1. Dr. Hugh Blanton ENTC 3331

2. Plane-Wave Propagation

3. Dr. Blanton - ENTC 3331 - Wave Propagation 3 Electric & Magnetic fields that vary harmonically with time are called electromagnetic waves:

4. Dr. Blanton - ENTC 3331 - Wave Propagation 4 RealComplexMeasurement In order to simplify the mathematical treatment, treat all fields as complex numbers.

5. Dr. Blanton - ENTC 3331 - Wave Propagation 5 The mathematical form of the Maxwell equations remains the same, however, all quantities (apart from x,y,z,t) are now complex.

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

7. Dr. Blanton - ENTC 3331 - Wave Propagation 7 The Maxwell equations (in differential form) can thus be expressed as: In a vacuum (space) In air (atmosphere)

8. Dr. Blanton - ENTC 3331 - Wave Propagation 8 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.

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

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

11. Dr. Blanton - ENTC 3331 - Wave Propagation 11 0

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

13. Dr. Blanton - ENTC 3331 - Wave Propagation 13 The previous two equations are called wave equations because their solutions describe the propagation of electromagnetic waves wave equation

14. Dr. Blanton - ENTC 3331 - Wave Propagation 14 In one dimension: If this describes an electromagnetic wave, it may also hold for a single photon.

15. Dr. Blanton - ENTC 3331 - Wave Propagation 15 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

16. Dr. Blanton - ENTC 3331 - Wave Propagation 16 physics of the macroscopic world Wave Equation Maxwell’s equations (Newtons laws) Schrodinger’s Equation (Postulates of Quantum Mechanics physics of the microscopic world strict derivation heuristic analogy particles and waves particles-wave duality

17. Dr. Blanton - ENTC 3331 - Wave Propagation 17 What are the solutions of the electromagnetic wave equations?

18. Dr. Blanton - ENTC 3331 - Wave Propagation 18 Perform the Laplacian

19. Dr. Blanton - ENTC 3331 - Wave Propagation 19 That is:

20. Dr. Blanton - ENTC 3331 - Wave Propagation 20 Consider a uniform plane wave that is characterized by electric and magnetic fields that have uniform properties at all points across an infinite plane.

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

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

23. Dr. Blanton - ENTC 3331 - Wave Propagation 23 The most general solutions of are where and are constants determined by boundary conditions.

24. Dr. Blanton - ENTC 3331 - Wave Propagation 24 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.

25. Dr. Blanton - ENTC 3331 - Wave Propagation 25 Let us assume that consists of a wave traveling in the +z-direction only

26. Dr. Blanton - ENTC 3331 - Wave Propagation 26 Magnetic field, ? We must fulfill the Maxwell equation: But

27. Dr. Blanton - ENTC 3331 - Wave Propagation 27

28. Dr. Blanton - ENTC 3331 - Wave Propagation 28

29. Dr. Blanton - ENTC 3331 - Wave Propagation 29

30. Dr. Blanton - ENTC 3331 - Wave Propagation 30

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

32. Dr. Blanton - ENTC 3331 - Wave Propagation 32 This is possible if Electric and magnetic field vectors are perpendicular! x z y

33. Dr. Blanton - ENTC 3331 - Wave Propagation 33 Transversal electromagnetic wave (TEM)

34. Dr. Blanton - ENTC 3331 - Wave Propagation 34 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  (mV/m) and for t = 0, z = 50 m. Obtain the expression for and.

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

36. Dr. Blanton - ENTC 3331 - Wave Propagation 36

37. Dr. Blanton - ENTC 3331 - Wave Propagation 37

38. PLANE WAVE PROPAGATION POLARIZATION

39. Dr. Blanton - ENTC 3331 - Wave Propagation 39 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. x y

40. Dr. Blanton - ENTC 3331 - Wave Propagation 40 Wave Polarization The locus of, may have three different polarization states depending on conditions: Linear Circular Elliptical

41. Dr. Blanton - ENTC 3331 - Wave Propagation 41 Polarization A uniform plane wave traveling in the +z direction may have x- and y- components. where

42. Dr. Blanton - ENTC 3331 - Wave Propagation 42 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.

43. Dr. Blanton - ENTC 3331 - Wave Propagation 43 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 . Thus,  is the phase-difference between the y- component of and its x-component. where a x and a y are the magnitudes of E x0 and E y0

44. Dr. Blanton - ENTC 3331 - Wave Propagation 44 Polarization The total electric field phasor is and the corresponding instantaneous field is:

45. Dr. Blanton - ENTC 3331 - Wave Propagation 45 Intensity and Inclination Angle The intensity of is given by: The inclination angle ψ

46. Dr. Blanton - ENTC 3331 - Wave Propagation 46 Linear Polarization A wave is said to be linearly polarized if E x (z,t) and E y (z,t) are in phase (i.e.,  ) or out of phase (  ). At z = 0 and  or ,

47. Dr. Blanton - ENTC 3331 - Wave Propagation 47 Linear Polarization (out of phase) For the out of phase case:  t = 0 and That is, extends from the origin to the point (a x,  a y ) in the fourth quadrant.

48. Dr. Blanton - ENTC 3331 - Wave Propagation 48 Linear Polarization (out of phase) For the in phase case:  t = 0 and That is, extends from the origin to the point (a x, a y ) in the first quadrant. x y

49. Dr. Blanton - ENTC 3331 - Wave Propagation 49 The inclination is: If a y = 0,  or , the wave becomes x- polarized, and if a x = 0,  or , and the wave becomes y-polarized.

50. Dr. Blanton - ENTC 3331 - Wave Propagation 50 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

51. Dr. Blanton - ENTC 3331 - Wave Propagation 51 Linear Polarization +z E B Can make any angle from the horizontal and vertical waves

52. Dr. Blanton - ENTC 3331 - Wave Propagation 52 Linear Polarization Looking up from +z x-polarized or horizontal polarized a y =0 ψ=0° or 180° y-polarized or vertical polarized a x =0 ψ=90° or -90°

53. Dr. Blanton - ENTC 3331 - Wave Propagation 53 Circular Polarization For circular polarization, a x = a y. For left-hand circular polarization, . For right-hand circular polarization, .

54. Dr. Blanton - ENTC 3331 - Wave Propagation 54 Left-Hand Polarization For a x = a y = a, and , and the modulus or intensity is

55. Dr. Blanton - ENTC 3331 - Wave Propagation 55 The angle of inclination is:

56. Dr. Blanton - ENTC 3331 - Wave Propagation 56 At a fixed z, for instance z = 0,  t. The negative sign means that the inclination angle is in the clockwise direction.

57. Dr. Blanton - ENTC 3331 - Wave Propagation 57 Right-Hand Circular For a x = a y = a, and ,, The positive sign means that the inclination angle is in the counter clockwise direction.

58. Dr. Blanton - ENTC 3331 - Wave Propagation 58 A RHC polarized plane wave with electric field modulus of 3 (mV/m) is traveling in the +y-direction in a dielectric medium with  ,  , and . The wave frequency in 100 MHz. What are and

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

60. Dr. Blanton - ENTC 3331 - Wave Propagation 60

61. Dr. Blanton - ENTC 3331 - Wave Propagation 61

62. Dr. Blanton - ENTC 3331 - Wave Propagation 62

63. Dr. Blanton - ENTC 3331 - Wave Propagation 63 Elliptical Polarization In general, a x  0, a y  0, and . 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, .

64. Dr. Blanton - ENTC 3331 - Wave Propagation 64 Elliptical Polarization The polarization ellipse has a major axis, a  along the  -direction and a minor axis a  along the  -direction.

65. Dr. Blanton - ENTC 3331 - Wave Propagation 65 Elliptical Polarization The rotation angle  is defined as the angle between the major axis of the ellipse and a reference direction, chosen below to be the x-axis.

66. Dr. Blanton - ENTC 3331 - Wave Propagation 66 Elliptical Polarization  is bounded within the range:

67. Dr. Blanton - ENTC 3331 - Wave Propagation 67 Elliptical Polarization The shape and the handedness are characterized by the ellipticity angle, . + implies LH rotation - implies RH rotation

68. Dr. Blanton - ENTC 3331 - Wave Propagation 68 Elliptical Polarization is called the axial ratio and varies between 1 for circular polarization and  for linear polarization

69. Dr. Blanton - ENTC 3331 - Wave Propagation 69 Elliptical Polarization

70. Dr. Blanton - ENTC 3331 - Wave Propagation 70 Elliptical Polarization Positive values of  (sin  > 0)  LH Rotation Negative values of  ( sin  < 0)  RH Rotation Also

71. Dr. Blanton - ENTC 3331 - Wave Propagation 71 Example 7-3 Find the polarization state of a plane wave Change to a cosine reference:

72. Dr. Blanton - ENTC 3331 - Wave Propagation 72 Example 7-3 Find the corresponding phasor: Find the phase angles: Phase difference: Auxiliary:

73. Dr. Blanton - ENTC 3331 - Wave Propagation 73 can have two solutions: or Since cos  < 0, the correct value of  is .

74. Dr. Blanton - ENTC 3331 - Wave Propagation 74 Since the angle of  is positive and less than , The wave is elliptically polarized and The rotation of the wave is left-handed.

75. Dr. Blanton - ENTC 3331 - Wave Propagation 75 Polarization States The wave is traveling out of the slide!