In stimulated emission process, we always assume that the secondary excited photon, as a result of e-h recombination, has the same frequency, polarization, phase and direction with the incident photon. The first three terms can be understood from phase matching criteria (coherency).... but for the last term, how can we explain and be sure about the direction of secondary photon? Are there any mathematical or physical description?
what should left/right up/down mean in cosmos? on earth an up for me must be down for someone down-under.
Aidin> The first three terms can be understood from phase matching criteria (coherency).... but for the last term, how can we explain and be sure about the direction of secondary photon? Are there any mathematical or physical description?
In simple physical language, stimulated emission occurs because photons are social objects who like to cluster in the same quantum mode. This means that the emission amplitude is proportional to
sqrt(nk,r + 1),
where nk,r is the number of photons already present in the quantum mode with wave-vector k and polarisation r. It is this "stimulation factor" which is responsible for stimulated emission. I think your attempt to understand the mechanism in classical terms is missing this essential point.
When an atom is stimulated by an external electromagnetic field, with a frequency equal to the frequency between two stationary electronic atomic states , there is created an "electric dipole" with a proper frequency. This forces to produce a transition from the higher to a lower energy state creating a new additional photon with the same phase and direction as the incident photon. The reason is simple, the conservation of the linear momentum of the external electromagnetic field obligues to have the same linear momentum for the one created. Notice that this absolutely different to the spontaneous stimulation where the electromagnetic external field is zero and therefore the momenta are randomly behave.
Dear Kai,
Ok, the number of photons is not relevant, but I can try to explain the conservation of moment needed (and of course also energy although is not important for the question related with the direction of the photons) in a simple form.
Let us have one atom at rest in a laboratory L (thus at rest ) hit by an incident photon p. The atom must recoils in a direction exactly opposite to the incident photon.
Now let us consider the time inversion symmetry T for the physical process. Then under time reversal this process would appear to be a case of absorption in which two photons impinge upon a moving atom. One of the photons is absorbed and cancels the momentum. The other continues on its way. Thus you can see how the conservation of momentum is needed in a simple form is you don't want to waste energy and momentum in this physical process.
No more Tyndall effect?? Or, in other words, no imaginary component of the refraction index?
Why not in lasers? We can see the laser ray standing nearby.
An excited atom (ion) will emit, some time later, a light quantum in arbitrary direction, thus even the stimulated emission is never 100% efficient, isn't it? This will occur even in vacuum, where there are no scatterers present, like those floating in dirty air. My understanding of scattering is that this act takes no time, it is just a change of k vector after collision of photon with some obstacle.
``Why we assume that the direction of excited photon in stimulated emission is in the same direction as the parent?''
The short answer is: because we believe in the validity of the laws of energy and momentum conservation, and because we apply these laws in a pragmatic way.
In an interaction of a photon and an electron the assumption mentioned above cannot be exactly true because energy and momemtum of both electron and photon may change, while their sums remain unchanged. If energy and momentum changes of the electron are neglected, then the assumption is approximately correct (existence of the Lamb shift is a sign that it is an approximation).
In nonlinear optics it is possible that by the interaction a single photon may be ``split'' into two photons, which then may have directions differing considerably from that of the incoming photon.
Gert> But why would there just be a photon of type k,s be emitted, couldn't it be in any direction in that case?
It must depend on how the factor (nk,s + 1) (for the transition rate) varies with k and s. I find it disappointingly and surprisingly difficult to find typical values for this, but one Google source I came across mentioned nks order 108-1010 at saturation in a laser, when k is the wavevector of the "stimulating field. When this is the case the extra 1 becomes rather negligible.
I thought Dick Feynman was right when he claimed that light is taking ALL ways, so that one of the ways is (must be) the shortest way in time.
Daniel> When an atom is stimulated by an external electromagnetic field, with a frequency equal to the frequency between two stationary electronic atomic states , there is created an "electric dipole" with a proper frequency.
The derivation of the result, to lowest perturbative order, includes absolutely no such classical stimulation effects. Classically the atom is completely unaware of the presence of any stimulating beam. The preference to decay to a mode where there already are many photons present is a quantum effect caused by Bose statistics.
Daniel> the number of photons is not relevant
Indeed, the effect of conservation laws is independent of whether we have spontaneous or stimulated emission. They must be taken into account, but this has nothing to do with the stimulation effect.
Willem> The short answer is: because we believe in the validity of the laws of energy and momentum conservation,
Again, this has really nothing to do with stimulated emission. Just look at the Dirac emission formula provided by Gert. The only place where the presence of a stimulating field enters is through the factor sqrt(nk,r + 1) in the emission amplitude (as I mentioned far above), implying a factor (nk,r + 1) in the emission rate. How important this effect is will of cause depend on how this factor varies with k and r.
Dear Kare,
When I wrote electromagnetic field I didn't tried to say a "classical electromagnetic field " but a quantized electromagnetic field made by photons. It seems to me be out of the "reality" to speak of a photon scattering with an atom as if we were able to through it between states of an atom. I know that in photonics or in quantum optics is employed such kind of languages becase the there is a selection of photonic spectra (k and amplitude), but for me it is easy to figure it as a field containing such photons. We never take one only atom where we apply an independent photon. That is not real at all!
Dear Kare,
In any case, let me to add a link for explaining clearly the relationship of the electromagnetic fied as I said and the radiation stimulation
http://www.tcm.phy.cam.ac.uk/~bds10/aqp/lec19_compressed.pdf
Daniel> a link for explaining clearly
Yes, slide 13 illustrates the points I stressed very well. As the figure on that slide indicates, the external photons are just "passing by" without touching the atom. This is reflected in the formula, which further does not take momentum conservation into account. Momentum conservation is implicitly taken into account by the atom, or the material matrix in which it sits, with the resulting recoil energy assumed to be negligible.
Daniel> We never take one only atom where we apply an independent photon. That is not real at all!
But that is exactly what is assumed when deriving the (first order) formula. That is a sufficiently good description to describe reality rather well. Thinking of all kinds of complications, without including them in the mathematical model, only adds fog and confusions to the issue.
KO> one Google source I came across mentioned nks of order 108-1010 at saturation in a laser
Each photon in a red (700 nm) laser carry 3 x 10-17 Ws energy. For a 1W continuous laser of length 0.5 meter, with a perfect mirror in one end, and a 99% reflective mirror in the other, there will be about 1010 photons in the laser cavity. However, these may be distributed over many different modes.
Further, as seen from the slides linked by Daniel, real laser theory is more complicated than the above simple, semiclassical estimates. Which nevertheless should indicate the orders of magnitude involved.
Dear Kare,
I'm not sure to understand you. The spontaneous emission is clearly understood and explained in the link that I have sent you. In the Fock space the creation operator is associated to a transition rate which increases due to the external photon. In fact this can be considered as a dipolar approximation. If you don't have the external photon what you obtain is the spontaneous emission. All that was obtained by Einstein many years ago before where the atom was under an electromagnetic field radiation using his famous coefficients. Frankly I don't understand what was the criticism that you have made to my previous posts, although I understand that saying only electromagnetic field can be nowadays not too modern for people devoted to quantum optics.
In any case, this spontaneous emission is not the actual question made in this thread, that what is the reason of having the excited photon in the same direction as its parent. And the explanation is only due to the conservation of energy and momentum for the photons. This is the only reason!!! I have tried to showed it just using the inversion time operator T in the physical process and perhaps it is too simple. But this is the only one that I know. Please, can you explain it in another form?
This is the question and nothing more than this question, the rest is noice.!!!!
Daniel> the creation operator is associated to a transition rate which increases due to the external photon. In fact this can be considered as a dipolar approximation.
I don't understand you either: The dipole approximation has nothing to do with the presence of external photons or stimulated emission. It is a common approximation, valid when the wavelength of the emitted photon is much larger than the size of the emitting atom. This is a common feature of spontaneous and stimulated emission. So it explains nothing about the latter.
Daniel> the explanation is only due to the conservation of energy and momentum for the photons.
But enhanced emission into already exited modes has nothing to with energy-momentum conservation, which is a common feature of spontaneous and stimulated emission. So it explains nothing about the latter.
Daniel> I have tried to showed it just using the inversion time operator T in the physical process and perhaps it is too simple. But this is the only one that I know.
But this again is a common feature of spontaneous and stimulating emission. So it explains nothing about the latter.
Please, can you explain it in another form?
I have previously (twice) pointed to the exact origin of the stimulation effect, the factor (nk,r+1), which is correlated to the bosonic property of photons. In layman terms: Photons are social entities; they prefer to jump into the pool where there is a lot of them from before. The factors (nk,r+1) give the relative weight of this preference, ignoring everything else (effects that may depend on k, r but not nk,r).
Dear Kare,
You are right, it seems that we don't understand each other. I'll try to answer you step by step
1. K> The dipole approximation has nothing to do with the presence of external photons or stimulated emission.
D> The dipole approximation simplifies the transition obtained by the Fermi's Gold Rule between the two states associated to the emitted photon in the stimated emission. Thus I don't agree that is not related with the stimulated emission (although we are again absolutely out of the question of this thread).
2. K> But enhanced emission into already exited modes has nothing to with energy-momentum conservation,
D> Obviously. This is only related with the question of this thread about the direction of the emitted photon.
3. K> But this again is a common feature of spontaneous and stimulating emission. So it explains nothing about the latter.
D> Obviously.
I have finished my last post with what was relevant for the question and the rest was nothing more than explanations for telling you that, although I'm very grateful for your explanations of the electromagnetic field was not related with the spontaneous emission and so on (perhaps a question of language being a good boy with you). Therefore it is logic that you find my answers nothing more than the usual formalism.
Let me repeat you the end of my last post.
In any case, this spontaneous emission is not the actual question made in this thread, that what is the reason of having the excited photon in the same direction as its parent. And the explanation is only due to the conservation of energy and momentum for the photons. This is the only reason!!! I have tried to showed it just using the inversion time operator T in the physical process and perhaps it is too simple. But this is the only one that I know. Please, can you explain it in another form?
This is the question and nothing more than this question, the rest is noice.!!!!
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