Finite Difference Time Domain (FDTD)

The FDTD [2, 4, 6, 66, 67] technique is a time-domain differential numerical method. It calculates the electric and magnetic fields everywhere in the computational domain as they evolve in time. FDTD is robust, efficient, versatile and adaptable to complex gemoetries. It is easy to understand and easy to implement method [4]. FDTD can be implemented in general algorithm that can be applied to any problem. It allows modelling of complex inhomogeneous configurations and performs better modelling of unbounded problems than FE method [1]. FDTD provides direct solution to Maxwell’s equations, which means that analyist can examine the radiation, propagation, and scattering of electric and magnetic fields as the computations proceed and the field distribution within the problem space can be monitored at any time step [68-74]. FDTD is used for a variety of applications such as microstrip [75-77], waveguides [78-80], antennas [77, 81], optoelectronics devices [82] and biological interactions [83, 84]. However, it exhibits problem of slow convergence for resonance problems and requires large memory requirements for inhomogeneous waveguide structures [4]. Standard FDTD has the inability to handle curved boundaries which led to the development of the method over non-orthogonal and unstructured meshes [85-

95]. The FDTD includes more complicated approach to terminate the problem space by introducing artificial termination conditions including PML to absorb incident and scattered fields. Care must be taken to minimize the error associated with such boundaries which dictate the accuracy of the FDTD [4]. Also, far-field calculations force the computational domain to be excessively large and requires extended amount of pre-processing [96].

The FDTD method is directly derived from discretizing the differential form of the Maxwell’s equations. Maxwell’s equations are solved by using the central difference technique to express the derivative over a spatial grid known as Yee cell [6, 68, 69]. The region being modeled is represented by two interleaved grids of discrete points. One grid contains the points at which the magnetic field is evaluated and the second grid contains the points at which the electric field is evaluated as shown in Figure 2.3 [6]. Note that i, j and k are integers in the x, y and z directions, respectively. The electric (Ex, Ey, Ez) and magnetic (Hx, Hy, Hz) field vectors are assigned to each grid cell. Each

facet of the Yee’s cubic cell is defined by two electric field components tangential to the facet and a magnetic field component normal to the facet in the third direction.

Figure 2.3: Yee cube [96]

The Yee algorithm [6] is based on finite difference approximations of the space derivatives and time derivatives. Consider any field component f as a function of space and time evaluated at a discrete point in the space lattice (x, y, z) and at a discrete point in time t, then u can be written as [69],

(2.50) where the i, j, and k indices represent the spatial positions and the n index represents time. ∆x, ∆y and ∆z are the spatial increments in x, y and z directions, respectively and t is the time increment.

Then Yee’s expression for the first spatial derivative of f in the x-direction, evaluated at a fixed time is [6],

(2.51)

with space finite difference over x/2. O[(x)2_{] is the truncation error in Taylor series}

expansion, which is proportional to the squared spatial discretization.

The Yee’s expression for the first time derivative of f, evaluated at the fixed space point (i,j,k) follows by analogy [6],

(2.52)

For electromagnetic problems, Yee’s algorithm is applied to the space and time derivative of Maxwell’s equations.

TM modes are expressed by Ez, Hx and Hy field components as expressed in Equations

(2.7). The field components are distributed in 2D Yee’s Cell as shown in Figure 2.4, where Hx and Hy components are directed along the x and y direction, respectively, and

Ez is normal to the x-y plane.

The derivative of Equations (2.7) can be expressed as,

lim lim _(2.53.a)

lim lim lim _(2.53.c) (i,j) (i+1,j) (i+1,j+1) (i,j+1)

H

H

E

Figure 2.4: 2D Yee’s cell formulation for TM mode

Discretizing space and time around a point (x,t), then applying central differences to Equation (2.53.a) to relate the derivatives of the neighbouring discrete fields we get,

(2.54)

or in other terms,

(2.55)

The value of the H field at point x and at time t+ 1/2 is approximated by two E-field spatial values at x+ 1/2 and x-1/2 and depends on the previous H-field temporal value at t-1/2.

Similarly, applying the central difference for (2.53.b) leads to,

And finally, the central difference for Equation (2.53.c) leads to

(2.57) The resolution of the grid is in practice determined by the dimensions of the smallest feature in the problem. In one time step, any point on the wave must not travel more than one cell to its nearest neighbours. The condition for numerical stability in one dimension is that [6]

(2.58)

where c is the free space velocity and ∆x is cell size over one direction. In 2D case and if the discretization length is the same in both directions, the condition becomes,

(2.59)

To guarantee numerical stability in the general 3D case, Courant condition [6] must be satisfied,

(2.60)

where ∆x, ∆y and ∆z are the discretization length in x, y and z directions, respectively. For open boundary problems, the problem is placed within an infinite free-space volume which is not practical to numerically discretise. The numerical problem must be truncated in such a way as to model open space and not introduce artificial reflections due to terminations. An artificial boundary conditions is applied to terminate the cells by absorbing the incident and scattered fields.

The first most widely used Absorbing Boundary Conditions (ABC) was devised by Mur in 1981 [97]. This boundary condition is derived from a one-way wave equation, which allows only outgoing waves as possible solutions. However, the attenuation of incident waves on the ABC boundary degrades as the incidence angle increases until at the grazing angle the boundary becomes perfectly reflecting.

In 1994, Berenger derived a new boundary condition referred to as a Perfectly Matched Layer (PML) which reduces reflections from boundaries [98]. The PML is placed around the boundaries of the FDTD domain. The presence of the PML ensures that a wave propagating in FDTD domain is fully absorbed once it hits the PML region at any incident angle, in other words the plane wave will be fully transmitted in the PML region without reflections back into the FDTD domain as shown in Figure 2.5.

PML Region FDTD Domain Incident wave Completely transmitted wave z x

Figure 2.5: Perfectly matched layer (PML) terminating FDTD domain

PML uses a modified set of Maxwell’s equations in which fields at the boundary layer interface are split into two components and an artificial anisotropic material is introduced to attenuate outgoing waves. The result is a PML wave impedance perfectly matched to the simulation space and independent of incidence angle. Incident waves are attenuated in the direction normal to the layers as they propagate through the artificial medium. Reflection coefficients as low as –80 dB have been demonstrated [99, 100] for both 2D and 3D FDTD simulations.

Although the PML technique provides an excellent boundary condition, its split field formulation requires a two-fold increase in computational memory and run-time in the regions containing the PML material relative to the conventional FDTD algorithm. To avoid this additional burden, a PML boundary condition based on a Maxwell formulation has been derived [101] and shown to be equivalent in effectiveness to the Berenger PML formulation. This alternate algorithm has the advantages of enhancing computational efficiency within the PML regions and eliminating the need for FDTD update equations modified for the split field formulation.

FDTD method that uses structured mesh, although a proven powerful numerical technique, has some limitations; one is involved with simulating problems with curved structures which need to be modelled using staircase approximations. In order to reduce the discretization error with staircasing, the mesh must be highly refined. This issue have motivated the development of new techniques based in unstructured meshes as discussed in Section 2.7.