1 Electromagnetic Sensing for Space-borne Imaging Lecture 3 Review of Maxwell’s equations, EM wave propagation

2 Maxwell’s equations ( in the “Gaussian” or “cgs” units)

3 The Wave Equation Assume the electric permittivity and the magnetic permeability are constant Take the curl of both sides of Faraday’s law of induction and use the Ampere/Maxwell law:  Using :  Most methods of measuring or recording the electromagnetic field interact primarily with the electric field so we concentrate on the dynamics of the electric field. With no current sources and charges, we obtain the wave equation:

4 Waves! Waves! Waves!

5 Complex Notation, Plane Waves

6 Plane Waves, Continued v=c/n k=2 

7 Space & time variation of the plane wave

8 Plane Waves, Continued So, we’ve rediscovered that one of the most striking phenomena predicted by Maxwell’s equations is the propagation of waves! We also see that the complex notation is somewhat more efficient. Another example in the same vein:

9 Specialization to scalar signals The vector is an example of a polarization vector. It is this vector that characterizes the electric field as a vector field. While polarization effects are very important phenomena, much of our study of EM radiation and imaging can ignore the transverse character of the electric field and focus on each individual component of the electric field. Note that each component of the electric field (in any Cartesian coordinate system) obeys the wave equation. Thus, in the following, we let U(x,t) stand for any one of the components of the electric field. The basic idea is that we show how to analyze any one component and then combine results at the end. Further, we use complex notation, so U(x,t) is a complex-valued function of position and time. U(x,t) generally satisfies the scalar wave equation:

10 Representing a function as a superposition of simple functions We can use this “top hat” function to represent any piece-wise continuous function to any desired degree of accuracy (depending on  ) x f(x) x

11 Is there a better way to represent functions? What if we could compose the top hat function with waves?

12 The more waves the better!

13 …Eventually, enough waves of different frequencies give us a good approximation to the top hat function

14 If the top hat is a combination of waves, then so is any function! x f(x)

15 The Fourier Transform

16 Decomposition into Quasi-monochromatic Signals

17 Let’s revisit the Top Hat Function… x f(x) x0x0 x

18 The Two-Dimensional Fourier Transform We’ve seen how the one-dimensional Fourier transform can be used to decompose waves into separate time-dependent oscillations – essentially getting rid of the time variable in the wave equation. But Fourier analysis is also useful in representing 2-D patterns, e.g. images. To do this, we extend the Fourier transform to two-dimensions:

19 The Corrugation Interpretation x y  v -1 u -1 q -1

20 Some Notation for Transforms

21 Examples of Transform Pairs : Delta Function (1-D)

22 Examples of Transform Pairs : Delta Function (2-D)

23 Examples of Transform Pairs : Symmetric Delta Functions

24 Examples of Transform Pairs : 2-D Gaussian

25 Examples of Transform Pairs : Square Box

26 Examples of Transform Pairs : Pillbox

27 Examples of Transform Pairs : Shah Function

28 Theorems on 2-D Fourier Transforms

29 Theorems on 2-D Fourier Transforms – Cont’d

30 Learn all this and you’ll make a big splash!