8-9 Normal Incidence at Multiple Dielectric Interfaces Antireflection coating for eye glasses. Radome (for the protection of radars of ships) Reflected wave Medium 3 (ε3 , μ3) z=0 z=d Medium 2 (ε2 , μ2) Medium 1 (ε1 , μ1) x Er Hr anr Unknowns : Transmitted wave Incident wave Boundary conditions at z=0 , z=d Hi Ei ani The procedure is straightforward and purely algebraic.

8-9.1 Wave Impedance of the total field (includes the effect of medium 2) O z In an unbounded medium , for +z direction wave In an unbounded medium, for -z direction wave η: intrinsic impedance Medium 1 Medium 2 z=0 which is the same as the input impedance of a transmission line of length l that has a characteristic impedance and terminates in a short circuit.

8-9.2 Impedance Transformation with Multiple Dielectrics We can transform to an arbitrary value of by choosing and when , There is no reflection. Ex. 8-12 p405

8-10 Oblique Incidence at a plane Dielectric Boundary Reflected wave x Refracted wave anr ant O’ Snell’s Law of refraction -The angle of reflection is equal to the angle of incident) A’ B A θr θt θi O z ani Incident wave n : refractive index Medium 1 (ε1,μ1) Medium 2 (ε2,μ2) z=0

8.10-1 Total Reflection What happen in Medium 2 when ? x z z=0 ε1 Medium 2 ε2 (Total reflection) Critical Angle θt What happen in Medium 2 when ? θi O z z=0 Evanescent wave along z Propagation in x direction tightly bound to the surface ⇒ surface wave If medium 2 is not thick enough, there is transmitted wave even though TIR condition is met! z (no power transmission)

8-10.2 Perpendicular Polarization x For given incident wave, θr θi O Find z • z=0

8-10.2 Perpendicular Polarization (2) x θr θi O z We know that • z=0

8-10.2 Perpendicular Polarization (3) Assume that we found the solutions, Fresnel’s Equation

8-10.3 Parallel Polarization x For given incident wave, x θt θi O Find • z=0

8-10.3 Parallel Polarization (2) Boundary conditions Z=0

8-10.3 Parallel Polarization (3) (No reflection) find such Snell’s law (assume μ1= μ2)

8-10.3 Parallel Polarization (4) - (1) (1)

8-10.3 Parallel Polarization (5) Brewster Angle What about perpendicular polarization? However this is the rare situation in nature!

8-10.3 Parallel Polarization (6)