Photon lifetime (cavity lifetime)

The photon lifetime is how long light lasts inside a resonator. If a cavity has a total photon loss rate $\kappa$, the stored energy decays as $e^{-t\mathrm{/}{\tau }_p}$ with ${\tau }_p=1\mathrm{/}\kappa$ once the source is switched off, and every way of describing a resonator's sharpness is this time in different clothing. The resonance linewidth is its Fourier transform, the quality factor is its ratio to the optical period, and the finesse is its ratio to the round-trip time:

At 1550 nm the conversion from $Q$ is ${\tau }_p[\mathrm{p}\mathrm{s}]=Q\mathrm{/}1215$: a $Q$ of ${10}^4$ is 8.2 ps, ${10}^6$ is 820 ps, ${10}^8$ is 82 ns, and the $4\times {10}^{10}$ of a good reference cavity is 33 µs.

For a Fabry-Perot cavity

Light loses the fraction $1-R_1{R}_2{e}^{-2{\alpha }_{i}L}$ of its power on each round trip of duration $T_{rt}=2n_{g}L\mathrm{/}c$, so the decay time is

with ${\alpha }_i$ the internal loss, ${\alpha }_m=\ln (1\mathrm{/}{R}_1{R}_2)\mathrm{/}2L$ the mirror loss written as a distributed coefficient and $v_g=c\mathrm{/}{n}_g$ the group velocity. This is the exact ring-down time of the round-trip picture and the form used in laser rate equations; the linewidth form $1\mathrm{/}(2\pi \mathrm{\Delta }\nu )$, with $\mathrm{\Delta }\nu$ the exact Airy width, agrees with it to within 1 % for a finesse above about 10, and for a cleaved laser chip (300 µm, $R=0.31$, ${\alpha }_i=10$ cm⁻¹, $\mathcal{F}=1.95$) gives 1.93 ps against the ring-down 2.45 ps, about 21 % shorter, because fringes that broad are not Lorentzian. The Fabry-Perot Cavity Calculator shows both for any cavity.

Why it matters

In a laser the photon lifetime is the loss the gain has to beat: the threshold condition is that the modal gain equal $1\mathrm{/}(v_g{\tau }_p)$, so a shorter lifetime means a higher threshold and, through the mirror-loss share, more of the light coupled out. It is also half of the laser's dynamics. The relaxation-oscillation frequency of a diode laser is

in the simplest rate-equation model, with ${\tau }_n$ the carrier lifetime, and the intrinsic modulation bandwidth grows as the photon lifetime shrinks, which is one reason short cavities and low-reflectivity facets make fast lasers and why a high-Q ring or DBR laser modulates slowly. In a passive resonator the lifetime is the sensitivity: the light circulates for ${\tau }_p\mathrm{/}{T}_{rt}$ round trips, so a cavity-enhanced absorption measurement sees the sample that many times, and cavity ring-down spectroscopy reads the absorption directly from the change in ${\tau }_p$ when the sample is present. The lifetime also sets how fast a resonator can respond: a filter or a ring modulator cannot change its transmission in less than about ${\tau }_p$, so a 10 GHz ring modulator needs a $Q$ below about $2\times {10}^4$, however good its phase shifter.

Measuring it

For lifetimes above a few nanoseconds the direct method is ring-down: excite the cavity, switch off or detune the source in less than ${\tau }_p$, and fit the exponential decay of the transmitted light; supermirror cavities with lifetimes of tens of microseconds are characterized this way because their kilohertz linewidths are hard to scan. For shorter lifetimes the linewidth is measured instead, with a swept laser for rings and etalons or from the below-threshold Fabry-Perot fringes of a laser chip, and ${\tau }_p$ follows from $1\mathrm{/}(2\pi \mathrm{\Delta }\nu )$.

References: A. E. Siegman, Lasers (University Science Books, 1986), chapter 11. L. A. Coldren, S. W. Corzine, and M. L. Mašanović, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012), chapters 2 and 5. G. Berden, R. Peeters, and G. Meijer, "Cavity ring-down spectroscopy: experimental schemes and applications," International Reviews in Physical Chemistry 19, 565 (2000).