1 EEE 431 Computational Methods in Electrodynamics Lecture 13 By Dr. Rasime Uyguroglu

2 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) Numeric Dispersion

3 FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical Dispersion) A dispersion relation gives the relationship between the frequency and the speed of propagation. Consider a plane wave propagating in the positive z direction in a lossless medium. In time harmonic form the temporal and spatial dependence of the wave are given by

4 FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical Dispersion) Where is the frequency and is the phase constant. The speed of the wave can be found by determining how fast a given point on the wave travels. Let =constant

5 FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical Dispersion) Setting this equal to a constant and differentiating with respect to time gives:

6 FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical Dispersion) Where is the speed of the wave, propagating in the z-direction. Therefore the phase velocity yields:

7 FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical Dispersion) This is apparently a function of frequency, but for a plane wave the phase constant is given by. Thus, the phase velocity is: Where c is the speed of light in free space.

8 FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical Dispersion) Note that in the continuous world for a lossless medium, the phase velocity is independent of frequency. The dispersion relationship is

9 FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical Dispersion) Since c is a constant, as for any given material, all frequencies propagate at the same speed. Unfortunately this is not the case in the discretized FDTD world—different frequencies have different phase speeds.

10 FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical Dispersion) Governing Equations Define the spatial shift-operator And the temporal shift-operator Let a fractional superscript represent a corresponding fractional step.

11 FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical Dispersion) For example:

12 FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical Dispersion) Using these shift operators the finite- difference version of : Can be written;

13 FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical Dispersion) Rather than obtaining an update equation from this, the goal is to determine the phase speed for a given frequency. Assume that there is a single harmonic wave propagating such that:

14 FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical Dispersion) Where is the phase constant which exists in the FDTD grid and are constant amplitudes. We will assume that the frequency is the same as the one in the continuous world. Note that one has complete control over the frequency of excitation, however, one does not have control over the phase constant, i.e., the spatial frequency.

15 FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical Dispersion) Assume, the temporal shift-operator acting on the electric field:

16 FINITE DIFFERENCE TIME DOMAIN METHOD( Numerical Dispersion) Similarly, the spatial shift-operator acting on the electric field yields

17 FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical Dispersion) Thus, for a plane wave, one can equate the shift operators with multiplication by an appropriate term:

18 FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical Dispersion) Now the scalar equation:

19 FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical Dispersion) Employing Euler’s formula to convert the complex exponentials to trigonometric functions:

20 FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical Dispersion) Canceling the exponential space-time dependence which is common to both sides produces:

21 FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical Dispersion) Solving for the ratio of the electric and magnetic field amplitudes yields:

22 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD-ABC’s) Another equation relating can be obtained from;

23 FINITE DIFFERENCE TIME DOMAIN METHOD Expressed in terms of shift operators, the finite-difference form of the equation:

24 FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical Dispersion) As before, assuming plane-wave propagation, the shift operators can be replaced with multiplicative equivalents. The resulting equation is:

25 FINITE DIFFERENCE TIME DOMAIN METHOD Simplifying and rearranging yields:

26 FINITE DIFFERENCE TIME DOMAIN METHOD (Numeral Dispersion) Equating equations and cross- multiplying:

27 FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical Dispersion) Taking the square root: This equation gives the relation between Which is different than the one obtained for the continuous case.

28 FINITE DIFFERENCE TIME DOMAIN METHOD (Numeral Dispersion) However, the two equations do agree in the limit as the discretization gets small.

29 FINITE DIFFERENCE TIME DOMAIN METHOD (numerical Dispersion) The first term in the Taylor series expansion of: for small Assume that the spatial and temporal steps are small enough so that the arguments of trigonometric functions are small in the dispersion relation.

30 FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical Dispersion) From this: Which is exactly the same as in the continuous world. However, this is only true when the discretization goes to zero. For finite discretization, the phase velocity in the FDTD grid and in the continuous world differ.

31 FINITE DIFFERENCE TIME DOMAIN METHOD ( Numerical Dispersion) In the continuous world: In the FDTD world the same relation holds

32 FINITE DIFFERENCE TIME DOMAIN METHOD For the one dimensional case a closed form solution for is possible. A similar dispersion relation holds in two and three dimensions, but there a closed-form solution is not possible.

33 FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical Dispersion) For a closed form solution: The factor Where,

34 FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical Dispersion) Thus

35 FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical Dispersion) Consider the ratio of the phase velocity in the grid to the true phase velocity:

36 FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical Dispersion) For the continuous case:

37 FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical Dispersion) The ratio becomes: This equation is a function of the material, the Courant number, and the number points per wavelength.

38 FINITE DIFFERENCE TIME DOMAIN METHOD (Numerical Dispersion)