Wave interference - coherence and wavelength

[McCannKay]

I know to produce an observable inteference pattern two waves must be coherent(same frequency/phase differnce). Do coherent waves therefore always have the same wavelength? Is it possible to have interference between two waves with the same frequency and different wavelengths, and if so what whould that look like?

 

[pixel]

The speed of a wave is dependent on the medium and is equal to the product of the wavelength and the frequency. So two interfering waves with the same frequency that are in the same medium will necessarily have the same wavelength.

Two waves of different frequency (and therefore different wavelengths) will interfere in a technical sense, but the pattern will change over time, and do so very quickly if the frequencies are very dissimilar. A stable interference pattern requires that the two waves have the same frequency.

 

[sophiecentaur]

Do coherent waves therefore always have the same wavelength?

That is part of the definition of Coherence, which requires the phase relationship between the two waves to be constant at any location. Interference is basically the vector sum of the fields at a particular point. When the waves are coherent, the pattern will be stationary.
You can, however, observe a non-stationary interference pattern when the frequency difference between two sources is 'small'. Two radio transmitters with nominally the same frequency but with an offset of, say 1Hz, will produce a 'beat' of 1Hz in the areas where the signal levels are nearly equal (Mush area) The interference pattern from the two transmitters will have the same shape as it would be from synchronised transmitters but it is racing across the country at one fringe spacing per second.
The optical equivalent is visible when you are moving one source (or slit or reflecting surface), which is producing a rate of change of phase. This will cause the pattern to move. Personally, I like RF examples because you can actually move the knobs on a piece of equipment to vary frequency and amplitude and you can have individual antennae which behave just like the stylised diagrams we use.
But I am from a generation when lasers were weird and wonderful things about which even the University Tutors were not too happy to discuss the finer points.

 

[Andy Resnick]

I know to produce an observable inteference pattern two waves must be coherent(same frequency/phase differnce). Do coherent waves therefore always have the same wavelength? Is it possible to have interference between two waves with the same frequency and different wavelengths, and if so what whould that look like?

Yes- if the two fields are mutually coherent, they will interfere. This technique (heterodyne) is commonly used in signal processing.

 

[sophiecentaur]

This technique (heterodyne) is commonly used in signal processing.

Is there not an essential difference here, between the linear process of interference and the non-linearity involved in frequency mixing (heterodyning)? Similar formulae involved in each, tho'.

 

[Andy Resnick]

Is there not an essential difference here, between the linear process of interference and the non-linearity involved in frequency mixing (heterodyning)? Similar formulae involved in each, tho'.

I don't think so- but let me expand a bit. Consider sum- or difference-frequency generation in an optical crystal. While this process (3 wave mixing) requires a nonlinear description of the medium, the requirement of phase matching relies on an interference pattern within the optical crystal.

 

[sophiecentaur]

I don't think so- but let me expand a bit. Consider sum- or difference-frequency generation in an optical crystal. While this process (3 wave mixing) requires a nonlinear description of the medium, the requirement of phase matching relies on an interference pattern within the optical crystal.

Is that so that you have regions of high field where the non linearity works most? In an RF scenario, you would probably use a resonant cavity or even an LC circuit. I don't have any experience with Parametric Amplifiers so it's all a bit arm waving for me.

 

[Andy Resnick]

Is that so that you have regions of high field where the non linearity works most? In an RF scenario, you would probably use a resonant cavity or even an LC circuit. I don't have any experience with Parametric Amplifiers so it's all a bit arm waving for me.

Not exactly- phase matching describes how the generates wave has a fixed phase relationship with the nonlinear polarization, leading to efficient transfer of energy from the incident fields to the generated field.

https://en.wikipedia.org/wiki/Nonlinear_optics#Phase_matching