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I have been recently wondering, if I take a powerful enough energy source (photon) and I have an perfect mirror exactly in front of it and assume an "emitter" shot the light towards the mirror.

As perfect mirrors absorb no energy of ANY kind from photons, should this mean that the perfect mirrors would never move due to transfer of momentum of the light?

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    Depends on the mass of the mirror, of course. Your perfect mirror would have to have an infinite mass, in which case it could absorb the momentum change, without absorbing any energy. A mirror of finite mass will absorb some energy in a collision that will change the energy, and thus the wavelength of the photon. There is no contradiction here. – CuriousOne Aug 25 '14 at 20:54

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One thing that is mentioned in the previous discussion is the frequency shift experienced by light reflected from a perfect mirror. The previous answers have been dancing around this, and essentially you need to conserve both energy and momentum, resulting in a Doppler shift of the reflected light frequency. Allow me to spell this out a little more, given that this may be a point of confusion.

The key to understanding this is to think of the reflection of a photon as two separate events: absorption and emission. Assume for simplicity that the photon (energy $\hbar\omega_0$) is normally incident to the mirror. The first step is the absorption process, which by conservation of energy and momentum will accelerate the mirror backwards (mirror velocity = $v_0$, momentum = $mv_0$, and kinetic energy = $\frac{1}{2}mv_0^2$) and excites the internal degrees of freedom of the electrons in the mirror to an energy of $\hbar\omega_0$. The photon will then be re-emitted in the opposite direction, reducing the internal energy of the mirror, as well as further accelerating it to velocity $v$. The energy and momentum balance for this emission process is described in [1] and is based on Fermi's original treatment from 1932, which I will reproduce here.

Assume the mirror is moving in the positive direction, while the new photon (energy $\hbar\omega$ and momentum $\hbar\omega/c$) is emitted in the negative direction. Equating the energy and momentum before and after the emission process, we obtain the following equations: \begin{equation} \textrm{Conserve energy:}\qquad \hbar\omega_0+\frac{1}{2}mv_0^2=\hbar\omega+\frac{1}{2}mv^2 \end{equation} \begin{equation} \textrm{Conserve momentum:}\qquad mv_0=mv-\hbar\omega/c \end{equation} Rearrange the equations and divide them to eliminate $m$ and solve for $\omega$: \begin{equation} \frac{v'-v}{2c}=\frac{\omega-\omega_0}{\omega} \end{equation} \begin{equation} \Longrightarrow\qquad \omega_0=\omega(1+\frac{v+v_0}{2c}) \end{equation}

This shows that the frequency of the emitted photon will indeed be red-shifted by a factor of $(1+\frac{v+v_0}{2c})^{-1}$, which is just the Doppler shift ($v\ll c$), with the relevant velocity being the average of the initial and final velocities. In fact, a complete treatment of the reflection problem would include another factor of the Doppler shift from the initial absorption process as well.

Although the mirror has not permanently absorbed the energy of the photon, it has acquired some kinetic energy from the recoil of the reflection process since momentum must be conserved. To conserve energy as well, this small kinetic energy causes a red-shift of the reflected photon by the Doppler factor.

[1] Barnett, S. Journal of Modern Optics, 57, 1445 (2010).

Gilbert
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Since the photon reflects, its momentum changes: $p_{ph}'=-p_{ph}$. But total momentum of the system is conserved: $p_m+p_{ph}=p_m'+p_{ph}'$. Thus, the mirror will change its momentum.

But, if the mirror has large mass, then it'll get very small energy from the collision. For zero-mass particle (photon) falling onto the mirror with mass $m_2$, the energy of this particle after reflection is$^\dagger$:

$$\mathscr{E}'_1=\frac{m_2}{1-\cos\theta_1+\frac{m_2}{\mathscr E_1}},$$ where $\theta_1$ is angle of scattering of the particle, $\mathscr E_1$ is its initial energy. We can see that in the limit $m_2\to\infty$, we get $\mathscr E_1'=\mathscr E_1$. What happens with the momentum? Simple: for even tiny mirror velocity its momentum is infinite. So, adding twice the momentum of photon doesn't change the momentum of the mirror.

All this means that such mirror indeed doesn't change its velocity after the collision.

$^\dagger$ See Landau, Lifshitz, "The Classical Theory Of Fields", equation (13.9)

Ruslan
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  • The problem here is that the mirrors don't absorb any energy? –  Aug 25 '14 at 20:37
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    The fact that it does not absorb helps. Loosely speaking, the incoming photon "stops" (not really, but stay with me) transferring all its forward momentum to the mirror. But then the photon goes back the way it came. In order for that to happen the mirror has to give up some "backwards" momentum ... which is the same as saying it gains momentum equal to the momentum of the original photon. With a mirror, you get two times the momentum boost compared to a perfect absorber. – garyp Aug 25 '14 at 20:43
  • @garyp Does this mean that the mirror moves or not? –  Aug 25 '14 at 20:46
  • The mirror will move if there's nothing in the way. – garyp Aug 25 '14 at 20:49
  • @garyp even if it is an perfect mirror? Assuming its possible. –  Aug 25 '14 at 20:51
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    This is the mechanism for getting propulsion with a solar sail. – Jon Custer Aug 25 '14 at 22:14
  • @RohanVijjhalwar I've updated my answer to address the case of more ideal mirror. – Ruslan Aug 26 '14 at 10:52
  • The mirror will move even if it is ideal. The more ideal (that is, the limit of no absorption and specular reflection) the more acceleration. But as this answer points out, if the mass of the mirror is infinite, or the mirror is attached to something of infinite mass, it will not accelerate. Infinite mass is not possible, of course, but if the mass is large enough it would be impossible to detect any acceleration, so it would be effectively motionless. – garyp Aug 26 '14 at 12:14
  • @garyp The only way to prevent absorption I see is taking infinte mass. What is your ideal finite-mass mirror? How does it achieve no-absorption? – Ruslan Aug 26 '14 at 12:18
  • Help me understand your argument; I don't see it yet. My mirror is 100% reflecting, and perfectly specular. In terms of photons, that means that every photon is reflected (perhaps with a frequency shift) and the angle of incidence equals the angle of reflection. – garyp Aug 26 '14 at 13:03
  • I stared at your formula from Landau and Lifshitz. It seems to imply no frequency shift for normal incidence. However, that analysis is done in the center of mass frame, not the frame in which the mirror is initially at rest. – garyp Aug 26 '14 at 13:06
  • @garyp if your mirror shifts frequency of the photon, then it absorbs some energy, which is not what OP wanted. As for formula, no, for normal incidence you have $\theta_1=\pi$ (it's scattering angle, not falling), so $1-\cos\theta_1=2$ in the denumerator. So, frequency does shift there for finite $m_2$. And no, the part of chapter 13 until that formula is done in laboratory frame, not center of mass frame. – Ruslan Aug 26 '14 at 13:50
  • Reflection with a frequency shift is not absorption and re-emission. The phase coherence of the incident light, the material polarization, and the reflected light is maintained. My interpretation of the OP is that he considers absorption to mean destruction of a photon with the energy ending up in internal degrees of freedom. I no longer have L&L in front of me, but at the start of the analysis a few pages prior, he specifies the "C-frame", which is the frame with zero total momentum. I'll have to look at that angle ... – garyp Aug 26 '14 at 14:15
  • In any case, if frequency changes, this means $\mathscr E'_1\ne\mathscr E_1$, which by conservation of energy means partial absorption of particle's energy by the mirror. The OP says "absorb no energy of any kind", so it seems clear that frequency must not get shifted, so the only solution for this is $m_2\to\infty$. As for Landau, he explicitly says that the calculations for following formulas (i.e. (13.4)-(13.9)) are in "L-system" — directly under formula (13.3). – Ruslan Aug 26 '14 at 14:20
  • @Ruslan Does this mean the angle of scattering is $\pi$ for a non-infinite mass mirror? And does the formula you submitted in your answer apply to non-infinite mass mirror? –  Aug 27 '14 at 16:09
  • @RohanVijjhalwar angle of scattering depends on incidence angle, as usual. The formula works for any $m_2$. I just take the limit $m_2\to\infty$ to find the result for infinite-mass mirror. – Ruslan Aug 27 '14 at 17:19
  • @Ruslan Thank you, what if the incidence angle is perpendicular to the mirror (normal incidence) then will the scattering be $\pi$ or simply $0$ as there is no change in the scattering and thanks for the great formula. –  Aug 27 '14 at 17:28
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    @RohanVijjhalwar if the incidence angle is $0$, then of course scattering angle is $\pi$ since the photon must reflect, not penetrate the ideal mirror. – Ruslan Aug 27 '14 at 17:49
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'The mirror is given a momentum twice that of the incoming photon. As a mirror is typically quit heavy, lets say one gram. Its kinetic energy due to momentum it received will be extremely small. However, the photon will actually change its energy by the same amount, thus its wavelength changes, but not much.

Per Arve
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