How Lasers Work — Question 5

Question 5

Round-trip intensity balance determines the laser threshold gain

A linear cavity of length L is filled by a uniform medium with intensity gain coefficient g and distributed intensity-loss coefficient $\alpha$. Mirror intensity reflectivities are $R_1,R_2$. One pass multiplies intensity by $e^{(g-\alpha)L}$.

1. Derive the threshold gain.

2. Evaluate it for $L=.250\,\mathrm m$, $\alpha=.100\,\mathrm{m^{-1}}$, $R_1=.99$ and $R_2=.95$.

Question 5 - Solution

Round-trip intensity balance determines the laser threshold gain

(a) Method and calculation

$$\mathcal G_{\mathrm{rt}}=R_1R_2e^{2(g-\alpha)L};\qquad\mathcal G_{\mathrm{rt}}=1\text{ at threshold}.$$$$\boxed{g_{\mathrm{th}}=\alpha+\frac{1}{2L}\ln\frac1{R_1R_2}}.$$

(b) Interpretation and further calculation

$$\ln[1/(.99\times.95)]=.0613436.$$$$\boxed{g_{\mathrm{th}}=.100+.0613436/.500=.222687\,\mathrm{m^{-1}}}.$$

Physical check

All gain, loss, and reflectivity quantities are defined for intensity, avoiding field-amplitude factors of two. The gain medium is assumed to fill the entire stated cavity length.