QB_PH_SEC6_020: Lasers and Quantum Optics

For a laser cavity of length $L$ bounded by mirrors of power reflectivities $R_1$ and $R_2$ with internal distributed loss coefficient $\alpha_s$, the threshold gain coefficient $\gamma_{\text{th}}$ of the active medium is given by

Correct Answer

### Step-by-Step Derivation & Verification: 1. **Round-Trip Amplitude Balance**: Consider an initial wave of intensity $I$ propagating from mirror 1 to mirror 2 and back over total path $2L$: $I_{\text{round-trip}} = I \cdot R_1 \cdot R_2 \cdot \exp[2(\gamma - \alpha_s)L]$ 2. **Threshold Criterion**: The net round-trip amplification must exactly compensate for all cavity transmission and distributed scattering losses: $R_1 R_2 \exp[2(\gamma_{\text{th}} - \alpha_s)L] = 1$ 3. **Logarithmic Isolation**: $2(\gamma_{\text{th}} - \alpha_s)L = -\ln(R_1 R_2) = \ln\left(\frac{1}{R_1 R_2}\right)$ $\gamma_{\text{th}} = \alpha_s + \frac{1}{2L} \ln\left(\frac{1}{R_1 R_2}\right)$ 4. **Conclusion**: Option (A) is correct.