Second harmonic generation vs interference

[dziech]

Hello,

I have a question about the mathematical expression of interference and second harmonic generation

If we have two waves $E_{1,2}(\Theta ,t) = E_{1,2}e^{j(\omega t + \Theta)} = \frac{1}{2}(E_{1,2}e^{j(\omega t + \Theta)} + c.c. )$

According to the Euler's formula. Now - both interference and second harmonic generation is connected to squaring the sum of the field and c.c. Interference is connected to the phase difference (I saw somewhere $\omega t$ being omitted when calculating the interference pattern) and if I square the $\omega t$ part, we have indeed second harmonic and some interference pattern because of that phase part.

Can anyone point to, or right down the second harmonic generation sort of equation also with phase information (interference) part ? I am a bit lost in this and would like that someone clear that for me.

 

[Achy47]

Hello,

Thank you for your question. The mathematical expression for interference and second harmonic generation can be written as:

Interference: $E_{int} = E_1 + E_2 + 2\sqrt{E_1E_2}\cos(\phi)$

where $E_1$ and $E_2$ are the amplitudes of the two waves, and $\phi$ is the phase difference between them.

Second harmonic generation: $E_{SHG} = 2E_1E_2\cos^2(\phi)$

where $E_1$ and $E_2$ are the amplitudes of the two waves, and $\phi$ is the phase difference between them.

As you can see, both expressions involve the cosine of the phase difference, which represents the interference pattern. In the case of interference, this interference pattern results from the superposition of the two waves, while in the case of second harmonic generation, it results from the nonlinear interaction between the two waves.

I hope this helps to clarify the mathematical expressions for interference and second harmonic generation. If you have any further questions, please let me know.