Lasers and Laser Light — Question 5

Question 5

Longitudinal laser modes are spaced by the inverse round-trip time

A nondispersive linear cavity has physical length $L=.300\,\mathrm m$ and uniform refractive index $n_r=1.50$. Ignore mirror phase changes. The standing-wave condition is $2n_rL=q\lambda_0$, where $\lambda_0$ is vacuum wavelength. Use $c=3.00\times10^8\,\mathrm{m/s}$.

1. Derive the mode frequencies and their spacing.

2. A gain interval of full width 1.00 GHz is centered exactly on a cavity mode. Count the modes strictly inside it.

Question 5 - Solution

Longitudinal laser modes are spaced by the inverse round-trip time

(a) Method and calculation

$$\nu_q=c/\lambda_0=\boxed{qc/(2n_rL)}.$$$$\boxed{\Delta\nu_{\mathrm{FSR}}=c/(2n_rL)=333.3\,\mathrm{MHz}}.$$

(b) Interpretation and further calculation

$$|j|\Delta\nu_{\mathrm{FSR}}<.500\,\mathrm{GHz}\Rightarrow |j|<1.50.$$$$j=-1,0,1\Rightarrow\boxed{3\text{ modes}}.$$

Physical check

A mode lying within the gain interval is a candidate, not a guarantee of lasing. Gain competition and losses determine which candidates actually oscillate.