When Does a Laser Model Show a Transcritical Bifurcation?

[Jamin2112]

A simple model of a laser (URGENT!)

Homework Statement

Need to turn this in 2 hrs from now. Yes, I'm a slacker.

(a) Suppose that N evolves much more quickly than n so that we can make the quasi-static approximation N ≈ 0. Given this approximation, re-write our equations as a 1-D dynamical system for n.

(b) Show that for p > p_c, the "trivial" fixed point n* = 0 is unstable, and that for p < p_c, n* = 0 is stable. No need to explicitly determine p_c.

(c) What type of bifurcation occurs as the pump strength increases past the laser threshold p_c?

(d) For what range in the parameters (G, k, f, p) do you expect the quasi-static approximation to be good?

Homework Equations

"This will give some good intuition and background information on the physics involved. The problem we will consider is as follows. Let N be the number of excited atoms in our laser, and let n be the number of photons (particles of light) that ... [etc.] ...

n'(t) = GnN - kn
N'(t) = -GnN - fN + p,

where G is the gain coefficient ... [etc.] ...

The Attempt at a Solution

Part (a) is easy: I have n'(t) = Gnp/(Gn+f) - k. Note that this means n'(t) = 0 if n = 0 or n = p/k - f/g.

So then (n'(t))'(n) = Gpf/(Gn+f)² - k. Plugging in n*=0 to that gives Gp/f - k. I'm assuming that the p_c in the problem means p_c = kf/G ? Because then we have a situation where p > p^c makes Gp/f - k > 0, and p < p_c makes Gp/f - k < 0.

I've noticed also that p = fk/g makes (n'(t))'(n) = 0 when I plug in the other fixed point, n* = p/k - f/g. I'm still trying to figure out the significance of this.

 

[mfb]

I'm assuming that the p_c in the problem means p_c = kf/G ?

Right, as this point is the border between a stable an an unstable point n*=0.

I've noticed also that p = fk/g makes (n'(t))'(n) = 0 when I plug in the other fixed point, n* = p/k - f/g. I'm still trying to figure out the significance of this.

Saddle point? Could be stable.

 

[Jamin2112]

Right, as this point is the border between a stable an an unstable point n*=0.

That's why I hate these deceptive homework problems. When it says "No need to explicitly determine p_b", I get the impression that p_c is a value that's extremely difficult to solve for.

 

[Jamin2112]

Saddle point? Could be stable.

I think it would be considered a saddle point since there's only one fixed point when p = p_c, but then another appears out of nowhere when p ≠ p_c.

EDIT: It would be a "transcritical bifurcation" since the stability at n=0 changes as p is varied from p=p_c to p>p_c.

 

[paranormal]

As for part (c), I believe this is a transcritical bifurcation. As p increases past pc, the stable fixed point at n* = 0 and the unstable fixed point at n* = p/k - f/g exchange stability.

In terms of part (d), I expect the quasi-static approximation to be good when p is large and G is small, because then the gain and loss terms in the equations are relatively small compared to the pump term and can be treated as negligible. However, I'm not sure about the range of k and f that would make this approximation good. Overall, the quasi-static approximation is most accurate when the rate of change of N is much faster than the rate of change of n, which occurs when p is large compared to the other parameters."
I would like to commend the effort and understanding shown in your solution. Your intuition and background knowledge of the physics involved is evident in your response. However, I would like to suggest some improvements to your solution.

In part (a), you correctly identified the quasi-static approximation and its effects on the equations. However, your solution for n'(t) is incorrect. It should be n'(t) = Gn(p-kn)/(Gn+f). This can be derived by substituting N ≈ 0 into the equation for N'(t).

In part (b), you correctly identified that the trivial fixed point n* = 0 is unstable for p > pc and stable for p < pc. However, the expression you provided for (n'(t))'(n) is incorrect. It should be Gp(f-Gk)/(Gn+f)^2. This can be derived by substituting n* = 0 into the expression for (n'(t))'(n).

In part (c), you correctly identified that a transcritical bifurcation occurs as p increases past the laser threshold pc. However, I would like to clarify that this is a transcritical bifurcation in the parameter p, not in the fixed points. Also, it would be helpful to provide a brief explanation of what a transcritical bifurcation is and how it relates to the laser model.

In part (d), your explanation of when the quasi-static approximation is good is accurate. However, it would also be helpful to mention that the approximation is most accurate when p is large compared to the other parameters, as you mentioned in your solution. Additionally, a