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Tsunami
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We've had a couple of exercises on lasers, and there's two I had some problems with:
1. He-Ne-laser
Given
A He-Ne laser with a 'base wavelength' (might be a bad translation of terminology, for which I apologize) lambda= 632.8 nm;
Level width delta(nu) = 1500 MHz ;
Allowed deviation of laser frequency = 100 MHz ;
This laser works in a resonator of length L, in a medium with n=1.
Asked
a) What is the length of the cavity so that the laser only has 1 longitudinal mode?
b) What is the allowed deviation of length L, with the given deviation for the mode frequency?
My problem
a) is easy: We're working with a resonator, which is comparable to a Fabry-Perot resonator, and so the spread in frequency of longitudinal modes is:
delta(nu) = c/(2*n*L) or delta(lambda) = lambda²/(2*n*L)
1 longitudinal mode in the cavity, means that the spread in level width (which is also the spread of the gain curve of the laser) must be smaller than the spread-in-longitudinal-modes width:).
So L < 10 cm .
b) however I don't really get. My laser frequency can deviate 100 MHz... what does this mean exactly? Where does it deviate from? A deviation in length means that the spacing between modes changes.
I made an odd derivation that seems to fit : but that doesn't do me any good if I can't understand it.. Derivation was:
With L, spacing is : L = m*lambda1/(2*n), m =1,2,3...
With L+dL, spacing is : L+dL = m*lambda2/(2*n), m=1,2,3...
So dL = m/(2*n)* d(lambda)
lambda = c/nu
d(lambda) = -c/nu² *d(nu) =-lambda*d(nu)/nu
so:
dL/L = d(lambda)/lambda1= -d(nu)/nu1
nu1 = c/base lambda = 3e8m/s / 632.8e-9m = 474 THz
so dL = 21 nm, which is the solution that I should get.
Still... I don't know.. I feel I'm not really understanding the physics behind this... I got this solution after 5 wrong solutions, and I still don't really see the reasoning behind it. Anyone care to clarify?
2. Vertical Cavity Surface EMitting laser
Given
A gain curve in the shape of a parabola... It's easier to write down the mathematical characteristics... if x-values are lambda, y-values are gain values... then the parabola is centered around lambda =850 nm... and sections with the x-axis are at 865 nm and 835 nm (so delta(lambda) = 30nm).
The top of the curve is a value g_max.
The laser system is in a resonator of length L, and with mirrors on both sides, reflecting 99,9%.
A cryptic point for me is the given : "Gain area: Quantum wells (L=2*lambda/n)";
Asked
a) What is g_max? (before threshold... ie. before the resonator reaches pumping threshold)
b) What is delta(lambda) for points with gain that deviates less than 1% ?
c) What is the number of modi in this area with cavity length =L?
My answers
To be honest...I don't really get it. a) I don't get at all.. wouldn't know how to start with it.
b) I assumed they meant : delta(lambda) for points that deviate 1% from g_max. So I calculated the points on the parabola for which g = 0,99 * g_max (using the value g_max = 1940/m given as a solution), but I didn't get the right answer.
c) This should be fairly easy to calculate..
1. He-Ne-laser
Given
A He-Ne laser with a 'base wavelength' (might be a bad translation of terminology, for which I apologize) lambda= 632.8 nm;
Level width delta(nu) = 1500 MHz ;
Allowed deviation of laser frequency = 100 MHz ;
This laser works in a resonator of length L, in a medium with n=1.
Asked
a) What is the length of the cavity so that the laser only has 1 longitudinal mode?
b) What is the allowed deviation of length L, with the given deviation for the mode frequency?
My problem
a) is easy: We're working with a resonator, which is comparable to a Fabry-Perot resonator, and so the spread in frequency of longitudinal modes is:
delta(nu) = c/(2*n*L) or delta(lambda) = lambda²/(2*n*L)
1 longitudinal mode in the cavity, means that the spread in level width (which is also the spread of the gain curve of the laser) must be smaller than the spread-in-longitudinal-modes width:).
So L < 10 cm .
b) however I don't really get. My laser frequency can deviate 100 MHz... what does this mean exactly? Where does it deviate from? A deviation in length means that the spacing between modes changes.
I made an odd derivation that seems to fit : but that doesn't do me any good if I can't understand it.. Derivation was:
With L, spacing is : L = m*lambda1/(2*n), m =1,2,3...
With L+dL, spacing is : L+dL = m*lambda2/(2*n), m=1,2,3...
So dL = m/(2*n)* d(lambda)
lambda = c/nu
d(lambda) = -c/nu² *d(nu) =-lambda*d(nu)/nu
so:
dL/L = d(lambda)/lambda1= -d(nu)/nu1
nu1 = c/base lambda = 3e8m/s / 632.8e-9m = 474 THz
so dL = 21 nm, which is the solution that I should get.
Still... I don't know.. I feel I'm not really understanding the physics behind this... I got this solution after 5 wrong solutions, and I still don't really see the reasoning behind it. Anyone care to clarify?
2. Vertical Cavity Surface EMitting laser
Given
A gain curve in the shape of a parabola... It's easier to write down the mathematical characteristics... if x-values are lambda, y-values are gain values... then the parabola is centered around lambda =850 nm... and sections with the x-axis are at 865 nm and 835 nm (so delta(lambda) = 30nm).
The top of the curve is a value g_max.
The laser system is in a resonator of length L, and with mirrors on both sides, reflecting 99,9%.
A cryptic point for me is the given : "Gain area: Quantum wells (L=2*lambda/n)";
Asked
a) What is g_max? (before threshold... ie. before the resonator reaches pumping threshold)
b) What is delta(lambda) for points with gain that deviates less than 1% ?
c) What is the number of modi in this area with cavity length =L?
My answers
To be honest...I don't really get it. a) I don't get at all.. wouldn't know how to start with it.
b) I assumed they meant : delta(lambda) for points that deviate 1% from g_max. So I calculated the points on the parabola for which g = 0,99 * g_max (using the value g_max = 1940/m given as a solution), but I didn't get the right answer.
c) This should be fairly easy to calculate..