We investigate supercontinuum generation in photonic crystal fibers under femtosecond single and dual wavelength pumping experimentally and by numerical simulations. Details about the expansion of the blue but also the red side of the continuum due to cross-phase modulation (XPM) and transfer of energy to dispersive waves are revealed and experimentally confirmed. Additionally, simple guidelines are given to predicte the maximum bandwidth of supercontinuum generation only by the use of the dispersion curve of the fiber.
©2005 Optical Society of America
The design freedom of photonic crystal fibers can be used to tailor and extend the range of optical parameters like dispersion and nonlinearity . Their applications in the field of ultrafast optics have made a big impact . By exploiting this new range of properties especially laser and amplifier systems could be scaled . Another exciting feature is the combination of engineered anomalous dispersion with either high or low nonlinear interaction of ultrashort optical pulses propagating inside such waveguides. On the one hand, air-core bandgap fibers have been fabricated, which exhibit anomalous dispersion with nonlinearity close to that of air, thus supporting solitons at power levels not obtained before . On the other hand, solid core photonic crystal fiber can be designed with enhanced waveguide dispersion due to small cores, which shifts the zero dispersion wavelength of the fiber down to the visible. Consequently, anomalous dispersion is available for ultrashort laser systems working below 1.3 μm. The high nonlinearity of these small cores in combination with the anomalous dispersion leads to nonlinear phenomena, such as supercontinuum generation .
The physical background of supercontinuum generation has been described theoretically  and experimental verifications of the model have been reported many times [7–11]. It describes supercontinuum generation as a process where a higher-order soliton is formed by a short pump pulse in the anomalous dispersion region of a fiber above, but close to a zero dispersion wavelength. This higher-order soliton breaks up into red-shifted fundamental solitons due to perturbations. These perturbations are higher order nonlinear effects like self-steepening and Raman scattering, which lead to different group velocities of the fundamental constituents of the higher order soliton . Additionally, due to the proximity to the zero dispersion wavelength, higher order dispersion leads to fission of a higher order soliton by splitting the initial spectrum into a red-shifted soliton part and dispersive waves [6,12]. The spectral position of the dispersive wave is determined by a phase-matching condition . Due to the dispersion profile of the fibers used for supercontinuum generation and the choice of the pump wavelength close to the lower zero-dispersion wavelength, the dispersive wave usually lies in a spectral region far away from the soliton wavelength, especially on the lower wavelength side of the zero dispersion wavelength. On the other hand, the soliton self-frequency shift extends the broadening to the infrared side of the spectrum. If a second zero dispersion wavelength exists above the pump wavelength, the Raman-shifted solitons will stop shifting due to its decay initiated by transfer of energy to dispersive waves on the red side of this second zero dispersion point . Such a second zero dispersion wavelength can be used to further enhance the bandwidth of the supercontinuum . The processes leading to creation of dispersive waves when perturbing solitons has indeed been described before , but due to the unusual dispersion properties in photonic crystal fiber octave spanning spectra have been obtained with pulse energies in the nJ range. To obtain a dispersive wave at a certain wavelength phase-matching has to be fulfilled as well as an overlap with the breathing spectrum of the higher-order soliton [6,16]. However, the experimentally observed location of these dispersive waves could not be completely explained by this phase-matching alone and although pulse trapping has been suggested [17,18], it has not been proven so far [17–19]. In contrast, it has been shown theoretically and experimentally that the blue dispersive wave can be further shifted by cross-phase modulation (XPM) initiated by the infrared soliton . Additionally, a theoretical suggestion has been given to increase the bandwidth in the visible by co-propagating the pump pulse with a pulse in the visible and thus, exploiting the XPM between the two . Although some experiments have been done using two spectrally distinct pulses to achieve additional broadening , the XPM induced frequency shifts in dual pumped femtosecond supercontinuum generation has not yet been confirmed experimentally.
In this contribution, we investigate the XPM induced shift in the case of supercontinuum generation numerically and experimentally. We demonstrate numerically XPM red and blue shifts of the dispersive waves crossing the infrared solitons temporally and give a simple condition to predict the appearing wavelengths. The same shifting is demonstrated in the case of a fiber with two zero dispersion wavelengths and therefore with a dispersive wave above the second zero dispersion point as suggested in . It is shown that the shifts depend on the temporal delay, initial power of the infrared seed pulse and the group delay matching and a simple equation to predict the XPM shifted wavelengths is given. Furthermore, we demonstrate simultaneous transfer of energy to dispersive waves in the blue and infrared normal dispersion region. In particular, the infrared dispersive wave is not arising from the soliton decay but from the initial soliton fission. This effect can be predicted theoretically [6,10,22], but is usually neglected and has not been observed so far. Moreover, all the effects are confirmed experimentally by femtosecond single and dual wavelength pumping. Based on all of these scenarios we predict the maximum spectral bandwidth for femtosecond as well as longer pulses based only on the dispersion of the fiber.
Pulse propagation in fibers is well described by the extended nonlinear Schrödinger Eq. (NLSE, Eq. (1)). It includes the linear effect of gain/loss g and dispersion βn , and self phase modulation (SPM), the self-steepening and the fractional contribution of the delayed Raman response function with the parameters fR and hR(t) as nonlinear effects scaling with the nonlinearity parameter γ. It describes the temporal and longitudinal dependency of the slowly varying pulse envelope A(z, T) in the retarded time frame T.
Equation 1 is capable to reproduce the supercontinuum generation described before to a great extend, even if an octave spanning spectrum is created. In our investigations, Eq. (1) has been solved numerically by use of an adaptive Split-Step algorithm [12,23] using a commercially available software (www.fiberdesk.com).
The NLSE implicates the fundamental nonlinear processes and describes their dependence on the actual parameters. These processes can be separated in soliton generation and their self-frequency shift, the fission of higher-order solitons and the transfer of energy to dispersive waves, dispersive cross-phase modulation and four-wave mixing. All of these processes have been analyzed in detail [e.g. 6,12–15]. The basic features can be described by simple equations, which are summarized in the following paragraph.
Nonlinear pulse propagation in presence of anomalous dispersion is a well-known process in fibers, where a short pulse can excite a soliton, which is characterized by its soliton number N with N2=γP0/β2 (P 0 - input peak power, T 0 - input pulse duration). The higher-order soliton (N>1), which actually consists of N fundamental solitons, reshapes its original form after a certain distance periodically. Each of the fundamental solitons has a peak power and pulse duration according to Eq. (2) .
Each fundamental soliton experiences the soliton self-frequency shift (SSFS) due to the overlap of its spectrum with the Raman gain spectrum. An approximation relates this shift to the pulse duration Tk  according to Eq. (3).
Thus, the first soliton (k=1) emitted has the highest peak power and is faster shifted toward the infrared spectral region. During the breakup of the higher-order soliton energy is emitted in form of a dispersive wave at a frequency ωDW where the necessary momentum conservation of Eq. (4) is fulfilled (ωS is the center frequency of the soliton).
Roughly speaking, cross-phase modulation is the change of the refractive index due to nonlinearity induced by another temporally overlapping pulse, e.g. at another wavelength or polarization state. The instantaneous XPM induced phase change (Eq. (5)) scales with the length dz and is twice that of self phase modulation. As XPM induces just a phase shift no energy transfer takes place.
The effects described so far will be used in the following sections to give an interpretation of the numerical results and their experimental confirmation. Therefore, one of the most important parameters is the relative group delay β1 of the different spectral components, because XPM depends on the walk-off length (interaction length) and due to the SSFS the solitons will change their group delay during propagation.
The two photonic crystal fibers under investigation exhibit two different zero dispersion wavelengths. The first fiber (PCF1) has zero dispersion points at 770 nm and 1600 nm. The second fiber (PCF2) has a similar first zero dispersion point at 750 nm but a second one at 1250 nm. Both fibers have almost equal core sizes translating into a nonlinearity coefficient of γ∼84(Wkm)-1. A Taylor series fit of the measured dispersion curves has been done to obtain the dispersion coefficients βn (n≥2) around 1030 nm up to 12th order (see Appendix). By integrating this β2(ω) function, the relative group delay β1(ω) (shown in Fig. 1) and propagation constant β(ω) can be calculated without the knowledge of the full effective refractive index profile. Throughout the paper, the relative group delays and the wavelength position of the dispersive wave will be important. Therefore, Fig. 2 shows the group delay matching β1(λm)=β1(λ0) for both fibers, where above the infrared curve (GDM-IR) the group delays are smaller, and below the low wavelength curve (GDM-VIS) the group delays are higher than at any other wavelength. The other curves show the predicted wavelength λm of the dispersive wave that is created during the soliton fission with the soliton wavelength λ0 according to Eq. (4) assuming vanishing nonlinear phase contribution (only the interesting region of anomalous dispersion is displayed).
3. Numerical results
3.1 Dispersive XPM in PCF1
In Ref.  the action of XPM of the infrared soliton on the dispersive wave has been described to explain the observed blue shift. There, a dispersive wave is created in the visible due to the fission of the higher-order soliton and separates in time from the soliton because the dispersive wave has a lower group velocity. As the SSFS shifts the soliton to higher wavelengths, it slows down and is finally caught by the dispersive wave. As it walks through the dispersive wave, the tail of the soliton induces a blue shift by XPM, which is directly proportional to the soliton peak power and a certain interaction length (Eq. (5)). Therefore, the blue shift is limited by the amount of group velocity compensation of the SSFS compared to the group velocity change due to XPM. If the difference in group-velocity of the soliton wavelength and the maximum shifted wavelength is too large, they start to separate temporally. For the spectral components, which have not separated temporally, the leading edge of the soliton compensates the XPM induced shift by the tail. This does not happen to the escaping components and a blue-shift remains even if the soliton walk-through was complete. In the non-dispersive case such a permanent frequency shift can only occur if the walk-through is not complete .
In Ref.  it is suggested, that this dispersive XPM shift can be used to create additional bandwidth by the use of an additional pulse, which in principle replaces the need of the dispersive wave created by the soliton fission process.
From an experimental point of view, this additional pulse can be easily created by second harmonic generaten (SH) of an intense infrared pulse (IR). An optical delay line adjusts the delay of the two pulses.
The situation of two co-propagation pulses in PCF1 is numerically solved. The initial green pulse at 515 nm has a pulse duration of 300 fs and a pulse energy of 100 fJ, which in all cases was small enough not to affect the infrared pulses. In Fig. 3 the propagation of this green pulse together with a 100 pJ, 150 fs sech2-pulse at a center wavelength of 1030 nm in 1.5 m of PCF1 is shown in a spectrogram view. The green pulse initially traveling 5 ps in front of the pump pulse is caught by the faster soliton, which has been created by the infrared pulse. In contrast to the scenario described above, now the leading edge of the soliton shifts the spectrum of the SH pulse due to XPM. As can be seen, this leads to new components above 515 nm up to 550 nm, which are not shifted back by the XPM of the tail of the soliton because they escape the interaction region due to dispersion. In Fig. 3(b) only the visible part of the evolution is shown and highlights the XPM induced shift. The observed large XPM shifts (in our case ∼35 nm) have been described before [14,15], but have not been checked by a rough estimation of the possible XPM shift.
The XPM induced red frequency shift for the dispersive wave can be evaluated to first order according to Eq. (6), thus replacing the need for evaluating the full NLSE. It can be derived from Eq. (5) easily by integrating the frequency shift that is induced at a certain position of the dispersive wave as the front of the soliton walks through (τ is the retared time at a certain position of the dispersive wave with frequency ωDW, from which the soliton S(τ) is delayed by β1(ωDW)-). Therefore it is clear that the maximum blue shift should be calculated by changing the limits of the integration from 0 to infinity (trailing part). The cascading effects of XPM are identified by the last term of Eq. (6): If the partial frequency shift is in direction of lower relative delay, the following partial shift is increased.
The condition in Eq. 6(a) has to be used for the evaluation of Eq. (6). If this condition is fulfilled, XPM has shifted the frequency to a position, where the delay relative to the soliton is opposite compared to the initial dispersive wave. This simply means that this component can now escape the XPM action of the soliton and stop further shifting.
Of course, the interaction length of the soliton (with a FWHM pulse duration of 1.76∙T0 and group delay ) and the dispersive wave zI is then given by the width of the soliton over the relative speed of the waves (Eq. (8)) assuming the pulse durations to fulfill TDW≫T0.
The combination of Eq. (7) and (8) is an estimate for the upper limit of the XPM shift. For instance, at a propagation distance of z=0.56 m (Fig. 3(b)) the following values have been obtained for the soliton by spectral filtering: TFWHM=51fs (T0=29 fs), λS=1117 nm, =8.74 ps/m. Using the group delay at 515 nm of 14.3 ps/m, the interaction length can be evaluated to zI=5.2 mm (the actual negative sign of zI determines the direction of the shift in Eq. (7)). With the soliton peak power of 1.23 kW, the XPM shift is then calculated to Δν=4.5 THz. An iterative evaluation of Eq. (6) yields Δν=7.0 THz, proving that the change of group delay during the XPM interaction yields a significant larger shift. The observed shift by fully evaluating the NLSE is much higher (37 THz ∼ 35 nm) and seems to be contradictory to the approximations. The explanation of this can be seen in Fig. 3(b): The XPM shift does actually generate some photons much further away than the shift calculated. Thus, these photons can escape at a frequency according to condition 6(a), which of course still holds. Once escaped, the instantaneous Kerr-nonlinearity, which is responsible for XPM, supplies enough new photons so that a significant portion of the SH pulse can be shifted to the new frequency during walk-through. With the group delays at 1117 nm and 515 nm, Eq. 6(a) predicts a wavelength of 550.6 nm to escape, which agrees very well with the observed wavelength of 550 nm, again obtained by spectrally filtering the numerical data at z=0.56 m.
The scenario in Fig. 3 can be influenced very easily. For the calculation in Fig. 4 the pulse energy of the infrared pulse is increased to 200 pJ. Therefore the first soliton that is created shifts faster due to the higher pulse peak power (Eq. (2)). The temporal delay is decreased to 3 ps to enable the dispersive wave to overlap the soliton during the 1.3 m propagation. This simulation reveals two things: Firstly, in the first part of the propagation XPM induces a red shift because the green pulse starts to fall behind the soliton. Meanwhile, the soliton slows down and in the last part of the simulation the situation changes and the soliton falls behind the green pulse starting to blue shift the trailing part of it. Secondly, a second soliton emerges and starts to blue shift the red-shifted components. Of course, the amount of shift differs due to different group velocities matching and peak power.
In Fig. 5 visible spectra after propagation through PCF1 are compared for different pulse delay and energy settings. In Fig 5(b) the delay was 8 ps compared to 5 ps for Fig. 5(c). The soliton is further shifted to the infrared in Fig. 5(b) at the distance, where it starts to catch the dispersive wave and therefore Eq. 6(a) predicts a higher wavelength to escape due to the larger relative speed of the dispersive wave and the soliton. The red shift is comparable to the one obtained in Fig. 5(d), where the parameters of Fig. 4 are used. There, the group velocity mismatch changed sign and as can be seen a blue shift starts. Propagating further, this blue shift is enhanced (Fig. 5(e)). Again, Eq. 6(a) can be used to predict the observed wavelength now for the blue shift: the predicted wavelength λpred, the numerical obtained wavelength λnum and the soliton wavelength λS, at which the XPM shifted wave starts to escape, are λpred=499.8 nm, λnum=499 nm, λS=1202 nm for Fig. 5(d) and λpred=492 nm, λnum=492 nm, λS=1219 nm for Fig. 5(e). We tested condition 6(a) with all other simulations and found it to predict the shifts almost perfectly.
Figure 6 compares the evolutions by just varying the pulse energy of the infrared pulse. The separation is again 5 ps and a complete walk-through can be observed during the propagation of 1.5 m. In Fig. 6(a), no spectral shift of the SH pulse can be observed due to the fact that the XPM shift is not strong enough to shift to a frequency range, where these components can escape. The situation changes in Fig. 6(b) where the soliton shifts faster to the infrared due to higher input power. The corresponding XPM shift is just enough to let a small amount of red shifted wavelengths to escape, which are not shifted back. In Fig. 6(c) the peak power of the first soliton is high enough to lead to the situation described above, leading to a permanent red-shift of the spectrum of the SH pulse.
3.2 Dispersive XPM and soliton breakup in PCF2
The second fiber under investigation has two zero dispersion points, which are closer to each other compared to PCF1. Figure 7(a) shows the propagation of a 300 fs, 200 pJ pulse at 1030 nm in 1.5 m of this fiber. In the initial propagation, the oscillatory behavior of the higher-order soliton is observed, followed by the first fundamental soliton shifting to the infrared. It soon reaches the point, where it decays at the second zero dispersion point at 1250 nm. This decay leads to the transfer of energy to dispersive waves around 1350 nm and stops the SSFS. Interestingly, the second soliton acts on this dispersive infrared waves by a XPM induced blue shift just as discussed before. It is also clear that no further XPM red shift is possible on this infrared side.
In Fig. 7(b) the spectral evolution is again displayed now up to -75 dB. Clearly, the initial breathing of the soliton spectrum can be seen. At the highest temporal compression, the spectrum reaches above the second zero dispersion and can transfer energy to a dispersive wave at a wavelength around ∼1750 nm. The series of compression is accompanied by slightly different phase-matched wavelength due to the Raman shift of the soliton wavelength. Nevertheless, such a dispersive wave across the second zero dispersion wavelength is important to explain the experimental data discussed in the next section. This effect is to distinguish from the decay of the self-frequency shifted solitons as it is related to the initial fission of the soliton, just like the well-known creation of blue dispersive waves [6–8,14–16]. The proximity of the second zero dispersion point fulfills the two requirements of observing a dispersive wave: phase-matching and spectral overlap with the soliton spectrum [6,16]. To prove this occurrence of dispersive waves across the second zero dispersion point we simulated a pulse with a higher pulse energy of 1 nJ (300 fs), where the soliton compression should lead to an increased spectral overlap. The result is shown in Fig. 7(c). The soliton compression is indeed strong enough to lead to the simultaneous transfer of energy to two dispersive waves: on the blue side of the first zero dispersion wavelength at ∼450 nm and on the red side of the second zero dispersion point around 1750 nm. Fig. 7(d) compares again the infrared spectra obtained by the two different pulse energies. The red curve (Fig. 7(d)) highlights the dispersive waves from the initial fission process (DWI) by stopping the propagation at 6 cm of the high energetic pulse. The black curve shows the spectrum after 60 cm of the low energetic pulse, showing mainly the dispersive wave due to the decay (DWD) of the soliton near the zero dispersion point.
4. Experimental confirmation
In this section the simulated XPM induced red and blue shifts and the transfer of energy to a dispersive wave across the second (higher) zero dispersion point of PCF2, will be confirmed experimentally by femtosecond single and dual wavelength pumping. The fiber lengths are longer than in the simulations but the basic results are not affected because most of the interactions occur already in the first cm of the fiber.
For the experimental setup (Fig. 8) we used a Yb:KGW oscillator delivering transform-limited 380 fs, sech2 pulses at a repetition rate of 9.8 MHz and a pulse energy up to 250 nJ. The pulses were frequency doubled in a 17 mm long LBO crystal with an efficiency up to 60%. The fundamental and second harmonic beams were then separated with a dichroich mirror reflecting 1030 nm and transmitting 515 nm. Each beam was re-collimated and sent through appropriate telescopes in order to ensure proper mode matching at the position of the fiber tip. The IR beam was sent through a delay stage before it was recombined with the SH and coupled into the PCF. After the fiber an achromatic microscope objective was used to collimate the super continuum.
Initially, coupling to the PCF was made with an achromatic lens designed for the involved wavelengths but due to its small aperture, coupling to the 1.7 μm core fibers was limited to about 10%. By making telescopes for both beams it was possible to use a regular aspheric lense (C230TM from Thorlabs) with a focal length of 4.5 mm and get approximately 30% of both colors through the fiber simultaneously. For both beams a half-wave plate and a polarizer were used to control polarization and the coupled power. As the fibers used in the experiments are strongly birefringent the polarizers were adjusted such that both beams were coupled into the same main axis of the fiber.
Temporal overlap was conveniently found by monitoring the spectrum diffracted off a grating onto a white surface. As the delay stage was scanned strong visible components appeared in the spectrum when the pulses overlapped in the fiber. Depending on the position of the stage red or blue frequency components could be emphasized as illustrated in Fig. 10. The output was analyzed using a power meter and an optical spectrum analyzer (ANDO AQ-6315-A).
The measured spectra after propagation in 5 m of PCF1 are shown in Fig. 9(a) for only the fundamental wavelength and with the SH pulse in Fig. 9(b). Each slice represents a spectrum at the fiber output when the input power is increased. In Fig. 9(c) selected slices of Fig. 9(b) are plotted additionally to show the exact spectral content. In Fig. 9(a) the power is increase up to a point, where no distinctions of the fundamental solitons due to their different wavelength is possible. Until that point, no blue dispersive wave is observed but the formation of the individual solitons and their SSFS can clearly be seen. The SSFS stops at ∼1580 nm, where the solitons keeps its central frequency even if the power is increased. This confirms the stop of the SSFS by the second zero dispersion point at 1600 nm .
Above this wavelength the dispersive waves created due to the decay of the soliton are observed at ∼1720 nm, which was close to the limit of the scanning range of the used spectrometer. Due to the long fiber, the first soliton can be separated as a femtosecond pulse tuneable from 1030 nm to 1600 nm, as suggested before .
In Fig. 9(b) the same measurement is done again with the additional SH pulse launched into the fiber but with lower resolution in the input power steps of the infrared pulses. The delay is chosen to maximize the bandwidth of the visible spectrum at the highest infrared power (Fig. 9(c), black curve). A white box highlights the measurement from Fig. 9(a). The power of the green pulses is kept constant at a low power of 6 mW (0.67 nJ). No changes in the infrared spectra are observed due to this additional pulse. In contrast, the spectrum of the green pulse is influenced significantly. As soon as a soliton is emitted (slice 7, Fig. 9(b) and red curve in Fig. 9 (c)) the spectrum of the green pulse starts to extend to the red. If the power is increased further, the peak power and therefore the speed of the SSFS is increased, thus changing the speed of the soliton, which then is caught by the green pulse and induces an XPM blue shift (blue curve in Fig. 9(c)) as described before. At the highest power, the cascaded action of all solitons results in a maximum blue shift down to ∼420 nm.
A similar measurement is done for fiber PCF2 and is shown in Fig. 9(d) and 9(e), whereas Fig. 9(f) again highlights selected slices from Fig. 9(e). Again, the fundamental solitons are shifted up to 1200 nm, where they are stopped by the existence of the zero dispersion point at 1250 nm. The decay creates dispersive waves at 1350 nm. Interestingly, at higher power (slice 35-40, Fig. 9(d) and slice 9-18, Fig. 9(e)) a second dispersive wave appears, which is the experimental prove of the dispersive wave created by the inital soliton fission process discussed in the section before. An additional indication to that is, that it does not appear if only one or two solitons propagate in the fiber. From Fig. 9(d) a soliton number of N∼3 is at least required to observe this wavelength, which correspond to the fact, that N∼3 is required to obtain a soliton compression, which overlaps with this wavelength region.
In Fig. 9(e) the additional SH pulse is launched. At slice 9 of this figure (also see Fig. 9(f)), the green spectrum starts to extend more than 100 nm to the red, in contrast to the weak blue shift starting with slice 13. The delay was again set to obtain the maximum bandwidth at highest power in the infrared. As predicted by the numerical calculations, red and blue XPM shifts are observed experimentally.
To prove that the delay can influence the crossing of the soliton with the SH pulse and therefore tailor the XPM shift to be red or blue shifted, two additional spectra have been measured for PCF1. The first one, shown in Fig. 10(a), is obtained by a delay, where the blue part is enhanced. With the same settings but only changing the delay, the red part of the XPM modulated SH spectrum can be enhanced, shown in Fig. 10(b). It has to be mentioned that also the blue region of the infrared part of the spectrum (700 nm -1060 nm) seems to be affected, which we refer to as a back interaction of the not completely negligible power of the green pulse. Nevertheless, even at this power level, only the mentioned red-expansion of the green pulse spectrum due to XPM is observed. For both of these measurements, the infrared power was increased to a level where a small amount of a blue dispersive wave at 408 nm could be observed even in the absence of SH pulses - as can be seen in both figures.
5. Further discussion and prediction of the supercontinuum bandwidth
The extension of the supercontinuum spectrum may be calculated by Eq. (4) to predict the wavelength of the dispersive wave created during the soliton fission process and then use condition 6(a) to account for XPM shifts. If a certain dispersive wave is present in the visible, which is crossed by a soliton during propagation, condition 6(a) predicts the wavelength remaining after complete walk-through, provided that the XPM induced shift is strong enough. The maximum shift is obtained if the group delays of the dispersive wave and the soliton have a large difference, that is, if the soliton has shifted to the highest wavelength when the XPM shift starts. For instance, assuming a dispersive wave is at 515 nm (β1=20 ps/m) and a soliton at a maximum wavelength of ∼1560 nm (β1=45 ps/m) the escaping wavelength has to have a group delay of 70 ps/m. From the curve in Fig. 1, this is fulfilled at a wavelength of ∼419 nm. If the group delay difference is higher no significant blue shift is observed anymore due to the high slope of the group delay below 500 nm (Fig. 1). On the other hand, if the difference is low, the soliton can slow down with propagation and again get in contact with the blue shifted dispersive wave. In this case it is clear that the cascaded XPM shift finally cannot lead to wavelength below the lowest wavelength in curve GDM-VIS. This argumentation can be proved by our experimental results. In PCF1, a strong blue shift is observed (Fig. 9(b) or Fig. 9(c)) with a minimum wavelength of 424 nm, in good agreement with the value calculated above. As can be seen in Fig. 9(e) it is difficult to get a XPM blue shift in PCF2, even if the delay is changed. This can be explained by the limited possible amount of SSFS in this fiber. Actually, the soliton at the highest wavelength has a lower group delay than the wavelength at 515 nm, which means that no blue shift can occur. Nevertheless, dispersive waves due to the initial soliton fission process can be generated down to 470 nm, if the input power is increased (100 mW), see Fig. 11.
On the infrared side of the spectrum, no red shift can occur due to the relative group delays. There, the limit is given by dispersive waves generated in the inital soliton fission process in the case of a fiber with two zero dispersion wavelengths. If there is no second zero dispersion point, the SSFS and therefore the extension of the spectrum is limited by the decreasing nonlinearity and absorption.
The difference between experimentally obtained and predicted shifts can be explained by the fact, that the measured dispersion we used still contains measurement uncertainties and also is an average of two polarization states . Nevertheless, using the dispersion curves and Eq. (4) and 6(a) a good prediction of the bandwidth of the continuum can be made.
For longer pulses, the initial propagation differs from the soliton description but the modulation instability observed there is again the initial point of soliton formation. Indeed, the spectra observed for longer pulses have the same spectral bandwidth as those obtained by femtosecond pulses at the highest power levels . Even for CW continua, modulation instability will lead to a train of short pulses [12,28] finally limiting the bandwidth due to the same mechanisms described here.
We focused on the effects of dispersive XPM and dispersive wave generation during supercontinuum generation by numerically solving the extended NLSE for two different fibers exhibiting two zero dispersion wavelengths. We demonstrated XPM red and blue shifts of the dispersive waves crossing the infrared solitons temporally and showed the dependence on input power and temporal delay. Additionally, dispersive waves have been obtained in one of the fibers above the second zero dispersion point arising not from the soliton decay but from the initial soliton fission, which has not been observed before. All these effects have been confirmed experimentally by femtosecond single and dual wavelength pumping for the first time to our knowledge. The condition for predicting the observable wavelength due to dispersive XPM have been used to predict the maximum bandwidth of the supercontinuum spectrum only by the values of the fiber dispersion, which will be a usefull tool to design highly nonlinear photonic crystal fibers and choosing the right pump wavelength to maximize the bandwidth.
We would like to thank the German Research Foundation for financial support within the program: Spatial-temporal nonlinear dynamics in dissipative and discrete optical systems, FOR 532. The NKT Academy is acknowledged for financing the work of Thomas V. Andersen and Thomas Schreiber.
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