370 likes | 467 Views
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. imaginary. x+iy. Argand Diagram. y. real. x.
E N D
Feb. 7, 2011Plane EM WavesThe Radiation Spectrum: Fourier Transforms
For solutions to the 3-Dimensional wave equation, use complex notation where
Before we go further, let’s review complex numbers imaginary x+iy Argand Diagram y real x 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 vector
Solution to the wave equations: The waves travel in direction or surfaces of constant phase travel with time in direction
These must also satisfy Maxwell’s Equations Recall the definition of the divergence: So
Also Require k>0 and ω>0 ω = c k Hence, E0=B0
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, vgroup= c
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 see Bracewell’s book: FT and Its Applications 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) 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 F.T. Suppose you have a complex function: Recall Euler’s formula: FT(F(x)) = Notes: When F(x) is real (FI=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 x x1 Note:
Amplitude of F.T. of delta function = 1 (constant with sigma) Phase = 2πxiσ linear function of sigma
(4) x +x1 -x1 0 So, cosine with wavelength transforms to delta functions at +/ x1
(5) -x1 x x1 0
Summary of Fourier Transforms