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We report on measurements of optical activity in reflection in the conical mount from two plasmonically resonant nanostructures; a sub-wavelength silver meshed grid and a fishnet metamaterial. The square-centimeter size of the materials, formed by nano-imprint lithography, allows reliable investigation of such materials by plane-wave techniques with minimal focusing. For both materials we observe strong polarization conversion (s- to p-polarization, and vice versa) in generalized ellipsometry measurements. We compared the spectra to analytical predictions using surface plasmon polariton (SPP) theory and find good agreement for the meshed grid. The spectra for the meshed grid are also well modeled using the rigorous coupled wave analysis (RCWA) technique. Simulated results for the more complicated fishnet layer showing qualitative agreement are also presented. We then probe the validity of describing the observations using homogenous parameters such as dichroism and birefringence, by examining the calculated reflection of nominally polarized incident light using simulated and measured Mueller matrices. The results show that the cross-polarization that we observe is primarily related to linear birefringence and dichroism, although circular effects are indeed present.
©2013 Optical Society of America
1. Introduction
The reflective and refractive interaction of polarized plane waves with periodic nanostructures and metamaterials is of topical interest [1–6]. Recent results demonstrate strong conversion of linear polarization states (s-polarized to p-polarized, and vice-versa) and selective reflection and transmission of right- or left-circular polarization [1,2]. Optical activity (OA) in nanostructured planar metamaterials has in particular aroused great interest [3–6]. OA occurs due to a variety of phenomena [7] including propagation through linear anisotropic and gyrotropic materials. Natural optical activity is observed in solutions of chiral molecules and solids with rotated crystal planes (e.g. quartz) and is attributed to spatial dispersion [8]. Artificial gyrotropy has been known for over a century [9,10] and is attributed to circular Bragg phenomena [11]. Many current descriptions of artificial gyrotropy are a miniaturization of these original experiments.
Recent reports reveal OA and chirality in symmetric periodic structures under oblique-incidence and under azimuthal rotation in transmission [12] and reflection [6]. These results have strong analogy with measurements of diffraction gratings in the conical mount, i.e. where the incident plane is not-perpendicular to the direction of the grating groove. The name stems from the observation that the diffracted orders lie on a cone whose axis is parallel to the grooves [13]. The orders contain significant proportions of cross-polarized light [14]. If the period of the grating is sub-wavelength (wavelength-to-period ratio, λ/P > 1) then only the zeroth order is non-evanescent. The optical properties are then well approximated by classical form birefringence described by an anisotropic effective refractive index, which is well known to produce significant cross-polarization under azimuthal rotation. Metallic gratings show unusually strong cross-polarization at surface plasmon polariton (SPP) frequencies in the conical mount [15]. Metallic meshed grids (also called inductive grids [16]) and sub-wavelength hole arrays (SWHAs) also exhibit unusual transmission [17] and polarization properties along the high symmetry planes [18], and magnetic-type resonances in stacked layers (also called fishnet metamaterials [19]).
Given the recent emergence of numerous periodic sub-wavelength nanostructures, both chiral and achiral, and their associated optical activity, it is timely to consider the quantification and differentiation of the physical origins of the observable polarization conversion properties, particularly under polar and azimuthal rotation. Adhering to the concept of metamaterials as effectively homogenous materials, many recent results have been interpreted using the traditional material properties of circular dichroism (CD) and birefringence (CB) [6,20]. This follows the trend of describing metamaterials using bulk material parameters such as effective refractive index and effective permeability, the use of which has been enthusiastically debated [21]. In this work we first demonstrate optical activity in oblique-incidence reflection from periodic metamaterials under azimuthal rotation. Using generalized ellipsometry (GE) in the conical mount we measure the reflective polarization properties of two samples; a meshed grid and a fishnet metamaterial. We are interested in the polarizing properties of the SPPs, including the so-called “gap-SPP” which provides the conditions for negative refraction in fishnet metamaterials. We compare the predictions of the SPP- Bloch wave theory, which is often used to describe such materials [22]. We explore the relative merits of the Jones and Mueller formalisms to analyze our data, and investigate their potential for identifying chiral effects. The attribution of homogenous parameters such as circular dichroism and birefringence is explored and discussed.
2. Experimental section
The material design is a periodic square mesh array shown schematically in Fig. 1(a) . Two samples, each with total dimensions of 1 cm2, were fabricated by nanoimprint lithography, similar to the process described in [23]; (a) a single-layer silver meshed grid and (b) a fishnet metamaterial consisting of three layers of Ag/SiO2/Ag. Both samples were deposited on a silicon substrate with native oxide. The nominal dimensions of the structures are period d = 365 nm, hole side length a = ax = ay = 205 nm, film thickness h = 40 nm (Ag) and s = 20 nm (SiO2), in all structures. From the scanning electron microscope image of the fishnet sample [Fig. 1(b)] we observe that the side lengths of the holes are larger than the nominal values due to the fabrication process, in particular those of the upper layers. The actual values measured from the images are a = 240 ± 5 nm for the bottom layer and a = 310 ± 5 nm for the upper layer.
Fig. 1 Schematic of (a) the silver meshed grid and (b) the fishnet metamaterial. Both are supported by a Si substrate. (c) Scanning electron micrograph of the fishnet metamaterial. The scale bar is 500 nm.
We measured the samples using spectroscopic ellipsometry (SENresearch and J.A.Woollam). The samples are large enough (cm2) to be reliably investigated by plane-wave techniques with minimal focusing. We tested the equivalence of using both an unfocused probe spot (diameter ca. 1mm) and focusing optics (diameter ca. 200 µm). We found the difference to be primarily in the noise level of the measurement. This also allowed us to confirm the homogeneity of the sample at different positions. We recently described [24] the incidence-angle dependence of SPP modes in these materials measured along the optical axes, and identified a magnetic resonance in the fishnet metamaterial near 780 nm. The real effective refractive index is negative between 750 and 790 nm, determined using normal incidence reflection and transmission measurements and the standard retrieval method [25]. Here we present results of measurements in the conical mount at an incident angle of 45° and azimuths from 0°-45° at 5° intervals. The optical properties of the samples were symmetric upon azimuthal rotation through 90°.
The results of previous works on polarization conversion in metallic gratings [15] were presented as the intensity of reflected s-polarized light detected when p-polarized light was incident. The results were normalized to the direct intensity of the p-polarized light. This is similar to a generalized ellipsometry [26] measurement where the elements of the Jones matrix are measured. In the Jones formalism one defines the polarization states as orthogonal electric field components, Ep and Es. Reflection from a surface of a propagating light ray is expressed by the Jones matrix [27]
where the matrix elements are the reflection coefficients and the subscripts r and i denote the reflected and incident rays, respectively. In isotropic materials, or along high symmetry axes, the off-diagonal elements, rps and rsp , are zero. Standard ellipsometry measures the ratio of the two diagonal elements, which may be expressed in terms of the ratio amplitude, Ψ, and phase difference, Δ, as:Generalized ellipsometry (GE) [28,29] measures the 3 complex elements of the normalized Jones matrix (often normalised by rss), giving the additional measured parameters, Ψps/ss, Δps/ss, Ψsp/ss and Δsp/ss. For all measurements performed in this study we observed that within the experimental noise, Ψps/ss = Ψsp/ss and Δ ps/ss = Δ sp/ss.The Jones vector cannot describe depolarized or partially polarized light. In contrast the Stokes-Mueller formalism defines real parameters which are directly measureable as irradiance, I. The Stokes vector, S = [S0, S1, S2, S3]T contains the Stokes parameters S0 = Ix + Iy, S1 = Ix - Iy, S2 = I+45 - I-45, S3 = IR - IL, where the subscripts denote polarization in the orthogonal x and y directions, at 45° to x and y, and right- and left-circularly polarized, respectively. The 4 x 4 Mueller matrix (MM)
represents the transformation of an incident Stokes vector Si to a vector Sr reflected from a sample surface. A complete or partial measurement of the MM using ellipsometric concepts is referred to as Mueller matrix ellipsometry (MME). In the case that the light is completely polarized, there is a direct equivalence between the Jones and Stokes-Mueller formalism [30].Primary causes of depolarization include; inhomogeneity of the film thickness and/or material refractive index; finite spectral resolution; backside reflection from a weakly absorbing substrate; incoherent superposition of the light from two or more materials; large sample surface roughness causing scattering of the probe beam (e.g. imperfect periodic structures); variation in the incident angle of the probe beam (e.g. caused by focusing optics) [31]. In our experiment we consider that only the last two examples make any significant contribution to depolarization. Our measurement system uses only limited focusing to keep depolarization effects comparable with other sources such as the spectral resolution. If the periodic sample is perfectly fabricated then reflection and diffraction of a plane wave should cause no depolarization, even in the conical mount [31].
3. Experimental results of the meshed grid
Figure 2 shows the tanΨpp/ss and tanΨps/ss data of the meshed grid measured in reflection at an incident angle of 45° as a function of rotation angle (0°-45°) and photon energy (0.75 - 3.3 eV). The results are presented as grey scale maps, and the white dashed lines show the predictions of the analytical formulas used to describe the excitation of SPPs by periodic gratings [22]. The SPP wavevector,, of a SPP propagating at the interface of a dielectric(permittivity εa) and a metal (permittivity εm) is given by the dispersion relation
where is the vacuum wavevector. A Bloch-wave (BW)-SPP may be excited on a grating if matches an in-plane lattice vector, , with additional momentum from the in-plane component of the plane wave , such thatwhere i and j are integers and θ is the angle makes with the normal. For εm we used a Drude model with ε(ω) = εinf - A/(ω2 + iΓ.ω) where εinf = 4.411, A = 85.3 eV and Γ = 0.018 eV. Values for εa were 1 for air and 11 for the silicon substrate [32].Fig. 2 Generalized ellipsometry parameters (a) tanΨpp/ss and (b) tanΨps/ss measured under azimuthal rotation for the meshed grid. The grey scale values are displayed at the top of each element. The predictions of Eqs. (4) and (5) are also shown as dashed lines in (a) for the Ag-Si interface (bottom modes below 1.5 eV) and Ag-air interface (top modes, above 1.5 eV).
The model predictions match well the dispersion of modes in the experimental data. The strongest modes are the Ag-Air (−1,0) and (0,-1) SPPs. At 45° azimuth the (−1,-1) branch is clearly observed. The weaker Ag-Si interface modes are also clearly observed at lower energies, in good agreement with the model. In the tanΨpp/ss data, the (−1,0) mode is more efficiently coupled at 0° azimuth, while the (0,-1) and (−1, −1) modes are more efficiently coupled at 45°. In the tanΨps/ss data the modes are not observed along the high symmetry directions (0° and 45°) but are present at other azimuths, reaching a maximum at 22.5° for the (−1, −1) mode (this is not clearly seen in the grey-scale plot but can be easily observed in line plots). This is in agreement with the measurements of ref [6] for SWHAs and in contrast to the 1-D grating case of ref [15] where the maximum was at 45°. Both the Ag-Air and Ag-Si (−1,0) modes show a maximum close to 30° azimuth in the tanΨps/ss data.
The results show that in the low-symmetry directions a significant amount of cross-polarization occurs, while in the high-symmetry directions the cross-polarization is negligible. This effect is well documented for 1D gratings [14] where diffraction and reflection in the conical mount produce strong cross-polarization effects. The introduction of a second orthogonal grating direction in a 2D grating introduces another symmetry axis. BW-SPPs are excited at all azimuths, however only along the high symmetry directions do they maintain their polarized state upon re-emission.
Figure 3 shows simulations using RCWA in the commercial software package RSoft of the meshed grid. For both the 0° and 45° azimuths, where the cross-polarization is zero, the positions of the resonances are in good agreement with the measurements. The simulated polarized reflectances |rpp|2 and |rss|2 are also shown, identifying the contributions of the p- and s-polarized components. Note that the s-polarization couples more efficiently at 45° than at 0°, in particular for the degenerate (−1,0) and (0,-1) modes. The (−1,-1) Si-Ag mode at 45° is at almost exactly the same energy as the degenerate (1,0) and (0,1) modes. However we observe that the Air-Ag mode is stronger and distinct from the degenerate (1,0) and (0,1) modes, which occur close to the effective plasma frequency εeff = 0. At this frequency we observe a strong resonance due to polarization of the silver film in the out-of-plane direction. This is the Ferrell mode; the plasmonic variant of the Berreman mode [33]. We also show the location of the localized surface plasmon resonance (LSPR) (dipolar Mie resonance) which is due to the polarization of the metal wires.
Fig. 3 Comparison of RCWA simulations of the tanΨ spectra (black line) with the measured values (red line) at azimuths of a) 0° and c) 45° for the meshed grid. The reflectances of the p- and s-polarizations are shown from the RCWA simulation for b) 0° and d) 45°.
4. Experimental results of the fishnet metamaterial
Figure 4 shows the tanΨ data for the fishnet metamaterial in the energy range from 0.75 to 1.9 eV measured at 45° incidence angle. The observed features arise from SPPs excited at the Ag-Si interface and the gap-SPP which is a coupled mode at the two internal Ag-SiO2 interfaces. For the gap-SPP an effective εa = 5 was used [24]. We noted in our previous study [24] that the assumption of an effective εa for the gap mode did not completely agree with the observed angular dispersion. This is also observed here under azimuthal rotation, particularly at 45° azimuth near 1.1 eV.
Fig. 4 Generalized ellipsometry parameters (a) tanΨpp/ss and (b) tanΨps/ss measured under azimuthal rotation for the fishnet metamaterial. The grey scale values are displayed at the top of each element. The predictions of Eqs. (4) and (5) are also shown in (a) for the Ag-Si interface (bottom modes below 1.125 eV) and gap-SPP (top modes above 1.125 eV).
The dark band near 1.1 eV at 0° azimuth is clearly the (−1,0) mode of the gap-SPP, as we showed previously [24]. According to the theory this mode should be the lowest energy mode of the gap-SPP and should increase in energy under azimuthal rotation. We do not observe this. The mode stays at almost the same energy under rotation. In the tanΨps/ss data this same mode shows the strongest coupling in the low-symmetry directions, with a maximum at 22.5° azimuth. At 45° azimuth the (−1,0) mode of the gap-SPP evolves into two bright bands either side of 1.1 eV. This appears to be the (−1,0) mode of the gap-SPP split by the (0,1) mode of the Ag-Si interface. The change from dark to bright in the figure implies that the coupling switches from p- to s-polarization. The simple SPP-Bloch wave analysis requires further refinement to properly predict the modes under azimuthal rotation.
Of interest is the magnetic-type resonance arising from the (−1, ± 1) gap-SPP resonance (ca. 1.62 eV at 0° azimuth), and its behavior under azimuth rotation. The tanΨps/ss data shows significant cross-polarization from this mode along low-symmetry azimuths. We clearly observe the mode splitting into the (−1,1) and (−1,-1) modes that are degenerate at 0° azimuth. The maximum cross-polarization occurs at around 10° azimuth for these modes.
Figure 5 shows the tanΨpp/ss along the high symmetry axes. Values of tanΨ pp/ss greater than 1 (|rss|2 < |rpp|2) are a signature that the (−1, ± 1) mode is associated with an SPP strongly coupled to s-polarized light. Note that in Figs. 3(b) and 3(d) |rss|2 is always greater than |rpp|2 and therefore tanΨ is less than 1. The tanΨpp/ss are greater than 1 at the frequency of the magnetic-type resonance at 0° azimuth. This provides a quick method to search for potential magnetic-type resonances. At 45° we observe a double peak with tanΨ > 1 [Fig. 5(b)] near 1.1 eV, the origin of which was discussed above. Whether this mode corresponds to an anti-symmetric SPP with anti-parallel group and phase velocities requires further investigation.
Fig. 5 Comparison of RCWA simulations of the tanΨ spectra (black line) with the measured values (red line) at azimuths of a) 0° and c) 45° for the fishnet metamaterial. The reflectances of the p- and s-polarizations are shown from the RCWA simulation for b) 0° and d) 45°.
The fishnet metamaterial is more complex in design than the meshed grid. Figure 5 shows the RCWA simulations using nominal dimensions, with sidewall angles introduced to account for the observed differences in the widths of the structures (the difference between dimensions d and a in Fig. 1). The simulations show qualitative agreement with the measurements, but the strength and position of the resonances is not well matched. Our model requires further improvement to match the measured data. Aside from small differences expected in the dielectric functions of the materials, the most likely cause of the discrepancy is a difference between the nominal dimensions of the nanostructures and the actual dimensions. We have observed that a small difference in these values can significantly affect the optical spectra.
5. Discussion
Conical diffraction of the zeroth order in sub-wavelength 1-D gratings is well modeled using effective medium theory [34]. The observed polarization conversion is then equivalent to that which occurs upon reflection from an anisotropic crystal when the wave vector is not parallel to the crystal optical axis [35]. For 2-dimensional periodic materials one cannot use an effective dielectric tensor since the symmetry is no longer such that a biaxial description will suffice, as noted by Gompf et al. [6]. As well as the high symmetry directions along the 0° and 90° azimuths there are now high symmetry directions along the 45° and 135° azimuths, as we observe here. At azimuths away from these symmetry planes we observe strong cross-polarization. In the absence of an effective medium model we require a physically consistent description of the origin of this cross-polarization.
References [6] and [12] describe the cross-polarization as chiral, or gyroscopic, in origin. In reference [6] the circular dichroism is extracted directly from one element of the Mueller matrix. It is of interest to explore this further. Under the assumption that the sample is infinitely thin, one may read directly from the transmission Mueller matrix the linear and circular birefringence and dichrosim [36], which are directly connected to the diattenuation and retardance [37]. The polarization properties are resolved into eight basic types of optical behavior, namely; isotropic refraction (n) and absorption (A), linear dichroism (LD) and birefringence (LB) (along the xy co-ordinate axes), LD' and LB' (along the bisectors of the xy axes), and CD and CB. Real samples have finite thickness which leads to a MM that is a complex convolution of the optical behaviours [38]. Only if LB, LD, LB' and LD' are zero can the CD and CB values be determined directly from the anti-diagonal MM elements, as is performed in commercial CD instruments for organic liquids [39]. It is important to note that LB and LD will contribute significantly to apparent CD and CB signals, causing so-called chiroptical artifacts [40]. Only in the special case that the linear elements are zero is the reflected polarized light circularly polarized; non-zero values of the linear elements imply an elliptical polarization state [41].
In reflection the definition of CD and CB is not established. In fact, for normal incidence reflection, reciprocity dictates that OA should not be observed [42]. However oblique incidence reflection from a sub-wavelength hole array results in an optical response that is equivalent to CD and CB, even though these terms are defined for normal incidence transmission [42]. Arwin et al. recently presented MM ellipsometry measurements of chirality-induced polarization effects in beetles, which clearly demonstrate that true chiral effects can be observed in oblique incidence reflection measurements [41]. For technological applications, one need not distinguish between a reflected or transmitted wave as long as the observed effect is the same.
In principle chirality may contribute to the cross-polarization observed in our experiment. To gauge its contributions requires distinguishing the linear and circular contributions to the observed OA. A standard experiment for chirality is the measurement of the difference of the transmitted or reflected intensities of right- (RCP) and left-circularly polarized (LCP) incident plane waves. We performed this comparison by simulating the reflection from the meshed grid sample at polar 45° and azimuth 20°. Figure 6(b) shows the difference in total reflectance [(|rss|2 + |rpp|2] for incident RCP and LCP plane waves (i.e. in Eq. (3), the difference between Sr(S0) for Si = [1,0,0,1]T and [1,0,0,-1]T). Particularly at the Ag-Si interfacial SPP frequencies there is notable difference in the reflected power for the two circular polarizations. However, comparison with the difference in total reflectance for incident orthogonal linear polarizations (i.e. the difference in Sr(S0) for Si = [1,1,0,0]T and [1,-1,0,0]T) shown in Fig. 6(a) reveals that the effect on the circular polarizations is weaker by more than an order of magnitude. Thus we conclude that chirality is indeed “mimicked” by conical diffraction in planar isotropic 2D plasmonic metamaterials, although the effect is very weak compared to the OA caused by linear anisotropy in the conical mount.
Fig. 6 Simulated difference in the reflectance of incident plane waves from the meshed grid for (a) orthogonal linear polarizations and (b) left- and right-circular polarizations. Note the difference in scale of the ordinate axes.
A similar analysis may be performed using the MM of the sample to calculate the polarizing properties of the sample for an unpolarized incident plane wave. Since our experimental arrangement only allows the measurement of the first three rows of the MM, we simulated the MM for the meshed grid for the same polar and azimuthal conditions above. The MM elements, normalized to m11, are shown in Fig. 7 . Firstly we note the very strong off-diagonal elements, particularly m13, m31, m24 and m42, which denote the conversion of S1 into S2 and vice versa. Additionally we may make the following statements about the symmetry of the MM for materials with in-plane square symmetry; m12 = m21; m13 = -m31; m14 = m41; m23 = -m32; m24 = m42; m34 = -m43. Although m44 appears similar to m33, it differs by an amount proportional to m22, namely; m44 = m33 - m22 + 1. The validity of these statements is easily verified theoretically by noting that Ψps/ss = Ψsp/ss and Δps/ss = Δsp/ss and using conversion rules for Jones-Mueller matrices [30].
Fig. 7 Simulated MM for the meshed grid using RCWA at polar 45° and azimuth 20° showing the symmetry of the elements.
Using the above relations it is straightforward to generate the final row of the MM for our measured data for the fishnet metamaterial. We now use this to calculate the polarizing properties of the sample for an unpolarized incident plane wave (i.e. Si = [1,0,0,0]T). This is equivalent to using the first column of the MM to define the so-called “polarizance vector” in the Lu-Chipman polar decomposition method [37]. Figure 8 shows the reflected Stokes parameters S1, S2 and S3. The values denote the proportion of reflected light that is polarized by the fishnet metamaterial. Positive values denote linear 0° (p-pol.), linear + 45° and RCP, respectively. We observe that at the SPP resonance frequencies the linear polarized elements, S1 and S2, show large non-zero values, while the circularly polarized element S3 is much weaker. These results suggest that there is appreciable anisotropy but very weak “handedness” in the sample response. Similar to the conclusions above, this indicates that the cross-polarization effects that we observe are predominantly due to linear birefringence and dichroism and cannot be described as being solely due to gyrotropy.
Fig. 8 Reflected Stokes parameters for incident unpolarized light, calculated from the measured Mueller matrix for the fishnet metamaterial at a polar angle of 45° and azimuth of 20°.
6. Conclusion
We presented polarized spectra from generalized ellipsometry measurements of a silver meshed grid and a fishnet metamaterial. We observe strong cross-polarization in the conical mount. We compared the spectra to analytical predictions using the SPP theory and found good agreement for the meshed grid. The spectra are also well modeled using the RCWA technique. The fishnet metamaterial shows qualitative agreement to the simulations but requires refinement of the model. Finally, we simulated the reflection of circular- and linear-polarized light, and used the measured Mueller matrix to predict the reflection properties of nominally polarized light from the materials. The results lead to the question; what is the best physical description of the origin of the cross-polarization? If the metamaterial optical properties are assumed to be effectively homogenous, then the terms dichroism and birefringence, both linear and circular, are valid. Under this assumption the results suggest that the cross-polarization that we observe is predominantly due to effects related to linear dichroism and birefringence and only weakly related to circular dichroism and birefringence.
By characterizing plasmonically resonant nanostructures using polarized light we have laid the foundation to exploit the wealth of polarization phenomena observed in the reflected spectra. The Jones and Mueller matrix formalisms provide complementary frameworks with which to fully describe these phenomena. A detailed understanding may lead to incorporation of these materials in polarization dependent devices, such as the holographic Stokesmeter [43] and grating division-of-amplitude photopolarimeter [44], which exploit conical diffraction to avoid the use of moving parts, thereby offering the potential of extremely fast data acquisition. Additionally, the rapidly growing field of polarized X-ray crystallography has direct analogy with many of the phenomena observed here [45], and the two fields should provide valuable cross-pollination.
Acknowledgments
The authors acknowledge funding by the European Community’s 7th Framework Programme under grant agreement no. 228637 NIM NIL (www.nimnil.org). The financial support by the Senatsverwaltung für Wissenschaft, Forschung und Kultur des Landes Berlin, the Bundesministerium für Bildung, Wissenschaft, Forschung und Technologie is gratefully acknowledged. Stimulating discussions with H. Wormeester are acknowledged.
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