The finite difference time domain method, which is a rigorous solution to Maxwell’s equation, is convenient for dealing with complicated geometry structure with a finite size. And the Drude model makes some assumptions for the electrons in the metal to simplify the analysis for the optical properties of metals.
2.4.1 Finite Difference Time Domain Method
Since the exact analytical solution to the practical electromagnetic problem are often formidably difficult, it is necessary to apply numerical calculations. Finite difference time domain (FDTD) method is one of the most powerful approaches due to its rigorous calculation without any physical assumptions or premises. Its main thought is to discretize Maxwell equations by using central difference operators to replace differential operators in both space and time variables. [16]
When E and H fields are represented by the discrete values on the spatial grids and are advanced in time steps ofΔt, the field vectors are staggered so that every components of the E field is surrounded by four circulating H components and vice versa, as described in Fig. 2-9. The contour integrals of E and H along the cell’s edge in Faraday’s and Ampere’s law circulate around the corresponding components at the cell face’s center. The algorithm solves for both E and H fields in a systematic approach as follow: all the E computations in the modeled space are completed and stored in computer memory for a particular time-point using previously stored H data.
Then all H computations in the space are completed using the E data just computed.
Inside a loop, this process continues until time-stepping is concluded. The computer simulation runtime increases with the size of the model and total number of time-steps.
It must be emphasized that the FDTD method directly solves the electromagnetic
model in both the space and time domains meaning that the system under excitation will exhibit a transient response before going to the steady state.
Fig. 2-9 Unit cell of FDTD mesh, components of E and H field are shifted by a half-pixel in x, y, z directions so that each E field component normal to the cell face is
surrounded by the circulation of those H field components defined on the cell edges.
In 3D simulations, at least six field components must be stored and updated at each grid, which leads to considerable memory consumption and CPU requirement.
Fortunately, the time update of any field component involves nearby fields one or two cells away from the grid. This translation from physical space into computer memory access allows for efficient implementation of FDTD on shared and distributed memory parallel platforms.
A general setup of the FDTD space is divided into tree regions, as shown in Fig.
2-10.
Fig. 2-10 FDTD simulation space setup
In the Region A, fields contain both incident waves and waves scattered by the interesting object. The scattered wave propagates to the Region B and is finally absorbed in the Region C. The incident wave is set around the boundary between the Region A and Region B. The region C, absorbing boundary, is purposed to model open region problems and to avoid non-physical noise which would corrupt the results in the main grid. It is used to truncate the lattice grid since the tangential components of the E and H fields along the outer boundary of the problem space cannot be updated using the normal time-stepping relations.
In most cases, low-reflection boundary conditions, such as the “Mur” and perfectly matching layer (PML) boundary condition [17], are popularly applied for those problems requiring open boundaries to simulate propagation into infinity.
Therefore, combining all these three regions will allow accurate evaluation of the incident, reflected, and the transmitted waves throughout the computational domains
Additionally, several used material-dispersion models such as Debye, Drude, and Loretz can be readily incorporated with the time-dependent formulation [17] via either a linear recursive relation [18] or auxiliary differential equation methods [19].
Drude model what we adapted in this thesis will be explained briefly at the next section.
Although the numerical discretization in FDTD method does not introduce additional dissipation into the physical problem, it contributes to the numerical dispersion error in mathematics. In the commonly used implementation of FDTD, this error is proportional to the cell size h square, as a second order error function
.In this matter, a grid with about 30 points per wavelength is anyhow desired practically in order to keep the numerical dispersion errors under control.
) (h2 O
2.4.2 Drude Model
The Drude model [20] for the dielectric function assumes that a metal can be approximated by a gas of free electron. It means that the valence electrons of atoms can move freely and independently in a solid. The complex relative permittivity in conductors can be written in the form as:
*
dissipation, absorption, and dispersion; Re[εr*] representing the index of refraction and the phase velocity comes from the real part of ε plus the imaginary part of σ . r At sufficiently high frequencies ωτ >>1 corresponding to the visible light range and ultra-violet, where τ is the electron scattering time, εr ~1 (the bound
, where ωp is the plasma frequency, n is the electron density, m is the effective mass
of the conduction electrons. In the range0<ω<ωp, the dielectric constant is negative and no modes are allowed in the bulk of the metal. This accounts for the fact that metals are extremely opaque at optical frequencies. However, it becomes effectively transparent forω>ωp which is in contradiction with the observed reddish color for copper and yellowish color for gold. The reason is that the metal band structure can no longer be neglected and the interband absorption effect arises for high frequency.