IEO5373 Introduction to Electromagnetics for Metamaterials
Course Part II, Homework #1
Due: May 9, 2016
1. As shown in the following figure, let us consider an electron bound in a potential
with an effective restoring force m02x(t), where m is the mass of the electron,
0 is the natural frequency of the oscillation, and x(t) is the position of the
electron. In addition to this restoring force, the electron also experiences a
damping force mv(t)/ which is proportional to its speed v(t) [v(t)= dx(t)/dt] and
characterized by a damping time . The electron has a charge e and is driven by an
electric field E(t)=Re[()exp(jt)] oscillating with a phasor amplitude () at a
frequency .
E(t)=Re[()ejt]
mv(t)/ e
m02x(t)
x(t)
x
(a) Use Newton’s second law, write down the equation of motion (a partial
differential equation in time) for the position x(t) of the electron.
(b) If we write x(t)=Re[X()exp(jt)] and v(t)= Re[V()exp(jt)] in their phasor
forms with the corresponding phasors X() and V(), respectively, find the
expressions of the phasors X() and V() in terms of ().
(c) The induced polarization density can be written as p(t)=enbx(t), where nb is the
density (per unit volume) of theses bound electrons. If we write p(t)=
Re[P()exp(jt)] in its phasor form with an amplitude P(), find the
expression of P() in terms of (). Next, express the phasor P() as
P()=0()(), where () is the susceptibility of this interaction. Find the
expression of ().
(d) Plot Re[()], Im[()], and |()| as functions of the normalized frequency
=/0. Set =10/0 and e2nb/(0m02)=1. At which frequency does Re[()]
turn from positive to negative? At which frequency does Im[()] have a
maximum peak? What is the full-width-at-half-maximum (FWHM) lineiwdth
of Im[()] (in the unit of 0)?
Note: In fact, the relative dielectric constant d,r() is related to the susceptibility ()
derived here as d,r()=bg,r()+(), where bg,r() is the background contribution
from other transitions not involved here. The model presented here is called Lorentz
model for bound electrons involved in a certain transition with a natural frequency 0.
This model should be compared with the Drude model discussed in the class for free
conduction electrons in absence of the restoring force.
2. Let us consider the transmission and reflection of the TM (p) wave at the interface
of two media, as shown in the following figure.
( 1,r , 1, r )
x
( 2, r ,  2, r )
z
(a) The TM wave has the following magnetic fields in different regions


 yˆ h e  j ( k x x  k z1z )  h y ,r e  j ( k x x  k z1z ) , z  0,
H  yˆ h y   y ,i
yˆ h y , t e  j ( k x x  k z 2 z ) , z  0.

Obtain the expressions of the electric field E in these two regions.
(b) Match the boundary conditions of the fields at z=0 and calculate the TM
reflection coefficient rh and transmission coefficient th:
rh 
h y ,r
h y ,i
, th 
h y,t
h y ,i
.
(c) Calculate the Poynting vectors at the two regions and match their z
components. Obtain the relation between the TM reflection coefficient rh and
transmission coefficient th.
(d) The Brewster angle B is the angle at which the TM reflection does not occur.
Obtain the condition in which the Brewster angle exists. What is its expression?
In the TE case, under what condition does an analogous angle without any
reflections exist?