Mathematical description of EM waves
Mathematical description of EM waves
? Use of complex numbers to represent EM waves
? The complex refractive index
? Scattering = real part ? Absorption = imaginary part
? Absorption and skin depth
? Beer's Law
1
What is a wave?
A wave is anything that moves.
To displace any function f(x) to the right, just change its argument from x to x-a, where a is a positive number.
If we let a = v t, where v is positive and t is time, then the displacement will increase with time.
f(x) f(x-2) f(x-1) f(x-3)
So f(x - v t) represents a rightward, or forward, propagating wave.
Similarly, f(x + v t) represents a leftward, or backward, propagating wave, where v is the velocity of the wave.
0 1 2 3x
For an EM wave, we could have E = f(x ? vt)
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The one-dimensional wave equation
The one-dimensional wave equation for scalar (i.e., non-vector) functions, f:
?2 f
1 ?2 f
?x2 - v2 ?t2 = 0
where v will be the velocity of the wave. The wave equation has the simple solution:
f (x,t) = f (x ? vt)
where f (u) can be any twice-differentiable function.
3
What about a harmonic wave?
E = E0 cos k(x - ct)
E0 = wave amplitude (related to the energy carried by the wave).
k = 2 = 2~ = angular wavenumber
~ ( = wavelength; = wavenumber = 1/)
Alternatively:
E = E0 cos(kx - t) Where = kc = 2c/ = 2f = angular
frequency (f = frequency)
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What about a harmonic wave?
E = E0 cos k(x - ct); = k(x - ct)
The argument of the cosine function represents the phase of the wave, , or the fraction of a complete cycle of the wave.
In-phase waves
Line of equal phase = wavefront = contours of maximum field
Out-of-phase waves
6
The Phase Velocity
How fast is the wave traveling? Velocity is a reference distance divided by a reference time.
The phase velocity is the wavelength / period: v = l / t
Since f = 1/t :
v=lf
In terms of k, k = 2p / l, and the angular frequency, w = 2p / t, this is:
v =w/k
7
3
The Group Velocity
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This is the velocity at which the overall shape of the wave's amplitudes, or the wave `envelope', propagates. (= signal velocity) Here, phase velocity = group velocity (the medium is non-dispersive) 8
Dispersion: phase/group velocity depends on frequency
Black dot moves at phase velocity. Red dot moves at group velocity. This is normal dispersion (refractive index decreases with increasing ) 9
4
Normal dispersion of visible light
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Shorter (blue) wavelengths refracted more than long (red) wavelengths. Refractive index of blue light > red light. 10
Dispersion: phase/group velocity depends on frequency
Black dot moves at group velocity. Red dot moves at phase velocity. This is anomalous dispersion (refractive index increases with increasing ) 11
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