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Microscopic origin of the chiroptical response of optical media

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Science Advances  11 Oct 2019:
Vol. 5, no. 10, eaav8262
DOI: 10.1126/sciadv.aav8262
  • Fig. 1 Generalized coupled oscillator model space.

    (A) Representation of an arbitrarily oriented incident plane-wave of wave vector k=k(âxsin θ0 cos ϕ0+âyk sin θ0 sin ϕ0+âzk cos θ0) originating from a source placed at infinity. (B) Molecular unit cell consisting of two oscillators u1 and u2 located at distances δr1 and δr2, respectively, from the molecular center of mass, O′, which is located at a distance r0 from the origin O. Each oscillator is arbitrarily oriented with respect to the other. (C) Coordinate system with the origin (O′) corresponding to the molecular center of mass. The oscillator displacement from O′ is given by δri=δri(âxsin ξi cos ψi+âysin ξi sin ψi+âzcos ξi) for i = 1,2. (D) The origin here corresponds to oscillator center of mass (O″), which is positioned at a distance δri from the molecular center of mass (O′). The orientation of each oscillator is described by the unit vector ûi=âxsin θi cos ϕi+âysin θi sin ϕi+âzcos θi for i = 1,2.

  • Fig. 2 Dependence of the CO response of nanocuboid bi-oscillator system on source angles θ0 and ϕ0.

    (A) Relative orientation of the incident light of wave vector k with respect to the two nanocuboid oscillators. The two oscillators, represented by u1 and u2, are oriented parallel to the x-y plane (θ1 = θ2 = π/2) with azimuth angles ϕ1 = 90° and ϕ2 = 45°, respectively. The nanocuboids are located at d1, z = d2, z = 100 nm with d1, y = d2, x = 100 nm, and for simplicity, d1, x = d2, y = 0 nm was assumed. The nanocuboid parameters were chosen such that they exhibit resonance at wavelengths of λ1 = 750 nm and λ2 = 735 nm, with coupling strengths ζ1,21) = ζ2,12) = 1.6 × 1029 s−1. (B) The calculated ΔAϵ,ϵ response at source angles θ0 = 0° and 180° (note that ϕ0 is undefined at these values of θ0) exhibits a onefold symmetric line shape and experiences an inversion in sign when the incident angle is changed from 0° to 180°. a.u., arbitrary units. (C) The corresponding ΔAΓ,ϵ response calculated under the same conditions exhibits a twofold symmetric line shape and does not experience an inversion in sign for a θ0 change from 0° to 180°. (D) The total CO response ΔA = ΔAϵ,ϵ + ΔAΓ,ϵ for the two source angles does not show any symmetry in the spectral line shape due to the presence of competing contributions from both ΔAϵ,ϵ and ΔAΓ,ϵ response types. (E to G) CO response for the oscillator configuration and orientations in (A) calculated at θ0 = 45° for two azimuth angles ϕ0 = 0° and 180°. (E) The calculated ΔAϵ,ϵ response does not change sign when the incident angle ϕ0 is changed from 0° to 180°. (F) The corresponding ΔAΓ,ϵ response, however, exhibits an inversion in sign for a 180° change in the source azimuth. At these source angles, ΔAϵ,ϵ exhibits a onefold symmetric line shape, whereas ΔAΓ,ϵ is asymmetric. (G) The total CO response ΔA = ΔAϵ,ϵ + ΔAΓ,ϵ also exhibits an asymmetric line shape due to the presence of both ΔAϵ,ϵ and ΔAΓ,ϵ contributions.

  • Fig. 3 Dependence of the CO response of nanocuboid bi-oscillator system on oscillator parameters.

    CO response of the two oscillators, under normal incidence excitation (θ0 = 0°), oriented parallel to the x-y plane (θ1 = θ2 = π/2) and arranged in a planar arrangement with d1, z = d2, z = 0 nm and d1, y = d2, x = 100 nm. In this planar configuration at normal incidence, ΔAΓ, ϵ = 0. (A) Dependence of ΔA = ΔAϵ, ϵ on the angular orientation between the two oscillators in the x-y plane calculated by varying ϕ2 at ϕ1 = 90°. The oscillators are designed to exhibit resonance at wavelengths of λ1 = 750 nm and λ2 = 735 nm, and assuming ζ1,2(ω) = ζ2,1(ω), the peak ΔAϵ, ϵ response is shown to occur at ϕ2 = 45°. (B) Orientation angle of the second oscillator ϕ2 (at ϕ1 = 90°) at which ΔAϵ, ϵ is maximized for a nonzero difference in coupling coefficients, ζ1,2 − ζ2,1, plotted here at ω = 2.43 × 1015 s−1. (C) ΔAϵ, ϵ dependence on the normalized difference in resonant frequencies (Δω)/γ at ζ1,2 − ζ2,1 = − 5.2 × 1028 s−2 corresponding to ϕ2 = 52°. A peak ΔAϵ, ϵ response is achieved at (Δω)/γ = 0.74. (D) ΔAϵ, ϵ dependence at normal incidence on a geometrically achiral system (l1 = l2) for oscillators of the same metal corresponding to γ1 = γ2 (red line) and of dissimilar metals corresponding to γ1 ≠ γ2 (blue line).

  • Fig. 4 Experimental characterization of the CO response of two-dimensional planar Au nanocuboids.

    (A) Simplified ΔAϵ,ϵ and ΔAΓ,ϵ relations, calculated from Eqs. 8.1 and 8.2, for three planar nanocuboid configurations. Top row: The two oscillators are aligned orthogonal to each other (ϕ1 = 90° and ϕ2 = 0°) and are assumed to be of different lengths (l1l2), corresponding to ω1 ≠ ω2 and ζ1,2(ω) ≠ ζ2,1(ω). In such a system, it is expected that both ΔAϵ,ϵ and ΔAΓ,ϵ contributions are present. Middle row: Same as above except with l1 = l2 resulting in ω1 = ω2 = ω0, ζ1,2 = ζ2,1. In this configuration, ΔAϵ,ϵ contribution is expected to be absent for excitation at any arbitrary angle of incidence. Bottom row: Same as above (l1 = l2) except that the two oscillators are oriented parallel to each other (ϕ1 = 90° and ϕ2 = 90°). Ignoring any optical resonance along the width of the nanocuboid, the model predicts both ΔAϵ,ϵ and ΔAΓ,ϵ to be absent, for excitation at any arbitrary angle of incidence. (B to D) Corresponding experimental CDA measurements for an array of planar Au nanocuboid bi-oscillators, illuminated with free-space light between wavelengths of λ0 = 500 and 1000 nm, as a function of incidence angle (varying ϕ0 at a fixed θ0 = 45°) for the three configurations shown in (A). Top-down scanning electron microscopy (SEM) images of unit cells consisting of the two Au nanocuboid oscillators, overlaid with the coordinate system and orientation of the in-plane wave vector of the incident light (kxy) along the x-y plane, are shown at the top of each column. Scale bar, 120 nm in the SEM images. (B) Experimentally measured (solid lines) and model-calculated (dashed lines) CDA spectra for a sample consisting of Au nanocuboids of unequal lengths (l1 = 120 nm and l2 = 100 nm) oriented orthogonal to each other (ϕ1 = 90° and ϕ2 = 0°) at various ϕ0. The spectra at ϕ0 = 0°, 90°, and 135° (blue plots) and at 180° offset from these angles (solid red plots) show an inversion in the sign, which is absent for excitation at ϕ0 = 45° (225°). The CDA model plots were calculated assuming ζ2,11) = 6.4 × 1029 s−2 and ζ1,22) = 8.1 × 1029 s−2 at λ1= 750 nm and λ2 = 720 nm, respectively. (C) Equivalent CDA measurements and model calculations for a device with Au nanocuboids of equal lengths (l1 = l2 = 120 nm). As expected, the CDA response is absent from this device for excitation at ϕ0 = 45° (225°). Moreover, the response at other ϕ0 angles exhibits a twofold symmetric spectral line shape [absent from measurements in (B)], indicating the CDA to only result from ΔAΓ,ϵ contribution. Model parameters used in the calculations are ζ2,10) = ζ1,20) = 8.1 × 1029 s−2 at λ1 = λ2 = 745 nm. (D) Same as (C) except that the two Au nanocuboids are oriented parallel to each other (ϕ1 = 90° and ϕ2 = 90°). The CDA spectra at ϕ0 = 0° (180°) and 90° (270°) show no response, whereas the spectra at ϕ0 = 45° (225°) and 135° (315°) show a pronounced signal of the ΔAϵ,ϵ type (no sign inversion for ϕ0 rotation by 180°). The CDA response at latter angles, though not expected from the model predictions in (A), can be attributed to the coupling to optical resonances along the cuboid widths (w1 = w2 = 60 nm), acting as additional orthogonally oriented oscillators (u1 and u2) in the system.

  • Fig. 5 Origin of the CO response from parallel nanocuboid oscillators through coupling along orthogonal oscillator dimensions.

    (A to D) Top-down SEM images of the device consisting of an array of Au nanocuboid oscillators oriented parallel to each other. Overlaid are the constitutive elliptical eigenmodes (dashed yellow curves) and the projected in-plane source electric field (Exy), indicated by a red vector arrow that traces the red elliptical path for a circularly polarized light at non-normal incidence. Scale bar, 125 nm in the SEM images. (A and B) Orientation of the two eigenmodes relative to the source electric field at ϕ0 = 0° (180°) and 90° (270°), illustrating that they can be accessed equally. (C and D) Same as above, except at source azimuths ϕ0 = 45° (225°) and 135° (315°), illustrating that only one of the two eigenmodes can be accessed. (E) Dependence of ∣ΔAϵ,ϵ∣ on ϕ0 for the parallel nanocuboid oscillator configuration studied here. The orientation of the long- and short-axis oscillators (ui and ui, respectively) corresponding to the length (li) and width (wi) of the two nanocuboids relative to ϕ0 is shown for clarity. (F) Top: Schematic illustrations of the two coupled oscillator contributions that result in a far-field CO response from parallel nanocuboid oscillators of equal lengths (l1 = l2) and widths (w1 = w2) upon illumination at θ0 = 45° and ϕ0 = 45° (225°) or 135° (315°). Note that u1=u2 and u1=u2 in this configuration leads to ζ1,2 = ζ2,1 as well as ζ1,2′ = ζ2,1′ and ζ2′,1 = ζ1′,2, resulting in ΔAϵ,ϵ response to be doubled (from Eq. 8.1, bottom). However, because of the inversion of the spatial dispersion term k(δr1δr2) of Eq. 8.2, the ΔAΓ,ϵ contributions between these two configurations become equal and opposite, canceling each other out.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/10/eaav8262/DC1

    Section S1. Generalized coupled oscillator model parameter definitions

    Section S2. Current density response calculation

    Section S3. Homogeneous material parameters

    Section S4. CO response calculation

    Section S5. Dependence of the κ on the difference between oscillator frequencies

    Section S6. CO response for two identical, orthogonally oriented nanocuboids at finite dz

    Section S7. Device fabrication

    Section S8. Comparison between calculated CDA and ΔA spectral response

    Section S9. CO response of 45° oriented cuboids of equal lengths

    Section S10. Asymmetric transmission CO response of all-dielectric media

    Fig. S1. Arrangement of bi-oscillator molecular unit cells in a representative volume of optical media.

    Fig. S2. Dependence of the multiplication factor κ on the difference between oscillator frequencies.

    Fig. S3. CO response of orthogonally oriented identical nanocuboids in a three-dimensional arrangement.

    Fig. S4. Nanofabrication process steps.

    Fig. S5. Comparison between calculated CDA and ΔA spectral response.

    Fig. S6. Experimental measurements of the CO response of 45° oriented cuboids of equal lengths.

    Fig. S7. Isofrequency surfaces and Poynting vectors for the eigenmodes of a biaxial medium.

    References (39, 40)

  • Supplementary Materials

    This PDF file includes:

    • Section S1. Generalized coupled oscillator model parameter definitions
    • Section S2. Current density response calculation
    • Section S3. Homogeneous material parameters
    • Section S4. CO response calculation
    • Section S5. Dependence of the κ on the difference between oscillator frequencies
    • Section S6. CO response for two identical, orthogonally oriented nanocuboids at finite dz
    • Section S7. Device fabrication
    • Section S8. Comparison between calculated CDA and ΔA spectral response
    • Section S9. CO response of 45° oriented cuboids of equal lengths
    • Section S10. Asymmetric transmission CO response of all-dielectric media
    • Fig. S1. Arrangement of bi-oscillator molecular unit cells in a representative volume of optical media.
    • Fig. S2. Dependence of the multiplication factor κ on the difference between oscillator frequencies.
    • Fig. S3. CO response of orthogonally oriented identical nanocuboids in a three-dimensional arrangement.
    • Fig. S4. Nanofabrication process steps.
    • Fig. S5. Comparison between calculated CDA and ΔA spectral response.
    • Fig. S6. Experimental measurements of the CO response of 45° oriented cuboids of equal lengths.
    • Fig. S7. Isofrequency surfaces and Poynting vectors for the eigenmodes of a biaxial medium.
    • References (39, 40)

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