Prof. David R. Jackson

1. ECE 3317 Prof. David R. Jackson Notes 18 Reflection and Transmission of Plane Waves [Chapter 4]

2. General Plane Wave Consider a plane wave propagating at an arbitrary direction in space z z Denote y x so

3. General Plane Wave (cont.) Hence z z y x Note: or (wavenumber equation)

4. General Plane Wave (cont.) We define the wavevector: z z y x (This assumes that the wavevector is real.) The k vector tells us which direction the wave is traveling in.

5. TM and TE Plane Waves The electric and magnetic fields are both perpendicular to the direction of propagation. • There are two fundamental cases: • Transverse Magnetic (TMz ) Hz = 0 • Transverse Electric (TEz) Ez = 0 S S z z H TMz TEz H E E y y x x Note: The word “transverse” means “perpendicular to.”

6. Reflection and Transmission As we will show, each type of plane wave (TEzand TMz) reflects differently from a material. reflected incident qi qr #1 x qt #2 transmitted z

7. Boundary Conditions Here we review the boundary conditions at an interface (from ECE 2317). ++++ Note: The unit normal points towards region 1. No sources on interface: The tangential electric and magnetic fields are continuous. The normal components of the electric and magnetic flux densities are continuous.

8. Reflection at Interface Assume that the Poynting vector of the incident plane wave lies in the xz plane (= 0). This is called the plane of incidence. First we consider the (x,z) variation of the fields. (We will worry about the polarization later.)

9. Reflection at Interface (cont.) Phase matching condition: This follows from the fact that the fields must match at the interface (z = 0).

10. Law of Reflection Similarly, Law of reflection

11. Snell’s Law We define the index of refraction: Snell's law Note: The wave is bent towards the normal when entering a more "dense" region.

12. Snell’s Law (cont.) The bending of light (or EM waves in general) is called refraction. reflected Acrylic block incident transmitted http://en.wikipedia.org/wiki/Refraction

13. Snell’s Law (cont.) Example air Given: water Note that is going from a less dense to a more dense medium, the wavevector is bent towards the normal. Note: If the wave is incident from the water region at an incident angle of 32.1o, the wave will exit into the air region at an angle of 45o. Note: At microwave frequencies and below, the relative permittivity of water is about 81. At optical frequencies it is about 1.7689.

14. Critical Angle qi = c reflected qi qr incident #1 qt x #2 qt = 90o transmitted z qi < c The wave is incident from a more dense region onto a less dense region. reflected incident qi qr #1 x qt #2 transmitted z At the critical angle:

15. Critical Angle (cont.) water reflected qc qr incident #1 x #2 air transmitted z Example

16. Critical Angle (cont.) At the critical angle: qi = c reflected qi qr incident #1 qt x #2 qt = 90o transmitted z There is no vertical variation of the field in the less-dense region.

17. Critical Angle (cont.) Beyond the critical angle: qi > c incident reflected qi qr #1 x #2 z There is an exponential decay of the field in the vertical direction in the less-dense region. (complex)

18. Critical Angle (cont.) Beyond the critical angle: qi > c incident reflected qi qr #1 x #2 z The power flows completely horizontally. (No power crosses the boundary and enters into the less dense region.) This must be true from conservation of energy, since the field decays exponentially in the lossless region 2.

19. Critical Angle (cont.) Example: "fish-eye" effect air water The critical angle explains the “fish eye” effect that you can observe in a swimming pool.

20. TEz Reflection Ei Hi qi qr #1 qt x #2 z Note that the electric field vector is in the y direction. (The wave is polarized perpendicular to the plane of incidence.)

21. TEz Reflection (cont.)

22. TEz Reflection (cont.) Recall that the tangential component of the electric field must be continuous at an interface. Boundary condition at z = 0:

23. TEz Reflection (cont.) We now look at the magnetic fields.

24. TEz Reflection (cont.) Recall that the tangential component of the magnetic field must be continuous at an interface (no surface currents). Hence we have:

25. TEz Reflection (cont.) Enforcing both boundary conditions, we have The solution is:

26. TEz Reflection (cont.) Transmission Line Analogy incident

27. TMz Reflection Ei Hi qi qr #1 qt x #2 z Note that the electric field vector is in the y direction. (The wave is polarized parallel to the plane of incidence.) Word of caution: The notation used for the reflection coefficient in the TMz case is different from what is in the book. (We use reflection coefficient to represent the reflection of the electric field, not the magnetic field.)

28. TMz Reflection (cont.)

29. TMz Reflection (cont.) We now look at the electric fields. Note that TM is the reflection coefficient for the tangential electric field.

30. TMz Reflection (cont.) Boundary conditions: Enforcing both boundary conditions, we have The solution is:

31. TMz Reflection (cont.) Transmission Line Analogy incident

32. Power Reflection qi qr #1 qt x #2 z

33. Power Reflection Beyond Critical Angle qi > c incident reflected qi qr #1 x #2 z All of the incident power is reflected.

34. Example Given: qi qr #1 qt x #2 z Find: % power reflected and transmitted for a TEz wave % power reflected and transmitted for a TMz wave Snell’s law:

35. Example (cont.) First look at the TMz case:

36. Example (cont.) Next, look at the TEz part:

37. Example Given: qi qr #1 qt x #2 sea water z Find: % power reflected and transmitted for a TEz wave % power reflected and transmitted for a TMz wave We avoid using Snell's law since it will give us a complex angle in region 2!

38. Example (cont.) First look at the TMz case:

39. Example (cont.) Next, look at the TEz part:

40. Brewster Angle Consider TMz polarization Assume lossless regions Set

41. Brewster Angle (cont.) Hence we have

42. Brewster Angle (cont.) Assume m1=m2:

43. Brewster Angle (cont.) q i Geometrical angle picture: Hence

44. Brewster Angle (cont.) This special angle is called the Brewster angle b. • For non-magnetic media, only the TMz polarization has a Brewster angle. • A Brewster angle exists for any material contrast ratio (it doesn’t matter which side is denser).

45. Brewster Angle (cont.) Example air water

46. Brewster Angle (cont.) Polaroid sunglasses polarizing filter (blocks TEz) TEz TMz+TEz sunlight puddle of water The reflections from the puddle of water (the “glare”) are reduced.