Multimode nonlinear pulse propagation within a Ta2O5 rectangular rib waveguide has been numerically simulated. The study provides information relating to both the localized spectral evolution along the waveguide and the transverse spectral variation across the guide. The results explain measurements from our previous near-field scanning microscopy experiments that were designed to map continuum generation along and across such waveguides, and that deviated significantly from simple theory. The simulations predict an increased nonlinear phase modulation compared to that occurring in nonlinear single-mode waveguides, due to intermodal nonlinear effects such as cross-phase modulation, leading to an enhanced spectral broadening.
©2007 Optical Society of America
Continuum generation sources (CGS) are becoming increasingly important to many scientific applications such as optical coherence tomography (OCT) , optical frequency metrology [2, 3] and femtosecond carrier-envelope phase stabilization . So far the sources are mainly based on either photonic crystal or tapered fibers, in which high nonlinearity is induced by strong confinement of pulsed laser light, enabling high peak powers. Recently, several alternative silicon-based nonlinear rectangular waveguides have been fabricated [5, 6, 7], having a nonlinear refractive index one or two orders of magnitude higher than that of silica glass, and suitable for lower-power nonlinear applications. Moreover, their silicon-based compactibility provides a future prospect for them to be applied to optical integrated circuits. We have previously experimentally investigated one such novel waveguide by utilizing an adapted near-field scanning optical microscope (NSOM) to enable visualization of continuum generation along and across the guide to a subwavelength spatial resolution, via its evanescent field . The device, which was fabricated from a dielectric thin film of Ta2O5 on a silicon substrate, was able to generate a π self-phase modulation (SPM) phase shift with pulse energies of just a few hundred pJ for 1 cm propagation length. The experimental data reveals several interesting aspects of continuum generation such as the small-scale variation of the spectrum across the width of the waveguide and the spectral growth rate along the waveguide, which cannot be fitted by simple modeling based on the nonlinear Schrödinger (NLS) equation. These discrepancies were suspected to result from the multimode nature of the waveguide since higher-order modes not only induce spatial variations in the field across a waveguide, but also effects due to temporal walk-off between modes by group velocity dispersion (GVD). The presence of the latter phenomenon was supported by Fourier analysis of the experimentally measured spectra, showing the separation of several peaks in time along the waveguide’s length, relating to the incident laser pulse modal separation .
In this paper, we build on previous work by investigating some of the unique characteristics of multimode nonlinear propagation with simulations based on the NLS equation, adapted for multimode fields. The simulations are designed to model local variations in the field along and across the waveguide in a manner which replicates the resolving power of NSOM. The waveguide dispersion, self- and cross-phase modulation (XPM) are the main contributions to spectral broadening in the study. The simulation results provide a good qualitative understanding of some of the previously acquired data on the basis of these few selected fundamental processes. In particular, the simulations demonstrate the importance of the inter-modal nonlinear effects, by showing, for example, how the XPM can become the dominating term, leading to enhanced broadening over single-mode propagation.
2. Waveguide characteristics
The waveguide in the simulation is a ridge of Ta2O5, 0.5 μm high, 4.2 μm wide, with length 6 mm, on a layer of SiO2 on a Si wafer, as displayed in Fig. 1(a). The dispersion constants of the propagating modes are determined by the effective index method  in which the wavelength-dependent refractive indices of Ta2O5 and SiO2 are given by the Sellmeier equation whose coefficients are provided by the literature [11, 12]. The laser wavelength for the study is 800 nm, and the polarization is such that the electric field aligns along the y–axis. Each mode, denoted by TMmn, has indices m and n which identify the mode field distribution along the x– and y–axes respectively. In total, the guide is able to support around 20 modes for these waveguide parameters. Some examples of modeled mode intensity profiles are shown in Fig. 1(b). Due to the fact that the guide’s width is ~ 8 times greater than its height, there are many more variations of the mode field distribution along the x-axis than along the y-axis, which consists only the first symmetric (n = 0) and antisymmetric modes (n = 1).
Figures 1(c) and 1(d) show the calculated group velocity dispersion β2 for some symmetric and antisymmetric modes respectively. As can be seen, at the laser wavelength 800 nm, all the symmetric modes are in the anomalous regime (β2 < 0) except for the fundamental mode TM00, whereas the antisymmetric modes are in the normal regime (β2 > 0).
3. Simulation results
The multimode nonlinear pulse propagation in this study can be described by the adapted NLS equation for an N-mode field which can be written as
where A (p) (z,t), β(p) 1, and β(p) 2 are the slowly varying field envelope, group velocity parameter, and GVD parameter at the central wavelength for mode p, respectively. The values of β(p) 1 and β(p) 2 for the lowest-order modes are shown in Table 1. The first term on the right of Eq. (1) is related to the relative phase change of each mode to the fundamental mode TM00 caused by their different group velocities, whereas the second term results in the group velocity dispersion of the pulse. Any inaccuracy introduced by the truncation of high-order dispersion terms, especially for symmetric TM modes which have their zero dispersion wavelength in the proximity of the pump wavelength, is negligible. This is due to the fact that the second-order dispersion length is comparable to the waveguide length for the high-dispersion modes of Table 1, whereas the third-order dispersion length is significantly longer for all modes. The nonlinear term in Eq. (1) retains only self- and cross-phase modulations, which are shown to be the most important nonlinear effects in our system. Other nonlinearities such as the Raman effect and four-wave mixing are neglected since the input power in the study is far below the Raman threshold, and the phase-matching condition is not satisfied in the waveguide.
The coupling nonlinear parameter γ(p)(q) in Eq. (1) depends on the nonlinear refractive index n 2 of the guide material, which is 7.23×10-19 m2/W , and on overlapping integrals of the transverse mode intensities .
The constant h (p)(q) in Eq. (2) is 1 for p = q and 2 for p ≠ q. This gives the integral term the value of each mode’s effective area and the intermodal effective area respectively. The calculated values of γ(p)(q) are in the range 3 – 5 W-1/m for SPM and in the range 2 – 6 W-1/m for XPM. The simulation is performed by the split-step Fourier algorithm with the number of discretized frequency points 212 and the step size less than 10 μm to ensure the accuracy of the calculation. The input pulse characteristics are identical to previous experimental work , i.e. a Gaussian profile with duration 86 fs at the wavelength 800 nm. The pulse energy is 2.1 nJ with a waveguide coupling efficiency of 0.03. The resultant spectrum is calculated from the integration of the evanescent field at each location over the diameter of a 100nm NSOM tip, at the height of 20 nm above the surface of the guide.
The spectral evolution along the length of the waveguide of various multimode pulses is shown in Fig. 2(a) with details of the corresponding modal intensity ratio given in Fig. 2(b). The simulated position along the x-axis of the NSOM measuring probe is x = 200 nm, in order to avoid a zero contribution from the field of the odd modes along the central axis of the guide (x = 0 nm). The intensity is normalized in order to compare spectral shapes. Unlike the SPM spectral profile of a single-mode (dotted red curve), the multimode spectra in Fig. 2(a) lacks symmetry, exhibiting a more complex structure which is similar to results gained from our continuum measurement experiments (Fig. 1 in ). Indeed the fine details of interference become more complicated with increasing modal contribution, and distance traveled. This is due to the interference of individual modes with different linear and nonlinear phase shifts. Note, however, that the exact features of experimentally observed spectra may also be affected by other minor nonlinear effects as well as by fluctuations of experimental parameters.
The temporal evolution of the pulse for 7-mode mixing is shown in Fig. 3(a). The time profile in this example shows the pulse separating into its various modes along the length of the waveguide. As the pulses progress along the waveguide, their modal separation becomes clear, each mode delayed relative to the others according to their individual group velocity parameters β1 shown in Table 1. The noticeable fluctuation of peak height is due to the alternating intensity experienced by the NSOM probe as can be seen in Fig. 3(b). The simulated data shows the integrated intensity over time at different locations along the guide. The huge intensity variation especially within the first few millimeters of the propagation is caused by the mode field interference which becomes less predominant when a greater number of higher-order modes separate themselves from the main pulse. Consequently the curve tends to settle toward the end of the guide.
The root-mean-square (RMS) width of the simulated spectral data as it develops along the waveguide is shown in Fig. 4(a). The results agree qualitatively with our previous multimode NSOM experiments (Fig. 2 in ) in which the growth of the spectrum is slower than expected, and cannot be fitted by a simple single-mode, SPM calculation. At the beginning of the propagation when mode separation is insignificant, the strength of XPM between the modes can cause multimode broadening stronger than the single-mode case as can be observed in the 3-mode and 5-mode curves, before the 1 mm point. In order to strengthen the effect, a higher amount of energy must be concentrated in higher-order modes. The evidence can be viewed by comparing both 3-mode curves (violet and green) which have differing amounts of energy concentrated within the higher-order modes. The spectral growth of the violet 3-mode curve, which initially has the greatest rate of growth, diminishes due to the separation of TM20 at around the distance of 1.5 mm. After this point, the main contribution of spectral broadening is due to the interaction of TM00 and TM10. Consequently, the green curve which has higher energy within these modes, continues broadening at a rate such that it overtakes the violet curve. The 5-mode (blue) and 7-mode (orange) also show the retardation of nonlinear phase modulation caused by mode separation. In the 5-mode case the rate is reduced to less than that of the single-mode case at a shorter distance due to the early separation of TM30 and TM40, whereas broadening in the 7-mode system cannot be higher than in the single-mode case owing to the temporal walk-off of higher-order modes at the very beginning of the propagation. To clarify the effect of contributing modes on the overall spectrum, the spectral widths of individual modes for the case of three-mode mixing (green curve in Fig. 4(a)) are also displayed in Fig. 4(b). In the first millimeter, the spectrum of each individual mode grows rapidly, in addition to the total spectrum. Note that the width of the curves TM10 and TM20 are greater than that of TM00 at this stage because XPM in the higher-order modes is fed by the main contribution of intensity from the TM00 mode. Conversely, only a fraction of 30% is contributing to TM00 XPM from these higher order modes. At the distance when TM20 starts to separate, the decrease in the slope of the TM20 spectral growth is apparent because now only the SPM of TM20 itself contributes to its nonlinear phase modulation. The same phenomenon is also true for TM10 at a later stage. After this, the overall spectral growth is totally governed by the SPM of the fundamental mode.
So far, propagation losses have been neglected in our analysis. However, based on the above discussion, we can easily predict the spectral modifications if higher-order modes exhibit significant losses. If losses are small on the length scales where temporal walk-off occurs, no significant spectral changes are expected since all broadening due to XPM occurs at an early stage of pulse propagation. On the other hand, if higher-order mode losses are large over much shorter distances, no significant XPM takes place and the observed spectra will resemble the single-mode spectrum of Fig. 2(a).
The variation in the spectrum across the waveguide (x-direction) that has been observed in our experimental work (Fig. 4 in ) is also confirmed in this theoretical study. Three-mode mixing spectral variations across the central line (x = 0) of the waveguide after 4 mm propagation distance are shown in Fig. 5. The individual modal contributions are also displayed. Modal interference is clearly apparent in Fig. 5(a) where both the shift in wavelength range and fluctuations of the spectral linewidth can be observed. In contrast, the spectral features across the guide for the individual modes as seen in Fig. 5(b) – Fig. 5(d) appear constant. Note that the shift of individual modal spectra from the central frequency is owing to their phase shift in time, relative to the fundamental mode pulse.
In conclusion, we have presented a numerical study of nonlinear spectral broadening in a Ta2O5 rectangular multimode waveguide. The results explain the origin of effects recorded in previous experiments where we have mapped continuum generation along a nonlinear multimode waveguide with NSOM. Experimentally observed phenomena such as modal interference which causes a complex non-uniform spectrum to evolve along the guide, and small-scale spatial variations in the spectrum across the guide, have been explained with our model. The model also elucidates the significant deviation from simple theory in the rate of continuum growth along the waveguide’s first few millimeter regime, recorded in our NSOM experiments. Moreover, the model demonstrates the importance of recognizing group velocity dispersion across the various modes when designing nonlinear waveguides of this type, since it is the rate of temporal separation between modal laser pulses along the guide which determines the contribution to the important XPM nonlinear effect. In the regime of small modal pulse separation, spectral broadening in the multimode case can exceed that of single-mode waveguides where SPM alone is the dominating nonlinear feature.
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