Rutgers Science Review -- Spring 2013

Rutgers Science Review Volume 2, Issue 2 Spring 2013

The Life of a Star Curiosity on Mars: The Journey to Mt. Sharp

An Interview With: Dr. Edward Castner

Cover Art: Bo Tang

Table of Contents Curiosity on Mars: The Journey to Mt. Sharp

pg 5

The Life of a Star

pg 12

An Interview With Dr. Edward Castner

pg 16

Momentum Effects on European Call Options Pricing – Questioning the Markov Chain Assumption

pg 19

Design and Fabrication of Multilayer Thin Film Coated Hollow Waveguides for Enhanced Infrared Radiation Delivery

pg 35

Ice Caves in Mutnovsky Volcano, Kamchatka, Russia

About

The Rutgers Science Review (RSR) biannually publishes student-written scientific features, opinions, and research papers.

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Staff

Editor-In-Chief Jonathan Shao Features Editors Alexandra DeMaio Deepak Gupta Scott Kilianski Arvind Konkimalla Stephanie Marcus Robert Svechin

Design Editors Lynn Ma Courtney Connolly Lauren Fish Bo Tang Business Manager Chris Xia Web Developer Lawrence Xie Faculty Advisor: Dr. Steven Brill

Nyiragongo Volcano, Congo

Articles

Features

Curiosity on Mars:

The Journey to Mt. Sharp

By Margaret Morris

Nine months ago on August

Martian air in search of methane,

55 days after it first landed on the

6, 2012, NASA successfully landed the

and — very recently — has drilled

planet, or Sol 55 (a Sol, equivalent to

Mars Science Laboratory (MSL) — also

beneath the planet’s surface rock,

1.027 Earth days, is the time it takes

known as Curiosity — at Gale Crater

collecting information in its scoop that

for Mars to revolve around the Sun)

on Mars. Its mission was to answer a

was once unobtainable. Many of these

(Figure Sol). The Rocknest area was

single captivating question: Could

analytical techniques have never been

chosen for its sand-like material, which

Mars have once harbored life?

used outside of planet Earth1. With

is the optimal size for two analytical

Since its landing, Curiosity

its drilling in February, Curiosity has

instruments called CheMin and SAM

has travelled roughly one third of a

successfully used every tool it was

(see chart). Here, the rover would take

mile, stopping to explore geologically

sent to Mars for at least once2. The rest

several scoops from the Martian soil:

interesting sites on the way to its final

of its mission will employ the same

some for practice, some for cleaning

destination, Mount Sharp. It currently

laser-shooting, scooping, and drilling

out the instruments, and two for actual

resides at Yellowknife Bay, still over 2.5

operations to determine if the Red

analysis3.

miles from the mountain (Figure Path).

Planet has ever provided the proper

Curiosity is armed with an impressive

conditions for life.

Chemistry and Mineralogy instrument,

suite of analytical instruments. It has

In early October, Curiosity

or CheMin, was the first completed and

fired lasers at rocks for analysis, gulped

arrived at Rocknest. This marked

on October 30, the world was shown

The

analysis

from

the

Spring 2013 | Rutgers Science Review | 5

Features the very first X-ray diffraction of minerals on Mars. The data gained from CheMin are essentially identical to data obtained using X-ray diffraction much closer to home. X-ray diffraction determines the crystalline mineral make-up of a soil or regolith (Figure Regolith).

The small-grained soil is shaken 2000 times per second in the CheMin instrument, while a laser shoots a beam of X-ray light – light of a shorter wavelength, and therefore greater energy, than visible light – through the rapidly vibrating sample. When this beam Figure Path: Curiosity landed at Bradbury Landing on August 6, 2012, and the white line tracks its path up until November.The CheMin results from October were obtained at Rocknest, and the drilling in February was done at Yellowknife Bay.

hits each crystalline mineral, it bends at a distinct angle and a sensor behind the soil sample records the diffraction of the laser beam as the crystals bounce around in the instrument4.

Since these minerals are shaken into

different positions, the diffracted X-rays form a circle. This creates a fingerprint identification

system

that

allows

each

mineral to be identified by its unique diffraction pattern5. Figure CheMin shows the fingerprints of a scoop of Rocknest soil on Mars. Among the most abundant minerals Figure Sol: A Sol is a Martian day. It takes Mars 24 hours and 40 minutes to turn on its axis, and 667 Sols (~687 Earth days) to complete an orbit around the Sun.

are feldspar, pyroxene, and olivine: minerals commonly found in rocks on Earth that have formed through volcanic processes. Although there are similar rocks on Earth, the rocks on Mars might not necessarily have formed the same way and more information from Curiosity’s tests should further illuminate Mars’ geological history6.

Figure Regolith: Regolith is like soil, but without microbes – all the ‘soil’ on Mars is technically regolith, but it is commonly referred to as both, even by NASA scientists.

6 | Rutgers Science Review | Spring 2013

One week later, MSL scientists

completed the first Curiosity analysis of the Martian atmosphere with the Sample

Features Analysis at Mars instruments (SAM)7.

from microbes, which turned carbon

SAM

Because

a

planet’s

atmosphere

dioxide and hydrogen into water and

they found, as shown in Figure SAM.

provides

for

a

cycle,

UV

methane (CO2 + 4H2 -> 2H2O + CH4).

While there were no definitive traces

absorption, and the greenhouse effect,

Other non-biological processes that

of methane, this does not rule out the

analysis of Mars’ atmosphere (only 1%

could produce methane on Mars —

current or past presence of microbial

the size of Earth’s)8 provides valuable

volcanoes, residue from meteors, and

life. Curiosity will continue its analysis

information regarding its habitability.

various chemical interactions — would

of the atmosphere and, over the next

produce negligible amounts compared

few months, conclude with much more

two instruments in the SAM suite to

to microbial life-forms9.

certainty whether or not methane has

answer two different questions: First,

This means that finding methane on

been found11.

what is the chemical composition

Mars could indicate that microbes

of the Martian atmosphere? And

once lay beneath Curiosity’s wheels.

the ratio of isotopes of certain elements

second, what is the isotope frequency

Current methane production on Earth

— can tell us what happened to Mars’

distribution of each chemical?

from microbes and animals combined

atmosphere over time. An isotope is a

The first question would tell

is responsible for only a millionth

version of an element determined by

us very little about the possibility of

of a percent of our atmosphere,

its number of neutrons. Every element

life, except in one scenario: if Curiosity

and it would be far less abundant if

has several isotopes to its name, and

found methane. In the first few billion

biological processes were not present10.

their average mass is what is shown on

years of the Earth’s existence, single-

Therefore, Curiosity would need only

the periodic table.

celled organisms were the only form

to detect the slightest trace of methane

of life. The only organic waste came

for the result to be significant. The

found that heavier isotopes were

water

The Curiosity scientists used

scientists

summarized

what

The second question — finding

A mass spectrometer in SAM

Figure CheMin (left): These X-ray diffraction patterns from the CheMin instrument identify feldspar, pyroxene, and olivine, which are commonly found in volcanic rocks here on Earth.The sample was taken in October from a site called Rocknest.

Figure SAM (above): These are the five most abundant gases and their percentage of abundance in the Martian atmosphere as determined by a mass spectrometer in SAM, the Sample Analysis at Mars suite of instruments in October.

Spring 2013 | Rutgers Science Review | 7

Features significantly more abundant than the

break, the rover explored the bay,

January, John Grotzinger, Curiosity’s

lighter isotopes for carbon and oxygen.

following commands it had been given

Project

This indicates one very important

before the break. When the scientists

attraction of such a site:

observation about the history of Mars’

and engineers returned to work in

atmosphere: it is shrinking12. Mars

January, they planned to drill in an area

is that…water moved, it percolated

used to have a liquid molten iron

named “John Klein” in honor of the

through these rocks, through these

core, which created a magnetic field

MSL Deputy Project Manager who had

fractured networks, and then minerals

as it spun with the planet — just as

worked on multiple Mars missions,

precipitated to form the white material

Earth has to this day. When the iron

including earlier stages of Curiosity,

which

core cooled, Mars lost its magnetic

before passing away in 2011. The Mars

Camera) has concluded is very likely

field, which had protected it from

site John Klein yields a diverse offering

a Calcium Sulfate… So this is the first

particles released from the Sun. The

of sedimentary rocks, exactly what was

time in this mission that we have seen

unhindered particles were then free to

needed for Curiosity to begin digging

something that is not just an aqueous

strip off molecules from the top layer

into Mars’ past15.

environment, but one that also results

of the atmosphere — a process which

Curiosity’s

disproportionately affects the lighter

of John Klein and other areas of

isotopes. Future tests from Curiosity

Yellowknife Bay reveal something vital

designed to analyze the make-up of

will attempt to determine the rate at

about the significance of the drilling

Martian rocks from a distance, using

which the atmosphere is thinning out,

site: it probably

indicating how thick it was and thus its

once lay under

compatibility with life at certain points

running water16.

in Mars’ history13.

Figure John Klein

Around

Sol

130,

Earth’s

is an image of

mid-December, Curiosity arrived at

a possible spot

Yellowknife Bay. The area was selected

within the site

by the Curiosity team to be the first

for

drilling target. For the first time,

to

said Louise Jandura, Sample System

channels known

Chief Engineer at the Jet Propulsion

as

Laboratory (JPL)14, we would see

found

“beyond the surface layer of the rock,

the rocks, and a

unlocking a time capsule of evidence

few visible, small

about the state of Mars going back 3 or

white

4 billion years.”

filled

For much of December, while

In a publicized

the Curiosity team went on holiday

teleconference in

8 | Rutgers Science Review | Spring 2013

photographs

Scientist,

explained

the

“What these vein fills tell us

ChemCam

(Chemistry

and

in precipitation of minerals.”17 ChemCam is an instrument

Curiosity drill.

veins

Small

were

between

streaks them

in.

Figure John Klein: This photograph taken of a possible drilling site became one of great interest to the Curiosity team because of veins that are visibly filled with a white substance.They indicate the past presence of running water and precipitation of minerals. ChemCam has determined that the white substance is most likely Calcium Sulfate.

Features Figure Results: The sample was divided up and delivered to different instruments on Curiosity.

The fourth portion was heated up in SAM, the Sample Analysis at Mars instruments, and the gases released were identified by SAM’s mass spectrometer.

The chemicals found and shown above suggest the presence of hydrated minerals, carbonates, perchlorates, sulfates and sulfides, and clays in the drilling sample.

pulses from a focused laser beam. The

is 6.4 centimeters deeper into Mars

in a neutral pH environment, a result

lasers are so intense that they bore

than anyone has ever seen19. However,

that no other previous observations on

holes in the targeted rocks. The energy

shortly after the powder collected

Mars had suggested before21. No iron

of the laser beam is then absorbed

from drilling was placed into the

sulfates were found, indicating that

by electrons in the rock molecules,

rover, a computer glitch in the rover’s

the iron oxidization did not occur far

sending them into an excited state.

main computer system delayed the

below the surface, and that the sample

The electrons, wanting to return to

sample’s analysis. The team has not

was likely not acidic. “The big story is

the ground state, release this energy

yet discovered the cause of the glitch,

in the powder that’s generated” from

back as photons to be picked up by a

or how to fix it, but quickly switched

the drilling, said Grotzinger. “We get to

detector on the rover; the recorded

to one of three back-up computers

see the new Mars, the grey Mars.”

frequencies of those waves are then

stored on-board20. The sample was

used to identify the type of molecules

then delivered to CheMin and SAM for

Mars’ found evidence of water, carbon

that were in those rocks. ChemCam

analysis.

dioxide, sulfates (SO42-) and sulfides

can do this from as far as 23 feet away18.

Finally on March 12, nearly one

(S2-): all more than enough ingredients

After 25 APXS analyses, more

month after the sample was collected,

to say that Mars was at some point in its

than 100 MAHLI images, over 12,000

four key members of the MSL team

history, and in some places, habitable22

laser shots from ChemCam, and weeks

held a news conference at NASA’s

(Figure Results). Here on Earth, sulfates

of testing the drilling apparatus, a site

headquarters in Washington, DC to

and sulfides would have been energy

within John Klein called ‘Drill’ was

announce the long-awaited results.

sources for early organisms, which

officially selected as Curiosity’s first

At this conference, Grotzinger stated

used the charge difference between

drilling point. Curiosity drilled 6.4

that beneath the iron-crusted surface,

the negatively-charged sulfates and

centimeters into Martian rock, which

Curiosity found clays that had formed

the neutral environment as a source of

The results from ‘the new

Spring 2013 | Rutgers Science Review | 9

Features energy – just like a common battery23.

For the rest of its tenure, Curiosity will

The combined findings of the water,

be analyzing various sites for their

sulfates, and a neutral pH environment

habitability from John Klein to Mt.

answer, at last, the question of whether

Sharp, and taking more samples from

Mars could have ever harbored (very

the atmosphere. Although it is officially

small) life: the answer is yes24.

set for one Martian year, or 687 Earth

The month of April is a quiet

days, the mission can be extended by

one for Curiosity. The orbits of Earth

the rover’s 14-year battery27. And the

and Mars are situated so that they

Curiosity scientists and engineers will

are on direct opposite sides of the sun

want to use that extra time. “We all had

from each other. This is called a solar

intended Curiosity to be a discovery-

conjunction, and happens about once

driven

every two years for Earth and Mars25

Vasavada, Curiosity’s Deputy Project

(Figure

this

Scientist28. “It’s not that we like Mt.

time, communications with the rover

Sharp any less, it’s just that studying

will be cut off, and the Curiosity team

this area on the floor has become a lot

will take a break from Mars and work

more interesting than we expected.”

Conjunction).

During

mission,”

said

Ashwin

Figure Conjunction: Once every 26 months, the orbits of Earth and Mars line up so that the Sun is almost directly in between them. Since the Sun will interfere with communications to the rover, during this time Curiosity will be following commands that were given to it in March.

NASA will be able to communicate with the rover again in late April.

on Earthly tasks for Curiosity. When communication is restored at the end of April, Curiosity will take another drilling sample and repeat the tests to confirm the initial results26.

Now nine months into the

mission, Curiosity has tested out all of its sampling and analytical instruments.

10 | Rutgers Science Review | Spring 2013

These are several of the instruments employed on the Curiosity rover, and the abbreviations that NASA uses for them.

Features Works Cited 1. NASA. Jet Propulsion Laboratory. NASA Mars Rover News: Oct. 30. Ustream.

N.p., 30 Oct. 2012. Web. <http://www.ustream.tv/ recorded/26549125>.

2. NASA. Jet Propulsion Laboratory. NASA Mars Rover News: Jan. 15, 2012. Ustream.

N.p., 15 Jan. 2013. Web. <http://www.ustream.tv/ recorded/28512078>.

3. c.f. 1 4. c.f. 1 5. “Chemistry & Mineralogy X-Ray Diffraction (CheMin).” Mars Science Laboratory. NASA JPL, n.d. Web. <http://mars. jpl.nasa.gov/msl/mission/ instruments/spectrometers/ chemin/>.

6. c.f. 1 7. Webster, Guy, Nancy Neal Jones, and Dwayne Brown. “NASA Rover Finds Clues to Changes in Mars’ Atmosphere.” Mars Science Laboratory. NASA, 2 Nov. 2012. Web. <http://www. nasa.gov/mission_pages/msl/ news/msl20121102.html>.

8. NASA. Jet Propulsion Laboratory. NASA Mars Rover News: Nov. 15. Ustream.

N.p., 20 Feb. 2013. Web. <http://www.ustream.tv/ recorded/27047602>.

9. NASA. Jet Propulsion Laboratory. NASA Mars Rover News: Nov. 2. Ustream.

NASA JPL, 2 Nov. 2012. Web. <http://www.ustream.tv/ recorded/26637100>.

10. “Methane.” EPA. Environmental Protection Agency, 22 June 2010. Web. 10 Mar. 2013. <http://www.epa.gov/ outreach/scientific.html 11. c.f. 9 12. c.f. 9 13. “Dr. Ashwin Vasavada, MSL Deputy Project Scientist.”

Telephone interview. 4 Apr. 2013.

14. NASA. Jet Propulsion Laboratory. NASA Mars Rover News: Feb. 20, 2013. Ustream.

N.p., 20 Feb. 2013. Web. <http://www.ustream.tv/

recorded/29432878>.

15. c.f. 2 16. c.f. 2 17. c.f. 2 18. “Chemistry & Camera (ChemCam).” Mars Science Laboratory: Curiosity Rover.

NASA JPL, n.d. Web. <http://mars.jpl.nasa.gov/ msl/mission/instruments/ spectrometers/chemcam/>.

19. c.f. 14 20. Webster, Guy. “Computer Swap on Curiosity Rover.” Mars Science Laboratory. NASA, 28 Feb. 2013. Web. <http://www.nasa. gov/mission_pages/msl/news/ msl20130228.html>.

21. NASA. Jet Propulsion Laboratory. Mars Curiosity Rover Update from the Lunar and Planetary Science Conference. Ustream.

N.p., 18 Mar. 2013. Web. <http://www.ustream.tv/ recorded/30076916>.

22. c.f. 21 23. c.f. 21 24. Agle, DC, and Dwayne Brown.

“NASA Rover Finds Conditions Once Suited for Ancient Life on Mars.” Mars Science Laboratory. NASA, 12 Mar. 2013. Web. <http://www. nasa.gov/mission_pages/msl/ news/msl20130312.html>.

25. Webster, Guy. “Sun in the Way Will Affect Mars Missions in April.” Jet Propulsion Laboratory. NASA, 20 Mar. 2013. Web. <http://www. jpl.nasa.gov/news/news. php?release=2013-108>.

26. c.f. 21 27. Haar, Judy. “Curious About Curiosity? Getting Around Mars with Nuclear Power.” Decoded Science.

Wordpress, 14 Aug. 2012. Web. <http://www. decodedscience.com/nuclearpowered-curiosity-onmars/16864>.

28. c.f. 13

Image Sources Path http://www.nasa.gov/mission_pages/ msl/multimedia/pia16577.html Sol http://mars.jpl.nasa.gov/msl/ multimedia/images/?ImageID=3550 Regolith http://www.nasa.gov/mission_pages/ msl/multimedia/pia16174.html CheMin http://www.nasa.gov/mission_pages/ msl/multimedia/pia16217.html SAM http://www.nasa.gov/mission_pages/ msl/multimedia/pia16460.html John Klein http://www.nasa.gov/mission_pages/ msl/multimedia/pia16705.html Results http://www.nasa.gov/mission_pages/ msl/multimedia/pia16817.html Conjunction http://solarsystem.nasa. gov/multimedia/display. cfm?Category=Planets&IM_ID=16185

For Information and News on the Curiosity Rover, visit: mars.nasa.gov/msl

Spring 2013 | Rutgers Science Review | 11

Features

The Life of a Star by: Aditya Parikh Astronomy, the study

gravitational

attraction

of celestial objects, began

alone is not strong enough

two thousand years ago

to trigger this process,

in the era of Ptolemy.

physicists reasoned that

With the emergence of

there must be another

new technology and the

factor at work.

development of quantum

Quantum

mechanics,

mechanics

astronomers

easily resolved this issue

have since been equipped

with the hypothesis of

to

explain

quantum

of

exotic

Despite

a

variety

phenomena.

tunneling.

Classically,

the

addressing

electromagnetic repulsion

events on the scale of

between protons creates

picometers,

a

it

aptly

explains observations of astrophysical giants such

Image taken with NASA’s Hubble Space Telescope of Sirius A and Sirius B, a white dwarf.

as stars, black holes, and

the energy of any system reaches its

supernovae. In the late 20th century, quantum mechanics and astrophysics were amalgamated in the study of stellar evolution. These new ideas explained a number of observations that

were

deemed

classically

impossible, and shed great insight into the birth of a star.

The formation of a star begins with the condensation of celestial clouds of hydrogen gas. This occurs because

absolute minimum in the collapsed phase where all particles fall to one point (arXiv article). As the cloud condenses, its internal temperature rises – but not to the extent necessary for atomic fusion to occur. Classically, proton-proton

fusion

can

only

result when the particles overcome electromagnetic repulsion and get close enough for the strong force to take over (within approximately 3 fm). Because

12 | Rutgers Science Review | Spring 2013

potential

which

barrier

prevents

them

from getting too close.

Quantum

mechanical

analysis, however, reveals that there is a small nonzero probability (as defined by a proton’s wave function) that a proton will appear on the other side of the barrier — close enough for strong interaction to trigger fusion. Although tunneling is a rare event, much fusion can occur through this process due to the sheer number of particles in a gas cloud. The energy released heats the cloud until its temperature and

Features pressure are great enough for fusion

and the outer layers are ejected in a

blows the star apart: a supernova

to occur without quantum tunneling.

planetary nebula. After the red giant

explosion. The constituents of the core

Quantum tunneling still proceeds,

phase, massive stars (larger than about

are crushed together. The ultimate fate

but at this point, the particles have

5-8 solar masses) form progressively

of the star is one of three possibilities,

enough kinetic energy to overcome

heavier elements. Fusion continues

determined by the coreâ&#x20AC;&#x2122;s mass. If

electromagnetic repulsion on a large

until the star reaches an iron core. At

the core is less than about 1.4 solar

scale, effecting a continuous fusion

this point, the binding energy per

masses, it will become a white dwarf:

cycle. This marks the birth of the star.

nucleon is highest, and additional

a dense body consisting of electrons

Fusion continues until a starâ&#x20AC;&#x2122;s

fusion is energetically unfavorable (see

squeezed together until they are nearly

hydrogen reserves run out, at which

Figure 1). Although fusion continues

forced to occupy the same quantum

point gravity once again takes over and

in the outer layers of the star, the main

state. The Pauli Exclusion Principle

causes the core to contract. The excess

reactions in the core slow down and

prevents fermions (particles with half-

heat that is generated causes the outer

stops; the star begins to die.

integer spin) from occupying the same

layers of the star to expand and cool.

After fusion stops, the outward

quantum state causing, which results

The core heats up, and helium fuses

radiation pressure that the reactions

in a phenomenon known as electron

to produce carbon. The fate of the star

had supplied is no longer present to

degeneracy pressure. This outward

is now inextricably linked to its mass.

counteract gravity. The star collapses

pressure halts the inward gravitational

For low mass stars, the gravitational

inward, but the iron atoms repel each

collapse and stabilizes the white

pull of the core is not strong enough

other, causing a shockwave which

dwarf. If the core is between 1.4 and

Figure 1: Binding Energy vs. Mass Number Spring 2013 | Rutgers Science Review | 13

Features 3.2 solar masses, electron degeneracy pressure is not strong enough to halt gravitational collapse and the

Submit to the RSR!

star continues to collapsing inward until electrons and protons fuse to produce neutrons. These neutrons are also subject to the Pauli Exclusion Principle and therefore analogously exhibit neutron degeneracy pressure.

This pressure, which is greater than electron degeneracy pressure, halts the gravitational collapse and produces a neutron star. If the core is greater than 3.2 solar masses, its gravitational collapse continues without succumbing to

degeneracy

pressure,

and

its

ultimate fate is to become a singularity,

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also known as a black hole.

Work Cited: Destri, C. and de Vega, H.J. â&#x20AC;&#x153;Dilute and Collapsed Phases of the SelfGravitating

Gasâ&#x20AC;?

Nuclear

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Interview An Interview With:

Dr. Ed Castner Conducted by Stephanie Marcus Dr. Edward Castner is a Professor of Chemistry at Rutgers University. His research group’s focus involves ionic liquids, and is funded in part by the U.S. Department of Energy and the National Science Foundation.

Q: What was your first experience in research, and how

in sodium chloride? Because it is a crystal. You would

did you get involved?

have to melt it; and sodium chloride melts at about 801 degrees. NaCl makes a beautiful, tightly packed rock

A: I was a junior at the University of Rochester and my

salt structure through strong interactions. If you were to

physical chemistry professor, David Perry, suggested that

replace sodium with larger molecules such as ammonium,

I might do research in his lab over the summer. He had

tertramethylammonium, or tetrabutylammonium,

just built a sophisticated laser device called an optical

the average distance between the ionic centers would

parametric oscillator, so we had tunable frequencies

increase and the attractions would weaken. By making

of infrared light. I decided to build an acousto-optic

these larger molecules asymmetric and flexible, the

cell in order to detect the absorption of infrared light

compound will crystalize poorly or not at all. Most of the

and molecular vibrations by “hearing” the molecules. I

ionic liquids that we make are liquid at or below room

remember contacting a very high level scientist at AT&T

temperature.

Bell Labs. He was stunned that this junior was calling him

Ionic liquids are of great interest for a variety of

up, but I just went ahead and asked him all the details,

applications such as next gen lithium batteries, solar-

which were crucial in building my device, since he had

photoelectrochemical cells, and ultra-capacitors for

the best published material at the time.

temporary charge storage. Think about a solar cell on a grid like the Livingston campus. What happens when

Q: Your lab is currently focused on studying ionic

the sun sets and the solar cells stop producing power?

liquids. Can you tell us what those are and some of

Does everyone stop using their computers and washing

their uses?

machines? No. We switch back to the other grid. You need to have energy stored up, and an ultra-capacitor is

A: Ionic liquids are molten salts that are liquid at room

how you do that. It is also how you regenerate the kinetic

temperature. Think about salts. What is our favorite salt?

energy in your Prius. Current capacitors work, but we

Sodium chloride. Why don’t we do chemical reactions

can make them three, even four times more efficient by

16 | Rutgers Science Review | Spring 2013

Interview creating better ionic fluids.

Ionic fluids can also help produce exceptionally clean

be an editor.

The greatest thing about being an editor is that I get

fuels by dissolving biomass. Have you ever tried to

the first look at the latest research. It is the most fantastic

dissolve cellulose? It is pretty hard. How about silk?

thing when you see something that will change the way

Ionic liquids can dissolve silk and cellulosic biomass on

we think about a field. Not only do we get to read the

contact. They also permit enzymes to maintain their

papers first, but we also alert the American Institute of

folded, 3D structures; often encouraging the enzymes to

Physics, our publisher, and say, “This is really hot, I

work with greater efficiency and with higher catalytic

want you to call the guy at The New York Times science

turnover. This allows you to dissolve biomass and

Tuesday and see if you can get it in there.” We get to

then use enzymes, like cellulase, to break it down into

publicize it as well as publish it.

something that you are ready to make fuels with.

Q: What advice do you have for students interested in Q: How have undergraduates contributed to your lab

starting research or who are currently in laboratories?

and what directions have they pursued?

A: You’ve got to steel yourself for real life. You’re going A:

Undergraduates have worked on a number of

to have to apply for not one or two or three jobs, but eight

different things here. Mostly our students want to

or ten or twenty. Make a list of different faculty members

get their hands wet with ionic liquids. Many of these

whose groups you might be interested in joining and talk

students go on to graduate or professional programs;

to all of them. Send them an email, with a CV. Make sure

quite a few go on to PhD programs in chemistry. We have

everything is spelled correctly. Think of it as cold calling

some at Rutgers as well as other state universities nearby;

for a job that hasn’t been advertised, but recognize that

Maryland, Penn State, and Delaware. A number go to the

you are at a research university. We love having students,

most prestigious research institutions. There are students

but at the same time, we are only able to take so many

at Stanford, Harvard, and Penn right now. One 2003

students for various reasons. You have to start early and

alumn became a professor at The College of New Jersey.

ask often.

There are two ways that people can think about

Q: You are also an editor for the Journal of Chemical

research. One is that it can be an aggressive jump-start on

Physics. How did you become involved with the

their professional career, while others make use of a more

journal, and what is your favorite aspect of this role?

diversified approach. Becoming involved in different fields or labs from summer to school year has helped

A: I was invited by one of the long-standing editors,

some students become incredibly successful in graduate

Don Levy, who is now Vice-President for Research

school. When they arrive at Stanford or Harvard they

and for National Laboratories at The University of

know exactly what they want to do and they can make a

Chicago. I know Don because of my graduate studies in

strong case to their prospective advisors.

Chicago. One night during the summer of 2006, I was at a conference in Yokohama, Japan. I got a phone call and I knew I wasn’t going to get far with my two-dozen words of Japanese. Well, it turned out Don Levy tracked me down in Yokohama and was calling to ask me if I would

Spring 2013 | Rutgers Science Review | 17

Research

Research Le, Lee & Patel Rutgers University

Momentum Effects on European Call Options Pricing – Questioning the Markov Chain Assumption Timothy Le, Daniel J. Lee, Parth Patel Advised by: Professor Rong Chen

Rutgers Aresty Research, Rutgers Science Review Rutgers University, New Brunswick

ABSTRACT We examine the mathematical Black-Scholes pricing model used by public corporations to value European-style call options issued to employees. European call options are financial securities that give holders the right but not the obligation to purchase a specified stock on a specified future date at a specified price. The Black-Scholes model assumes that stock prices have Markov-chain properties of “memorylessness”—the price at time t is directly based on the price at time t-1 only. We question this assumption by investigating whether the implied log-normal distribution of stock returns given by the Black-Scholes model changes in the presence of stock returns momentum, here defined as consecutive positive (or consecutive negative) returns for the time periods t-1, … , t-n for some number of time periods n > 1. An analysis of daily returns from SPY, a tradable security whose performance mirrors that of the S&P 500 index of U.S. stocks, strongly suggests that the distribution is indeed different from that implied by the BlackScholes model in cases with n = 3 days momentum. We also built an alternative Monte Carlo valuation model that accounted for the effects of momentum to value SPY European call options for various valuation dates. We compared the “true value” of the SPY options using known postvaluation date returns with the valuations given by our model and the Black-Scholes model. Our results indicated that on average, the Black-Scholes model still yielded a more accurate valuation, though our model was on average only 1.74 % less accurate.

1 Spring 2013 | Rutgers Science Review | 19

Research Le, Lee & Patel Rutgers University I.

INTRODUCTION

Background: For many publicly-listed companies based in the United States as well as overseas, stock call options comprise a significant portion of employee compensation packages. While there are numerous different types of options, each with their own nuances, employee compensation options often closely resemble European options.

European call options are essentially contracts granting the holder of the options the right but not the obligation to purchase a specified number of shares of a specified publicly-traded company at a specified price (called the strike price) at an expiration date, which is some specified date in the future, but not before or after that date. Thus, the option has a positive value on expiration day if the price at which the financial markets are trading the stock on said day is higher than the contractual strike price, since this enables the holder of the option contract to exercise the option—he may purchase the stock at the strike price and sell it immediately in the open market at the higher market price, realizing the difference in the prices as profit. European options thus have no maximum bound on their value, and have a minimum of zero value, which occurs when the market price on expiration day is lower than the strike price. In this case, the holder can choose to simply do nothing, not exercising his right to buy, as it would make no rational sense to purchase the stock at a higher price than that which the market is currently trading the stock at.

Due to the nature of such options, their value is obviously positively correlated with the price of a company’s stock, and the bestowment of options to employees as part of their compensation packages thus serves to incentivize them into making the company more prosperous, which will likely result in a higher stock price, and consequently a higher value of their options.

However, as a discouragement against “window-dressing”, or the practice of implementing company policies that result in a short-term increase in company profitability at the expense of long-term profitability (eg. firing large numbers of employees to decrease short-term costs and increase short-term profits), most firms have lock-up periods. These are periods of time, for instance, three to five years from the issuance of the call options, during which the employee cannot exercise or sell his options. This serves as a check against window-dressing, and an incentive for the employees to take actions which result in long-term growth and prosperity for the company, since their options heavily depend on the stock price three or five years from the date of issuance. Hence, such options resemble European call options, which allow holders to exercise only on some future expiration date.

Black-Scholes Pricing Model: Since European call options are dependent upon the price of a stock at some future date, which is of course unknown due to the uncertainty of the financial markets, they must be valued according to some expected value of the stock price in the future. To this end, professors Fischer Black, Myron Scholes, and Robert Merton formulated the Black-Scholes options pricing model, a mathematical model that gives the expected value of a European call option on the date of 2 20 | Rutgers Science Review | Spring 2013

Research Le, Lee & Patel Rutgers University issuance under certain assumptions. Due to the similarity of employee compensation options to European call options, the Black-Scholes model is also currently widely used in the corporate world to price the options issued to employees.

While this paper will not extensively detail the intricacies of the Black-Scholes model, we will examine certain implications and proposed limitations of the model.

The model was derived from the idea that one could, under certain assumed conditions, use a combination of stocks and bonds to fully replicate the effect of a European option. A careful processing of this idea yields a partial differential equation which relates an option’s value to the value or price of the underlying stock that it gives the holder the right to purchase. Further analysis yields the following Black-Scholes formula for the price of a European call option: (

)

(

) (

)

(

)

( ) ∫

√ (N(a) is the cumulative distribution function of a standard normal random variable) ( )

(

√

√

)

(

√

)

From the derivation of Eq. 1.1, which we largely omit here, the following assumptions are made: 1. Constant parameters (r, ), which we can obtain from historical data prior to time 0 2. Markov chain “memoryless” properties of stock prices The second property, the stock price being Markov chain “memoryless” can be defined as: (

|

)

is a random variable denoting the stock price at time n and where value that the price took on.

(

|

)

represents some actual

3 Spring 2013 | Rutgers Science Review | 21

Research Le, Lee & Patel Rutgers University This property results directly from the model for a stock price, which assumes the price can be modeled as a Brownian motion. This property is also the quintessential definition of a Markov chain in probability theory. In this paper, we will question this assumption, and propose that market momentum has an effect on the returns of a stock, and hence its price.

We can define momentum for n days in the following way: For a stock price at time t, the stock exhibits positive momentum if

Similarly, for a stock price at time t, the stock exhibits negative momentum if

The Black-Scholes model implies that the probability distribution of stock prices is as follows:

where

((

)

)

(

(

)

)

So the distribution of stock returns is: (

)

((

)

)

Since the distribution of stock returns as implied by Black-Scholes is derived from the implied distribution of stock prices, if momentum were to have effects that altered the distribution of stock returns, it would also necessarily alter the distribution of stock prices, and the implied distribution of stock prices as assumed by Black-Scholes would be incorrect. Thus, the BlackScholes valuation model would be incorrect, and the Markov chain assumption would no longer 4 22 | Rutgers Science Review | Spring 2013

Research Le, Lee & Patel Rutgers University be true, since the price at time t would now be influenced by returns, and hence prices, at times t1, t-2, … , t-n as well.

Research Question(s): Our primary research question reflects our aforementioned considerations of the Black-Scholes model. We specifically wish to see if the presence of momentum does indeed affect the probability distribution of stock returns such that the distribution is markedly different from the implied distribution of the Black-Scholes model. This could very well be the case, since momentum trading is a widely-used trading strategy in the financial markets. Positive or upward momentum may cause traders to follow the trend up by being more likely to buy a stock, or perhaps cause traders to predict a reversal and therefore make them more likely to sell the stock.

In any case, it may render the distribution different from that implied by Black-Scholes.

We will also devise our own valuation model which incorporates the effects of momentum on the probability distribution of returns. We will then compare this model to the Black-Scholes model and attempt to make conclusions regarding the increased accuracy or lack thereof of our model vis-à-vis the Black-Scholes model.

Relevance of Research: Due to its simplicity, Black-Scholes is used by almost all corporations to find the fair value of employee stock options. Improving the accuracy of the Black-Scholes model would benefit said firm’s shareholders and management alike by increasing the transparency and accuracy of financial statements. More specifically, transparency in employee compensation will be increased as employee stock options are expensed and listed on publicly-available financial statements at their fair value.

However, the Black-Scholes model does have its limitations. The model assumes that stock prices are log-normally distributed. While this is not completely unrealistic, empirical evidence suggests that the above distribution underestimates tail variance. The true distribution can be said to have fat tails because of the group-think mentality. Abnormal events cause the markets to follow itself on a downward or upward trade, thus resulting in a steeper change in the stock price than otherwise assumed by a log-normal distribution at the tail-ends. The model also assumes constant volatility. In the very short run this may be true, but volatility in the market is never constant over a reasonable period of time. For instance, with respect to the volatility of the S&P 500, historically, periods of high volatility have been followed by periods of calm and vice versa.

This phenomenon is often called volatility clustering, and leads to sharp changes in volatility, thereby violating our assuming of constant volatility.

Most of the literature on the model mention the above limitations, but do not examine the limitation of the Markov chain assumption, whereby the effects of momentum on the distribution of stock returns violates the assumption that stock prices are a Markov chain. We therefore will investigate the validity of this limitation. For simplicity’s sake, we assume that the Black5 Spring 2013 | Rutgers Science Review | 23

Research Le, Lee & Patel Rutgers University Scholes model’s implied distributions of stock returns and stock prices are accurate whenever momentum is not present. However, in cases of momentum, we will use an alternative distribution that more accurately captures the alterations in the distribution brought about by stock returns momentum.

II.

DATA

For the purposes of this paper, we will restrict our analysis to European call options of the underlying publicly-traded financial security: SPY, which is a representative ticker symbol of the SPDR S&P 500 exchange-traded fund. SPY is in essence a tradable financial security much like a stock, whose performance and returns closely follow that of the S&P 500 index, which is a weighted index of the stocks of 500 different companies. This makes the S&P 500 a good representation of a large portion of the U.S. financial markets, and thus a suitable candidate for analysis.

For our analysis, we used daily adjusted prices as of the 4:00 PM EST U.S. market close time of SPY obtained from Yahoo! Finance for every trading day within the period 2/1/1993 – 12/31/2012. It should be noted that our adjusted close prices account for dividends paid by stocks by subtracting them from the actual closing prices. This exactly replicates the theoretical effects of dividend payment on stocks. The original Black-Scholes model assumes that no dividends are paid on the underlying stock, but our adjusted data ensures that the effects of dividend payments are fully taken into account, and therefore does not jeopardize our use of the Black-Scholes model. Moreover, our adjusted close prices factor in any stock splits and reverse-splits (certain companies have historically implemented policies that split each unit of a stock into multiple units of proportionately lesser value, or policies that reverse-split each unit by merging multiple units into one new unit of a stock) to ensure that our historical data is consistent across time.

The Black-Scholes model also takes in the risk-free rate (the rate of return that investors will receive if they took no risk, i.e. depositing their funds in government-insured banks or U.S. government debt) as a parameter. We have therefore gathered data on annual risk-free rates, measured as the market yields on 1-year U.S. Treasury bonds which are internationally perceived to be financial fixed income instruments with zero default risk and hence function as a suitable representation of the risk-free rate for U.S. investors. The data on the yields is taken from the U.S. Federal Reserve Bank’s database of U.S. Treasury historical yields. The yields were gathered for the same period as our closing price dataset: 2/1/1993 – 12/31/2012.

III.

DATA ANALYSIS

As aforementioned, the Black-Scholes model assumes and implies that stock returns are lognormally distributed with the following distribution: (

)

((

)

) 6

24 | Rutgers Science Review | Spring 2013

Research Le, Lee & Patel Rutgers University

)

(

We used our data on closing prices to calculate day-on-day returns, which were then used to find estimates of the parameters of the above distribution. Our parameters were as such: Risk-free rate = 0.0014 Volatility = 0.2366 δt = 0.0027 Thus our distribution for the full set of returns is: (

)

)

(

ln (daily return + 1)

A QQ-plot of our full dataset of all daily returns within our time period against the above distribution is shown below:

Normal Quartiles It is evident from the above plot that the distribution implied by Black-Scholes for returns is a rather good fit for the actual data, except for extreme values of returns. The distribution underestimates the fatness of the tails of the distribution, a problem which, as previously mentioned in our Introduction section, has been frequently expounded upon in Financial Mathematics and Economics literature.

7 Spring 2013 | Rutgers Science Review | 25

Research Le, Lee & Patel Rutgers University Using momentum for the case where the number of time periods n = 3 days, we graphically investigated the goodness of fit of the above distribution to the data on returns in the specific case of upward and downward momentum for n = 3. We chose n = 3 days as this provided sufficient historical instances corresponding to this case of momentum for us to analyze, while also providing a significant influence on the distribution of stock returns.

ln (daily return + 1)

For cases without upward or downward momentum, the QQ-plot of the daily returns against the implied distribution is shown below:

Normal Quartiles As can be seen above, the implied distribution is also a good fit for cases without n = 3 upward or downward momentum. It possesses the same problems in that it underestimates the fatness of the tails of the distribution. However, we can assume that in general, stock returns for cases without momentum follow the overall distribution implied by Black-Scholes, whose parameters we estimated from our entire dataset. Thus, we can use the same distribution in cases without momentum.

A QQ-plot of the data in cases with upward momentum for n = 3 against the above implied distribution is shown below:

8 26 | Rutgers Science Review | Spring 2013

Research

ln (daily return + 1)

Le, Lee & Patel Rutgers University

Normal Quartiles From the above plot, it is obvious that the implied distribution is indeed a bad fit for the returns in cases with upward momentum. The points vacillate about the line, indicating that an alternative distribution must be used to better model the returns in cases with upward momentum.

ln (daily return + 1)

A QQ-plot of the data in cases with downward momentum for n = 3 against the above implied distribution is shown below:

Normal Quartiles

9 Spring 2013 | Rutgers Science Review | 27

Research Le, Lee & Patel Rutgers University The above plot shows a slightly poorer fit of returns in cases with downward momentum to the implied distribution compared to returns in cases with no momentum. However, the fit appears to be significantly better than the fit of returns in cases with upward momentum.

Thus, we can conclude that the presence of momentum does visibly affect the distribution of stock returns for SPY and renders the distribution different from the one implied by the BlackScholes model, at least for the case of n = 3 days momentum, and especially in the case of upward momentum.

Hence, the Markov-chain assumption of the Black-Scholes model whereby the price of a stock at time t is only determined by the model’s implied distribution of stock returns and the price at time t-1 is flawed, since the distribution of stock returns is influenced by momentum, which involves returns and hence prices from periods t-1, t-2, … , t-n for n-period momentum, the periods being days in our case. Ergo, the random walk of the stock price is not a Markov-chain in the strict sense that the Black-Scholes model assumed it to be.

Therefore, we will attempt to formulate an alternative valuation model which will take into account the presence of momentum. We will use the implied distribution of Black-Scholes for stock returns in cases without momentum since our QQ-plots have shown it to be a reasonable fit except for its misjudgment of kurtosis (fat tails), but we must find a different distribution that better fits the stock returns in cases with upward and downward momentum.

IV.

MONTE CARLO MODEL

In light of the above results regarding our investigation into the effects of momentum, we will formulate a valuation model which will take into account the differing distributions in cases of momentum.

From here on in, we conducted all simulations and iterative computations using programs that we wrote in Java (version 1.7).

We have very little theoretical basis for postulating an exact distribution of stock returns given upward or downward momentum, so we simply examine historical returns given the respective condition of momentum and use the data to get a probability density function by taking √ equal intervals over our range of m stock returns in the case of a certain specified momentum condition (eg. n = 3 days upward momentum). We tabulate the frequency of occurrence for each interval and consequently form a histogram-like piecewise-continuous probability density function for the stock returns in the specified case of momentum. To make our possible stock returns continuous, when selecting a random return from this distribution, we first select a random interval from our √ intervals, with the probability of any interval being selected proportionate to its relative frequency. Then, we randomly generate a stock return between the minimum and maximum return of that chosen interval, using a random generator that accords uniform probability to any stock return possibly generated within that interval. Thus, we have a pseudo-probability density function for cases of momentum, which is derived from historical data. This is instrumental to our model.

10 28 | Rutgers Science Review | Spring 2013

Research Le, Lee & Patel Rutgers University Our model values one European option with specified parameters by taking in the following arguments: Valuation_Date: The date of issuance of the option, i.e. the present date. We want to value the option as of this date.

Days_to_Expiry: The number of days till the option expires, starting from the Valuation_Date.

Historical_Years: The number of years going back from Valuation_Date that we will gather data from to generate our probability density function for cases with momentum.

Momentum_Days: The number of days that we will use to define momentum. For instance, if Momentum_Days = 3, then we define upward momentum as a case where we have positive returns for time periods t-1, t-2, t-3 when deciding which distribution to use to generate a random return for time period t.

Momentum_Days applies to both upward and downward momentum.

Strike: The contractual strike price of the option at which a holder can purchase the stock on expiration date if he so wishes to.

Our model uses the following algorithm to obtain a value of a European call option for the SPY tradable financial security. Our algorithm is inspired by the Monte Carlo options valuation method, which uses Monte Carlo simulations to simulate numerous possible paths of future stock prices and then takes the average to get the expected price of the stock at expiration date. The Law of Large Numbers in probability theory ensures that our average will approach the actual expectation of the distribution if our simulation involves a large number of simulated paths.

Algorithm Steps: 1) Examine all stock returns from the Valuation_Date going back Historical_Years number of years. (Our model obviously assumes that we do not know any returns for days after Valuation_Date since that is in the retrospective future.) Select all stock returns for which there are cases of upward momentum. In other words, if the returns on day t-1, t-2, ... , t-n were all positive, select the return for day t and add it to our dataset of stock returns in the case of upward momentum for Momentum_Days = n.

2) Use the method described above to obtain an approximate pseudo-probability density function from the stock returns in the case of upwards momentum. This distribution will be used to generate random returns in the case of upwards momentum. Label this distribution fupward.

11 Spring 2013 | Rutgers Science Review | 29

Research Le, Lee & Patel Rutgers University 3) Repeat steps 1 and 2, this time looking at cases with downward momentum for n = Momentum_Days instead. Label the resultant probability density function for cases of downward momentum fdownward.

4) For t = Valuation_Date to t = Expiration_Date (Expiration_Date = Days_to_Expiry number of days after Valuation_Date), do: i)

Check whether day t is a case of upward momentum, downward momentum, or no momentum by looking at the returns (or randomly-generated returns) of days t1, t-2, … , t-n where n = Momentum_Days.

ii)

If t is a case of upward momentum, use fupward to generate a random return for day t. If t is a case of downward momentum, use fdownward to generate a random return for day t. Otherwise, use the implied distribution of Black-Scholes with the parameters estimated by the stock returns from the Valuation_Date going back Historical_Years to generate a random return.

iii)

The randomly generated closing price at day t is then Pricet-1*(1 + return), where return was our randomly generated return. Go through the rest of the loop, repeating these steps for the rest of the days until expiration, storing the price on the expiration date.

5) After the loop, we will have one possible price of the stock at the expiration date. Repeat step 4 a large number of times (we used 10,000 iterations in our model) to get numerous possible prices of the stock at expiration. Take the average of those prices as our expected price of the stock at expiration. Label this Priceaverage.

6) Calculate the value of the European option by taking (Priceaverage – Strike) and discounting that to the present by the 1-year U.S. Treasury market yield (that we have data for) as of the Valuation_Date. This is the value of our option. If the outputted value is < 0 , then set the value to 0, since options cannot have negative value. In this case, the implication is simply that the option is worthless and has no value.

V.

MONTE CARLO MODEL VS. BLACK-SCHOLES

We are now ready to analyze the accuracy of our model with that of the Black-Scholes model.

We do this by comparing our model’s valuation and the Black-Scholes’ valuation with the “true value” of the option, here calculated as (actual price of the stock at expiration date – strike price) discounted back by the 1-year U.S. Treasury market yield as of the date of valuation / issuance of the option.

We measure the accuracy of our model with that of the Black-Scholes model by valuing SPY European-style call options with valuation dates starting from 1/3/2000 to 1/3/2012 (3020 options in total) with both our models and then calculating the sum of the absolute differences 12 30 | Rutgers Science Review | Spring 2013

Research Le, Lee & Patel Rutgers University between a model’s outputted value and the true price of the option for the all the 3020 options.

The sum for our model is compared with the sum for the Black-Scholes model to determine which is more accurate. The larger the sum, the more inaccurate the model is as it has produced valuations with greater differences from the true price. The true price is bound from below by 0, since options cannot be negative in value.

We used 200 days and 110 days as our Days_to_Expiry argument in our Monte Carlo model and used Historical_Years = 3, 4, and 5 to generate our probability density function. In each of these six cases, we used Momentum_Days = 3 for our definition of momentum, and valued all 3020 options with valuation dates from 1/3/2000 to 1/3/2012 using both of our models. We then did a comparison for each of these six cases of our model with the Black-Scholes model using the comparison metric defined above.

The results are shown in the following table: No. of years used in probability density function generation (Historical_Years)

3

4

5

3

4

5

No. of days till option expiration (Days_to_Expiry)

200

200

200

110

110

110

-0.45 %

-1.54 %

-3.35 %

-1.22 %

-1.79 %

-2.07 %

% more accurate than Black-Scholes using sums of differences (> 0 for more accurate, < 0 for less accurate)

The results from our simulations, as represented in the above table, unfortunately show that our Monte Carlo model did not yield a more accurate valuation for SPY European options for any of our six cases. However, our valuations were not far off either, and our model was never more than 3.35 % less accurate than Black-Scholes on average for the six cases. Thus, though our model to some reasonable extent took into consideration the effects of momentum, it did not yield more accurate valuations.

It may be interesting to note that our model seemed to perform slightly better during periods of recession in the U.S. economy. Running our simulations and comparing accuracy of valuations for the periods 1/3/2000 – 12/31/2001 and 1/2/2008 – 12/31/2008 with those of Black-Scholes using the same metric as we did in our above analysis indicated that our model was a slight 0.50 % more accurate than the Black-Scholes model on average over the six cases. The period 1/3/2000 – 12/31/2001 contains the dot-com bubble burst and the September 11 attacks on the U.S., both of which contributed to a tumultuous economic period. The period 1/2/2008 – 12/31/2008 witnessed the subprime mortgage crisis, which eventually led to a deep recession 13 Spring 2013 | Rutgers Science Review | 31

Research Le, Lee & Patel Rutgers University towards the end of 2008. Thus, the two periods contain economic recessions, and lend themselves to an analysis of our model’s performance under times of financial stress.

The specific results are shown below for the two time periods: 1/3/2000 – 12/31/2001 No. of years used in probability density function generation (Historical_Years)

3

4

5

3

4

5

No. of days till option expiration (Days_to_Expiry)

200

200

200

110

110

110

3.94 %

-9.67 %

-12.47 %

6.42 %

-0.93 %

-1.47 %

% more accurate than Black-Scholes using sums of differences (> 0 for more accurate, < 0 for less accurate)

1/2/2008 – 12/31/2008 No. of years used in probability density function generation (Historical_Years)

3

4

5

3

4

5

No. of days till option expiration (Days_to_Expiry)

200

200

200

110

110

110

4.11 %

2.99 %

-5.99 %

10.40 %

9.66 %

-0.94 %

% more accurate than Black-Scholes using sums of differences (> 0 for more accurate, < 0 for less accurate)

While the 0.50 % increased accuracy of our model over Black-Scholes has not been tested for statistical significance, it is an interesting result to note. Since we have not yet established our result as statistically significant, we will avoid commenting further on it or speculating on possible causes of this increased accuracy.

14 32 | Rutgers Science Review | Spring 2013

Research Le, Lee & Patel Rutgers University VI.

CONCLUSION

In conclusion, our graphical analysis of daily returns of the SPY exchange-traded fund strongly suggested that upward momentum as defined with n = 3 days significantly changed the probability distribution of SPY returns from the implied distribution that the Black-Scholes model assumed. Downward momentum with n = 3 also seemed to alter the distribution, but not by as much. Cases without momentum seemed to conform reasonably to the implied distribution, with the exception of having more tail-end variance than the implied distribution, though this shortcoming was present even in fitting the data of all daily returns against the implied distribution, and is a limitation of the Black-Scholes model that has previously been espoused in the literature.

We devised a valuation model that took into account the effects of upward and downward momentum on SPY returns, but were unable to produce more accurate valuations than the BlackScholes model for the six cases that we covered (varying the optionâ&#x20AC;&#x2122;s days to expiration and the number of years of historical data used to generate the probability density function for cases with momentum).

It is difficult to say if this implies that on average, accounting for momentum will not significantly increase the accuracy of the valuation or not for several reasons. Firstly, we only conducted tests for the case of n = 3 days momentum, and so have not reasonably exhaustively tested all cases of momentum. Secondly, our model may have been limited in its exactitude and precision. For instance, we could have used more or less than â&#x2C6;&#x161; intervals for the range of daily returns when formulating our probability distribution for cases of momentum. We could also have increased our iterations in the random walk simulation (where we get a possible price for SPY on expiration day) when running our Monte Carlo model. Altering some of these features may make our model more or less accurate.

Notably, our model performed slightly better in its valuations during periods of economic recession and stress. However, we did not ascertain whether our increase in accuracy was statistically different from 0.

On a final note, there is an alternative to our Monte Carlo valuation model. Instead of using Monte Carlo to take into account the effects of momentum, we could have used Markov chain concepts. While we showed that momentum violates the Markov chain assumption of stock prices and returns, we can transform a non-Markov process into a Markov process using vectors.

Therefore, if we defined momentum with n = 3 days and were willing to assume that the returns from t-4, t-5, â&#x20AC;Ś onwards had no effect on the distribution of stock returns, then we could use a 3entry vector Yt where Yt = (St , St-1 , St-2) to transform our random walk into a Markov chain.

Here, Yt would depend only on Yt-1 and would still account for the effects of momentum. Thus, we have a modified Markov chain, and we can use the Chapman-Kolmogorov theorem of Markov chains (which involves multiplication of transition probability matrices) to find the expected stock price on the expiration date of the European option. However, in terms of runtime efficiency for our Java program, this does not significantly increase efficiency, nor does it necessarily increase accuracy as compared to a Monte Carlo simulation with 10,000 iterations.

15 Spring 2013 | Rutgers Science Review | 33

Research Le, Lee & Patel Rutgers University We leave all the aforementioned unaddressed issues to further studies in this topic.

VII.

ACKNOWLEDGEMENTS

McWilliams N. Pricing American Options Using Monte Carlo Simulation. University of Edinburgh (United Kingdom), 2005.

Hull J. Options, futures, and other derivatives (5th Edition). Prentice Hall International, 2003.

Lamberton D. & Lapeyre B. Introduction to stochastic calculus applied to finance. Chapman & Hall, 1996.

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Design and fabrication of multilayer thin film coated hollow waveguides for enhanced infrared radiation delivery Carlos M. Bledt*a,b, Jeffrey E. Melzer a, and James A. Harrington a a Dept. of Material Science & Engineering, Rutgers University, Piscataway, NJ 08855 b School of Engineering, Brown University, Providence, RI 02912 ABSTRACT Metal coated Hollow Glass Waveguides (HGWs) incorporating single dielectric thin films have been widely used for the low-loss transmission of infrared radiation in applications ranging from surgery to spectroscopy. While the incorporation of single dielectric film designs have traditionally been used in metal/dielectric coated HGWs, recent research has focused on the development of alternating low/high refractive index multilayer dielectric thin film stacks for further transmission loss reduction. Continuing advances in the deposition of optically functional cadmium sulfide and lead sulfide thin films in HGWs have allowed for the simultaneous increase in film quality and greater film thickness control necessary for the implication of such multilayer stack designs for enhanced reflectivity at infrared wavelengths. This study focuses on the theoretical and practical considerations in the development of such multilayer stack coated waveguides and presents novel results including film growth kinetics of multilayer stack thin film materials, IR spectroscopic analysis, and IR laser attenuation measurements. The effects of incorporating progressive alternating cadmium sulfide and lead sulfide dielectric thin films on the optical properties of next generation dielectric thin film stack coated HGWs in the near and mid infrared regions are thoroughly presented. The implications of incorporating such dielectric multilayer stack coatings based on metal sulfide thin films on the future of IR transmitting hollow waveguides for use in applications ranging from spectroscopy, to high laser power delivery are briefly discussed.

Keywords: Infrared fiber optics, hollow waveguides, multilayer dielectric thin film designs, optical materials

1. INTRODUCTION Hollow glass waveguides (HGWs) have experienced widespread success in the low-loss, broadband delivery of infrared radiation across the Îť = 2 â&#x20AC;&#x201C; 12 Îźm region and have thus been used in a variety of applications ranging from photothermal imaging to infrared spectroscopy.[1] Traditionally, the functional structure of HGWs has relied on light propagation within a hollow fused silica capillary of predetermined, constant dimensionality whose inner surface is coated with a silver film followed by a single dielectric thin film of adequate thickness to enhance reflectivity of the inner HGW surface, thus allowing for low-loss propagation of infrared radiation along the waveguide length. In HGWs, the loss achieved depends on a number of factors, most importantly the waveguide bore size, the surface quality of the deposited metal and dielectric films, the transparency of the dielectric thin film material, the film thickness of the dielectric thin film, and the degree of bending that the waveguide is subjected to. Theoretically and experimentally, the propagation loss in HGWs has been shown to increase as 1/a3 where a is the bore radius and upon bending as 1/R where R is the radius of curvature.[1,2] While current Ag/AgI HGWs are capable of achieving acceptable transmission losses for most applications, as is the case in any transmissive system, it is desirable to further decrease propagation losses of passive components, in this case the waveguide. In order to decrease the waveguide loss, one must first look at the waveguide components which influence the attenuation. Eq. 1 is the geometrical optics equation for the power attenuation coefficient of meridional rays propagating in a straight, constant bore HGW.[3] ( )

( ) ( )

( )

* carlos_bledt@brown.edu; Phone 862-485-9289; irfibers.rutgers.edu

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where a is the bore radius, θ is the angle of propagation of the light ray, and R(θ) is the reflectance of the inner HGW surface, which of course depends on the angle of propagation, the film structure, and the polarization of light. While either the bore size can be increased, or the propagation angle may be decreased to lower the loss, in practice it is often not desirable to do so for various reasons (ie. to achieve a desired propagating mode or to enable the waveguide to have flexibility). Therefore, the loss must be decreased in another manner, namely in increasing the surface reflectivity. [1-3] In considering the possibilities, perhaps the most practical manner to modify the HGW surface structure so as to increase reflectivity is through the incorporation of multilayer dielectric thin film stacks with layers composed of alternating low and high refractive index materials in place of the traditional single dielectric layer, as shown in Fig. 1.

Figure 1 – Representative cross-sections of a) single dielectric layer and b) multilayer dielectric stack HGWs

By optimizing the thin film thickness of each of the dielectric layers depending on the desired wavelength range of operation, it is theoretically possible to increase the reflectivity, and thus lower the transmission loss, to a level beyond that attainable through the use of any single dielectric layer. To increase the reflectivity of such a multilayer structure, one can either increase the number of total layers or increase the refractive index mismatch between the two dielectric materials for a given number of layers. The dielectric materials must thus be carefully selected and must be transparent at the wavelength(s) of operation. To date, our laboratory has had considerable success in the deposition of metal sulfide, particularly CdS and PbS thin films, in HGWs using fabrication methods similar to those used for the fabrication of Ag/AgI HGWs.[4-6] The combined IR transparency, mechanical and chemical compatibility, and considerable index mismatch (nCdS ≈ 2.28 / nPbS ≈ 4.00) between CdS and PbS make these two materials ideal for use in the fabrication of enhanced reflectivity next generation multilayer dielectric stack HGWs. [1,7,8]

2. THEORETICAL CONSIDERATIONS 2.1 Theory and Design of Multilayer Film Structures Multilayer dielectric stacks consisting of a periodic arrangement of alternating low and high refractive index materials are capable of producing high reflectivity while simultaneously exhibiting minimal absorption. Such designs have been widely used in a number of applications ranging from selective filters to high efficiency dielectric mirrors to 1-D photonic bandgap devices.[8,9] A representative diagram of such a multilayer dielectric stack is given in Figure 2.

Figure 2 – Representative diagram of periodic multilayer dielectric stack

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In order to achieve the desired reflectivity, the physical dimensions of each individual dielectric layer, δ L / δH, must be such that its optical thickness is a quarter-wavelength thick depending on the target wavelength and angle of incidence (compliment of propagation angle). By controlling the individual film thicknesses and obtaining the desired layer dimensions, coherent scattering from each subsequent interface results in constructive interference of reflected light and thus an enhancement in reflectance. As previously mentioned, the reflectance may be increased in any of several different manners including increasing the total number of layers, N, and/or selecting dielectric materials with a larger refractive index mismatch. When optimizing the individual layer thicknesses for λ = 1.0 μm, a CdS film thickness of 122 nm (nCdS = 2.28 at λ = 1.0 μm) and a PbS film thickness of 58 nm (n PbS = 4.43 at λ = 1.0 μm) are necessary.[1,2,8,9] 2.2 Multilayer Stack Calculations Calculation of the reflectance achieved by a stack of dielectric thin films can be calculated in several methods. For this particular study, reflectance was calculated using the ray transfer matrix method due to its versatility in accounting for varying influential parameters including the polarization of light, the angle of incidence, the total number of layers, the optical properties of each of the materials, and the thickness of each individual layer. The ray transfer matrix method is a numerical method for determining the optical response of such a system and its fundamental function lies with Eq. 2. [

]

∏[

][

]

( )

where E is the amplitude of the electric field, H is the amplitude of the magnetic field, A, B, C, D are the characteristic matrix coefficients, and N is the total number of layers.[9] Using this method, the electric and magnetic field amplitudes at any point z along the axis of propagation (normal to interface) may be calculated and compared to the initial amplitudes to determine the reflection and transmission of the system. Calculations carried out in this manner yielded the optical responses presented in Figure 3 for the particular multilayer stack system under investigation with increasing number of total layers in the stack.

Figure 3 – IR reflectance as a function of total number of layers optimized for λ =1.0 μm with a) N = 2, b) N = 4, c) N = 6, d) N = 8

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From Figure 3, it can be seen that the reflectance does indeed increase at the target wavelength (λ = 1.0 μm), as the total number of CdS/PbS alternating layers increases. For this calculation, the ideal case was assumed with no surface roughness related scattering losses and; nL = 2.28, nH = 4.43, δL = 122 nm, δH = 58 nm, and θi = 87.5° (θp = 2.5°).[7] If surface roughness can be neglected and the number of layers continues to maintain its ideal periodicity, the structure will eventually lead to the emergence of a photonic bandgap at the design wavelength as the number of layers is increased. [8]

3. FABRICATION METHODOLOGY The fabrication of metal / dielectric HGWs involves the controlled deposition of the desired material on the inner fused silica capillary surface from precursor containing aqueous solutions via dynamic liquid phase deposition (DLPD).

DLPD is a versatile technique similar in nature to chemical bath deposition (CBD) in which synthesis of a desired material on the substrate surface is achieved via electroless chemical reaction between precursor containing aqueous solutions. However, unlike CBD, DLPD has the advantages of a higher degree of control over the thickness of the deposited thin film and, given that the precursor solutions are continuously being simultaneously pumped through the fused capillary, film growth is not limited by depletion of reactive species in solution. The DLPD process involves simultaneous pumping of precursor containing solutions at equal rates through the silica capillary using a peristaltic pump. The quality and thickness of the films depends on several parameters including solution concentrations, solution temperature, reactivity of precursor species, total procedure time, and fluid flow velocity. In practice, it is desirable to minimize variables during the fabrication process and thus the solution concentration and temperature, as well as the fluid flow speed are kept constant regardless of the dimensions of the silica capillary or type of procedure being carried out. Experimentally, optimal quality thin films have been deposited through the entire waveguide length at fluid flow speeds around 70 cm/s and thus the pump rate is programmed to allow for the corresponding volumetric flow rate to be achieved for any HGW bore size and length. In this particular study all experimental samples had a bore size of 700 μm and were initially approximately 220 cm long, corresponding to an optimal volumetric flow rate of 16.2 mL/min. Using this standard volumetric pump rate, the silver film was deposited, followed by either a cadmium sulfide (CdS) or lead sulfide (PbS) dielectric thin film and so on.

3.1 Deposition of Silver Films As previously mentioned, the reflective Ag film is deposited on the inner fused silica capillary via DLPD as a result of an electroless red-ox reaction between a complexed silver ion and a reducing solution. Prior to depositing the Ag layer, samples undergo a simple sensitization procedure which involves flowing an acidic tin (II) chloride solution having a concentration of 0.35 g/L [Sn(II)Cl2· 2H2O] for a total of five minutes. This allows replacement of surface hydroxyl species and adsorption of active tin (II) ion species on the silica surface, thus improving the film quality of the subsequently deposited Ag film and drastically reducing the procedure time necessary for obtaining a silver film greater than 200 nm thick. After the sensitization procedure, reducing dextrose and ammonia-complexed silver ion solutions are simultaneously pumped through the silica capillary, depositing a silver film on the silica surface substrate. The concentrations of the precursor solutions involve 0.56 g/L of dextrose for the reducing solution and 2.44 g/L of silver (I) nitrate along with approximately 15 mL/L of NH4OH and 44 mL/L of NaOH for the active silver ion solution. The procedure is continued for as long as necessary, keeping in mind that too thin of a silver film will lead to light penetration through the film while an excessively thick film will result in unnecessary and unwanted increase in surface roughness and related scattering losses. In practice, the silver film deposition procedure is carried out at 20 °C, resulting in a necessary deposition time of 15 min to achieve silver films thicker than 200 nm while simultaneously minimizing surface roughness and thus its detrimental effects.

3.2 Deposition of Metal Sulfide Dielectric Films After deposition of the Ag film, the reflectivity enhancing dielectric film, or as in the case of this study films, is/are deposited via DLPD methods. As for the silver deposition procedure, the metal ion precursor solution must be complexed in solution so as to achieve a high degree of control over the rate of film growth. [10,11] In the case of CdS deposition this involves preparation of an ammonia-complexed cadmium (II) ion solution at a standard concentration of 4.55 g/L of Cd(NO3)2·4H2O along with 150 mL/L of NH4OH. In the case of PbS deposition this then involves preparation of a hydroxide-complexed lead (II) ion solution at a standard concentration of 1.80 g/L of Pb (NO 3)2 along with 9 g/L of NaOH. The reducing solution for both procedures involves a thiourea solution at concentrations of 11.418 g/L and 4.141 g/L, when depositing CdS and PbS, respectively. All solutions are heated to 30 °C and kept at this

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temperature (± 1 °C) in a water bath throughout the entire deposition procedure. Holding all other fabrication parameters constant, the dielectric thin film thickness is controlled by modulating the total deposition procedure time, with thicker films being deposited as deposition time increases. Thus, in order to achieve a high degree of control over the deposited thin film thickness, the film deposition kinetics (in particular the film growth rate), must be known. Depending on the overall film structure desired and the wavelength of operation, the metal sulfide deposition procedure is repeated as many times as desired, with each metal sulfide (CdS or PbS) being deposited every other deposition procedure in order to achieve the alternating low / high refractive index profile. In practice, it should be noted that the fabrication of a larger number of layers leads to decreased film quality and performance as a result of increasing surface roughness and its exponentially detrimental effect on light propagation. The same is true of very thick single CdS or PbS films, thus limiting their practical use to SWIR and MWIR wavelengths. In this study, prior to fabrication of multilayered CdS / PbS HGWs, the deposition kinetics for each procedure was experimentally studied in order to achieve the desirable CdS and PbS film thicknesses and will be presented in the proceeding section.

4. MEASUREMENTS AND OPTIMIZATION 4.1 Determination of Thin Film Growth Kinetics Initial analysis involved experimental determination of the thin film growth kinetics of the different procedures involved in the proposed fabrication of CdS / PbS multilayer thin film stacks. As such, four 220 cm long Ag coated HGWs were prepared and either CdS, PbS, or both were deposited using the standard solution concentrations and temperatures given in the preceding section. The first of these guides was coated with CdS for a total of 210 min with 12 cm segments being cut and collected from the output end of the waveguide at 15 min intervals in order to investigate the deposition kinetics of CdS on an Ag coated HGW (Ag substrate). The second of these guides was coated with PbS for a total of 70 min with 12 cm segments being cut and collected from the output end of the waveguide at 5 min intervals in order to investigate the deposition kinetics of PbS on an Ag coated HGW (Ag substrate). The third sample was then coated with PbS for 25 min in its entirety after which CdS was deposited for a total of 150 min with 12 cm segments being cut and collected from the output end of the waveguide at 15 min intervals in order to investigate the deposition kinetics of CdS on an Ag/PbS coated HGW (PbS substrate). The final sample was then coated with CdS for 155 min in its entirety after which PbS was deposited for a total of 37 min with 12 cm segments being cut and collected from the output end of the waveguide at 4 min intervals in order to investigate the deposition kinetics of PbS on an Ag/CdS coated HGW (CdS substrate). All fabricated samples where then analyzed using a Bruker Tensor 37 FTIR in conjunction with a cryogenic Teledyne / Judson MCT detector in order to obtain the spectral response from λ = 2.0 – 15.0 μm. As expected, the spectral response shifted to longer wavelengths with increasing total deposition time (film thickness). In order to obtain the film growth kinetics, the dielectric film thickness of each sample was calculated using Eq.3. √

( )

where λp is the centroid wavelength of the first interference peak and nF is the refractive index of the dielectric material at λ = λp. Through this methodology, the deposited film thickness as a function of deposition time for each of the four procedures was calculated and the resulting measurements, along with the linear fit in each case is given in Figure 4.

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Figure 4 – Film Thickness as a function of time for a) CdS on Ag, b) CdS on PbS, c) PbS on Ag, and d) PbS on CdS

From the film growth data presented in Figure 4 several things can be noted. First, it is interesting to note that there is considerable difference (particularly for PbS) in the film growth rates depending on the substrate with a difference of 0.9 nm/min and 4.7 nm/min for CdS and PbS, respectively. Furthermore, it can be seem that a higher degree of variability in film thickness occurs when the substrate is the alternate metal sulfide, rather than silver. It can be seen that the fastest film growth occurs for the case of PbS on CdS at 11.0 nm/min, while the slowest film growth occurs for the case of CdS on Ag at 2.5 nm/min. The considerably higher deposition times necessary for depositing CdS arises from the two step reaction process of converting deposited cadmium hydroxide to CdS rather than the direct deposition of PbS from aqueous solutions.[10,11] Consequently, optimization of CdS thin films for longer wavelengths requires thicker films, longer deposition times, and a greater probability of low film quality and increased surface roughness. Thus, in practice, the overall film structure, and thus the wavelength of optimization, is restricted by the limitations of CdS films. After analyzing the deposition kinetics data, it was decided that the film structure for this particular study would be optimized for λ = 1.0 μm, requiring individual film thicknesses of 122 nm and 58 nm for CdS and PbS, respectively. These desired film thicknesses are predicted to require deposition times of 60 min for CdS on Ag, 100 min for CdS on PbS, 12 min for PbS on Ag, and 8 min for PbS on CdS. 4.2 Spectroscopic Analysis The metal sulfide multilayer dielectric stack coated HGWs analyzed in this study consisted of individual CdS and PbS film thicknesses optimized for λ = 1.0 μm, requiring a CdS film thickness of 122 nm and a PbS film thickness of 58 nm. The 220 cm multilayer sample fabricated in this study was first coated with a silver layer using the aforementioned standard silver film deposition procedure. The CdS / PbS multilayer stack was then fabricated one additional film at a time, alternating between CdS and PbS and having have had started with CdS for the first film. After each subsequent deposition procedure, the straight loss at λ = 10.6 μm was tested and a 12 cm segment was taken off the end of the waveguide for spectroscopic analysis. The next coating procedure was then carried out on the longer remaining segment and this methodology was continued for as long as possible until a final waveguide length of less than 80 cm. The spectral response from λ = 2.0 – 15.0 was taken with an FTIR spectrometer to track changes in the optical response as a function of additional layers as well as to determine the actual film thickness of the last film deposited. Using this methodology, a total of 10 (N = 10) alternating CdS/PbS layers were deposited before the optical response in this region became too low for qualitative and quantitative analysis. The spectral response for select total number of layers is presented in Figure 5.

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Figure 5 – IR optical response as a function of deposited layers for a) N = 2, b) N = 4, c) N = 6, and d) N = 8

From the spectral response of these CdS/PbS multilayer stack HGWs it can be seen that high quality, optically functional dielectric film stacks have been successfully deposited. The ability to deposit up to 10 layers while maintaining a high degree of film quality denotes a great deal of improvement over previous attempts, which have been successful in depositing up to 5 to 6 films before film quality degraded to the point of failing to obtain a clear spectral response. This achievement shows that through the proposed methodology, a greater degree of control over the film thickness can be achieved while maintaining high quality films for up to a total of 10 layers.

4.3 Attenuation Measurements As mentioned earlier, in addition to obtaining the spectral response of the sample after each subsequent metal sulfide deposition, the straight waveguide loss was also measured at λ = 10.6 μm using a Synrad I-series 15 W max output CW CO2 laser. Obtaining the attenuation in this manner allowed for straight transmission loss as a function of total layers deposited to be obtained at this important IR wavelength. The resulting attenuation measured for up to a total of 10 layers is given in Figure 6.a for analysis.

Figure 6 – a) Attenuation and b) composite film thickness as a function of total number of layers

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As given in Figure 6.a, it is seen that the attenuation does indeed decrease with an increase in total number of layers as would be expected for the case of an ideal multilayer dielectric stack. The loss is the lowest at a total of N = 4 alternating CdS/PbS layers but then rises once more, jumping considerably at N = 6. It is then seen to decrease at a rather steady rate until N = 9 before going back up at N = 10. From the spectral response, the substantial increase in attenuation around N = 6 can be attributed to the first absorbance peak, which pushes the loss up as it shifts in the vicinity of λ = 10.6 μm. The increased loss for a larger number of total layers N > 6 could be attributed to a decrease in film quality resulting from an increase in surface roughness and related scattering losses. It is important however to note that for N = 2 – 5, the measured loss drops significantly and is lower than that obtained with any single dielectric film for this particular bore size. As such, increasing the number of layers can be seen to successfully decrease transmission losses by a considerable amount (nearly by a factor of 2 at N = 4).

At the same time, we can see from Figure 6.b that the measured composite film thickness deviates from the optimized composite film thickness as the measured thickness is consistently higher than ideal and as would be logical, continues to deviate from the ideal as the number of total layers increases. The difference between measured and ideal film thicknesses is seen to be small up to N = 4 after which point it continues deviating further, resulting in a considerable difference for a larger number of layers. This deviation from the ideal composite film thickness can be expected to further contribute to higher losses with a larger number of films as the structure begins to deviate from having the desired periodicity. As such, a greater degree of control over the individual film thicknesses must be obtained in order to truly optimize the thin film structures for any given wavelength. Furthermore, while unable to do so in this particular study, losses should be measured at the target wavelength (λ = 1.0 μm) rather than at a different wavelength.

Albeit this variation, it is noteworthy to see that through the use of CdS/PbS multilayer stack HGWs the loss at λ = 10.6 μm can in fact be decreased relative to using a single dielectric layer, whether it be CdS, PbS, or AgI.

5. SUMMARY AND CONCLUSIONS This study was successful in further investigating the theoretical and experimental results of CdS/PbS multilayer dielectric stacks in HGWs at infrared wavelengths. Specifically, the argument for incorporating multilayer dielectric stacks in HGWs in place of single dielectric films for enhanced performance next generation HGWs was made, and the proposed structure was analyzed through numerical methods through use of the transfer matrix method. Calculations based on theory confirmed that the loss could indeed be expected to decrease as reflectance was enhanced with less than a dozen layers. Experimental work focused on the initial determination of an empirical film growth kinetics relationship for the deposition of CdS and PbS films on Ag and the alternate metal sulfide as substrate. The film growth rate was seen to be nearly linear for the deposition times selected and was determined to be highly dependent on the species being deposited as well as the substrate species. Following empirical derivation of the metal sulfide film growth kinetics, deposition of CdS/PbS multilayer stack designs in HGWs were attempted using the appropriate deposition times suggested by the empirical film growth relationships. The deposition of metal sulfide multilayer stacks in HGWs was shown to be successful as supported by the high quality of the spectral response obtained (up to N = 10), and the acceptable losses measured. Furthermore, the multilayer design presented in this study achieved losses lower than previously recorded for any single dielectric layer HGW for certain number of total films (N = 3 – 4). This study has set the foundation for more in depth study and optimization of next generation CdS/PbS multilayer stack HGW designs.

Further study of these novel designs in HGWs will involve further control of the deposition procedure in order to more closely approach the ideal individual film thicknesses as well as focusing measurements at the target wavelength.

6. ACKNOWLEDGEMENTS The authors would like to thank Dr. Jason M. Kriesel and Opto-Knowledge Systems, Inc. (OKSI) of Torrence, CA, USA for their support in the success of this research initiative.

REFERENCES [1] Harrington, J. A., [Infrared Fiber Optics and Their Applications], SPIE Press, Bellingham, WA, (2004).

[2] Miyagi, M. and Kawakami, S., "Design theory of dielectric-coated circular metallic waveguides for infrared transmission,“ IEEE Journal of Lightwave Technology. LT-2, 116-126 (1984).

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[3] Miyagi, M., “Waveguide-loss evaluation in circular hollow waveguides and its ray-optical treatment,” IEEE Journal of Lightwave Technology. LT-3, 303 – 307 (1985).

[4] Gopal, V., Harrington, J. A., “Deposition and characterization of metal sulfide dielectric coatings for hollow glass waveguides,” Optics Express, 11, 24 (2003).

[5] C. M. Bledt, D. V. Kopp, and J. A. Harrington, “Dielectric II-VI and IV-VI Metal Chalcogenide Thin Films in Silver Coated Hollow Glass Waveguides (HGWS) for Infrared Spectroscopy and Laser Delivery”, in Advances and Applications in Electroceramics II: Ceramic Transactions, Volume 235 (eds K. M. Nair and S. Priya), John Wiley & Sons, Inc., Hoboken, NJ, USA, (2012) [6] C. M. Bledt, J. A. Harrington and J. M. Kriesel, "Multilayer silver / dielectric thin-film coated hollow waveguides for sensor and laser power delivery applications", Proc. SPIE 8218, 82180H (2012).

[7] Palik, E. D. and Ghosh, G., Handbook of optical constants of solids, (Academic, London, 1998).

[8] Joannopoulos, J. D., Johnson, S. G., Mead, R. D., and Winn, J. N., Photonic Crystals: Molding the Flow of Light, Second Edition, Princeton Univ. Press, (2008) [9] Heavens, O. S., [Optical Properties of Thin Film Solids], First Edition, Dover Publications, Inc., (1991) [10] Niesen, T. P., De Guire, M. R., “Review: Deposition of Ceramic Thin Films at Low Temperatures from Aqueous Solutions.” Journal of Electroceramics, 6, 169 – 207 (201).

[11] Mane, R. S. and Lokhande, C. D., “Chemical deposition method for metal chalcogenide thin films,” Mat. Chem.

Phys. 65, 1-31 (2000).

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