The efficiency of the wave mixing process decreases as $|\bigtriangleup k|L$ increases( although there are some fluctuations). This is becuase when L gets greater than $\frac{1}{\bigtriangleup k}$ the harmonic wave can get out of phase with incident beam and power can flow from the $\omega_{2}$ back into the 2 $\omega_{1}$ waves. The coherence length of the interaction is $L_{c}=\frac{2}{\bigtriangleup k}$ so the phase mismatch factor can be written as $sinc^{2}(\frac{L}{L_{c}})$
From the phase mismatch plot we see a big decrease in efficiency when $\bigtriangleup k \neq 0$ is not satisfied. This is quite difficult to obtain in labs because the refractive index of materials that are lossless in the range $\omeg_{1}$ to $\omega_{2}$ have normal dispersion when $\frac{dn}{d\lambda}<0$. The refractive index is an increasing function of frequency. For the case of second harmonic $n\omega_{1} = n\omega_{2}$, this is not possible since $n(\omega)$ increases with $\omega$. So what is generally used is birefringence of crystals ie.the dependence of the refractive index on the direction of polarization of the optical radiation. This slows down out of phase waves to get a perfect mismatch.
This post makes 2nd harmonic generation really accessible to undergraduate students. You successfully modeled the phase matching. I would have been really interested in taking these simulations past the presentations in Robert Boyd’s text. How does the phase matching work physically? What do researchers do in the lab do to achieve phase matching? Do they pick crystals with certain properties? Are the crystals treated? If so in what way?
Which of your graphs and/or equations show that the efficiency of the wave mixing process decreases as
increases? Your conclusions provide a look at the practical efficiencies to expect in real situations, but perhaps you could further explain why second harmonic generation is so difficult to achieve, as well as more applications. Overall, though, I think you made significant progress in your project, and you could potentially continue this project in the future!
The derivations were extremely helpful, but looked very difficult to do (very impressive!), to understanding exactly how second harmonic generation occurs so now it’s not just some mysterious phenomenon. I guess I had two questions that are focused on the more experimental aspects. First, I was just wondering if you knew what a “good” efficiency was in a lab setting. Second, I was curious about some of the applications of second harmonic generation. What types of experiments use second harmonic generation?