# Interference fringe of light beams of different frequencies

1. Dec 21, 2011

### abcrelativity

I have a question. If you have two light beams of different frequencies, would you observe interference fringes?

2. Dec 21, 2011

### cortiver

Re: Interference fring of light beams of different frequencies

This is an interesting question. It turns out that the answer is no. This is pretty straightforward to see mathematically, but for the benefit of intuition it's instructive to consider the case when the light beams have two frequencies f1 and f2 which are very close, but not quite equal. Then when adding these two light beams you have a beat frequency fb = f1 - f2. If the two light beams are in phase at time t = 0 (so you have constructive interference), then at time t = pi/fb the light beams will be 180 degrees out of phase (so you have destructive interference). However, unless f1 and f2 are really close (say within 10Hz of each other), then the beat frequency will be too fast the human eye to resolve. In that case the light intensity that we will see is the average of constructive interference and destructive interference, which amounts to no interference at all.

3. Dec 21, 2011

### technician

Re: Interference fring of light beams of different frequencies

With 2 different frequencies of sound interference shows itself as 'beats'
With radio waves of different frequencies beats are also produced. This is used in the superheterodyne principle of radio reception.
I am not aware of any similar effect with light.
Your question is a good question !!!

4. Dec 21, 2011

### abcrelativity

Hi Cortiver, you seem to answer my question. This is for two light beams so frequency difference is really small (much less than 10 Hz). It is a difference of frequency of 1 part per 10'000 with respect to the average frequency of daylight. What I am not sure to understand is what would be the beat frequency? Would it increase when the difference in frequency is getting smaller or the reverse? Is there a way to quantify the beat frequency? Maybe if you have a reference of a book with this problem would be usefull..

5. Dec 21, 2011

### Staff: Mentor

He told you in his post:

The frequency of visible light is around 1015 Hz. One part per 10000 of that would give you a beat frequency of 1011 Hz which is much too rapid to see visually. I don't know whether electronic detection equipment is good enough to "see" it.

Last edited: Dec 21, 2011
6. Dec 21, 2011

### cortiver

Beat frequency is a pretty basic concept in physics so you should be able to find lots of good explanations on the web -- you could start with the Wikipedia article. The beat frequency is calculated as the difference in the frequencies of the two light beams, fb = f1 - f2. So the smaller the difference between the frequencies of the two beams, the smaller the beat frequency (so the beats are slower and hence easier to detect). In the case that the frequencies are actually the same, f1 = f2, then the beat frequency is zero -- so that the interference fringes don't change with time at all.

I don't know how you're getting "1 part per 10,000". The frequency of visible light is measured in hundreds of terahertz, so a difference in 10Hz is more like one part in 10^14, which is pretty minuscule. That's why beats are much harder (impossible?) to observe with light waves than, say, sound waves.

7. Dec 22, 2011

### Cthugha

Are you sure it is really 10 Hz? That is nothing in the optical domain. Even rather tiny polarization splittings of 10 microeV which are already hard to resolve in the spectral domain coreespond to a difference of 10^9 Hz. Also are we talking about two monochromatic beams or two spectrally broad beams with different center frequency? In the latter case beatings are washed out anyway.

You can see the beatings using a Michelson interferometer. If you record an interferogram of the combined beams you can directly see the beats in the signal. Depending on how good (and expensive) your translation stages are, you can resolve pretty fast beatings. If you use good piezos, temporal resolution in the upper attosecond range is possible.

8. Dec 22, 2011

### e.bar.goum

Re: Interference fring of light beams of different frequencies

I think you're mixing up sound and light. The ear can resolve beat frequencies at 10Hz seperation. Visible light is many orders of magnitude higher in frequency (THz). In addition, considering that a linewidth of 10 Hz is an extremely good laser, such a restriction would mean that in practice we would never see interference fringes. (And certainly not before a decade or so ago).

9. Dec 22, 2011

### cortiver

Re: Interference fring of light beams of different frequencies

There's no difference between sound and light, in principle. Yes, in practice it is much harder to observe beats with visible light due to the high frequencies, as I already noted.