ECSE 354 - Laboratory 7 - Wave dispersion - Solved

 

In this laboratory, we will explore wave propagation in lossless media with different dispersion relations.

 

The instantaneous electric field amplitude of a pulse of electromagnetic waves travelling along the z-axis at a carrier frequency ω0 is represented by the following sum of ( 2N + 1 ) travelling wave components at different frequencies:

 

𝐸(𝑧, 𝑡) =    𝐸 cos(𝜔 '𝑡 − 𝛽 '𝑧 ) ' 𝑁

 

where 𝜔 ' = 𝜔    + 𝑛'𝛿𝜔 is the frequency of each forward travelling wave, with a corresponding phase constant 𝛽 ' = 𝛽(𝜔            ) that is a function of frequency. E0 is a constant amplitude for each component and the bandwidth is Nδω. Note that the Fourier-like sum above could be expressed with complex exponentials by applying Euler’s theorem.

 

1. Pulse propagation in vacuum

 

Consider vacuum, where the dispersion relation is given by 𝛽 = 𝜔/𝑐. We will further take the following parameters: E0 = 1 V/m, ω0 = 2π x 109 rad/s, N = 20, δω = ω0/60. The simulation will take place over a time interval 0 ns < t < 33 ns in steps of 0.3 ns, with field displayed on the spatial interval -2 m < z < 10 m with 0.01 m resolution.

 

A MATLAB code implementing the above simulation is given below, producing an animation of the field E(z,t) versus position z as time t evolves:

 

close all; clear all;

 

eps0 = 8.854e-12; mu0 = 4*pi*1e-7; c = 1/sqrt(mu0*eps0);

 

t = (0:0.3:33)*1e-9; z = -2:0.01:10;

 E0 = 1; omega0 = 2*pi*1e9; domega = omega0/60; for n = 1:41;

    omega(n) = omega0+(n-21)*domega; end;

 

% vacuum dispersion beta = omega/c;

 for m=1:length(t);     E=zeros(1,length(z));     for n = 1:41;         Etemp=E0*cos( omega(n)*t(m) - beta(n)*z );

        E=E+Etemp;     end;    plot(z, E);

   axis( [ -2 +10 -50 +50 ] );    xlabel("z [m]");    ylabel("E(z,t)[V/m]");

   title("instantaneous field, \beta = \omega / c");  

   M(m)=getframe; end;  

movie2gif(M,"vacuum.gif")

 

Copy the above code into a script and execute. Notice how the phase velocity vp and group velocity vg are equal.

 

Observe the distance Δz that the pulse travels in the elapsed simulation time Δt = 33 ns. What is the group velocity vg = Δz/Δt ? Use the mid-point of the pulse to approximate Δz. How does vg compare with c, the speed of light?

 

№ 1: Show your results to the teaching assistant.

 

 

2.  Pulse propagation in a medium with constant refractive index n = 1.5

 

Consider now the propagation of the pulse through a lossless medium with a constant refractive index n = 1.5. The modified dispersion relation is 𝛽 = 𝜔/𝑐 × 1.5 .

 

Use the code of exercise 1 with a phase constant 𝛽 = 𝜔/𝑐 × 1.5, and calculate E(z,t) to simulate pulse propagation.

 

Observe the distance Δz that the pulse travels in the elapsed simulation time Δt = 33 ns. What is the group velocity vg = Δz/Δt ? Use the mid-point of the pulse to approximate Δz.

Is the phase velocity vp equal to the group velocity vg?

Why does the pulse appear compressed versus the distance z as compared to the pulse in vacuum?

 

№ 2: Show your results to the teaching assistant.

 

 

3.  Pulse propagation in a medium with a dispersive refractive index n(ω) = 1.5 ( 1 + ω / Ω )

 

Consider now the propagation of the pulse through a lossless medium with a frequency dependent refractive index 𝑛(𝜔) = 1.5 ( 1 + 𝜔/𝛺 ) where the frequency Ω = 2π x 8 x 109 rad/s. The refractive index increases with frequency. The modified dispersion relation is 𝛽 = 𝜔/𝑐 × 1.5 ( 1 + 𝜔/𝛺 ) .

 

Use the code of exercise 1 with a phase constant 𝛽 = 𝜔/𝑐 × 1.5 ( 1 + 𝜔/𝛺 ) and calculate E(z,t) to simulate pulse propagation.

 

Observe the distance Δz that the pulse travels in the elapsed simulation time Δt = 33 ns. What is the group velocity vg = Δz/Δt ? Use the mid-point of the pulse to approximate Δz.

Is the phase velocity vp equal to the group velocity vg?

Observe the pulse very carefully, and you will see that the pulse broadens, and the apparent value of  𝛽 (the spatial periodicity) is different at the leading edge and trailing edge of the pulse. How can this be explained in terms of n(ω) ?

 


 

Optional: Is vg larger or smaller than vp?

 

 

 

4. Pulse propagation in a medium with a dispersive refractive index n(ω) = 1.5 ( 1 - ω / Ω )

 

Consider now the propagation of the pulse through a lossless medium with a frequency dependent refractive index 𝑛(𝜔) = 1.5 ( 1 − 𝜔/𝛺 ) where the frequency Ω = 2π x 8 x 109 rad/s. The refractive index increases with frequency. The modified dispersion relation is 𝛽 = 𝜔/𝑐 × 1.5 ( 1 − 𝜔/𝛺 ) .

 

Use the code of exercise 1 with a phase constant 𝛽 = 𝜔/𝑐 × 1.5 ( 1 − 𝜔/𝛺 ) and calculate E(z,t) to simulate pulse propagation.

 

Observe the distance Δz that the pulse travels in the elapsed simulation time Δt = 33 ns. What is the group velocity vg = Δz/Δt ?

Is the phase velocity vp equal to the group velocity vg?

Observe the pulse very carefully, and you will see that the pulse broadens, and the apparent value of  𝛽 (the spatial periodicity) is different at the leading edge and trailing edge of the pulse. How can this be explained in terms of n(ω) ?

 
.

 

Optional: Is vg larger or smaller than vp?

 

 

 

A cautionary note: The causality of physical laws imposes strict bounds on physically realizable dispersion relations and the associated frequency dependent attenuation (which we have neglected in this exercise).