Appendix D: Exact Solutions to the Coupled Amplitude Equations

Exact solutions to the coupled amplitude equations for three monochromatic plane waves were first presented by Armstrong et al. (1962). The following parallels their development. The coupled amplitude equations from Section 4.1.4 are repeated for convenience:

$$\frac{\text{d}{A}_{1}}{\text{d}z}=\text{i}\frac{{\omega}_{1}}{{n}_{1}c}{d}_{\text{eff}}{A}_{3}{A}_{2}{\text{e}}^{-\text{i}\Delta kz},\left(\mathrm{D.1}\right)$$

$$\frac{\text{d}{A}_{2}}{\text{d}z}=\text{i}\frac{{\omega}_{2}}{{n}_{2}c}{d}_{\text{eff}}{A}_{1}{A}_{3}^{*}{\text{e}}^{\text{i}\Delta kz},\left(\mathrm{D.2}\right)$$

$$\frac{\text{d}{A}_{3}}{\text{d}z}=\text{i}\frac{{\omega}_{3}}{{n}_{3}c}{d}_{\text{eff}}{A}_{1}{A}_{2}^{*}{\text{e}}^{\text{i}\Delta kz}.\left(\mathrm{D.3}\right)$$

Instead of carrying various constants through the calculation, we start by writing the coupled amplitude equations in “normalized” form. The amplitudes are normalized by rewriting them in magnitude and phase form,

$$A=\alpha u{\text{e}}^{\text{i}\varphi},\left(\mathrm{D.4}\right)$$

where u and ϕ are real quantities and α is a real normalizing term. Note that from the definition of ...

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