Laboratory 7: Wave dispersion

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Laboratory 7: Wave dispersion
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௡ᇱ) 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;
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 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.
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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) ?
№ 3: Show your results to the teaching assistant.
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) ?
№ 4: Show your results to the teaching assistant.
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).