Topic 3a Electromagnetic Waves & Polarization · 2018-03-18 · Electromagnetic Waves & Polarization Slide 1 EE 4347 Applied Electromagnetics Topic 3a Electromagnetic Waves & Polarization

9/13/2017

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Electromagnetic Waves & Polarization Slide 1

EE 4347

Applied Electromagnetics

Topic 3a

Electromagnetic Waves & Polarization These notes may contain copyrighted material obtained under fair use rules. Distribution of these materials is strictly prohibited

Course InstructorDr. Raymond C. RumpfOffice: A‐337Phone: (915) 747‐6958E‐Mail: [email protected]

Lecture Outline


• Maxwell’s Equations

• Derivation of the Wave Equation

• Solution to the Wave Equation

• Intuitive Wave Parameters

• Dispersion Relation

• Electromagnetic Wave Polarization

• Visualization of EM Waves

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Maxwell’s Equations


Recall Maxwell’s Equations in Source Free Media

Curl Equations

E j B

H j D

Divergence Equations

0

0

D

B

Constitutive Relations

D E

B H

In source‐free media, we have and .Maxwell’s equations in the frequency‐domain become

0J

0v

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The Curl Equations Predict Waves

E j H

After substituting the constitutive relations into the curl equations, we get

H j E

A time‐harmonic magnetic field will induce a time‐harmonic electric field circulating about the magnetic field.

A time‐harmonic circulating electric field will induce a time‐harmonic magnetic field along the axis of circulation.

A time‐harmonic electric field will induce a time‐harmonic magnetic field circulating about the electric field.

A time‐harmonic circulating magnetic field will induce a time‐harmonic electric field along the axis of circulation.

An H induces an E. That E induces another H. That new H induces another E. That E induces yet another H. Ando so on.


Visualization of Curl & Waves

Electric Field

Magnetic Field

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Derivation of the Wave Equation


Wave Equation in Linear Media (1 of 2)

E j H

Since the curl equations predict propagation, it makes sense that we derive the wave equation by combining the curl equations.

H j E

Solve for H

11H E

j

11E j E

j

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Wave Equation in Linear Media (2 of 2)

The last equation is simplified to arrive at our final equation for waves in linear media.

1 2E E

This equation is not very useful for performing derivations. It is typically used in numerical computations.

Note: We cannot simplify this further because, in general, the permeability is a function of position and cannot be brought outside of the curl operation.

21E E


Wave Equation in LHI Media (1 of 2)

In linear, homogeneous, and isotropic media two important simplifications can be made.

First, in isotropic media the permeability and permittivity reduce to scalar quantities.

1 2E E

Second, in homogeneous media is a constant and can be brought to the outside of the curl operation and then brought to the right‐hand side of the equation.

2E E

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Wave Equation in LHI Media (2 of 2)

We now apply the vector identity

2

2 2

2 2

E E

E E E

E E E

2A A A

In LHI media, the divergence equation can be written in terms of E.

0

0

0

0

D

E

E

E

2 2 0E E


Wave Number k and Propagation Constant We can define the term as either

2 2k

This let’s us write the wave equation more simply as

2 2 0E k E

2 2 2k or

2 2 0E E

or

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Solution to the Wave Equation


Components Decouple in LHI Media

We can expand our wave equation in Cartesian coordinates.

2 2

2 2 2 2 2 2

2 2 2 2 2 2

0

ˆ ˆ ˆ ˆ ˆ ˆ 0

ˆ ˆ ˆ 0

x x y y z z x x y y z z

x x x y y y z z z

E k E

E a E a E a k E a k E a k E a

E k E a E k E a E k E a

We see that the different field components have decoupled from each other.

All three equations have the same numerical form so they all have the same solution.

Therefore, we only need the solution to one of them.

2 2 0E k E

2 2

2 2

2 2

0

0

0

x x

y y

z z

E k E

E k E

E k E

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General Solution to Scalar Wave Equation

Our final wave equation for LHI media is

2 2 0E k E

This could be handed off to a mathematician to obtain the following general solution.

0 0jk r jk rE r E e E e

forward wave backward wave


General Solution to Vector Wave Equation

Given the solution to scalar wave equation, we can write solutions for all three field components.

jk r jk rx x x

jk r jk ry y y

jk r jk rz z z

E r E e E e

E r E e E e

E r E e E e

We can assemble these three equations into a single vector equation.

0 0

ˆ ˆ ˆx x y y z z

jk r jk r

E r E r a E r a E r a

E e E e


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General Expression for a Plane Wave

The solution to the wave equation gave us two plane waves. From the forward wave, we can extract a general expression for plane waves.

, cos

jk rE r Pe

E r t P t k r

Frequency‐domain

Time‐domain

We define the various parameters as

ˆ ˆ ˆ position

total electric field intensity

polarization vector

x y zr xa ya za

E

P

wave vector

2 angular frequency

time

k

f

t


Magnetic Field Component

Given that the electric field component of a plane wave is written as

jk rE r Pe

The magnetic field component is derived by substituting this solution into Faraday’s law.

jk rPe j H

1

jk rH k P e

E j H

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Solution in Terms of the Propagation Constant The wave equation and it solution in terms of is

2 2 0 E k E 0 0r rE r E e E e


, cos

rE r Pe

E r t P t r

Frequency‐domain

Time‐domain

The general expressions for a plane wave are

The magnetic field component is

1 rH P ej

jk

The wave vector and propagation constant are related through


Visualization of an EM Wave (1 of 2)

We tend to draw and think of electromagnetic waves this way…

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Visualization of an EM Wave (2 of 2)

However, this is a more realistic visualization. It is important to remember that plane waves are also of infinite extent in all directions.


Intuitive Wave Parameters

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Fundamental Vs. Intuitive Parameters

Fundamental Parameters Intuitive Parameters

These parameters are fundamental to solving Maxwell’s equations, but it is difficult to specify how they affect a wave. This is because all of they all affect all properties of a wave.

Magnetic Permeability, Electric Permittivity, Electrical Conductivity,

These parameters collect specific information about a wave from the fundamental parameters.

Refractive index, nImpedance, Wavelength, Velocity, vWave Number, kPropagation Constant, Attenuation Coefficient, Phase Constant,


Wave Velocity, v

The scalar wave equation has been known since the 1700’s to be

22 0

v

wave disturbance

angular frequency

wave velocityv

If we compare our electromagnetic wave equation to the historical wave equation, we can derive an expression for wave velocity.

2 2

22

0

0

E E

v

1

vv

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Speed of Light in Vacuum, c0

In a vacuum, = 0 and = 0 and the velocity becomes the speed of light in a vacuum.

0

0 0

1 1 299,792, 458 m sv c

When not in a vacuum, = 0r and = 0r and the velocity is reduced by a factor n called the refractive index.

0

0 0 0 0

1 1 1 1

r r r r

cv

n

r rn


Frequency is Constant, Wavelength Changes

When a wave enters a different material, its speed and thus its wavelength change.

0cv

n

Frequency is the most fundamental constant about a wave. It never changes in linear materials.

0

n

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Speed, Frequency & Wavelength

The speed of a wave, its frequency, and its wavelength are related through

v f We are now in a position to derive an expression for wavelength.

1 1 2 f

f


Wavelength & Wave Number k

Recall that we defined the wave number as

Substituting this into the first equation gives

0 r 0 r 0 0 r r0

1k n

c

The angular frequency is related to wavelength through the ordinary frequency f.

0 0

0

2 2 2c c

fn

0

0 0

1 1 22

ck n n

c n c

2k

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Wave Vector,

The wave vector conveys two pieces of information: (1) Magnitude conveys the wavelength inside the medium, and (2) direction conveys the direction of the wave and is perpendicular to the wave fronts.

k

k

ˆ ˆ ˆx x y y z zk k a k a k a


Magnitude Conveys Wavelength

Most fundamentally, the magnitude of the wave vector conveys the wavelength of the wave inside of the medium.

1 2

11

2k

22

2k

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Magnitude May Convey Refractive Index

When the frequency of a wave is known, the magnitude of the wave vector conveys refractive index.

1 2

11 0 1

0

2 nk k n

22 0 2

0

2 nk k n

00

2k


Material Impedance, (1 of 3)

Impedance is defined as the relationship between the amplitudes of E and H.

0 0E H Recall the relationship between E and H.

1 and jk r jk rE r Pe H k P e

We can derive an expression for impedance by collecting all of the amplitude terms together in our expression for H.

00

1 ˆ ˆˆ ˆjk r jk rE kH kk E P e k P e

This term is the amplitude of H

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From the last slide, the amplitude of H is

0 00

ˆ ˆ jk rE k E kH k P e H

Dividing both sides of our expression by E0 gives an expression for impedance .

0

0

E

H k

Since , the final expression for impedance isk

k



We can now revise our expression for the electric and magnetic field components of a wave as

0

0

E

H

0

00

376.73011346177

Vacuum Impedance

where

ˆ

and jk r jk rk PE r Pe H e

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Dispersion Relation


Derivation in LHI Media

We started with the wave equation.

2 2 0E k E

We found the solution to be plane waves.

jk rE r Pe

If we substitute our solution back into the wave equation, we get an equation called the dispersion relation.

2

22 2 2 20

0x y z

nk k n k k k

c

The dispersion relation relates frequency to wave vector. For LHI media, it fixes the magnitude of the wave vector to be a constant.

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Index Ellipsoids

From the previous slide, the dispersion relation for a LHI material was:

This defines a sphere called an “index ellipsoid.”

The vector connecting the origin to a point on the surface of the sphere is the k‐vector for that direction. Refractive index is calculated from this.

For LHI materials, the refractive index is thesame in all directions.

Think of this as a map of the refractiveindex as a function of the wave’s directionthrough the medium.

2 2 2 2 20x y zk k k k n

ˆxa

ˆya

ˆza

index ellipsoid0k k n


What About Anisotropic Materials?

Isotropic Materials2 2 2 2 2

0a b ck k k k n

Uniaxial Materials2 2 2 2 2 2

2 20 02 2 2

0a b c a b c

O E O

k k k k k kk k

n n n

Biaxial Materials2 2 2

2 2 22 2 2 2 2 20 0 0

1a b c

a b c

k k k

k k n k k n k k n

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ElectromagneticWave Polarization


What is Polarization?

Polarization is that property of a radiated electromagnetic wave which describes the time‐varying direction and relative magnitude of the electric field vector.

Linear Polarization (LP) Circular Polarization (CP)

Left‐Hand Circular Polarization (LCP)

To determine the handedness of CP, imagine watching the electric field in a plane while the wave is coming at you. Which way does it rotate?

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Orthogonality and Handedness

We get from the curl equations that

E H

From the divergence equations, we see that

and E k H k

We conclude that , , and form an orthogonal triplet. In fact, they follow the right‐hand rule.

E

H

k

E

H

k

k

a

b

ˆâ bp a bP p


Possibilities for Wave Polarization

Recall that so the polarization vector must fall within the plane perpendicular to .

We can decompose the polarization into two orthogonal directions,and .

E k

P

k

a b

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Explicit Form to Convey Polarization

Our electromagnetic wave can be now be written as

ˆˆjk r jk ra bE r Pe p a p b e

pa and pb are in general complex numbers in order to convey the relative phase of each of these components.

a bj ja a b bp E e p E e

ˆ ˆˆ ˆ b aa b ajj j jjk r jk ra b a bE r E e a E e b e E a E e b e e

Substituting pa and pb into our wave expression gives

We interpret a as the phase common to both pa and pb.

b a a

We interpret b – a as the phase difference between pa and pb.

ˆˆ j j jk ra bE r E a E e b e e

The final expression is:


Determining Polarization of a Wave

To determine polarization, it is most convenient to write the expression for the wave that makes polarization explicity.


âmplite along

âmplite along

phase difference

common phase

a

b

E a

E b

Polarization Designation Mathematical Definition

Linear Polarization (LP) = 0°

Circular Polarization (CP) = ± 90°, Ea = Eb

Right‐Hand CP (RCP) = + 90°, Ea = Eb

Left‐Hand CP (LCP) = - 90°, Ea = Eb

Elliptical Polarization Everything else

We can now identify the polarization of the wave…

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Linear Polarization

A wave travelling in the +z direction is said to be linearly polarized if:

ˆ ˆ, , sin coszjk zE x y z Pe P x y

P is called the polarization vector.

For an arbitrary wave,

ˆˆsin cos

ˆˆ

jk rE r Pe

P a b

a b k

k k

a

bAll components of P have equal phase.


Linear Polarization

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Circular Polarization

A wave travelling in the +z direction is said to be circularly polarized if:

ˆ ˆ, , zjk zE x y z Pe P x jy

is called the polarization vector.

For an arbitrary wave,

ˆˆ

ˆˆ

jk rE r Pe

P a jb

a b k

k k

LCP

RCPj

j

The two components of have equal amplitude and are 90 out of phase.

P

P


LPx + LPy = LP45

A linearly polarized wave can always be decomposed as the sum of two linearly polarized waves that are in phase.

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LPx + jLPy = CP

A circularly polarized wave is the sum of two orthogonal linearly polarized waves that are 90° out of phase.


RCP + LCP = LP

A LP wave can be expressed as the sum of a LCP wave and a RCP wave. The phase between the two CP waves determines the tilt of the LP wave polarization.

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Circular Polarization (1 of 2)

EngineeringRight‐Hand Circular Polarization (RCP)

Physics/OpticsLeft‐Hand Circular Polarization (LCP)

x

y

z


Circular Polarization (2 of 2)

EngineeringLeft‐Hand Circular Polarization (LCP)

Physics/OpticsRight‐Hand Circular Polarization (RCP)

x

y

z

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Poincaré Sphere

The polarization of a wave can be mapped to a unique point on the Poincaré sphere.

Points on opposite sides of the sphere are orthogonal.

See Balanis, Chap. 4.

RCP

LCP

+45° LP0° LP

90° LP‐45° LP

Why is Polarization Important?

• Different polarizations can behave differently in a device

• Orthogonal polarizations will not interfere with each other

• Polarization becomes critical when analyzing devices on the scale of a wavelength

• Focusing properties of lenses are different

• Reflection/transmission can be different

• Frequency of resonators

• Cutoff conditions for filters, waveguides, etc.


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Example – Dissect a Wave (1 of 9)

The electric field component of a 5.6 GHz plane wave is given by:

573.0795 330.8676 240.8519

573.0795 330.8676 240.8519

573.0795 330.8676 240.8519

ˆ, 0.4915 0.8550

ˆ 1.4224 0.4702

ˆ 0.7844 1.3885

j x j y j zx

j x j y j zy

j x j y j zz

E r t a j e e e

a j e e e

a j e e e

1. Determine the wave vector.2. Determine the wavelength inside of the medium.3. Determine the free space wavelength.4. Determine refractive index of the medium.5. Determine the dielectric constant of the medium.6. Determine the polarization of the wave.7. Determine the magnitude of the wave.



The standard form for a plane wave is

jk rE r Pe

Solution – Part 1 – Determine Wave Vector

Comparing this to the expression for the electric field shows that

573.0795 330.8676 240.8519

ˆ ˆ ˆ0.4915 0.8550 1.4224 0.4702 0.7844 1.3885x y z

jk r j x j y j z

P a j a j a j

e e e e

The polarization vector will be use again later. The wave vector is determined from the second expression above to be

P

k

1ˆ ˆ ˆ573.0795 330.8676 240.8519 mx y zk a a a

573.0795 330.8676 240.8519yx zjk yjk x jk zjk r j x j y j ze e e e e e e

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The wavelength inside the medium is related to the magnitude of the wave vector through

2 2 k

k

Solution – Part 2 – Wavelength inside the medium

The magnitude of the wave vector is

2 2 2

2 2 21 1 1

1

573.0795 m 330.8676 m 240.8519 m

704.239 m

x y zk k k k

The wavelength is therefore

1

28.9224 cm

704.239 m



The free space wavelength is

80

0 0 0 9 1

3 10 m s 53.5344 cm

5.6 10 s

cc f

f

Solution – Part 3 – Free space wavelength

It follows that the refractive index of the medium is

Solution – Part 4 – Refractive index

0 0 53.5344 cm 6.0

8.9224 cmn

n

Alternatively, we could determine the refractive index through k

8 10

0 9 10 0

3 10 m s 704.239 m 6.0

2 2 5.6 10 s

k k c kk k n n

k c f

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Assuming the medium has no magnetic response,

22r r 6.0 36n n

Solution – Part 5 – Dielectric constant

Solution – Part 6 – Wave Polarization

To determine the polarization, the electric field is written in the form that makes polarization explicit.


The choice for and is arbitrary, but they most both be perpendicular to a b k

k

a

b

ˆâ bp a bP p



Solution – Part 6 – Wave polarization (cont’d)

We determine a valid choice for by first picking any vector that is not in the same direction as

ˆ ˆ ˆ1 2 3x y zv a a a

The cross product will give us a vector perpendicular to

ak

k

ˆ ˆ ˆ ˆ0.2896 0.8381 0.4622x y z

k va a a a

k v

We determine a valid choice for using the cross product so that it is perpendicular to both and

ba k

ˆˆ ˆ ˆ ˆ0.5038 0.2771 0.8182ˆ

x y z

k ab a a a

k a

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To determine the component of the polarization vector in the and directions using the dot product.

ˆ 1.6971 V m

ˆ 1.6971 V m

a

b

p P a

p P b j

a bP

We can now write Ea and Eb from pa and pb by incorporating the phase difference into the parameter .

1.6971 V m

1.6971 V m

90

a

b

E

E

The common phase between pa and pb is simply 0°.

0




Finally, we have

From this, we determine that we have circular polarization (CP) because Ea = Eb

and = ±90°.

1

ˆˆ

1.6971 V m

1.6971 V m

90

0

ˆ ˆ ˆ573.0795 330.8676 240.8519 m

j j jk ra b

a

b

x y z

E r E a E e b e e

E

E

k a a a

More specifically, this is left‐hand circular polarization (LCP) because = -90°.

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Solution – Part 7 – Magnitude of electric field

The magnitude of the wave is simply the magnitude of the polarization vector

2 2

2 21.6971 V m 1.6971 V m

2.4 V m

a b

E r P

E E

Visualization ofEM Waves

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Waves in Materials (1 of 3)

Waves in Vacuum

• H is 377× smaller than E.

• E and H are in phase

• E H

• Amplitude does not decay

00

0

376.73 E

H

Im 0

H k P

0


Waves in Materials (2 of 3)

Waves in Dielectric

• H is larger now, but still smaller than E.

• E and H are still in phase

• E H

• Amplitude still does not decay

1

Im 0

H k P

0

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More Realistic Wave (E Only)

It is important to remember that plane waves are of infinite extent in the x and

y directions.


More Realistic Wave (E & H)

It is important to remember that plane waves are of infinite extent in the x and

y directions.

Documents

Topic 3a Electromagnetic Waves & Polarization · 2018-03-18 · Electromagnetic Waves & Polarization Slide 1 EE 4347 Applied Electromagnetics Topic 3a Electromagnetic Waves & Polarization