eit, electrical and information technology - eten05 … · 2009. 9. 17. · e jkz!e jkr assuming...

ETEN05 Electromagnetic Wave PropagationLecture 6: Oblique propagation, beams, and

FDTD

Daniel Sjoberg

Department of Electrical and Information Technology

September 18, 2009

Outline

1 Introduction

2 Oblique propagation

3 Paraxial approximation: beams

4 Finite differences in the time domain: FDTD

5 Conclusions

Daniel Sjoberg, Department of Electrical and Information Technology

Oblique propagation

I So far, we have chosen the z direction as the propagationdirection.

I We shall see how this generalizes, and how to describe wavepropagation when the preferred direction is not the purepropagation direction.

I Understanding of this situation helps analyzing reflection onsurfaces at different angles.

I It also helps analyzing more general wave types than planewaves (today we look at beams).


Numerical methods

Some problems are too complicated to do by hand, such as apacemaker in the human body. Numerical methods help.


Outline

1 Introduction




5 Conclusions


Generalized propagation factor

For a wave propagating in an arbitrary direction, the propagationfactor is generalized as

e−jkz → e−jk·r

Assuming this as the only spatial dependence, the nabla operatorcan be replaced by −jk since

∇(e−jk·r) = −jk(e−jk·r)

Writing the fields as E(r) = E0e−jk·r, Maxwell’s equations forisotropic media can then be written{

−jk ×E0 = −jωµH0

−jk ×H0 = jωεE0⇒

{k ×E0 = ωµH0

k ×H0 = −ωεE0


Properties of the solutions

Eliminating the magnetic field, we find

k × (k ×E0) = −ω2εµE0

This shows that E0 does not have any components parallel to k,and the BAC-CAB rule implies k× (k×E0) = −E0(k · k). Thus,

k2 = k · k = ω2εµ

It is further clear that E0, H0 and k constitute a right-handedtriple since k ×E0 = ωµH0, or

H0 =k

ωµ

k

k×E0 =

1ηk ×E0


Preferred direction

What happens when k is not along the z-direction (which could bethe normal to a plane surface)?

I There are then two preferred directions, k and z.

I These span a plane, the plane of incidence.

I It is natural to specify the polarizations with respect to thatplane.

I When the H-vector is orthogonal to the plane of incidence,we have transverse magnetic polarization (TM).

I When the E-vector is orthogonal to the plane of incidence, wehave transverse electric polarization (TE).


TM and TE polarization

From these figures it is clear that the transverse impedance is

ηTM =ExHy

=A cos θ

1ηA

= η cos θ

ηTE = −EyHx

=B

1ηB cos θ

=η

cos θ


Interpretation of transverse wave vector

The transverse wave vector kt = kxx corresponds to the angle ofincidence θ as

kx = k sin θ

This interpretation holds also for the vector kt in the fundamentaleigenvalue equation

β

k0

(Et

η0Ht

)=(

0 −z × Iz × I 0

)·[(εtt ξtt

ζtt µtt

)−A(kt)

]︸︷︷︸

=W

·(Et

η0Ht

)

in the completely general bianisotropic case. The transverseimpedance is

Et = Zr · (η0Ht × z), η0Zr = η cos θxx+η

cos θyy︸︷︷︸

isotropic case


Outline

1 Introduction




5 Conclusions


The plane wave monster

So far we have treated plane waves, which have a seriousdrawback:

I Due to the infinite extent of e−jβz in the xy-plane, the planewave has infinite energy.

However, the plane wave is a useful object with which we can buildother, more physically reasonable, solutions.


Finite extent in the xy-plane

We can represent a field distribution with finite extent in thexy-plane using a Fourier transform:

Et(r, ω) =1

(2π)2

∞∫∫−∞

Et(kt, ω)e−jkt·r−jβz dkx dky

Et(kt, ω) =

∞∫∫−∞

Et(r, ω)ejkt·r dx dy

Defining the (square of the) total wavenumber ask2 = |kt|2 + β2 = k2

x + k2y + β2 = ω2εµ implies

β = (k2 − |kt|2)1/2


Initial distribution

Assume a Gaussian distribution in the plane z = 0

Et(x, y, z = 0, ω) = A(ω)e−(x2+y2)/(2b2),

The transform is itself a Gaussian

Et(kt) = A(ω)2πb2e−(k2x+k2

y)b2/2


Paraxial approximation

The field in z ≥ 0 is then

Et(r) =12πAb2

∞∫∫−∞

e−(k2x+k2

y)b2/2−j(kxx+kyy)−jβz dkx dky

The exponential makes the main contribution to come from aregion close to kt ≈ 0. This justifies the paraxial approximation

β = (k2 − |kt|2)1/2 = k(1− |kt|2/k2)1/2

= k

(1− 1

2|kt|2

k2+ O(|kt|4/k4)

)= k − |kt|2

2k+ · · ·


Computing the field

Inserting the paraxial approximation in the Fourier integral implies

Et(r) ≈12πAb2

∞∫∫−∞

e−(k2x+k2

y)( b2

2−j z

2k)−j(kxx+kyy)−jkz dkx dky

= · · · = A

1− jξ(z, ω)e−(x2+y2)/(2F 2)−jkz

where F 2(z, ω) = b2 − jz/2k = b2(1− jξ(z, ω)) and ξ = z/(kb2) isa real quantity.


Beam width

The power density of the beam is proportional to

e−(x2+y2)Re(1/F 2)

and the beam width is then

B(z) =1√

Re 1F 2

= · · · = bξ√

1 + ξ−2

where ξ = z/(kb2). For large z, the beam width is

B(z)→ bξ(z) =z

kb, z →∞


Beam width

The beam angle θb is characterized by

tan θb =B(z)z

=1kb

Small initial width compared to wavelength implies large angle.Daniel Sjoberg, Department of Electrical and Information Technology

Outline

1 Introduction




5 Conclusions


Finite difference approximation

To make numerical simulations, we approximate the derivatives inthe wave equation

∂2E

∂z2− 1c2∂2E

∂t2= 0

with finite differences (where the field is evaluated in grid pointsE|nr = E(r∆z, n∆t):

E|nr+1 − 2E|nr + E|nr−1

(∆z)2− 1c2E|n+1

r − 2E|nr + E|n−1r

(∆t)2= 0

We expect the error in the approximation to decrease as ∆z and∆t become small.


Time stepping

By solving for the field at time step n+ 1, we find

E|n+1r = 2E|nr − E|n−1

r +(c∆t

∆z

)2

(E|nr+1 − 2E|nr + E|nr−1)

Thus, the solution at time t can be found if the solution at twoprevious time steps is known.


Leap frog stencil

-

6

0

0

z

t

∆z

∆t

r∆z

n∆t

b b b b b b bb b b b b b bb b b b b b bb b b b b b bb b b b b b b

jjj jj

Initial conditions: E|0r = fr and E|1r = gr (known functions).Boundary conditions: E|n0 = 0 and E|nR = 0 (metal walls).


Leap frog stencil

-

6

0

0

z

t

∆z

∆t

r∆z

n∆t


j

jj jj



Leap frog stencil

-

6

0

0

z

t

∆z

∆t

r∆z

n∆t


jj

j jj



Leap frog stencil

-

6

0

0

z

t

∆z

∆t

r∆z

n∆t


jjj j

j



Leap frog stencil

-

6

0

0

z

t

∆z

∆t

r∆z

n∆t


jjj jj



Dispersion relation

A harmonic wave propagating through the lattice can be written

ejωn∆t−jkr∆z

Inserting this into the difference approximation we obtain

ejωn∆t−jkr∆z

(e−jk∆z − 2 + ejk∆z

(∆z)2− 1c2

ejω∆t − 2 + e−jω∆t

(∆t)2

)= 0

which can be rewritten as(sin

ω∆t

2

)2

=(c∆t

∆z

)2(sin

k∆z

2

)2

This relates the spatial frequency k to the temporal frequency ω,and is called the dispersion relation.


Time step and stability: R = c∆t/∆z

R = 1: Magic time step, ω = ±ck, no dispersion.

R < 1: Stable solution, ω 6= ±ck, different frequencies travel withdifferent speed (lower than c).

R > 1: Unstable solution, growing exponentially.


Example: square wave

R = 1

R < 1

R > 1


Example: smooth wave


Time and frequency

Even though FDTD solves problem in the time domain, the resultscan be transformed to the frequency domain. A common situationis the computation of resonance frequencies in a cavity:

I Discretize the cavity with finite differences.

I Set up PEC boundary conditions.

I Give random data as initialization in order to excite allfrequencies.

I Run simulation for a long time.

I Fourier transform the field, sampled at a point.

I The resulting spikes correspond to the resonances of thecavity.


Outline

1 Introduction




5 Conclusions


Conclusions

I Oblique propagation with respect to some preferred directionz generates a plane of incidence.

I The TM and TE polarizations are natural decompositions ofthe fields.

I A beam can be composed by many plane waves travellingalmost parallel to the axis.

I The beam broadens as it propagates. Narrow beam, largeangle.

I FDTD is a simple numerical scheme for time domain.

I It is vital to choose a smaller discretization in time than infrequency. The relation between time and space determinesthe numerical dispersion.


Next week

I Three lectures next week, lots to read!

I Overall theme is pulse propagation and dispersion.

I The second handin is distributed on Tuesday, and will dealwith numerical experiments.


eit, electrical and information technology - eten05 … · 2009. 9. 17. · e jkz!e jkr assuming...

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