Backward waves and negative refraction are shown to exist in plasmonic crystals whose lattice cell size is a very small fraction of the vacuum wavelength (less than 1/40th in an illustrative example). Such ``quasi-homogeneity'' is important, in particular, for high-resolution imaging. Real and complex Bloch bands are computed using the recently developed finite-difference calculus of ``Flexible Local Approximation MEthods'' (FLAME) that produces linear eigenproblems, as opposed to quadratic or nonlinear ones typical for other techniques. FLAME dramatically improves the accuracy by incorporating local analytical approximations of the solution into the numerical scheme.
Deep Dive into Quasi-Homogeneous Backward-Wave Plasmonic Structures: Theory and Accurate Simulation.
Backward waves and negative refraction are shown to exist in plasmonic crystals whose lattice cell size is a very small fraction of the vacuum wavelength (less than 1/40th in an illustrative example). Such quasi-homogeneity'' is important, in particular, for high-resolution imaging. Real and complex Bloch bands are computed using the recently developed finite-difference calculus of Flexible Local Approximation MEthods’’ (FLAME) that produces linear eigenproblems, as opposed to quadratic or nonlinear ones typical for other techniques. FLAME dramatically improves the accuracy by incorporating local analytical approximations of the solution into the numerical scheme.
Backward waves (Poynting vector opposite to phase velocity [29]) and the closely related phenomenon of negative refraction have been extensively studied in recent years (e.g. [9,12] and references there) due to the intriguing physical effects and potential applications in imaging and other areas.
Backward waves in periodic structures may in general exist only if the lattice cell size, as a fraction of the vacuum wavelength λ 0 , is above certain thresholds derived recently in [23]. Plasmonic crystals [4,14] are an interesting exception because in the vicinity of a plasmon resonance the constraints on the cell size are relaxed or removed, as explained below, and thus it may be possible to reduce the lattice cell size and approach an ideal homogeneous negative-index medium.
Bloch modes play a central role in the analysis of electromagnetic waves in periodic structures. While analytical expressions for these modes are available only in special one-dimensional cases [11,22,26], there exist a variety of computational techniques: Fourier transforms (plane wave expansion, PWE) [7,11,22], scattering matrices and lattice summation [2], finite difference [25,27] and finite element [1,4,5,8,17] analysis, semi-analytical methods [28], and more.
Bloch wave problems have three (in 2D) or four (in 3D) scalar eigenparameters: frequency ω and the Cartesian components of the Bloch vector K. Solving for all these parameters, and the respective eigenmodes, simultaneously is impractical. The usual approach is to look for the values of ω for any given K. The differential operator of the problem, and hence the respective matrices in the numerical computation, contain the permittivity ǫ and therefore for dispersive media (ǫ = ǫ(ω)) depend on the frequency in a complicated way. Consequently, the resulting eigenvalue problems with respect to ω are nonlinear. Several solution methods have been proposed [16,19] but are not simple, and convergence is not guaranteed.
A more elegant approach, where frequency is treated * Electronic address: igor@uakron.edu as a given parameter and components of the Bloch vector as unknown eigenvalues, has been explored relatively recently in PWE [13] and in FEM [4,17]. Quadratic eigenproblems with respect to the Bloch number usually arise and can be converted to linear ones by introducing auxiliary unknowns either on the continuous level (e.g. solving for both fields E, H instead of just one) or, alternatively, on the linear algebra level [18]. This conversion doubles the number of unknowns; the computational cost, typically proportional to the cube of the system size, increases about eightfold.
In the recently developed generalized finite-difference (FD) calculus of Flexible Local Approximation MEthods (FLAME [21,22,24]) high accuracy is achieved by replacing the Taylor expansions of standard FD analysis with much better approximating functions, e.g. plane waves or cylindrical harmonics. In FLAME, ω is a natural “independent variable” because the approximating functions are derived for a fixed value of ω. FLAME has two clear advantages in the computation of (real or complex) Bloch modes: (i) it dramatically improves the accuracy by incorporating local analytical approximations of the solution into the numerical scheme (see examples below); (ii) it produces linear eigenproblems.
Let us consider band structure calculation in a photonic crystal formed by an infinite lattice of rectangular cells L x × L y in the xy-plane. In a very common case, each cell contains a dielectric cylindrical rod with a radius r rod and the relative dielectric permittivity ǫ rod . The medium outside the rod has permittivity ǫ out . At optical frequencies, all media are intrinsically nonmagnetic. The governing wave equation for the TE mode (one-component magnetic field phasor
where the relative dielectric permittivity ǫ = ǫ(x, y) is periodic over the lattice. The exp(-iωt) convention is used for complex phasors. The H field is sought as a Bloch-Floquet wave [11] with a (yet undetermined) Bloch vector K B :
where H PER is periodic over the lattice.
In the space of Bloch vectors K B , the first Brillouin zone is There are two general options: solving for the periodic factor H PER (x, y) or, alternatively, for the full H-field of (1). In the first case, standard periodic boundary conditions apply, but the differential operator is more complicated than in the second case. The boundary conditions for the full H-field are “scaled-periodic” due to the Bloch exponential exp(iK • r):
H (a/2, y) = exp(iK x a) H (-a/2, y) ; |y| ≤ a/2 (3) with similar conditions at the boundaries y = ±a/2.
Accurate local analytical approximations that FLAME relies on are available for the full H-field formulation and involve Bessel / Hankel functions [20,21,22]. Two Bloch conditions -one for the H field and another one for the E field -need to be imposed on the cell boundaries; the implementation details are described in [24]. In matrix-vector for
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