Sensitive gas absorption coefficient measurements based on Q reduction in an optical cavity. 1) Pulsed laser ring-down time measurements 2) Chopped CW laser resonant excitation of cavity and measurement of ring-down time 3) Continuous CW laser resonant excitation of cavity and measurement of cavity Q from optical power detection 1) Basic time-domain analysis, laser emitting one large pulse of short duration (and thus not so narrowband). T 1 <= 1-R 1 T 2 <= 1-R 2 P 1 T 1 T 2 e -t/  P1P1 Three measurement methods to consider:

Fine Piezo control LASER Photo detector Resonant cavity: model Assume all transmissions T and losses L <<1 E1E1 E5E5 E2E2 E3E3 E4E4 |E 4 | 2 |E 5 | 2 Mirror 1: T 1 = power transmission R 1 = 1 – T 1 – L 1 L 1 = power loss Mirror 2 T 2, R 2, L 2, Additional round-trip losses: L 0 = various power losses (diffraction, Rayleigh scattering, etc.) 2 D  = absorption loss (what we're trying to measure!) Net cavity loss: L CAV = T 1 + T 2 + L 0 + L 1 + L D   = (2D / c) / L CAV (“Photon lifetime” = ring-down time const.) Q = 2  f opt  BW = f opt / Q F = 2   / (2D / c) = 2  / L CAV (Finesse)

E2E2 E3E3 E4E4 E1E1 E5E5 Intra-cavity waves E 2 and E 3 (at position of mirror 1 surface): E 3 * R 1 1/2 = (1 – L CAV / 2) e j 2kD E 2 = (1 – L CAV / 2) e j  E 2 Resonance when  = 0 (that is, 2D = N ) Find that amplitude increased inside cavity at resonance: E 2 / E 1 = T 1 1/2 * 2 / L CAV On general principles: Since detection of sample absorption depends on loss of energy (photons) passing through sample, increasing the intracavity power | E 2 | 2 increases the potential detectivity. D

E1E1 E5E5 E2E2 E3E3 E4E4 T1L1T1L1 All sources of cavity loss: T2L2T2L2 L 0 2 D  Net cavity loss: L CAV = T 1 + T 2 + L 0 + L 1 + L D  Output power relative to incident laser power: | E 4 | 2 / | E 1 | 2 = 4 T 1 T 2 / L 2 CAV (at resonance) Thus most sensitive to  when T 1, T 2, L 0, L 1, L 2 reduced (high Q cavity). Photo detector Computer etc. Direct measurement of cavity Q

E1E1 E5E5 E2E2 E3E3 E4E4 T1L1T1L1 T2L2T2L2 L 0 2 D  Output power relative to reflected power (at resonance) : | E 4 | 2 / | E 5 | 2 = 4 T 1 T 2 / (L CAV - 2T 1 ) 2 We assume that T 1 accounts for less than ½ of L cav – otherwise |E 5 | 2 will not be monotonically reduced with reduced L cav (E 5 will actually go through zero and reappear in opposite phase!). This will always be assured when using two identical mirrors M 1 and M 2 (and most other realistic cases). Again, we get the best sensitivity to  by reducing losses L 0, L 1, L 2 (of course) and also reducing the mirror transmitances T 1 and T 2 when they are a large part of L cav (after reducing the actual loss terms). Photo detector Computer etc. Photo detector |E 5 | 2 |E 4 | 2 Better scheme: Measure peak ratio (thus at resonance) of transmitted power |E 4 | 2 to reflected power |E 5 | 2

| E 4 | 2 / | E 1 | 2 | E 5 | 2 / | E 1 | 2 This computation (if I didn't make any mistake!) plots the transmission through the etalon and reflection at the input vs. frequency. This is for a very short etalon: D=.1mm, T 1 = T 2 =.001, L 0 =.0005., so L CAV =.0025, measured around =1 micron (300 THz).

E1E1 E5E5 E2E2 E3E3 T1L1T1L1 T 2 =0 R = 1-L 2 L 0 2 D  Computer etc. Photo detector |E 5 | 2 Cavity ring-down time measurement using a highly reflecting mirror 2 in order to further reduce L CAV (increasing the cavity Q) Net cavity loss: L CAV = T L 0 + L 1 + L 2 + 2D  ---> Higher Q Method: 1) Computer dithers piezo to point of minimum reflected power |E 5 | 2 2) Laser beam is interrupted. 3) Ring-down time constant is measured at mirror 1 cavity output. Again, we are assuming that T 1 < ½ L CAV

Polarizing Beamsplitter LASER E1E1 E2E2 E3E3 E5E5 Photo detector Quarter wave 45 o Circularly polarized waves Practical (and energy efficient) implementation of separation of incident laser beam and reflected wave from cavity Shutter