Phase fluctuation optical heterodyne spectroscopy of gases

Christopher C. Davis; Samuel J. Petuchowski

doi:10.1364/AO.20.002539

Applied Optics
Vol. 20,
Issue 14,
pp. 2539-2554
(1981)
•https://doi.org/10.1364/AO.20.002539

Phase fluctuation optical heterodyne spectroscopy of gases

Christopher C. Davis and Samuel J. Petuchowski

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Christopher C. Davis and Samuel J. Petuchowski, "Phase fluctuation optical heterodyne spectroscopy of gases," Appl. Opt. 20, 2539-2554 (1981)

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Abstract

The theory underlying the use of the phase fluctuation optical heterodyne (PFLOH) technique in studies of molecular relaxation, thermal conduction, and extremely weak absorptions in the gas phase is presented. Several new solutions to the heat conduction equation, with and without mass diffusion, which are appropriate to pulsed, cw, and modulated-cw laser modulation of weakly absorbing gases, are presented. Experiments employing pulsed and modulated-cw excitation have borne out the essential predictions of the theory. PFLOH systems for use in a trace detection application have been built with a demonstrated sensitivity, in terms of measurable absorption by the trace material, below 10⁻¹⁰ cm⁻¹.

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Corrections

Christopher C. Davis and Samuel J. Petuchowski, "Phase fluctuation optical heterodyne spectroscopy of gases: errata," Appl. Opt. 20, 4151-4151 (1981)
https://opg.optica.org/ao/abstract.cfm?uri=ao-20-24-4151

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a. V → T relaxation much faster than thermal conduction or mass diffusion
TOP HAT POPULATION DISTRIBUTION
$\begin{matrix} \begin{array}{l} N (r, 0) & = N_{0} \\ = 0 \end{array} & \begin{array}{l} 0 ⩽ r ⩽ a \\ r > a \end{array} \end{matrix}$	$\begin{array}{l} T (0, t) & = T_{0} [1 - exp (- a^{2} / 4 κ t)] t ≫ τ \\ T (r, t) & = T_{0} [1 - exp (- a^{2} / 4 κ t)] t ≫ τ, r^{2} / 4 κ \\ T (r, t) & = T_{0} a^{2} / 4 κ t t ≫ τ, r^{2} / 4 κ, a^{2} / 4 κ \end{array}$
GAUSSIAN POPULATION DISTRIBUTION
N(r,0) = N₀ exp(−2r²/w²)	$T (r, t) = \frac{T_{0} w^{2}}{(8 κ t + w^{2})} exp [- 2 r^{2} / (8 κ t + w^{2})]$
b. V → T relaxation during thermal conduction and/or mass diffusion
1. TOP HAT DISTRIBUTION INITIALLY	$\begin{array}{l} T (0, t) & = T_{0} [1 - exp (- t / τ)] [1 - exp (- a^{2} / 4 κ t)] τ ≪ a^{2} / 4 κ, a^{2} 4 D \\ T (r, t) & = T_{0} [1 - exp (- t / τ)] P (\frac{a}{\sqrt{2 κ t}}, \frac{r}{\sqrt{2 κ t}}) τ ⩽ a^{2} / 4 κ, a^{2} / 4 D \\ T (r, t) & = T_{0} [1 - exp (- t / τ)] (1 + r^{2} / 4 κ t) exp (- r^{2} / 4 κ t) t ≪ a^{2} / 4 κ, r < a, τ ≪ a^{2} / 4 κ \\ T (r, t) & = T_{0} [1 - exp (- t / τ)] [1 - exp (- a^{2} / 4 κ t)] t ≫ r^{2} / 4 κ \\ T (r, t) & = T_{0} a^{2} / 4 κ t t ≫ τ, r^{2} / 4 κ, a^{2} / 4 κ \end{array}$
2. No mass diffusion
GAUSSIAN DISTRIBUTION	$\begin{array}{l} T (r, t) & = \frac{T_{0} w^{2}}{(8 κ t + w^{2})} [1 - exp (- t / τ)] exp [- 2 r^{2} / (8 κ t + w^{2})] τ ≪ w^{2} / 8 κ \\ T (r, t) & = T_{0} exp {- 2 r^{2} / w^{2} [1 - exp (- t / τ)] (1 - 8 κ t / w^{2})} τ, t ≪ w^{2} / 8 κ \end{array}$
3. With mass diffusion
GAUSSIAN DISTRIBUTION	$\begin{array}{l} T (r, t) & = \frac{T_{0} w^{2}}{(8 κ t + w^{2})} [1 - exp (- t / τ)] exp (- 2 r^{2} / (8 κ t + w^{2}) & τ ≪ w^{2} / 8 κ, w^{2} / 8 D \\ T (r, t) & = T_{0} w^{2} [\frac{1}{(8 κ t + w^{2})} - \frac{exp (- t / τ)}{(8 D t + w^{2})}] t ≫ τ \begin{matrix} and / or \\ 8 κ + w^{2} ≫ r^{2} \end{matrix} & τ ≪ w^{2} / 8 κ \\ T (r, t) & = T_{0} w^{2} / (8 κ t + w^{2}) t ≫ τ, 8 κ t + w^{2} ≫ r^{2} \end{array}$

	Configuration I	Configuration II (no cell)
Available He–Ne power	1.08	0.68	mW
Signal power at detector	13	19	μW
Sample length	29	29	cm
PZT sensitivity	0.018 ± 0.001	0.028 ± 0.003	rad/V
SNR with 1-v P-P sinusoid applied to PZT	275	7900	Hz^−1/2
	(Acoustic noise lim)	(Detector noise lim)
Theoretical noise limit	2.9 × 10⁻⁷	2.4 × 10⁻⁷	rad/Hz^1/2
Phase resolution obtained	6.5 × 10⁻⁵	5.3 × 10⁻⁶	rad/Hz^1/2
Index resolution	2.2 × 10⁻¹¹	1.8 × 10⁻¹²	Hz^−1/2
I₀α resolution	4.2 ×10⁻⁶	3.5 ×10⁻⁷	W/cm³/Hz^1/2
α_min resolution	4.8 × 10⁻⁷	1.8 × 10⁻⁹	cm⁻¹/W

a. V → T relaxation much faster than thermal conduction or mass diffusion
TOP HAT POPULATION DISTRIBUTION
$\begin{matrix} \begin{array}{l} N (r, 0) & = N_{0} \\ = 0 \end{array} & \begin{array}{l} 0 ⩽ r ⩽ a \\ r > a \end{array} \end{matrix}$	$\begin{array}{l} T (0, t) & = T_{0} [1 - exp (- a^{2} / 4 κ t)] t ≫ τ \\ T (r, t) & = T_{0} [1 - exp (- a^{2} / 4 κ t)] t ≫ τ, r^{2} / 4 κ \\ T (r, t) & = T_{0} a^{2} / 4 κ t t ≫ τ, r^{2} / 4 κ, a^{2} / 4 κ \end{array}$
GAUSSIAN POPULATION DISTRIBUTION
N(r,0) = N₀ exp(−2r²/w²)	$T (r, t) = \frac{T_{0} w^{2}}{(8 κ t + w^{2})} exp [- 2 r^{2} / (8 κ t + w^{2})]$
b. V → T relaxation during thermal conduction and/or mass diffusion
1. TOP HAT DISTRIBUTION INITIALLY	$\begin{array}{l} T (0, t) & = T_{0} [1 - exp (- t / τ)] [1 - exp (- a^{2} / 4 κ t)] τ ≪ a^{2} / 4 κ, a^{2} 4 D \\ T (r, t) & = T_{0} [1 - exp (- t / τ)] P (\frac{a}{\sqrt{2 κ t}}, \frac{r}{\sqrt{2 κ t}}) τ ⩽ a^{2} / 4 κ, a^{2} / 4 D \\ T (r, t) & = T_{0} [1 - exp (- t / τ)] (1 + r^{2} / 4 κ t) exp (- r^{2} / 4 κ t) t ≪ a^{2} / 4 κ, r < a, τ ≪ a^{2} / 4 κ \\ T (r, t) & = T_{0} [1 - exp (- t / τ)] [1 - exp (- a^{2} / 4 κ t)] t ≫ r^{2} / 4 κ \\ T (r, t) & = T_{0} a^{2} / 4 κ t t ≫ τ, r^{2} / 4 κ, a^{2} / 4 κ \end{array}$
2. No mass diffusion
GAUSSIAN DISTRIBUTION	$\begin{array}{l} T (r, t) & = \frac{T_{0} w^{2}}{(8 κ t + w^{2})} [1 - exp (- t / τ)] exp [- 2 r^{2} / (8 κ t + w^{2})] τ ≪ w^{2} / 8 κ \\ T (r, t) & = T_{0} exp {- 2 r^{2} / w^{2} [1 - exp (- t / τ)] (1 - 8 κ t / w^{2})} τ, t ≪ w^{2} / 8 κ \end{array}$
3. With mass diffusion
GAUSSIAN DISTRIBUTION	$\begin{array}{l} T (r, t) & = \frac{T_{0} w^{2}}{(8 κ t + w^{2})} [1 - exp (- t / τ)] exp (- 2 r^{2} / (8 κ t + w^{2}) & τ ≪ w^{2} / 8 κ, w^{2} / 8 D \\ T (r, t) & = T_{0} w^{2} [\frac{1}{(8 κ t + w^{2})} - \frac{exp (- t / τ)}{(8 D t + w^{2})}] t ≫ τ \begin{matrix} and / or \\ 8 κ + w^{2} ≫ r^{2} \end{matrix} & τ ≪ w^{2} / 8 κ \\ T (r, t) & = T_{0} w^{2} / (8 κ t + w^{2}) t ≫ τ, 8 κ t + w^{2} ≫ r^{2} \end{array}$

	Configuration I	Configuration II (no cell)
Available He–Ne power	1.08	0.68	mW
Signal power at detector	13	19	μW
Sample length	29	29	cm
PZT sensitivity	0.018 ± 0.001	0.028 ± 0.003	rad/V
SNR with 1-v P-P sinusoid applied to PZT	275	7900	Hz^−1/2
	(Acoustic noise lim)	(Detector noise lim)
Theoretical noise limit	2.9 × 10⁻⁷	2.4 × 10⁻⁷	rad/Hz^1/2
Phase resolution obtained	6.5 × 10⁻⁵	5.3 × 10⁻⁶	rad/Hz^1/2
Index resolution	2.2 × 10⁻¹¹	1.8 × 10⁻¹²	Hz^−1/2
I₀α resolution	4.2 ×10⁻⁶	3.5 ×10⁻⁷	W/cm³/Hz^1/2
α_min resolution	4.8 × 10⁻⁷	1.8 × 10⁻⁹	cm⁻¹/W

Abstract

Corrections

Cited By

Figures (13)

Tables (2)

Equations (154)

Applied Optics