Download Sensitive gas absorption coefficient measurements based on Q re

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 = 2fopt 
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