## Abstract

Multimode nonlinear pulse propagation within a Ta_{2}O_{5}
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

## 1. Introduction

Continuum generation sources (CGS) are becoming increasingly important to many
scientific applications such as optical coherence tomography (OCT) [1], optical frequency metrology [2, 3] and femtosecond carrier-envelope phase stabilization [4]. 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 [8]. The device, which was fabricated from a dielectric thin
film of Ta_{2}O_{5} 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 [9].

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 Ta_{2}O_{5}, 0.5
*μ*m high, 4.2 *μ*m wide,
with length 6 mm, on a layer of SiO_{2} on a Si wafer, as displayed in Fig. 1(a). The dispersion constants of the propagating modes
are determined by the effective index method [10] in which the wavelength-dependent refractive indices of
Ta_{2}O_{5} and SiO_{2} 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 TM_{mn}, 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
β_{2} 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 (β_{2} < 0) except
for the fundamental mode TM_{00}, whereas the antisymmetric modes are in the
normal regime (β_{2} > 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 TM_{00} 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} m^{2}/W [7], and on overlapping integrals of the transverse mode
intensities [13].

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 2^{12} 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 [8], 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 [8]). 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 β_{1} 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 [8]) 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 TM_{20}
at around the distance of 1.5 mm. After this point, the main contribution of
spectral broadening is due to the interaction of TM_{00} and
TM_{10}. 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 TM_{30} and TM_{40}, 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 TM_{10} and TM_{20} are greater than that of
TM_{00} at this stage because XPM in the higher-order modes is fed by
the main contribution of intensity from the TM_{00} mode. Conversely, only a
fraction of 30% is contributing to TM_{00} XPM from these higher order
modes. At the distance when TM_{20} starts to separate, the decrease in the
slope of the TM_{20} spectral growth is apparent because now only the SPM of
TM_{20} itself contributes to its nonlinear phase modulation. The same
phenomenon is also true for TM_{10} 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 [8]) 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.

## 4. Conclusion

In conclusion, we have presented a numerical study of nonlinear spectral broadening
in a Ta_{2}O_{5} 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.

## References and links

**1. **I. Hartl, X. D. Li, C. Chudoba, R. K. Hganta, T. H. Ko, J. G. Fujimoto, J. K. Ranka, and R. S. Windeler, “Ultrahigh-resolution optical
coherence tomography using continuum generation in an air-silica
microstructure optical fiber,” Opt. Lett. **26**, 608–610
(2001). [CrossRef]

**2. **S. A. Diddams, D. J. Jones, J. Ye, S. T. Cundiff, and J. L. Hall, “Direct link between microwave and
optical frequencies with a 300 THz femtosecond laser
comb,” Phys. Rev. Lett. **84**, 5102–5105
(2000). [CrossRef] [PubMed]

**3. **R. Holzwarth, T. Udem, T. W. Haensch, J. C. Knight, W. J. Wadsworth, and P. S. J. Russell, “Optical frequency synthesizer for
precision spectroscopy,” Phys. Rev. Lett. **85**, 2264–2267
(2000). [CrossRef] [PubMed]

**4. **D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-Envelope Phase Control of
Femtosecond Mode-Locked Lasers and Direct Optical Frequency
Synthesis,” Science **288**, 635–639
(2000). [CrossRef] [PubMed]

**5. **S. Spalter, H.Y. Hwang, J. Zimmermann, G. Lenz, T. Katsufuji, S.-W. Cheong, and R. E. Slusher, “Strong self-phase modulation in
planar chalcogenide glass waveguides,”
Opt. Lett. **27**, 363–265
(2002). [CrossRef]

**6. **Y. Ruan, W. Li, R. Jarvis, N. Madsen, A. Rode, and B. Luther-Davis, “Fabrication and characterization of
low loss rib chalcogenide waveguides made by dry
etching,” Opt. Express **12**, 5140–5145
(2004). [CrossRef] [PubMed]

**7. **C-Y. Tai, J. S. Wilkinson, N. M. B Perney, M. C. Netti, F. Cattaneo, C. E. Finlayson, and J. J. Baumberg, “Determination of nonlinear
refractive index in a Ta2O 5 rib waveguide using self-phase
modulation,” Opt. Express **12**, 5110–5116
(2004). [CrossRef] [PubMed]

**8. **J. D. Mills, T. Chaipiboonwong, W. S. Brocklesby, M. D. B. Charlton, M. E. Zoorob, C. Netti, and J. J. Baumberg, “Observation of the developing
optical continuum along a nonlinear waveguide,”
Opt. Lett. **31**, 2459–2461
(2006). [CrossRef] [PubMed]

**9. **J. D. Mills, T. Chaipiboonwong, W. S. Brocklesby, M. D. B. Charlton, C. Netti, M. E. Zoorob, and J. J. Baumberg, “Group velocity measurement using
spectral interference in near-field scanning optical
microscopy,” Appl. Phys. Lett. **89**,
051101–1–051101–3
(2006). [CrossRef]

**10. **K. S. Chiang, K. M. Lo, and K. S. Kwok, “Effective-index method with built-in
perturbation correction for integrated optical
waveguides,” J. Lightwave Technol. **14**, 223–228
(1996). [CrossRef]

**11. **D. Smith and P. Baumeister, “Refractive index of some oxide and
fluoride coating materials,” Appl. Opt. **18**, 111–115
(1979). [CrossRef] [PubMed]

**12. **M. Jerman, Z. Qiao, and D. Mergel, “Refractive index of thin films of
SiO_{2}, ZrO_{2}, and HfO_{2} as a function of
the films’ mass density,”
Appl. Opt. **44**, 3006–3012
(2005). [CrossRef] [PubMed]

**13. **G. P. Agrawal, *Nonlinear Fiber Optics*
(Academic Press, San Diego,
2001).