Question

In a He-Ne laser system the starting intensity is 0.1 picowatts with a gain coefficient of 0.1 (1/cm). If R1 = .98 and R2 = .99, with a cavity length of 50 cm, what is the intensity of the beam inside AND outside the cavity after five round trips? Assume the starting beam is from the second mirror R2, and that there is no saturation mechanism involved.

Answer 1

Let initial intensity = 0.1 pW (p ~ pico)

Gain coefficient times the intial intensity before travelling the length of the laser, give the total intensity with added photons from traversing the optical medium.

Thus,

for every pass of the tube, we obtain an increase in intensity by a factor of the gain coefficient times the length travelled. This is followed by the decrease in intensity by the reflection from the mirror.

To showcase: let us run the first round trip in detail:

Initially at 2nd mirror: Io = 0.1pW.

On reaching the 1st mirror, by passing through the medium the intensity before reflection from this mirror is:

;

; (this is per traversal of the tube)

After reflection at mirror 1:     

This again travels to mirror 2 gaining intensity given as:

On reflection at mirror 2: .

This completes 1 round trip.

Thus, intensity after one round trip can be simply written as:

;

where I1RT = intensity after 1 round trip; Io = initial intensity at the start of the trip.

Thus, extrapolating it to 5 round trips: the equation simply cascades to the next as follows:

;

,..... and so on.

Thus:

Thus, as the 5th round trip ends:

we consider the light inside the cavity to have completed reflection at mirror 2; while the percentage transmitted to be the one outside (assuming light goes out of mirror 2)

Inside the cavity:

where .

Outside the cavity: (the transmitted part of light)

We divide by R2 to remove the reflection component, and multiply with (1-R2) to calculate the transmitted part.