The document discusses omnidirectional reflection from a one-dimensional photonic crystal structure. It presents the following key points: 1) A one-dimensional photonic crystal, such as a multilayer film, can exhibit complete reflection of light within a frequency range for all incident angles and polarizations, even without a full photonic bandgap. 2) The criterion for omnidirectional reflection is that there exists a frequency range where the projected band structures of the photonic crystal and surrounding medium do not overlap, rather than there being no propagating states within the crystal itself. 3) As an example, a multilayer film with refractive indices of n1 = 1.7 and n2 = 3.4

1. New folder/elec425_2016_hw5.pdf
Mar 25, 2016
ELEC 425 Spring 2016 HW 5 Questions
due in class on Tue Mar 31, 2016
1) Read Sec. 1.11 from the textbook. Use the conventions
plotted on Fig. 1.42 to derive the TM
matrix in Eq. 1.253.
2) The file Tmatrix.m is a Matlab script that evaluates the
reflection and transmission coefficients
for TE and TM polarizations. Analyze the code, and write a
script that uses Tmatrix.m to
generate Fig. 3 from Winn1998.pdf file. When the output from
the Matlab code is overlaid with
Fig. 3 from the paper, they should match exactly as shown
below. Note the dB scale in the
figure.
3) Read the following tutorial from the Lumerical website.
https://kb.lumerical.com/en/diffractive_optics_stack.html

2. First, run and verify the tutorial. Then, modify the tutorial files
so that you simulate 0° and 45°
results from Fig. 3 of the Winn1998.pdf paper as shown above.
The structure is composed of a
total of 12 layers: air on the entrance and exit sides, and five
repetitions of two quarter wave
(�1 + �2 =
�1
4
+
�2
4
= �) layers of refractive index �1 = 1.7 and �2 = 3.4 and
thicknesses �1
and �2. Export your simulation results, import them into
Matlab, and plot the output from part
2) with the output from Lumerical FDTD on the same plot.
Verify that FDTD code results in a
similar set of results.
Please hand in your derivations, your plots and the relevant
code used to generate the plots all
stapled together.

3. You can find the required files under the Handouts section on
the course website at:
http://courses.ku.edu.tr/elec425
https://kb.lumerical.com/en/diffractive_optics_stack.html
http://courses.ku.edu.tr/elec425
New folder/PhotonicsLaserEngineering.pdf.part

42. New folder/TMatrix.m
% Calculates the reflection coefficient for a plane wave
impinging on a
% multilayered thin film coating.
% The formulation assumes exp(+j*w*t) convention
% TMatrix takes a list of inputs that determine the system to be
simulated
% and outputs the reflection and transmission coefficient
% Formulation is based on Sec 1.11 of A. Sennaroğlu,
Photonics and Laser
% Engineering, McGraw-Hill 2010, ISBN: 978 007 160 6080
function [r,t]=TMatrix(nList,dList,thetain,lambda,pol)
% nList is the list of refractive indices
% dList is the list layer thicknesses in micrometers

43. % thetain is the input angle in radians
% lambda is the wavelength in micrometers
% pol is either "TE" or "TM"
% r is the complex reflection coefficient
% t is the complex transmission coefficient
%{
Notes on data structure:
------------------------
We assume that the first element of nList is the layer where the
plane wave
is incident, and the last element is where the plane wave comes
out.
The first and last layers are assumed to have infinite
thicknesses.
All elements of dList are in microns.
d1 = dLast = infinite
There is an M matrix associated with each layer. The first and
the last M
matrices are not used.
n1 | n2 | n3 | ... | nLast
d1 | d2 | d3 | ... | dLast
M1 I1 M2 I2 M3 I3 ... | MLast
"I" stands for the interface between layers. I1, I2 stand for total
fields
at the interface. They are related by M matrices as
I1 = M2*I2, I2 = M3*I3 etc.
%}

44. % Check inputs
numLayers = length(nList);
if length(dList) ~= numLayers
error('Length mismatch, exiting');
end
if length(thetain) ~= 1
error('Length mismatch, exiting');
end
if length(lambda) ~= 1
error('Length mismatch, exiting');
end
if ~(strcmp(pol,'TE') || strcmp(pol, 'TM'))
error('Undefined polarization');
end
% Set the first and last elements of dList to inf
dList(1) = inf;
dList(end) = inf;
% Calculate the propagation angle in each layer
sinList = zeros(numLayers,1); % list of sines of angles
cosList = zeros(numLayers,1); % list of cosines of angles
PhiList = zeros(numLayers,1); % list of Phi values, see eqn
1.247 on p. 68
MList = zeros(numLayers,2,2); % list of matrices, eqn. 1.252 &
1.253, p. 69
% Apply Snell's Law
dum = nList(1)*sin(thetain);
for ii=1:numLayers
sinList(ii) = dum / nList(ii);
cosList(ii) = sqrt(1-sinList(ii)^2);
if imag(cosList(ii)) > 0

45. cosList(ii) = -cosList(ii);
end
PhiList(ii) = 2*pi/lambda*nList(ii)*dList(ii)*cosList(ii);
end
% Create the M matrices
if strcmp(pol,'TE')
for ii=1:numLayers
MList(ii,:,:) = ...
[cos(PhiList(ii)),
1i*sin(PhiList(ii))/(nList(ii)*cosList(ii)) ; ...
1i*sin(PhiList(ii))*(nList(ii)*cosList(ii)),
cos(PhiList(ii))];
end
elseif strcmp(pol,'TM')
for ii=1:numLayers
MList(ii,:,:) = ...
[cos(PhiList(ii)), -
1i*sin(PhiList(ii))/(nList(ii)/cosList(ii)) ; ...
-1i*sin(PhiList(ii))*(nList(ii)/cosList(ii)),
cos(PhiList(ii))];
end
end
% Calculate the overall matrix
MFinal = [1 0;0 1];
for ii=2:(numLayers-1)
MFinal = MFinal * squeeze(MList(ii,:,:)) ;
end
m11 = MFinal(1,1);
m12 = MFinal(1,2);
m21 = MFinal(2,1);

46. m22 = MFinal(2,2);
% Calculate the reflection and transmission coefficients
if strcmp(pol,'TE')
n0 = nList(1)*cosList(1);
ns = nList(end)*cosList(end);
r = (m11*n0+m12*n0*ns-m21-
m22*ns)/(m11*n0+m12*n0*ns+m21+m22*ns);
t = 2*n0 /(m11*n0+m12*n0*ns+m21+m22*ns);
end
if strcmp(pol,'TM')
n0 = nList(1)/cosList(1);
ns = nList(end)/cosList(end);
r = (m11*n0-m12*n0*ns+m21-m22*ns)/(m11*n0-
m12*n0*ns-m21+m22*ns);
t = 2*nList(1)/cosList(end) /(m11*n0-m12*n0*ns-
m21+m22*ns);
end
end
New folder/Winn1998.pdf
October 15, 1998 / Vol. 23, No. 20 / OPTICS LETTERS 1573
Omnidirectional reflection from a one-dimensional
photonic crystal
Joshua N. Winn, Yoel Fink, Shanhui Fan, and J. D.
Joannopoulos
Massachusetts Institute of Technology, 77 Massachusetts
Avenue, Cambridge, Massachusetts 02139
Received July 7, 1998

47. We demonstrate that one-dimensional photonic crystal
structures (such as multilayer f ilms) can exhibit
complete ref lection of radiation in a given frequency range for
all incident angles and polarizations. We
derive a general criterion for this behavior that does not require
materials with very large indices. We
Optical Society of America
OCIS codes: 230.4170, 230.1480.
Low-loss periodic dielectrics, or photonic crystals, allow
the propagation of light to be controlled in otherwise
diff icult or impossible ways.1 – 4 In particular, a pho-
tonic crystal can be a perfect mirror for light from any
direction, with any polarization, within a specif ied fre-
quency range. It is natural to assume that a necessary
condition for such omnidirectional ref lection is that the
crystal exhibit a complete three-dimensional photonic
bandgap, that is, a frequency range within which there
are no propagating solutions of Maxwell’s equations.
Here we report that this assumption is false — in fact a
one-dimensional photonic crystal will suff ice. We in-
troduce a general criterion for omnidirectional ref lec-
tion for all polarizations and apply it to the case of a
dielectric multilayer f ilm. Previous attempts to attain
high ref lectance for a wide range of incident angles in-
volved dielectric f ilms with high indices of refraction,
high special dispersion properties, or multiple contigu-
ous stacks of f ilms.5 – 9
A one-dimensional photonic crystal has an index
of refraction that is periodic in the y coordinate and
consists of an endlessly repeating stack of dielectric
slabs, which alternate in thickness from d1 to d2 and
in index of refraction from n1 to n2. Incident light

48. can be either s polarized (E is perpendicular to the
plane of incidence) or p polarized (parallel). Because
the medium is periodic in y and homogeneous in x and
z, the electromagnetic modes can be characterized by a
wave vector k, with ky restricted to 0 # ky # pya. We
may suppose that kz - 0, kx $ 0, and n2 . n1 without
loss of generality. The allowed mode frequencies vn
for each choice of k constitute the band structure of the
crystal. The continuous functions vnskd, for each n,
are the photonic bands.
For an arbitrary direction of propagation, it is conve-
nient to examine the projected band structure, which is
shown in Fig. 1 for a quarter-wave stack with n1 - 1
and n2 - 2. To make this plot we f irst computed the
bands vnskx, ky d for the structure, using a numeri-
cal method to solve Maxwell’s equations in a periodic
medium.10 (In fact, for the special case of a multi-
layer f ilm, an analytic expression for the dispersion
relation is available.11) Then, for each value of kx,
the mode frequencies vn for all possible values of ky
were plotted. Thus in the gray regions there are elec-
tromagnetic modes for some value of ky , whereas in
0146-9592/98/201573-03$15.00/0
the white regions there are no electromagnetic modes,
regardless of ky .
One obvious feature of Fig. 1 is that there is no
complete bandgap. For any frequency there exists
some electromagnetic mode with that frequency — the
normal-incidence bandgap is crossed by modes with
kx . 0. This is a general feature of one-dimensional
photonic crystals.
However, the absence of a complete bandgap does
not preclude omnidirectional ref lection. The criterion

49. is not that there be no propagating states within
the crystal; rather, the criterion is that there be
no propagating states that can couple to an incident
propagating wave. As we argue below, the latter
criterion is equivalent to the existence of a frequency
range in which the projected band structures of the
crystal and the ambient medium have no overlap.
The electromagnetic modes in the ambient medium
obey v - cskx 2 1 ky 2d1/2, where c is the speed of light in
the ambient medium, so generally v . ckx . The whole
region above the solid diagonal light lines v - ckx is
f illed with the projected bands of the ambient medium.
If a semi-inf inite crystal occupies y , 0 and the
ambient medium occupies y . 0, the system is no longer
periodic in the y direction and the electromagnetic
modes of the system can no longer be classif ied by a
single value of ky . These modes must be written as
Fig. 1. Projected band structure for a quarter-wave stack
with n1 - 1 and n2 - 2. Electromagnetic modes exist only
in the shaded regions. The s-polarized modes are plotted
to the right of the origin, and the p-polarized to the left.
The dark lines are the light lines v - ckx . Frequencies
are reported in units of 2p cya.
1574 OPTICS LETTERS / Vol. 23, No. 20 / October 15, 1998
a weighted sum of plane waves with all possible ky .
However, kx is still a valid symmetry label. The angle
of incidence u upon the interface at y - 0 is related to
kx by v sin u - ckx .

50. For there to be any transmission through the semi-
inf inite crystal at a particular frequency, there must
be an electromagnetic mode available at that frequency
that is extended for both y . 0 and y , 0. Such a
mode must be present in the projected photonic band
structures of both the crystal and the ambient medium.
(The only states that could be present in the semi-
inf inite system that were not present in the bulk
system are surface states, which decay exponentially
in both directions away from the surface and are
therefore irrelevant to the transmission of an external
wave). Therefore, the criterion for omnidirectional
ref lection is that there exist a frequency zone in which
the projected bands of the crystal have no states
with v . ckx .
In Fig. 1, the lowest two p bands cross at a point
above the line v - ckx , preventing the existence of such
a frequency zone. This crossing occurs at the Brewster
angle uB - tan21sn2yn1d, at which there is no ref lection
of p-polarized waves at any interface. At this angle
there is no coupling between waves with ky and 2ky , a
fact that permits the band crossing to occur.
This diff iculty vanishes when we lower the bands of
the crystal relative to those of the ambient medium by
raising the indices of refraction of the dielectric f ilms.
Figure 2 shows the projected band structure for the
case n1 - 1.7 and n2 - 3.4. In this case there is a
frequency zone in which the projected bands of the
crystal and ambient medium do not overlap, namely,
from the f illed circle svay2pc - 0.21d to the open
circle svay2pc - 0.27d. This zone is bounded above
by the normal-incidence bandgap and below by the
intersection of the top of the f irst gray region for
p-polarized waves with the light line.

51. Between the frequencies corresponding to the f illed
and open circles there will be total ref lection from
any incident angle for either polarization. For a f inite
number of f ilms the transmitted light will diminish ex-
ponentially with the number of f ilms. The calculated
transmission spectra for a f inite system of ten f ilms
(f ive periods) are plotted in Fig. 3 for various angles
of incidence. The calculations were performed with
transfer matrices.12 The stop band shifts to higher
frequencies with more-oblique angles, but there is a re-
gion of overlap that remains intact for all angles.
The graphic criterion for omnidirectional ref lection
is that the f illed circle be lower than the open circle
(the second band at kx - 0, ky - pya). Symbolically,
vp1
µ
kx -
vp1
c
, ky -
p
a
∂
, vp2
µ
kx - 0, ky -

52. p
a
∂
,
(1)
where vpnskx , ky d is the p-polarized band structure
function for the multilayer f ilm. Note that the left-
hand side is a self-consistent solution for frequency
vp1. The difference between these two frequencies is
the range of omnidirectional ref lection.
We calculated this range (when it exists) for a
comprehensive set of f ilm parameters. Since all the
mode wavelengths scale linearly with d1 1 d2 - a, we
need consider only three parameters for a multilayer
f ilm: n1, n2, and d1ya. To quantify the range of om-
nidirectional ref lection sv1, v2d in a scale-independent
manner, we report the range – midrange ratio, which is
def ined as sv2 2 v1dy1/2sv2 1 v1d.
For each choice of n1 and n2yn1, there is a value of
d1ya that maximizes the range – midrange ratio. That
choice can be computed numerically. Figure 4 is a
contour plot of the ratio, as n1 and n2yn1 are varied,
for the maximizing value of d1ya.
An approximate analytic expression for the optimal
zone of omnidirectional ref lection can be derived:
Dv
2c

53. -
a cos
µ
2
r
A 2 2
A 1 2
∂
d1n1 1 d2n2
2
a cos
µ
2
r
B 2 2
B 1 2
∂
d1
p
n12 2 1 1 d2
p
n22 2 1
,

54. (2)
Fig. 2. Projected band structure for a quarter-wave stack
with n1 - 1.7 and n2 - 3.4, with the same conventions as
in Fig. 1.
Fig. 3. Calculated transmission spectra for a quarter-
wave stack of ten f ilms sn1 - 1.7, n2 - 3.4d for three angles
of incidence. Solid curves, p-polarized waves; dashed
curves, s-polarized waves. The overlapping region of high
ref lectance s.20 dBd corresponds to the region between the
open and f illed circles of Fig. 2.
October 15, 1998 / Vol. 23, No. 20 / OPTICS LETTERS 1575
Fig. 4. Range – midrange ratio for omnidirectional ref lec-
tion, plotted as contours. For the solid contours the opti-
mal value of d1ya was chosen. The dashed curve is the
0% contour for the case of a quarter-wave stack. For the
general case of an ambient medium with index n0 fi 1,
the abscissa becomes n1yn0.
where
A ;
n2
n1
1
n1
n2
, B ;
n2

55. p
n12 2 1
n1
p
n22 2 1
1
n1
p
n2 2 2 1
n2
p
n1 2 2 1
.
(3)
In deriving Eq. (2) we assumed that the optimal f ilm
is approximately a quarter-wave stack. Numerically
we f ind this to be an excellent approximation for
the entire range of parameters depicted in Fig. 4; the
frequencies as predicted by this approximation are
within 0.5% of the exact frequencies. As a result the
optimization of d1ya results in a range – midrange ratio
very close to that which results from a quarter-wave
stack with the same indices: d1ya - n2ysn2 1 n1d.
In Fig. 3, the 0% contour for quarter-wave stacks
is plotted as a dashed curve, which is very close
(always within 2% in the indices of refraction) to the
numerically optimized contour.

56. It can be seen from Fig. 4 that, for omnidirectional
ref lection, the index ratio should be reasonably high
s.1.5d and the indices themselves somewhat higher
(by .1.5) than that of the ambient medium. The
former condition increases the band splittings, and
the latter depresses the frequency of the Brewster
crossing. An increase in either factor can partially
compensate for the other. The materials should also
have a long absorption length for the frequency range
of interest, especially at grazing angles, where the path
length of the ref lected light along the crystal surface
is long.
Although we have illustrated our arguments by
use of multilayer f ilms, the notions in this Letter
apply generally to any periodic dielectric function
ns yd. What is required is the existence of a zone of
frequencies in which the projected bands of the crystal
and ambient medium have no overlap.
However, the absence of a complete bandgap does
have physical consequences. In the frequency range
of omnidirectional ref lection there exist propagating
solutions of Maxwell’s equations, but they are states
with v , ckx and decrease exponentially away from the
crystal boundary. If such a state were launched from
within the crystal, it would propagate to the boundary
and ref lect, just as in total internal ref lection.
Likewise, although it might be arranged that the
propagating states of the ambient medium do not
couple to the propagating states of the crystal, any
evanescent states in the ambient medium will couple to
them. For this reason, a point source of waves placed
very close sd , ld to the crystal surface could indeed
couple to the propagating state of the crystal. Such
restrictions, however, apply only to a point source, and

57. one can easily overcome them by simply adding a low-
index cladding layer to separate the point source from
the f ilm surface.
This work was supported by the U.S. Army Research
Off ice under contract /grant DAAG55-97-1-0366.
J. N. Winn thanks the Fannie and John Hertz
Foundation for its support. We acknowledge useful
discussions with Pochi Yeh and Hermann Haus.
J. D. Joannopoulos’ e-mail address is [email protected]
mit.edu.
References
1. E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
2. J. D. Joannopoulos, R. D. Meade, and J. N. Winn,
Photonic Crystals (U. Princeton Press, Princeton, N.J.,
1995).
3. J. Pendry, J. Mod. Opt. 41, 209 (1994).
4. J. D. Joannopoulos, P. R. Villeneuve, and S. Fan,
Nature (London) 386, 143 (1997).
5. P. Baumeister, Opt. Acta 8, 105 (1961).
6. J. D. Rancourt, Optical Thin Films User Handbook
(SPIE Optical Engineering Press, Bellingham, Wash.,
1996), pp. 68 – 71.
7. P. Yeh, Optical Waves in Layered Media (Wiley, New
York, 1988), pp. 161 – 163.
8. K. V. Popov, J. A. Dobrowolski, A. V. Tikhonravov, and
B. T. Sullivan, Appl. Opt. 36, 2139 (1997).

58. 9. M. H. MacDougal, H. Zhao, P. D. Dapkus, M. Ziari, and
W. H. Steier, Electron. Lett. 30, 1147 (1994).
10. R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D.
Joannopoulos, Phys. Rev. B 77, 8434 (1993).
11. P. Yeh, A. Yariv, and C.-S. Hong, J. Opt. Soc. Am. 67,
423 (1977).
12. E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading,
Mass., 1987), pp. 373 – 378.
THE UNIVERSITY OF NEWCASTLE: SCHOOL OF
ENGINEERING
Assignment Cover Sheet
All assignments are the responsibility of the student. The
School of Engineering cannot
accept responsibility for lost or unsubmitted assignments. You
are therefore advised to
keep a copy.
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Date submitted: -------------------------------------------------

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for the Prevention and
Detection of Plagiarism Main Policy Document, which is
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Date: ------------------------------
MECH2450 Engineering Computations 2
Assignment #2 (2016) Singapore
Due Date: 9:30 pm, Thursday 31st March 2016.
Show all working clearly.
Q.1 (25 marks)
The size y (in millimetres) of a crack in a Pontiac Trans Am’s
front sub-frame weld is described by a
random variable X with the following PDF:

60. elsewhere
y
yy
yf X
0
3
1
3
3
1
1,3/1
3
1
10,16/3
)(

61. 2
(a) Sketch the PDF and CDF. (5 marks)
(b) Determine the mean crack size. (5 marks)
(c) What is the probability that a crack will be smaller than 3
mm? (5 marks)
(d) Determine the median crack size. (5 marks)
(e) Suppose there are four cracks in the weld. What is the
probability that only one of the four cracks
is larger than 3 mm? (5 marks)
Q.2 (25 marks)
The average speed of vehicles on a freeway is being studied.
Assume that the standard deviation of
vehicle speed is known to be 8 km/h.
(a) Suppose observations on 120 vehicles yielded a sample
mean of 105 km/h. Determine two-sided
99.5% confidence intervals of the mean speed. (Assume a
normal distribution). (6 marks)
(b) In part (a), how many additional vehicles’ speed should be
observed such that the mean speed
marks)
(c) Suppose Jason and Britney are assigned to collect data on
the speed of vehicles on this highway.
After each person has separately observed 60 vehicles, what is

62. the probability that Jason’s
km/h? (7 marks)
(d) Repeat part (c) if each person has separately observed 120
vehicles instead. (6 marks)
Q.3 (25 marks)
The fuel consumption of a certain make of car may not be
exactly that rated by the manufacturer.
Suppose ten cars of the same model were tested for combined
city and highway fuel consumption, with
the following results:
Car No. Observed fuel consumption
1 7.1 litres per 100km
2 6.4 litres per 100km
3 6.8 litres per 100km
4 6.2 litres per 100km
5 7.8 litres per 100km
6 6.1 litres per 100km
7 6.6 litres per 100km
8 7.8 litres per 100km
9 6.4 litres per 100km
10 7.4 litres per 100km
(a) Estimate the sample mean and sample standard deviation of
the actual fuel consumption of this
particular model of car. (10 marks)
(b) Suppose that the manufacturer’s stated fuel consumption of
this particular model of car is 7 litres
per 100km; perform a hypothesis test to verify the stated fuel
consumption with a significance
level of 2%. (15 marks)

63. Q.4 (25 marks)
The occurrence of bushfires in the Port Stephens area may be
modelled by a Poisson process. The
(event A1), 20 (event A2) or 25 (event A3)
times a year. The probability of A2 is the same as the
probability A1 and the probability of A3 is half of
A1.
(a) Determine the probability that there will be 20 occurrences
of bushfire in the next year.
(10 marks)
(b) If there are exactly 20 bushfires in the next year (event B),
what will be P(A1|B)? Determine
P(A2|B) and P(A3|B). (15 marks)
Q.5 (Additional Question – No need for submission)
Measurements of the velocity in a fully developed flow in a
circular pipe whose wall radius is R, gave
the following data:
r/R 0.0 0.102 0.206 0.412 0.617 0.784 0.846 0.907 0.963 1.0
u/Uc 1.000 0.997 0.988 0.959 0.908 0.847 0.818 0.771 0.690
0.000
where r is the radius at which the velocity is measured, u is the
velocity, and Uc is the centerline
velocity. For R = 12.35 cm, and Uc = 30.5 m/sec, estimate the
average velocity, Uave, in the pipe as
defined by:
2 0 0

64. 2
2
R R
ave c
c
u
U urdr U d
R U
(a) Explain why the Trapezoidal Rule is the most appropriate
rule to use for the integration and why
it is likely to under-estimate the value of Uave. Give your
answer to two decimal places.
(b) Check your answer by writing a Matlab program to perform
the trapezoidal integration. The data
is given in the data file pipeflow.dat. The easiest way to read a
data file into a Matlab program is to use
the ‘load’ command to fill an array with the data for u/Uc and