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CHAPTER 9ELECTROMAGNETIC WAVES

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Outlines

1. Waves in one dimension

2. Electromagnetic Waves in Vacuum

3. Electromagnetic waves in Matter

4. Absorption and Dispersion

5. Guided Waves

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Skip 9.1.1 and 9.1.2 Wave on a string & Sinusoidal waves.

, ,

where v is the velocity of the wave ,

The solution to the above differential equation is given by

f(z,t)=Acos[(kz- t)+

We can express the wave function in the complex form

, ≡ where

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Boundary conditions: Reflection and transmission

For waves travel on a string that has different density, the wave reflects and transmits at the boundary.

, For z < 0

For z > 0

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The wave equation

· , (3) ,

(2) · , (4) 0.

Maxwell equations in free space:

Take the Curl of eq. (3) we have

·

(5)

Similarly the B field also satisfy the wave equation.

(6)

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These are called wave equations. We can see that both and fields satisfy identical partial differential equation.

One of the greatest achievements of classical physics was the realization that physical phenomena which can be represented by fields cab be expressed in terms of partial differential equations.

In theoretical physics, one of the major tasks is to relate a physical phenomenon to a partial differential equations. Once we found the PDE governing the physical phenomena, we assume that the phenomenon is understood.

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For example: Wave on a flexible string (pages 383-385)

T ------- tension on the string,------- mass per unit length

Heat diffusion

T(r,t) ---- temperature fieldα ---------- thermal diffusivity

Schrodinger equation

, , ,

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Reading assignment: Section 9.1

Summary

The solution of wave equation has the following form

,

And one common form of f(z,t) can be

,

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A more general form of the sinusoidal wave can be expressed in the complex form:

,

Or a combination of many different terms with different

,

If k is continuous, the above can be expressed as an integral

,

The frequency-time domain also has similar property. This is related to the Fourier transformation, or Fourier series.

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One dimensional monochromatic plane wave

, , , .

The wave is traveling in one direction (z-direction), with only one frequency (one color), and the wave front is a plane (plane wave).

Plane wave Spherical wave

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As we can see in section 9.1, wave equation is a more general physical phenomenon. It can be used to describe not just the electromagnetic wave, but also can describe waves on a string. The electromagnetic waves need to satisfy the Maxwell equations also.

For example:

·

So the EM plane waves that satisfy the Maxwell equations have to be a transverse wave!

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From Faraday’s law

From eq. (9) and (10), we can see that

and

We can combine the above two equations using vector notation

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So the electric and magnetic fields are perpendicular to each other in an EM wave and they are in phase. The k is the wave number or called wave vector and is defined as

the ω is the angular frequency

In general, the wave number can be viewed as a vector which is pointing at the direction of traveling of the EM wave.

We further define as the direction of the electric field, then we can write down the solution of the wave equation as:

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, · ·

, · ·

,

Where --------Polarization direction-------- Propagation direction

·

⋅

EM wave is transverse wave.

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Energy and Momentum in EM waves

From Chapter 8, we know that the energy stored in the electric and magnetic field per unit volume is

For EM waves in free space, and /

The energy flux density transported by the EM wave is given by

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For monochromatic plane wave, the Poynting vector is

· ·

The momentum density stored in the EM field is given by

℘

The energy-momentum relationship for EM waves

℘

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For visible light, the frequency is about to Hz, so we usually measure the time-average quantities, since time average of is one half,

℘

We define the intensity of the light (EM waves) as the average power per unit area transported by the EM waves

Since EM waves also carry momentum, light will exert a radiation pressure on the surface it shines

∆∆

℘ ∆∆

(21)

(22)

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Propagation through linear medium (no source)

Inside a linear medium, we need to use μ and ε for the medium, so the Maxwell’s equations become

· , ,

· , 0.

The velocity of the EM wave is given by

√

And n is called “index of refraction”

≅

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When EM waves pass from one medium to another, we expect to get some reflected and some transmitted waves just like other types of waves. The details depend on the boundary conditions we derived in chapter 7, page 343.Assuming that there is no sources (free charges, or free current), the boundary conditions are as follow:

,

∥ ∥ ∥ ∥

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Reflection and Transmission at normal incidence

For normal incidence, the EM wave is very much like a wave on a string. Transmitted wave travels in the same direction as the incident wave, while the reflected wave travels in the opposite direction.

, ̂

, ̂

Incident wave

Reflected wave

,

, ̂

Transmitted wave

, ̂

, ̂

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Since this is a normal incident situation, no normal component at the interface. We only have to match the parallel components.

∥ ∥

∥ ∥ (1)

Let

≅

(1)

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In general the permeability μ of non-magnetic material is very close to the permeability of the free space , so let .

The equations on previous page become

If n1 > n2 , (V2>V1) reflected wave has the same phase as the incident wave,

If n1 < n2, (V2<V1) reflected wave is out of phase.

www.youtube.com/watch?v=9OpL3OFuVXo

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The intensity of the EM wave is given by eq. 9.(73)

We define the reflection and transmission coefficients as follow:

,

(29)

For the transmission coefficient, it is more difficult because the two media have different v and ε.

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Since

For ≅ ,

Now if we add T and R together, we can see that

This is just conservation law!

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Reflection and transmission at oblique incidence

Here we have a plane wave incident on xy plane, part of it reflected and part of it transmitted into the second medium on the right.

, · , ,

, · , ,

,t) = · , ,

X

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The boundary conditions must be satisfied at all points on the interface and at all time. This means that the phase factors of these expressions have to be the same. This implies that the frequency of the incident, reflected, and transmitted waves have to be the same.

The spatial part of the phase factors also have to be the same at the interface:

· · ·

Let point in the direction

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Eq. (32) leads to the three so-called kinematics properties of waves.

Since

Snell’s law

We can also choose a particular such that · , from eq. (32) this means

· ·

If , and are all perpendicular to the same vector ,that implies that implies all three wave vectors are all in the same plane!!!

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Next we will study the dynamic properties at the interface, namely the amplitude, intensity of the reflected and transmitted waves and the phase and polarization changes.

Polarization is parallel to the plane of incidence (TM mode)

= (35)

(36)

z-component

x-component

y-component

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Re-arrange eq. (34)

and

Re-arrange eq. (35), we have

and

Combine the above two equations, we obtain

Theses are Fresnel’s equations.

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The transmitted waves are always in phase with the incident waves, but the reflected waves will depend on the values of αand β.

For , is in phase with For , is out of phase with

A plot of the ratio of amplitudes as a function of incident angle is given below.

,

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/

/

And /

(39)

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The power per unit area striking the interface is · . Thus the incident intensity is

The reflected and transmitted intensities are

and

The reflection and transmission coefficients are given as follow:

≡

≡

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Reflection and transmission coefficients as functions of the incident angle.

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Total internal reflection

From the Snell’s law

If no problem

If , , leads to internal reflection.

When °

This is called critical angle, the angle where the internal reflection starts.

(40)

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For and , we have total internal reflection. That means 100% of the light reflected back to medium 1 and nothing transmitted to medium 2.

The Snell’s law will also break down. (No propagation of energy to medium 2.)

Not allowed for real

This implies that , so could be an imaginary number.

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·

··

The wave travels in the x-direction is called evanescent wave. It is just travels at the interface. In the y-direction, it is not a traveling wave any more. It is just an exponential decay function which does not carry energy at all.

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Absorption and Dispersion

When a medium start to absorb the energy of EM waves, things start to become more complicated. This type of medium is called dissipative medium. The propagation vector, k is a complex number and the material-dependent quantities such as ε, μ, σ, and n will depend on the frequency, ω.

We start with a conductor, there are free charge density ,

and free current density, . From continuity equation:

· ·

· will go to zero at equilibrium.

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So the free charge density inside a conductor will decay and reach zero at equilibrium.

· , ,

· ,

·

The solution still can be a plane wave, but the wave vector is a complex number now.

(41)

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Substitute into the modified wave equations on previous page, we obtain the following expression

So is a complex number and

(43)

(42)

2

Solve the above equations, we end up with:

∓ ∓

/

(43-1)

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Now if we put eq. (43) back into eq.(41), we will see that

, · · (44)

There is an attenuation term and there is the propagation term. We define skin depth as / .

All quantities associated with wave propagation are related to the term.

, ,

For good conductor

≫

For “poor” conductor

≪ ,

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Because of Maxwell equations, there are further constraints imposed on E field and B field. (See eq.(11))

,

,

The wave vector is a complex number and can be expressed as:

where the complex wave number, is its magnitude, and is phase angle.

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From eqs. (47) and (48), we can see that the exponential parts are the same for E and B field, the main differences are in the complex amplitudes. Let

So the and fields are no longer in phase.

So the magnetic field is lagging behind the E field by a phase angle of and

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The figure on the right showed the attenuation of the field and also the magnetic field lags behind the electric field.

Skip 9.4.2 Reflection at a conducting surface

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The frequency dependence of permitivity

Dispersion is the phenomenon that waves travel at different speed at different frequency.

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A simple model for :

We want to develop a simple theory that can explain the frequency dependence of the permittivity. We ask the question “How an electron responds to the excitation of EM waves?”

1. Use SHO model. Assume electron is attached to a spring. ( -- natural frequency)

2. Assume there is a damping force. (Friction)

3. EM wave is the driving force.

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Use Newton’s second law:

Re-arrange the above equation, we end up with

Leads to

and E(t) =

/

The dipole moment is given by

Let

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The polarization is defined as the # of dipole moment per unit volume

=∑

(57)

Now compare (54), (55) and (56), we end up with

and

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The dielectric constant can be viewed as a response of the medium to an external field

Medium, ,

, , , ′

In free space

,

, ,

This is based on linear response theory. It is assumed that the medium is linear.

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Linear response theory

A linear system will produce an output O(t), when it is subjected to an input I(t) through the following eq.

′

R(t-t’) is the response function of the medium and causality requires that .

The eq.(61) can be viewed as a convolution of an input function I(t’). Two important implications of eq.(61):

1.

2. certain differential equation.

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A dispersive medium

We will use the complex permittivity derived from eq.(59) to find the absorption coefficient and the index of refraction of such medium.

, ≡

The solution to the above equation is given below.

,

≅

(assume x is small so 1 ≅ 1 )

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≅

≅

As ω near , the absorption approach a maximum value, and n decreases as frequency increases, this is called anomalous dispersion.

We can see that there is a “close” relationship between n and α.

~

~

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If we agree to stay away from the resonance, the damping term can be ignored.

For transparent dielectric materials, the resonances lie in the ultraviolet, so for visible light,

≅

Substitute into (64’)

This is known as Cauchy’s formula in optics.

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Kramers-Kroniq relation

Physical measurements only deal with real quantities. But through Kramers-Kronig relation, we can obtain phase information from measurements of a real quantity.

Review of Complex Analysis

If f(z) is an analytical function in domain D and C is a contour in D where a is enclosed by C, then

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If a is on C, then

.

Now if we let f(z) be the complex absorption

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Re-write eq.(66) separately in real part and in imaginary part

Further more, eq.(66) times (s+ω) then can be written as

These equations are called Kramers-Kronig relationships.

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Waveguides

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Wave Guides

Here we study guided waves confined inside a “waveguide” and propagate down the waveguide. Assume the waveguide is made of a perfect conductor, then at the boundary:

∥

Because inside a perfect conductor, both E and B fields are zero.The field propagate down the z-axis can be described as

, , , , , , , ,

Because of the boundary conditions in a wave guide, the confined waves are not necessarily transverse anymore

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Substitute into , we have

Use eq. (73) and (75) to eliminate By

⁄

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Similarly, (72) + (76) and eliminate

⁄

Again, starts with (72) and (76), but eliminates

⁄

Starts with (73) and (75), but eliminates

⁄

All , , , are expressed in terms of

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Substitute (77) and (78) into · , we end up with

Similarly, if we substitute (79) and (80) into · , we obtain

Equations (77) through (82) determine the type of waves that propagate through the waveguide. Typically we start with (81) and (82) to find the z-component of E and B fields then substitute back into (77) through (80).

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IfTE wave

TM wave

For hollow waveguide, TEM mode can not exist. (See page 427)

or

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TE waves in a rectangular waveguide

Let

, ·

Substitute into eq. (82), we have

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The solution to

leads to

with boundary conditions at x = 0 and x = a.

for m = 0, 1, 2, 3, …

Similarly in the y-direction

for n = 0, 1, 2, 3, …

,

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let

,

We can re-write the equation (13) as

,

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Significance of ,

Because of the boundary conditions imposed on the solution of the wave equation, we can see that waves with a certain frequencies can not propagate inside a waveguide. Equation (86) on page 63 defined a cutoff frequency, below the cutoff frequency, the wave can not propagate, because the wave number k will be purely imaginary.

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We can re-write equation (87) as follow:

We find that the group velocity and the phase velocity of the guided wave are:

Phase velocity

Group velocity

·

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,

We can substitute eq. (85) into eqs. (77 - 80) to find the other components

⁄

⁄

⁄

⁄

The wave travels in the z-direction, but has non-zero component of B field in the z-direction.

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One way to visualize the situation is to imaging the EM wave is bouncing back and forth inside the waveguide:

Here x-direction is pointing up and y-direction is out of the paper. The wave-vector is k’

Modes

n=0 n=1 n=2

m=0 ***** 01 02

m=1 10 11 12

m=2 20 21 22

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Transmission line Wave guide

Two or more insulated conductors Metal waveguide consists of one enclosed conductor.

Operates at TEM mode Operates at TE or TM modes

No cutoff frequency Operates above cutoff frequency

Significant signal attenuation Lower attenuation at high frequency

Typically transmit at low power level.

Metal waveguides can transmit at high power level.

Comparison of waveguide and transmission lines

Skip 9.5.3 The co-axial transmission line