Fig. 3. $S_{\text{sol}}$ is the ratio of the energy required for forming a gap soliton in a resonator gap to the energy required for forming the same gap soliton in a Bragg gap with the same gap width relative to its center frequency.
consequently have a smaller spatial width. We define a quantity $S_{\text{sol}}(\bar{\delta}\omega) = \epsilon_{mk}^{\text{solitonのres}}/\epsilon_{mk}^{\text{solitonのBragg}}$, where $\epsilon_{mk}^{\text{solitonのres(Bragg)}}$ is the energy required for exciting a gap soliton in a resonator (Bragg) gap; $S_{\text{sol}}$ is a measure of how much easier it is to form a gap soliton in a resonator gap than in a Bragg gap. To make this comparison we consider one system in which $\omega_{uk}^-$ corresponds to the upper band edge of a resonator gap and another in which the same frequency $\omega_{uk}$ corresponds to the upper band edge of a Bragg gap. The overlap integrals that we use to determine the nonlinear coefficient, $\Gamma_{mk}$, are roughly equal at the gaps, so $S_{\text{sol}} \approx (\omega_{\text{res}''}/\omega_{\text{Bragg}'})^{1/2}$. We use physical parameters defined above but vary the values of $\sigma$ and $d$ to achieve different gap widths and center frequencies.
In Fig. 3 we plot the value of $S_{\text{sol}}$ as a function of the gap width ($\bar{\delta}\omega/\omega_{uk}$). For a small gap width, $(\bar{\delta}\omega/\omega_{uk}) = 10^{-6}$, $S_{\text{sol}} = 10^{-4}$; for gap width $(\bar{\delta}\omega/\omega_{uk}) = 10^{-4}$, which is more realistic, $S_{\text{sol}} = 10^{-2}$. Of course, material and mode dispersion, both neglected in our calculations, will set a lower bound on $S_{\text{sol}}$. The low energy requirements for gap solitons in a resonator gap are balanced by a much longer soliton
formation length,⁸ but for switching applications this restriction is not so important. A pulse with a form similar to Eq. (7) but with a much lower amplitude will be unable to propagate, because all its frequencies lie within the gap. By contrast, if the pulse has the correct amplitude, it will form into a soliton while it propagates. If the initial pulse is close to a soliton, then reshaping should be minimal.
In conclusion, we have investigated optical propagation in a two-channel SCISSOR structure with a weak Kerr nonlinearity. We have presented a NLSE that accurately describes propagation near the band edges of a resonator gap if the light is propagating in only one mode of the system. The energy required for forming a gap soliton is much smaller than in a Bragg gap of similar width. We note, too, that whereas the one-channel SCISSOR structure investigated by Heebner et al.¹ supports solitons that can travel with a small group velocity, that velocity can never vanish; furthermore, that structure possesses no gap, so a true gap soliton could not be launched. We intend to extend the analysis to coupled gap solitons and to discuss the issues involved in experimentally launching and observing gap solitons.
This research was supported by the Natural Science and Engineering Research Council of Canada and by Photonics Research Ontario. S. N. Pereira's e-mail address is pereira@physics.utoronto.ca.
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