# What is the reason for the equality of gain and losses in CW lasers?

• Niles
In summary: Where is my reasoning wrong?Your reasoning might be wrong if you think that gain and loss are the only things that could influence the number of photons added coherently to the field, but that's not what I am saying. What I am saying is that if we continue pumping energy to fluoresce to this level then the number of photons added coherently will start to become 15.12 eV making the system (italics) potentially (italics) favor another energy level. Uhm, I must have misunderstood a concept along the line, because according to you the gain cannot be equal to anything but the losses, which I don't see. OK, I'll write how I have understood it, then perhaps you can point out
Thanks. I thought I understood it, until I read through the derivation of the gain in a 3-level system and saw the attached graph -- note that in the derivation they do not take into account a lossy cavity. But still I believe that the graph contradicts the main point we have been talking about: That the gain is clamped. Don't you agree with me on this one?

The gain increases with pump, until lasing starts. All good so far. But after lasing starts (gain > 0), then gain = loss, but I don't see that *anywhere* on the graph. According to them, gain increases linearly for low pump power, and saturates at some point. So they talk about it as if they have command over the specific value of the gain, we didn't in our discussion -- there it was either nothing or equal to losses.

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Niles said:
Thanks. I thought I understood it, until I read through the derivation of the gain in a 3-level system and saw the attached graph -- note that in the derivation they do not take into account a lossy cavity. But still I believe that the graph contradicts the main point we have been talking about: That the gain is clamped. Don't you agree with me on this one?

Well, I cannot say much without knowing under what assumptions the result was derived. Which book is the graph from? Just two comments: First, a lossless cavity is unphysical and therefore more of academic or pedagogical interest. Second, a lossless cavity usually does not attain a steady state as there is no real way to decrease the intracavity photon number (and thus also no gain clamping). At some point reabsorption becomes prominent and the growth of the intracavity photon number will decrease. You may get a steady state if the probability flow is interrupted for some interaction parameter (see e.g. chapter 23 (quantum theory of the laser) written by Bergou, Englert, LaxScully, Walther and Zubairy in handbook of optics. However this is a very non-standard issue usually not treated in introductory textbooks and also not treated in most detailed textbooks.

Niles said:
The gain increases with pump, until lasing starts. All good so far. But after lasing starts (gain > 0), then gain = loss, but I don't see that *anywhere* on the graph. According to them, gain increases linearly for low pump power, and saturates at some point. So they talk about it as if they have command over the specific value of the gain, we didn't in our discussion -- there it was either nothing or equal to losses.

Difficult to say. What is the definition of gain used here? Usually it is given in units of 1/m and denotes the amplification of a light field per distance traveled through the amplifying medium. Some textbooks simply use the intracavity power divided by the pump power and call this gain. This is all difficult to say without seeing the derivation and which factors are considered and which are not.

Cthugha said:
Well, I cannot say much without knowing under what assumptions the result was derived. Which book is the graph from? Just two comments: First, a lossless cavity is unphysical and therefore more of academic or pedagogical interest. Second, a lossless cavity usually does not attain a steady state as there is no real way to decrease the intracavity photon number (and thus also no gain clamping). At some point reabsorption becomes prominent and the growth of the intracavity photon number will decrease. You may get a steady state if the probability flow is interrupted for some interaction parameter (see e.g. chapter 23 (quantum theory of the laser) written by Bergou, Englert, LaxScully, Walther and Zubairy in handbook of optics. However this is a very non-standard issue usually not treated in introductory textbooks and also not treated in most detailed textbooks.

Difficult to say. What is the definition of gain used here? Usually it is given in units of 1/m and denotes the amplification of a light field per distance traveled through the amplifying medium. Some textbooks simply use the intracavity power divided by the pump power and call this gain. This is all difficult to say without seeing the derivation and which factors are considered and which are not.

It is from this book, chapter 5: http://books.google.com/books?id=uA...:"Emmanuel Desurvire"&source=gbs_similarbooks

But this is a generel criticism I have of this field: The textbooks contain so many different examples - both physical and unphysical - that it gets very confusing.

Ah, ok. That book is about Erbium doped fiber amplifiers. These are rather special as they are single pass amplifiers. You just put some signal and some pump laser into the fiber. The pump laser excites the dopant ions to a higher level from which stimulated emission to the signal wavelength can occur, so you get some amplified signal at the signal wavelength (usually 1550 nm). The amplification here is just single pass. You just send your signal through the fiber once and it gets amplified due to the presence of the pump beam pumping the fiber. As there are no mirrors, there are also no mirror losses (or better: you have 100% loss at the end of the fiber).

Erbium doped fiber amplifiers are basically lasers without mirrors and you have no feedback mechanism and it does not make sense to define a steady state in the same way you would do for lasers. Just to make the difference more clear: in lasers you just have a pump beam and the signal builds up due to spontaneous and stimulated emission caused by the pump beam and gets amplified. In EDFAs pump and signal are different beams and the signal just gets amplified.

Thanks for clarifying that. I have a few questions left, and I believe I know the answers. If you would like to, you are very welcome to confirm them (I lack confidence to believe I am correct).

1) In the fiber-book, they mention right below the figure of the gain that: "The experimentally determined saturation output power is defined as the signal output power at which the gain has been reduced (compressed) by 3dB". Just to be clear: When they say compressed, are they referring to the stagnation of the gain in the figure from my previous post? Because from that figure, we never see the gain decrease its size, and originally I thought this was the decrease they referred to.2) A clarifying question regarding 1 + flux / fluxsat = g0/g: When we have steady state lasing, and increase the pump and wait for steady state to occur, then what has changed is the intracavity flux on the LHS. On the RHS g is still given by the same ratio, but g0 has increased, right? 3) In Milonni/Eberly, they derive the following equation for the gain of a three level laser (homogeneously broadened), which they ultimately write as (chapter 10.11)

$$g(\nu) = g(\nu_{21}) \frac{1}{(\nu_{21}-\nu)^2 / (\delta \nu_{21})^2 + 1 + \phi_\nu / \phi^{sat}_{21}}$$

Then they go on to show a figure of g/g0 as a function of (ν21-ν)/δν21 for different fluxes, and it gives a Lorentzian with decreasing ampltitude with increasing flux (they illustrate saturation). Just to be clear: That graph is nothing more than pedagogical interest, since in a physical laser we have a constant gain because of gain-clamping, right?

I really appreciate your help. I've said it a few times already, but I am sitting with 3-4 laser books around me, and none of them really give answers to the questions I have. Niles.

Niles said:
1) In the fiber-book, they mention right below the figure of the gain that: "The experimentally determined saturation output power is defined as the signal output power at which the gain has been reduced (compressed) by 3dB". Just to be clear: When they say compressed, are they referring to the stagnation of the gain in the figure from my previous post? Because from that figure, we never see the gain decrease its size, and originally I thought this was the decrease they referred to.

Yes, gain compression usually refers to the differential slope ofd the gain curve.

Niles said:
2) A clarifying question regarding 1 + flux / fluxsat = g0/g: When we have steady state lasing, and increase the pump and wait for steady state to occur, then what has changed is the intracavity flux on the LHS. On the RHS g is still given by the same ratio, but g0 has increased, right?

If the quantities are defined as I think, that should be the case.

Niles said:
3) In Milonni/Eberly, they derive the following equation for the gain of a three level laser (homogeneously broadened), which they ultimately write as (chapter 10.11)

$$g(\nu) = g(\nu_{21}) \frac{1}{(\nu_{21}-\nu)^2 / (\delta \nu_{21})^2 + 1 + \phi_\nu / \phi^{sat}_{21}}$$

Then they go on to show a figure of g/g0 as a function of (ν21-ν)/δν21 for different fluxes, and it gives a Lorentzian with decreasing ampltitude with increasing flux (they illustrate saturation). Just to be clear: That graph is nothing more than pedagogical interest, since in a physical laser we have a constant gain because of gain-clamping, right?

Well, this is not necessarily only of pedagogical interest. The main point is to show that the saturation behavior will differ at different energies for broadened (and therefore realistic) transitions. Second, although the gain will be clamped in the lasing regime, g/g0 may still vary as g0 may differ.

Thanks, it was tough, but I learned a lot from this!

Cthugha said:
[...] The main point is to show that the saturation behavior will differ at different energies for broadened (and therefore realistic) transitions. [..]

I have been thinking about that for some time now, and I am actually not quite sure what different behaviors you refer to. At pump < pumpthreshold, g = g0 and it increases with pump. When g0>losses, lasing starts and g = losses. Where does the frequency come into play with regards to g?

Niles said:
I have been thinking about that for some time now, and I am actually not quite sure what different behaviors you refer to. At pump < pumpthreshold, g = g0 and it increases with pump. When g0>losses, lasing starts and g = losses. Where does the frequency come into play with regards to g?

OK, now I think I know what you were referring to earlier. Thanks. Regarding the gain saturation figure I posted earlier just above, then I believe there is a contradictory behavior. We have been talking about 1 + flux/flux_sat = g0/g quite alot. So when e.g. lasing begins in a cavity, the gain decreases its value (saturates) because of increased intracavity flux. On the other hand, you confirmed that when talking of e.g. a fiber, gain saturation is when the differential slope doesn't increase as much with pump (as we see in the figure).

So in the latter case, gain increases (but not as much), and in the first case the gain decreases its value. What is the difference here, why can't 1 + flux/flux_sat = g0/g be used in the case of single pass?Niles.

OK, I just realized what the answer is. Formulating the question is actually quite useful at times! Thanks.

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