How Do Laser Resonator Length and Deviations Affect He-Ne and VCSEL Lasers?

[Tsunami]

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..

 

[Claude Bile]

With regard to question 1, part 2:

Okay, the maths first. You got it right in a roundabout way, which may contribute to why you are confused with the physics. A more succinct way of working out the solution would be;

\frac{dL}{L_0}=\frac{d\nu}{\nu_0}

Then rearranging for dL;

dL=\frac{Ld\nu}{\nu_0}

Plugging in the numbers gives the correct result.

All you are doing is working out the minimum shift in dL required to change the laser frequency by the maximum allowed amount from the central frequency. Central frequency is usually defined as the centre frequency of a 3 db gain bandwidth.

Claude.

 

[thepatientmental]

but again, I don't really understand the given information and how to use it to calculate the number of modes.


In response to the content on the laser resonator exercise, it seems like there are some misunderstandings about the physics behind the equations and concepts involved. Let's break down the problems and try to clarify them.

1. He-Ne laser

a) The question asks for the length of the cavity so that the laser only has 1 longitudinal mode. This means that the laser will only emit light at one specific frequency, rather than multiple frequencies (which is what happens when there are multiple longitudinal modes). The spread in frequency of longitudinal modes is given by delta(nu) = c/(2*n*L) or delta(lambda) = lambda^2/(2*n*L). This means that for 1 longitudinal mode, the spread in frequency (or wavelength) must be smaller than the allowed deviation of the laser frequency, which is 100 MHz. Using the given values, we can solve for the length of the cavity (L) and it should be less than 10 cm.

b) The allowed deviation of length (dL) can be calculated using the same equation as in part a), but this time solving for dL instead of L. This will give us the maximum deviation in length that is allowed for the laser to maintain only 1 longitudinal mode.

2. Vertical Cavity Surface Emitting laser

a) The question asks for the value of g_max, which is the maximum gain before the laser reaches its pumping threshold (the point at which it starts emitting light). The given information about the parabolic gain curve can be used to calculate this. The top of the curve is at a value g_max, so we can use this and the given values for the x-axis (wavelength) to solve for g_max.

b) The question asks for the value of delta(lambda) for points with gain that deviate less than 1%. This means we need to calculate the wavelength values for which the gain is 1% less than g_max. This can be done by setting up an equation using the given values and solving for delta(lambda).

c) The number of modes in the cavity with length L can be calculated using the equation m = 2*L/lambda, where m is the number of modes, L is the cavity length, and lambda is the wavelength. However, the given information about "gain area" and "quantum