The investig a tion of optic a l sc a ttering from n a nostructures shows a growing interest due to current trends for novel n a nometric surf a ce a n a lysis techniques. A useful phenomenon in this ende a vour is the ev a nescent light inter a ction with n a nofilms deposited on a surf a ce. 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 experiment a l d a t a 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 phenomenon 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 optic a l contr a st vs. d a rk b a ckground, the DELI technique provides convenient nanometre thickness z-profiling using a rather simple optic a l microscopy densitometry technique, a s comp a red to SEM or TEM microscopies which require complex and expensive electron be a m systems a nd v a cuum ch a mbers. The DELI surface morphology observ a tion technique is a lso much e a sier to perform, f a ster, and non-destructive for n a nometre films profiling a s comp a red 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] .

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 is the image histogram i.e., the number of pixels per gray level. In this work, to convert the NIOD to a bsolute thickness v a lues we used the known sputtering r a tes to calculate the mean thicknesses of the Pd s a mples. 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.

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 c a n be considered th a t in every n^th l a yer, the ev a nescent light intensity decre a ses exponenti a lly 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 is the deposited material atom volume density, in (cm⁻³), is the crystalline unit cell dimension assumed identical for every layer, in cm, and is the evanescent extraction cross section of each atom, in (cm²). I₀ denotes the longitudinal propagating optical field intensity below the waveguide/material interface and is the evanescent photon extraction parameter of the material, where d_m is an “effective” evanescent depth. To simplify matters, we a ssume th a t 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, taking also into account the optical absorption coefficient of the material:

Here we a ssumed th a t in the f a r field c a se, only h a lf of the tot a l sc a ttered intensity from the l a yer prop a g a tes in the upw a rd direction as obtained in Ref. [3] . Using (2) in (3) and denoting a scattering p a r a meter by, we obt a in:

where denotes the sum of N terms of a geometrical series, i.e., , with, and:

From Equation (4) and Equation (5) the normalized evanescent light intensity defined as a fractional scattered evanescent intensity, is given by:

For two layers with thicknesses h₁, h₂ we obtain from Equation (6):

and (7)

We also assume that the experimentally measured NIOD is proportional to from Equations (6) and (7), hence the ratio for two NIOD’s and, for two thicknesses h₁, and h₂ is given approximately by:

Thus, given the absolute thickness h₂ at one point we can calculate also other thicknesses values h_i of the nanoprofile from Equation (8) provided γ is known and is obtained from the optical density measurements of NIOD [10] -[13] . For absolute thicknesses v a lues determin a tion, the a ver a ge layers thicknesses c a n be c a libr a ted by independent techniques such a s material deposition r a tes, spectrometry, SEM, AFM or mech a nic a l n a noprofilometry.

The evanescent photon extraction parameter γ of the material and the “effective” thickness parameter can be determined from the NIOD experimental data and the theoretical model of Equation (6). Equ a tion (8) is a tr a nscendental equ a tion, i.e., a n a lytic a lly unsolv a ble. One approximation procedure to calculate γ from Equation (8) is to measure any two NIOD for known thicknesses where NIOD₂ > NIOD₁, and using the optical absorption constant for the material. In our case for Pd, is calculated from [18] using the complex value of the dielectric parameter of Palladium thin films given in Ref. [19] at. 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 and we obtain the experimental ratio ~7, hence from Equation (8) we have:

A solution of Equation (9) can be obtained considering that the term can be neglected as. In this case the solution of Equation (9) gives. We can use this approach to solve Equation (9) because the ratio of the omitted term i.e., with respect to for all positive γ is smaller than 1/7 = 14%. This can also be used as an estimation of the error in this approximation:. Using the value for γ obtained above, it is about.

Another approximate numerical solution of Equation (9) can be sought for the case. Using here only

the first four terms in the series expansion we obtain for γ the equation:

The only solution for this equation making sense physically (due to > 0) is.

We check next the validity of the two different values comparing them to the experimental data obtained from the NIOD. For comparison we used the normalized scattered intensity function for from Equation (6),

defined by the ratio of the function to its maximum, i.e., , where, (with

), is the maximum value at a thickness of. This function is plotted in Figure 3 using the two values obtained above, i.e., for curve (a), and for curve (b). The continuous curves (a) and (b) in Figure 3 show a maxima at the thickness given by, giving for the value and for the value.

The asterisks in Figure 3 are the experimental points from the Palladium films optical densitometry data

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 h₁ = 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 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.

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 N₀ = 6.022 × 10²³ [atoms/mol] is the Avogadro number, is the material density in [gr/cm³] and M is the material molecular mass in [gr/mol]. For Pd = 12.02 gr/cm³ and M = 106.42 gr/mol, so that N_a = 0.68 × 10²³ atoms/cm³.

The m a teri a l optic a l a bsorption coefficient is usu a lly rel a ted to the optical a bsorption cross-section and the concentr a tion of the light a bsorbing molecules N _a by the following equ a tion:

Analogously, γ can be also considered as a product of the volume concentration of the light scattering dipoles, each with an effective evanescent scattering cross-section 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 = 1.287 × 10⁻⁹ µm².

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 = 532 nm light for incident angles near 90˚, gave light scattering cross sections of about 10⁻⁹ μm². Although the metal is different in [2] , this result is quite comparable to our results. This strengthens our a ssumption th a t optic a l sc a ttering from the deposited n a nop a rticles is the mech a nism for the extr a ction of the ev a nescent w a ves by the material on top of the w a veguide.

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), the evanescent 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 c a se, being the w a venumber, i.e., when the p a rticle r a dius is, it w a s shown th a t for a n s-pol a rized incident w a ve the sc a ttering di a gr a m in the sp a ce in the direction perpendicul a r to the interf a ce between the two medi a h a s a m a ximum 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).

A discrete layers evanescent model was developed to evaluate the evanescent light extraction parameter 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.