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Lecture (9):
Round trip gain with losses |
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درسنا في محاضرات سابقة (محاضرة 6) الحصيلة الناتجة عن
دورة كاملة لليزر خلال المادة وعلمنا أن انقلاب التعداد شرط اساسي 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 الاقل 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:
-
Transmission at
the mirrors
-
Absorption and
scattering by the mirrors
-
Absorption in
the laser medium
-
Diffraction
losses at the mirrors
All these
losses will contribute to reduce the effective gain coefficient to (go-
k)
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.
 Gain
(G) Without Losses
From
lecture (6) we found that the intensity after one round trip is given by
the equation
I5 = R1* R2*G2*I1
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 (a),
and using it we can define absorption factor k:
k = exp(-2aL)
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).
a = Loss coefficient (in units of 1 over
length).
2L = Path Length, which is twice the
length of the cavity.
Adding the loss factor (k) to the equation of I5:
I5 = R1* R2*GA2*I1*k
From this we can calculate the
round trip gain:
G = I5/I1 = R1* R2*GA2
*k
As we assumed uniform distribution of the loss coefficient (a),
we now define gain coefficient (g),
and assume active medium gain (GA) as distributed uniformly along
the length of the cavity.
GA = exp(+gL)
Substituting the last equation in the Loop Gain:
G = R1* R2* exp(2(g-a)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
GA2k = R1* R2*
exp(2(g-a)L)
 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 30
cm. Loss coefficient is: a = 1.34*10-4
cm-1. Calculate:
-
The loss factor k.
-
The round trip gain G.
-
The gain coefficient (g).
Solution
1. The loss factor k:
k = exp(-2aL) = exp[-2(1.34*10-4)*30]
= 0.992
2. The Loop gain G:
G = R1R2GA2k =
0.999*0.95*1.052*0.992 = 1.038
Since GL > 1, this laser operates above
threshold.
3. The gain coefficient (g):
G = exp(gL)
Ln G = gL
g
= Ln G/L = ln(1.05)/30 = 1.63*10-3 [cm-1]
The gain coefficient (g)
is greater than the loss coefficient (a), as
expected.
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:
-
The loss factor k.
-
The loss coefficient a.
Solution
Since the laser operates in threshold condition, G = 1.
Using this value in the round trip gain:
G = 1 = R1R2GA2k
1. The loss factor k:
k = 1/( R1R2GA2) =
1/(0.999*0.97*1.022) = 0.9919
As expected, k < 1.
Since G > 1, this laser operates above threshold.
2. The loss coefficient (a) is
calculated from the loss factor:
k = exp(-2aL)
lnk = -2aL
a = lnk/(-2L) = ln(0.9919)/(-100) =
8.13*10-5 [cm-1]
Attention:
If the loss factor was less than 0.9919, then
G < 1, and the oscillation condition was not fulfilled.
Example
Reflection coefficients of the mirrors are: 0.999, and 0.95.
All the losses in round trip are 0.6%. Calculate the
active medium gain.
Solution
For finding the active medium gain G, the loss factor (k) must
be found.
All the losses are 1-k.
1-k = 0.006
k = 0.994
Using this value in the threshold loop gain:
Gth = 1 = R1R2GA2
k
(GA)th = 1/sqrt( R1R2k)
= 1/sqrt(0.999*0.95*0.994) = 1.03
The active medium gain must be at least 1.03 for creating
continuous output from this laser.
Summary
- G = round trip Gain,
determines if the output power of the laser will increase, decrease,
or remain constant. It include all the losses and amplifications that
the beam have in a complete round trip through the laser.
GL = R1R2GA2k
R1, R2
= Reflection coefficients of the laser mirrors.
GA = Active medium
gain as a result stimulated emission.
GA = exp(+gL)
g
= Gain coefficient.
L = Active Medium length.
k = Optical Loss Factor in a round trip path in
the laser cavity.
k = exp(-2aL)
a = Loss
coefficient.
- When G = 1, The laser operate in a
steady state mode, meaning the output is at a constant power.
This is the
threshold condition for lasing,
and the active medium gain is:
(GA)th = 1/sqrt( R1R2k)
The
round trip Gain
is:
GL = R1* R2*
exp(2(g-a)L)
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End of Lecture (9)
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