A Discrete Model of the Evanescent Light Emission from Ultra-Thin Layers

A discrete model of the Differential Evanescent Light Intensity (DELI) technique was developed to calculate and map 3D nanolayers thicknesses from the evanescent light intensity captured from optical waveguides. The model was used for ultra-thin Pd nanometric layers sputtered on glass substrates. The layers thickness profiles were displayed in 3D and 1D profiles plots. The total thickness profiles of the ultra-thin Pd films obtained in the range of 1 10 nm were validated using AFM measurements. Based on the model developed the evanescent photon extraction parameter of the material was estimated.


Introduction
The investigation of optical scattering from nanostructures shows a growing interest due to current trends for novel nanometric surface analysis techniques.A useful phenomenon in this endeavour is the evanescent light interaction with nanofilms deposited on a surface.Many models have been proposed to describe the phenomenon of evanescent waves in nano-optics but at present there is still no conclusive physical picture for it [1]- [9].More experimental data in this field becomes thus of both theoretical and practical interest.
The optical microscopy technique named Differential Evanescent Light Intensity (DELI) was used in the past decade to investigate nanostructures profiles of various materials films with small thicknesses obtaining information also about the optical photon scattering parameters [10]- [16].The DELI technique is based on the phe-nomenon of Total Internal Reflection (TIR) [4] [5] at interfaces where evanescent waves and optical tunnelling may occur when peculiar boundary conditions are given.Due to its excellent optical contrast vs. dark background, the DELI technique provides convenient nanometre thickness z-profiling using a rather simple optical microscopy densitometry technique, as compared to SEM or TEM microscopies which require complex and expensive electron beam systems and vacuum chambers.The DELI surface morphology observation technique is also much easier to perform, faster, and non-destructive for nanometre films profiling as compared to electron beam microscopy (EBM) or even the simpler atomic force microscopy (AFM).It is also better suited for nanostructures morphology mapping of large areas as required for instance in industrial applications [10]- [16].
In this work we describe a discrete model for DELI which in addition to the evanescent phenomenon takes into account also the optical absorption.Then this discrete model is utilized to analyse and construct the thickness profiles of sputtered Pd films with ultra-thin thicknesses in the range of 1 -10 nm and the results are compared in Section 3 with the previously used model in Ref. [15].

General DELI Theory
In the past, nanometric thin films of various materials [10]- [16] and with thicknesses in the range of 1 -200 nm were deposited on top of glass substrates which also served as light waveguides.Then, they were observed from the top by an optical microscope, obtaining the 3D spatial profiles and characteristic parameters of the evanescent field using the DELI phenomenological model [10]- [13].To evaluate the relative and true surface nanoprofile thicknesses of the various samples from the evanescent captured images, the mean Normalized Integrated Optical Density (NIOD), defined in Equation ( 1) is obtained experimentally from the evanescent optical density of images from the deposited zones: where D (x, y) is the gray level value of each pixel (0 -255), S is the sampled image area and ( ) H D is the image histogram i.e., the number of pixels per gray level.In this work, to convert the NIOD to absolute thickness values we used the known sputtering rates to calculate the mean thicknesses of the Pd samples.From AFM surface scans, see Figure 1(c) and the experimental sputtering rate the mean thickness of the thickest Pd sample was estimated as ~10 nm.

A DELI Discrete Model
We present here a discrete model for the captured evanescent light intensity.Let the nanometric film of thickness h be discretized into N layers (n = 1, 2, N), each with a thickness equal to the crystalline unit cell dimension a of the deposited material.It can be considered that in every n th layer, the evanescent light intensity decreases exponentially in the z-direction.Assuming that the light beam is scattered by the atoms in the nano layers, the perpendicular light intensity scattered by the n th layer can be described by: where a N is the deposited material atom volume density, in (cm −3 ), a is the crystalline unit cell dimension assumed identical for every layer, in cm, and ew σ is the evanescent extraction cross section of each atom, in (cm 2 ).I 0 denotes the longitudinal propagating optical field intensity below the waveguide/material interface and 2 m d γ = is the evanescent photon extraction parameter of the material, where d m is an "effective" evanescent depth.To simplify matters, we assume that ( ) h γ is constant at every h.Considering every layer as a local source of scattered light and neglecting multiple scattering we obtain the following expression for the upward evanescent scattered integrated light intensity from a layer of overall thickness h N a = ⋅ , taking also into account the optical absorption coefficient α of the material: Here we assumed that in the far field case, only half of the total scattered intensity from the layer propagates in the upward direction as obtained in Ref. [3].Using (2) in ( 3) and denoting a scattering parameter by , , , , , , , we obtain: ( ) 2 where N S denotes the sum of N terms of a geometrical series From Equation (4) and Equation ( 5) the normalized evanescent light intensity defined as a fractional scattered evanescent intensity η , is given by: For two layers with thicknesses h 1 , h 2 we obtain from Equation ( 6): We also assume that the experimentally measured NIOD is proportional to η from Equations ( 6) and ( 7), hence the ratio k for two NIOD's and i η , 1, 2 i = for two thicknesses h 1 , and h 2 is given approximately by:

, e e h h h h NIOD h
Thus, given the absolute thickness h 2 at one point we can calculate also other thicknesses values h i of the nanoprofile from Equation (8) provided γ is known and k is obtained from the optical density measurements of NIOD [10]- [13].For absolute thicknesses values determination, the average layers thicknesses can be calibrated by independent techniques such as material deposition rates, spectrometry, SEM, AFM or mechanical nanoprofilometry.
The evanescent photon extraction parameter γ of the material and the "effective" thickness parameter 2 m d γ = can be determined from the NIOD experimental data and the theoretical model of Equation (6).Equation ( 8) is a transcendental equation, i.e., analytically unsolvable.One approximation procedure to calculate γ from Equation ( 8) is to measure any two NIOD for known thicknesses 2 1 h h > where NIOD 2 > NIOD 1 , and using the optical absorption constant α for the material.In our case for Pd, .From the experimental observation (see Figure 2) it follows that the scattered light intensity captured from thicker layers is greater than that from the thinnest.
For 2 10 nm h = and 1 1 nm h = we obtain the experimental ratio 126 18 k = ~7, hence from Equation (8) we have: A solution of Equation ( 9) can be obtained considering that the term 10 e γ − can be neglected as 0.091 γ  .In this the solution of Equation ( 9) gives .We can use this approach to solve Equation ( 9) because the ratio of the omitted term i.e., 10  e γ − with respect to e γ − for all positive γ is smaller than 1/7 = 14%.This can also be used as an estimation of the error δ in this approximation:  The sampled 2D zones have an area of (520 × 520) µm 2 .Below the 2D images, the optical density gray value of a 1D profile plot averaged across the sampled areas is shown.The last black square is the control zero intensity value.obtained for the NIOD's from Figure 2. The highest NIOD obtained is for the greatest thickness point obtained from the sputtering rates measurements giving h 1 = 10 nm, corresponding well with the AFM profilometry, see Figure 1(c).Since curve (a) is clearly a better fit for the experimental points, we can assume that the value 1 0.156 nm γ − = is the more acceptable result for the evanescent photon extraction parameter γ of the material.In [15] we obtained in a simpler model without absorption being taken into account giving γ = 0.1 nm −1 .We also observe in this paper that γ is larger than the absorption coefficient, which is

Discussion
Regarding the mechanism of the evanescent wave extraction, many investigators assume that the material molecules dipoles deposited on the waveguide interface are responsible for the evanescent photons scattering [1]- [9].The maximum number of dipole molecules per unit volume N a can be estimated in a simple model by the following equation [18]: where N 0 = 6.022 × 10 23 [atoms/mol] is the Avogadro number, ρ is the material density in [gr/cm 3 ] and M is the material molecular mass in [gr/mol].For Pd ρ = 12.02 gr/cm 3 and M = 106.42gr/mol, so that N a = 0.68 × 10 23 atoms/cm 3 .
The material optical absorption coefficient α is usually related to the optical absorption cross-section a σ and the concentration of the light absorbing molecules N a by the following equation: Analogously, γ can be also considered as a product of the volume concentration of the light scattering dipoles a N , each with an effective evanescent scattering cross-section ew σ given by: Then, using Equation ( 12) and the number density of molecule dipoles per unit volume obtained from Equation (10), we obtain an estimation for the "effective" evanescent scattering cross-section of the Pd nanofilms molecules ew σ = 1.287 × 10 −9 µm 2 .
In reference [2], typical optical scattering properties of Ag nano-particles deposited on a substrate were inves- tigated by the discrete source method.The numerical evaluations for Ag spheres with diameter D = 48 nm on Ag films of thickness d = 49 nm irradiated with λ = 532 nm light for incident angles i θ near 90˚, gave light scattering cross sections sc σ of about 10 −9 μm 2 .Although the metal is different in [2], this result is quite com- parable to our results.This strengthens our that optical scattering from the deposited nanoparticles is the mechanism for the extraction of the evanescent waves by the material on top of the waveguide.The results of evanescent experiments [10]- [16] including the current work for extremely thin layers of Pd in the range 1 -10 nm show that such layers located in the evanescent field zone contribute to a sensitive observation of optical density variation due to thickness variations, which can be recorded easily using optical microscopes equipped with a camera.The contribution of the evanescent waves due to the scattered field was discussed in detail in [1] and it has been shown by the method of angular spectrum representation that for a scattering medium which is located near the sources (up to a distance of the light half wavelength 2 λ ), the evanes- cent field becomes an observable optical field [1].Also, far field scattering optical spatial distributions for different polarized evanescent waves from particles with several sizes were calculated in [3].For the case 0.1 r β < , β being the wavenumber, i.e., when the par- ticle radius is 0.1 2π 8 nm r λ < ≈ , it was shown that for an s-polarized incident wave the scattering diagram in the space in the direction perpendicular to the interface between the two media has a maximum in the +z direction and one in the -z direction.This suggests that approximately only half of the scattered intensity goes upward as assumed in Equation (3).

Conclusion
A discrete layers evanescent model was developed to evaluate the evanescent light extraction parameter γ for Pd ultra-thin nanolayers with thicknesses below 10 nm.The experimental z-profiling optical microscopy method called DELI, confirmed to be extremely sensitive, enabling simple and effective optical microscopy observation of thickness variations in ultra-thin Pd films in the range of 1 -10 nm.

Figure 1 .
Figure 1.3D perspective and 1D profile from a sampled area with a mean thickness of ~10 (nm): (a) DELI image of an (520 × 520) µm 2 sampled area; (b) AFM image of an (5 × 5) µm 2 sampled area and (c) 1D-AFM sample thickness profile of (b).Note the different scale of images (a) and (b).
using the complex value of the dielectric parameter 2 k of Palladium thin films given in Ref.[19] at 555 nm λ = solution of Equation (9) can be sought for the case 2 1 h γ < .Using here only the first four terms in the series expansion + − + we obtain for γ the equation:

=Figure 3 Figure 2 .
Figure 2. 2D images of 6 Pd sampled areas with mean thicknesses in the range 1 -10 nm.The sampled 2D zones have an area of (520 × 520) µm2 .Below the 2D images, the optical density gray value of a 1D profile plot averaged across the sampled areas is shown.The last black square is the control zero intensity value.

Figure 3
Figure 3. Plots of