Feb. 7, 2011 Plane EM Waves The Radiation Spectrum: Fourier Transforms

Vector Wave Equations for E and B:

For solutions to the 3-Dimensional wave equation, use complex notation where

Before we go further, let’s review complex numbers Argand Diagram imaginary real y x x+iy Complex number Complex conjugate x = real part of z y = imaginary part of z

In polar coordinates so The Euler Formula implies r = magnitude of z θ = phase angle of z Re(z) = real part of z = rcosθ Im(z) = imaginary part of z = r sinθ

Since exponentials are so easy to integrate and differentiate, it is convenient to describe waves as Where A is a real constant To get the physically meaningful quantity, which must be a real number, one solves the wave equation and then takes the REAL part of the solution. This is OK, since the wave equation is linear, so that the real part of Ψ and its imaginary part are each separately solutions.

So for example you can write then vector

Solution to the wave equations: The waves travel in directionor surfaces of constant phase travel with time in direction

These must also satisfy Maxwell’s Equations Recall the definition of the divergence: So

“Can show” the other equations are

Also Require k>0 and ω>0  ω = c k Hence, E 0 =B 0

Qualitative Picture: For “one” wave with one λ In real situations, one wants to consider the superposition of many waves like this – and the more general case where the direction of E (and hence B) is random as the wave propagates.

Phase Velocity v. Group Velocity The speed at which the sine moves is the phase velocity The group velocity is This is usually discussed when you have several waves superimposed, which make a modulated wave: the modulation envelope travels with the group velocity In a dispersive medium ω=ω(k) so However, in a vacuum, v group = c

Group and Phase Velocities

The Radiation Spectrum Joseph Fourier

The Radiation Spectrum The spectrum depends on the time variation of the electric field (or, equivalently, the magnetic field) It is impossible to know what the spectrum is, if the electric field is only specified at a single instant of time. One needs to record the electric field for some sufficiently long time. The spectrum (energy as a function of frequency) is related to the E-field (as a function of frequency) through the Poynting Vector. The E-field (as a function of frequency) is related to the E-field (as a function of time) through the Fourier Transform Likewise, ω = angular frequency

Fourier Transforms A function’s Fourier Transform is a specification of the amplitudes and phases of sinusoidals, which, when added together, reproduce the function Given a function F(x) The Fourier Transform of F(x) is f(σ) see Bracewell’s book: FT and Its Applications The inverse transform is note change in sign

Not all functions have Fourier Transforms. F.T. sometimes called the “power spectrum” e.g. search for periods in a variable star

Visualizing the Fourier Transform:

Visualizing the F.T. Suppose you have a complex function: Recall Euler’s formula: FT(F(x)) = Notes: When F(x) is real (F I =0) the fourier transform f(σ) can still be complex. For fixed σ, these integrals involve multiplying F by a sine (or cosine) with period 1/σ and summing the area underneath the result. Changing the frequency of the sines and cosines and repeating the process gives f(σ) at a second value of σ, and so on.

Some examples (1) F.T. of box function

“Ringing” -- sharp discontinuity  ripples in spectrum When ω is large, the F.T. is narrow: first zero at other zeros at

(2) Gaussian F.T. of gaussian is a gaussian with narrower width Dispersion of G(x)  β Dispersion of g(σ) 

(3) delta- function Note: x x1x1

Amplitude of F.T. of delta function = 1 (constant with sigma) Phase = 2πx i σ  linear function of sigma

(4) So, cosine with wavelength transforms to delta functions at +/ x 1 x 0 -x 1 +x 1

(5) x -x 1 x1x1 0

Summary of Fourier Transforms