Abstract
In this chapter we consider the propagation of continuous-wave (cw) laser beams in a linear medium.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
- 2.
- 3.
To simplify the notations, in this chapter (only) we denote the inhomogeneous linear index of refraction by \(n(\cdot )\), and its value in the absence of inhomogeneities by \(n_0\).
- 4.
This is the linear analog of a weakly-nonlinear homogeneous Kerr medium, i.e., when the contribution of the nonlinear Kerr effect to the index of refraction is small, see (1.29).
- 5.
This ansatz can be viewed as a generalization of the plane-wave solution (1.12).
- 6.
See e.g., [210].
- 7.
In geometrical optics the characteristics are also called rays, because they correspond to the curves followed by light rays.
- 8.
I.e., \(\Sigma _1 = \{ \mathbf{x}(\sigma _1) \, | \, \mathbf{x}(0) \in \Sigma _0 \}\).
- 9.
Not to be confused with \(n\), the index of refraction.
- 10.
This property is called whole-beam collapse (Sect. 7.7).
- 11.
Here the angle is defined with respect to the interface, hence the use of cosines instead of sines.
- 12.
We use the notations \(S^{(\!E)}\) and \(S^{(\psi )}\) whenever we want to distinguish between the phases of the Helmholtz and Schrödinger solutions, respectively.
- 13.
Recall that \(S^{(\!E)}\) increases in the direction of propagation of the ray (Sect. 2.1.1).
- 14.
Even in a linear homogeneous medium, if we do not apply the geometrical optics approximation, the lens does not focus the solution to a point (Sect. 2.10).
- 15.
This book is mainly concerned with singular NLS solutions. Therefore, understanding the origin of singularities in the linear case will help us to determine whether NLS singularity is a linear or a nonlinear phenomenon.
- 16.
The discussion here is very informal. The formulation of boundary conditions for the Helmholtz equation in the positive half space will be discussed in Sect. 34.7.
- 17.
That this is the correct expression will follow from Galilean invariance of the Schrödinger equation.
- 18.
The quadratic phase term (2.35) for the effect of a lens in the Schrödinger model is also not equal to \(S^{(\!E)}-z\), see (2.34). It is, however, \(O\!\!\left( f^2\right) \) equivalent to it, since under the paraxial approximation \( \frac{r}{z-F} = O(f)\), see (2.64). We “prefer” approximation (2.35) over the “exact” expression (2.34), because (2.35) agrees with the lens transformation of the linear Schrödinger equation (Corollary 8.1).
- 19.
- 20.
We already derived this expression by applying the geometrical optics approximation and then the paraxial approximation, see (2.35). In the derivation here the order of the approximations is reversed.
- 21.
The subscript go emphasizes that \(\psi _\mathrm{go}\) is not an exact solution of the Schrödinger equation, as it is obtained under the geometrical optics approximation.
- 22.
The assumption that the input beam has a Gaussian profile is common in optics.
- 23.
This is not the case in nonlinear propagation of Gaussian input beams (Sect. 3.5). Even in linear propagation, this aberrationless propagation property holds only for Gaussian profiles. Thus, for example, if the input beam has a sech profile, the beam does not maintain a sech profile during linear propagation. The reason why this aberrationless property only holds for linear Gaussian beams will become clear in Sect. 2.15.3.
- 24.
This is possible because the transverse profile of the solution is “known” to be a rescaled Gaussian.
- 25.
See Sect. 2.11.
- 26.
Indeed, under the rescaling \(\tilde{z} = z/L_\mathrm{diff}\), (2.51a) reads \(L_{\tilde{z}\tilde{z}} = L^{-3}\), and so \([L_{\tilde{z}\tilde{z}}] = O(1)\).
- 27.
This approach has the advantage that it applies also to non-Gaussian beams.
- 28.
- 29.
This is the dimensionless analog of \(\psi _{zz} \ll k_0 \psi _z\).
- 30.
I.e., in the linear Schrödinger model.
- 31.
See Sect. 2.14 for definitions of \(L^p\) and \(H^1\) norms and spaces.
- 32.
The dispersive character of the Schrödinger equation in optics is not due to the paraxial approximation, since the linear Helmholtz equation is also “mathematically dispersive”. Indeed, the dispersion relation for the linear modes \(E = e^{-i \omega z+ i \mathbf{k} \cdot \mathbf{x}}\) of the Helmholtz equation (2.22) is \(\omega = \sqrt{k_0^2 - |\mathbf{k}|^2}\), and so \( {\omega }/{|\mathbf{k}|} \not = constant\). This dispersion is also mathematical and not optical, since the Helmholtz equation models the propagation of a single temporal frequency.
- 33.
This reversibility property also holds for the rays of the eikonal equation (Lemma 2.2).
- 34.
To see that the Gaussian beam is collimated at \(z_{\min }\), note that by (2.49), it can be written as \( \psi =A(z,\mathbf{x}) e^{ i \frac{L_z(z)}{L} \frac{|\mathbf{x}|^2}{2}+ \zeta (z)}, \) where \(A\) is real. Since \(L(z)\) attains its minimum at \(z_{\min }\), then \(L_z(z_{\min })=0\), and so (2.83) holds. Intuitively, the Gaussian beam is collimated at \(z_{\min }\), since it is focusing for \(z< z_{\min }\) and defocusing for \(z> z_{\min }\).
- 35.
Whether the nonlinearity is focusing or defocusing depends on the relative signs of nonlinearity and diffraction (Sect. 5.9).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Fibich, G. (2015). Linear Propagation. In: The Nonlinear Schrödinger Equation. Applied Mathematical Sciences, vol 192. Springer, Cham. https://doi.org/10.1007/978-3-319-12748-4_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-12748-4_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-12747-7
Online ISBN: 978-3-319-12748-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)