# The Theoretical Concept of Polarization Reflectometric Interference Spectroscopy (PRIFS): An Optical Method to Monitor Molecule Adsorption and Nanoparticle Adhesion on the Surface of Thin Films

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. The Reflectometric Interference Spectroscopy Principle

_{2}> n

_{1}, and the resultant of the double phase shift is zero. The reflection and transmission amplitude parameters in Equation (2) can be calculated by the following equations:

#### 2.2. The Polarization Reflectometric Interference Spectroscopy Principle

_{01}, r

_{12}, t

_{01}and t

_{12}amplitudes are calculated in different ways depending on the polarization state (s: senkrecht and p: parallel) [34].

_{0}> 0°), otherwise the reflection amplitudes in Equations (7) and (8) become equal to each other and to the coefficient in Equation (4) and the polarization effect ceases (see Figure A1b). These reflection amplitude values are included in the I

_{R}equations, similarly to in Equation (7):

_{1}= 1.38, respectively, using a substrate with n

_{2}= 1.7, in air, ε

_{0}= 35°).

## 3. Results and Discussion

_{R}) depends on several explicit and implicit parameters which are included in the components of the summation. While the explicit parameters are the film thickness (d), the refraction angle (ε

_{1}) and the refractive index of the film (the sensing layer) (n

_{1}), the implicit parameters are included in the reflection and transmission amplitudes (in conventional and polarized cases, Equations (4)–(6), (8)–(11), respectively), namely the refractive indices of the medium (n

_{0}), the film (n

_{1}) and the substrate (n

_{2}). The dependence of the reflected intensity on these parameters are demonstrated in Section 3.1, Section 3.2 and Section 3.3, while two simulated, time-resolved adsorption experiments are presented in Section 3.4 and Section 3.5.

_{s}and R

_{p}curves are hardly different (see Figure 2A,B) (except near the Brewster angle, where R

_{p}≈ 0, see Figure A1, Figure A2, Figure A3 and Figure A4 in Appendix A), only the R

_{S}and R

_{P}curves are shown in Figure 3, Figure 4 and Figure 5 for the sake of visual clarity. For similar reasons, the refractive indices of the thin film and the substrates are indicated as constant values (average value in the λ = 400–800 nm wavelength range); however, in the calculations, the characteristic n

_{2}(λ) (dispersion) functions are used (fused silica [35] and SF10 [36] glasses). Assuming that the thin film materials are weakly or non-absorbent in the investigated (visible) wavelength range (e.g., ZnO, TiO

_{2}), the refractive indices of the thin films are considered to be real. In the case of absorbing materials, the refractive index should be extended to the complex value (η), introducing k as the extinction coefficient (η = n + ik). On the other hand, it should be noted that the low effective refractive index (RI) value (1.1 < n

_{1}< 1.6) is a typical feature in the case of polymer or porous nanostructured thin films in sensorial applications, while the higher RI values are characteristic to sputtered layers, for example in antireflective and low-E coatings.

#### 3.1. The Film Thickness (d)

_{1}= 1.21 and different film thicknesses, d = 110, 135 and 160 nm (Figure 3A,B, Figure 3C,D and Figure 3E,F, respectively).

_{0}= 30°, the RI of the medium is n

_{0}= 1 (gas phase) and the model substrate is fused silica glass (n

_{2}= 1.46), since these are typical gas phase VOC sensor conditions [6], except the low layer thickness which is much lower than the suitable value in RIfS technique. It can be observed in Figure 3A,C,E that the interference pattern barely appears because of the low sensing layer thickness, compared to Figure 2B (d = 800 nm). In contrast, when the ratio of the s- and p-polarized components is measured, the obtained reflectance curve has a definite extreme value (edge). These curves (corresponding to different layer thicknesses) have the similar shape, contrast and FWHM (full width at half maximum) with the increase in layer thickness d, while only the edge shifts toward the higher values (red shift) (see Figure 3B,D,F). The wavelength shift, and thereby the wavelength of the edge can be precisely tuned: the scaling factor is λ/d = 4.5 nm/nm.

#### 3.2. The Refractive Index of the Medium (n_{0}), the Thin Film (n_{1}) and the Substrate (n_{2})

_{0}= 1), the optimal thin film refractive index is n

_{1}= 1.21 when fused silica substrate (n

_{2}= 1.46) is used (Figure 4A,B). Similarly, n

_{1}= 1.32 or n

_{1}= 1.52 thin film refractive indices are the optimal values in the case of SF10 substrate in air (Figure 4C,D) or in aqueous phase (Figure 4E,F), respectively. Furthermore, it can be observed that the contrast and the FWHM of the curves are determined by the refractive index difference between two adjacent media (n

_{i}and n

_{j}in Equation (4), which can be n

_{0}and n

_{1}, or n

_{1}and n

_{2}, as the refractive indices of the medium and the thin film, or the thin film and the substrate, respectively). It can be established that the higher refractive index difference results in greater contrast (ΔR = 0.55) and FWHM (Δλ = 100 nm), when the R

_{S}and R

_{P}curves are almost parallel in the region of interest (see Figure 4C, zoom), compared to the lower RI difference with diverging R

_{S}and R

_{P}curves (ΔR = 0.4 and Δλ = 70 nm, Figure 4B,D). The further advantage of the proper RI matching, i.e., the geometric mean, is that the Brewster angle (Θ

_{B}) will be the same at the medium/film and film/substrate interfaces [Θ

_{B}= arctan(n

_{1}/n

_{0}) = arctan(n

_{2}/n

_{1})]. In Section 3.4 and Section 3.5, these optimal conditions will be examined in simulated absorption experiments: for sensorial measurements in gas phase, thin films with low refractive index are applicable, while in aqueous phase a relatively high (n

_{1}~1.5) RI is required. The RI increase can be achieved by applying materials with higher refractive index (e.g., SiN, TiO

_{2}) or by decreasing the porosity of the sensing layer.

#### 3.3. The Angle of Incidence (ε_{0})

_{0}, n

_{1}, n

_{2}values were investigated. This section focuses on another important (certainly the most important) parameter, namely, the angle of incidence. The selected parameters are n

_{0}= 1, n

_{1}= 1.21, n

_{2}= 1.46 and d = 110 nm, while the angles are ε

_{0}= 35, 41.5 and 50°. The results are presented in Figure 5. At the interface of the medium and the thin film the Brewster angle is Θ

_{B}= 50.4° with these conditions (see Figure 5A,B and Figure A1), where the parallel component of the reflectance is zero and near this angle the PRIfS curve has a maximum extreme value (Figure 5G,H). By decreasing the incident angle to 35°, where the R

_{p}/R

_{s}ratio is 0.2, the edge shows a red shift, as expected, and the curve has a minimum extreme value (Figure 5C,D). Between these two angles (around ε

_{0}= 41°), there is a transition region where the contrast of the curve is extremely unfavorable (Figure 5F) and the transition between the minimum and maximum can be observed by zooming (Figure 5F inset). This region is important for two reasons: (1) sensorial applications based on polarized reflectometric interference spectroscopy cannot be carried out in this wavelength range; (2) the optical responses (ΔI and Δλ) of PRIfS technique are 4–7 times higher near this domain, depending on the type of the medium (gas or liquid). Both statements will be verified in the next subsections. These observations and findings are the same in an aqueous medium, with the exception that the Brewster angle is Θ

_{B,AQ}= 48.8°.

#### 3.4. Simulating an Immobilization Measurement in Aqueous Phase

_{0}= 1.333 (aq. medium), d = 110 nm, n

_{1}= 1.52, n

_{2}= 1.73 and ε

_{0}= 42.2° (Figure 6).

_{0}(near the surface of the sensing layer) is increasing from 1.333 to 1.3333 (Δn

_{0}= 0.0003); 2. the adsorption/immobilization process occurs, so n

_{1}increases (from 1.52 to 1.526, Δn

_{1}= 0.006), indicating the (permanent) presence of the surface excess; 3. the measurement cell is rinsed by pure solvent, n

_{0}decreases from 1.3333 to 1.333, only the immobilized molecules remain.

_{1}= 0.006, the wavelength shift of the conventional technique is much lower than the Δλ values of the polarized method, as well as the intensity conditions of the latter one being more favorable. However, it can be observed that the contrast of the PRIfS curve is significantly decreased during the process.

_{RIfS}= 3.4 nm, while Δλ

_{PRIfS}= 23.8 nm, which is a seven-fold difference. The calculated sensitivities for RIfS and PRIfS methods are 566 and 3966 nm/RIU (refractive index unit), respectively.

_{0}= 42.2°. Figure 8 shows the ΔI and the Δλ values as the function of ε

_{0}corresponding to step#9 when Δn

_{1}= 0.006 and Δn

_{0}= 0.0003. The similar efficiency of the two methods can be observed at low angle of incidence values and near the Brewster angle (Figure 8A), as well as, the significant improvement of the PRIfS technique near the transition region (see Figure 5F) and the two orders of magnitude difference in ΔI values (Figure 8B). Δλ values in the transition domain are unrealistically high (~100 nm); thus, the simulations cannot be carried out. However, it should be noted that the model experiments are running by assuming ideal attributes, but the real laboratory conditions and the quality of the prepared thin films (such as surface roughness, layer thickness and refractive index inhomogeneity, spot size of the light, etc.) may reduce the ideal nature of the curves. The simulated Δλ sensorgrams corresponding to ε

_{0}= 42.2, 43 and 45° incident angles (near the transition region) are presented in Figure 9A,B (RIfS and PRIfS, respectively).

#### 3.5. Simulating an Absorption Measurement in Gas Phase

_{0}= 1 (air), d = 110 nm, n

_{1}= 1.21, n

_{2}= 1.46 and ε

_{0}= 44°, as presented in Figure 10. Similar to the aqueous medium, the comparison of RIfS and PRIfS curves revealed that the wavelength shift of the conventional technique is much lower than the Δλ values of the polarized method. In addition, it can be observed, that the contrast and the FWHM of the PRIfS curve was not changed significantly during the simulated adsorption process compared to the test carried out in the aqueous phase. The resulting peak values were Δλ

_{RIfS}= 2.2 nm and Δλ

_{PRIfS}= 7.3 nm (see Figure 11C); therefore, the sensitivities were 550 and 1825 nm/RIU for the conventional and the polarized method, respectively, which is almost a four-fold difference. Furthermore, there is a three orders of magnitude difference between the two methods in the ΔI values (see Figure 11A).

_{2}) flow, n

_{0}(near the surface of the sensing layer) increases from 1 to 1.0002 (Δn

_{0}= 0.0002); 2. the adsorption process occurs, so n

_{1}increases (from 1.21 to 1.214, Δn

_{1}= 0.004) because of the reversible physisorbed test molecules; 3. the measurement cell is rinsed by the pure carrier gas, n

_{0}decreases from 1.0002 to 1; while 4. the weakly physisorbed test molecules leave the surface, and the pores (n

_{1}decreases from 1.214 to 1.21), finally the original state is restored, and there is no permanent surface excess.

_{0}= 44 and 45°. The results are similar to the aqueous phase experiments: the PRIfS method is significantly more efficient than the conventional technique near the transition domain (see Figure 12A,B), while at low incident angle values and near the Brewster angle, the efficiency of the two methods is comparable (see Figure 12C).

## 4. Conclusions

^{−3}nm, a sensitivity of 10

^{−5}–10

^{−6}RIU can be achieved by the polarization reflectometric interference technique, which is comparable with the sensitivity of the spectroscopic ellipsometry and surface plasmon resonance methods.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Figure A1.**(

**A**) The calculated (s- and p-polarized) reflectance curves, (

**B**) their zoom into R

_{01}= 0–0.1 range and (

**C**) their ratio as the function of the incident angle (ε

_{0}) for a gas/thin film interface with n

_{0}= 1 and n

_{1}= 1.21; (

**D**) presents the used thin film model.

**Figure A2.**The calculated reflectance curves from a thin film with a thickness d = 110 nm and an effective refractive index n

_{1}= 1.21 (using fused silica substrate with n

_{2}= 1.46, in air, ε

_{0}= 35°): (

**A**) The conventional reflectance (RIfS) curve, calculated by Equation (7); (

**B**) The s- and p-polarized components of (

**A**), calculated by Equations (12) and (13) and (

**C**) the polarized reflectance curve, which is defined as the ratio of s- and p-polarized components, calculated by Equation (14).

**Figure A3.**The calculated reflectance curves from a thin film with a thickness d = 110 nm and an effective refractive index n

_{1}= 1.21 (using fused silica substrate with n

_{2}= 1.46, in air, ε

_{0}= 50°, near the Brewster angle): (

**A**) The conventional reflectance (RIfS) curve, calculated by Equation (7); (

**B**) The s- and p-polarized components of (

**A**), calculated by Equations (12) and (13) and (

**C**) the polarized reflectance curve, which is defined as the ratio of s- and p-polarized components, calculated by Equation (14).

**Figure A4.**(

**A**) The calculated (s- and p-polarized) reflectance curves, (

**B**) their zoom into R

_{01}= 0–0.1 range and (

**C**) their ratio as the function of the incident angle (ε

_{0}) for a liquid/thin film interface with n

_{0}= 1.333 and n

_{1}= 1.52; (

**D**) presents the used thin film model.

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**Figure 1.**The schematic diagram of a typical reflectometric interference setup and thin film model for the calculations: n, ε, r and t mark refractive index, refraction angle, reflection and transmission amplitudes (calculated by Fresnel equations), and the indices 0, 1 and 2 represent medium, thin film and substrate, respectively. The curves show the spectrum of the light source and the measured reflectance (which is defined here as the ratio of the s- and p-polarized component).

**Figure 2.**The calculated reflectance curves from a thin film with a thickness d = 800 nm and an effective refractive index n

_{1}= 1.38 (using SF10 substrate with n

_{2}= 1.7, in air, ε

_{0}= 35°): (

**A**) The conventional reflectance (RIfS) curve with the interference pattern, calculated by Equation (7); (

**B**) The s- and p-polarized components of (A) and their average, calculated by Equations (12) and (13); (

**C**) The polarized reflectance curve, which can be measured as the ratio of s- and p-polarized components, calculated by Equation (14).

**Figure 3.**The calculated reflectance curves from thin films with an effective refractive index of n

_{1}= 1.21 and different film thicknesses (

**A**,

**B**), (

**C**,

**D**) and (

**E**,

**F**), d = 110, 135 and 160 nm, respectively (using fused silica substrate with n

_{2}= 1.46, in air, ε

_{0}= 30°): (

**A**,

**C**,

**E**) The s- and p-polarized components of the conventional RIfS curves (Equations (12) and (13)) and the zoom in the region where R

_{S}and R

_{P}curves get closer; (

**B**,

**D**,

**F**) the ratio of the s- and p-polarized components (Equation (14)).

**Figure 4.**The calculated polarized reflectance curves from thin films with a film thickness of d = 110 nm and different effective refractive indices (using fused silica substrate with n

_{2}= 1.46, in air, ε

_{0}= 35°): (

**A**,

**C**,

**E**) The s- and p-polarized components of the conventional RIfS curves (from Equations (12) and (13)); and (

**B**,

**D**,

**F**) The ratio of the s- and p-polarized components (from Equation (14)) in the case of n

_{1}= 1.21, 1.32 and 1.52 thin film refractive indices, respectively; n

_{1}values are optimized and calculated as the geometric mean of n

_{0}and n

_{2}.

**Figure 5.**(

**A**) The calculated (s- and p-polarized) reflectance curves and (

**B**) their ratio as the function of the incident angle (ε

_{0}) for a gas/thin film interface with n

_{0}= 1 and n

_{1}= 1.21; (

**C**,

**E**,

**G**): The calculated (s- and p-polarized) reflectometric interference curves and (

**D**,

**F**,

**H**) their ratios in the case of an n

_{1}= 1.21 and d = 110 nm thin film on fused silica substrate (n

_{2}= 1.46), in air, with different incident angles, ε

_{0}= 35, 41.5 and 50°, respectively.

**Figure 6.**The calculated reflectometric interference curves for (

**A**) the conventional and (

**B**) the polarization method, in the case of an n

_{1}= 1.52 and d = 110 nm thin film on SF10 substrate (n

_{2}= 1.73), in aqueous medium, by applying an incident angle of ε

_{0}= 42.2°; during the measurement ΔR and Δλ values are monitored as the function of Δn

_{1}; (

**C**,

**D**) The region of interest zooms to the red shift of the RIfS and PRIfS curves, while n

_{1}increases from 1.52 to 1.526.

**Figure 7.**The results of the simulated model experiment: (

**A**) The ΔI curves for the conventional (RIfS ΔI) and the polarized (PRIfS ΔI) cases due to Δn

_{0}= ± 0.0003 (steps 1. and 3.) or Δn

_{1}= 0.006 (adsorption, step 2.); (

**B**) Schematic drawings of the steps:

**1.**the dilute solution of the analyte reaches the measurement cell,

**2.**the immobilization process,

**3.**rinsing of the measurement cell; (

**C**) The Δλ curves for the conventional (RIfS Δλ) and the polarized (PRIfS Δλ) cases due to Δn

_{0}= ± 0.0003 (ranges 1. and 3.) or Δn

_{1}= 0.006 (adsorption, range 2.).

**Figure 8.**Maximal (

**A**) Δλ and (

**B**) ΔI responses of RIfS (yellow) and PRIfS (blue) techniques due to Δn

_{0}= 0.0003 and Δn

_{1}= 0.006 refractive index changes as the function of the incident angle ε

_{0}, in the case of a thin film with d = 110 nm and n

_{1}= 1.52 on SF10 substrate, in aqueous medium.

**Figure 9.**Δλ sensorgrams of (

**A**) RIfS and

**(B)**PRIfS techniques at different incident angles (ε

_{0}= 42.2, 43 and 45°), due to Δn

_{0}= ± 0.0003 (steps #1–3 and #10–12) and Δn

_{1}= 0.006 (steps #4–9) refractive index changes, in the case of a thin film with d = 110 nm and n

_{1}= 1.52 on SF10 substrate, in aqueous medium.

**Figure 10.**The calculated reflectometric interference curves for (

**A**) the conventional and (

**B**) the polarization method, in the case of an n

_{1}= 1.21 and d = 110 nm thin film on fused silica substrate (n

_{2}= 1.46), in air, by applying an incident angle of ε

_{0}= 44°; during the measurement, ΔR and Δλ values are monitored as a function of Δn

_{1}; (

**C**,

**D**) The region of interest zooms to the red shift of the RIfS and PRIfS curves, while n

_{1}increases from 1.21 to 1.214.

**Figure 11.**The results of the simulated model experiment: (

**A**) The ΔI curves for the conventional (RIfS ΔI) and the polarized (PRIfS ΔI) cases due to Δn

_{0}= ±0.0002 (steps 1. and 3.) or Δn

_{1}= ±0.004 (adsorption, step 2.; desorption, step 4.); (

**B**) Schematic drawing of the steps: 1. the tested molecules reach the measurement cell by the carrier gas (e.g., N2) flow; 2. the adsorption process occurs; 3. the measurement cell is rinsed by the pure carrier gas; while 4. the weakly physisorbed test molecules leave the surface and the pores; (

**C**) The Δλ curves for the conventional (RIfS Δλ) and the polarized (PRIfS Δλ) cases due to Δn

_{0}= ±0.0002 (ranges 1. and 3.) or Δn

_{1}= ±0.004 (adsorption, range 2.; desorption, range 4.).

**Figure 12.**Δλ sensorgrams of (

**A**) RIfS and (

**B**) PRIfS techniques at different incident angles (ε

_{0}= 44, 45 and 51°), due to Δn

_{1}= ±0.004 refractive index changes (steps #4–9), in the case of a thin film with d = 110 nm and n

_{1}= 1.21 on fused silica substrate, in air; (

**C**) Maximal Δλ responses of RIfS (yellow) and PRIfS (blue) techniques due to Δn

_{0}= 0.0002 and Δn

_{1}= 0.004 refractive index changes as the function of the incident angle ε

_{0}.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Janovák, L.; Dékány, I.; Sebők, D.
The Theoretical Concept of Polarization Reflectometric Interference Spectroscopy (PRIFS): An Optical Method to Monitor Molecule Adsorption and Nanoparticle Adhesion on the Surface of Thin Films. *Photonics* **2019**, *6*, 76.
https://doi.org/10.3390/photonics6030076

**AMA Style**

Janovák L, Dékány I, Sebők D.
The Theoretical Concept of Polarization Reflectometric Interference Spectroscopy (PRIFS): An Optical Method to Monitor Molecule Adsorption and Nanoparticle Adhesion on the Surface of Thin Films. *Photonics*. 2019; 6(3):76.
https://doi.org/10.3390/photonics6030076

**Chicago/Turabian Style**

Janovák, László, Imre Dékány, and Dániel Sebők.
2019. "The Theoretical Concept of Polarization Reflectometric Interference Spectroscopy (PRIFS): An Optical Method to Monitor Molecule Adsorption and Nanoparticle Adhesion on the Surface of Thin Films" *Photonics* 6, no. 3: 76.
https://doi.org/10.3390/photonics6030076