DOING PHYSICS WITH MATLAB

[1D] FDTD electromagnetic wave simulations of propagating waves in non-magnetic and non-lossy dielectric media

INTERFENCE EFFECTS WITH THIN FILMS

Ian Cooper

Please email any corrections, comments, suggestions or additions

matlabvisualphysics@gmail.com

View ELECTROMAGNETISM USING THE FDTD METHOD

Matlab Download Directory

ft_04.m

Download and run the script ft_04.m. Carefully inspect the script to see how the FDTD method is implemented. Many variables can be changed throughout the script, for example, type of excitation signal, boundary conditions, time scales, properties of the medium.

We can also model the interference effects on the propagation of EM waves on encounter a thin film.

Firstly, we can consider an EM sinusoidal wave (wavelength ) propagating in free space that strikes a non-magnetic and non-lossy dielectric slab (dielectric constant or refractive index ) of width d. The strength of the electric wave that is transmitted into free space after passing through the dielectric slab depends upon the width d as a function of the wavelength .

The EM wave will be partially transmitted through the slab and there will be multiple partial reflections from the front and back of the dielectric slab. At an interface if the refractive index increases, the reflected wave will have a rad change in phase and if the refractive index decreases there be no change in phase (0 rad phase change). Taking into account the phase changes upon reflection, the width d as a function of wavelength , the multiple paths of the EM wave through the dielectric slab may interfere constructively or destructive to give a maximum or minimum in the intensity of the transmitted wave respectively. In our example, there is a 0 rad change in phase in the reflections at the front and back interfaces of the dielectric slab. The path lengths of the reflected EM through the slab are multiples of 2d. So, if the condition

is satisfied, then the waves will interfere constructively (figure 1).

If

is satisfied, then the waves will interfere destructively (figure 2).

The script ft_04.m can be used to model the interference effects of the dielectric slab.

Model parameters:

Nz = 150 lambda = 80 source: sinusoidal boundary conditions: absorbing

eR1 = 1 S1 = 0 eR2 = 4 S2 = 0

dz = lambda / 40 40 Z-grid spacings is equal to one wavelength

For eR2 = 4, the refractive index n of the dielectric slab is

If (40 grid spacing) then the wavelength in the dielectric slab is (20 grid spacings). The width d of the slab is given by the number of grid spacing specified by the variable indexR.

Constructive interference indexR = 50:50+20+10 equals 10 grid spacings

Destructive interference indexR = 50:50+20+5 equals 5 grid spacings

Fig. 1. The width of the dielectric slab is and the transmitted wave has a maximum amplitude (constructive interference). The electric field is measured at three probe positions.

Fig. 2. The width of the dielectric slab is and the transmitted wave has a minimum amplitude (destructive interference).

We can observe the phase changes occurring at the interfaces by using a pulse for the EM wave as shown in figure 3.

Fig. 3. A pulse striking the dielectric slab. Carefully inspect the electric field as measured by the three probes.