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PHYS 408
Applied Optics
(Midterm Review)
Jan-April 2016 Edition
Jeff Young
AMPEL Rm 113
Q#1
Q1 [Total Mark of 10, one each for each question]
1a)
“Light” refers to waves of electromagnetic fields:
T or F
1b)
A light wave can consist of only a time varying electric field: T or F
1c)
The polarization of a light wave is well defined in vacuum and in a homogeneous
isotropic dielectric medium:
T or F
1d)
The polarization density of the vacuum is unity:
T or F
1e)
The Poynting vector has units of energy:
T or F
Q#1
1f)
The time-averaged Poynting vector for a plane wave in vacuum depends on position:
T or F
1g)
The time-averaged Poynting vector for a spherical wave in vacuum depends on position:
T or F
1h)
The Fresnel equations describe how much phase is accumulated as a wave propagates
through a material:
T or F
1i)
The reflectivity of a plane wave incident on a thin dielectric film with a fixed refractive
index depends on the frequency of the lightwave: T or F
1j)
When calculating the electromagnetic fields generated when a plane wave is incident on
a thin dielectric film, there are two unknown wave amplitudes that must be determined using
boundary conditions:
T or F
Q#2
Your boss tells you that the electric field in some uniform region has been determined to be
 
E (r , t )  E0 exp i ( kr t ) , where E0  10 V / m2 , k  9.94e4 cm 1 ,   1.49e15 Hz and asks you to
determine i) what is the corresponding magnetic field, and ii) what the velocity of light is in that region?
You go away and start to figure out the answers, but quickly realize that there are a number of
inconsistencies in the information provided, and there is insufficient information, given these
inconsistencies, for you to proceed. You think carefully what to ask your boss in order to clarify the
inconsistencies and obtain enough information to answer the questions.
Q#2
Q2a) [10]
What questions would you pose in order to be
able to properly answer question i) asked by your boss?
List as many relevant questions you can think of. Marks
will be given for consistent, relevant questions that would
quickly resolve the issue.
1)Are the units for the Electric field not V/m?
2)Are the units for  not radians/sec?
3)Please indicate whether k is in fact a vector, or is it really a scaler?
4)If it is a vector, please provide it’s components and the orientation of the electric field associated with
the plane wave.
5)If it is a scalar, please provide the orientation of the electric field in the equatorial plane of the dipole
field.
6)Does the material in this region have a magnetic response?
Q#2
Q2b) [10]
Could you come up with a likely answer to part of
question ii), assuming your boss has just made a “silly
mistake”? Thoroughly explain your reasoning, and provide
your answer.
Yes, since the speed s is s=/k, if we assume the value of  given by the boss is in fact in radians per second,
s turns out to be half the speed of light in vacuum. If you assume that you had to multiply the  given by
the boss by 2 p to correct a silly mistake, you would get a speed larger than the speed of light, so most likely
it was a unit issue, and the refractive index of the material is ~ 2, a reasonable number.
Q#3
A plane wave is incident on a thin, homogeneous, nonmagnetic dielectric thin film as shown in Fig. 1.
Q#3
Q3a) [10]What are the numerical values of the Fresnel
coefficients r , r , t t
01
10
01, 10
The Fresnel’s equations at normal incidence are:
𝑟1→2 =
𝑛1 − 𝑛2
,
𝑛1 + 𝑛2
𝑡1→2 =
2𝑛1
𝑛1 + 𝑛2
Therefore,
1
2
1
4
𝑟01 = − 3 , 𝑡01 = 3, 𝑟10 = 3, 𝑡10 = 3
Q#3
Q3b) [10]What is the amount of phase that the wave
travelling to the right inside the film acquires going from
just inside the film (z=0+), to just inside the film at z=d-?
𝜔
𝑐
𝜙 = 𝑛𝑘0 𝑑 = 2 d
Q#3
Q3c) [10]Write expressions for the total electric field in
regions 0, 1 and 2. You do not have to solve for the
unknown coefficients in this question, just express the
functional form of the fields in each region, knowing that
some of the complex field amplitudes still need to be
evaluated somehow.
𝐸0+ = 𝐸0+ 𝑒 −𝑗𝜔𝑧/𝑐 , 𝐸0− = 𝐸0− 𝑒 𝑗𝜔𝑧/𝑐 ; 𝐸0 = 𝐸0+ +𝐸0−
𝐸1+ = 𝐸1+ 𝑒 −𝑗𝑛𝜔𝑧/𝑐 , 𝐸1− = 𝐸1− 𝑒 𝑗𝑛𝜔𝑧/𝑐 ; 𝐸1 = 𝐸1+ +𝐸1−
𝐸2+ = 𝐸2+ 𝑒 −𝑗𝜔𝑧/𝑐 = 𝐸2
Q#3
Q3b) [20]Solve the problem, using boundary conditions
and the equations in Q3c, and write down the full field
expressions in terms of the refractive index and thickness
of the film, and the incident field amplitude 𝐸0 .
Boundary conditions are:
Interface 01:
𝐸0+ + 𝐸0− = 𝐸1++𝐸1− (parallel electric field)
𝐸0+ − 𝐸0− = 𝑛(𝐸1+ − 𝐸1− ) (parallel magnetic field)
Interface 12:
𝐸1+ 𝑒 −𝑗𝑛𝜔𝑑/𝑐 + 𝐸1− 𝑒 𝑗𝑛𝜔𝑑/𝑐 = 𝐸2+ (parallel electric field)
n (𝐸1+ 𝑒 −𝑗𝑛𝜔𝑑/𝑐 − 𝐸1− 𝑒 𝑗𝑛𝜔𝑑/𝑐 ) = 𝐸2+ (parallel magnetic field)
Q#3
Q3b)
4 unknowns 𝐸0− , 𝐸1+, 𝐸1−, 𝐸2+ and 4 equations, amplitudes can be solved:
𝐸0−
3𝑒 −𝑗2𝜔𝑑/𝑐 − 3𝑒 𝑗2𝜔𝑑/𝑐
= 𝐸0
,
−𝑒 −𝑗2𝜔𝑑/𝑐 + 9𝑒 𝑗2𝜔𝑑/𝑐
6𝑒 𝑗2𝜔𝑑/𝑐
𝐸1+ = 𝐸0
−𝑒 −𝑗2𝜔𝑑/𝑐 + 9𝑒 𝑗2𝜔𝑑/𝑐
𝐸1−
2𝑒 −𝑗2𝜔𝑑/𝑐
= 𝐸0
,
−𝑒 −𝑗2𝜔𝑑/𝑐 + 9𝑒 𝑗2𝜔𝑑/𝑐
𝐸2+ = 𝐸0
8
−𝑒 −𝑗2𝜔𝑑/𝑐 + 9𝑒 𝑗2𝜔𝑑/𝑐
Q#4
Q4a) [10]Write down the individual M matrices and the net M
matrix (just show the correct order of the multiplication of the 3
individual matrices, you don’t have to carry out the matrix
multiplication) that propagates the field amplitude vector on the
input side of the structure in Figure 1, to the output field
amplitude vector. The matrix elements should be functions of
the refractive index and thickness d.
𝑀𝑛𝑒𝑡 = 𝑀01 𝑀11 𝑀10
1 𝑛+1
=
2𝑛 𝑛 − 1
𝑛 − 1 𝑒 −𝑗𝑛𝜔𝑑/𝑐
𝑛+1
0
0
𝑒 𝑗𝑛𝜔𝑑/𝑐
1 1+𝑛
2 1−𝑛
1−𝑛
1+𝑛
Q#4
Q4b) [10] Show that the inverse of the first and third M matricies
in Q1, [Mi,j]-1, are actually equal to Mj,i where i is not equal to j.
This can be most easily shown by directly multiply the two matrices and see
if the result is identity.
𝑀01 𝑀10
1 𝑛+1
=
2𝑛 𝑛 − 1
1
=4𝑛
=
1
0
𝑛−1 1 1+𝑛
𝑛+1 2 1−𝑛
(𝑛 + 1)2 −(𝑛 − 1)2
0
0
1
1−𝑛
1+𝑛
0
(𝑛 + 1)2 −(𝑛 − 1)2