Prof. David R. Jackson Dept. of ECE Fall 2013 Notes 19 ECE 6340 Intermediate EM Waves 1

Critical Angle Snell’s law: n1n1 y z ii tt n2n2 Assume lossless materials 2

Critical Angle n1n1 y z cc tt n2n2 At the critical angle (  i =  c ) : so  t = 90 o 3

Critical Angle  The critical angle applies to any polarization.  The critical angle is only defined for lossless materials.  A critical angle exists only when going from a higher to a lower density medium ( n 1 > n 2 ). Notes: 4

Beyond Critical Angle so Lets' examine the transmitted angle: Assume 5

Beyond Critical Angle (cont.) y z Region 2 Power flow Note: The power flow in the upper region is horizontal, and is independent of z. No power crosses the boundary Note: The power flow in the lower region is horizontal, and decays with z. Region 1 6

Beyond Critical Angle (cont.) 7 All of the incident power is reflected

Use Determine the transmitted angle  t : Let or Not possible must use +sign Beyond Critical Angle (cont.) (real) 8

Hence Beyond Critical Angle (cont.) 9

Choose + sign to obtain correct value for k z2 : The + sign is chosen to obtain a decaying wave. Beyond Critical Angle (cont.) 10

Hence Practical note: When dealing with inhomogeneous plane waves (complex angles), it is usually easier to avoid working with angles and use the separation equation instead. Beyond Critical Angle (cont.) Requires complex angle Does not requires complex angle 11

Brewster Angle TM z No reflection y z  tb n1n1 EiEi n2n2 Lossless materials 12

Brewster Angle (cont.) Perfect match: 13

Assume  1  =  2 =  : Brewster Angle (cont.) 14

Hence  b Brewster Angle (cont.) 15

 For non-magnetic media, only the TM z polarization has a Brewster angle.  A Brewster angle exists for any material contrast ratio. Notes: 16 Brewster Angle (cont.)

Hence Brewster Angle (cont.)  b  tb From Snell's law: 17 This means that  tb is the angle shown in this figure.