Absorption and Gain for Lasers

[PH^S!C5]

Homework Statement

If a round trip gain in a 56.8 cm long laser is 5.16%, what is the net gain coefficient (g-$\alpha$).

g is the small signal gain coefficient
$\alpha$ is the absorption coefficient
L-length of cavity

Homework Equations

The Round Trip Power Gain: Gr= R1*R2* exp[(g-$\alpha$)*2L]
where I have to solve for (g-$\alpha$)

The Attempt at a Solution

I have set R1=R2=1 assuming high reflectivity but I am not sure if this is alright.
First, solve as follows: ln(Gr)/2L = (g-$\alpha$)
Then, substitute the values for L and Gr:
ln(0.0516)/(2*56.8 cm) = -0.0260936057 = (g-$\alpha$)

However, I am not sure what this represents in terms of laser net gain coefficient and if I have correctly solved this (assuming R1=R2=1 and where Gr=0.0516 instead of 5.16% form.
I would appreciate some revision and explanation. I have read the textbooks from Milonni (Laser Physics) and Silfvast (Laser Fundamentals) but they don't really explain the meaning of this in detail. I have searched on google and this ebook helped a bit but I really can't say I understand completely... Any help is appreciated! THANKS!

Google ebook link: http://books.google.ca/books?id=DlW...EUQ6AEwAw#v=onepage&q=round trip gain&f=false

 

[TSny]

Would the round trip power gain be Gr = 1 + .0516 = 1.0516? (Instead of .0516)

 

[PH^S!C5]

Truthfully, I am not certain. I have read over the textbook again and the only possible explanation for adding a one is to have a net gain per round trip increase through the amplifier?
I have tried your suggestion and I obtained a value of 4.43x10^-4 cm^-1... at least it's not negative. Do you know if this is correct? Why? Thanks again!

 

[TSny]

I'm not real certain. But "gain" is often defined as the ratio of the output power to the input power. See http://en.wikipedia.org/wiki/Amplifier.

So, in your case, I would think the round trip gain would be the power after one round trip divided by the power at the start of the round trip. So, if $P_o$ is the power at the start of the round trip and if the power increases by 5 % in one round trip, then the power after one round trip would be $P_o + .05P_o = (1+.05)P_o = 1.05 P_o$. So the gain would be $\frac{1.05P_o}{P_o} = 1.05$

 

[paranormal]

Your solution seems to be correct. The net gain coefficient (g-\alpha) represents the overall gain or amplification achieved in the laser cavity. It takes into account both the small signal gain coefficient (g) and the absorption coefficient (\alpha). In other words, it represents the net effect of the gain and loss mechanisms in the laser cavity.

By setting R1=R2=1, you are assuming that the laser cavity has perfect mirrors with 100% reflectivity. This is an ideal case and in reality, there will always be some amount of loss at the mirrors due to imperfections. This is why the value of Gr is not exactly 5.16%, but rather 0.0516. So your solution is valid for this particular case, but keep in mind that in real lasers, the reflectivity of the mirrors will have to be taken into account.

To better understand the concept of net gain coefficient, it might be helpful to think of it in terms of a simple analogy. Imagine you have a bucket with a hole at the bottom and you are filling it with water from a tap. The tap represents the gain mechanism and the hole represents the loss mechanism. The net gain coefficient (g-\alpha) will be like the overall flow rate of water into the bucket, taking into account both the flow rate from the tap and the leakage from the hole. Similarly, in a laser, the net gain coefficient represents the overall amplification achieved, taking into account both the gain and absorption processes.

I hope this helps clarify the concept. Keep in mind that the net gain coefficient can vary depending on the specific laser system and its parameters.