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### PLANE WAVE PROPAGATION

Electric & Magnetic fields that vary harmonically with time are called electromagnetic waves:

Dr. Blanton - ENTC 3331 - Wave Propagation 3

In order to simplify the mathematical treatment, treat all fields as complex numbers.

Dr. Blanton - ENTC 3331 - Wave Propagation 4

The mathematical form of the Maxwell equations remains the same, however, all quantities (apart from x,y,z,t) are now complex.

Dr. Blanton - ENTC 3331 - Wave Propagation 5

The Maxwell equations (in differential form) can thus be expressed as:

- In a vacuum (space)
- In air (atmosphere)

Dr. Blanton - ENTC 3331 - Wave Propagation 7

Thus, the Maxwell equations (in differential form) and in air can be expressed as:

- The Maxwell equations are fundamental and of general validity which implies
- It should be possible to derive a pair of equations, which describe the propagation of electromagnetic waves.

Dr. Blanton - ENTC 3331 - Wave Propagation 8

Dr. Blanton - ENTC 3331 - Wave Propagation 11

The previous two equations are called wave equations because their solutions describe the propagation of electromagnetic waves

wave equation

Dr. Blanton - ENTC 3331 - Wave Propagation 13

In one dimension:

- If this describes an electromagnetic wave, it may also hold for a single photon.

Dr. Blanton - ENTC 3331 - Wave Propagation 14

For a photon, is significant at the current location of the photon.

- The probability of finding a photon at location x is .
- This implies:

Schrodinger’s equation

Dr. Blanton - ENTC 3331 - Wave Propagation 15

heuristic analogy

Schrodinger’s Equation (Postulates of Quantum Mechanics

physics of the macroscopic world

Maxwell’s equations (Newtons laws)

physics of the microscopic world

Wave Equation

particles and waves

particles-wave duality

Dr. Blanton - ENTC 3331 - Wave Propagation 16

What are the solutions of the electromagnetic wave equations?

Dr. Blanton - ENTC 3331 - Wave Propagation 17

Perform the Laplacian

Dr. Blanton - ENTC 3331 - Wave Propagation 18

That is:

Dr. Blanton - ENTC 3331 - Wave Propagation 19

Consider a uniform plane wave that is characterized by electric and magnetic fields that have uniform properties at all points across an infinite plane.

Dr. Blanton - ENTC 3331 - Wave Propagation 20

The most general solutions of

- are
- where and are constants determined by boundary conditions.

Dr. Blanton - ENTC 3331 - Wave Propagation 23

For mathematical simplification rotate the Cartesian coordinate system about the z-axis until

- The plane wave is
- The first term represents a wave with amplitude traveling in the +z-direction, and
- the second term represents a wave with amplitude traveling in the –z direction.

Dr. Blanton - ENTC 3331 - Wave Propagation 24

Let us assume that consists of a wave traveling in the +z-direction only

Dr. Blanton - ENTC 3331 - Wave Propagation 25

Magnetic field, ?

- We must fulfill the Maxwell equation:
- But

Dr. Blanton - ENTC 3331 - Wave Propagation 26

Recall

Dr. Blanton - ENTC 3331 - Wave Propagation 31

z

y

- This is possible if
- Electric and magnetic field vectors are perpendicular!

Dr. Blanton - ENTC 3331 - Wave Propagation 32

Transversal electromagnetic wave (TEM)

Dr. Blanton - ENTC 3331 - Wave Propagation 33

Electromagnetic Plane Wave in Air

- The electric field of a 1-MHz electromagnetic plane wave points in the x-direction.
- The peak value of is 1.2p (mV/m) and for t = 0, z = 50 m.
- Obtain the expression for and .

Dr. Blanton - ENTC 3331 - Wave Propagation 34

The field is maximum when the argument of the cosine function equals zero or multiples of 2p.

- At t = 0 and z =50 m

Dr. Blanton - ENTC 3331 - Wave Propagation 35

POLARIZATION

x

Wave Polarization- Wave polarization describes the shape and locus of tip of the vector at a given point in space as a function of time.
- The direction of wave propagation is in the z-direction.

Dr. Blanton - ENTC 3331 - Wave Propagation 39

Wave Polarization

- The locus of , may have three different polarization states depending on conditions:
- Linear
- Circular
- Elliptical

Dr. Blanton - ENTC 3331 - Wave Propagation 40

Polarization

- A uniform plane wave traveling in the +z direction may have x- and y- components.
- where

Dr. Blanton - ENTC 3331 - Wave Propagation 41

Polarization

- and are the complex amplitudes of and , respectively.
- Note that
- the wave is traveling in the positive z-direction, and
- the two amplitudes and are in general complex quantities.

Dr. Blanton - ENTC 3331 - Wave Propagation 42

Polarization

- The phase of a wave is defined relative to a reference condition, such as z = 0 and t = 0 or any other combination of z and t.
- We will choose the phase of as our reference, and will denote the phase of relative to that of , as d.
- Thus, d is the phase-difference between the y-component of and its x-component.

where axand ayare the magnitudesof Ex0and Ey0

Dr. Blanton - ENTC 3331 - Wave Propagation 43

Polarization

- The total electric field phasor is
- and the corresponding instantaneous field is:

Dr. Blanton - ENTC 3331 - Wave Propagation 44

Intensity and Inclination Angle

- The intensity of is given by:
- The inclination angleψ

Dr. Blanton - ENTC 3331 - Wave Propagation 45

Linear Polarization

- A wave is said to be linearly polarized if Ex(z,t) and Ey(z,t) are in phase (i.e., d = 0) or out of phase (d = p).
- At z = 0 and d =0 or p,

Dr. Blanton - ENTC 3331 - Wave Propagation 46

Linear Polarization (out of phase)

- For the out of phase case:
- w t = 0 and
- That is, extends from the origin to the point (ax ,ay) in the fourth quadrant.

Dr. Blanton - ENTC 3331 - Wave Propagation 47

Linear Polarization (out of phase)

- For the in phase case:
- w t = 0 and
- That is, extends from the origin to the point (ax ,ay) in the first quadrant.

y

x

Dr. Blanton - ENTC 3331 - Wave Propagation 48

The inclination is:

- If ay = 0, y = 0 or 180, the wave becomes x-polarized, and if ax = 0, y = 90 or -90 , and the wave becomes y-polarized.

Dr. Blanton - ENTC 3331 - Wave Propagation 49

Linear Polarization

- For a +z-propagating wave, there are two possible directions of .
- Direction of is called polarization
- There are two independent solution for the wave equation

Dr. Blanton - ENTC 3331 - Wave Propagation 50

E

+z

B

Can make any angle from the horizontal and vertical waves

Dr. Blanton - ENTC 3331 - Wave Propagation 51

Linear Polarization

Looking up from +z

x-polarized or horizontal polarized

ay=0 ψ=0° or 180°

y-polarized or vertical polarized

ax=0 ψ=90° or -90°

Dr. Blanton - ENTC 3331 - Wave Propagation 52

Circular Polarization

- For circular polarization, ax = ay.
- For left-hand circular polarization, d = p/2.
- For right-hand circular polarization, d = -p/2.

Dr. Blanton - ENTC 3331 - Wave Propagation 53

Left-Hand Polarization

- For ax = ay = a, and d = p/2,
- and the modulus or intensity is

Dr. Blanton - ENTC 3331 - Wave Propagation 54

The angle of inclination is:

Dr. Blanton - ENTC 3331 - Wave Propagation 55

At a fixed z, for instance z = 0, y = -wt.

- The negative sign means that the inclination angle is in the clockwise direction.

Dr. Blanton - ENTC 3331 - Wave Propagation 56

Right-Hand Circular

- For ax = ay = a, and d = p/2,

,

- The positive sign means that the inclination angle is in the counter clockwise direction.

Dr. Blanton - ENTC 3331 - Wave Propagation 57

A RHC polarized plane wave with electric field modulus of 3 (mV/m) is traveling in the +y-direction in a dielectric medium with e = 4eo, m = mo, and s = 0.

- The wave frequency in 100 MHz.
- What are

and

Dr. Blanton - ENTC 3331 - Wave Propagation 58

z

- Since the wave is traveling along the y-axis, its field components must be along the z-axis and x-axis.

w

Dr. Blanton - ENTC 3331 - Wave Propagation 59

Elliptical Polarization

- In general,
- ax 0,
- ay 0, and
- d 0.
- The tip of traces an ellipse in the x-y plane.
- The wave is said to be elliptically polarized.
- The shape of the ellipse and its handedness (left-hand or right-hand rotation) are determined by the values of the ratio and the polarization phase difference, d.

Dr. Blanton - ENTC 3331 - Wave Propagation 63

Elliptical Polarization

- The polarization ellipse has a major axis, ax along the x-direction and a minor axis ah along the h-direction.

Dr. Blanton - ENTC 3331 - Wave Propagation 64

Elliptical Polarization

- The rotation angle g is defined as the angle between the major axis of the ellipse and a reference direction, chosen below to be the x-axis.

Dr. Blanton - ENTC 3331 - Wave Propagation 65

Elliptical Polarization

- The shape and the handedness are characterized by the ellipticity angle, c.

+ implies LH rotation

- implies RH rotation

Dr. Blanton - ENTC 3331 - Wave Propagation 67

Elliptical Polarization

is called the axial ratio and varies between 1 for circular polarization and for linear polarization

Dr. Blanton - ENTC 3331 - Wave Propagation 68

Elliptical Polarization

Dr. Blanton - ENTC 3331 - Wave Propagation 69

Elliptical Polarization

Positive values of c (sind > 0) LH Rotation

Negative values of c (sind< 0) RH Rotation

Also

Dr. Blanton - ENTC 3331 - Wave Propagation 70

Example 7-3

- Find the polarization state of a plane wave
- Change to a cosine reference:

Dr. Blanton - ENTC 3331 - Wave Propagation 71

Example 7-3

- Find the corresponding phasor:
- Find the phase angles:
- Phase difference:
- Auxiliary:

Dr. Blanton - ENTC 3331 - Wave Propagation 72

can have two solutions:

- or
- Since cosd < 0, the correct value of g is -69.2.

Dr. Blanton - ENTC 3331 - Wave Propagation 73

Since the angle of c is positive and less than 45,

- The wave is elliptically polarized and
- The rotation of the wave is left-handed.

Dr. Blanton - ENTC 3331 - Wave Propagation 74

Polarization States

The wave is traveling out of the slide!

Dr. Blanton - ENTC 3331 - Wave Propagation 75

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