EE530 Lecture 3

1
EE530 Lecture 3

• Review of Electromagnetics
• Please add your own figures

2
Maxwells equations

• We use cgs units in this course. Maxwells
  equations are

rfree charge, Jcurrent
One major difference in this course from standard
electromagnetic books is that the dielectric
function e or the magnetic permeability m is a
function of position and have spatial
derivatives We will not use the polarization P or
magnetization M. Instead we use E,H, e, m
3
Wave equation

• Assume no free current or free charge r0, J0
  (note that antennas require a current source)
• Maxwells equations are

EM waves in uniform media Assume e(r)e m(r)m
(constant values) Take the curl of the curl E
equation to get the wave equation, which
decouples the fields
This is the wave equation for each component of E
and H. The wave has velocity v c/vem c/n
c3108 m/s in vacuum n vem is the refractive
index. Wave traverses more slowly in a dielectric
medium by the factor 1/n.
4
Electromagnetic waves

• Plane electromagnetic waves have solutions
• E(x,t)E0e ik.x-iwt
• H(x,t)H0e ik.x-iwt
• Wavevector k vem w/c nw/c is the amount of
  momentum in the wave
• Use plane wave solution in Maxwells equation to
  relate E and B (or H)

E, B, k form a right-handed triad of vectors
k.E0 k.B0 (transverse waves) Wave traveling
along the z direction Wavelength
l2p/kv/f Impedance of the mediumratio of
electric magnetic field Impedance is a
characteristic property of the medium In free
space Z377 ohm in mks units or Z1 (cgs units)
5
Energy flow

• Poynting vector S describes time averaged flow
  of energy
• S is real and in the direction of k
• Intensity IE02
• Larger Z reduce the energy flow similar to
  reducing the current flow when the resistance is
  larger

6
Reflection and refraction of waves
Z0

• General case wave incident from media em to
  semi-infinite medium e m

Boundary conditions must be satisfied at all
times on the plane z0 Spatial and time variation
of the fields must be the same at z0 (k.x)
(k.x) (k.x) at z0 k sini k sinr
ksinr rr Snells law Will later see
differences for negative refractive index
materials
7
Boundary conditions

• Continuity at interface of
• Normal components of D (from divergence equation)
• Normal components of B
• Tangential components of E (from curl equations)
• Tangential components of H

8
S-polarized TE mode

• Electric field perpendicular to plane of
  incidence
• From continuity of tangential components (3), (4)
  should obtain two equations relating E and E to
  E

TE
These can be solved for the reflection
coefficient R and transmission coefficient T
9
Normal incidence

• Simplifies result for reflection and transmission
• Same result for both polarizations

Go from air with n1, (or less dense medium) to
more dense medium find that ratio E/E is
negative Phase change on reflection For a very
dense medium (n large) or for a metal, the
reflected wave cancels out the incident wave so
that no wave goes inside the medium and boundary
conditions are satisfied Glass n1.5 R4, T96
10
P-polarized TM mode

• Magnetic field H is perpendicular to plane of
  incidence
• Use continuity of tangential E and H fields

Solve two equations for two unknowns to get
reflection and transmission
For normal incidence should get the same results
as TE polarization
11
Total internal reflection

• From higher index to lower index material
• The refracted angle r90º when the incident angle
  is at a critical value
• i0 sin-1(n/n)sin-1(n) (n1)
• Total reflection when igti0
• Field is attenuated in the z direction
  evanescent field at interface
• Principle of optical fibers dense core (glass
  n1.5) surrounded by lower index cladding
  material
• Light enters from fiber end and is transported by
  total internal reflection
• Entering light has to be incident at less than
  the critical angle

12
Fiber optics

• Attenuation vs wavelength shows lowest loss at
  1.55 m and low loss at 1.3 m
• There is OH absorption at l1.4 m from residual
  water in SiO2(OH stretch vibration at very high
  frequency)
• Transmission bands
• Long wavelength band l1.55m lowest loss 0.26
  dB/km
• Medium wavelength band l1.3m loss 0.4 dB/km
• Short wavelength band l800-900 nm high
  absorption
• Typical fiber dimensions
• Core 50 m diameter
• Cladding 125 m diameter
• Fiber is 50/125
• Bend fibers Need bending radius R to be several
  wavelengths l, to achieve low loss around bend.
  Typical optical fiber has few mm bend
• This problem can be solved by using photonic
  crystals with small radius bends