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Sensitive gas absorption coefficient measurements based on Q reduction in an optical cavity. Three measurement methods to consider: 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 P1 1) Basic time-domain analysis, laser emitting one large pulse of short duration (and thus not so narrowband). P1T1T2 e-t/ T1 <= 1-R1 T2 <= 1-R2 Resonant cavity: model Assume all transmissions T and losses L <<1 Fine Piezo control LASER |E5 |2 E1 E2 E5 E3 Photo detector E4 Photo detector |E4|2 Mirror 2 T2 , R2 , L2 , Additional round-trip losses: Mirror 1: T1 = power transmission R1 = 1 – T1 – L1 L1 = power loss L0 = various power losses (diffraction, Rayleigh scattering, etc.) 2 D = absorption loss (what we're trying to measure!) Net cavity loss: LCAV = T1 + T2 + L0 + L1 + L2 + 2 D = (2D / c) / LCAV (“Photon lifetime” = ring-down time const.) Q = 2fopt BW = fopt / Q F = 2 / (2D / c) = 2 / LCAV (Finesse) D E1 E2 E5 E3 E4 Intra-cavity waves E2 and E3 (at position of mirror 1 surface): E3 * R11/2 = (1 – LCAV / 2) ej 2kD E2 = (1 – LCAV / 2) ej E2 Resonance when = 0 (that is, 2D = N ) Find that amplitude increased inside cavity at resonance: E2 / E1 = T11/2 * 2 / LCAV On general principles: Since detection of sample absorption depends on loss of energy (photons) passing through sample, increasing the intracavity power |E2|2 increases the potential detectivity. Direct measurement of cavity Q All sources of cavity loss: E1 E2 E5 E3 T1 L1 E4 L0 2D T2 L2 Net cavity loss: LCAV = T1 + T2 + L0 + L1 + L2 + 2 D Photo detector Computer etc. Output power relative to incident laser power: |E4|2/ |E1|2 = 4 T1 T2 / L2CAV (at resonance) Thus most sensitive to when T1, T2, L0, L1, L2 reduced (high Q cavity). Better scheme: Measure peak ratio (thus at resonance) of transmitted power |E4|2 to reflected power |E5|2 |E5 |2 Photo detector E1 E2 E5 E3 T1 L1 E4 L0 2D T2 L2 Output power relative to reflected power (at resonance): Photo detector |E4|2 Computer etc. |E4|2/ |E5|2 = 4 T1 T2 / (LCAV - 2T1)2 We assume that T1 accounts for less than ½ of Lcav – otherwise |E5|2 will not be monotonically reduced with reduced Lcav (E5 will actually go through zero and reappear in opposite phase!). This will always be assured when using two identical mirrors M1 and M2 (and most other realistic cases). Again, we get the best sensitivity to by reducing losses L0, L1, L2 (of course) and also reducing the mirror transmitances T1 and T2 when they are a large part of Lcav (after reducing the actual loss terms). |E4|2/ |E1|2 |E5|2/ |E1|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, T1 = T2 =.001, L0 =.0005., so LCAV =.0025, measured around =1 micron (300 THz). Cavity ring-down time measurement using a highly reflecting mirror 2 in order to further reduce LCAV (increasing the cavity Q) |E5 |2 Photo detector E1 E2 E5 E3 T1 L1 L0 2D T2=0 R = 1-L2 Computer etc. Net cavity loss: LCAV = T1 + 0 + L0 + L1 + L2 + 2D ---> Higher Q Method: 1) Computer dithers piezo to point of minimum reflected power |E5|2 2) Laser beam is interrupted. 3) Ring-down time constant is measured at mirror 1 cavity output. Again, we are assuming that T1 < ½ LCAV Practical (and energy efficient) implementation of separation of incident laser beam and reflected wave from cavity Polarizing Beamsplitter LASER Circularly polarized waves E2 E1 Shutter E5 Photo detector Quarter wave plate @ 45o E3