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**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..