Abstract
In this chapter we study the effect of nonparaxiality on collapsing solutions.
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Notes
- 1.
In fact, \(\epsilon \ll 1\) even if \(r_0 = \lambda \), since then \(\epsilon = \frac{1}{16 \pi ^2} \sim \frac{1}{160}\).
- 2.
Simulations of the NLH as a genuine boundary value problem were performed many years later (Sect. 34.8).
- 3.
An informal a posteriori justification for treating nonparaxiality as a small perturbation is given in Conclusion 34.4.
- 4.
In Conclusion 31.5 we saw that generically \((f_1)_z \ll f_2\). We could not Conclusion 31.5 here, however, because its proof is based on the assumption that \( \left[ \text{ Re } \left\{ F (\psi _{R^{(0)}}) e^{-iS} \right\} \right] = \left[ \text{ Im } \left\{ F (\psi _{R^{(0)}}) e^{-iS} \right\} \right] , \) and hence that \( \left[ \text{ Re } \int F (\psi _{R^{(0)}}) e^{-iS} (\rho {R^{(0)}})_\rho \,\rho d\rho \right] = \left[ \text{ Im } \int F(\psi _{R^{(0)}}) e^{-iS}\right. \left. {R^{(0)}}\, \rho d\rho \right] .\) By (34.12), (34.16), and the \(\beta \) principle, however, we have that
$$\begin{aligned} \text{ Re } \left\{ F (\psi _{R^{(0)}}) e^{-iS} \right\} \sim \frac{1}{L^5} \gg \frac{1}{L^3 [Z]} \sim \text{ Im } \left\{ F (\psi _{R^{(0)}}) e^{-iS} \right\} \!. \end{aligned}$$(34.17)Nevertheless, we do have that \((f_1)_z \ll f_2\), because the contribution of the \(L^{-5}\) term vanishes in the integration, see (34.13).
- 5.
More precisely, nonparaxiality is a nonconservative perturbation in the NLS model. The NLH power \(P_\mathrm{NLH}\), however, is conserved (Sect. 34.8).
- 6.
The motivation for this ansatz will become more apparent in Sect. 34.6.
- 7.
The boundary value problem (34.26) can be rescaled as \(\tilde{E} = E/{E_0^\mathrm{inc}}\) and \(\tilde{\epsilon } = \epsilon |E_0^\mathrm{inc}|^2\). Under this rescaling \(\tilde{E}_0^\mathrm{inc}=1\), and variations of \(\tilde{\epsilon }\) correspond to variations of the input power.
- 8.
Note that as \(|E(Z_{\max })|=|T|\), a choice of \(T\) is equivalent to a choice of \(E\) at \(\left. z=Z_{\max }\right. \).
- 9.
See Sect. 28.1.
- 10.
A typical rule of thumb is to have at least 10 points per wavelength. In some cases, finer resolutions are required.
- 11.
This equation is analyzed in Sect. 33.1.
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Fibich, G. (2015). Nonparaxiality and Backscattering (Nonlinear Helmholtz Equation). In: The Nonlinear Schrödinger Equation. Applied Mathematical Sciences, vol 192. Springer, Cham. https://doi.org/10.1007/978-3-319-12748-4_34
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DOI: https://doi.org/10.1007/978-3-319-12748-4_34
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