Photothermal spectroscopy of scattering media

Zafer A. Yasa; Warren B. Jackson; Nabil M. Amer

doi:10.1364/AO.21.000021

Applied Optics
Vol. 21,
Issue 1,
pp. 21-31
(1982)
•https://doi.org/10.1364/AO.21.000021

Photothermal spectroscopy of scattering media

Zafer A. Yasa, Warren B. Jackson, and Nabil M. Amer

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Zafer A. Yasa, Warren B. Jackson, and Nabil M. Amer, "Photothermal spectroscopy of scattering media," Appl. Opt. 21, 21-31 (1982)

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Abstract

We present a general unified theoretical analysis of the role of scattering in photothermal spectroscopy, e.g., photoacoustic and photothermal deflection. We show that while the photothermal signal is significantly affected in the case of highly scattering media, it is independent of scattering for optically thin samples. Numerical estimates of the scattering contribution and comparison with experimental results are given. We also elucidate the relationship between photothermal and diffuse reflectance spectroscopies.

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Method	T_r(ω) (transducer sensitivity)	T_t(ω)	G(z,ρ,ϕ)	Description of parameters (f, fluid; s, sample)
Sample–fluid–transducer (SFT)	V/dyn/cm²	$\frac{β_{f}}{V_{e} K_{T f}}$	$\frac{δ (z)}{k_{f}} + \frac{β_{s}}{β_{f}}$	K_T = isothermal compressibility
				β = coefficient of thermal expansion
				k = (1 + i)(ωρC_p/2κ)^1/2
				ω = modulation frequency
				ρC_p = heat capacity/unit volume at constant pressure
				V_e = effective cell volume = V_f + V_sK_Ts/K_Tf
				ρ = density
				κ = thermal conductivity

Fluid-transducer (FT)	V/dyn/cm²	$\frac{β}{K_{T} V}$	1	a = cell radius
(a) nonresonant		$\frac{i 2 β}{Q_{j} π {V K}_{T} J_{0} (π α_{j})}$	$J_{0} (\frac{π}{a} α_{j} ρ) \to 1$	$\frac{d}{d_{α}} J_{0} (π α) ∣_{α = α_{j}} = 0$
(b) resonant				Q_j=cavity Q(jth mode)

Solid–transducer (ST)	V/unit strain	β(1 + ν)	$1 \pm 3 (1 - \frac{2 z}{l})$	l = sample length
Solid–transducer (ST)	V/unit strain	β(1 + ν)	$1 \pm 3 (1 - \frac{2 z}{l})$	ν = Poisson’s ratio

Transverse photothermal deflection (PDS)	V/rad	$\frac{1}{n_{0}} \frac{d n}{d T} \sqrt{\frac{2}{π}} \times k_{f}^{2} \frac{exp (k_{f} z_{0})}{w_{0}}$	$\frac{\partial (z)}{k_{f}}$	w_o = pump beam radius
Transverse photothermal deflection (PDS)	V/rad		$\frac{\partial (z)}{k_{f}}$	$\frac{1}{n_{0}} \frac{d n}{d T} = relative change of refractive index with temperature$

Parameter	Effective parameter	Remarks
K_T	K_eΩ	Ω = volume fraction of fluid (porosity) Accounts for thermal and mechanical properties of the fluid-sample system
β	β_eΩ
κ	κ_e
ρC_p	(ρC_p)_e

Method	T_r(ω) (transducer sensitivity)	T_t(ω)	G(z,ρ,ϕ)	Description of parameters (f, fluid; s, sample)
Sample–fluid–transducer (SFT)	V/dyn/cm²	$\frac{β_{f}}{V_{e} K_{T f}}$	$\frac{δ (z)}{k_{f}} + \frac{β_{s}}{β_{f}}$	K_T = isothermal compressibility
				β = coefficient of thermal expansion
				k = (1 + i)(ωρC_p/2κ)^1/2
				ω = modulation frequency
				ρC_p = heat capacity/unit volume at constant pressure
				V_e = effective cell volume = V_f + V_sK_Ts/K_Tf
				ρ = density
				κ = thermal conductivity

Fluid-transducer (FT)	V/dyn/cm²	$\frac{β}{K_{T} V}$	1	a = cell radius
(a) nonresonant		$\frac{i 2 β}{Q_{j} π {V K}_{T} J_{0} (π α_{j})}$	$J_{0} (\frac{π}{a} α_{j} ρ) \to 1$	$\frac{d}{d_{α}} J_{0} (π α) ∣_{α = α_{j}} = 0$
(b) resonant				Q_j=cavity Q(jth mode)

Solid–transducer (ST)	V/unit strain	β(1 + ν)	$1 \pm 3 (1 - \frac{2 z}{l})$	l = sample length
Solid–transducer (ST)	V/unit strain	β(1 + ν)	$1 \pm 3 (1 - \frac{2 z}{l})$	ν = Poisson’s ratio

Transverse photothermal deflection (PDS)	V/rad	$\frac{1}{n_{0}} \frac{d n}{d T} \sqrt{\frac{2}{π}} \times k_{f}^{2} \frac{exp (k_{f} z_{0})}{w_{0}}$	$\frac{\partial (z)}{k_{f}}$	w_o = pump beam radius
Transverse photothermal deflection (PDS)	V/rad		$\frac{\partial (z)}{k_{f}}$	$\frac{1}{n_{0}} \frac{d n}{d T} = relative change of refractive index with temperature$

Parameter	Effective parameter	Remarks
K_T	K_eΩ	Ω = volume fraction of fluid (porosity) Accounts for thermal and mechanical properties of the fluid-sample system
β	β_eΩ
κ	κ_e
ρC_p	(ρC_p)_e

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Equations (45)

Applied Optics