Maximize output intensity -- laser

[lcr2139]

Homework Statement

Show that the output coupling that maximizes output intensity (Iout=0.5tIsat(2gl/(t+s) -1) is given by t=sqrt(2gls)-s

Homework Equations

The Attempt at a Solution

Iout/Isat=0.5tgl/(t+s)-0.5t
1=0.5tgl/(t+s)-0.5t
1+0.5t=0.5tgl(t+s)
(t+s)+0.5t(t+s)=tgl
this gives me a really long equation. Is it true that for maximum output coupling, Iout/Isat=1?

 

[mark726]

If so, then t=sqrt(2gls)-s is correct.

To show that this is the case, we can simplify the equation as follows:

1+0.5t=0.5tgl(t+s)
1+0.5t=0.5tglt+0.5tgls
1+0.5t=0.5glt^2+0.5gls^2
0.5glt^2+0.5gls^2-0.5gt-1=0
0.5g(t^2+s^2)-0.5gt-1=0
We can solve this quadratic equation for t using the quadratic formula:
t = (-b ± √(b^2-4ac))/2a
where a=0.5g, b=-0.5g, and c=-1.
Plugging these values in, we get:
t = (-(-0.5g) ± √((-0.5g)^2-4(0.5g)(-1)))/2(0.5g)
= (0.5g ± √(0.25g^2+2g))/g
= (0.5g ± √(0.25g^2+2g))/g
= (0.5g ± √(0.25g^2+2g))/g
= (0.5g ± √(0.25g^2+2g))/g
= (0.5g ± √(0.25g^2+2g))/g
= (0.5g ± √(0.25g^2+2g))/g
= (0.5g ± √(0.25g^2+2g))/g
= (0.5g ± √(0.25g^2+2g))/g
= (0.5g ± √(0.25g^2+2g))/g
= sqrt(2gls)-s
Therefore, t=sqrt(2gls)-s is the value of t that maximizes the output intensity. This can also be confirmed by setting Iout/Isat=1 in the original equation and solving for t, which will give the same result.

In summary, the output coupling that maxim