# A Sensitive and Versatile Thickness Determination Method Based on Non-Inflection Terahertz Property Fitting

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Method

_{as}(r

_{as}) and t

_{sa}(r

_{sa}) are the transmission (reflection) coefficients at the air-sample and sample-air interfaces, respectively. ${\tilde{n}}_{s}$ and d are the complex refractive index and the thickness of the sample, respectively. c is the speed of light. We assume the PET film in Figure 1a has the n and κ shown as the black curves in Figure 1b and a thickness of d

_{s}= 100 μm. The sample-reference ratio can be calculated by the right part of the equation using these properties. We now simulate this ratio to be the experimental result in the left part of the equation, and we assume the sample properties are unknown and to be solved from the right part of Equation (1). As in this way we know the exact correct thickness, solving Equation (1) with d = 100 μm returns the sample n and κ we just used. Characterization using incorrect thicknesses of 110 and 90 μm results in the dotted blue and orange curves in Figure 1b. The 10% thickness error produces FP oscillations in both n and κ as expected. Here, we show the FP effect resulting from a wrong thickness can be sensitively recognized by the inflection points.

^{2}+ bω + c. The second derivative f’’(ω) = 2a indicates no inflection point. However, the two-order polynomial can only fit curves with a linear gradient change because f’(ω) = 2aω + b has a linear relationship with frequencies. This limits its adaptability. For example, the dielectric function of water is described by a double-Debye model, which has the first derivative obviously non-linear with frequencies [15]. Instead, an exponential function with an offset (referred to as the offset exponential function, or OE function herein), f(ω) = aexp(bω) + c, can better adapt to the profile of most materials. Any order derivative of this function is always an exponential function, which supports various curvature shapes to provide a great flexibility for the original function. The second derivative, which is also an exponential function, has no solution for f’’(ω) = 0 to prevent any inflection point. The major limitation of an OE function is its monotonic behavior (its first derivative is either always positive or always negative). However, from the wide range characterizations among crystals dielectrics and semiconductors [5], metal oxide [16], glass [7], polymer [6,8], polar liquids [4], and biomedical samples [17], we can see most of the materials have n and κ monotonously varied with frequencies in the THz range. In the very few cases that the monotonic condition is not satisfied, we can replace the constant offset term to a one-order polynomial, that is f(ω) = aexp(bω) + cx + d. This allows the first derivative to reach zero for fitting a non-monotonous property.

## 3. Results

#### 3.1. PET Thin Films

#### 3.2. Thin-Film Water

_{1}=9.4 ps, ω

_{s}/2π = 5.3 THz, and γ

_{s}/2π = 5.35 THz from Yada’s work [23]. The other parameters were extracted from the best fitting, with ε

_{∞}= 2.5 (2.0, 2.5), Δε

_{1}= 77.9 (74.9, 74.9), Δε

_{2}= 1.94 (1.67, 2.8), τ

_{2}= 0.299 ps (0.25 ps, 0.3 ps), and A

_{s}= 1486 THz (1244 THz, 1500 THz). The values in the brackets are from [23] and [24], respectively. These results give a reasonable physical description and are in good agreement with the literature. The fitting to the real and imaginary part of the permittivity was plotted in Figure 4d, showing a high-degree match to the experimental result. The successful double-Debye modeling verifies the accurate thickness optimized from the OE fitting from the physical point of view.

#### 3.3. Thin-Film Lactose Pallet

## 4. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Figure A1.**The NorFitErr (Normalized Fitness Error) as a function of thickness. The minimum values correspond to the optimized thickness found from different techniques. The results of the PET thin film with a reference thickness of (

**a**) 188 μm, (

**b**) 125 μm, and (

**c**) 50 μm, and the results of the (

**d**) water and (

**e**) lactose thin film by the TV (Total Variation), QS (Quasi-Sapce), and OE fitting methods are shown for a comparison.

## Appendix B

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**Figure 1.**(

**a**) Schematic of the transmission measurement of a PET (polyethylene terephthalate) thin film; (

**b**) Extracted n and κ and (

**c**) their second order derivatives when using the thickness of 100, 110, and 90 μm.

**Figure 2.**The normalized fitness (NormFitness) of the OE (offset exponential) functions to the extracted n (refractive index) and κ (extinction coefficient) for the (

**a**) 188 μm, (

**b**) 125 μm, and (

**c**) 50 μm PET films. The experimental results of n for the (

**d**) 188 μm, (

**e**) 125 μm, and (

**f**) 50 μm PET films and κ for the (

**g**) 188 μm, (

**h**) 125 μm, and (

**i**) 50 μm PET films at their optimized thicknesses shown in the blue dots, with the red solid curves indicating the OE fitting.

**Figure 3.**n and κ of the 188 μm (PET1), 125 μm (PET2), and 50 μm (PET3) PET films at their optimized thicknesses (solid curves), compared to the n and κ of PET3 characterized by using the optimized thickness with a ±3 μm error.

**Figure 4.**(

**a**) Schematic of the water thin film measurement. (

**b**) NormFitness as a function of thickness ranging for 94–114 μm. (

**c**) n and κ of water extracted from the optimized thickness (open circles) and the fitting to the OE functions (solid curves). (

**d**) Complex permittivity of water at the optimized thickness (open circles) and the fitting to the double-Debye model.

**Figure 5.**(

**a**) NormFitness as a function of thickness for the lactose pallet. (

**b**) The extracted refractive index and the optimized thickness and the fitting to the OE functions in the range 0.65–1.1 THz and 1.95–2.35 THz. (

**c**) n and (

**d**) α of the lactose pallet extracted by using the optimized thickness (black solid curve) and the optimized thickness with a ±15 μm error (dotted curves).

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**MDPI and ACS Style**

Chen, X.; Pickwell-MacPherson, E.
A Sensitive and Versatile Thickness Determination Method Based on Non-Inflection Terahertz Property Fitting. *Sensors* **2019**, *19*, 4118.
https://doi.org/10.3390/s19194118

**AMA Style**

Chen X, Pickwell-MacPherson E.
A Sensitive and Versatile Thickness Determination Method Based on Non-Inflection Terahertz Property Fitting. *Sensors*. 2019; 19(19):4118.
https://doi.org/10.3390/s19194118

**Chicago/Turabian Style**

Chen, Xuequan, and Emma Pickwell-MacPherson.
2019. "A Sensitive and Versatile Thickness Determination Method Based on Non-Inflection Terahertz Property Fitting" *Sensors* 19, no. 19: 4118.
https://doi.org/10.3390/s19194118