## Abstract

Gold nanoparticles deposited on self-organized nano-ripple quartz substrates have been studied by spectroscopic Mueller matrix ellipsometry. The surface was found to have biaxial anisotropic optical properties. For electric field components normal to the ripples the periodic and disconnected nature of the in plane nanowires gives rise to an optical response dominated by the localized plasmon resonance. In the direction parallel to the ripples the gold nanoparticles are aligned closely leading to localized plasmon resonances in the infrared. As Au was deposited at an angle oblique to the surface normal, the gold nanoparticles were formed on the side of the ripples facing the incoming evaporation flux. This makes the gold particles slightly inclined, correspondingly the principal coordinate system of the biaxial dielectric tensor results tilted. The anisotropic plasmonic optical response results in a strong polarizing effect, making it suitable as a plasmonic nanowired grid polarizer.

© 2013 Optical Society of America

## 1. Introduction

The optical excitation of collective oscillations of free electrons from noble metal
nanoparticles, known as localized surface plasmon resonances, are known to have spectroscopic
properties resulting in *e.g.* various color effects. One example is the well known
“Lycurgus cup” [1], in which
silver nanoparticles distributed in glass provides a different color to it depending if illumination
is performed in transmission or in reflection. It is known that nanoparticle size, spacing and
substrate, affects the plasmonic resonance frequency [2], and is an effective way to design selective optical properties. A recent wave of
interest in plasmonics is motivated by the proven increase in photon absorption and thus in
efficiency of photovoltaic devices [3] caused
by the strong localization of the electric field and by enhanced scattering from the metal
nanoparticles. On the fundamental level the strongly anisotropic plasmonic nanostructures are used
to form metamaterials [4] with possible
applications to e.g. negative refractive materials in the visible [4] or in non-linear applications [5–7].

The anisotropic optical response of plasmonic nanoparticles and metamaterials can favourably be
studied by spectroscopic generalized ellipsometry, as recently reviewed by Oates *et
al.* [8]. In particular, several
studies of the optical properties of in plane silver nanowires on various substrates have been
reported [9–11], where in particular the effective dielectric tensor of silver nanoparticle
arrays on a silicon substrate was determined with spectroscopic Mueller Matrix Ellipsometry
[10]. Also ellipsometric studies of isotropic
and anisotropic silver and gold island films have been reported [12, 13].

In this work, we are exploring the enhanced sensitivity of Mueller matrix spectroscopic ellipsometry from the ultra violet to the near-infrared, combined with azimuthal rotation of the sample around the sample normal, using multiple angles of incidence, in order to determine the complex biaxial properties of a plasmonic layer of gold nanoparticles supported on a nano-patterned quartz substrate. Such nano-patterned plasmonic wires have many similarities to the standard infrared wire grid polarizer [14]. However, due to the localized plasmons, an inverse polarizing effect can be observed in the visible spectral range using polarized transmission spectroscopy at normal incidence [5, 15]. It has been observed that the anisotropic localized surface plasmonic properties change the polarizing properties of the sample from transverse electric (TE) to transverse magnetic (TM) [16], but the details of the dielectric tensor for such a complex plasmonic system based on aligned and partially connected gold nanoparticles have so far not been reported. The development of systematic optical methods to reveal the dielectric function of such gold based nanoplasmonic samples is further of fundamental interest due to the common use of gold nanoparticles in applications of plasmonics and metamaterials.

The samples studied in this paper were prepared by shadow deposition of gold at grazing incidence onto a quartz self-organized nano-ripple surface produced by ion beam sputtering [15], similar to samples in [17, 18].

## 2. Experimental

The anisotropic gold nanopatterned surfaces were prepared in a two stage process combining
self-organized ion beam sputtering (IBS) to produce ripples on the surface of the quartz substrate,
and gold deposition by thermal evaporation. IBS is a low cost nano-fabrication process used to
pattern a range of materials from metallic to dielectric [19, 20]. A spatial modulation of the surface
profile is induced by a combination of an erosive instability induced by the ion beam and energy
relaxation dominated by the thermally activated diffusion and hyperthermal mobility induced by the
ion beam [21,22]. Variations of the surface morphology is made possible by changing the
irradiation parameters such as ion energy, incidence angle, gas species, and sample temperature
[23]. A clean quartz substrate was irradiated
by Ar^{+} ions in an ultra high vacuum (UHV) system at an incidence angle of
45°. The ions are ejected from a gridded multi aperture source having an energy of 800 eV at
a constant flux of 4.0 × 10^{15} ions/cm^{2}s. The IBS process results in a
surface having a well defined ripple pattern, facing the ion beam. An atomic force microscope (AFM)
micrograph of such a surface is shown in Fig. 1(a). The
ripples have a period of approximately Λ = 70 ± 5 nm and an amplitude around
6 nm.

Gold was evaporated in the same UHV system onto the surface at a grazing incidence angle (80°), forming nanoparticles on the surface. The spatial distribution of the nanoparticles is locally modulated by the shadowing of the nanoripple ridges, such that more material is deposited onto the facing ridges. The shadowed ridge is then mainly uncovered by Au. During the deposition the distribution of particles are limited in the direction normal to the ripples. In the direction along the ripples, the particles are partially connected, forming in some cases elongated planar nanowires. Figure 1(b) shows an AFM micrograph of the nanopatterned surface.

A schematic of the cross-section of the sample system is shown in Fig. 2, where the coordinate system is aligned with the *x*–axis in
the long direction of the nanowires, and the *y* − *z* plane
rotated by an angle *θ* so that the *y*–axis is in the
plane of the gold nanoparticle-substrate interface. *θ* is in the following
regarded as the tilt angle of the biaxial system. Cross-section electron microscope micrographs were
not easily obtained due to the charging of the dielectric substrate. A geometrical model of the
nanowires based on the AFM topographies and on the deposited amount of gold, allows to estimate the
local height of the Au nanowires (*h* = 29.5 nm) measured along the
*z*-axis and their width (*w* = 72 nm), measured along the
y-axis.

For the optical characterization a variable angle multichannel dual rotating compensator Mueller matrix ellipsometer (RC2) from JA Woollam Company was used. The instrument has a collimated 150 W Xe source and operates in the spectral range from 210 nm (5.9 eV) to 1700 nm (0.73 eV), using a combination of silicon and indium gallium arsenide spectrographs having a resolution of 1 nm below 1000 nm and 2.5 nm above. The initial collimated beam has a waist of approximately 3 mm, but in the present work focusing and collection lenses with a focal length of 80 mm were applied, allowing a normal incidence spot size of 150 μm. This spot size allowed us to study a reasonably spatially homogeneous region of the sample [18].

The spectroscopic Mueller matrix was measured for the incidence angles 50° to 75° in steps of 5°. Full azimuthal rotation of the sample (360°) in steps of 5° was performed for each angle of incidence in order to fully map the anisotropy of the sample. When using focusing optics the sample alignment upon rotation is very sensitive, and was therefore adjusted prior to the measurement of each incidence angle. The same instrument was also used to measure the spectroscopic transmission Mueller matrix of the sample.

## 3. Theory

With ellipsometry, the polarization nature of light is used to indirectly measure extrinsic and
intrinsic properties of *e.g.* thin films, nanostructures or bulk materials
[24–26]. In specific, the change of polarization state of monochromatic light upon
reflection from a smooth surface can be formulated by the 2 × 2 complex Jones matrix
transforming the incoming polarization state to the reflected by the Fresnel coefficients
(*r _{pp}*,

*r*, etc.) by [27]

_{ps}*E*and

_{p}*E*are orthogonal plane wave electric field components, where

_{s}*E*is parallel and

_{p}*E*perpendicular to the incidence plane. In practical applications, in Mueller matrix polarimetry and ellipsometry in particular, the polarization state is commonly described using the four element Stokes vector

_{s}*s*

_{0},

*s*

_{1},

*s*

_{2}and

*s*

_{3}are time averages over electric field components resulting in the total intensity (

*s*

_{0}), the intensity difference between

*p*and

*s*polarized light (

*s*

_{1}), +45° and −45° linearly polarized light (

*s*

_{2}), and the right and left polarized part of the light (

*s*

_{3}).

The change of polarization upon interaction with a sample using the Stokes vector is described using the 4 × 4 element Mueller matrix. For reflection measurements of an isotropic surface the Mueller matrix is

*r*= tan Ψ exp(

_{pp}/r_{ss}*i*Δ), see

*e.g.*Azzam and Bashara [27]. In Mueller matrix ellipsometry the Mueller matrix is measured directly, and may, for anisotropic samples, have no elements that are zero. Then the coupling between the

*s*and

*p*polarized light becomes important, and the corresponding Mueller matrix expressed using the Fresnel reflection coefficients from the Jones matrix (Eq. (1)) becomes [28]

*s*

_{0}and

**M**

_{11}) element.

The spectroscopic Mueller matrix can through the appropriate modelling be used to invert for the
dispersive optical properties of thin plasmonic layers/plasmonic surfaces. Commonly available
effective medium theories, such as Yamaguchi [29,
30], anisotropic Bruggeman [31] and anisotropic Maxwell-Garnett [32], do not well capture the plasmonic response of nanoparticles on a
substrate. Furthermore, tabulated reference optical properties of metallic nanoparticles are
uncertain. On the other hand, recent rigorous numerical approaches [2, 33, 34] require the particles to be of regular shape and regularly distributed, and in
[2] not in direct contact with the substrate.
However, the latter approach supplies useful physical insight into the line shape of the dielectric
function. An appropriate and practical approach to extract the intrinsic optical response of the
nano-plasmonic layer, is to make an anisotropic parametric dispersion model based on oscillators in
order to capture the plasmonic, interband and free electron response, in addition to appropriate
Euler angles to capture the possibility of a biaxial dielectric tensor with principal axes tilted
away from the substrate normal. This model allows to simulate and compare to the large Mueller
matrix data set. By trial an error, a reasonable set of starting parameters can be found, and
finally fitted to the full data-set. As the nanowires are anisotropic, the dielectric function is a
tensor with at most three orthogonal axes with different properties when assuming an orthogonal
coordinate system. The inherent tilt and the truncation of the gold nanoparticles on the ripple
surface, suggested a *biaxial* tensor with appropriate Euler angles. The anisotropic
dispersion model approach is here used to determine the biaxial dielectric tensor and the Euler
angles for the gold nanowires deposited on the quartz ripple substrate.

## 4. Results and discussions

Figure 3 shows the fascinating information captured by the full spectroscopic Mueller matrix recorded at 50° incidence for a complete azimuthal rotation of the sample. The Mueller matrix is here presented as a polar color map, where the wavelength is mapped linearly to the radial direction, and the incidence plane orientation is mapped to the polar angle. The color map shows the numerical value of the Mueller matrix element at a particular incidence plane and wavelength, at the given incidence angle. The incidence plane has initially been rotated so that 0° and 90° correspond to the directions where the block off-diagonal elements are at a minimum, a close to pseudo isotropic orientation where the Mueller matrix may be approximated using Eq. (3), i.e. the incidence plane is coinciding with the long and short axis of the nanowires.

In Fig. 4 the Mueller elements
*m*_{12}, *m*_{33} and
*m*_{34}, *i.e.* the standard ellipsometric parameters
*N*, *C* and *S*, respectively, are plotted for these
two incidence planes (*ϕ* = 0° and 90°) for the
incidence angles *θ* = 50°, 60° and 70°, as a
function of photon energy. All three elements shows large differences between the two incidence
planes. In particular, below 3 eV the difference is largest, while for higher energies the data is
similar.

The fabrication method employed leads to the formation of gold nanowires which are preferentially aligned along the side of the quartz nano ripples illuminated by the gold atom flux during evaporation, as illustrated in Fig. 2. For the optical model, localized surface plasmon resonances from strictly monodisperse noble metal nanoparticles in a well defined infinite regular array can be modelled by a Lorentzian line shape given by [35]

where*A*is the amplitude,

_{k}*E*energy location,

_{k}*γ*a broadening parameter and

_{k}*E*the photon energy. Such a Lorenzian model may be readily understood also in terms of Maxwell-Garnett theory for particles within a host matrix. Normal to the local plane surface which is supporting the gold particles (z-direction), the model also includes a standard Drude dispersion term [35] where

*E*is the plasma energy and

_{p}*γ*is the broadening parameter.

_{k}The self-organized formation of the nano ripples recurring to a stochastic process of sputtering
may result in small variations of the plasmon resonance energy. This can be represented by a sum of
Lorentzians distributed around a center energy *E _{k}*. The resulting line
shape may for the imaginary part of the dielectric function then more simply be expressed by a
Gaussian line shape

*γ*is the broadening,

_{k}*A*the amplitude and

_{k}*E*the center energy position. Equation (7) is a sum of two Gaussians with positive and negative center energy making it an odd function which is needed for Kramers-Kronig consistency [36]. The real part of the dielectric function is calculated using Kramers-Kronig relations, and results [36]

_{k}We let the localized plasmons be described by one or several Gaussians, where each localized
plasmon is denoted *ε*_{Loc}. Several localized plasmons was found
necessary to account for a distribution of nano particle sizes, and a distribution of connectivity
between the particles. The Drude contribution *ε _{D}* is the special
case of near bulk gold behaviour such as expected for completely connected nanowires. For simplicity
of the model, the interband contribution are accounted for by a single Gaussian (Eq. (7)), denoted

*ε*. The dielectric function for the three tensor components

_{IB}*q*=

*x*,

*y*,

*z*is then proposed described by:

A total of 9 oscillators were needed in order to have an acceptable mean square error between the
simulated and measured data. Three oscillators for each direction. The complex inverse problem was
found to be most easily solved using an iterative process. First, the common ellipsometric
parameters (*N*, *C* and *S*) were used to determine an
approximate solution to the biaxial effective dielectric tensor by assuming that the principal axes
of the tensor were in the sample plane. All incidence planes and the different incidence angles were
used in order to increase the sensitivity to the effective properties normal to the surface
(*z*-direction). The (*N*, *C* and *S*)
parameters were also found to be most sensitive to the effective layer thickness, such that the
first analysis supplied an estimate of the effective film thickness in addition to a first estimate
to the dielectric tensor.

Upon rotation of the incidence plane, it is observed from the polar plots in Fig. 3 that the block-diagonal Mueller elements have a 180° symmetry.
The off block-diagonal elements, which are probing the cross polarization *i.e.* the
anisotropy (cf. Eq. (4)) show a more complex
behaviour. It is particularly observed that they are oscillating with a different amplitude for the
maxima and mimima. Further, for *ϕ* = 0° (incidence plane
normal to the nanowires) the Mueller matrix is pseudo-isotropic (i.e. a diagonal Jones matrix).

The metallic nano particles on a substrate indicate that the dielectric tensor has principal axes
aligned with, and perpendicular to the local surface normal. The local slope
(*θ* in Fig. 2) is approximated to be
the Euler rotation angle for the dielectric tensor. The *z*–axis of the
tensor is then no longer orthogonal to the global sample plane and does also have a component in the
*y*–direction, in principle one could expect that the plasmonic resonance may
also be weakly observed in this part of the dielectric tensor. The off-diagonal elements were
therefore used to fit the tilt angle *θ* by applying an Euler rotation of the
dielectric tensor. This process was repeated until convergence.

The tilt angle converged to 12.8° and the effective thickness of the nanoparticle film amounts to 28 nm. The parameters were found to have an accuracy within ±4 nm and ±2°. The parameters of the dielectric tensor is summarized in Table 1, while Figure 5 shows the real and imaginary part of the dielectric functions for the three principal axes, where we have used the fitted dispersion model parameters in Table 1.

The mean square error for the final model was MSE = 9.7 [38] when evaluating 32 free parameters, using 50 incidence planes, 6 incidence angles and 1067 wavelengths. In the calculation of the MSE the uncertainty in each data point in the Mueller matrix was estimated to 0.001.

The most striking feature in Fig. 5 is the localized
plasmonic resonance peak at 1.58 eV for the *ε _{y}* component (normal
to the wires, but in the sample plane), although it also appears to contain an additional localized
plasmonic feature around 0.9 eV. The

*ε*component appears to contain strong contributions from two localized plasmons located in the IR region of the spectrum. The exact locations of these plasmons are uncertain, and may be further revealed in upcoming work using IR Mueller matrix ellipsometry. As previously mentioned, it is speculated that several such localized plasmons may be the result of incomplete connectivity within the chains of particles making up the ”nanowires” along the x-axis. A completely connected chain should thus be represented by the Drude model, while the reduced chain lengths may result in the IR localized plasmons. For comparison a typical dielectric function for Au is included in Fig. 5. The interband transitions in the plasmonic film are strongly attenuated compared to Au, and the plasmon resonances in

_{x}*ε*,

_{x}*ε*and

_{y}*ε*makes

_{z}*ε*

_{1}increasing in the infrared.

An interband contribution with center energy 3.75 eV and 3.57 eV was found for both the
*ε _{x}* and

*ε*components. The

_{y}*ε*component was found to be dominated by a weak Drude component in the near infrared and some weak interband contribution with center energy 4.83 eV. Another weak localized plasmon contribution at 2.33 eV appeared in

_{z}*ε*. The blue shift for the out of plane resonance follows the results of polarizability calculations for silver hemispherical islands on MgO substrate by Lazzari and Simonsen [33]. The

_{z}*ε*component also has a Drude component, which was also found for silver hemispherical nanowires supported on silicon substrate in [10], while the effective film appears much less dense compared to bulk Au properties. The blue shifted out of plane resonance predicted by Lazzari and Simonsen was, however, not reported by Oates

_{z}*et al.*[10]. The effective thickness of the layer is probably much larger than the truncated particle thickness on the ”hills” of the ripple along the

*z*-axis, i.e. the local surface normal.

In Fig. 6 the measured spectroscopic Mueller matrix at 50° incidence for the incidence planes 0°, 45°, 135°, 180°, 225° and 270° is plotted together with the simulated data using the fitted parametric model. The simulated data is also plotted as dashed lines for different incidence angles in Fig. 4. The figure of merit was calculated using the root mean square error of the entire data set, where the weighting on all Mueller matrix elements and measured wavelengths in nm are the same.

In order to verify the optical model, a direct analysis of the slope distribution of the AFM image in Fig. 1(a) was performed. Fig. 7 shows a histogram of the slope in the image. It is found that the most dominant inclination is 12°–13°, which much verify the tilt angle found by Mueller matrix ellipsometry.

The polarizing properties of the sample was investigated by spectroscopic transmission Mueller
matrix measurements. Figure 8 shows the measured data at
normal incidence as a solid curve and the simulated data using the model from above, where only the
relative orientation of the sample was refitted. The simulations reproduce all features of the
measurement, except a very small Gaussian like “bump” (amplitude 0.002) in the
*m*_{14/41} elements indicating an induced circular dichroism. The Mueller
matrix is largely block-diagonal, meaning that the optical axis of the sample coincide with the axes
of the instrument, only small deviations of a few per cent origin from sample orientation
misalignment of approximately 3°.

The linearly polarizing properties are directly observed in the
*m*_{12/21} elements, and through the model, we can now observe that the main
features are related to the plasmon resonances in *ε _{y}* and

*ε*.

_{x}For the *y* direction the transmission measurements show a peak at 1.84 eV that is
the maximum in the extinction coefficient *κ* =
*ε*_{2}/2*n*, where *n* is the real
part of the refractive index ${n}^{2}=\frac{1}{2}{\epsilon}_{1}+\frac{1}{2}\sqrt{{\epsilon}_{1}^{2}+{\epsilon}_{2}^{2}}$. In the infrared, the *m*_{12/21} elements are
negative due to the resonance in the *ε _{x}*. This anisotropy is of
inverse polarizing character [39, 40], where the polarization shifts spectrally.

The features in the lower right 2 × 2 matrix comes from the spectral birefringence
(Δ*n* = *n _{x}* −

*n*). A polar decomposition [41] of the Mueller matrix supplies the magnitude and orientation of the retarding and diattenuating (polarizing) properties. Figure 9(a) shows the linear retardance (

_{y}*δ*) and the linear diattenuation, the corresponding orientation of the slow axis (

*θ*) and the orientation of the transmission axis (

_{δ}*θ*) are plotted in Fig. 9(b). By analysing Fig. 9 in detail, it is observed that the sample works as a polarizer with transmission axis along the

_{D}*y*-axis in the infrared range. At approximately 1.4 eV the sample is a pure retarder (minimum diattenuation and maximum retardance), while at approximately 2 eV the sample is a polarizer with transmission axis along the

*x*-direction. These properties can in principle be shifted spectrally by varying the period of the ripple structure and consequently the distance between the nanowires [2]. It is also expected that the density of the particles forming the nanowires would change the resonance frequency, from a pure Au response for perfect wires, to a response dominated by localized surface plasmon resonances for separated particles.

## 5. Conclusion

The localized plasmonic optical properties of in plane gold nanowire array deposited at grazing incidence on a nano-ripple quartz substrate has been determined by variable angle spectroscopic Mueller matrix ellipsometry (MME) with complete azimuthal rotation of the sample. The sensitivity to the anisotropy is strong in all Mueller elements, including the elements measured in standard ellipsometry. The off block-diagonal elements show a lower symmetry, suggesting that the shadowing effects during deposition leaves the nano-wires tilted with respect to the surface normal. The dielectric tensor axes are proposed tilted by the same angle, as was found nondestructively by MME. The three components of the dielectric tensor were determined through parametric dispersion models for each component. The extracted dielectric functions complete the understanding of the observed wire-grid and inverse polarizing properties. The parametric dispersion models extracted from the effective thin surface layer composed of aligned gold nanoparticles on the nano-ripple glass surface, is expected to be a useful model starting point for many similar nano-plasmonic systems. Finally, a systematic approach was proposed to attack the complex modelling issue with such a large MME data-set.

## Acknowledgments

L.M.S.A acknowledges financial support from “The Norwegian Research Center for Solar Cell Technology” (project num. 193829).

This work has been partly supported by MAE under program Italia-Polonia, and by MSE in the framework of the Operating Agreement with ENEA for research on the Electric System.

## References and links

**1. **F. Wagner, S. Haslbeck, and L. Stievano, “Before striking gold in gold-ruby
glass,” Nature **407**, 691–692 (2000). [CrossRef] [PubMed]

**2. **P. A. Letnes, I. Simonsen, and D. L. Mills, “Substrate influence on the plasmonic response of clusters
of spherical nanoparticles,” Phys. Rev. B **83**, 075426 (2011). [CrossRef]

**3. **H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic
devices.” Nat. Mater. **9**, 205–213 (2010). [CrossRef] [PubMed]

**4. **D. Smith and D. Schurig, “Electromagnetic Wave Propagation in Media with Indefinite
Permittivity and Permeability Tensors,” Phys. Rev. Lett. **90**, 077405 (2003). [CrossRef] [PubMed]

**5. **A. Belardini, M. C. Larciprete, M. Centini, E. Fazio, C. Sibilia, M. Bertolotti, A. Toma, D. Chiappe, and F. Buatier de Mongeot, “Tailored second harmonic generation from self-organized
metal nano-wires arrays.” Opt. Express **17**, 3603–3609 (2009). [CrossRef] [PubMed]

**6. **A. Belardini, M. C. Larciprete, M. Centini, E. Fazio, C. Sibilia, D. Chiappe, C. Martella, A. Toma, M. Giordano, and F. Buatier de Mongeot, “Circular Dichroism in the Optical Second-Harmonic Emission
of Curved Gold Metal Nanowires,” Phys. Rev. Lett. **107**, 257401 (2011). [CrossRef]

**7. **V. Robbiano, M. Giordano, C. Martella, F. D. Stasio, D. Chiappe, F. B. de Mongeot, and D. Comoretto, “Hybrid Plasmonic–Photonic Nanostructures: Gold
Nanocrescents Over Opals,” Adv. Opt. Mat. **1**, 389–396 (2013). [CrossRef]

**8. **T. Oates, H. Wormeester, and H. Arwin, “Characterization of plasmonic effects in thin films and
metamaterials using spectroscopic ellipsometry,” Prog. Surf.
Sci. **86**, 328–376 (2011). [CrossRef]

**9. **T. W. H. Oates, A. Keller, S. Facsko, and A. Mücklich, “Aligned Silver Nanoparticles on Rippled Silicon Templates
Exhibiting Anisotropic Plasmon Absorption,” Plasmonics **2**, 47–50 (2007). [CrossRef]

**10. **T. W. H. Oates, M. Ranjan, S. Facsko, and H. Arwin, “Highly anisotropic effective dielectric functions of silver
nanoparticle arrays,” Opt. Express **19**, 2014–2028 (2011). [CrossRef] [PubMed]

**11. **S. Camelio, D. Babonneau, D. Lantiat, L. Simonot, and F. Pailloux, “Anisotropic optical properties of silver nanoparticle
arrays on rippled dielectric surfaces produced by low-energy ion erosion,”
Phys. Rev. B. **80**, 1–10 (2009). [CrossRef]

**12. **M. Lončarić, J. Sancho-Parramon, and H. Zorc, “Optical properties of gold island films—a
spectroscopic ellipsometry study,” Thin Solid Films **519**, 2946–2950 (2011). [CrossRef]

**13. **A. J. de Vries, E. S. Kooij, H. Wormeester, A. Mewe, and B. Poelsema, “Ellipsometric study of percolation in electroless deposited
silver films,” J. Appl. Phys. **101**, 053703 (2007). [CrossRef]

**14. **E. Hecht, *Optics* (Addison Wesley,
2002).

**15. **A. Toma, D. Chiappe, D. Massabò, C. Boragno, and F. Buatier de Mongeot, “Self-organized metal nanowire arrays with tunable optical
anisotropy,” Appl. Phys. Lett. **93**, 163104 (2008). [CrossRef]

**16. **L. Anghinolfi, R. Moroni, L. Mattera, M. Canepa, and F. Bisio, “Flexible Tuning of Shape and Arrangement of Au
Nanoparticles in 2-Dimensional Self-Organized Arrays: Morphology and Plasmonic
Response,” J. Phys. Chem. C **115**, 14036–14043 (2011). [CrossRef]

**17. **A. Toma, D. Chiappe, C. Boragno, and F. Buatier de Mongeot, “Self-organized ion-beam synthesis of nanowires with
broadband plasmonic functionality,” Phys. Rev. B **81**, 165436(2010). [CrossRef]

**18. **L. M. S. Aas, I. S. Nerbø, M. Kildemo, D. Chiappe, C. Martella, and F. Buatier de Mongeot, “Mueller matrix imaging of plasmonic polarizers on
nanopatterned surface,” Proc. SPIE **8082**, 80822W (2011). [CrossRef]

**19. **A. Toma, F. Buatier de Mongeot, R. Buzio, G. Firpo, S. Bhattacharyya, C. Boragno, and U. Valbusa, “Ion beam erosion of amorphous materials: evolution of
surface morphology,” Nucl. Instrum. Methods **230**, 551–554 (2005). [CrossRef]

**20. **D. Chiappe, A. Toma, and F. Buatier de Mongeot, “Tailoring resisitivity anisotropy of nanorippled metal
films: Electrons surfing on gold waves,” Phys. Rev. B **86**, 045414 (2012). [CrossRef]

**21. **R. M. Bradley and J. M. E. Harper, “Theory of ripple topography induced by ion
bombardment,” J. Vac Sci. Technol. A **6**, 2390–2395 (1988). [CrossRef]

**22. **W. L. Chan and E. Chason, “Making waves: Kinetic processes controlling surface
evolution during low energy ion sputtering,” J. Appl. Phys. **101**, 121301 (2007). [CrossRef]

**23. **U. Valbusa, C. Boragno, and F. Buatier de Mongeot, “Nanostructuring by ion beam,”
Mater. Sci. Eng. C **23**, 201–209 (2003). [CrossRef]

**24. **I. Nerbø, S. L. Roy, M. Foldyna, E. Søndergård, and M. Kildemo, “Real-time in situ Mueller matrix ellipsometry of GaSb
nanopillars: observation of anisotropic local alignment,” Opt.
Express **19**, 571–575 (2011). [CrossRef]

**25. **M. Schubert, “Generalized ellipsometry and complex optical
systems,” Thin Solid Films **313–314**, 323–332
(1998). [CrossRef]

**26. **D. Schmidt, B. Booso, T. Hofmann, E. Schubert, A. Sarangan, and M. Schubert, “Generalized ellipsometry for monoclinic absorbing
materials: determination of optical constants of Cr columnar thin films.”
Opt. Lett. **34**, 992–994 (2009). [CrossRef] [PubMed]

**27. **R. Azzam and N. Bashara, *Ellipsometry and Polarized light*
(North-Holland, 1977).

**28. **P. S. Hauge, R. H. Muller, and C. G. Smith, “Conventions and formulas for using the Mueller-Stokes
calculus in ellipsometry,” Surf. Sci. **96**, 81–107 (1980). [CrossRef]

**29. **T. Yamaguchi, S. Yoshida, and a. Kinbara, “Optical effect of the substrate on the anomalous absorption
of aggregated silver films,” Thin Solid Films **21**, 173–187 (1974). [CrossRef]

**30. **T. Yamaguchi, H. Takahashi, and A. Sudoh, “Optical behavior of a metal island
film,” J. Opt. Soc. Am. **68**, 1039 (1978). [CrossRef]

**31. **J. Spanier and I. Herman, “Use of hybrid phenomenological and statistical
effective-medium theories of dielectric functions to model the infrared reflectance of porous SiC
films,” Phys. Rev. B **61**, 10437–10450 (2000). [CrossRef]

**32. **G. A. Niklasson and C. G. Granqvist, “Optical properties and solar selectivity of coevaporated
Co-Al2O3 composite films,” J. Appl. Phys. **55**, 3382 (1984). [CrossRef]

**33. **R. Lazzari and I. Simonsen, “GranFilm: a software for calculating thin-layer dielectric
properties and Fresnel coefficients,” Thin Solid Films **419**, 124–136 (2002). [CrossRef]

**34. **I. Simonsen, R. Lazzari, J. Jupille, and S. Roux, “Numerical modeling of the optical response of supported
metallic particles,” Phys. Rev. B **61**, 7722–7733 (2000). [CrossRef]

**35. **S. A. Mayer, *Plasmonics, Fundamentals and Applications*
(Springer, 2007).

**36. **D. De Sousa Meneses, G. Gruener, M. Malki, and P. Echegut, “Causal Voigt profile for modeling reflectivity spectra of
glasses,” J. Non-Cryst. Solids **351**, 124–129 (2005). [CrossRef]

**37. **E. D. Palik, *Handbook of Optical Constants of Solids I*
(Academic, 1985).

**38. **J. Woollam, B. D. Johs, J. N. Herzinger, M. Craig, J. Hilfiker, R. A. Synowicki, and C. L. Bungay, “Overview of variable-angle spectroscopic ellipsometry
(VASE): I. Basic theory and typical applications,” Proc.
SPIE **CR72**, 3–28
(1999).

**39. **M. Honkanen, V. Kettunen, M. Kuittinen, J. Lautanen, J. Turunen, B. Schnabel, and F. Wyrowski, “Inverse metal-stripe polarizers,”
Appl. Phys. B. **68**, 81–85 (1999). [CrossRef]

**40. **A. Drauschke, B. Schnabel, and F. Wyrowski, “Comment on the inverse polarization effect in metal-stripe
polarizers,” J. Opt. A. **3**, 67–71 (2001). [CrossRef]

**41. **S. Lu and R. Chipman, “Interpretation of Mueller matrices based on polar
decomposition,” J. Opt. Soc. Am. A. **13**, 1106–1113 (1996). [CrossRef]