Does specular reflection impart a force?

[Bob The Tough]

I am currently doing a PhD on the subject of solar sailing (if you don't know what that is and are curious enough, a quick google search on "Solar sailing" will give you a plethora of websites that will explain it for you). Solar sails depend crucially on forces generated when a photon is specularly reflected of a surface. However, recently I have started to wonder a) how such forces are generated and even b) if such force exist at all.

Suppose a photon is specularly reflected of a mirror (specularly in the optical sense means angle of incidence is angle of reflection, but I guess with a single photon that does not have much meaning). The momentum of the photon changes in this system and therefore a force (or more precisely, an impulse) has been induced on the mirror. This is just a simple application of conservation of momentum.

(Minor Issue: The photon that comes out is not the same as the photon that goes in. Yes, I accept that, but that does not change the fact that before a photon was moving towards the mirror, and afterwards a photon is moving away from the mirror, and therefore a momentum change has occurred.)

Therefore conservation of momentum leads us to believe that the mirror experiences a force. However, no let's consider conservation of energy. If the photon is reflected at the same frequency as it originally had, then the photon's energy has not changed. Therefore no force can act on the mirror, for if such a force acted and the mirror was free to move, the mirror would accelerate, thus increasing it's Kinetic Energy (KE), and therefore violating conservation of total energy of the system. So conservation of energy leads us to believe that no force acts on the mirror.

So - what is going on here? Assuming energy and momentum arguments given above are not erronous, I see the following possibilities:

1: The mirror does experience a force, but total energy is conserved by the mirror losing internal energy (this is, as I see it, the only other source of energy in the system). Therefore the mirror would cool, but move away. This seems just insane to me, but it would satisfy the above two arguments for energy and momentum conservation.

2: The frequency of the photon emitted is less than the frequency of the original photon. The difference in energy account for the increase in the KE of the mirror. This would mean that any photon reflected from a mirror MUST experience a drop in frequency. This also seems wrong to me, but not as wrong as option 1.

I can't see any other resolutions to this problem. All the solar sailing papers I have read so far adopt the attitude of ignoring the conservation of energy argument, and just looking at the momentum one. But surely, somebody out there in the physics community know the correct resolution of this problem and can finally let this issue be resolved. (I would have thought that any laser physicist would know the answer to this)

Thanks for reading,

Bob The Tough

 

[ZapperZ]

2: The frequency of the photon emitted is less than the frequency of the original photon. The difference in energy account for the increase in the KE of the mirror. This would mean that any photon reflected from a mirror MUST experience a drop in frequency. This also seems wrong to me, but not as wrong as option 1.

The "reflected" photons came from the mirror or metallic surface. Since the surface is moving back, then the photons are doppler-shifted to a larger wavelength, and thus, smaller energy.

Zz.

 

[Bob The Tough]

The "reflected" photons came from the mirror or metallic surface. Since the surface is moving back. So the photons are doppler-shifted to a larger wavelength, and thus, smaller energy.

Zz.

Hmm. Interesting. Gone through the maths, and taking into account the doppler shift of the photon seems to satisfy both energy and momentum conservation, at least in the initial situation when the mirror starts to move.

Hmm - cool. Well, I guess maybe it was an easy question then :) I guess the thing that is annoying now is how come no one in any of the solar sailing papers have explained this subtlety. They all just quote the conservation of momentum law.

Interesting how the change in momentum (and therefore the impulse) depends on the velocity of the sail. I'll have to look at this more carefully, but this definitely seems to be the thing I was missing. If I get stuck again, be sure to expect another post :)

Thanks Zapper.