Nonlinearity and Dispersion

b3824855

Hello All,

My question is "What is the relationship between nonlinearity and dispersion?"

I know that all materials are dispersive in nature, but keeping that aside for a moment and thinking of an ideal material, can I have a nonlinear (2nd order to be exact) material which is nondispersive? (Refractive index does not change with frequency)?

Iv been trying to implement such a material in a simulator, but Im not getting any second order nonlinear interactions such as SHG or OR when I use this material. The same simulation with dispersive materials gives SHG and OR.

Can some one please tell me the reason for this? Thank you very very much.

Studiot

Are you clear as to the difference between dispersive and dissipative?

Andy Resnick

AFAIK, they are independent from each other. I'm not sure how your simulator works, but second harmonic generation requires phase matching, so that may be why omitting dispersion causes problems. I don't know what OR stands for.

b3824855

Hi,

@Studiot -> Im referring to a material whose refractive index does not change with frequency. i believe this is called non -dispersive, (i dont think that this has anything to do with dissipative)

@ Andy -> Yes, I thought that the non linearity and the Dispsersion were independent too. Thats why I was baffled with the simulator results.

As you said, Second Harmonic Generation requires phase matching between the individual frequency components ω and 2ω. Which means both these frequencies must travel at the same velocity inside the material. When the refractive index does not change with freq, the velocity of both freq would be c/n inside the material. Where n is the refractive index. So, I had thought that this would be a perfect phase matching scenario!! Wouldnt it be?

What do you think?

Thanks for the replies. Btw, OR stands for Optical Rectification. (its a similar process to SHG, but the generated freq, is not at 2ω. but at 0).

Studiot

Sorry but my knowledge of optics, whether classic, wave or quantum, is limited.

From what you say you do not want to include a dispersive term (ω is therefore constant) to make your equations non linear.

Another way to make you equation non linear is to introduce a non linear dissipative term.

Therefore I suggest you consider Duffing's equation.

Chapter 7 of

Non linear Ordinary Diffrential Equations by Jordan and Smith

may well cover your needs.

Andy Resnick

I don't know how your simulator works, so I can't really comment on why you are getting odd results.

In any case, scanning Boyd's 'Nonlinear Optics', assuming perfect phase matching (which seems to be similar to a dispersionless medium) results in a monotonically increasing second harmonic signal. Optical rectification only requires a single frequency (other than a zero-frequency) and should also behave normally for a dispersion-free medium.

b3824855

Thanks, I'll have a look in Boyd's nonlinear optics.