How Does Degenerate Four-Wave Mixing Create a Frequency Comb?

[marco1235]

Hi all, I've some questions for you which challenge my brain a lot!

I'm dealing with a Photonic course and our professor introduced us a little bit of Non-Linear Optics. We spoke about Four-Wave-Mixing (4wm for brevity) and we said that when we shine a material with ##\chi ^{3}## with two frequencies ## \omega_1## and ## \omega_2 ## they mix to produce a downshifted (Stokes) wave at ## \omega_3 = 2\omega_1 - \omega_2 ## and an upshifted (anti-Stokes) wave at ## \omega_4 = 2\omega_2 - \omega_1 ##. The frequencies have equal spacing, that is, ## \omega_1 - \omega_3 = \omega_2 - \omega_1 = \omega_4 - \omega_2 ##. That's ok like in fig1 of the attached files. Now in our discussion in class the professor showed the second picture, where the process is a little bit different in my opinion (is this the well-known Degenerate Four-Wave-Mixing??). He sad that also this second process is a 4wm, and that we must think that ## \omega_1## and ## \omega_2 ## are now the same frequency ## \omega_p ## of the pumping laser. But if they are identical I don't get ##\omega_3## and ##\omega_3## as from the formula above, because if ## \omega_1 = \omega_2 = \omega_p ## then by substituting I get ##\omega_3 = \omega_4 = \omega_p ##. There's something wrong in my head of course! XD

The last question is how is it possible with a degenerate-4wm to create a frequency comb!
I think that if I could undestand the first part, everything it would be (a little bit) more clear!
Thank you in advance and have a beautiful day!

Greetings from Italy!

 

[eljose79]

Hello there,

Thank you for sharing your questions and thoughts on the topic of Non-Linear Optics and Four-Wave-Mixing. I can understand why you may find this topic challenging, as it involves some complex concepts and equations.

To address your first question, the process shown in the second picture is indeed known as Degenerate Four-Wave-Mixing. In this process, the two frequencies ## \omega_1## and ## \omega_2 ## are the same frequency ## \omega_p ## of the pumping laser. This is why you are getting ##\omega_3 = \omega_4 = \omega_p ## when you substitute in the formula. However, it is important to note that in this process, the two frequencies are not exactly the same. There is a slight difference in their frequencies, which is known as the frequency detuning. This detuning is what allows for the creation of the downshifted and upshifted waves at ## \omega_3 = 2\omega_1 - \omega_2 ## and ## \omega_4 = 2\omega_2 - \omega_1 ##, respectively. This detuning also explains why the spacing between the frequencies is not exactly equal, as you have noticed.

As for your question about creating a frequency comb, this can be achieved by using a technique called phase-matching. In phase-matching, the phase velocities of the interacting waves are matched, which results in constructive interference and the creation of a frequency comb. This technique is used in non-linear optics to control and manipulate the frequencies of the generated waves.

I hope this helps to clarify some of your questions and I wish you all the best in your studies of Non-Linear Optics. Have a beautiful day as well!