Dr Hugh Blanton ENTC 3331 PlaneWave Propagation Electric

Dr. Hugh Blanton ENTC 3331

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. Real Complex Dr. Blanton - ENTC 3331 - Wave Propagation Measurement 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

• For • It follows that Dr. Blanton - ENTC 3331 - Wave Propagation 6

• 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

• We expect solutions like: • How do we get from • to Dr. Blanton - ENTC 3331 - Wave Propagation 9

• Recall that • and apply to both sides of • but Dr. Blanton - ENTC 3331 - Wave Propagation 10

0 Dr. Blanton - ENTC 3331 - Wave Propagation 11

wave number =k 2 wave equation Dr. Blanton - ENTC 3331 - Wave Propagation 12

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

strict derivation physics of the macroscopic world Maxwell’s equations (Newtons laws) heuristic analogy Wave Equation particles and waves Dr. Blanton - ENTC 3331 - Wave Propagation Schrodinger’s Equation (Postulates of Quantum Mechanics physics of the microscopic world particles-wave duality 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

no component in the zdirection x wave crescents y “up” Dr. Blanton - ENTC 3331 - Wave Propagation z 21

• Consequently, • simplifies to Dr. Blanton - ENTC 3331 - Wave Propagation 22

• 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 zaxis 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

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• Recall Dr. Blanton - ENTC 3331 - Wave Propagation 31

x • This is possible if z y • 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. 2 p (m. V/m) and for t = 0, z = 50 m. • Obtain the expression for Dr. Blanton - ENTC 3331 - Wave Propagation and . 34

• The field is maximum when the argument of the cosine function equals zero or multiples of 2 p. • At t = 0 and z =50 m Dr. Blanton - ENTC 3331 - Wave Propagation 35

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PLANE WAVE PROPAGATION POLARIZATION

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. y x • The direction of wave propagation is in the zdirection. 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 complex quantities. and Dr. Blanton - ENTC 3331 - Wave Propagation are in general 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 ycomponent of and its x-component. where ax and ay are the magnitudes of Ex 0 and Ey 0 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 angle ψ 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: y • w t = 0 and x • That is, extends from the origin to the point (ax , ay) in the first quadrant. Dr. Blanton - ENTC 3331 - Wave Propagation 48

• The inclination is: • If ay = 0, y = 0 or 180 , the wave becomes xpolarized, 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 polarization . is called • There are two independent solution for the wave equation Dr. Blanton - ENTC 3331 - Wave Propagation 50

Linear Polarization E B +z 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 Dr. Blanton - ENTC 3331 - Wave Propagation ψ=90° or -90° 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 (m. V/m) is traveling in the +y-direction in a dielectric medium with e = 4 eo, m = mo, and s = 0. • The wave frequency in 100 MHz. • What are and Dr. Blanton - ENTC 3331 - Wave Propagation 58

• Since the wave is traveling along the y-axis, its field components must be along the z-axis and x-axis. x w z Dr. Blanton - ENTC 3331 - Wave Propagation 59

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Elliptical Polarization • In general, • ax 0, • ay 0, and • d 0. • The tip of plane. traces an ellipse in the x-y • The wave is said to be elliptically polarized. • The shape of the ellipse and its handedness (lefthand 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 • g is bounded within the range: Dr. Blanton - ENTC 3331 - Wave Propagation 66

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