Round trip gain with losses

Round trip gain with losses
درسنا في محاض ا رت سابقة الحصيلة الناتجة عن دورة كاملة لليزر خلال المادة، وعلمنا
أن انقلاب التعداد شرط أساسي لكي يستمر الحصول على ليزر ولكن الفقد الناتج عن
عدة عوامل يسبب في نقصان الحصيلة. ولكي نحصل على ليزر فإن الحصيلة لكل دورة
يجب أن تكون على الأقل كبر من الخسارة في كل دورة. في هذه المحاضرة سنركز على
العلاقة بين الحصيلة والخسارة.
The total losses of the laser system is due to a number of
different processes these are:
1. Transmission at the mirrors
2. Absorption and scattering by the mirrors
3. Absorption in the laser medium
4. Diffraction losses at the mirrors
All these losses will contribute to reduce the effective gain
coefficient to (?o- k)
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Round trip Gain (G)
Figure below show the round trip path of the radiation through
the laser cavity. The path is divided to sections numbered by 1-5,
while point “5” is the same point as “1”.
Round trip path of the radiation through the laser cavity.
By definition, Round trip Gain is given by:
G = I5/ I1
G = Round trip Gain.
I1 = Intensity of radiation at the beginning of the loop.
I5 = Intensity of radiation at the end of the loop.
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Gain (G) Without Losses
From lecture (9) we found that the intensity after one round trip
is given by the equation
I5 = R1 * R2* G2 * I1
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Gain (G) With Losses
We assume that the losses occur uniformly along the length of the cavity
(L). In analogy to the Lambert formula for losses, we define loss
coefficient (?), and using it we can define absorption factor k:
k = exp(-2?L)
k = Loss factor, describe the relative part of the radiation that remain in
the cavity after all the losses in a round trip loop inside the cavity.
All the losses in a round trip loop inside the cavity are 1-k (always less
than 1).
? = Loss coefficient (in units of 1 over length).
2L = Path Length, which is twice the length of the cavity.
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Adding the loss factor (k) to the equation of I5:
I5 = R1* R2* GA
2 *I1* k
From this we can calculate the round trip gain:
G = I5/I1 = R1* R2*GA
2 * k
As we assumed uniform distribution of the loss coefficient (?), we
now define gain coefficient (?), and assume active medium gain
(GA) as distributed uniformly along the length of the cavity.
GA = exp(+?L)
Substituting the last equation in the Loop Gain:
G = R1* R2* exp(2(????L)
k = exp(-2?L)
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G = R1* R2* exp(2(????L)
When the loop gain (G) is greater than 1 (G > 1), the beam intensity will
increase after one return pass through the laser.
When the loop gain (G) is less than 1 (G < 1), the beam intensity will
decrease after one return pass through the laser. laser oscillation decay,
and no beam will be emitted.
Conclusion:
There is a threshold condition for amplification, in order to create
oscillation inside the laser.
This Threshold Gain is marked with index “th”.
For continuous laser , the threshold condition is:
Gth = 1 = R1 R2 GA
2 k = R1* R2* exp(2(????L)
Gth = 1
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Example
Active medium gain in a laser is 1.05. Reflection coefficients of
the mirrors are: 0.999, and 0.95. Length of the laser is
30cm. Loss coefficient is: ? = 1.34*10-4 cm-1.
Calculate:
1. The loss factor k.
2. The round trip gain G.
3. The gain coefficient (?).
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Solution
1. The loss factor k:
k = exp(-??L) = exp[-2(1.34*10-4)*30] = 0.992
2. The Loop gain G:
G = R1R2GA
2k = 0.999*0.95*1.052*0.992 = 1.038
Since GL > 1, this laser operates above threshold.
3. The gain coefficient (?):
G = exp(?L)
Ln G = ?L
????Ln G/L = ln(1.05)/30 = 1.63*10-3 [cm-1]
The gain coefficient (?) is greater than the loss coefficient (?), as
expected.
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Example
Helium Neon laser operates in threshold condition. Reflection
coefficients of the mirrors are: 0.999, and 0.97. Length of the
laser is 50 cm. Active medium gain is 1.02.
Calculate:
1. The loss factor k.
2. The loss coefficient ?.
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Solution
Since the laser operates in threshold condition, G = 1.
Using this value in the round trip gain:
G = 1 = R1R2GA
2k
1. The loss factor k:
k = 1/( R1R2GA
2) = 1/(0.999*0.97*1.022) = 0.9919
As expected, k < 1.
Since G > 1, this laser operates