Model for continuous-wave laser-induced thermal lens spectrometry of optically transparent surface-absorbing solids

Dmitry A. Nedosekin; Mikhail A. Proskurnin; Mikhail Yu. Kononets

doi:10.1364/AO.44.006296

Applied Optics
Vol. 44,
Issue 29,
pp. 6296-6306
(2005)
•https://doi.org/10.1364/AO.44.006296

Model for continuous-wave laser-induced thermal lens spectrometry of optically transparent surface-absorbing solids

Dmitry A. Nedosekin, Mikhail A. Proskurnin, and Mikhail Yu. Kononets

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Dmitry A. Nedosekin, Mikhail A. Proskurnin, and Mikhail Yu. Kononets, "Model for continuous-wave laser-induced thermal lens spectrometry of optically transparent surface-absorbing solids," Appl. Opt. 44, 6296-6306 (2005)

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Abstract

A theoretical model for cw laser-induced thermal lens spectrometry of optically transparent surface-absorbing solids is developed. In the model, the sample is represented as a set of discrete layers with certain thicknesses and light absorptivities. The bloomed thermo-optical element in the sample is described with a summation of heat-flux functions for all the layers. The model employs simple mathematical expressions and can be used for both steady-state and time-resolved thermal lens experiments. Good coincidence of the experimental and theoretically predicted signal dependences is achieved. This model is verified for volume-absorbing samples (colored optical glasses) and used successfully to calculate absorbances and concentrations for various surface-absorbing samples.

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Constituents of the Model	Solution of Differential Thermal-Diffusion Equations^a	Mirage Spectroscopy Model for Layered Solids^b
Sample
Optically homogeneous samples	Exact solution in analytical form Eq. (20)	Description of the sample with any light absorption structure
Optically inhomogeneous samples and layered solids	Numerical estimation
Boundary conditions. Heat transfer to the coupling medium.	Exact solution for homogeneous samples^c and numerical estimation for layered solids	Use of special amends in heat-transfer equations for border layers
Heating–cooling cycle. Duration.	Any from 0 up to infinity	Limited due to use of instant energy-transfer functions (from 0 up to 0.5 ms)

Spectrometer Unit	Parameter	Value
Excitation laser	λ_e (nm)	488.0 and 514.5
	Focusing lens focal length, f (mm)	300
	Confocal distance, z_0,_e (mm)	6.05 and 6.38
	Spot size at the waist, ω_e₀ (μm)	61.3 and 64.6
	Laser power at the sample, P_e (mW)	50–475 and 75–500
Probe laser	λ_p (nm)	632.8
	Focusing lens focal length, f (mm)	185
	Confocal distance, z_0,_p (mm)	0.90
	Laser power at the sample, P_p (mW)	5
	Spot size in the sample, ω₁ (μm)	109 ± 1
	Distance between the waist of the probe beam and the sample, z_w_,_p (mm)	3.53
	Spot size at the waist, ω_0,_p (μm)	27 ± 1
Other constants	Optical path in the sample (mm)	1–3
	z_d (cm)	180
	Spectrometer time resolution (ms)	36
	Chopper frequency (Hz)	2.00 ± 0.01

Parameter	Value
Density, ρ (g cm⁻³)	2.38
Thermal-diffusion coefficient [measured experimentally with Eqs. (5) and (6) (n = 16, P = 0.95)], α (m² s⁻¹)	(4.3 ± 0.2) × 10⁻⁸
Characteristic time of the thermal lens in colored optical glass, t_c(s)	22.5 ± 0.5
Thermal conductivity, k (W m⁻¹ K⁻¹)	1.25
Isobaric specific heat, C_p (J g⁻¹ K⁻¹)	858.0
Thermal expansion coefficient, γ_P (K⁻¹)	1.04 × 10⁻⁵
Refractive index, n₀	1.53

Specimen	A (λ_e = 488 nm)	β₄₈₈, m⁻¹	n₀	dn/dT Eq. (16)
ZhS-10	0.064 ± 0.001	9.26	1.523	30.63 ± 10⁻⁵
ZhS-11	0.072 ± 0.001	10.42	1.523	30.63 ± 10⁻⁵
ZhS-12	0.075 ± 0.001	10.85	1.523	30.63 ± 10⁻⁵
ZhS-16	0.115 ± 0.001	16.64	1.523	30.63 ± 10⁻⁵
ZhS-17	0.28 ± 0.01	41.39	1.523	30.63 ± 10⁻⁵
ZhS-20	0.97 ± 0.01	140.97	1.540	30.67 ± 10⁻⁵
ZhS-18	1.81 ± 0.01	262.41	1.523	30.63 ± 10⁻⁵

			Surface Concentration of the Absorbing Substance (mol/cm²)

Sample	Average Steady-State Thermal Lens Signal	Calculated Absorbance, Eqs. (5)–(18)	Calculated	Anticipated	Reference
Ion aggregates adsorbed of chromate ions and 2,10-ionene on the quartz glass	−(2 ± 1) × 10⁻³	6 × 10⁻⁵	(7 ± 2) × 10⁻⁹	1.5 × 10⁻⁹^a	32
Complex of 4-(2-piridylazo) resorcinol with chemisorbed vanadium (V) on the glass	−(6 ± 2) × 10⁻³	3 × 10⁻⁴	(8 ± 1) × 10⁻¹¹	1 × 10⁻¹⁰^b	30
Reactivrot 5A dye grafted to the glass surface	−(4 ± 1) × 10⁻³	2 × 10⁻⁴	(2.2 ± 0.2) × 10⁻¹¹	1.6 × 10⁻¹⁰^b	29

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Applied Optics