Mathematical Optics

where

$\begin{array}{l}\text{τ}(\text{ξ})=\pm 2\sqrt{a^2+2c}\left(\text{ξ}+\frac{ab+d}{a^2+2c}\right),\\ \text{β}(\text{λ})=2\text{λ}-b^2+\frac{{(ab+d)}^2}{a^2+2c}\pm \sqrt{a^2+2c}.\end{array}$

The occurrence of the double sign in the preceding expressions makes evident that the transformation from (10.9) to (10.26) is not unique (Nikiforov and Uvarov 1988); the specific choice may eventually be suggested by the problem at hand.

10.3.2.1.2.2.1 Plane Waves, Parabolic Waves, and Gaussian Beams  If τ′(ξ) = 0, which for the specific form of τ(ξ) amounts to τ(ξ) = 0 as well, and hence to the relations a² + 2c = 0 and ab + d = 0 for the various coefficients, Equation (10.26) reduces to the normal form

$\left(\frac{d^2}{d{\text{ξ}}^2}+2\text{λ}-b^2\right)v_{\text{λ}}(\text{ξ})=0,$

yielding $v_{\text{λ}}(\text{ξ})=v_0{e}^{\pm i\sqrt{2\text{λ}-b^2}\text{ξ}}$. Indeed, it can readily be seen that, under the aforementioned relations for the coefficients, Equation (10.9) ultimately yields the ...