Laser Physics: Gain Clamping

Hi

Here they explain the concept of gain clamping in steady state lasers: http://books.google.com/books?id=x5...DYQ6AEwAzge#v=onepage&q=gain clamping&f=false. They say:

"The decrease in gain stops once the gain in the medium γ(v) exactly balances out the cavity losses α resulting from parasitic lasing and mirror losses."

However they never say why the gain stops decreasing at that point, and I haven't been able to find the explanation in any book so far. I know that gain = losses in CW-lasers -- otherwise they wouldn't be CW, but that doesn't explain why we have an equality. Does anybody know the reason?

Best,
Niles.

The saturating gain drops as the light intensity increases, until the point that losses equal any further gains and the light intensity doesn't increase any more.

The saturating gain drops as the light intensity increases, until the point that losses equal any further gains and the light intensity doesn't increase any more.

Thanks. That is also what the book says, but how does one argue that the saturated gain exactly equals losses at that point? I mean, look at figure 4.6 from the above link: The gain keeps dropping as the photon flux is increased -- who says that the gain is clamped at the losses?

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Thanks. That is also what the book says, but how does one argue that the saturated gain exactly equals losses at that point?

You're losing me, how could it not? And what do you mean by exactly - are you concerned about the number of significant figures of the equality? If either gain or loss dominates, the light intensity changes up or down - but it's not doing that if the terms are equal, it's remaining constant. Note that gain generally varies in time and space, so simply setting two numbers equal to each other glosses over a lot of detail.

You're losing me, how could it not? And what do you mean by exactly - are you concerned about the number of significant figures of the equality? If either gain or loss dominates, the light intensity changes up or down - but it's not doing that if the terms are equal, it's remaining constant. Note that gain generally varies in time and space, so simply setting two numbers equal to each other glosses over a lot of detail.

Thanks. 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 where I am wrong? Here goes:

In order to have lasing, we need a gain larger than the losses such that the number of photons added coherently to the existing field is positive. Generally, if gain>losses, then the signal increases (exponentially), and likewise gain<losses will make the signal decrease.

So at the small-signal (low intensity) we start out with a gain larger than the losses, so N2-N1 increases (as in figure 4.7a). But with increasing pump intensity, the gain also starts to decrease. In the extreme limit of an infinite pump photon flux, the gain reaches a constant value determined by what system we are dealing with (2, 3 or 4 level).

Our gain follows the curve shown in figure 4.6 with increasing pump intensity, so with my reasoning so far, nothing prevents the gain from becoming lower than losses.

Where is my reasoning wrong?

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Your reasoning ins't exactly wrong. Think of magnetic flux. There's a point at which a turning magnetic field will offset the electric field component of force applied to a particle. Since with lasers we are dealing with an EM field we can think of the photons excited to fluorescence as part of a population N-N sub i where N sub i is the number of photons which collide to non-fluorescing energy states. Since to give off energy the state of a photon must change either the state of another needs to change or a new one has to be created. There is a chance for the new one to also fluoresce. Since this population is included in N, the number of fluorescing photons, we do not add a new term for it. I'll further the example now.

The ionization energy of a molecule of Argon is approximately 15.12 eV. I'm estimating that from a chart from my college quantum mechanics book: though, most should have a similar table. We excite an electron to this level and it fluoresces lets say. If we continue pumping energy to fluoresce to this level then the average energy of N will start to become 15.12 eV making the system (italics) potentially (italics) favor another energy level. We assume for theory, since it work, that the energy levels will fall back to the macroscopic metastable state where fluorescence isn't occurring at categorical laser levels. This is part of the reason why we use short impulses, or discharges, of light to excite photoelectons to release photons of light at the desired range of fluorescing states which is variable for different elements. We have now fufilled that the Population P of lasing photoelectrons is comprised of N-N sub i. Because the light is pulsed there are already iterations where the energy level of the material being flashed against returns to its rest energy metastable state. When we pulse again we expect similar results and also due to fourier optics manifest the continuance of further parts of the wave group since there are variable wavelengths and frequencies comprised of both packets and beats. Packets being the spacing of a sinusoidal or cosinic (I made that word lol) component and beats being the more locally haphazard arrangement of wave crests and troughs within that. So in effect it is because we started lasing that the laser continues and also because we partially reset the laser, disturbing the metastable state, that we continue to observe the fluorescence of photoelectrons.

In the extreme limit of an infinite pump photon flux, the gain reaches a constant value determined by what system we are dealing with (2, 3 or 4 level).

Our gain follows the curve shown in figure 4.6 with increasing pump intensity, so with my reasoning so far, nothing prevents the gain from becoming lower than losses.

Here: In the limit of a photon flux that is not infinite, the gain reaches a constant value. This flux is the point where the gain is suppressed to the point where it equals losses. Note we're not talking about pump photons in an optically-pumped laser, if that's what you mean to say - the photon flux is laser photons.

Here: In the limit of a photon flux that is not infinite, the gain reaches a constant value. This flux is the point where the gain is suppressed to the point where it equals losses. Note we're not talking about pump photons in an optically-pumped laser, if that's what you mean to say - the photon flux is laser photons.

Yes, my error -- I was referring to cavity photons. But I still don't get the argument when I think about it physically: Many cavity photons should saturate the gain (which it does), and that it reaches a constant value I can believe. But that this constant value is the loss, I still don't see why. What obvious detail am I lacking here?

Why is it that the gain approaches the value of losses when the cavity photon density gets larger? How do you see that from figure 4.6? http://books.google.com/books?id=x5...DYQ6AEwAzge#v=onepage&q=gain clamping&f=false (just press the link and it will take you to the figure).

What do you mean by approaches the losses?

That it approaches the value of the losses.

Actually I just read in 3 different books on lasers on Google Books that when the pump is equal to or above threshold, the gain is clamped at the threshold gain, and the mechanism for keeping the gain at its threshold value is that the gain is saturated. But none of them answer why the saturated gain is equal to the threshold gain. So my question is still open to anyone wanting to chip in.

I guess I'm not sure what else to say, it seems clear. Maybe if you stop thinking about it in terms of gain, and think about it in terms of whether or not the laser intensity continues to increase forever (obviously it can't) or if it stops increasing at some point. That point is when the gain (which is always positive and trying to increase the laser intensity) is equal to the loss (which is always negative and trying to drop the laser intensity). The net gain (amplification minus loss) is therefore zero, so the laser intensity doesn't change any more.

I guess I'm not sure what else to say, it seems clear. Maybe if you stop thinking about it in terms of gain, and think about it in terms of whether or not the laser intensity continues to increase forever (obviously it can't) or if it stops increasing at some point. That point is when the gain (which is always positive and trying to increase the laser intensity) is equal to the loss (which is always negative and trying to drop the laser intensity). The net gain (amplification minus loss) is therefore zero, so the laser intensity doesn't change any more.

I agree 100% on your explanation -- from that point of view it is obvious. This is also the viewpoint explained in e.g. Eberly and Milonni. But as far as I can tell, both you and I can't really see why the decrease in gain stops when the gain in the medium exactly balances out the cavity losses, as said in http://books.google.com/books?id=x5...DYQ6AEwAzge#v=onepage&q=gain clamping&f=false

But OK, its not super important, I just thought it would be nice to know why. But thanks for helping -- both of you.

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Cthugha
But as far as I can tell, both you and I can't really see why the decrease in gain stops when the gain in the medium exactly balances out the cavity losses, as said in http://books.google.com/books?id=x5...DYQ6AEwAzge#v=onepage&q=gain clamping&f=false

Somehow I do not get your point. What makes you think the decrease in gain stops at some point in curve 4.6 shown in the book? 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. In fact, it will go slightly below the steady state intracavity photon number again. However, in this region gain is larger than the loss rate is, so the intracavity photon number will increase again and will again show some overshoot (although smaller than the first time). These are typical relaxation oscillations. This occurs also when you perturb a laser which is operating under steady state conditions.

So in some sense the gain must equal the losses because at the point where both are equal, there is an efficient feedback mechanism that drives the intracavity photon number back to the steady state one if a laser is brought to a regime where losses and gain are not equal.

The steady state intracavity photon number of course depends on the pump rate. However, the loss rate is a quantity typical for the cavity used and does not depend on the pumping rate. Accordingly, as the steady state condition is that the loss rate is equal to the gain rate, the gain must stay constant if the pump rate is increased as the loss rate also does not change once the lasing regime is reached.

Sorry if this reply is too obvious, but, why would the gain stop when it becomes equal to the total loss of the cavity? The gain should exceed the loss or there would be no laser light output. What stops then gain must be a finite amount of electrons and states for stimulated emission events. An analogy, in a transistor oscillator circuit, the gain exceeds the loss and the circuit begins to oscillate with growing amplitude. The nonlinear, large signal behavior and limited power supply then clamps the level of the output oscillations.

But as far as I can tell, both you and I can't really see why the decrease in gain stops when the gain in the medium exactly balances out the cavity losses, as said in....

Sorry, I'm not understanding your point - isn't it obvious from all of the above? What fails to be obvious? Is there some deeper, more profound meaning you looking for in the word why, beyond mathematics and physical insight?

Somehow I do not get your point. What makes you think the decrease in gain stops at some point in curve 4.6 shown in the book?

I don't think that, but that is what it says in the book. On the page before figure 4.6: "The decrease in gain stops once the gain in the medium γ(v) exactly balances out the cavity losses α resulting from parasitic lasing and mirror losses." My problem is that don't understand why they write that. But if - as you say - the gain keeps decreasing wth increasing cavity photon flux, then we aren't dealing with a steady state laser since at some point loss>gain. So no matter what side I choose to believe, I don't understand it.

Sorry if this reply is too obvious, but, why would the gain stop when it becomes equal to the total loss of the cavity? The gain should exceed the loss or there would be no laser light output. What stops then gain must be a finite amount of electrons and states for stimulated emission events.

Thanks, and no, don't apologize. But if we are dealing with a steady state laser, then the gain cannot exceede the losses of the cavity (or decrease it).

Sorry, I'm not understanding your point - isn't it obvious from all of the above? What fails to be obvious? Is there some deeper, more profound meaning you looking for in the word why, beyond mathematics and physical insight?

Cthugha said that the gain doesn't stop decreasing when the cavity photon flux increases.

I think the easiest way is for me to write the following: Say I am looking at a laser setup in the lab. Nothing fancy, maybe a simple fiber laser. I increase the pump power by increasing the current on the power supply starting from 0 A. This increases the inversion in my gain medium according to figure 4.7a. At the same time the gain saturates according to figure 4.6. At the same time I have a power meter measuring the output of my setup, and I plot the output power as a function of the power supply current.

To begin with, all the data points will lie on an approximately horizontal line (we are below threshold!). But at some point, the pump power is so large that I will begin to see the output power growing as a function of current, i.e. as a function of pump power (we are above threshold!).

When this happens, I know that I have hit the threshold value shown in figure 4.7a and b. So far so good. During all of this the gain keeps decreasing with increasing current according to figure 4.6 (it saturates). But what I see to my big surprise is that for some arbitrary current (larger than the threshold value), the power meter shows a constant output power, i.e. it is a CW laser!

So this raises the following question: Where along the line did the gain stop decreasing and instead attained a constant value equal to the losses of my cavity? Because this has to be the case, since we have a steady output. In other words, where during my above lab process did the gain get clamped to the threshold value shown by the horizontal dotted line in figure 4.6 instead of saturating further? As far as I can tell, this question has not been answered so far.

I appreciate all your help so far.

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Cthugha
I don't think that, but that is what it says in the book. On the page before figure 4.6: "The decrease in gain stops once the gain in the medium γ(v) exactly balances out the cavity losses α resulting from parasitic lasing and mirror losses." My problem is that don't understand why they write that. But if - as you say - the gain keeps decreasing wth increasing cavity photon flux, then we aren't dealing with a steady state laser since at some point loss>gain. So no matter what side I choose to believe, I don't understand it.

Ok, to be precise one should distinguish between the case where the photon flux is increased as a consequence of the steady state pumping being increased and the case where the photon flux is increased as a consequence of a perturbation without changing the pump rate. In the latter case the gain will decrease and this is the case I referred to before.

However, I do not understand your second point. Why should a decrease in gain keep the laser from being steady state? The photon flux will adjust accordingly to steady state situations.

Did you check the description of gain saturation in the book of Saleh/Teich? Maybe that helps you as there the gain is discussed and calculated explicitly for small photon fluxes and in the saturation regime separately.

...the gain doesn't stop decreasing when the cavity photon flux increases.

Look at the equation for gain as a function of flux, above 4.26a - in the denominator you see 1 + flux/saturation_flux. Increasing cavity flux causes the gain to decrease once the flux reaches an appreciable fraction of the saturation flux. Flux keeps increasing slowly, gain keeps dropping slowly, until something stops the process, which is when the gain is so low that it only just compensates losses.

Ok, to be precise one should distinguish between the case where the photon flux is increased as a consequence of the steady state pumping being increased and the case where the photon flux is increased as a consequence of a perturbation without changing the pump rate. In the latter case the gain will decrease and this is the case I referred to before.

I see. Good, we agree there then.

Did you check the description of gain saturation in the book of Saleh/Teich? Maybe that helps you as there the gain is discussed and calculated explicitly for small photon fluxes and in the saturation regime separately.

I just borrowed the book from the library today. Their description raises the same question as in my post #17, because they have the same figures at the Optoelectronics-book, i.e. figure 4.7a and b. They say gain decreases with increasing pump intensity, and that at some point it equals losses, and stays at that value irregardless of the pump intensity. At this point and forward any increase in pump intensity goes into the amplitude of the signal. This confirms all the things we know.

Look at the equation for gain as a function of flux, above 4.26a - in the denominator you see 1 + flux/saturation_flux. Increasing cavity flux causes the gain to decrease once the flux reaches an appreciable fraction of the saturation flux. Flux keeps increasing slowly, gain keeps dropping slowly, until something stops the process, which is when the gain is so low that it only just compensates losses.

In the equation 1 + flux/saturation_flux = small_signal_gain/losses then if flux increases, so does small_signal_gain have to.

I don't mean to be rude to any of you, quite the contrary I think its really nice of you to display such patience towards (basically) a stranger on the WWW. But I can't formulate my question any more explicitly (and physically) than I have done in the last part of post #17. It relies on all the statements we have from the Optoelectronics-book, but none of the replies in this thread have answered it.

I think your point of confusion is that you are thinking of the actual total energy loss as some constant number. The actual total loss depends on the number of photons present. If there are more photons present, there are more that can be lost. The gain gets clamped because this is the balance point. If you were to somehow raise the gain beyond the total losses, you would be increasing the number of photons in the cavity. But with more photons present, there will be more photons lost per given time period, so the total loss will momentarily increase. A higher loss means less photons, which means less stimulated emission, which means less gain, so that the gain would return to original clamped value. So the clamped value is a point of stable equilibrium. Any attempt to move the system away from point will simply see the system return itself to this point because of this feedback process.

I think your point of confusion is that you are thinking of the actual total energy loss as some constant number. The actual total loss depends on the number of photons present. If there are more photons present, there are more that can be lost. The gain gets clamped because this is the balance point. If you were to somehow raise the gain beyond the total losses, you would be increasing the number of photons in the cavity. But with more photons present, there will be more photons lost per given time period, so the total loss will momentarily increase. A higher loss means less photons, which means less stimulated emission, which means less gain, so that the gain would return to original clamped value. So the clamped value is a point of stable equilibrium. Any attempt to move the system away from point will simply see the system return itself to this point because of this feedback process.

Thanks, that is not a bad explanation, but you are referring to the specific case of intensity-dependent cavity losses. Gain clamping happens both when losses depend on intensity and when they don't, and your explanation is only valid for the former case.

(By the way, I checked out the notes for your E&M courses, and I bookmarked them. I'll check them out later, they look good!)

Cthugha
In the equation 1 + flux/saturation_flux = small_signal_gain/losses then if flux increases, so does small_signal_gain have to.

Ehm, at first it is supposed to be 1 + flux/saturation_flux = small_signal_gain/gain. That of course will turn into your version for gain=loss. More important: small_signal_gain is the "cold cavity" gain value. That is the gain if there are no amplified photons in the cavity. Accordingly it does not depend on photon flux and will not increase with it.

Cthugha
Thanks, that is not a bad explanation, but you are referring to the specific case of intensity-dependent cavity losses. Gain clamping happens both when losses depend on intensity and when they don't, and your explanation is only valid for the former case.

(By the way, I checked out the notes for your E&M courses, and I bookmarked them. I'll check them out later, they look good!)

Another answer as I cannot edit my last post anymore: This is not a specific case. In usual systems loss rates never depend on intensity. Losses always do. In laser cavities the loss rate is given by the cavity quality factor and just gives the fraction of photons per unit time or per round trip that will leave the cavity on average. This value is determined by the cavity and there is no way to change that. Accordingly the losses are always this fraction of the intensity inside the cavity and must necessarily always depend on intensity. Could you give me an example of a simple lasing system with losses that do not depend on the intensity inside the cavity?

There are only few exceptions and they are not common. For example you could pump a cavity so hard that the refractive index of the materials the cavity is made of changes, thereby changing the cavity quality factor and the loss rate. Then it might show a loss rate depending on intensity. However, that is rare and not standard textbook stuff.

Ehm, at first it is supposed to be 1 + flux/saturation_flux = small_signal_gain/gain. That of course will turn into your version for gain=loss. More important: small_signal_gain is the "cold cavity" gain value. That is the gain if there are no amplified photons in the cavity. Accordingly it does not depend on photon flux and will not increase with it.

I see, my bad. I was a little too quick there. OK, I think I'm beginning to appreciate gain clamping now, but I still have some uncertainties left when I go through my daily lab routine:

1) When I turn the knob on my power supply and increase the pump rate, I increase the number of photons in the cavity, and hence the loss. But the gain is already saturated at this point, so how is clamping satisfied?

2) So gain = loss is only valid for cavities where loss depends on intensity, right?

3) Say a mirror in my cavity has a tunable transmission, such that we can get an output. Now I increase the transmission of the mirror, because I want to measure something. This will increase the loss of my cavity or - equivalently - make the gain smaller. But using the same line of throught as chrisbaird we can argue that the gain will just increase to the value of the new losses, right?