Round-trip laser threshold condition

If $g$ and ${\alpha }_i$ are intensity gain and loss per length, a linear cavity has round-trip intensity multiplier $R_1{R}_2{e}^{2(g-{\alpha }_i)L}$. Threshold occurs when it equals one, giving $g_{\mathrm{th}}={\alpha }_i+\log [1/(R_1{R}_2)]/(2L)$. Phase resonance and the gain bandwidth determine which modes can oscillate.