Fundamentals of Nonlinear Optics, 2nd Edition

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:

$\frac{d A_{1}}{d z} = i \frac{ω_{1}}{n_{1} c} d_{eff} A_{3} A_{2} e^{- i Δ k z}, (D.1)$

$\frac{d A_{2}}{d z} = i \frac{ω_{2}}{n_{2} c} d_{eff} A_{1} A_{3}^{*} e^{i Δ k z}, (D.2)$

$\frac{d A_{3}}{d z} = i \frac{ω_{3}}{n_{3} c} d_{eff} A_{1} A_{2}^{*} e^{i Δ k z} . (D.3)$

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,

$A = α u e^{i ϕ}, (D.4)$

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

Get Fundamentals of Nonlinear Optics, 2nd Edition now with the O’Reilly learning platform.

O’Reilly members experience books, live events, courses curated by job role, and more from O’Reilly and nearly 200 top publishers.

Start your free trial

Fundamentals of Nonlinear Optics, 2nd Edition by Peter E. Powers, Joseph W. Haus

Don’t leave empty-handed

It’s yours, free.

Check it out now on O’Reilly