated by the full field energy. Analytically, the spectral shift can be estimated from the threshold conditions for the cw radiation and the ultrashort pulse. The former is ( ) ( ) 0; the latter is ( ) ( ) 0 (in the case of full saturation of the effective fast absorber). The resultant shifts do not exceed 10 nm. This value is a good approximation for the cw s spectral position, whereas the numerically obtained value is a good approximation for the observed spectral shift of the pulses with the largest duration ( 100 fs). Thus the explanation for the spectral shift of the sub- 100 fs pulses has to take into account more-subtle causes. 5. CONTRIBUTION OF HIGHER-ORDER DISPERSION An essential feature of the Cr 4 :YAG laser is the presence of relatively strong TOD, which in our case is only partially compensated by chirped mirrors. It is known that TOD leads to distortion of the pulse spectrum, resulting in a shift of the spectrum s maximum and in the appearance of sidebands. 32,33 The dispersive conditions in our case are similar to those for soliton propagation in the vicinity of the ZDW. 19 In the latter case the pulse spectrum shifts towards the region of anomalous GDD (frequency downshift) and the dispersive wave s peak appears in the vicinity of the ZDW. The shift of the soliton spectrum from the ZDW depends on the pulse duration 19 : nm m /t p ps, where 0 is the ZDW and t p is the FWHM of the pulse. For example, the experimental parameters are nm and t p 26 fs, which result in 130 nm; i.e., the spectrum s maximum has to be located at 1567 nm. This is quite a large shift, which would be able to explain the experimental data. However, we have to take into account that the conditions of ultrashort-pulse propagation in the laser differ from those in the fiber. Therefore the numerical simulations on the basis of Eq. (1) that contains the non-hamiltonian part are required. In the beginning we exclude SRS and higher than TOD terms from Eq. (1) and neglect the dependence of on. The main results are shown in Figs Figure 4 presents the pulse width, its energy, and the dependence of the spectrum s maximum on the GDD for various TODs. Because the generation of ultrashort pulses is possible for the different values of that form the finite zone of KLM (Fig. 5), we averaged the pulse parameters over for each value of 2. In the absence of TOD there is a monotonic decrease in the average pulse width as a result of 2 0 [curves A in Figs. 4(a) and 4(d)]. The approach to zero GDD constricts the region of the stable single-pulse generation on and shifts it in the direction of larger saturation parameters [Figs. 5(a) and 5(d)]. Together with the pulse shortening for 2 0, the pulse energy decreases for fixed [Figs. 4(b) and 4(e)]. In as much as the net gain is constant, simultaneous decrease of the pump intensity [see Eq. (3)] is required for suppression of multiple pulse generation in the vicinity of the zero GDD. TOD notably affects the pulse parameters (curves B and C in Fig. 4). The minimum pulse width increases in comparison with 3 0, 34 and it is achieved for larger 2 [Figs. 4(a) and 4(d)]. As the relative contribution of TOD to the net dispersion increases as a result of 2 0, the pulse-width increase that is due to 3 0 is most visible in the vicinity of zero GDD. The values of minimum pulse duration obtained and the values of GDD required for the shortest pulse generation are close to the experimentally observed values (Figs. 2 and 9). At the relatively large negative GDD, TOD can lead to even shorter pulses than when 3 0. We explain this effect as being a result of extra broadening of the pulse spectrum caused by TOD (see below). The growth of the anomalous GDD cancels the TOD-related influence. Pulse energy generally decreases in the presence of TOD [curves B and C in Figs. 4(b) and 4(e)]. This results from the growing spectral loss that is due to the generation of sidebands. However, in the vicinity of the zero GDD the pulse energy exceeds that for 3 0. A possible explanation of this energy growth is the shift of the sidebands toward the main part of the pulse spectrum when 2 0. In combination with the narrowing of the pulse spectrum owing to growth of the pulse width [curves B and C in Figs. 4(a) and 4(d)], this shift reduces the spectral loss. The TOD destabilizes the ultrashort pulse for the relatively large anomalous GDD fs 2 in Figs. 5(b), 5(c), and 5(e) and fs 2 in Fig. 5(f)]. The source of the pulse destabilization produced by the TOD is the excitation of the perturbations that are attached to the ultrashort pulse [bounded perturbations; see Figs. 6(a) and 6(b) and Refs. 18, 21, and 28]. The growth of TOD for the fixed laser parameters increases these bounded perturbations and causes pulse splitting [Fig. 6(c)]. A similar effect appears also in the absence of TOD as a result of the growth of, providing an increase in pulse energy and, as a consequence, amplification of the bounded perturbation. 27 This destabilization mechanism forms the upper boundary on of the mode-locking region. In the vicinity of zero GDD there is some increase in

5 2088 J. Opt. Soc. Am. B/ Vol. 20, No. 10/ October 2003 Kalashnikov et al. the maximum in the presence of TOD [cf. zones in Fig. 5: Fig. 5(c) versus Fig. 5(a) and Fig. 5(f) versus Fig. 5(d)]. We assume that in this case the generation of sidebands provides the mechanism for passive negative feedback, which can stabilize the pulse. The sidebands as well as the cw radiation provide the channel through which to discharge the superfluous energy stored in the pulse and cause pulse splitting (see Ref. 27). The cw-like perturbation that discharges the superfluous energy is clearly visible in Fig. 7 (labeled by arrows), where the wide pedestal in the time domain [Fig. 7(b)] corresponds to the spike in the vicinity of the gain band maximum [Fig. 7(a)]. The most impressive manifestation of TOD is the shift in the ultrashort-pulse spectrum, which depends strongly on the value of the GDD [Figs. 4(c) and 4(f)]. The pulse spectrum shifts in the region of the anomalous GDD, where the conditions of quasi-soliton propagation are satisfied for the given pulse energy and width. 7 In the presence of TOD, this corresponds to the redshift for 2 0 and to the blueshift for the relatively large 2. The growth of TOD or increases the spectral shift and, for the smaller modulation depth, a pronounced dependence of this shift on the net-gain value appears. The value of the maximum redshift is comparable with the experimentally observed value (Fig. 2) but smaller than that derived from the soliton model (see above). This is understandable, because the soliton model assumes zero gain, whereas in reality the decrease of gain outside its peak (at 1450 nm) counteracts the TOD action. Nevertheless, there are some disagreements with the experimental data. The model that takes into account only TOD as a source of the frequency shift predicts toorapid disappearance of the frequency shift and even a change of its sign as a result of the increase in 2 [Figs. 4(c) and 4(f)]. The experiment demonstrates a clear redshift even for the long pulses and suggests additionally that SRS inside the active medium be taken into account (see Section 6 below). Now let us consider distortion of the pulse spectrum induced by high-order dispersion. A typical simulated pulse spectrum is shown in Fig. 8 as the lighter curve. Besides the frequency downshift, a strong asymmetry of the pulse spectrum appears: an intensive blue side tail and blue sidebands. 33,35 This effect is obviously visible in the experiment, too (the darker curve in Fig. 8). As a result, the spectrum is additionally broadened. For the high-reflective optics the spectrum can range approxi- Fig. 4. Dependence on GDD of (a), (d) pulse width; (b), (e) energy; and (c), (f) spectrum shift averaged over saturation parameter. The TOD parameters are 0 (curves A), 2550 (curves B), and 5100 fs 3 (curves C). (a) (c) for 0.05; (d) (f) for

6 Kalashnikov et al. Vol. 20, No. 10/October 2003/J. Opt. Soc. Am. B 2089 Fig. 5. Single-pulse stability zones (shaded regions). The TOD parameters are (a), (d) 0; (b), (e) 2550; and (c), (f) 5100 fs 3. (a) (c) 0.05; 0.048; (d) (f) 0.025; mately from 1.25 to 1.65 nm; i.e., a range of 400 nm on the 20-dB level markedly exceeds the gain bandwidth. The nature of such extrabroadening is similar to that for the generation of spectral continua in fibers in the vicinity of the ZDW 19 : The redshift of the ultrashort-pulse spectrum induced by the TOD is accompanied by the excitation of the dispersive spectral components in the blue region that are phase locked with the pulse and are supported by the energy shed by the soliton (see also Ref. 35). Together with the long shoulder on the blue side there are also blue and red resonant sidebands. As is known, the source of the sidebands is the resonant radiation stimulated by the dispersion. 19,21,36 38 The corresponding resonance condition is 38 m 3 1 m 0 sb m m m! 2 2t p 2 2 j, j 0, 1,..., (5) Fig. 6. Field intensities for 0.05, 0.04, fs 2, 7, and (a) , (b) 6800, and (c) 12,000 fs 3. Dashed curves, correspond to the sech-shaped pulse. where 0 is the frequency of the pulse-spectrum maximum, t p is the pulse width (a sech-shaped pulse profile was assumed), and sb is the resonant peak frequency. The numerical simulations demonstrate good agreement (within the few nanometers) with the analytical estimations. The growth of 3 drags the sidebands toward the pulse s center and deepens the hole on its blue side. As a rule, the GDD of real-world lasers depends nonlinearly on wavelength. For example, a compensation technique involving chirped mirrors can introduce an oscillatory behavior of the GDD that is especially pronounced in

7 2090 J. Opt. Soc. Am. B/ Vol. 20, No. 10/ October 2003 Kalashnikov et al. the IR region (see, for example, Ref. 39). This is not only a negative factor, because there is a possibility of using the GDD oscillation for additional pulse-spectrum widening. 27,34 Adding the oscillation to the simulated GDD demonstrates that the presence of local dispersion minima (caused, e.g., by the fourth-order dispersion term 4 ) could excite the additional spectral shoulders, depending on the amplitude and location of the GDD oscillations, and thus widen the pulse spectrum. In the vicinity of zero GDD such oscillations can support stable single-pulse generation even when the smooth GDD produces only multipulsing. The increase in the oscillation period and its amplitude acts as a spectral shifter, pushing the spectral components to the local minimum of the GDD and creating shoulders on the spectral side, depending on the value of the GDD and on the sign of its gradient. The nonzero fourth-order dispersion of the appropriate sign can cancel the frequency shift induced by TOD in the vicinity of the ZDW. In this case a second ZDW can exist for 0, and the pulse spectrum will be stabilized between the GDD zeros. Also, the existence of a multitude of sidebands within the region where the GDD Fig. 7. Quasi-cw perturbation (labeled by arrows) of (a) the generation spectrum and (b) the field intensity profile. 0.05, 0.05, fs 2, fs 3, and 0.3. Fig. 9. Spectral shifts resulting from the joint action of SRS, TOD, and frequency-dependent losses (solid curve) and from sole TOD action (dashed curve, fs 3 ). Data correspond to the shortest pulses for a given and Circles denote the experimentally observed shifts. gradient is large (see Ref. 16) can add an additional channel to discharge the superfluous pulse energy, causing multiple pulse operation (see above). Thus, taking TOD and the higher-order dispersions into account aids in understanding the basic spectral features of the femtosecond pulses in the laser system under consideration. In particular, there is a significant redshift of the spectrum (by as much as 70 nm), and an extra broadening of the spectrum (by as much as 400 nm) that exceeds the gain bandwidth. The appearance of these properties is governed by effects that are similar to those in the fibers in the vicinity of the ZDW. However, the values of the spectral shift are too sensitive to the GDD value. Moreover, the experiment demonstrates larger frequency downshifts of the pulse spectrum (cf. the darker and lighter curves in Fig. 8). These disagreements with the experimental data require that SRS be considered an additional mechanism for the spectral shift in the Cr 4 :YAG laser. Fig. 8. Experimentally observed (darker curve) and simulated (lighter curve) spectra. Simulation parameters are 0.05, 0.04, fs 2, 5, and fs 3. 0 is the ZDW; 0 corresponds to the maximum spectrum from soliton theory; sb is the wavelength of the dispersive wave, with j 1 [see Eq. (5)]. 6. STIMULATED RAMAN SCATTERING Taking into account the P SRS term in Eq. (1) allows the SRS contribution to the ultrashort pulse characteristics within a broad range of pulse durations to be considered. As was found numerically in Ref. 16, the main parameters that affect the self-frequency shift are the pulse s duration and energy. The former determines the intensity of the Stokes components produced by the SRS. If the pulse spectrum is broad enough that the Stokes components are located within it, the stimulated scattering efficiently amplifies the red part of the spectrum. However, if the spectrum is too broad the spectrum shift decreases because of a diminishing overlap area of the pulse spectrum and the Raman line. 24 Finally, the ultrashortpulse energy growth enhances the scattering effect by virtue of the nonlinear nature of the SRS. As it can be seen from Table 2, Cr 4 :YAG possesses a set of relatively intense and broad Raman lines that form a chain whose first line is already covered by the spec-

8 Kalashnikov et al. Vol. 20, No. 10/October 2003/J. Opt. Soc. Am. B 2091 trum of the pulse with 100-fs duration. Figure 9 (solid curve) shows the dependence of the spectrum shift for the shortest (for a given 2 ) pulse on GDD in the presence of SRS, TOD, and spectrally dependent losses. Although the increasingly negative GDD increases the pulse duration, the spectrum shift does not vanish, as it occurs for a sole TOD action (as is illustrated by the dashed curve in Fig. 9). The joint contribution of the factors considered increases the redshift for 2 0to 120 nm. The simulated shifts are in good agreement with those observed experimentally for fs 2 (open circles in Fig. 9). The TOD contribution for the relatively small anomalous GDD prevails over that which is due to SRS. However, the contribution of SRS increases for the relatively large negative GDD as a result of growth in the pulse energy. As a result, the decrease of the spectrum s redshift produced by the growing negative 2 becomes slower, in an agreement with experiment. Thus we can conclude that only the combined action of TOD and SRS explains the experimentally observed shifts of the femtosecond-pulse spectrum. However, there is some difference between experimental and simulated shifts in the region fs 2. The point is that the distributed model predicts more-rapid growth of the pulse width than in the experiment for the large anomalous GDD, which can reduce the spectrum shifts to a level that corresponds to that caused by reabsorption ( 20 nm; see Section 4). 7. CONCLUSIONS The features of the mode-locked Cr 4 :YAG laser with dispersion compensation by a prism pair are the essential ( 100-nm) redshift of the ultrashort-pulse spectrum, the asymmetry of the spectrum, and the extra broadening that is due to the generation of sidebands. A pronounced redshift of the ultrashort-pulse spectrum is typical also for other IR lasers in the pulse-width range fs (i.e., Cr:LiSGaF, Cr:LiSAF, and Cr:forsterite; see Ref. 16). In this research we systematically analyzed possible sources of distortions in the ultrashort-pulse spectrum of a Kerr-lens mode-locked Cr 4 :YAG laser. Using the numerical model and taking into account self-phase modulation, fast saturable absorption, high-order dispersion, and SRS, we identified the cause of the spectrum s shift and of the extra broadening. When the ZDW is located close to the gain maximum, the soliton part of the spectrum shifts toward anomalous dispersion (frequency downshift), whereas the phase-matched dispersive part of the spectrum acquires a frequency upshift. The key factor here is TOD, which exerts the strongest influence in the vicinity of the ZDW. We have demonstrated that TOD enhances the tendency to multipulsing for relatively large negative dispersion. However, in the intermediate region it provides some pulse stabilization as a result of the appearance of passive negative feedback caused by the generation of sidebands. The cost of such stabilization is an increase in the pulse duration. Nevertheless, the experimental pulse-spectrum shift cannot be explained only by dispersive effects. The experimentally observed dependence of this shift on GDD suggests larger shifts (of more than 100 nm) and a relatively slow approach of the shift to zero with the growth of anomalous dispersion. Agreement with the experiment can be reached when the SRS inside the active medium is taken into account. This process is intensified by the growth of pulse energy and by pulse shortening (to 20 fs in our analysis). As a result, the maximum spectrum shift exceeds 100 nm, and the value of its dependence on the GDD is close to the experimentally observed value. ACKNOWLEDGMENTS This research was supported by Austrian National Science Fund projects M688, T-64, F016, and P14704-TPH, and the Bundesministerium fuer Bildung, Wissenschaft und Kultur and International Science and Technology Center project B631. V. L. Kalashnikov s address is vladimir.kalashnikov@tuwien.ac.at. REFERENCES 1. N. B. Angert, N. I. Borodin, V. M. Garmash, V. A. Zhitnyuk, A. G. Okhrimchuk, O. G. Siyuchenko, and A. V. Scestakov, Lasing due to impurity color centers in yttrium aluminium garnet crystal at wavelengths in the range m, Sov. J. Quantum Electron. 18, (1988). 2. N. I. Borodin, V. A. Zhitnyuk, A. G. Okhrimchuk, and A. V. Shestakov, Oscillation of a Y 3 Al 5 O 12 :Cr 4 laser in the wavelength region of m, Bull. Acad. Sci. USSR, Phys. Ser. 54, (1990). 3. I. T. Sorokina, S. Naumov, E. Sorokin, and E. Wintner, Directly diode-pumped tunable continuous-wave roomtemperature Cr 4 :YAG laser, Opt. Lett. 24, (1999). 4. A. J. Alcock, P. Scorah, and K. Hnatovsky, Broadly tunable continuous-wave diode-pumped Cr 4 :YAG laser, Opt. Commun. 215, (2003). 5. P. M. W. French, N. H. Rizli, J. R. Taylor, and A. V. Shestakov, Continuous-wave mode-locked Cr 4 :YAG laser, Opt. Lett. 18, (1993). 6. D. J. Ripin, C. Chudoba, J. T. Gopinath, J. G. Fujimoto, E. P. Ippen, U. Morgner, F. X. Kärtner, V. Scheuer, G. Angelow, and T. Tschudi, Generation of 20-fs pulses by a prismless Cr 4 :YAG laser, Opt. Lett. 27, (2002). 7. S. Naumov, E. Sorokin, V. L. Kalashnikov, G. Tempea, and I. T. Sorokina, Self-starting five optical cycle pulse generation in Cr 4 :YAG laser, Appl. Phys. B 76, 1 11 (2003). 8. P. J. Conlon, Y. P. Tong, P. M. W. French, J. R. Taylor, and A. V. Shestakov, Passive mode locking and dispersion measurement of a sub-100-fs Cr 4 :YAG laser, Opt. Lett. 19, (1994). 9. Y. Ishida and K. Naganuma, Characteristics of femtosecond pulses near 1.5 m in a self-mode-locked Cr 4 :YAG laser, Opt. Lett. 19, (1994). 10. Y. P. Tong, J. M. Sutherland, P. M. W. French, J. R. Taylor, A. V. Shestakov, and B. H. T. Chai, Self-starting Kerr-lens mode-locked femtosecond Cr 4 :YAG and picosecond Pr 3 :YLF solid-state lasers, Opt. Lett. 21, (1996). 11. Zh. Zhang, T. Nakagawa, K. Torizuka, T. Sugaya, and K. Kobayashi, Self-starting mode-locked Cr 4 :YAG laser with a low-loss broadband semiconductor saturable-absorber mirror, Opt. Lett. 24, (1999). 12. A. Sennaroglu, C. R. Pollock, and H. Nathel, Continuouswave self-mode-locked operation of a femtosecond Cr 4 :YAG laser, Opt. Lett. 19, (1994). 13. I. T. Sorokina, E. Sorokin, E. Wintner, A. Cassanho, H. P. Jenssen, and R. Szipöcs, 14-fs pulse generation in Kerrlens mode-locked prismless Cr:LiSGaF and Cr:LiSAF la-

9 2092 J. Opt. Soc. Am. B/ Vol. 20, No. 10/ October 2003 Kalashnikov et al. sers: observation of pulse self-frequency shift, Opt. Lett. 22, (1997). 14. S. Uemura and K. Torizuka, Generation of 12-fs pulses from a diode-pumped Kerr-lens mode-locked Cr:LiSAF laser, Opt. Lett. 24, (1997). 15. C. Chudoba, J. F. Fujimoto, E. P. Ippen, H. A. Haus, A. Morgner, F. X. Kärtner, V. Scheuer, G. Angelow, and T. Tschudi, All-solid-state Cr:forsterite laser generating 14-fs pulses at 1.3 m, Opt. Lett. 26, (2001). 16. V. L. Kalashnikov, E. Sorokin, and I. T. Sorokina, Mechanisms of spectral shift in ultrashort-pulse laser oscillators, J. Opt. Soc. Am. B 18, (2001). 17. S. Kück, Laser-related spectroscopy of ion-doped crystals for tunable solid-state lasers, Appl. Phys. B 72, (2001). 18. G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, San Diego, Calif., 2001), Chap P. K. A. Wai, C. R. Menyuk, H. H. Chen, and Y. C. Lee, Soliton at the zero-group-dispersion wavelength of a singlemode fiber, Opt. Lett. 12, (1987). 20. F. W. Wise, I. A. Walmsley, and C. L. Tang, Simultaneous formation of solitons and dispersive waves in a femtosecond ring dye laser, Opt. Lett. 13, (1988). 21. P. K. A. Wai, H. H. Chen, and Y. C. Lee, Radiation by solitons at the zero group-dispersion wavelength of singlemode optical fibers, Phys. Rev. A 41, (1990). 22. T. Tomura and H. Petek, Effect of third-order dispersion on the phases of solitonlike Cr 4 :YAG-laser pulses characterized by the second-harmonic generation frequency-resolved optical gating method, J. Opt. Soc. Am. B 18, (2001). 23. P. C. Wagenblast, U. Morgner, F. Grawert, V. Scheuer, G. Angelow, M. J. Lederer, and F. X. Kärtner, Generation of sub-10-fs pulses from a Kerr-lens mode-locked Cr 3 :LiCAF laser oscillator by use of third-order dispersioncompensating double-chirped mirrors, Opt. Lett. 27, (2002). 24. H. A. Haus, I. Sorokina, and E. Sorokin, Raman-induced redshift of ultrashort mode-locked laser pulses, J. Opt. Soc. Am. B 15, (1998). 25. Zh. Li, L. Li, H. Tian, G. Zhou, and K. H. Spatschek, Chirped femtosecond solitonlike laser pulse form with selffrequency shift, Phys. Rev. Lett. 89, (2002). 26. M. Piché and F. Salin, Self-mode locking of solid-state lasers without apertures, Opt. Lett. 18, (1993). 27. V. L. Kalashnikov, E. Sorokin, and I. T. Sorokina, Multipulse operation and limits of the Kerr-lens mode locking stability, IEEE J. Quantum Electron. 39, (2003). 28. S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (Springer, New York, 1992), Chap R. H. Stolen, J. P. Gordon, W. J. Tomlinson, and H. A. Haus, Raman response function of silica-core fibers, J. Opt. Soc. Am. B 6, (1989). 30. C. Headley III and G. P. Agrawal, Unified description of ultrafast stimulated Raman scattering in optical fibers, J. Opt. Soc. Am. B 13, (1996). 31. D. Hollenbeck and C. Cantrell, Multiple-vibrational-mode model for fiber-optic Raman gain spectrum and response function, J. Opt. Soc. Am. B 19, (2002). 32. F. W. Wise, I. A. Walmsley, and C. L. Tang, Simultaneous formation of solitons and dispersive waves in a femtosecond ring dye laser, Opt. Lett. 13, (1988). 33. H. A. Haus, J. D. Moores, and L. E. Nelson, Effect of thirdorder dispersion on passive mode locking, Opt. Lett. 18, (1993). 34. A. Rundquist, C. Durfee, Z. Chang, G. Taft, E. Zeek, S. Backus, M. M. Murnane, H. C. Kapteyn, I. Christov, and V. Stoev, Ultrafast laser and amplifier sources, Appl. Phys. B 65, (1997). 35. T. Tomaru and H. Petek, Effect of third-order dispersion on the phases of solitonlike Cr 4 :YAG-laser pulses characterized by the second-harmonic generation frequency-resolved optical gating method, J. Opt. Soc. Am. B 18, (2001). 36. J. N. Elgin, Soliton propagation in an optical fiber with third-order dispersion, Opt. Lett. 20, (1992). 37. C. Spielmann, P. F. Curley, T. Brabec, and F. Krausz, Ultrabroadband femtosecond lasers, IEEE J. Quantum Electron. 30, (1994). 38. Q. Lin and I. Sorokina, High-order dispersion effects in solitary mode-locked lasers: side-band generation, Opt. Commun. 153, (1998). 39. R. Szipöcs, A. Köhazi-Kis, S. Lako, P. Apai, A. P. Kovacs, G. DeBell, L. Mott, A. W. Louderback, A. V. Tikhonravov, and M. K. Trubetskov, Negative dispersion mirrors for dispersion control in femtosecond lasers: chirped dielectric mirrors and multi-cavity Gires Tournois interferometers, Appl. Phys. B Suppl. 70, S51 S57 (2000).