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Electromagnetic Sensing for Space-borne Imaging

Lecture 3Review of Maxwell’s equations, EM wave propagation

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

Gauss' law for charge: 4

Gauss' law for magnetism: 0

1Faraday's law of induction:

4 1Ampere/Maxwell law:

c t

c c t

D

B

BE

DH J

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Electric field (esu per cm)

Magnetic field (emu per cm)

Electric displacement field (esu per cm )

electrical permittivity or dielectric constant of the material

Magnetic flux

E

H

D E

B

2

3

2

density (emu per cm )

magnetic permeability of the material

Free electric charge density (esu per cm )

Free current density (esu per cm )

H

J

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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:

4

c t c c t

E

E J

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:

2 E E E

2

2 42 2 2

4+

c t c t

J E

E

22

2 2

1, 0

/ Speed of propagation of EM waves

refractive index of the material

speed of light in a vacuum

=299,792.458 km s

v tv c n

cn

vc

EE E

4

22

2 2

1, 0

Speed of propagation of EM waves

refractive index of the material

speed of light in a vacuum

v tv

cn

vc

EE E

Waves! Waves! Waves!

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Complex Notation, Plane Waves

A mathematical note: In our study of EM theory and imaging, it proves extremely convenient

to write all solutions and their derivations in terms of complex numbers. At the very end of a

calcul

ation, we take the real part of the result to get the physically meaningful final answer.

This process is justified by the linearity of Maxwell's equations.

Example : Derivation of a plane wave sol

Assume depends only upon and , and at a given , it varies harmonically with time with

frequency .

Now instead of setting , cos sin , we use the mor

x t x

x t x t x t

ution of the wave equation.

E

E a b

-i t

e compact form:

exp

Substituting this into the wave equation and dividing by e gives:

x i t

E U

2 2

2 2 0

ˆ Also the condition 0 shows that 0. Thus the E-field executes a transverse wave.

We find a complex-valued solution to the equation:

d xx

d x v

UU

E x U

U

ˆ exp

The wavenumber

ˆ and is a unit vector in the y - plane and is

ix Ae ikx

k v

z

U s

s

a constant phase factor, and is a positive constant.A

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Plane Waves, Continued

In summary:

ˆ , exp

Finally, to get the physical electric field; call it , we merely take the real part:

ˆ , , cos

x t A i t kx

x t

x t real x t A t kx

E s

E

E E s Consider the plus sign on the term. Then , looks like:

kx x tE

s

x

v=c/n

k=2

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Space & time variation of the plane wave

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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:

2

2

2

2 2 2 2

2 2

0

0

The energy carried by the wave is proportional to the time average of :

1ˆ

1 cos cos sin sin

cos cos 2 sin cos sin cos1

sin

T

T

dtT

A kx t A kx t dtTA kx t A kx kx t t

T A

E

E sE

2

2

2

0

0

sin

1

2But, using the complex notation, we have simply:

1 1ˆ ˆ ˆ

2

T

T

T

dtkx t

A

dtT

E s E s E sE

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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:

s

2 22

2 2

n U

Uc t

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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)

1 12 2 ,

n

f x f n T x n

,T x

x

1

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

1

, cosN

n nn

n

T x a s x s

s n s

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The more waves the better!

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

Using complex notation:

, exp 2n nn

n

T x i s x s

s n s

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If the top hat is a combination of waves, then so is any function!

x

f(x)

12

1 12 2

212

,

Since for each top hat: , exp 2 , we find:

exp 2

Thus any (square integrable) function can b

m

n

m mm

i s n

m mm n

m

f x f n T x n

T x i s x s

f x f n e i s x s

0

e approximated by a sum of waves.

The relation becomes exact if we let the increments of s get smaller and smaller:

lim exp 2 exp 2

, where

m ms

m

m

f x i s x s F s i sx ds

F s m s s

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The Fourier Transform

In the relation:

exp 2

is called the shows us how to

synthesize from combinations of waves.

is unique to (one implies the other and

f x F s i sx ds

F s F s

f x

F s f x vice

Fourier transform.

).

Now, just as we used waves with coefficients of combination that

depend on to synthesize , we can synthesize with co-

efficients of combination that depend on . In fact this coefficient

f

versa

s f x F s

x

unction is nothing more than itself:

exp 2

where we use the complex conjugate waves to synthesize

f x

F s f x i sx dx

F s

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Decomposition into Quasi-monochromatic Signals

Consider , (which is any one component of the electric field, recall). Take

the Fourier transform of , with respect to time:

ˆ , , exp

U t

U t

U U t i t dt

x

x

x x

The frequency in rad/ s 2

ˆ , characterizes the portion of , that consists of waves of frequency

. We can always decompose the radiation field into sets of waves, each set con-

si

s

U U tx x

0 0 0sting of waves with frequencies in a narrow slice, , , .

Each set constitutes a field. For each quasi-monochromatic

ˆsegment, , , exp . It is extremely conve

quasi - monochromatic

U t U i tx x

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2

nient to analyze each

quasi-monochromatic component separately and combine results in the end. For

ˆsuch a component containing waves with frequencies very near , ,

and the wave equation

UU

t

2 2

becomes:

ˆ , 0,

This is the

k U kv

x

Helmholtz equation.

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Let’s revisit the Top Hat Function…

x

f(x)

x0

0 0 0

0 0

Notice that for all :

, 1

Also, for small :

,

When becomes aritrarily small, these two

equations give:

, 1, ,

Now, the ,

T x dx

f x T x x dx f x O f x

T x dx f x T x x dx f x

define delta function

0 0

, to be

a (somewhat fictitious) function that resembles

, when is arbitrarily small and that

gives the above results exactly:

1, ( property)

, ( prop

x

T x

x dx

f x x x dx f x

normalization

sifting erty)

,T x

x

1

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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:

, , exp 2

, , exp 2

Alternately, if we represent the point , by the 2-D vector and the point

F u v f x y i ux vy dxdy

f x y F u v i ux vy du dv

x y x

2

2

, in the plane by the vector , we can write the transforms as:

e

e

is ca

i

i

u v u v

F f d

f F d

u x

u x

u

u x x

x u u

u

x

u

lled the wave vector.

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The Corrugation Interpretation

Define a 'corrugation" as a surface generated as the locus of a level straight line that passes

through a sinusoid perpendicular to the plane containing that sinusoid. The the two-D

Fourier synthesis ca1

n be thought of as a superposition of corrugations having all possible

wavelengths, , and all possible orientations, , with appropriate amplitudes, where:

q

q

2 2 , tanu v vu u

x

y

v -1

u -1

q -1

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Some Notation for Transforms

Let's follow Ron Bracewell and denote the one-dimensional Fourier

transform relations:

exp 2

exp 2

by the notation:

f x F s i sx ds

F s f x i sx dx

1

where the "1" indicates one-dimension.

Similarly for the two-dimensional relationships;

, , exp 2

f x F s

f x y F u v i ux vy du dv

2

, , exp 2

we write: , ,

F u v f x y i ux vy dxdy

f x y F u v

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

0

0

Let's start with the one-dimensional delta function in one dimension.

Here, , and the Fourier transform is:

exp 2

But recall the property:

f x x x

F s x x i sx dx

sifting

0 0

0

The left-hand side matches the transform expression if we set

exp 2 . Sustituting this into the sifting property:

exp 2 e

f x x x dx f x

f x

i sx

x x i sx dx F s

0

1

0 0

0

xp 2

Thus, in our chosen notation:

exp 2

1 .

i sx

x x i sx

x

Thus, the delta function "selects" one particular wave having

wavelength

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

0 0

2

0 0 0 0

Likewise, for the 2-D delta function, , , ,

we apply the sifting property twice to get:

, exp 2

f x y x x y y

x x y y i ux vy

Thus, the 2 - D delta function "selects" one particul

0 0

2

Note that when we set 0 :

, 1

x y

x y

ar 2 - D wave

or corrogation.

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Examples of Transform Pairs : Symmetric Delta Functions

2

Applying the above results to the half-strength impulses at (-a,0) and (a,0)

gives:

1 , , cos 2

2x a y x a y au

Thus, this combination of functions "selects" one particular 2 - D,

c

osinusoidal corrogation.

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Examples of Transform Pairs : 2-D Gaussian

2

2 2

A 2-D Gaussian function is its own transform:

exp exp

Here, and are the radial polar coordinates in the spatial and wave number planes,

respective

r q

r q

2 2 2

2 2 2

ly:

r x y

q u v

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Examples of Transform Pairs : Square Box

Define the function as:

1, 1 2

0, 1 2

and for the 2-D case:

,

Then:

rect x

xrect x

x

rect x y rect x rect y

2

, sinc sinc sinc ,

where:

sin sinc

rect x y u v u v

uu

u

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Examples of Transform Pairs : Pillbox

2 2

2 2

22 2

Define the function as:

1, 1 2

0, 1 2

Then:

,

where:

rect r

x yrect r

x y

rect r jinc q q u v

1 2

J xjinc x

x

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Examples of Transform Pairs : Shah Function

Define the as:

III , ,

This consists of an array of unit-strength impulses distributed at unit spacings acr

n m

x y x n y m

shah function (or bed - of - nails)

2

oss

the x-y plane. Like the Gaussian, the Shah function is its own transform:

III , III ,

x y u v

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Theorems on 2-D Fourier Transforms

2 2

2

In the following: , , and g , , are assumed.

, ,

f x y F u v x y G u v

F x y f u v

Reversal of Roles Thm. :

Similarity Thm. :

2

22

2

1, ,

, ,

1, ,

1 1 1

i au bv

f ax by F u a v bab

f x a y b e F u v

u dv v buf x by dx y F

bd bd bd

Shift Thm. :

Shear Thm. :

Parseval's

, , , ,

f x y g x y dx dy F u v G u v du dv

Thm. :

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Theorems on 2-D Fourier Transforms – Cont’d

2 2

2 2

In the following: , , and g , , are assumed.

, 2 , , , 2 ,

f x y F u v x y G u v

f x y i uF u v f x y i vF u vx y

Derivative Thm. :

Separable Product Thm. :

2

2 2 22 2

2

1, , ,

, , , ,

where the

f x g y F u G v

dx dy r f x y du dv q F u v dx dy f x y

f x y g x y F u v G u v

Uncertainty Principle :

2 - D Convolution Thm. :

2-D convolution operator ,"**", is defined by:

, , , ,f x y g x y f x x y y g x y dx dy

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