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Investigation of aqueous alcohol and sugar solutions with reflection terahertz time-domain spectroscopy

Peter Uhd Jepsen, Uffe Møller, and Hannes Merbold

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Abstract

We give a detailed analysis of a general realization of reflection terahertz time-domain spectroscopy. The method is self-referenced and applicable at all incidence angles and for all polarizations of the incident terahertz radiation. Hence it is a general method for the determination of the dielectric properties of especially liquids in environments where transmission measurements are difficult. We investigate the dielectric properties in the 0.1–1.0 THz frequency range of liquids using reflection terahertz time-domain spectroscopy. We apply the technique for the determination of alcohol and sugar concentration of commercial alcoholic beverages and liquors. The special geometry of the experiment allows measurement on sparkling beverages.

©2007 Optical Society of America

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Figures (14)
Fig. 1.
Fig. 1. Beam path of the reflection spectrometer. The gray rectangles indicate the incidence planes of the four off-axis paraboloidal mirrors. The sample window is elevated from the main plane of the spectrometer by rotation of the incidence planes before and after the sample. The vectors shown in blue color illustrate the polarization of an initially vertically polarized THz field through the spectrometer.

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Fig. 2.
Fig. 2. Detailed view of the reflection geometry at the window which splits the input THz signal into a reference part and a sample part.

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Fig. 3.
Fig. 3. The THz signal trace recorded in a spectrometer with vertical input polarization (α = 90°) and elevation angle θ = 60°. The reference signal is the reflection from the air-silicon interface, and the sample signal is the reflection from the opposite interface between silicon and the sample material, in this case air.

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Fig. 4.
Fig. 4. (a) Amplitude ratio and (b) phase difference between the sample- and reference signal in Fig. 3. The phase difference has been corrected for the accumulated phase due to propagation through the window material. The thin, purple curves show three individual measurements, and the thick, blue curves show the average of these. The dashed lines indicate the theoretical amplitude ratio and phase difference.

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Fig. 5.
Fig. 5. Amplitude A(ν) and phase Δ(ν) of the calibration factor of our spectrometer with θ = 60° and α = 90° (vertical input polarization). The thin, purple curves show three individual measurements, and the thick, blue curves show the average of these.

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Fig. 6.
Fig. 6. The apparent (a) real and (b) imaginary part of the dielectric function of air, as measured with the reflection spectrometer. The error bars indicate the standard deviation of the average value of 5 sequential measurements.

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Fig. 7.
Fig. 7. The real (red symbols) and imaginary (blue symbols) part of the dielectric function of (a) water and (b) ethanol recorded at a temperature of 20°C. The fitted curves are described in the text.

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Fig. 8.
Fig. 8. The (a) real and (b) imaginary part of the dielectric function at room temperature of water-ethanol mixtures in the 0.1–1.1 THz range. The alcohol concentration is varied from 0% to 100% in 5%-steps.

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Fig. 9.
Fig. 9. The average value of the real and imaginary part of the dielectric function of water-ethanol mixtures in the 0.15–1.0 THz range, as function of the ethanol concentration. Solid curves are phenomenological fits to the experimental data, and the dashed lines show the ideal behavior of the mixtures.

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Fig. 10.
Fig. 10. The correlation between the known and the predicted concentration of ethanol in mixtures of deionized water and ethanol (black symbols) and commercial alcoholic beverages and liquors (blue, star-shaped symbols).

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Fig. 11.
Fig. 11. The real and imaginary part of the dielectric function at room temperature of (a) deionized water, carbonated mineral water, and mineral water with gas removed, and (b) deionized water, a typical carbonated softdrink, and the same softdrink with the gas removed.

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Fig. 12.
Fig. 12. The real and imaginary part of the dielectric function at room temperature of aqueous sucrose solutions, with sucrose concentrations from 0% to 75% by weight, measured in 5%-intervals.

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Fig. 13.
Fig. 13. The average value of the real and imaginary part of the dielectric function in the frequency interval 0.15–1.0 THz of aqueous sucrose solutions as function of the sucrose concentration. The solid curves are phenomenological fits, as discussed in the text.

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Fig. 14.
Fig. 14. The real and imaginary part of the dielectric function of pure water, a 20% sucrose solution, a 20% ethanol solution, and two solutions containing 10 and 20% sucrose and 20% ethanol.

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Tables (1)

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Table 1. Comparison with existing literature of Debye model parameters for water and ethanol. Values marked with * have been kept fixed in the fitting process. The reported uncertainties are the standard deviations of the estimated parameter values.

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Equations (30)

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( E ∥ ′ E ⊥ ′ ) = ( − r ∥ 0 0 r ⊥ ) ( E ∥ E ⊥ ) ,
M ( θ ) = ( cos ⁡ θ sin ⁡ θ − sin ⁡ θ cos ⁡ θ ) ,
E → out ref = 1 2 E in ( ( r 12 ∥ − r 12 ⊥ ) cos ⁡ α − ( r 12 ∥ + r 12 ⊥ ) cos ⁡ ( α − 2 θ ) ( r 12 ∥ − r 12 ⊥ ) sin ⁡ α + ( r 12 ∥ + r 12 ⊥ ) sin ⁡ ( α − 2 θ ) ) ,
E → out sam = 1 2 E in A cal exp ⁡ ( i Δ cal ) exp ⁡ ( 2 i n 2 ω d eff c ) × ( ( r ̂ 23 ∥ t 12 ∥ t 21 ∥ − r ̂ 23 ⊥ t 12 ⊥ t 21 ⊥ ) cos ⁡ α − ( r ̂ 23 ∥ t 12 ∥ t 21 ∥ + r ̂ 23 ⊥ t 12 ⊥ t 21 ⊥ ) cos ⁡ ( α − 2 θ ) ( r ̂ 23 ∥ t 12 ∥ t 21 ∥ − r ̂ 23 ⊥ t 12 ⊥ t 21 ⊥ ) sin ⁡ α + ( r ̂ 23 ∥ t 12 ∥ t 21 ∥ + r ̂ 23 ⊥ t 12 ⊥ t 21 ⊥ ) sin ⁡ ( α − 2 θ ) ) .
OP = 2 n 2 d Si cos ⁡ ϕ ′ − d air = 2 n 2 d Si cos ⁡ ϕ ′ ( 1 − sin ⁡ 2 ϕ ′ ) = 2 n s d Si ( 1 − sin ⁡ 2 ϕ n s 2 ) .
d eff = d Si ( 1 − sin ⁡ 2 ϕ n s 2 ) .
E out , x sam E out , x ref = r ̂ 23 , sam ∥ t 12 ∥ t 21 ∥ + r ̂ 23 , sam ⊥ t 12 ⊥ t 21 ⊥ r 12 ∥ + r 12 ⊥ A cal exp ⁡ ( i Δ cal ) exp ⁡ ( 2 i n 2 ω d eff c )
E out , y sam E out , y ref = r ̂ 23 , sam ∥ t 12 ∥ t 21 ∥ − 3 r ̂ 23 , sam ⊥ t 12 ⊥ t 21 ⊥ r 12 ∥ − 3 r 12 ⊥ A cal exp ⁡ ( i Δ cal ) exp ⁡ ( 2 i n 2 ω d eff c ) .
A cal exp ⁡ ( i Δ cal ) = E out air E out ref r 12 ∥ + r 12 ⊥ r 23 , air ∥ t 12 ∥ t 21 ∥ − r 23 , air ⊥ t 12 ⊥ t 21 ⊥ exp ⁡ ( 2 i n 2 ω d eff c )
A sam exp ⁡ ( i Δ sam ) ≡ E out sam E out ref = r ̂ 23 , sam ∥ t 12 ∥ t 21 ∥ + r ̂ 23 , sam ⊥ t 12 ⊥ t 21 ⊥ r 23 , air ∥ t 12 ∥ t 21 ∥ + r 23 , air ⊥ t 12 ⊥ t 21 ⊥ E out air E out ref
r 12 ∥ = − n 2 2 cos ⁡ ϕ + n 2 2 − sin ⁡ 2 ϕ n 2 2 cos ⁡ ϕ + n 2 2 − sin ⁡ 2 ϕ ,
r 12 ⊥ = cos ⁡ ϕ − n 2 2 − sin ⁡ 2 ϕ cos ⁡ ϕ + n 2 2 − sin ⁡ 2 ϕ ,
t 12 ∥ = 2 cos ⁡ ϕ n 2 cos ⁡ ϕ + 1 − sin ⁡ 2 ϕ n 2 2 ,
t 12 ⊥ = 2 cos ⁡ ϕ cos ⁡ ϕ + n 2 1 − sin ⁡ 2 ϕ n 2 2 ,
t 21 ∥ = 2 n 2 1 − sin ⁡ 2 ϕ n 2 2 n 2 cos ⁡ ϕ + 1 − sin ⁡ 2 ϕ n 2 2 ,
t 21 ⊥ = 2 n 2 1 − sin ⁡ 2 ϕ n 2 2 cos ⁡ ϕ + n 2 1 − sin ⁡ 2 ϕ n 2 2 ,
r ̂ 23 ∥ = − n ̂ 3 2 n 2 2 − sin ⁡ 2 ϕ + n 2 2 n ̂ 3 2 − sin ⁡ 2 ϕ n ̂ 3 2 n 2 2 − sin ⁡ 2 ϕ + n 2 2 n ̂ 3 2 − sin ⁡ 2 ϕ ,
r ̂ 23 ⊥ = n 2 2 − sin ⁡ 2 ϕ − n ̂ 3 2 − sin ⁡ 2 ϕ n 2 2 − sin ⁡ 2 ϕ + n ̂ 3 2 − sin ⁡ 2 ϕ .
A sam ⊥ exp ⁡ ( i Δ sam ⊥ ) = r ̂ 23 , sam ⊥ r ̂ 23 , air ⊥ E out air E out ref .
A sam ∥ exp ⁡ ( i Δ sam ∥ ) = r ̂ 23 , sam ∥ r ̂ 23 , air ∥ E out air E out ref .
n ̂ sam = ( 1 − r ̂ 23 , sam ) 2 n 2 2 + 4 r ̂ 23 , sam sin 2 ϕ 1 + r ̂ 23 , sam ,
ε ( ω ) = ε s − ε 1 1 − iω τ 1 + ε 1 − ε ∞ 1 − iω τ 2 + ε ∞ .
ε ( ω ) = ε s − ε 1 1 − iω τ 1 + ε 1 − ε 2 1 − iω τ 2 + ε 2 − ε ∞ 1 − iω τ 3 + ε ∞ ,
ε′ ¯ ( x ) = A′ + B′ exp ⁡ ( − C′ x ) ,
ε″ ¯ ( x ) = A″ + B″ exp ⁡ ( − C″ x ) ,
n ̂ ideal ( ν ) = c H 2 o mix c H 2 o neat n ̂ H 2 o ( ν ) + c EtOH mix c EtOH neat n ̂ EtOH ( ν ) ,
ε′ ( y ) = { D′ , y < 40 % − E′ y + F′ , y ≥ 40 %
ε ″ ( y ) = − E ″ y + F ″
ε′ ( x , y ) = A′ + B′ exp ⁡ ( − C′ x )
ε″ ( x , y ) = A″ + B″ exp ⁡ ( − C″ x ) − E″ y
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