Optical Waveguide Lightmode Spectroscopy (OWLS) is widely applied to monitor protein adsorption, polymer self-assembly, and living cells on the surface of the sensor in a label-free manner. Typically, to determine the optogeometrical parameters of the analyte layer (adlayer), the homogeneous and isotropic thin adlayer model is used to analyze the recorded OWLS data. However, in most practical situations, the analyte layer is neither homogeneous nor isotropic. Therefore, the measurement with two waveguide modes and the applied model cannot supply enough information about the parameters of the possible adlayer inhomogeneity and anisotropy. Only the so-called quasihomogeneous adlayer refractive index, layer thickness, and surface mass can be determined. In the present work, we construct an inhomogeneous adlayer model. In our model, the adlayer covers the waveguide surface only partially and it has a given refractive index profile perpendicular to the surface of the sensor. Using analytical and numerical model calculations, the step-index and exponential refractive index profiles are investigated with varying surface coverages from 0 to 100%. The relevant equations are summarized and three typically employed waveguide sensor structures are studied in detail. We predict the errors in the calculated optogeometrical parameters of the adlayer by simulating the OWLS measurement on an assumed inhomogeneous adlayer. We found that the surface coverage has negligible influence on the calculated refractive index below film thicknesses of 5 nm; the calculated refractive index is close to the refractive index of the adlayer islands. But the determined quasihomogeneous adlayer refractive index and surface mass are always underrated; the calculated quasihomogeneous thickness is heavily influenced by the surface coverage. Depending on the refractive index profile, waveguide geometry, and surface coverage, the thickness obtained from the homogeneous and isotropic modeling can even take negative and largely overestimated values, too. Therefore, experimentally obtained unrealistic adlayer values, which were dismissed previously, might be important indicators of layer structure.
Evanescent field-based optical techniques are popular in surface sensitive chemicals and biosensors. The intensity of this evanescent wave is the highest at the sensing surface and is exponentially decaying in the media above the sensor. Typical examples of these techniques are the surface plasmon resonance [
Among waveguide-based sensors, Optical Waveguide Lightmode Spectroscopy (OWLS) is one of the oldest and most popular methods, realized in reliable commercial devices [
One drawback of the traditional OWLS technique is that it does not have imaging capabilities, like imaging ellipsometry, imaging SPR [
In the present work, we investigate the OWLS signals when the adlayer covers the waveguide surface only partially and the adlayer refractive index is inhomogeneous perpendicular to the surface of the sensor. Using analytical and numerical model calculations, the step-index and exponential refractive index profiles are investigated with varying surface coverages from 0 to 100%. The relevant equations are summarized and three different typically employed waveguide sensor structures are studied in detail.
The structure of the waveguide sensor and the investigated adlayer models is visualized in Figure
The structure of the modeled OWLS waveguide chips with inhomogeneous adlayer. The modeled multilayered assembly consists of 4 layers: substrate, waveguide film, adlayer, and cover (see inset c). The adlayer covers the waveguide film surface in a certain percent (see inset b) and it has
During the model calculations, the thickness of the adlayer was varied between 0 and 500 nm. The adlayer was considered horizontally inhomogeneous with
The step-index profile is a widely applied model for a compact protein layer [
The optogeometrical parameters of the planar optical waveguide structures treated here are summarized in Table
The structural parameters of the modeled planar optical waveguide sensors.
OW2400 | 1.77 | 1.52 | 200 nm |
Ta2O5 | 2.12 | 1.52 | 150 nm |
Reverse | 1.575 | 1.2 | 150 nm |
The model calculations performed in the present study are schematically overviewed in Figure
The structure of the developed MATLAB algorithm. The input parameters are the parameters of the chip structure and the parameters of the modeled adlayer: the refractive index, the layer thickness, the surface coverage, and the vertical refractive index distribution. The algorithm has two main paths. In the upper one, the real surface mass is calculated with an integral. In the other path, the effective refractive indices were calculated from the input parameters, then these values were applied in the homogeneous and isotropic model resulting in quasihomogeneous surface mass, adlayer thickness, and refractive index. The output parameters of the algorithm are the quasihomogeneous refractive index, the layer thickness, and the ratio of the obtained masses.
The input parameters of the algorithm are
In order to simulate the isotropic and homogeneous thin adlayer model,
In short, the transmission and reflection coefficients of the waves of the TE and TM modes at each boundary are determined by using the Maxwell’s equations and the boundary conditions [
(For notations, see reference [
For the whole structure,
The effective refractive indices of the modes with
Then, the effective refractive indices from equations (
On the other hand, the surface mass of the adlayer with a certain
At the end of these calculations, the results are the quasihomogeneous optogeometrical parameters and the ratio of the two surface masses:
Moreover, we have calculated an averaged adlayer refractive index and thickness based on the following well-known equations, too [
The above formulae do not take the possible inhomogeneities in adlayer refractive index parallel to the surface (surface coverage) into consideration. Considering the surface coverage, the following equation was applied:
After straightforward calculations, we obtained
The results obtained for the step-index and exponential profiles are summarized in Figures
The quasihomogeneous adlayer refractive index in the function of surface coverage for seven different adlayer film thicknesses. The dashed line represents the average value of the refractive index,
The quasihomogeneous adlayer film thickness in function of the surface coverage for seven adlayer film thicknesses. For thicker layer thicknesses—100 nm, 200 nm, and 500 nm—overrated values and singularities appeared in case of the OW2400 and the reverse chip. The singularities are shown in the insets.
The lines of the adlayer thickness-surface coverage pairs where the singularities appear.
The ratio of the obtained masses in the function of the surface coverage for seven different adlayer film thicknesses.
Interestingly, the surface coverage has almost no influence on the obtained
By solving equations (
Therefore, for thin adlayers, the quasihomogeneous adlayer refractive index equals the refractive index of the modeled adlayer islands, independent of the surface coverage. The quasihomogeneous adlayer thickness linearly depends on the surface coverage.
It is also important to note that when
As pointed out, when
For simplicity, in Figure
It is important to emphasize that the negative adlayer thickness has no physical meaning, as it originates from the application of a wrong model. But, for a complicated adlayer, which is characterized by more than two independent parameters, the two independent modes (TE, TM) and the isotropic and homogeneous adlayer model, which are applied in OWLS experiments, do not give enough independent equations to calculate all of the parameters of the adlayer. Based on the presented results, however, the properties of the complicated adlayer can be deduced in some circumstances from the nonphysical adlayer parameters. For example, if the Ta2O5 sensor chip is used and negative quasihomogeneous adlayer thicknesses are obtained, it is a clear indication that the adlayer has a step-index profile and its thickness is larger than 50 nm, or it has an exponential profile with a thickness above 80 nm. Importantly, the calculations also show that the surface coverage has practically no role in modifying these parameters (see Figures
In Figure
In the present work, we investigated the OWLS signals when the adlayer covers the waveguide surface only partially and the adlayer refractive index is inhomogeneous perpendicular to the surface of the sensor. Using analytical and numerical model calculations, the step-index and exponential refractive index profiles were investigated by varying surface coverages from 0 to 100%. The relevant equations were summarized and three different typically applied waveguide sensor structures were studied in detail. We concluded that the surface coverage has negligible influence on the obtained quasihomogeneous adlayer refractive index for thin adlayers; the obtained index equals the refractive index of the adlayer islands. A simple analytical calculation supported the finding of the numerical simulations. The quasihomogeneous adlayer refractive index is always underrated for thicker adlayers, independent of surface coverage, waveguide sensor type, and refractive index profile. In special cases, the quasihomogeneous refractive index of the adlayer can be equal to the refractive index of the cover media. In this case, a singularity appears in the quasihomogeneous thickness. The obtained thickness can also be negative around the singularity. The conditions when the singularity appears were analyzed in detail. This behavior is very similar to what was obtained for positively birefringent adlayers [
The parameters of the employed sensor structures and the modeled inhomogeneous analyte layer used to support the findings of this study are included within the article.
We confirm that the publication does not lead to any conflict of interest.
This work was supported by the Lendület Program of the Hungarian Academy of Sciences and by the ERC_HU, KH_17 Programs of the Nemzeti Kutatási, Fejlesztési és Innovációs Hivatal.