ELEG 648 Plane waves II Mark Mirotznik, Ph.D. Associate Professor The University of Delaware

Uniform Plane Waves: Propagation in Any Arbitrary Direction E H y x z  

Uniform Plane Waves: Propagation in Any Arbitrary Direction E H y x z  

Uniform Plane Waves: Propagation in Any Arbitrary Direction E H y x z   Since E and  are at right angles from each other. whereand

Uniform Plane Waves: Propagation in Any Arbitrary Direction Observation 1. E, H and  vectors are pointing in orthogonal directions. Summary and Observations: Frequency Domain Time Domain Observation 2. E and H are in phase with each other, however, H’s magnitude is smaller by the amount of the wave impedance

Uniform Plane Waves: Propagation in 2D E H x y  Can we write this a bit more compact?

Uniform Plane Waves: Propagation in 2D E H x y  Can we write this a bit more compact?

Uniform Plane Waves: Propagation in 2D E H x y  What about the polarization of E?

Uniform Plane Waves: Propagation in 2D E H x y  What about the polarization of E?

Uniform Plane Waves: Propagation in 2D E H x y  What about the polarization of E?

Uniform Plane Waves: Propagation in 2D E H x y  What about the polarization of E? Two cases E H x y  Parallel Polarization Perpendicular Polarization

Uniform Plane Waves: Propagation in 2D E H x y  What about H?

Uniform Plane Waves: Propagation in 2D E H x y  What about the polarization of H? Two cases E H x y  Parallel Polarization Perpendicular Polarization

Reflection and Transmission Write down the electric fields in the two regions (2 unknowns, R and T)

Reflection and Transmission Next find the magnetic fields in each region

Reflection and Transmission Apply boundary conditions

Reflection and Transmission

Write down the E field in both regions (4 unknowns, R, T,  r and  t )

Reflection and Transmission Find the H field in both regions

Reflection and Transmission Apply boundary conditions 2 equations and 4 unknowns We need two more equations. How do we get them?

Reflection and Transmission

Angle of Incidence, Degrees Reflection Coefficient Example: Reflection from an Ocean Interface

Reflection and Transmission from Dielectric Slabs 1.Normal Incidence z=0 z=d Region I Region II Region III

Reflection and Transmission from Dielectric Slabs Region I:Region II: Region III: Boundary Conditions z=0 z=d

Reflection and Transmission from Dielectric Slabs Boundary Conditions z=0z=d Four equations and four unknowns Solution for the Reflection Coefficient:

Reflection and Transmission from Dielectric Slabs Special Cases I. Half Wavelength Thickness Slab z=0 z= 2 /2 Region I Region II Region III

Reflection and Transmission from Dielectric Slabs Special Cases II. Quarter Wavelength Thickness Slab z=0 z= 2 /4 Region I Region II Region III

Reflection and Transmission from Dielectric Slabs: Example z=0 z=  m Region I Region II Region III

Reflection and Transmission from Dielectric Slabs: Example Frequency, MHz |R|

How do we broaden the bandwidth around the zero reflection point? Frequency, MHz |R|

One Solution is Multiple Dielectric Layers

Reflection and Transmission from Dielectric Slabs 1.Oblique Incidence ( Parallel Polarization) z=0 z=d Region IRegion IIRegion III ii tt rr  II

Reflection and Transmission from Dielectric Slabs Region I: Region II:

Reflection and Transmission from Dielectric Slabs Region III: Boundary Conditions z=0 z=d Phase Matching Conditions

Reflection and Transmission from Dielectric Slabs Six Equations and Six Unknowns

Reflection and Transmission from Dielectric Slabs: Solution (parallel polarization) *note we have assumed all non-magnetic materials here

Reflection and Transmission from Dielectric Slabs: Solution (perpendicular polarization) *note we have assumed all non-magnetic materials here

Reflection and Transmission from Dielectric Slabs: Example z=0 z=  m Region IRegion II Region III 