# Can I choose non-resonant Modes deriving the semiclassical Laser Equations?

The Semiclassical Laser-Equations (also called "Laser Self-Consistancy Equations" ) are used to model a classical EM-Field that is driven by a given Polarisation in an active medium.

In Order to derive these equations, one chooses a full set of orthonormal functions $U_m(x,y,z)$ and expands the electric Field $E$ as a superposition of these functions. Each of the $u_m$ solves the wave equation, and is asigned to a specific frequency, which will then appear in the time evolution equation for the amplitude, by which the mode $u_m$ oscillates.

Now my question is: Why do we choose the set of $U_m$ in a way that the $U_m$ are eigenmodes of the laser cavity? Couldn't I choose abitrarily functions, as long as they form a full set, and satisfy the wave equations? And if I did so, wouldn't I see that off resonant modes are amplified less than resonant modes?

## 1 Answer

If you choose basis functions that aren't eigenmodes of the cavity, and your model is correct, what you should find is that your basis "modes" are coupled. That is, as the system evolves with time, energy in one of the basis functions transfers to other functions. The benefit of using the eigenmodes is that energy in these modes stays there, so that you can analyze the behavior of the system one mode at a time.

You'll see a related analysis when studying coupled cavities or waveguides. Rather than find the eigenmodes of the system of coupled cavities we can start with a basis set based on the eigenmodes of the individual cavities. Then we find how these "modes" are coupled, that is, how energy transfers from one cavity to the other.

• Can you elaborate on why the modes are coupled? Can I imagine that the reflection of the mode will yield a different mode? How do I couple Cavities, and what would be appropriate Literature to read into the subject? – Quantumwhisp Mar 11 '17 at 10:35
• Answer 1: Because they're not eigenmodes. Eigenmodes means modes that when they evolve in time you end up with the same mode afterwards. Any other field pattern, after evolution in time you'll end up with a different pattern. – The Photon Mar 11 '17 at 16:06
• Answer 2: In a coupled system, if their field patterns overlap, one quasi-mode is able to transfer energy to another quasi-mode. – The Photon Mar 11 '17 at 16:08
• Could I use this as an explanation to why only resonant modes are amplified in a laser? Let's say in both cases we start with one excited mode (one time resonant, other time off-resonant). While in the first case energy remains in this mode, it is distributed among many other modes in the second case, and because amplification works nonlinear, it is more big for the resonant mode? – Quantumwhisp Mar 11 '17 at 17:33
• That's kind of it, but not a very clear way to say it. What really happens is that (for a homogeneously broadened gain mechanism) one (actual) mode reaches threshold first and then the gain is "pinned" as all available excited specie are used to amplify that mode; due to gain pinning no other mode can reach threshold. (Unless of course there's some other mechanism at work and then you have a multi-mode laser) – The Photon Mar 11 '17 at 17:54