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

[Niles]

What confuses me is that you say

If you have a higher intracavity photon number without increasing the pump rate, this means that the increase happened due to noise or you added some photons to the cavity or something like that. In that case the number of photons that gain can add to the intracavity photon number per turn does not increase, but stays constant as you do not increase the pump rate. As gain is the amplification per photon already present in the cavity and you will not add the same amount as before, but have more photons in the cavity, gain will decrease.

and then state

What I said about photons added to noise was exactly that: the reaction of a steady state system when it is subject to external noise and slightly perturbed from the steady state. However, the process of switching a laser on is not fundamentally different. It is like adding a huge perturbation to the steady state by emptying the laser cavity and waiting for the system to recover to the steady state.

As far as I understand from those two quotes, perturbing the laser by changing the pump during steady state will increase/decrease the gain?

So if I have a laser which is operating at steady state, and I suddenly increase the pump, then then gain will decrease because the intracavity flux increases -- see figure 4.6 (http://books.google.com/books?id=x5...DYQ6AEwAzge#v=onepage&q=gain clamping&f=false). But then gain will then return to its previous value because of what you said here previously

In fact, when you turn on a real laser, you will notice some small overshoot of the intracavity photon number above the steady state value. However, in this region losses are larger than gain (as seen to the right of the steady state in the figure), so the intracavity photon number will reduce again.

The gain returns to the loss value, and I understand that 100% physically. But I can't see that from figure 4.6, because according to that, the gain will decrease with increasing flux, but then the gain has to increase to the value of the loss. But this value is at a lower flux? So gain will always return the its value it has at threshold flux, but the point is that by increasing the pump we open up the possibility to reach the same gain at a higher intracavity photon number. Am I correct here?

 

[Cthugha]

Two things:

So if I have a laser which is operating at steady state, and I suddenly increase the pump, then then gain will decrease because the intracavity flux increases

This is a bit tricky as two things are going on here while the laser has not already arrived at the steady state. A higher intracavity photon number decreases gain, but pumping harder also means more photons per cycle can be added, so this increases the possible gain. These effects balance and the system somewhen settles at the steady state.

So gain will always return the its value it has at threshold flux, but the point is that by increasing the pump we open up the possibility to reach the same gain at a higher intracavity photon number. Am I correct here?

Yes, exactly!

 

[Niles]

Thanks. I will have to study the replies in this thread for now and compare it to other references to see if I have truly understood it. I'll let you guys know later. Thanks for now.

 

[Niles]

Hi

I printed out the entire thread, and have read it since my last post. I believe there are still a few gaps missing. BTW, do you have a reference to all these things? Where do you all this from?

This is a bit tricky as two things are going on here while the laser has not already arrived at the steady state. A higher intracavity photon number decreases gain, but pumping harder also means more photons per cycle can be added, so this increases the possible gain. These effects balance and the system somewhen settles at the steady state.

1) So you say we have the two processes

*) "[...] pumping harder also means more photons per cycle can be added [...]"
**) "A higher intracavity photon number [...]"

Just to be perfectly clear: Doesn't (*) imply (**)? I mean, they are not two distinct processes.2) The above process describes what happens when going from a nonzero rate R1 (at steady state) to a nonzero R2 (at steady state). But if we go from a rate 0 to R1, does the
gain also increase due to * and decrease due to ** until it ultimately reaches value of losses?3) Why do people refer to the gain as being "saturated" when lasing occurs? I mean, it just attains the only value it can after small_signal_gain > losses, so its not like we have saturated it in any way.

 

[Cthugha]

BTW, do you have a reference to all these things? Where do you all this from?

Good question. I read a few books on this topic, but to be honest I am not sure which one covers these questions best. If you have a good library you might just want to have a look at some books and pick one that you like best. Examples include Siegman's book on lasers, Boyd's Non-linear optics. Maybe Meschede's book If you are interested in some special lasers maybe also the semiconductor laser book by Chow and Koch.

1) So you say we have the two processes

*) "[...] pumping harder also means more photons per cycle can be added [...]"
**) "A higher intracavity photon number [...]"

Just to be perfectly clear: Doesn't (*) imply (**)? I mean, they are not two distinct processes.

Yes, (*) will usually induce (**).

2) The above process describes what happens when going from a nonzero rate R1 (at steady state) to a nonzero R2 (at steady state). But if we go from a rate 0 to R1, does the
gain also increase due to * and decrease due to ** until it ultimately reaches value of losses?

Yes, there is basically no difference between starting from any steady state and starting from 0 pumping. However, at 0 and below threshold the stimulated emission photon number under steady state conditions will be very small and in many cases 0.

3) Why do people refer to the gain as being "saturated" when lasing occurs? I mean, it just attains the only value it can after small_signal_gain > losses, so its not like we have saturated it in any way.

Well, it takes on a constant value and stays there if you pump harder. It is not far off to call this saturation. If I remember correctly, the term comes from amplifier theory in general and was also used for lasers, but I could be wrong there.

 

[Niles]

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.

 

[Cthugha]

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.

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.

 

[Niles]

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.

 

[Cthugha]

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.

 

[Niles]

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 / flux_sat = 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)

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

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.

 

[Cthugha]

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.

2) A clarifying question regarding 1 + flux / flux_sat = 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.

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)

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

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.

 

[Niles]

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

 

[Niles]

[...] 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 < pump_threshold, 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]

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 < pump_threshold, 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 a lot. 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.

 

[Niles]

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