Prof. D. R. Wilton Notes 18 Reflection and Transmission of Plane Waves Reflection and Transmission of Plane Waves ECE 3317 [Chapter 4]

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

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

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

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 (TM z ) H z = 0  Transverse Electric (TE z ) E z = 0 TE z Note: The word “transverse” means “perpendicular to.” x y z E H S, k x TM z y z E H S, k

Reflection and Transmission As we will show, each type of plane wave (TE z and TM z ) reflects differently from a material. #1 x z ii rr tt #2 incident reflected transmitted

Boundary Conditions Here we review the boundary conditions at a dielectric interface (from ECE 2317) Note: The unit normal points towards region 1 or away from a PEC No sources on interface: The tangential electric and magnetic fields are continuous. The normal components of the electric and magnetic flux densities are continuous. Usually zero!

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

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

Law of Reflection Similarly, Law of reflection

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.

Snell’s Law (cont.) The bending of light (or EM waves in general) is called refraction. incident transmitted reflected Acrylic block

Given: Note that in 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 32.1 o, the wave will exit into the air region at an angle of 45 o. Snell’s Law (cont.) Example Note: At microwave frequencies and below, the relative permittivity of water is about 81. At optical frequencies it is about water air

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

Critical Angle (cont.) Example #1 x z water rr #2 incident reflected transmitted cc air

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

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

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

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

TE z Reflection Note that the electric field vector is in the y direction. (The wave is polarized perpendicular to the plane of incidence.) #1 x z ii rr tt #2 EiEi HiHi

TE z Reflection (cont.)

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

We now look at the magnetic fields. TE z Reflection (cont.)

In applying boundary conditons we deal only with tangential components, so it’s convenient to introduce a “wave impedance” relating tangential E and H. TE plane waves have the general form TE z Reflection (cont.)

We rewrite all the various plane wave fields in terms of wave impedances, introducing subscripts appropriate to the region in which they apply:

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

Enforcing both boundary conditions, we thus have The solution is: TE z Reflection (cont.)

Transmission Line Analogy incident TE z Reflection (cont.)

TM z Reflection Note that the magnetic field vector is in the y direction. (The wave is polarized parallel to the plane of incidence.) #1 x z ii rr tt #2 EiEi HiHi Word of caution: The notation used for the reflection coefficient in the TM z case is different from that of the Shen and Kong book. (We use reflection coefficient to represent the reflection of the electric field, not the magnetic field.)

TM z Reflection (cont.) Define a wave impedance

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

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

Transmission Line Analogy incident TM z Reflection (cont.)

Power Reflection #1 x z ii rr tt #2

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

Find: % power reflected and transmitted for a TE z wave % power reflected and transmitted for a TM z wave Example #1 x z ii rr tt #2 Given: Snell’s law:

First look at the TM z case: Example (cont.)

Next, look at the TE z part: Example (cont.)

Find: % power reflected and transmitted for a TE z wave % power reflected and transmitted for a TM z wave Example Given: #1 x z ii rr tt #2 sea water We avoid using Snell's law since it will give us a complex angle in region 2!

First look at the TM z case: Example (cont.)

Next, look at the TE z part: Example (cont.)

Consider TM z polarization Brewster Angle Set Assume lossless regions

Brewster Angle (cont.) Hence we have

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

Brewster Angle (cont.) Hence  i Geometrical angle picture: A Brewster angle exists for any ratio of real relative permittivities

Brewster Angle (cont.) Hence  i  t

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

Brewster Angle (cont.) Example water air

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