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The interaction of nanostructures, periodic or random, with polarized light creates very rich physics where scattering, diffraction and absorbance are linked to a variety of dispersive modes and coupling effects. Each of these excitations depends strongly on polarization, angle of incidence, azimuthal orientation of the sample and wavelength. The entire optical response can be obtained, independently from any model, by measuring the Mueller matrices at various k-vectors over a broad frequency range. This results in complex data hiding the underlying physics. Here we present a simple but versatile method to identify the physical properties present in the Mueller matrices. This method is applicable to a wide variety of photonic and plasmonic samples. Based on the simple example of a one-dimensional gold grating where the optical response is characterized not only by diffraction but also by a complex mixing of polarization, we present a very general procedure to analyze the Mueller matrix data using simple analytical tools. The calculated Mueller matrix contour plots obtained from an effective anisotropic layer model are completed by the presence of plasmonic modes, Rayleigh-Woods anomalies and the interband transition absorbance. A comparison of the so-constructed contour plots with the measured ones satisfactorily connects the optical properties of the grating to their physical origin. This straightforward procedure is very general and will be powerful for the analysis of complex optical nanostructures.
© 2017 Optical Society of America
Corrections
20 March 2017: A correction was made to the author affiliations.
1. Introduction
The interaction of polarized light and matter can be described by the 4x4 Mueller matrix (MM), which connects the incoming to the outgoing Stokes vectors, independently from any assumed model [1]. Mueller matrix spectroscopic ellipsometry (MMSE) is therefore a powerful and sensitive tool to fully characterize anisotropic and depolarizing samples. MMSE has been applied in different fields such as, for example, depolarization [2, 3], metrology [4, 5], plasmonic nanomaterials [6–11] as well as magnetic or biological materials [12, 13]. In particular it is a powerful method to characterize optical nanostructures, such as for instance plasmonic meanders for polarization control and depolarizers [9, 11], and reveal their optical properties. The focus of this article is to systematically correlate the observed polarization mixing in MMs to their underlying physical origin using a simple, versatile analytical method suitable to all samples exhibiting both anisotropy and dispersive modes. Further, the applicability can be extended to a wide variety of samples whether anisotropic or not, presenting localized or propagating plasmonic resonances, photonic modes -whether originating from waveguiding or scattering- or other dispersive modes at the only condition that an analytical expression for the dispersion of these modes exists. To demonstrate the power of this method and its implementation, we apply it to a very simple case of optical nanostructure: the one-dimensional plasmonic metallic grating. Moreover, so far, only few reports discuss the azimuthal angle dependence of the optical response of gratings in terms of MMs or the influence of diffraction orders in Mueller matrix elements (MMEs) [10, 14].
Surface plasmon polaritons (SPPs) are collective electron excitations coupled to the electromagnetic radiation propagating along metal-dielectric interfaces, evanescently confined in the perpendicular direction [15]. Owing to their unique dispersion and strong field confinement, SPPs have attracted attention during the last decades and have promising applications in integrated optics [16], field enhancement [17, 18], sensing [19–21] and imaging [22]. One-dimensional gratings are very efficient tools to excite SPP modes by matching their dispersion to that of light. When considering grating-based structures, it is well known that regular periodic nanostructures can show abrupt changes in the optical response, referred to as Rayleigh wood’s anomalies (RWAs) [23]. RWAs can result in very narrow plasmon resonances in regular plasmonic arrays of metallic nanoparticles, originating from the diffraction coupling of localized plasmons [24–28], or modify the reflectance of non-plasmonic metallic square arrays [29].
The Au grating sample is fabricated by following the procedure shown in Fig. 1(a). The schematic of the obtained sample is shown in Fig. 1(b). Figure 1(b) also shows the measurement configurations. α = 0° is defined as the classical mounting which means that the grating ridges are perpendicular to the plane of incidence of the incoming light, while α≠0° is defined as conical mounting. The excitation of SPPs in one-dimensional gratings follows either the classical mounting or the conical mounting. In grating coupling methods, p-polarized light in classical mounting is mostly used to excite the SPPs due to its high efficiency. In this geometry, s-polarized light cannot excite SPPs. However, with conical mounting, the symmetry is broken and both p- and s-polarized light can be used to excite SPPs [30, 31]. Even though SPPs on metallic gratings have been studied for a long time and RWA effects have also been well known for decades [32–36], a clear understanding of how SPPs and RWAs influence the polarization mixing of gratings is still lacking.
Fig. 1 (a) Schematic illustration of Au grating fabrication procedure. (b) Measurement configuration θ is the angle of incidence, α is the azimuthal angle with α = 0° for classical mounting and α≠0° for conical mounting.
Here, we compare both measured reflectance and full Mueller matrix data obtained from a one-dimensional gold grating in a broad wavelength range (210nm-1690nm), full azimuthal range and varied angles of incidence. The role of sample anisotropy, propagating SPPs, diffraction orders and material related absorbance is respectively identified by their different dispersion behavior and their interaction and coupling are highlighted. Finally the comparison of the measured MM with calculations based on a simple biaxial layer traces back the observed polarization mixing to its physical origin.
2. Sample description
For our study we use Au gratings produced by a self-assembly technique based on the creation of regular wrinkles on top of a surface modified elastomer, which allows the fabrication of large area gratings without the need of a lithographic process [37, 38]. For our samples, PDMS (polydimethylsiloxane) was prepared by mixing silicone elastomer base and curing agent (Sylgard 184) in weight ratio of 10:1, followed by degassing and curing at 100°C for 1 h. The two shorter sides of a rectangular PDMS slab (1mm thick, 13mm wide and 27mm long) were glued on a home-made stretching stage and the slab was linearly stretched up to 30%. Then, the stretched PDMS slab was treated by O2 plasma (90W for 10min) in order to modify the nature of the surface of the elastomer and as a consequence a 1D periodic grating was formed by slowly releasing the pre-strain. After the 1D periodic grating was formed, the sample was treated again by O2 plasma for 1 minute to increase the surface hydrophilicity and enable the deposition of a homogeneous metallic thin film. A 35nm thin gold film was finally evaporated (Univex 300) on the sample surface while keeping the time after plasma treatment as short as possible to avoid degradation of the surface treatment. An atomic force microscope (AFM) image over an area of 5µmx5µm is shown in Fig. 2(b). The AFM analysis reveals that the fabricated gratings have a regular grating period equal to p = 570nm. The depth of the grating is estimated to H = 100nm. Figure 2(c) shows five photographs taken at different angles from a fixed light source. The homogeneous colors of the patterned areas change from blue to red with increasing viewing angle indicating the high quality and excellent homogeneity of the grating.
Fig. 2 (a) Schematic drawing and (b) 3D AFM image of the sinusoidal Au grating with period p = 570nm, amplitude H = 100nm and Au thickness h = 35nm. (c) Five photographs taken from different reflection angles with sunlight coming from the incident side with fixed incident angle. The numbers of the photographs are related to the angles in the schematics in the right inset.
3. Experimental
Two different sets of measurements were performed on the sample: (1) angular dependent reflectance measurements with p-polarized and s-polarized light and (2) spectroscopic reflectance Mueller matrix measurements in the spectral range between 210nm and 1690nm at angle of incidence (AOI) varying between 25° and 65° in steps of 5° with full azimuthal rotation (α) from 0° to 360° in steps of 5° in order to fully map the anisotropy in the optical response of the sample. Both measurements were performed with a variable angle dual rotating compensator spectroscopic Mueller matrix ellipsometer (RC2) from J. A. Woollam Company.
4. Analytical expressions for SPPs and RWAs
The dispersion relation of SPPs propagating at the Au-air interface is described by the well know equation
where is the wave vector of the incoming light in vacuum, ε1 is the permittivity of air and ε2 is the real part of the Au permittivity. The wave vector mismatch between the in-plane momentum of the incoming photons and that of the SPP can be overcome by a grating structure with periodicity P. Then the conservation of momentum is obtained by setting equal to the sum of the projected wave vector of the incident light on the sample surface with an integer multiple of the grating vector:where Ki is the in plane momentum of the incoming photons, P is the period of the grating, G is the grating vector equal to , x and y are unit length vectors in x and y directions, m is an arbitrary integer, θ is the incident angle, and α is the azimuthal angle.Replacing Ksp by Eq. (1), we can obtain the relation:
Then the condition for SPP excitation is obtained by taking the square value of both sides of Eq. (3):The above relation for SPP excitation is not related to the incident polarization, which means that mixed s- and p-modes can be used to excite SPPs obeying Eq. (4). However, by tuning the incident polarization value, optimal coupling with SPPs can be obtained at a given incident angle [36]. The optimal coupling condition is given by:Here is the polarization angle (from −90° to 90°, = 0° and = ± 90° correspond respectively to p-polarization and s-polarization), is the angle of incidence and α is the azimuthal angle (α = 0° corresponds to grating ridges perpendicular to the plane of incidence).The RWAs are defined as the wavelengths where the different diffraction orders in transmission and reflection just become tangent to the sample interface. If the sample is rotated (conical mounting), the condition for RWAs is given by
Here, K// = sin θ is the component of the incident wave vector parallel to the sample surface, m is an integer indicating the order of the RWA, and n is the refractive index of the medium (air or PDMS). Solving this Eq. (6) gives the final condition for RWAswhere λm is the wavelength for RWAs at mth order, and n2 is the refractive index of air (reflection) or PDMS (transmission). The positive sign of the term in brackets corresponds to negative diffraction orders, whereas the negative sign corresponds to positive orders. The spectral dispersion of the refractive index of PDMS is very small and close to constant value of 1.4 for wavelengths higher than 300nm. The Eq. (7) is used for calculating RWAs in plots in the rest of the article.5. Experimental results
After recalling the analytical expressions for SPPs and RWAs modes as used in the remainder of the paper in the above section, we now present the intensity measurements and discuss both dispersion and polarization conversion. Then, we detail the results obtained by the measurement of the full MM.
5.1 Reflectance dispersion and angular dependence
Dispersion plots of reflectance are shown together with the expected positions (dashed lines) of the SPPs and RWAs in Figs. 3(a) and 3(b). Figure 3(a) shows the reflectance measured with p-polarized light at an azimuthal angle α = 0°. The dispersion from 822nm at AOI 25° to 1052nm at AOI 65° indicates the excitation of SPPs propagation along the sample surface. The line cut on the left side represents the values at AOI 45° along the vertical dotted lines. SPP dispersion relation is calculated and shown by a black dashed line and it follows the measured SPPs mode dispersion. The dispersion of the RWAs describes the wavelengths at which transmission and reflection orders just appear and become tangent to the sample surface. The air (−1) order RWA indicated by the red dashed curve is slightly shifted to lower wavelength from SPPs positions. Figure 3(b) shows the reflectance measured with s-polarized light at α = 90°. Interestingly, s-polarization also excites a surface plasmon resonance at α = 90° indicated by an resonance around 570nm for AOI 25° and around 515nm for AOI 65° and it also fits quite well with the calculated SPP line. The reflectance at AOI 45° with SPP excited around 540nm is shown in the line cut. However, instead of the large red shift as for p-polarization, the SPP excited by s-polarization blue shifts slightly with increasing AOI. Besides, the RWA air (−1) order at α = 90° does not follow the SPP anymore and strongly influences reflectance away from the SPP resonance.
Fig. 3 (a) Contour plot of the reflectance with p-polarized light between AOI 25° and 65° in steps of 5° at α = 0° in the spectral range between 210 and 1200nm. (b) Contour plot of the reflectance with s-polarized light between AOI 25° and 65° in steps of 5° at α = 90° in the spectral range between 210 and 1000nm. The dashed lines in the contour plots correspond to SPPs and RWAs. The line cuts in the left side of (a) and (b) are reflectance measured at AOI 45°, along the dotted lines, with p-polarized light at α = 0° and with s-polarized light at α = 90°, respectively.
In order to visualize the azimuthal dependence of the excited modes, Figs. 4(a) and 4(b) show the measured reflectance for p- and s-polarized light at AOI 45° over a complete azimuthal rotation in steps of 5° in the spectral range between 210nm and 1200nm together with SPPs, RWAs and interband transition lines. The interband transition of Au at 480nm [39] exhibits no dispersion and therefore corresponds to a circle in the polar plot, while the azimuthal dependent RWAs form arched curves. As expected, the reflectance shows a simple C2- symmetric behavior due to the symmetry of the grating. When a grating is used to launch SPPs, it is necessary for the incident light to have a polarization component that is perpendicular to the grooves [15]. So, SPPs cannot be excited near α = 90° with p-polarized light and not near α = 0° with s-polarized light, which is proved in Figs. 4(a) and 4(b). According to Eq. (5), the optimal azimuthal angles for SPP excitation with p- and s- polarized light is at α = 0° and α = 90°, respectively. The position of the excited resonances (i.e., reflectance dips) in Figs. 4(a) and 4(b) follows the calculated SPPs lines very well confirming their plasmonic origin. In Figs. 4(a) and 4(b) the influence of the RWAs is in general rather weak. In Figs. 4(c) and 4(d), the polarization conversion Rps (incoming p- into reflected s-polarized light) and Rsp (vice versa) are shown. As expected for a C2-anisotropic sample Rps and Rsp are identical with maximum values at α = 45° and two optical axes at α = 0° and 90°. The polarization conversions Rps and Rsp basically trace the calculated SPPs curve, indicating that polarization conversion is mainly caused by surface plasmons and that the contribution of the RWAs is small. The isotropic interband transition exhibits no polarization conversion.
Fig. 4 Contour plots of experimental reflectance with p-polarized (a) and s-polarized light (b) together with the SPPs, different order RWAs and the interband transition lines. The excitations are only plotted in the upper half-space to avoid masking of the raw data in the other half space. (c), (d) experimental polarization conversion Rps and Rsp with SPP lines. All the contour plots are at AOI 45° over a complete azimuthal rotation in step of 5° in the spectral range between 210nm and 1200nm. The polar axis represents the wavelength λ and the polar angle represents the azimuthal angle α.
5.2 Mueller matrix in reflection
To get a deeper insight in the influence of SPPs and RWAs on the complex optical behavior of the Au grating, MME measurements were carried out in reflection geometry in the spectral range between 210nm and 1200nm at AOI 45° over a complete azimuthal rotation (Fig. 5). All the MM elements are normalized to M11 element which represents the total reflectance of the sample. To visualize the huge amount of data accumulated in this kind of measurement, the elements of the MM are presented at a given incident angle as polar contour plots, where the azimuthal angle α is the polar angle and the radial axis represents the wavelength. In order to obtain a better visualization due to weak values in some MM elements, we scale some elements with multiplication factors. In general, as we can see from Fig. 5, all 16 elements exhibit complex patterns and depend on the azimuthal angle and the wavelength. In reflection, the MMEs show the expected symmetry with identical element pairs M12/M21, M14/M41, M24/M42 and inverted element pairs M13/M31, M23/M32 and M34/M43. All the patterns reflect the symmetry of the grating with optic axes at α = 0° and 90°.The off-block-diagonal elements, which are the upper right and the lower left 2x2 submatrices, represent the anisotropy and cross-polarization information of the sample. From the off-block-diagonal elements, one can see that the Au grating is strongly anisotropic and therefore mixes polarization states. Moreover, the off-block-diagonal elements show curved lobes with maxima around α = 45° and 135° at the excitation wavelengths of the dispersive SPP and RWA modes. The element M12 represents linear dichroism and reflects Rps and Rsp, which are equal [15]. The diagonal elements resemble the reflectance plots of Fig. 4. By multiplying the Mueller matrix by an input Stokes vector corresponding to p-polarized light Sin = (1 1 0 0) or s-polarized light Sin = (1 −1 0 0) we obtain a contour plot equal to that presented in Fig. 4 (not shown).
Fig. 5 MMEs measured in reflection at AOI 45° over the complete azimuthal rotation in the spectral range between 210nm and 1200nm. All MMEs are normalized to M11.
6. Modelling results and interpretation
6.1 Modelled reflectance
In principle it is of course possible to calculate the MMs for different angles of incidence, various azimuthal orientations and a broad range of frequencies by solving the Maxwell-equations under the boundary conditions given by the 3-dimensional geometry of the grating. But this approach is on the one hand time-consuming and computer intensive and on the other hand it does not really promote the physical understanding of the origin of the observed optical behavior. In order to correlate the observed MM pattern to the known properties of the grating, i.e., its periodicity, the material parameters and the SPP and RWA modes, we present here a much simpler 2-dimensional approach based on a cumulative method, starting with a simple anisotropic effective medium model based on the Fresnel equations to which we add the dispersive modes described above.
In a first step, we prepare an optical model able to reproduce the measured intensity taking into account the anisotropy of the sample. According to the symmetry of the grating, the sample has two optical axes along X and Y. Therefore, we model the reflectance along α = 0° and α = 90° using Gaussian oscillators with the ellipsometry software [40]. A flat, pure PDMS sample was measured at different angles of incidence by spectroscopic ellipsometry and the extracted permittivity is used for the substrate of 1 mm thickness in all models. A 35nm thick biaxial layer is placed on top of the substrate. Along Z direction normal to the layer interface, a Cauchy oscillator is used, while in X and Y directions we use general oscillator models. The parameters of the oscillators in X and Y direction are obtained by fitting the measured reflectance Rpp and Rss, along the azimuthal angles 0° and 90°. The model generated by fitting the measured Rpp (Rss) along its optical axis is called in this article “p-model” and “s-model” respectively. Using these models, we start by calculating both the p- and s-reflectance at AOI 45° over the whole azimuthal range and compare the result with the experimentally obtained reflectance plots in Fig. 6. The expected anisotropy of the reflectance is visible in both, the experimental and the modelled plots. The calculated and measured Rpp and Rss are very similar.
Fig. 6 Rpp (a) and Rss (b) measured at AOI 45° over a complete azimuthal rotation in the spectral range between 210nm and 1200nm. Rpp(c) and Rss(d) generated from the p- and s-biaxial models, respectively. All the plots are shown together with the calculated SPP lines in the top half space.
The second step of the method is to add the analytically calculated positions of the excited SPPs from Eq. (4), which are also shown in all graphs of Fig. 6. Since no dispersion effects are included in our biaxial models, the dispersive SPP modes are not accurately reproduced. In particular, in our simple model the plasmonic resonances are independent of the azimuthal orientation and therefore appear as part of a circle due to the simplification of the model. This is visible when one compares Fig. 6(c) with Fig. 6(a). However, this method allows us to determine what comes from the plasmonic mode from what is due to anisotropy. The outer feature of the measured reflectance is attributed to the plasmonic resonance, which follows very well the analytical curve in Fig. 6(a). The signal at lower wavelengths is mostly determined by anisotropy and non-dispersive effects, therefore both modelled and measured graphs are very similar in this wavelength range. We now turn our attention to Rss (Fig. 6(b) and d). Similarly to Rpp, the polar plots of Rss are reproduced by the biaxial model at shorter wavelengths while deviations are observed at the SPP wavelength. In particular the “V-shape” of the SPP around α = 90° is not reproduced by the pure biaxial model and can be identified only when the analytical mode dispersions are superimposed.
6.2 Modelled Mueller matrix
Once the models obtained for p- and s-polarized light reproduce reasonably well the intensity data, we use them to calculate the MMs at AOI 45° as shown in Figs. 7(a) and 7(b), respectively. In general, we can see that both p and s-models can reproduce the measured signal in the shorter wavelength region (210-690nm). However, in the longer wavelength region (890-990nm), which is mainly influenced by SPPs, the generated MME show deviations to the measurements. Here also the effects of dispersion are not included and all the features at longer wavelengths corresponding to SPPs follow circles instead of arcs. For example, the measured M12 shows SPP curved features near α = 0° and another SPP feature at lower wavelength near α = 90°. This can be understood by recalling that the p-model (s-model) considers as a simplification only the p-polarization (s-polarization respectively). However, the measured MME exhibit the full optical response and there s-polarization also plays an important role. Therefore, Figs. 7(a) and 7(b) need to be combined to explain the full measurements. We can see that the features near α = 90° are reproducing the measured features, which also proves that the SPPs near α = 90° are excited by s-polarization. However, the calculated M24, M23 and M34 at higher wavelength have opposite phase compared to the measurements, a feature obviously not captured by the simplifying model. From this preliminary result we can see that we obviously need both the pure anisotropy modelled by an effective medium approach for both s- and p-excitation, including the plasmonic effects, and the dispersion originating from the periodicity of the sample. In the next section, we will give more details on the physical interpretation of all the features as well as their interplay.
Fig. 7 Simulated Mueller matrix at AOI 45ᵒ in the spectral range between 210nm and 1200nm with full azimuthal rotation generated from P biaxial model (a) and S biaxial model (b). Multiplication factors are used to scale the data to [-1;1].
6.3 Identification of the optical features
For sake of simplicity, we now choose only the four MME (M12, M13, M24 and M34) for a more detailed analysis shown in Fig. 8. Our aim here is to illustrate the method of superposition of the separated physical effects through their respective analytical models. This superposition, taking into account the assumptions of each model, is helpful in comparison with the measured data in order to reveal the origin of the optical properties. The detailed comparison of all the elements of this figure gives us a complete description and interpretation of the complex pattern of the measured MM. Each row of Fig. 8 corresponds to one of the four chosen MME, all measured or generated at AOI 45°. The first column displays the measured data superposed with the analytical dispersion of the SPP modes. The second column compares the measured data with the expected positions of some RWA lines as well as the interband transition of gold. The last two columns illustrate the MME calculated by the anisotropic p- (resp. s-) models, together with the analytical positions of the SPPs, RWAs and the interband transition of gold.
Fig. 8 Measured and simulated Mueller matrix elements M12, M13, M24 and M34 together with the SPP, RWA and interband transitions draws in the upper half space at AOI 45°. Simulated p- and s-model means, that the Mueller matrices are calculated only from the anisotropic effective medium approach obtained from the s- and p-reflectance measurements. The multiplication factors give the enhancement factor in respect to the scale bar.
The first column, comparing the measured data with the expected position of the propagating SPP mode, reveals that the outer feature of the MMEs is strongly influenced by the plasmonic mode. However we can see that the measured signal follows the SPP lines only over a certain azimuthal angle range. Indeed, the excitation angle range is determined by the sample anisotropy and can be calculated by our anisotropic layer model (last two columns): the simulated MMEs, dependent on anisotropy, indicate the azimuthal range where the excitation of the features is allowed and its respective strength. Moreover the comparison of the p-model and s-model allows us to determine which measured feature is linked with which polarization. However, since the anisotropic layer model does not take dispersion into account, it is expected that the curvature of the SPP mode is not reproduced. As a result of a dispersionless model, the simulated elements M12 and M13 predict the SPP mode position correctly only at α = 0° (for p-model) and α = 90° (for s-model). For all azimuthal angles in between these values, the spectral position of the SPP modes deviates following the SPP analytical line. In other words, intentionally curving the simulated features with respect to the azimuthal angle following the SPP mode dispersion leads to similar patterns as those measured. Therefore the interpretation of the comparison shown in Fig. 8 is as follows: the simulated MMEs – last two columns – display the range of azimuthal angles α where spectral features due to anisotropy appear (in the respective MMEs), while the position of the dispersive modes interact with the anisotropy related features by either curving the branches or modifying their shape and width.
When we turn our attention to the features at the center (shorter wavelengths between 210nm and 690nm), we can see that the superposition works similarly to the case of the SPP modes. In this spectral region, the optical properties are influenced by anisotropy, diffraction orders (RWAs) and interband transitions. In particular, the direct visual comparison between the anisotropy related lobes, the position of the RWAs and the measured MMEs indicate clearly that the shape of the MMEs is produced by a modification of the anisotropic signal by all the physical phenomena present in this spectral range. If we first consider M12, along the 180°-0° azimuthal line, we can clearly see that the anisotropy lobe is modulated by the presence of the RWA air (−2) (green) and RWA PDMS (−2) (black) orders. The same is valid for M13 along the 235°-45° line, and for M34 at most azimuthal angles. The latter is, in addition to the influence of the RWAs, also influenced by the Au interband transition. Quite interestingly, if we compare M13 with Fig. 4, we can see that M13 is really similar to the cross-polarized reflectance Rps or Rsp over the whole measurement range, because M13 reflects the anisotropy effect of the sample, which strongly influences the polarization conversion.
The method presented here is based on the direct comparison of simple analytical models, each describing one distinct physical aspect: in our particular case, these are linked to anisotropy, SPP, dispersion, diffraction and interband transition. The influence of these four main aspects is easily identified. The interplay between modes can be found at the intersection between their expected positions. This physics-based approach is very general and can be applied to various nanostructures in order to predict and interpret the Mueller matrix contour plots. The separate role of anisotropies and photonic/plasmonic modes is at first distinguished: once the isotropic or anisotropic optical model is made, it is extended with the analytical expressions of the identified additional modes. These could be, for instance, localized plasmonic modes, photonic passive modes originating from waveguides or scattering, active modes related to emissive nanostructures, etc, under the condition that an analytical dependence of the optical dispersion can be given as a function of λ, θ and α. Beyond the simple interpretation, the exact attribution of every spectral feature (both in amplitude and in phase) together with its azimuthal dependence opens up the path to a possible tailoring of specific functionalities of nanostructures and therefore paves the way for a very precise control of the design and metrology of plasmonic nanostructures. Therefore the presented method is not only powerful in identifying the optical features of a given sample, but also for optimization of the design of novel structures and for characterization or metrology. The optimization of the structure design will be facilitated by the decomposition of complex patterns into its basic modes, which can be tailored individually.
7. Conclusion
In summary, we have demonstrated how the complex optical response of a simple Au grating can be decomposed into its physical ingredients. First, we measure the reflectance along the two optic axes of the grating, along and perpendicular to the grooves under s- and p-polarization. The reflectance is then modelled by a simple anisotropic effective medium approach using Drude-Lorentz oscillators. From this anisotropic model, the intensity plots (reflectance in our case) over the whole spectral and angular range are generated. Once the agreement between the generated and measured plots is insured, the Mueller-matrix plots can be calculated. On top of this calculated MME we superimpose the expected dispersive SPP and RWA modes, calculated from the known periodicity of the grating. Comparing this composed result with measured MMs gives a deep insight on how the different physical contributions originating from periodicity, anisotropy and material properties influence the complex polarization mixing. We have seen that SPPs can be excited by both p- or s-polarized light when the incident plane is perpendicular or parallel to the grating grooves. Both SPP modes are dispersive with the angle of incidence and follow the same phase matching condition. P- or s-polarized light can be converted to s- or p- polarized light via SPP excitation, and maximum polarization conversion occurs when the angle between incident plane and grooves is 45°. Additionally to the excitation of SPPs, the optical properties are influenced by geometric anisotropy, the RWAs related to the periodic grating structure and, to a lesser extent, the Au interband transition. The anisotropy, the interband transition and the non-dispersive approximation of the SPPs are understood in terms of an effective medium approach, obtained from fitting the measured reflectance. However, the dispersion of the SPP modes and the RWAs effects should be added (directly from their analytical expressions) on top of this model. This straightforward procedure is very general and it is applicable to the analysis of different optical nanostructures taking into account any anisotropy and dispersion of the supported optical modes. Mueller matrix ellipsometry is increasingly appreciated for its fast, non-invasive and precise metrology. In this perspective, the physical understanding of MM contour plots can be used as characterization tool to aim a particular physical feature. In conclusion, this study presents a general method to analyze complex MM patterns from a physics-based approach and it is a more powerful and straightforward method than the use of full Maxwell solvers: it needs much less computer power, it is very fast and it is directly linked to the physical interpretation.
Appendix Sample fabrication method
The fabrication procedure is detailed in Fig. 1(a) and described here:
- 1. The PDMS slab was linearly stretched up to 30%.
- 2. The stretched PDMS slab was treated by an O2 plasma (90W for 10min) inside a plasma etcher chamber at a pressure of 1.4 mbar (Pico low-pressure plasma system from Diener electronic company) in order to modify the nature of the surface of the elastomer.
- 3. 1D periodic grating was formed by slowly releasing the pre-strain.
- 4. The sample was treated again by an O2 plasma for 1 minute to increase the surface hydrophilicity and enable the deposition of a metallic thin film.
- 5. The sample was placed inside the evaporation chamber (Univex 300) as soon as possible (less than 1h) to avoid degradation of the surface treatment and a 35nm thin gold film was finally evaporated on the sample surface.
Funding
Chinese Scholarship Council (CSC) (201307040036); Carl Zeiss Stiftung.
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