Effective Radiative Properties of Tilted Metallic Nanorod Arrays Considering Polarization Coupling

With the advent of new nanomanufacturing techniques has come the rise of the field of nanophotonics and an increased need to determine optical properties of novel structures. Commercial software packages are able to estimate the behavior, but require large resources and heavy computational time. By combining coordinate transforms and Effective Medium Theory (EMT), an effective relative permittivity tensor is defined and further exploited to calculate the polarization-coupled Fresnel coefficients through Maxwell’s equations. A uniaxial simplification is made to show the case of tilted nanorod arrays. To demonstrate the flexibility of this system, the interfacial reflectance has been calculated for both s- and p-polarizations as well as the coupled case with the volume filling fractions of f = 0.10 and 0.30 for silver (Ag) and titanium (Ti) nanorods, and a scenario of a Ag nanorod array with polymethyl methacrylate (PMMA) as the surrounding medium. The exact results computed by the finite-difference time-domain method justify the validity of EMT with polarization coupling taken into account. The effects of incidence angle and azimuthal angle on reflectance are also discussed. The relatively simple nature of this approach allows for fast estimations of the optical properties of various nanostructures.


Coordinate transformation of relative permittivity tensor
The Fresnel coefficients, and thus the reflectivity and transmissivity of the uniaxial crystal interface, can be obtained by solving Maxwell's equations. A coordinate transform on the local permittivity tensor is applied to convert the anisotropic permittivity from a local coordinate system defined by the geometric orientation of the NRs to the global coordinate system for the ease of tracing the light propagation ( Figure 1) 1 .
For uniaxial materials, the relative permittivity tensor (the permittivity tensor divided by the vacuum permittivity) is given by the diagonal matrix: Note that for the NR arrays discussed above, the shape-induced anisotropy induces a highly metallic behavior along the extraordinary direction and a dielectric behavior along the ordinary direction. This leads to a hyperbolic feature of the array-like structure (i.e., ℜ(ε E ) < 0 and ℜ(ε O ) > 0) following a crossover point where the real portion of one component changes from positive to negative at a certain wavelength. This point is dependent on the material optical constants and geometry.
To transformε into the global coordinates, it is necessary to define these two coordinate systems. The relationship between the global & local coordinate systems is shown in Figure 1b. The uniaxial behavior of this material simplifies this calculation from requiring nine angles to a transform that requires only three angles. These Euler angles are often defined as cosines of angles to simplify the notation, and for this system are α = cos(ψ) sin(θ t ), β = sin(ψ) sin(θ t ), and γ = cos(θ t ), where θ t and ψ are the tilt angle and azimuthal angle, respectively (see Fig. 1). Applying the coordinate transform to the local, uniaxial permittivity tensor results in the 3 × 3 relative permittivity tensor: where ∆ε = ε E − ε O describes the difference between the material dielectric functions in response to the primary extraordinary and ordinary waves. Using the relationships discussed in Ref. [1], this permittivity tensor can be used to calculate Fresnel coefficients, including: r ss (the Fresnel coefficient corresponding to s-polarized light that is reflected as s-polarized light), r sp (s-polarized light that is reflected as p-polarized light), r ps (p-polarized light that is reflected as s-polarized light), and r pp (p-polarized light that is reflected as p-polarized light). Note that for isotropic cases, or cases where the optical axis is inside the plane of incidence (ψ = 0 • or 180 • ), the crosspolarization terms (r sp and r ps ) will reduce to zero. In these special situations, there is no coupling between electric fields for the ordinary and extraordinary cases, leading to decoupled cases. For these decoupled cases, β = 0, which causes the permittivity tensor to simplify to an admittance and impedance approach as described by Gaylord et al. 2 , resulting in r ss and r pp being equivalent to r 12s and r 12p as defined by Wang et al. 3 .

Calculation for the coupled Fresnel coefficient
The solution for Maxwell's equation in three-dimensional space for non-conducting, non-magnetic, and anisotropic materials can be shown as where k 0 = (ω 0 /c 0 ) is the wavevector of free space, k x = k 0 n 1 sin(θ i ) is the wavevector in the global x-direction, n 1 is the index of refraction of the material that the incoming light is traveling through prior to the NR array, k z is the wavevector in the z-direction, and E i is the electric field in the x, y, or z-direction. This solution results in eigenvalues that correspond to allowed values of k z and eigenvector that correspond to the allowed values of electric field. In an isotropic material, there exists only one unique solution for k z and the electric field, but for uniaxial materials, there are two allowed wavevectors, and thus, two allowed electric fields. These two solutions correspond to the ordinary and extraordinary waves discussed previously.
Ordinary wave propagation in the nanorod array can be described with the wavevector with the resulting electric field vector given by where N O is a normalization factor that normalizes the magnitude of the vector to unity. Note that this derivation does not require the absolute magnitudes to find the correct solutions. Mathematically, eigenvectors are normalized because any scalar multiple of the eigenvector is a correct solution, which is true here as well. The relative magnitude of each component is important, not the actual magnitude, for the determination of the Fresnel coefficients. The other eigenvalue corresponds to extraordinary propagation, and is given by with an electric field of These definitions can now be used to calculate the Fresnel coefficient of the nanorod surface. This can be achieved by tracking and arbitrary TE (only non-zero in the y-direction, s-polarization) or TM (only non-zero in the x and z-directions, p-polarization) polarized electric field of unit magnitude at a given θ i . Applying continuity boundary conditions results in a linear set of four equations with four unknowns (the Fresnel coefficients in transmission and reflection) for each polarization. r ss = [(k 0 n 1 cos(θ i ) − k ze )AE Ey − (k 0 n 1 cos(θ i ) − k zo )BE Oy ]/C r sp = [2n 1 k 0 (AE Ex − BE Ox ]/C r pp = (2 n 1 k 0 C cos(θ i ) )[(n 1 k 0 cos(θ i ) + k ze )E Ox E Ey − (n 1 k 0 cos(θ i ) + k zo )E Ex E Oy ] − 1 r ps = 2n 1 k 0 (k ze − k zo )E Oy E Ey /C A, B, and C are coefficients that appear multiple times and are defined as: A = (k 0 n 1 cos(θ i ) + k x tan(θ i ) + k zo )E Ox − k x E Oz B = (k 0 n 1 cos(θ i ) + k x tan(θ i ) + k ze )E Ex − k x E Ez C = (k 0 n 1 cos(θ i ) + k ze )AE Ey − (k 0 n 1 cos(θ i ) + k zo )BE Oy (9) 2/3