Prof. David R. Jackson ECE Dept. Spring 2014 Notes 3 ECE

Grounded Dielectric Slab Goal: Determine the modes of propagation and their wavenumbers. Assumption: There is no variation of the fields in the y direction, and propagation is along the z direction. x z h 2

Dielectric Slab TM x & TE x modes: x z H E TM x z E H TE x x 3

Surface Wave The internal angle is greater than the critical angle, so there is exponential decay in the air region. z x Exponential decay The surface wave is a “slow wave”. 4 h

Surface Wave The wave must also satisfy a “consistency condition”: This forces the angle  1 to be a discrete value, depending on n. or 5 z x h

TM x Solution Assume TM x B.C. : (see TM x -TE x tables in Appendix) Assume 6

Hence Denote TM x Solution (cont.) 7

Applying boundary conditions at the ground plane, Note: Since the surface wave is a slow wave, we have: TM x Solution (cont.) 8

Boundary Conditions BC 1) BC 2) 9

Boundary Conditions (cont.) These two equations yield: Divide second by first: or 10

Final Result: TM x This may be written as: This is a transcendental equation for the unknown wavenumber k z. Note: The choice of square root for k x1 is not important, but it is for k x0 : 11

Appendix 12 TM x Note: There is a factor  difference with the Harrington text.

Appendix (cont.) 13 TE x Note: There is a factor  difference with the Harrington text.