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

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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.

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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

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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

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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

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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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