Mid-infrared gyrotropy in split-ring resonators measured by Mueller matrix ellipsometry

Thomas W. H. Oates; Timur Shaykhutdinov; Tolga Wagner; Andreas Furchner; Karsten Hinrichs

doi:10.1364/OME.4.002646

1. Introduction

Understanding and controlling plasmonic resonances in metamaterials to produce designer optical properties, such as optical activity, is of topical interest for optical-sensor and photonics applications. Optical activity, also known as gyrotropy, is defined as a rotation of the plane of polarization of linearly polarized light upon transmission through, or reflection from, a material, when all the optical axes are parallel or perpendicular to the plane of incidence. Gyrotropy is observed in a range of plasmonically resonant metamaterials and molecular aggregates, originating from the interaction of light with both chiral and achiral arrangements of resonators. Conventional gyrotropy (also termed “intrinsic chirality” [1]) stems from light interaction with the chiral structure of the meta-atom or -molecule, for example a helical arrangement.

Recently, gyrotropy was reported upon transmission through [2] and reflection from [3] arrays of achiral meta-atoms. In relation to metamaterials this was dubbed “extrinsic chirality” [2], however the term “pseudochirality” has historically been used to describe the same effect in ordered molecular materials [1]. Gyrotropy is observed in ordered achiral materials with, for example, C_2v symmetry under oblique-incidence illumination, when the plane of incidence is not a mirror-plane [3]. Achiral gyrotropy requires a high degree of molecular order and is generally weak, having been observed in less than 40 achiral crystals [4]. In contrast, the strong coupling of plasmonic resonances to radiation means gyrotropic metamaterials offer the potential for cost-effective, tunable, and ultra-compact polarization-control elements for photonics applications. Developing these elements in the mid-infrared (MIR) spectral region is of particular interest, for example to enhance the optical activity of chiral organic molecules [5] in the fingerprint region using vibrational circular dichroism [6].

For a homogenous material, gyrotropy arises from (complex) circular anisotropy, of which the real and imaginary parts comprise the Kramers-Kronig related circular birefringence (CB) and circular dichroism (CD), respectively. CD measurements are a standard test of chirality in isotropic systems. CB is determined using optical rotary dispersion or may be calculated by Kramers-Kronig transformation of the CD spectrum [7]. When investigating chirality in the solid phase, one must be careful to consider the potential of chiral artefacts caused by linear anisotropy, with the associated linear birefringence (LB) and linear dichroism (LD). These are of vanishing intensity in measurements of chiral molecules in solution but are often strong in measurements of solids. The complete description of the polarization properties of the sample via measurement of the Mueller matrix (MM) allows one to identify these phenomena. Since the terms birefringence and dichroism are usually defined for normal incidence measurements in transmission, we therefore restrict ourselves to the term gyrotropy, as defined above. However, both circular and linear anisotropy are readily determined using reflection-based measurements [8,9].

In this work we use generalized ellipsometry to demonstrate the presence of gyrotropy in the MIR spectral region upon reflection from a planar array of achiral split ring resonators (SRRs). The reflection intensity through crossed polarizers under azimuthal rotation is first presented, identifying potential gyrotropy. We then present MM measurements of the SRRs. Careful consideration of the symmetry properties of the meta-atoms and the MM allows us to identify the gyrotropy and eliminate contributions from linear anisotropy. Comparing the results with rigorous coupled-wave analysis (RCWA) simulations, we show that the gyrotropy is observed at the first and third plasmonic resonant modes of the SRR. We then argue that the gyrotropy arises from coupling between the electric dipole and quadrupole moments in the SRR, as well as the regular electric dipole–magnetic dipole coupling associated with natural optical activity.

2. Simulated and experimental results

Gold SRRs on silicon substrates were fabricated using nanoimprint lithography [10] with a unit cell of p = 1 µm, side-length of D = 790 nm, side-width of d = 75 nm, height of h = 54 nm, and gap-width of g = 300 nm. We previously presented the polarization properties of these samples in standard ellipsometric geometry, and identified the plasmonic resonance modes using RCWA [11]. In this work we use the commercial RCWA optical simulation package RSoft [12]. A schematic of the unit cell and the measurement geometry is shown in Fig. 1(a).Also shown are the simulated absolute values (normalized) of the complex electric field vectors (near field) in the z direction, $| E_{z} |$ , at a location 17 nm inside the silicon substrate. The polar angle was zero in all cases (θ = 0°). The first three plasmonic resonance modes are shown in Figs. 1(b)-1(d). The first (800 cm⁻¹) and third (2200 cm⁻¹) are excited with p-polarization at the azimuthal angle ϕ = 0° and the second (1700 cm⁻¹) with p-polarization at ϕ = 90°.

Fig. 1 (a) Schematic of a unit cell of the SRR array showing the Cartesian axes and the polar (θ) and azimuthal (ϕ) co-ordinates. The simulated normalized modulus of the z-component of the electric field for the; (b) first resonance mode at 800 cm⁻¹; (c) second resonance mode at 1700 cm⁻¹; (d) third resonance mode at 2200 cm⁻¹. The sign of the phase at the maximum values are shown with “+” and “–“.

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To measure the polarization properties of the sample we upgraded an in-house fabricated FTIR ellipsometer [13]. To detect the small cross-polarization signal we installed two aligned wire-grid polarizers for each of the polarizer (P) and analyzer (A). Despite the reduction in intensity at the detector, double polarizers were required to increase the degree of polarization to above 0.9998 to avoid unpolarized light “leaking” through the analyzer. The ellipsometer was used in a step-wise rotating polarizer/sample/rotating analyzer (PSA) mode, without the use of a compensator. This provided measurements of the crossed-polarizers reflectance R_ps (P = 0°, A = 90°) and R_sp (P = 90°, A = 0°). With additional measurements of all combinations of P and A settings at −45°, 0°, 45° and 90° we gain access to nine of the sixteen MM elements (the first three elements of each row and column) [14]. The simulations provided the complex reflection coefficients ${\tilde{r}}_{pp}, {\tilde{r}}_{ps}, {\tilde{r}}_{sp}$ and ${\tilde{r}}_{ss}$ , from which we construct the reflection Jones matrix, and subsequently the MM (under the assumption that the sample is non-depolarizing).

A complete description of the polarization state of the incoming and outgoing light is provided by the 4x1 Stokes vectors, $S_{i n}$ and $S_{o u t}$ , respectively. The polarization dependent material properties described above are accessible by measuring the 4x4 Mueller matrix $M$ of the sample in transmission or reflection

S_{o u t} = {[\begin{matrix} S_{0} \\ S_{1} \\ S_{2} \\ S_{3} \end{matrix}]}_{o u t} = [\begin{matrix} m_{00} & m_{10} & m_{20} & m_{30} \\ m_{01} & m_{11} & m_{21} & m_{31} \\ m_{02} & m_{12} & m_{22} & m_{32} \\ m_{03} & m_{13} & m_{23} & m_{33} \end{matrix}] {[\begin{matrix} S_{0} \\ S_{1} \\ S_{2} \\ S_{3} \end{matrix}]}_{i n} = M S_{i n} .

Under idealized conditions (e.g. isotropic chiral liquids), gyrotropy can be directly observed in the off-diagonal elements (m₁₂, m₂₁, m₀₃, m₃₀) of the MM, or by matrix decomposition under more complex conditions [15]. The general case is far more complicated, with linear anisotropy mixing with circular anisotropy in all the off-diagonal MM elements [16].

Figure 2 compares R_ps from the RCWA simulations and the ellipsometry measurements for the SRR at polar angle θ = 50° for azimuth angle ϕ from −90° to + 95°. Both simulated and measured data display a strong peak, indicating a rotation of the polarization, at 1700 cm⁻¹ for ϕ = +/− 45°. The wavenumber of this peak corresponds to the second mode. The optical rotation (OR) stems from the asymmetry of the incident electric field with respect to the dipolar resonance at ϕ = +/− 45°, i.e. the in-plane electric dipole is oriented at 45° with respect to the plane of incidence. The rotation maximum occurs at wavenumbers less than the minimum Rayleigh wavenumber (ca. 2300 cm⁻¹ under these conditions [17]). This polarization rotation is also observed in square arrays of holes [18] and fishnet metamaterials [19] at the Rayleigh/SPP wavenumbers. The homogenization assumption is generally valid if the period of the structure is much smaller than the wavelength ( $p / λ ≪ 1$ ). However, for square-hole-arrays and metamaterials it is often the case that $p / λ \approx 1$ [19]. Under homogenization, the effective dielectric function of square-hole-arrays is in-plane isotropic. Then the OR cannot be attributed to classical form birefringence and such materials “mimic” chirality [18]. Thus homogenization is invalid and the effects of spatial dispersion cannot be neglected.

Fig. 2 Simulated (a) and measured (b) values of R_ps at polar angle θ = 50° for azimuth angle ϕ from −90° to + 95°. The measured values are normalized to R_pp from gold.

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Notably in Fig. 2, although the optical rotation is zero at all wavenumbers for ϕ = +/− 90° (i.e. the high-symmetry optical axis), it is non-zero at ϕ = 0° (i.e. the low-symmetry optical axis) at wavenumbers near 2200 cm⁻¹ (third mode). According to our definition in the introduction, this indicates the presence of gyrotropy. However, we should still rule out the possibility of linear birefringence contributing to the observed polarization rotation. We do this by considering the symmetry of the MM off-diagonal terms (m₀₂, m₀₃, m₁₂, m₁₃, m₂₀, m₂₁, m₃₀, m₃₁). These terms are non-zero only under conditions of birefringence and dichroism in specific geometries. In general, both circular and linear anisotropy are observed in all the off-diagonal terms [16].

For an orthorhombic system in which the optical axes are parallel or perpendicular to the sample surface, any contribution to the MM from linear anisotropy must be symmetric such that m₀₁ = m₁₀, m₀₂ = –m₂₀, m₀₃ = m₃₀, m₁₂ = –m₂₁, m₁₃ = m₃₁ and m₂₃ = –m₃₂ [8]. This is also the case for isotropic gyrotropy, of the type observed in solutions of chiral molecules [20]. Therefore, since our metamaterial with C_2v symmetry is orthorhombic with optical axes parallel or perpendicular to the sample surface, summing (or subtracting) the diagonally opposite off-diagonal elements listed above will remove all contributions from linear anisotropy and isotropic gyrotropy. Any remaining signal cannot be explained by these phenomena.

Figure 3 shows the simulated and experimental cases for the MM elements m₁₂, m₂₁ and for (m₁₂ + m₂₁)/2 at ϕ = 0°-90°. Despite a baseline offset, the measured and simulated data show good agreement. We attribute the observed signal in the sum plot (Figs. 3(c) and 3(f)) to anisotropic gyrotropy, of the type observed in crystals of the space group symmetry mm2 [4] and predicted in molecules of point group symmetry C_2v [21]. Note that at ϕ = 0° the sign and magnitude of m₁₂ and m₂₁ are equal. Thus at this azimuth the entire measured signal arises from anisotropic gyrotropy, linear anisotropy is absent, and no summation is required to identify the effect.

Fig. 3 Measured (a-c) and simulated (d-f) Mueller matrix elements m₁₂ and m₂₁, and (m₁₂ + m₂₁)/2, at ϕ = 0°-90° and θ = 50°. The anisotropic gyrotropic resonances at the first and third resonance mode wavenumbers are clearly resolved in (c) and (f). The resonances are maximum at the low-symmetry optical axis ϕ = 0°.

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We also simulated the polar angle θ dependence of the MM on the optical axes ϕ = 0° and 180°. Figure 4 shows m₁₂ ( = m₂₁) as θ varies from 20° to 80° in 20° steps, plus the orthogonal cases on the axes (θ = 0°, 90°). The gyrotropy reaches a maximum at the third resonance peak around θ = 60°. At θ = 0° and 90° it is zero. Of importance is the fact that the gyrotropy changes sign with an azimuthal rotation of 180°. A similar effect has been observed in a number of works on asymmetric transmission through metamaterials (e.g. see [22]). This might be expected in a monoclinic or triclinic material system. However, the C2v symmetry of our SRRs is orthorhombic. We discuss the origin of this below.

Fig. 4 The MM element m₁₂ as the polar angle θ is varied from 0° to 90°, for azimuths ϕ = 0° and 180°. A maximum in the gyrotropy is observed at the third resonance mode at around 60°.

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3. Discussion

The plasmonic resonance modes shown in Fig. 1 may be considered as combinations of dipoles in the “arms” and “base” of the SRR [3]. As shown in Fig. 5(a), the first resonance mode (800 cm⁻¹) has antiparallel dipoles in the arms (with an associated induced electric quadrupole transition moment, $Θ$ ) and a dipole in the “base” that is in phase with the dipoles in the arms (with an induced electric dipole transition moment, $μ$ , parallel to the base). These dipoles combine to create a circulating current with an induced magnetic dipole transition moment, $m$ , along the z-axis. The second mode (1700 cm⁻¹) has parallel dipoles in the arms and zero net dipole in the base, resulting in a $μ$ parallel to the arms and zero net $m$ and $Θ$ [Fig. 5(b)]. The third mode (2200 cm⁻¹) has antiparallel dipoles in the arms and a dipole in the base that is out of phase with those in the arms [Fig. 5(c)]. It has similar $m$ and $Θ$ character to the first mode, however the direction of $μ$ is now reversed.

Fig. 5 Schematic of the modes shown in Figs. 1(b)-1(d), built up from individual dipoles. (a) corresponds to the first resonance mode, (b) the second mode and (c) the third mode. The combination of dipoles creates transition moments of electric dipole, $μ$ , magnetic dipole, $m$ , and electric quadrupole, $Θ$ , character.

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The ratio of the SRR dimensions to the wavelength at the third resonance mode frequency is $D / λ = 0.18$ , which indicates that the long-wavelength (dipole) approximation is invalid and the spatial dependency of the electric field (spatial dispersion) must therefore be included. The theory is based on a series expansion of the multipole transition moments up to the first order. The induced transition moments $μ$ , $m$ and $Θ$ (with components $μ_{α}, m_{α}, Θ_{α β}$ using Einstein notation) are [23]

μ_{α} = α_{α β} E_{β} + \frac{1}{ω} {G^{'}}_{α β} {\dot{B}}_{β} + \frac{1}{3} A_{α β γ} E_{β γ} + ...,

m_{α} = χ_{α β} B_{β} - \frac{1}{ω} {G^{'}}_{β α} {\dot{E}}_{β} + ...,

Θ_{α β} = A_{γ α β} E_{γ} + ...,

where

E_{β}

and

B_{β}

are the electric and magnetic fields, and

{\dot{E}}_{β}

and

{\dot{B}}_{β}

are the corresponding time derivatives, all in the

β

direction.

E_{β γ} = \nabla_{β} E_{γ}

is the electric field gradient at the molecule. The effective fields are assumed to be the actual fields associated with the plane wave and are taken at the origin used to define the multipole moments. The

G^{'}

and

A

tensors describe how the electron density will respond to a time-varying electric/magnetic field and electric field gradient, respectively.

Quantum mechanical perturbation theory may be used [23] to determine $G^{'}$ and $A$ . At transparent frequencies, $G^{'}$ describes the dynamic coupling between the electric dipole and the magnetic dipole moments via

{G^{'}}_{α β} = - \frac{2}{ℏ} \sum_{j \neq n} \frac{ω}{ω_{j n}^{2} - ω^{2}} Im (〈 n | μ_{α} | j 〉 〈 j | m_{β} | n 〉),

where

| n 〉

and

| j 〉

are the ground and excited molecular electronic states. Similarly,

A

describes the coupling between the electric dipole and electric quadrupole moments via

A_{α β γ} = \frac{2}{ℏ} \sum_{j \neq n} \frac{ω_{j n}^{}}{ω_{j n}^{2} - ω^{2}} Re (〈 n | μ_{α} | j 〉 〈 j | Θ_{β γ} | n 〉) .

If either

G^{'}

or

A

is non-zero, OR may be observed upon transmission through the medium. The Buckingham-Dunn equation gives the symmetric OR tensor in an oriented sample as [23]

g_{α β} = - \frac{1}{2} i [{G^{'}}_{α β} + {G^{'}}_{β α} - \frac{1}{3} ω (ε_{α γ δ} A_{γ δ β} + ε_{β γ δ} A_{γ δ α})],

where

ε_{α γ δ}

is the Levi-Civita symbol. In isotropic chiral media, such as sugar molecules in solution, all terms in

A

and all off-diagonal components of

G^{'}

average to zero. Then only the trace of

G^{'}

influences the radiation scattering, leading to the well-known Rosenfeld expression for the optical rotation of light (propagating in z) in an isotropic chiral medium,

θ = - \frac{1}{2} μ_{0} N L ω [〈 {G^{'}}_{x x} 〉 + 〈 {G^{'}}_{y y} 〉],

where

μ_{0}

is the vacuum permeability,

N

is the number density,

L

is the path length.

In contrast, if the molecules are aligned then the spatial average of $A$ is non-zero. It follows that the coupling of $μ$ and $Θ$ also influences the radiation scattering, and OR is then a sum of the $μ$ - $m$ and $μ$ - $Θ$ coupling as in Eq. (7). Note that the second terms in both Eq. (2) and Eq. (3) are inversely proportional to the frequency. Thus although $A$ may be small, its contribution to OR dominates at high frequencies if the molecules are aligned. This helps explain why the $μ$ - $Θ$ mechanism is observed in measurements of X-ray natural CD (XNCD) [24]. In fact, XNCD has been observed in crystals of KTiOPO₄, which are of crystal class mm2, the space group analogue of the SRR point group C_2v [25].

Until now the $μ$ - $Θ$ coupling has been ignored as a mechanism for gyrotropy at optical and infrared frequencies in SRRs. Previous discussions [2,26] focused only on the $μ$ - $m$ interaction. Since $μ$ and $m$ are perpendicular in the SRR, the direction of the wave vector determines the sign and magnitude of the optical rotation. Only if the scalar projections of both $μ$ and $m$ onto the wave vector are non-zero will OR occur, as shown schematically in [2]. This explains why we observe OR at oblique incidence, but not at θ = 0°.

We argue here that, just as the $μ$ - $Θ$ coupling is important in oriented molecules [23], it undoubtedly contributes to the observed OR in metamaterials. The Buckingham-Dunn equation can now be used to determine the sign and magnitude of the optical rotation in SRRs. Since the point group symmetry of a SRR is the same as that for the water molecule (C_2v), we follow the discussion by Isborn et al. who present a theoretical work on the optical rotary power of water [21], and compare and contrast the results of the SRR. We note that both the 1st and 3rd modes of the SRR have the same symmetry as the B₂ transition in the H₂O molecule. The B₂ mode is an asymmetric mode, with the only non-zero element in Eq. (7) being

g_{z x} (B_{2}) = - \frac{1}{2} i [{G^{'}}_{x z} - \frac{1}{3} ω A_{x x y}] .

This explains why we observe gyrotropy at oblique incidence only, and why it is zero for ϕ = +/−90° and reaches a maximum for ϕ = 0° (zx-plane). This also explains why we observe Mueller matrix elements m₁₂ = m₂₁, and why their sign changes as we rotate ϕ by 180°, as observed in Fig. 4. This corresponds well with previous descriptions of the bianisotropic tensor components in SRRs [27,28]. The difference in the magnitude of gyrotropy for the 1st and 3rd modes may be attributed to the difference in magnitude of the

A

and

G^{'}

contributions, especially noting that the higher frequency 3rd mode should be more dependent on

A

due to the frequency dependence in Eq. (9).

In contrast the 2nd mode of the SRR has the same symmetry as the A₁ transition in H₂O which, being magnetic dipole forbidden, has the OR tensor contribution [21]

g_{z x} (A_{1}) = - \frac{1}{2} i [- \frac{1}{3} ω (- A_{y x x} + A_{y z z})] .

Note that the individual dipoles in the water molecule are not parallel but are offset by the bond angle of water (104.5°). Thus both

Θ_{x x}

and

Θ_{z z}

are non-zero and for the water molecule

g_{z x} (A_{1}) \neq 0

. However, unlike H₂O the second mode of the SRR consists of two parallel dipoles [Fig. 1(c)], and thus

Θ_{i j} = 0

. It follows from Eq. (10) that for the SRR

g_{z x} (A_{1}) = 0

, which explains why we observe vanishing OR at the frequency of the second mode of the SRR. It is important to note that the magnitudes of the contributions of

A

and

G^{'}

are not constant since they are origin dependent [24]. The change in one contribution on shifting the origin is cancelled by the change in the other. It remains a challenge to distinguish the two contributions.

4. Conclusion

Gyrotropy in ordered arrays of SRRs was measured using generalized ellipsometry. The gyrotropy occurs since the sample is composed of meta-atoms that are non-centrosymmetric and are ordered. In this case both the regular electric dipole–magnetic dipole interaction and the electric dipole–electric quadrupole interaction (which averages to zero in isotropic samples) contribute to the observed optical rotation, as described by the Buckingham-Dunn equation [Eq. (7)]. A determination of the spatially dispersive material polarizabilities may provide an analytical approach to simplify the optical characterization of the SRR array.

Acknowledgments

The authors acknowledge Iris Bergmair for production of the SRR samples funded by the European Community’s 7th Framework Programme under grant agreement no. 228637 NIM NIL (www.nimnil.org). We thank Ilona Engler for technical assistance with the ellipsometric measurements. The financial support by the Senatsverwaltung für Wirtschaft, Technologie und Forschung des Landes Berlin and the Bundesministerium für Bildung und Forschung is gratefully acknowledged.

References and links

1. V. K. Valev, J. J. Baumberg, C. Sibilia, and T. Verbiest, “Chirality and chiroptical effects in plasmonic nanostructures: fundamentals, recent progress, and outlook,” Adv. Mater. 25(18), 2517–2534 (2013). [CrossRef] [PubMed]

2. E. Plum, X. X. Liu, V. A. Fedotov, Y. Chen, D. P. Tsai, and N. I. Zheludev, “Metamaterials: optical activity without chirality,” Phys. Rev. Lett. 102(11), 113902 (2009). [CrossRef] [PubMed]

3. T. W. H. Oates, T. Shaykhutdinov, T. Wagner, A. Furchner, and K. Hinrichs, “Gyrotropy in achiral materials: the coupled oscillator model,” Adv. Mater. 26(42), 7197–7201 (2014). [CrossRef] [PubMed]

4. K. Claborn, C. Isborn, W. Kaminsky, and B. Kahr, “Optical rotation of achiral compounds,” Angew. Chem. Int. Ed. Engl. 47(31), 5706–5717 (2008). [CrossRef] [PubMed]

5. E. Hendry, T. Carpy, J. Johnston, M. Popland, R. V. Mikhaylovskiy, A. J. Lapthorn, S. M. Kelly, L. D. Barron, N. Gadegaard, and M. Kadodwala, “Ultrasensitive detection and characterization of biomolecules using superchiral fields,” Nat. Nanotechnol. 5(11), 783–787 (2010). [CrossRef] [PubMed]

6. C. Gautier and T. Bürgi, “Chiral N-isobutyryl-cysteine protected gold nanoparticles: preparation, size selection, and optical activity in the UV-vis and infrared,” J. Am. Chem. Soc. 128(34), 11079–11087 (2006). [CrossRef] [PubMed]

7. M. Krykunov, M. D. Kundrat, and J. Autschbach, “Calculation of circular dichroism spectra from optical rotatory dispersion, and vice versa, as complementary tools for theoretical studies of optical activity using time-dependent density functional theory,” J. Chem. Phys. 125(19), 194110 (2006). [CrossRef] [PubMed]

8. T. W. H. Oates, M. Ranjan, S. Facsko, and H. Arwin, “Highly anisotropic effective dielectric functions of silver nanoparticle arrays,” Opt. Express 19(3), 2014–2028 (2011). [CrossRef] [PubMed]

9. M. P. Silverman and J. Badoz, “Light reflection from a naturally optically active birefringent medium,” J. Opt. Soc. Am. A 7(7), 1163–1173 (1990). [CrossRef]

10. I. Bergmair, B. Dastmalchi, M. Bergmair, A. Saeed, W. Hilber, G. Hesser, C. Helgert, E. Pshenay-Severin, T. Pertsch, E. B. Kley, U. Hübner, N. H. Shen, R. Penciu, M. Kafesaki, C. M. Soukoulis, K. Hingerl, M. Muehlberger, and R. Schoeftner, “Single and multilayer metamaterials fabricated by nanoimprint lithography,” Nanotechnology 22(32), 325301 (2011). [CrossRef] [PubMed]

11. M. M. Jakovljević, G. Isic, B. Vasic, T. W. H. Oates, K. Hinrichs, I. Bergmair, K. Hingerl, and R. Gajic, “Spectroscopic ellipsometry of split ring resonators at infrared frequencies,” Appl. Phys. Lett. 100(16), 161105 (2012). [CrossRef]

12. R. Scarmozzino, “Simulation tools for devices, systems, and networks,” in Optical Fiber Telecommunications V Part B: Systems and Networks, I. P. Kaminow, T. Li, and A. E. Willner, eds. (Academic Press, 2008).

13. K. Hinrichs, A. Röseler, K. Roodenko, and J. Rappich, “Surface-enhanced infrared absorption: infrared ellipsometry of Au evaporated ultrathin organic films,” Appl. Spectrosc. 62(1), 121–124 (2008). [CrossRef] [PubMed]

14. T. W. H. Oates, H. Wormeester, and H. Arwin, “Characterization of plasmonic effects in thin films and metamaterials using spectroscopic ellipsometry,” Prog. Surf. Sci. 86(11–12), 328–376 (2011). [CrossRef]

15. O. Arteaga, Mueller matrix polarimetry of anisotropic chiral media (Universitat de Barcelona, Departament de Física Aplicada i Òptica, 2010).

16. J. Schellman and H. P. Jensen, “Optical spectroscopy of oriented molecules,” Chem. Rev. 87(6), 1359–1399 (1987). [CrossRef]

17. T. W. H. Oates, B. Dastmalchi, G. Isic, S. Tollabimazraehno, C. Helgert, T. Pertsch, E. B. Kley, M. A. Verschuuren, I. Bergmair, K. Hingerl, and K. Hinrichs, “Oblique incidence ellipsometric characterization and the substrate dependence of visible frequency fishnet metamaterials,” Opt. Express 20(10), 11166–11177 (2012). [CrossRef] [PubMed]

18. B. Gompf, J. Braun, T. Weiss, H. Giessen, M. Dressel, and U. Hübner, “Periodic nanostructures: spatial dispersion mimics chirality,” Phys. Rev. Lett. 106(18), 185501 (2011). [CrossRef] [PubMed]

19. T. W. H. Oates, B. Dastmalchi, C. Helgert, L. Reissmann, U. Huebner, E. B. Kley, M. A. Verschuuren, I. Bergmair, T. Pertsch, K. Hingerl, and K. Hinrichs, “Optical activity in sub-wavelength metallic grids and fishnet metamaterials in the conical mount,” Opt. Mater. Express 3(4), 439–451 (2013). [CrossRef]

20. O. Arteaga, “Useful Mueller matrix symmetries for ellipsometry,” Thin Solid Films. in press).

21. C. Isborn, K. Claborn, and B. Kahr, “The optical rotatory power of water,” J. Phys. Chem. A 111(32), 7800–7804 (2007). [CrossRef] [PubMed]

22. V. A. Fedotov, P. L. Mladyonov, S. L. Prosvirnin, A. V. Rogacheva, Y. Chen, and N. I. Zheludev, “Asymmetric propagation of electromagnetic waves through a planar chiral structure,” Phys. Rev. Lett. 97(16), 167401 (2006). [CrossRef] [PubMed]

23. A. D. Buckingham and M. B. Dunn, “Optical activity of oriented molecules,” J. Chem. Soc. A 0, 1988–1991 (1971). [CrossRef]

24. L. D. Barron, Molecular Light Scattering and Optical Activity (Cambridge University Press, 2004).

25. P. A. Thomas, I. J. Tebbutt, and A. M. Glazer, “Potassium titanyl phosphate, KTiOPO4. I. Experimental investigation of optical gyration, absolute optical chirality and twinning,” J. Appl. Cryst. 24(5), 963–967 (1991). [CrossRef]

26. M. Ren, E. Plum, J. Xu, and N. I. Zheludev, “Giant nonlinear optical activity in a plasmonic metamaterial,” Nat Commun 3, 833 (2012). [CrossRef] [PubMed]

27. R. Marqués, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B 65(14), 144440 (2002). [CrossRef]

28. W. J. Padilla, “Group theoretical description of artificial electromagnetic metamaterials,” Opt. Express 15(4), 1639–1646 (2007). [CrossRef] [PubMed]

Mid-infrared gyrotropy in split-ring resonators measured by Mueller matrix ellipsometry

Abstract

1. Introduction

2. Simulated and experimental results

3. Discussion

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (5)

Equations (10)

Optical Materials Express