Lecture #5+6 Helmholtz equation: quasi-monochromatic solution, with beta1 and losses

 We considered as a possible solution a quasi-monochromatic wavepacket, meaning a plane wave with a "carrier" frequency at w0 modulated by a slowing varying envelope. For the z dependence of the slowing varying part we factored out a plane wave-like term with a constant, real phase beta(w_0). Moving to frequency domain we obtained an equation for the envelope depending on beta(w), a complex, frequency dependent quantity. We expanded beta(w) around w_0 as we expect the spectrum of the solution to be "narrow" with respect to w_0. We considered the first (beta1) and second (beta2) orders of the rel part expansion, while we took the entire imaginary part alpha (responsible for the absorption). We went back to time domain. Retaining only beta1 and the absorption, we found for the envelope a solution propagating as a wave with velocity 1/beta1 (group velocity). Losses appear as an exponential decay in real space. As a reference, see the lecture notes, the first Agrawaal chapter 1.2.3 and second: 2.1 e 2.3.1 (Nonlinear Fiber Optics Govind P. Agrawal).