Hello,
I have a bit of an urgent question that I need an answer to, so I was hoping someone here might be able to help me out:

Earlier today I was taking measurements for an experiment in which I was measuring the width of a laser beam that had been focused down by an objective lens. The experiment involved stepping a surgical blade across the beam in 100 nm increments and measuring the intensity underneath of the blade using a photodiode. I had expected the intensity to drop off over about 10-20 microns, so after running it about 20 times and seeing it drop off over about 2 microns very consistently I was a bit surprised. I had thought that 2 microns would be about equal to the (fundamental, not experimental) uncertainty in the position of a photon from a laser, though I wasn't sure what that notion was based on so I decided to calculate what the uncertainty should be and ended up with a very strange answer.

According to the entry on wikipedia (yes I know not the most reliable source but it's all I can find on short notice) He-Ne lasers typically emit 632.991±0.002nm. For a photon, p=h/λ so Δp=hΔλ/λ²
Now, from the uncertainty principle, ΔpΔx>=ħ/2=h/4π. So the minimum value for Δx is then Δx=h/4πΔp=λ²/4πΔλ=16mm. Now I know my beam is smaller than 16mm across so the uncertainty in its position has to be significantly smaller than that, but that would require the uncertainty in the wavelength to be much larger and using the same calculations, my 2 micron beam would require Δλ to be about 16nm, 4 orders of magnitude higher than the 2pm indicated by the wikipedia article, and high enough to make me suspect that either my math is wrong or something fishy is going on.

So can anyone help me out and tell me where I'm going wrong here? Thanks a lot!

Responding to title...
Bub, when lasers are involved, you don't want to be uncertain!

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The uncertainty is in the time domain. for photons time = distance. 16mm corrisponds to about a nanosecond of uncertainty in when the photon passed thru the focus. Also there is uncertainty as to the direction it is going. If you were to chop the beam to pulses shorter than this, you would spread the frequencies. Read up on "ultra short pulse laser phenomina"

Ps You can make the spot even smaller by beam expanding before the lens or using a shorter focal length lens. It is easir to use the optical equations than the uncertainty principle a thurough resource is here. http://cord.org/cm/leot/course01_mod...01-09frame.htm
Both of these increase the uncertainty in direction. Uncertainty is in the 4-d space-time vector.

Well, the uncertainty in wavelength corresponds to uncertainties both in energy and momentum, which have uncertainty relationships with time and position respectively. However doing the same calculations using the energy/time relation yields the same result: E=hc/λ, ΔE=hcΔλ/λ², ΔtΔE=ħ/2, Δt=h/4πΔE=λ²/4πcΔλ, Δx=cΔt=λ²/4πΔλ, which is exactly what I had before. But I must be making a mistake somewhere because I know the results I get using that equation are incorrect.

So I guess what I'm getting at then, is how do I determine the uncertainty in x (ie the minimum possible beam width) from the range of wavelength values that the laser outputs?

I've merged the two threads and deleted the resulting double post by uniqueuponhim and vergentbill's post which now referred back to this thread.

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