1. Toward an Electrically-Pumped Silicon Laser: Modeling and
Optimization
by
Daniel B. Riley
Submitted in Partial Fulfillment
of the
Requirements for the Degree
Master of Science
Supervised by
Professor Philippe M. Fauchet
Department of Electrical and Computer Engineering
The College
School of Engineering and Applied Sciences
University of Rochester
Rochester, New York

2. Curriculum Vitae
Daniel B. Riley was born on October 15th
, 1982 in Rochester, NY. He attended the
State University of New York at Geneseo where he graduated with the Bachelor of Science
degree in Applied Physics and the Bachelor of Arts degree in Mathematics in May of 2005.
While in attendance, he was inducted into the Geneseo chapters of the Phi Beta Kappa,
Sigma Pi Sigma, and Phi Eta Sigma honor societies. He attended University of Rochester
from August 2005 – August 2007 where he pursued the Master of Science degree in
Electrical and Computer Engineering with a concentration in optoelectronics. He worked in
the research group of Dr. Philippe Fauchet and also served as a graduate teaching assistant
during his first year in attendance.
ii

3. Acknowledgements
I would like to first thank my advisor, Dr. Philippe Fauchet who never failed to
enlighten me with his insight and incredible knack for explaining even the most difficult
concepts in a matter of minutes.
I also thank all members of the Fauchet research group over the last year. You have
all provided me a constant source of support, knowledge and encouragement. In particular I
thank Yjing Fu and Jidong Zhang for valuable suggestions and vital discussion.
I am also deeply indebted to Vicki Heberling for her tireless efforts in support of
my work – especially with the color printer and all the adventures it presented.
I must also thank the members of all participating institutions in the MURI silicon
laser project for the continuous insight and enlightenment they have provided, as well as all
sources of funding for the project.
Finally, I thank the members of my Masters Thesis examination committee: Dr.
Miguel Alonso from the Institute of Optics and Dr. Thomas Hsiang from the Department of
Electrical and Computer Engineering.
iii

4. Abstract
A slot-confined, multilayer waveguide structure to be used as the resonant optical
cavity in a silicon (Si)-based laser intended for use in chip-scale, nano-photonic systems
has been modeled and optimized using both numerical and theoretical methods. The
structure is a rib waveguide and consists of alternating nanometer-thin layers of amorphous
Si (a-Si) and erbium (Er)-doped silicon dioxide (SiO2) surrounded by a cap layer and
substrate each composed of SiO2. A carefully chosen thermal budget during fabrication
will result in the formation of silicon nanocrystals in the Si layers producing
nanocrystalline Si (nc-Si). This creates a series of high-index-contrast interfaces which
enables high optical mode confinement and higher gain within the low index SiO2:Er layers
for quasi-TM modes. A series of full-vectorial, finite-difference time domain (FDTD)
simulations are performed on a range of optimized structures yielding confinement factors
equal to or higher than those currently reported for similar structures. Additional
simulations are performed with gain in the SiO2 layers and loss in the Si layers to model a
realistic cavity. Varying the ratio of gain coefficient to loss coefficient between simulations
and normalizing the output power yields a value for the minimum gain to loss ratio
required to achieve net gain within the cavity. This in turn leads to a reasonable
approximation for the lower limit on the threshold current density for electrical injection –
a critical parameter in the design of devices with low power dissipation and consumption.
All results are reported along with those of a different model known as the transfer matrix
method (TMM) with analysis and discussion of the consistencies and discrepancies.
Overall considerable agreement is found between the two approaches. This enhances the
understanding of performance expectations for such device structures.
iv

5. Table of Contents
Curriculum Vitae .............................................................................ii
Acknowledgements..........................................................................iii
Abstract............................................................................................ iv
Table of Contents ............................................................................. v
List of Figures.................................................................................vii
Foreword........................................................................................... 1
1 Introduction................................................................................... 2
Motivation.......................................................................................................2
MURI Silicon Laser Project........................................................................14
Project Overview..........................................................................................16
2 Theory .......................................................................................... 21
Maxwell’s Equations....................................................................................21
The Wave Equation......................................................................................23
Boundary Conditions at Dielectric Interfaces...........................................25
Field Polarization .........................................................................................28
Poynting Vector............................................................................................28
v

6. Total Internal Reflection .............................................................................29
Waveguides ...................................................................................................30
Planar Slab Waveguide ............................................................................................... 30
Slot Confinement Waveguide...................................................................................... 38
Power Confinement in Waveguides ........................................................................... 41
Lasers.............................................................................................................46
Introduction.................................................................................................................. 46
Three Processes: Absorption, Spontaneous Emission, Stimulated Emission......... 47
Population Inversion and Pumping Schemes............................................................ 49
Optical Resonators and Amplification....................................................................... 50
Gain Saturation and Lasing Threshold ..................................................................... 53
Electrical Charge Injection Pumping and Semiconductor Lasers.......................... 54
MURI Silicon Laser Project and Erbium Doped Fiber Amplifiers (EDFAs)........ 57
3 Simulation & Results .................................................................. 63
Introduction ..................................................................................................63
Structure .......................................................................................................66
Investigation of Eigenmodes Using BeamProp .........................................70
FDTD Simulations........................................................................................72
Gain and Loss Analysis................................................................................80
4 Conclusion.................................................................................... 85
Bibliography ................................................................................... 87
vi

7. List of Figures
Figure 1-1. Diagram showing various applications of different interconnect technologies.
Optical links are currently only used at longer distances (>100m) [2].................................... 4
Figure 1-2. Timeline of information carrying capacity for links in various electrical and
optical links. Those in optical systems have data rates roughly 2 orders of magnitude higher
than those in electrical systems [2].......................................................................................... 5
Figure 1-3. Relationship between processor and front side bus (FSB) clock speeds for
various models over the past 15 years. Note the significant improvement in processor
speeds over FSB speeds in recent years [2]............................................................................. 6
Figure 1-4. Diagram showing individual components of an optical transmitter-receiver
system. The transmitter is the top diagram and the receiver is the bottom [4]........................ 7
Figure 1-5. Wafer cost per unit area for various materials versus wafer size. Notice that
InP is currently only available in one wafer size and is prohibitively expensive compared to
a Si wafer of the same size [2]................................................................................................. 9
Figure 1-6. Hypothetical timeline for the transition between electrical and optical based
communication systems for various distances [2]. .................................................................. 9
Figure 1-7. Energy band diagrams for silicon (above) and gallium arsenide (below).
Notice that Si has an indirect bandgap while GaAs has a direct bandgap. This explains
why GaAs is a much better light emitter than Si [2].............................................................. 10
Figure 1-8. Cross section of the device designed by the Intel researchers. It is a ridge Si
waveguide surrounded by SiO2. The silicon on insulator (SOI) technology used allows for a
large mode confinement within the Si waveguide so that high enough Raman amplification
is reached. The p and n type Si on either side help to draw electrons generated by two
photon absorption out of the device and reduce parasitic losses such as free carrier
absorption (FCA) [11]............................................................................................................ 12
Figure 1-9. Scanning electron microscope (SEM) image of a p-i-n diode waveguide used
for Raman amplification and lasing experiments [12]........................................................... 12
Figure 1- 10. Outline of the actual structure (s) to be studied in this thesis. ........................ 17
Figure 1-11. (a) Complete diagram of the device. The upper left shows a top view of the
ring resonator which is used to accomplish frequency selective feedback and amplification
as well as output coupling. The bottom shows a cross section of the cavity. The upper right
shows a zoomed in picture of the gain medium and the surrounding layers. In this work, the
gain medium is actually composed of alternating layers of nc-Si and SiO2:Er as depicted in
vii

8. both Figure 1-10 and in (b). This particular diagram shows the horizontal slot ridge design.
Electrons and holes are injected by tunneling through the p and n contacts into the nc-Si
layers where they will form excitons. Also visible is the ring resonator structure used to
achieve frequency selective feedback and amplification as well as output coupling [13]. .. 18
Figure 2-1. A monochromatic plane wave incident upon a dielectric interface. The subscripts
‘i’, ‘r’, and ‘t’ denote the incident, reflected and transmitted portions of the wave. The
diagram shows the case of TE polarization meaning that the E field is directed out of the
page in all regions. 27
Figure 2-2. Representation of three possible scenarios for light incident upon a dielectric
interface for the case when n1>n2. At left is the case for θi<θc. There will be a transmitted
ray in this case. The figure in the center shows the case for θi=θc in which there will be a
transmitted ray along the interface itself. Finally TIR is shown at right when θi>θc. There
will be no transmission. This is the physical basis for waveguides....................................... 29
Figure 2-3. A basic asymmetrical planar slab waveguide. Note the orientation of the axes.30
Figure 2-4. A diagram illustrating the relationship between wavevector k, transverse
wavevector κ, and longitudinal wavevector β. ...................................................................... 32
Figure 2-5. Graphical solution of the transcendental equation (2.24) for an asymmetrical
waveguide. Each intersection of the red and blue graphs (aside from the vertical lines
where the value of tangent is infinite) represents a possible mode........................................ 34
Figure 2-6. Graphs (a) – (d) show pictures of the cross section of the E field for modes 0–3
respectively. The value of β for each mode is also given on the graphs. The evanescent
decay into the substrate and cladding is shown by the blue and green sections of the plots
respectively. It can be observed that higher order modes penetrate further into the substrate
and cladding........................................................................................................................... 36
Figure 2-7. Picture of the two dimensional slot confined waveguide proposed in [32]. It is
essentially two planar slab waveguides placed in close proximity in a low index SiO2
background [32, 33]............................................................................................................... 38
Figure 2-8. Graph of the x component of the transverse E field of the fundamental TM
mode across the regions of the slot confinement waveguide. The blue sections show the
evanescent decay into the cladding regions, the black sections show the distribution in the
waveguides and the red section shows the behavior within the slot. In this case the slot
width is 144nm while the guiding layers are each 180nm..................................................... 40
Figure 2-9. Figures (a) – (d) show graphs of the E field distribution of the transverse
fundamental TM mode for various slot widths and slot to guide layer thickness ratios.
Notice how as the slot thickness increases and consequently lowers the overall effective
index, a larger evanescent tail leaking into the cladding materials results. Thus the
confinement factor begins to decrease as the total amount of the low index material present
in the structure increases beyond a certain point. .................................................................. 45
viii

9. Figure 2-10. Graph showing the relationship between confinement factor and ratio of slot
width to waveguide width. The width of the guiding layers is kept constant at 180nm while
the slot width is chosen to fit the desired ratio. The CF increases up to a ratio of about 0.80
because the added low index material allows a larger amount of light in the desired region.
The values saturate between 0.80 and about 1.1.................................................................... 45
Figure 2-11. (a) Simplified representation of absorption, (b) spontaneous emission, and (c)
stimulated emission................................................................................................................ 48
Figure 2-12. A four level system. L3 and L1 are both unstable states so electrons rapidly
decay to L2 and L0, usually in a non-radiative transition...................................................... 50
Figure 2-13. (a) Simple Fabry-Perot cavity with mirrors on either end. Also shown is the
pumping (energy) source [9]. (b) Ring resonator structure which is the type of structure
used for the device in this work............................................................................................. 52
Figure 2-14. A simple energy band diagram for a bulk material (left) and a quantum-
confined nanocrystal (right)................................................................................................... 56
Figure 2-15. Diagram of the field-effect structure for achieving electrical injection [14].
The array of silicon nanocrystals exists in the light gray area directly beneath the gate....... 57
Figure 2-16. (a) Schematic energy level scheme of Er3+
showing the different types of
transitions that may occur. The energy levels are very sharp for the free ions. Doped into a
SiO2 lattice, the levels split into many discrete levels.(b) A more detailed view of the
4
I13/2
4
I15/2 transition. There are many discrete energy states in which electrons can exist
within each energy band. The electrons will obey Boltzmann statistics. The graphs on the
right show how the number of electrons (N(E)) changes with increasing energy (E). It can
be observed that absorption takes place from the most populated states in the lower band to
the least populated states in the upper band and vice versa for emission. This difference
accounts for the slightly different wavelength and frequency of photons involved in these
two processes. ........................................................................................................................ 59
Figure 2-17. Solid lines show the absorption spectrum of an Er-doped optical fiber for
wavelengths between 450-850nm. This shows the principal absorption bands within the
range of wavelengths. [36]..................................................................................................... 60
Figure 2-18. Photoluminescence (PL) emission spectrum for Er -doped (~5 x 1015
cm-2
)
silica film on a Si substrate when pumped with 488nm light. The wavelengths at which the
two peak intensities occur are labeled [37]............................................................................ 61
Figure 2-19. Diagram showing pumping of the 1.55µm transition in Er via Förster energy
transfer between nc-Si and Er. Non-radiative decay occurs from the I11/2 level to the I13/2
level. This is followed by radiative recombination from I13/2 I15/2 at 1.55µm..................... 62
Figure 3-1. Cross section of structure to be analyzed. T_C, T_Si and T_SiO2 denote the
thicknesses of the cladding, silicon and silicon dioxide respectively. The multilayer region
consists of layers between 15 and 30nm thick. The layers alternate between low index SiO2
ix

10. and higher index Si with Si forming both the top and bottom layers. It is flanked on top by
a cladding of SiO2 and is deposited on a substrate of the same material. The dimensions of
the cladding are set for all simulations while the slab height varies depending on layer
thicknesses and the SiO2 to Si ratio. Note the orientation of the x, y, and z axes…..67
Figure 3-2. Complete cross sectional diagram of the structure. The active gain medium is
actually composed of alternating layers of nc-Si and SiO2:Er. This particular diagram
shows a horizontal slot ridge waveguide design. Electrons and holes are injected through
the p and n contacts into the p and n type device layers. They then form excitons by
tunneling into the nc-Si layers. Also visible is the ring resonator structure used to achieve
frequency selective feedback and amplification............................................................... 68
Figure 3-3. (a) Slot confinement structure proposed by MIT [32]. The structure consists of
upper and lower high index cladding layers with alternating thin low index slot layers with
high index thicker guide layers in between. In this case, w is the width, tc, tH , and tl are the
thickness of the cladding, low index slots and high index layers respectively. The
refractive indices of the high and low index layers are nH and nl respectively while n0 is the
refractive index of the surrounding air. It operates on the principle of confinement within
low index slot layers but is much more difficult to fabricate owing to the extremely high
precision required to etch the sidewalls. (b) Cross section profile showing the quasi TM
modal distribution of the y component of the E field in the structure proposed in [32]. Also
shown is the distribution of the y component of the E field along the structure (on the right
side)................................................................................................................................... 69
Figure 3-4. Outline of the structure and picture of the major (y) and minor (x) components
of the E field of the fundamental TM eigenmode looking along the z direction found using
the BeamProp mode solver. This is a structure with N = 9 periods, T_Si = 20nm and a ratio
of 1.00. The slab height is 0.380µm. The high concentration of the field in the low index
layers is clearly visible...................................................................................................... 72
Figure 3-5. 3D picture of the power distribution of the TM mode in a structure with N = 9
periods, T_Si = 20nm and a ratio of 1.00. The high power confinement in the low index
layers is readily apparent here. Also noted is the relatively large tail showing a significant
amount of leakage of the mode into the low index substrate............................................ 74
Figure 3-6. 2D cross section at x=0 of the power distribution of the TM mode of the same
structure as in Figure 3-5. The SiO2 layers are labeled along with the other regions of the
structure. The long tail extending into the substrate is more discernable in this figure. .. 75
Figure 3-7. Cross section of the z component of power distribution of the TM mode across
the layers in the structure shown in Figure 3-1. Again the high confinement in the low
index layers is evident. Note also the oblong shape of the mode as it propagates down the
structure. The evanescent tail in the substrate is visible here as in the other two figures. 76
Figure 3-8. Confinement factor versus SiO2/Si thickness ratio of the TE and TM modes for
both the FDTD and TMM methods. The FDTD plots are shown as crosses and circles and
x

11. are for slab heights between 0.370µm and 0.405µm. The TMM plots are in dashed and
solid lines and are for a slab height of exactly 0.380µm................................................... 77
Figure 3-9. Confinement factor versus SiO2/Si thickness ratio of the TE and TM modes for
the TMM method. It exhibits a similar trend to Figure 3-8. The ideal ratios fall within the
same range as for the 0.380µm structure with the maximum being about 75% occurring at a
ratio of about 1.1............................................................................................................... 79
Figure 3-10. Plot of overall gain G (in cm-1
) versus gain to loss coefficient ratio for the
TMM method for both the TE and TM modes. The zero gain value for TE modes is about
2.45 and about 0.25 for TM modes................................................................................... 83
Figure 3-11. Plot of overall gain in the optical cavity, G (in arbitrary units) versus gain to
loss coefficient ratio for the FDTD method for both the TE and TM modes. The net gain
value for the TE modes is about 2.5 and about 0.25 for TM modes................................. 84
xi

12. Foreword
This thesis was undertaken as part of a multi university research initiative (MURI).
The project involves all aspects in the design and development of a complete electrically-
pumped silicon based laser for chip-scale nanophotonic systems. My research was
primarily concerned with the design of a waveguide for the optical cavity that would
provide high mode confinement, minimize losses and realize lower threshold current
densities. I worked closely with a fellow member of my research group, Yjing Fu. He
helped me with many of the design aspects with which I was unfamiliar. He also developed
the TMM algorithm and provided the data for comparison in this work. There is currently
an unpublished article based in part on this research. More complete details regarding this
article can be found in the Bibliography.
1

13. Chapter 1
Introduction
1.1 Motivation
The communications industry has in recent years seen a rapid rise in the demand for
high speed, high bandwidth and low loss connections at all levels of system integration.
This increased demand necessitates better performance, particularly at the chip-to-chip and
datacom levels which almost exclusively use electronic circuitry. Many experts have
predicted that Moore’s law will break down as device dimensions continue to shrink.
Resistor-capacitor coupling, noise, and other parasitic effects contribute to the bandwidth
bottleneck for interconnects. These effects are further agitated on smaller scales. As data
rates and performance demands continue to increase, this poses a serious problem.
Generating, transmitting, receiving, and processing data in the form of optical signals
provides a viable solution to this problem. The advantages of optical systems are abundant
and their continued development and implementation is held back by only a few remaining
obstacles – chief among them being cost. The ability to leverage the extensive knowledge
and manufacturing capabilities of a well developed and well established microelectronics
industry centered on silicon provides the most plausible path to success in this regard. The
major objective remains the development of systems based primarily on optical signals and
fabricated in silicon materials in such a way that a gradual transition or partitioning
between the two technologies can occur.
2

14. 3
It is appropriate to begin with an overview of the major advantages afforded by
optical solutions over their respective electrical counterparts. Two of the most obvious
improvements in an optical system are the reduction or complete elimination of
electromagnetic interference (EMI) and noise as well as immunity from short circuits.
These are inherent problems that have plagued electrical systems for years. Examples of
EMI include cross talk between adjacent cables in a bundle, stray wires or electromagnetic
radiation in the atmosphere. It is impossible to completely remove either of these from such
systems. Metallic shielding is one approach but it adds extra weight, cost and parasitic
capacitance which limits the bandwidth – all significant enough drawbacks to warrant
consideration as to whether or not it is necessary in the first place [1]. Another minor yet
critical benefit of optical fibers is that they are much more difficult to tap and therefore
more secure.
The most significant advantages offered by optical solutions are: 1) lower loss
transmission at higher frequencies, 2) larger bandwidth (multiplexing capabilities), and 3)
smaller size, weight and overall power consumption (at smaller interconnection distances).
As it has been previously noted, electrical interconnects are currently the better choice for
systems at the chip level but at higher data rates and larger distances optical links are far
superior. The hope is that this will be able to scale to the chip level as optical circuits based
in Si become more of a reality. There are a few other major factors hindering a full
transition. These are the need for: 1) wavelength independence over the entire
communication spectrum, 2) polarization independence and 3) higher optical power levels
– all of which could be addressed utilizing nonlinear effects in Si [1]. In order for this to
take place, many of the current problems confronting optoelectronic integrated circuits
(OIC) will require a variety of innovative solutions to meet the ever increasing data and
speed requirements of the rapidly evolving global communications network.
As the lowest level of integration, chip-to-chip optical circuitry will realize the
greatest benefit and also presents the greatest challenge due to the extreme conditions
inherent at smaller dimensional sizes such as high temperatures which create the problems
with resistor-capacitor coupling, bandwidth limitations and durability. Optical interconnects
become increasingly competitive at higher levels and longer transmission distances where
their electrical counterparts experience greater losses due to parasitic effects. For data rates

15. 4
in excess of 1Gb/s and distances of greater than 100m, the use of optical interconnects is
currently standard practice with the slightly higher manufacturing cost easily justified by the
large increase in performance. At more intermediate distances of 1 – 100m there are
solutions in both forms. A diagram showing the applications of different interconnect
technologies is presented in Figure 1-1 [2]. Figure 1-2 [2] is a timeline of the information
carrying capacity of various electrical and optical links. It is clear that optical systems afford
much higher data rates than even the best electrical systems.
Figure 1-1. Diagram showing various applications of different interconnect technologies. Optical links are
currently only used at longer distances (>100m) [2].

16. 5
Figure 1-2. Timeline of information carrying capacity for links in various electrical and optical links. Those
in optical systems have data rates roughly 2 orders of magnitude higher than those in electrical systems [2].
The need for higher data rates and bandwidths places increasing strain on current
electrical interconnects. The degradation of performance, particularly at the chip-to-chip
level, due to the trend towards smaller dimensions and higher temperatures has proven to
be a hindrance to the continued progress of the microelectronics. A diagram showing the
relationship between processor and front side bus (FSB) clock speeds for different models
within the past 15 years is shown in Figure 1-3 [2]. Recently, the potential of optical
integrated circuits (OICs) has been realized due to an increased need for higher
performance, especially on smaller scales. Despite their aforementioned drawbacks
however, electrical ICs remain the most cost effective and reliable solution particularly at
the chip-to-chip level largely due to highly developed fabrication methods and technologies
along with the readily accessible infrastructure that has been firmly established over the
years.

17. 6
Figure 1-3. Relationship between processor and front side bus (FSB) clock speeds for various models over
the past 15 years. Note the significant improvement in processor speeds over FSB speeds in recent years [2].
Until recently, most optical devices have been made from what are known as exotic
materials such as gallium arsenide (GaAs) or indium phosphide (InP) owing to their
excellent optical properties. These materials are both expensive and complicated to process,
however. Before integrated optics became a viable option, most devices were discrete
components which were hand-assembled and required extreme attention to detail as far as
aligning them to fiber connectors to maximize light coupling. Thus there is often a
significant cost disadvantage when working with optical systems and devices. Complete
integration of optical components fabricated in silicon would eliminate the need for such
precision by automation of this process with a subsequent reduction in cost. The hope is
that specific components can be fabricated to perform all the necessary functions in the
transmission and reception of optical data with the end result being a completely optical-
based system on a single chip. Each of the necessary functions in this system can be
thought of as an optical analog to a similar electrical function accomplished with transistors.
These functions include:
1) light generation
2) light guiding

18. 7
3) light detection
4) signal multiplexing
5) high speed (≥ 1 GHz) modulation
6) low cost assembly
7) high coupling efficiency to optical fibers.
A block diagram for a typical transmitter-receiver system is shown in Figure 1-4 [3, 4].
Figure 1-4. Diagram showing individual components of an optical transmitter-receiver system. The
transmitter is the top diagram and the receiver is the bottom [4].

19. 8
Si has been shown to be adept at all of these functions except for light generation – the
major barrier that must be conquered. Recent results by Intel have shown devices capable
of high speed modulation [5]. Si based components are great for light guiding solutions
because Si is transparent (its bandwidth is larger than the energy of the photons) at the
infrared wavelengths used in communication systems, namely 1.55µm. However, Si is
notorious for high free carrier absorption losses and Auger processes – problems which,
along with poor light emission, have plagued Si-based systems for years. This is part of the
tradeoff when utilizing a material with such excellent electrical properties. In order for
photonic components to take hold at the lowest level of integration, performance demands
must outweigh cost drawbacks as is the case with long haul telecommunications links. If a
transition to exclusively optical-based communication systems is to take place, it will most
likely take place through slow developments and advances in fabrication techniques and
device technologies as performance demands increase and costs decrease.
Demand for superior performance will continue to increase as it has in the past and
will continue to do so at a faster rate in the future. Global communication networks are
expanding rapidly every day and the need for large quantities of information in short
periods of time has never been greater. The burden thus falls on innovation of new
fabrication techniques and design technologies to help precipitate the transition,
particularly in terms of the information processing level (as opposed to information
transmission which already employs optical solutions). With the microelectronics industry
firmly centered on silicon and baseline CMOS fabrication processes, these innovations will
almost certainly involve Si-based components. Although hybrid integration involving III-
V-based semiconductors or other types of materials is possible, they are highly unlikely to
firmly take hold since Si is by far more plentiful, inexpensive and well understood [1, 2].
Evidence of how much cheaper fabrication in Si is than in III-V materials is shown in
Figure 1-5 [2]. At best they will provide alternative or customized solutions. A hypothetical
timeline showing the transition between electrical and optical-based systems for
communication over various distances is visualized in Figure 1-6 [2]. Some experts in the
area of silicon photonics instead foresee a merging of the two technologies and believe that
future systems will rely on parallel solutions using smart electronic-photonic partitioning.

20. 9
Figure 1-5. Wafer cost per unit area for various materials versus wafer size. Notice that InP is currently only
available in one wafer size and is prohibitively expensive compared to a Si wafer of the same size [2].
Figure 1-6. Hypothetical timeline for the transition between electrical and optical based communication
systems for various distances [2].

21. 10
The inability of Si to behave as an efficient light emitter is due to the indirect nature
of its band structure. The energy band diagrams for both Si and GaAs are shown in Figure
1-7 [2]. Notice that GaAs has a direct bandgap which helps explain why it is a very
efficient light emitter, while Si does not. Numerous attempts have been made over the
years to circumvent this problem, most with only varying or irreproducible success.
Bandgap engineering (controlling the band structure with alloys and Brillouin zone folding
in superlattices are two notable methods) and enhanced quantum confinement via quantum
wells, wires and dots are among the more successful and notable approaches in recent years.
See Chapter 1 in [2] and references therein for a complete overview of these techniques.
Figure 1-7. Energy band diagrams for silicon (above) and gallium arsenide (below). Notice that Si has an
indirect bandgap while GaAs has a direct bandgap. This explains why GaAs is a much better light emitter
than Si [2].

22. 11
It should be mentioned that while getting Si to emit light as efficiently as a direct
gap semiconductor is somewhat difficult and may never be achieved, light amplification
and lasing have in fact been achieved in Si, mostly as a result of the phenomenon of
stimulated Raman scattering (SRS). This phenomenon arises from the Raman Effect
whereby photons from a pump beam transfer their energy to longer wavelength photons
through interaction with vibrating atoms within a material. Spontaneous Raman scattering
and emission at 1540nm in Si waveguides was first observed and reported in 2002 by a
group at UCLA [6]. This was followed by a report of SRS and amplification in a 2003
publication [7]. Finally the first successful demonstration of lasing at 1675nm in a Si
waveguide [8] was given in 2004. One of the major drawbacks of the initial Raman laser
was that at the high pumping powers required for Raman amplification to achieve lasing,
the two photon absorption problem becomes more significant. Two photon absorption
(TPA) occurs when two incoming photons come together at the same exact spot in the
crystal lattice and provide enough energy to raise an electron from the valence band to the
conduction band. This is not possible for single photons at the infrared wavelengths
normally used in silicon photonic devices because they do not provide enough energy for
the electrons to cross the bandgap. Once in the conduction bands, the electrons then
become free carriers and contribute to the problem of free carrier absorption (FCA) at
higher power levels by absorbing light propagating through the waveguide and weakening
the overall signal strength in the device. At high pump intensities, TPA is enough to either
cancel out the Raman gain or prevent SRS altogether. Because of this, only pulsed-pump
operation was possible with the UCLA design. In 2006 a group of Intel researchers got
around this problem by surrounding the silicon waveguide with p and n type silicon to
effectively form a PIN diode [9-12]. When the diode is reverse biased, the free carriers
(electrons) generated by TPA are ‘swept’ away by the electric field. This reduces the free
carrier absorption losses enough to allow continuous operation. Diagrams are shown in
Figure 1-8 and Figure 1-9 [11, 12].

23. 12
Figure 1-8. Cross section of the device designed by the Intel researchers. It is a ridge Si waveguide
surrounded by SiO2. The silicon on insulator (SOI) technology used allows for a large mode confinement
within the Si waveguide so that high enough Raman amplification is reached. The p and n type Si on either
side help to draw electrons generated by two photon absorption out of the device and reduce parasitic losses
such as free carrier absorption (FCA) [11].
Figure 1-9. Scanning electron microscope (SEM) image of a p-i-n diode waveguide used for Raman
amplification and lasing experiments [12].

24. 13
The important thing to realize about these results however is that SRS is a non-
linear process that involves the emission or absorption of a phonon. This means that the
overall efficiency for this type of lasing will be much lower than that which relies primarily
on first order stimulated emission processes. Also, it requires high powered pumping from
an additional source of laser light (usually a mode-locked fiber laser) which complicates
the fabrication process and means that the device will not be exclusively based in Si, both
of which will increase cost. It also exacerbates problems like two photon absorption and
free carrier absorption as noted. Devices based on SRS are still being actively studied
however, and in all likelihood will play a role in the future of silicon photonics.
There is one other recent breakthrough that warrants mention. In 2006 a
collaborative team from both Intel and the University of California at Santa Barbara
designed and fabricated an electrically pumped hybrid Si laser. Being the first electrically
pumped Si laser, this was a major development. Instead of relying on SRS which
necessitates optical pumping, it achieves lasing by electrical pumping – injecting electrons
and holes into InP which is a direct bandgap material and an efficient light emitter. Light
generated in the InP is coupled to the Si waveguide layer which sits underneath. The device
is hybrid because it is actually InP that directly emits and amplifies the photons of light [3].
One of the main advantages of this technique is that it eliminates the need to align
external light sources with the Si chip by either pre-fabricated lasers on chip or optical
fibers off chip, both of which would not be practical for low-cost, high-volume production.
The novelty of the design is based on a new way of bonding together the InP and Si
substrates. The main problem with growing III-V materials like InP onto Si is that they
have a large lattice mismatch and dislocations develop which negatively impact device
performance and reliability. The new bonding technique uses a low temperature plasma
enhanced oxidation process that acts as a ‘glass glue’ between the two materials. It allows
InP substrates to be bonded directly to pre-patterned Si photonic chips without the need for
alignment. This technique may prove to be very successful as a solution for large scale
optical integration onto a Si platform by simplifying the manufacturing process and
reducing cost. It is a hybrid approach however, so it remains to be seen whether or not it
will be able to realize the low cost – high volume potential that a monolithic approach

25. 14
could simply because both InP and other III-V materials are more expensive and not as
abundant or as well understood as is Si [1].
The inability to achieve efficient light emission in Si based components and devices
– aside from the fact that a transition to technology based completely on hybrid materials
would not be practical in terms of cost – remains the major impeding factor to the
successful implementation of low cost-high volume optical integration at all system levels.
It will be seen that the use of nanocrystalline silicon and its novel energy transfer properties
with erbium ions in close proximity poses a very viable solution to this problem.
1.2 MURI Silicon Laser Project
A Multi University Research Initiative (MURI) is a large scientific project
composed of a collection of university research teams and possibly partners from national
or industrial labs. For this particular MURI project, the primary task at hand is the design
and development of practical and useful Si-based photonic components – particularly a
laser system. The institutions involved are listed in alphabetical order as follows:
Boston University
California Institute of Technology
Cornell University
Lehigh University
Massachusetts Institute of Technology
Stanford University
University of Delaware
University of Rochester
Also taking part are researchers from the McMaster University, University of
Toronto and the Army, Navy, and Air Force research laboratories.
The technical approach as stated in the executive summary reads:
“Our program will deliver a nanophotonic, silicon-based laser for monolithic, CMOS-
compatible integration with electronic and photonic chip-level circuits. The silicon-
based laser must be electrically pumped; it must have a high wallplug efficiency and
low heat dissipation; it must have a long operating lifetime; and it must emit at a carrier
frequency and output power that meet the expectation of a > 10Gbit/s optical

26. 15
communications link, operating at a Bit-Error Rate tolerance < 10-12
. Two principal
challenges frame the science and engineering research context of this program: high
lasing efficiency cavities and high gain efficiency media [13]”
The program is organized into three major tasks, each with supporting subtasks. The
first task is an extrinsic gain laser with the subtasks including a slot waveguide structure
with Er-doped oxide and a traditional index guided structure with Er-doped silicon nitride
and quantum dot doped oxides. The second task is an intrinsic gain laser with the subtasks
of a conventional edge emitter structure with bulk and quantum-confined germanium (Ge),
and a conventional edge emitter structure with strained and alloyed Ge. Finally, the third
task is the design of the application drivers. The subtasks are a CMOS compatible Er-
doped optical micro-amplifier for 1550 nm light, a CMOS-compatible optical signal
processor link and a CMOS-compatible, multi-wavelength optical power supply.
In particular this thesis deals with the design and optimization of a structure falling
under the first subtask of the first major task. A table summarizing the four laser design
approaches for this initiative with their compositional materials, device parameters,
projected performance and risk assessment is presented in Table 1-1 [13]:

27. 16
Table 1-1. The four design approaches for the MURI initiative with their specific parameter values.
1.3 Project Overview – Objectives, Approaches, Tools Used
This project is concerned with the design, modeling and optimization of a slot-
confined, multilayer waveguide structure to be used as the optical cavity for a silicon-based
laser system. It consists of alternating nanometer-thin layers of amorphous Si (a-Si) and
erbium (Er)-doped silicon dioxide (SiO2:Er) with a cap layer and substrate both composed
of SiO2. A carefully chosen thermal budget during fabrication will result in the formation
of silicon nanocrystals in the Si layers producing nanocrystalline Si (nc-Si). An outline of
the actual structure analyzed can be seen in Figure 1- 10. A more complete picture of the
overall system of which it is to be a part is shown in Figure 1-11[13]. There is one minor
clarification with this figure however: instead of a random array of nc-Si and Er3+
ions in a
SiO2 matrix, the gain medium (the central layer in the figure) consists of distinct multiple
layers of SiO2:Er and nc-Si [13]. This is also shown in Figure 1-11.

28. 17
1µm
Cap Layer50nm
T_C(SiO2)
Figure 1- 10. Outline of the actual structure (s) to be studied in this thesis.
(a)
Substrate
(SiO2)
nc-Si
SiO2:Er
T_SiO
T_Si
Multilayer Region
y
Slab Height
~370 – 400nm z
x
SiO2:Er
nc-Si

29. 18
(b)
SiO2:Er (low index)
nc-Si (high index)
Figure 1-11. (a) Complete diagram of the device. The upper left shows a top view of the ring resonator
which is used to accomplish frequency selective feedback and amplification as well as output coupling. The
bottom shows a cross section of the cavity. The upper right shows a zoomed in picture of the gain medium
and the surrounding layers. In this work, the gain medium is actually composed of alternating layers of nc-Si
and SiO2:Er as depicted in both Figure 1-10 and in (b). This particular diagram shows the horizontal slot
ridge design. Electrons and holes are injected by tunneling through the p and n contacts into the nc-Si layers
where they will form excitons. Also visible is the ring resonator structure used to achieve frequency selective
feedback and amplification as well as output coupling [13].
The high index contrast at each of the Si/ SiO2 interfaces allows for a large
concentration of light within the low index (SiO2) layers. The idea is to use field effect
injection to bring electrons and holes into the surrounding p and n type device layers and
then rely on tunneling to allow them to travel into the nc-Si layers and form excitonic pairs
[14]. The nc-Si will allow efficient excitation and energy transfer to the Er atoms through a
dipole-dipole energy coupling interaction. This dipole-dipole energy coupling phenomenon
is known as Förster energy transfer. Essentially the nanocrystals provide a convenient
means of exciting the Er. This gives rise to light emission from Er at 1550nm – the
operating wavelength for most currently active optical communications links in the world.
Erbium is chosen for this reason as well as the aforementioned energy coupling capabilities
with silicon nanocrystals, and because it is generally well understood based on its use in
erbium doped fiber amplifiers (EDFAs) since the advent of the fiber optic section of the
telecom industry. The ability to confine a high percentage of light within the SiO2 layers
containing Er ions enables the device to more easily generate the gain necessary to achieve

30. 19
lasing. In this way, the electronic properties of nanocrystalline silicon and the light emitting
capabilities of Er are each leveraged in such a way as to produce laser light emission in a
silicon based structure.
Due to the complex nature of the proposed structure, the use of numerical
simulation software becomes necessary in obtaining an accurate picture of the propagating
modes. These simulations are done using the RSoft Design Group, Inc. Photonics Suite and
in particular the BeamProp and FullWave tools. Both BeamProp and FullWave are
software packages designed to compute and model the propagation of light waves in
arbitrary geometries. Both are inherently based on finite difference computational methods.
BeamProp uses the Beam Propagation Method while FullWave uses another well-known
technique called the finite-difference time-domain (FDTD) method. Both techniques are
considered very robust and efficient at giving accurate pictures of how light propagates in
even the most complicated photonic devices. A full explanation with mathematical details
is not within the scope of this document but interested readers are referred to the numerous
publications on such methods [15-28]. An understanding of these numerical methods is not
critical to the full comprehension of the concepts presented but references are given for the
sake of completeness. It should be noted that the BeamProp package was used only for its
mode-solving capabilities while FullWave was used for actual simulations. Matlab was
used for data analysis and visualization purposes.
Simulations of the proposed structure are performed using different periods of
repeating Si/SiO2 layers and different thickness ratios between the Si and SiO2 layers. The
raw data from these simulations is then read into Matlab where it is used to calculate the
confinement factor (CF) within the SiO2 layers for each structure. By examining simpler
2D slot confined waveguide structures and something known as the effective index method,
it is possible to predict that the thickness ratio between the Si and SiO2 layers is the critical
parameter affecting the CF. This hypothesis is confirmed by analysis of simulation results.
Based on these results, a few optimum structures are chosen and a gain/loss
performance study is undertaken. This entails simulations of each structure, with loss
inserted into the Si layers to model the free carrier absorption effects in silicon, and gain
within the SiO2 layers to model the light emission from erbium. The ratio of the gain in the
SiO2 to loss in the Si is then varied over a range of values beginning with zero. For each

31. 20
simulation, light waves are propagated along the entire length of the structure (~10µm).
The maximum power at a point just shy of the end (~9.5µm) is recorded for simulations
with no gain or loss (P0) and then compared to those with both loss and gain in varying
ratios (Pmon). The net gain throughout the propagation (G) can be calculated from these
values. Repeating these steps for both transverse magnetic (TM) and transverse electric
(TE) polarizations and then plotting a graph of net gain (G) versus the gain/loss coefficient
ratio reveals a linear relationship. The value of the ratio for which G = 0 is the cutoff ratio
of gain in SiO2 to loss in Si for which net gain is observed. This in turn can be used to
realize a reasonable approximation for the minimum injection current density in the system
– a useful parameter in terms of minimizing the overall power consumption of the device
and improving its overall efficiency.

32. Chapter 2
Theory
Any theoretical description of propagating light waves begins with a discussion
of Maxwell’s equations and the solutions to the electromagnetic wave equation in an
isotropic, linear and source free medium. This is followed by explanations of the
boundary conditions at a dielectric interface and total internal reflection (TIR), and a
discussion of wave propagation in guiding media. Two simplified 2D waveguide
examples are then presented. First is a three layer slab waveguide with graphical
depictions of the solutions for the eigenmodes of propagation. Second is a slightly more
complicated slotted waveguide, also with graphical results. The idea of optical
confinement within the guiding layers is established alongside these examples. The
graphs are intended to give an indication as to both the advantages and limitations
inherent in the slotted structure. A very well-written and thorough introduction to this
area of electromagnetism can be found in chapter 9 of [29].
2.1 Maxwell’s Equations
The packaging of four previously derived relationships (Gauss’ Law, Gauss’
Law for magnetism, Faraday’s Law and Ampere’s Law) into a compact and consistent
form and the slight correction provided for one of them (Ampere’s Law) by James
Maxwell is without question among the crowning scientific achievements of the 19th
century. Maxwell’s equations are presented below in differential form. The integral
21

33. 22
forms can be found in almost any text on electromagnetism and can be easily derived
from the differential forms using either Gauss’ divergence theorem or Stokes’ theorem
[30].
t∂
∂−
=×∇
B
E (2.01a)
J
D
H +
∂
∂
=×∇
t
(2.01b)
0=⋅∇ B (2.01c)
ρ=⋅∇ D (2.01d)
In order from top to bottom they are: (a) Faraday’s Law, (b) Ampere’s Law with
Maxwell’s correction term, (c) Gauss’ Law for magnetism, and (d) Gauss’ Law. A table
summarizing the various quantities and their units in MKS form is given in Table 2-1:
Quantity Description Units (MKS)
E Electric field amplitude Volts/meter (V/m)
H Magnetic field amplitude Amps/meter (A/m)
D Electric flux density Coulombs/meter2
(C/m2
)
B Magnetic flux density Webers/meter2
(W/m2
)
J Current density Amps/meter2
(A/m2
)
ρ Charge density Coulombs/meter3
(C/m3
)
Q Charge Coulombs (C)
Table 2-1. Various quantities used in electromagnetics and their units expressed in MKS units [30].
The constitutive relationships between the flux densities (D and B) and the field
amplitudes (E and H) are determined by the properties of the medium through which the
waves are propagating. In simple linear and isotropic media1
, these are written as,
HB µ= and ED ε= (2.02)
1
This is only true for monochromatic fields.

34. 23
where ε is the electric permittivity of the medium in Farads/meter and µ is the magnetic
permeability of the medium in Henrys/meter. For reference, the accepted values for
permittivity and permeability in vacuum are given as
and (2.03)Farads/m10854.8 12
0
−
×=ε Henrys/m104 7
0
−
×= πµ
For the sake of simplicity and relevance, this section deals only with linear and isotropic
media – that is, materials in which the index of refraction does not depend on the
strength of the fields or on the direction of propagation within the material. Maxwell’s
equations in a linear, isotropic and source free medium (ρ = 0, J = 0) are given below.
t∂
∂−
=×∇
B
E (2.04a)
t∂
∂
=×∇
D
H (2.04b)
0=⋅∇ B (2.04c)
0=⋅∇ D (2.04d)
Note that equations (2.04a) and (2.04c) are unchanged. Also note the nice
symmetry that results in this case. These equations are now much easier to work with
and put into a more useful and insightful form.
2.2 The Wave Equation
Using equations (2.04) the wave equation can be derived in a fairly
straightforward manner. The approach involves decoupling the two curl equations for B
and D by first taking the curl of both sides of (2.04a). Substituting µH for B and further
realizing that the time derivative of H can come outside of the curl operation yields
)()( HE ×∇
∂
∂
−=×∇×∇
t
µ , (2.05)
assuming that µ and ε are both time- and position-invariant. Applying a vector identity
to the left hand side of this result and using (2.04b) and (2.02) gives
2
2
2
)()(
t∂
∂
−=∇−⋅∇∇=×∇×∇
E
EEE µε . (2.06)

35. 24
By further analysis the first term on the left hand side of the middle equation can be
shown to be effectively negligible [30]. This produces the standard form for the wave
equation:
02
=
∂
∂
−∇
t
E
E
2
µε . (2.07)
This same equation can be derived for H by starting with equation (2.04b) instead of
(2.04a).
It is also important to note that the wave equation reproduced here is actually a
vector equation which involves the vector Laplacian. It can be easily simplified to a
scalar equation for a rectangular coordinate system however. In a non-rectangular
coordinate system it is usually more difficult to simplify the vector equation because of
the need to break it into orthogonal components. Every structure presented in this work
lends itself readily to a rectangular coordinate system, so (2.07) is sufficient. A more
detailed derivation of the wave equation can be found in reference [30].
Solving the vector wave equation involves first choosing an appropriate
coordinate system, as mentioned above. For a simple rectangular system the scalar wave
equation can easily be found for each vector component and then solved by separation
of variables to get
)exp(),( tjj(t)(t) 0 ωψφψψ −⋅== rkrr , (2.08)
where ψ0 is the amplitude, ω is the temporal frequency and k is the wavevector in
radians/second. The magnitude of k is written as
µεω=k . (2.09)
This is known as the dispersion relationship for a light wave. The wavevector is defined
as k = 2π/λ, where λ is the wavelength of the light. It can be thought of as a ‘spatial’
frequency. The solutions to the scalar wave equation in a rectangular coordinate system
represent a sinusoidal wave in both space and time. Since this same equation can be
solved for H just as easily, it confirms the origin of the term electromagnetic radiation.
Maxwell’s equations can be decoupled to produce two distinct forms of the wave
equation – one each describing the magnetic and electric fields, each of which oscillates

36. 25
through space and time. It can further be proven that E and H are orthogonal to each
other2
[29].
Through examination of the wave equation, it can be shown that the velocity of
the wave in a medium with unique values for µ and ε is
µε
1
=v , (2.10)
where v is the phase velocity.
Using the values of permittivity and permeability for vacuum which are given in
(2.03), the value for the speed of light (2.99 x 108
m/s) can be calculated. This leads to a
definition for the index of refraction of a material as
000 ε
ε
εµ
µε
==n , (2.11)
when µ = µ0. Thus the index of refraction is exactly one in vacuum and is very close to
one in air. This quantity determines the factor by which the speed of light is decreased
when traveling through a particular medium. It will be seen in the next section that it
also determines the manner in which light bends or refracts when passing from one
medium into another.
2.3 Boundary Conditions at Dielectric Interfaces
Before taking on a discussion of light propagation in a guiding medium such as a
waveguide, it is essential that the behavior of light waves at a dielectric interface be first
understood. A dielectric interface occurs at the boundary between materials with two
different indices of refraction (n1 ≠ n2). The general rule for the wave solutions in the
two regions is that they must be continuous across the boundary in some way. Specific
boundary conditions, presented below, explain the exact nature of this continuity for
each type of field or flux quantity.
When electromagnetic radiation is incident upon an interface of any type, it will
undergo one of the following phenomena or some combination thereof:
Transmission – no interaction of light with the new material
2
This is only true when the fields are in a plane wave and does not apply in general.

37. 26
Absorption – energy lost as transmitted light interacts with impurities in the new
material
Reflection – light is re-directed back into the original material
Refraction – light passes into the new material but with an altered path
Technically there is no such thing as pure transmission. There is some degree of
absorption in almost any type of material and at different wavelengths of light.
Absorption becomes important when modeling the loss mechanisms within a waveguide
and is covered in a later section of this chapter. The two remaining phenomena will be
of primary concern in discussing the behavior of light at a dielectric interface.
The diagram in Figure 2-1 shows a simple case of a plane wave of light incident
on a dielectric interface with electric field (E), magnetic field (H), and wavevector (k)
all labeled. The subscripts ‘i’, ‘r’, and ‘t’ stand for incident, reflected and transmitted
portions of each field quantity respectively. The direction of the wavevector denotes the
direction of propagation for each ray and can be determined through calculation of the
cross product between E and H or more simply by the right hand rule [29]. In this
diagram, the E field points directly out of the page for all rays and the H field is oriented
perpendicular to this, as required by Maxwell’s equations. This is known as the
transverse electric (TE) polarization. Polarization is discussed in the next section.
Using the integral form of Maxwell’s Equations, the aforementioned boundary
conditions can be derived [29]. They are listed below for a source-free medium.
.0)(ˆ
,0)(ˆ
,0)(ˆ
,0)(ˆ
12
12
12
12
=−⋅
=−⋅
=−×
=−×
DD
BB
HH
EE
s
s
s
s
(2.12)
In this case, s is a unit vector that is normal to the surface. A simple way of
summarizing these rules is: “D-B normal, E-H tangential”, meaning that the normal
components of D and B are continuous across the boundary, as are the tangential
components of E and H.
ˆ
Using these continuity rules for the fields, three additional relationships can be
derived [29]. The first of these simply states that the incident, reflected, and transmitted
wave vectors form a ‘plane of incidence’ which also includes the normal to the surface

38. 27
(in this case, the z axis). The other two are also known as the law of reflection and the
law of refraction (Snell’s Law). These are given below and also visualized in Figure 2-1.
k
n n
E
k
E
θ θ
Ht
θ z
k
E
H
y
Figure 2-1. A monochromatic plane wave incident upon a dielectric interface. The subscripts ‘i’, ‘r’, and
‘t’ denote the incident, reflected and transmitted portions of the wave. The diagram shows the case of TE
polarization meaning that the E field is directed out of the page in all regions.
ri θθ = , (2.13a)
ti nn θθ sinsin 21 = . (2.13b)
These relationships, along with the original boundary conditions can further be
extended to derive general equations relating the reflected and transmitted field
amplitudes to the incident field amplitude for both E and H as a function of θi [30].
These equations are not given since they are not as pertinent to the nature of this work,
however they can be found in chapter 1 of [30]. The physical implications of these four
relatively simple boundary conditions on the electric and magnetic fields and flux
densities are astonishing – they essentially lead to the two most fundamental laws of
optics. They also have other important consequences as the remainder of this chapter
will address.

39. 28
2.4 Field Polarization
Another key point to hit on is the orientation of either the electric or magnetic
field with respect to the interface. If the electric field is perpendicular to the interface the
polarization is transverse electric (TE). If the magnetic field is perpendicular to the
interface the polarization is transverse magnetic (TM). The diagram used in the last
section showed a case of TE polarization. The four boundary conditions, the law of
reflection and Snell’s Law all remain the same regardless of the polarization, however
the behavior of the equations relating reflected and transmitted fields to the incident
ones is different [30].
For more complicated 3D geometries – specifically rectangular and circular
waveguides – these characterizations are meaningless so instead the terms quasi-TE and
quasi-TM are used to describe the polarization, depending on whether the particular
mode is primarily TE or TM respectively. There are some additional quirks in the
description of the polarization with regard to the simulation software and the particular
structure being analyzed. These will be referenced in Chapter 3.
2.5 Poynting Vector
An important quantity in determining the power incident on a surface from an
electromagnetic wave is called the Poynting vector. It is defined as
HES ×= , (2.14)
where S has units of Watts/meter2
(intensity). To get the total incident power it is
necessary to integrate S over the entire surface of interest.
Since S in general points in the direction of propagation, which in this case is
perpendicular to both E and H, it is often of interest to obtain the time average intensity
along this direction (in this case, z). This is shown below for propagation along the z
direction.
]ˆRe[
2
1 *
zSz ⋅×= HE . (2.15)
The factor of one half and the reason for the Re and complex conjugate operators
arises from taking the time average intensity for a sinusoidal waveform. This is derived

40. 29
fully in [31]. Again, the important point is that this quantity represents the power per
unit cross sectional area or instantaneous intensity for a wave and can be integrated over
the cross section of an interface to determine the total power flow [30].
2.6 Total Internal Reflection (TIR)
The critical process in any guided wave approach is that of total internal
reflection (TIR). A pictographic representation of TIR is seen in Figure 2-2 .
Examination of Snell’s Law and Figure 2-2 shows that when n1 > n2 and θi (referred to
as θ1 here) is greater than a certain critical angle θc, the incident ray is completely
reflected back into the first medium. The critical angle is a function of both n1 and n2
and can be written as
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
= −
1
21
sin
n
n
cθ (2.16)
using Snell’s Law (2.13b) and substituting 90° for θ2.
Based on this it can easily be seen how waveguiding becomes a possibility by
surrounding a high index medium with layers of a lower index medium. Light must be
coupled into the waveguide in such a way that its angle of incidence at the boundary
with the lower index layer is always greater than the critical angle. This forms the basis
for waveguides which will be presented in the next section.
Figure 2-2. Representation of three possible scenarios for light incident upon a dielectric interface for the
case when n1>n2. At left is the case for θi<θc. There will be a transmitted ray in this case. The figure in the
center shows the case for θi=θc in which there will be a transmitted ray along the interface itself. Finally
TIR is shown at right when θi>θc. There will be no transmission. This is the physical basis for waveguides.
θ
nn
n n
θ
θ
n
θ
n
θ

41. 30
2.7 Waveguides
2.7.1 Planar Slab Waveguide
Often the simplest way to introduce the concept of a waveguide is with the
example of a planar slab waveguide. A diagram of this structure is seen in Figure 2-3. It
can be seen from this that the structure is infinite in the y direction and therefore not
very realistic. The axis of propagation here, as in most cases, is taken to be the z
direction. It will be seen that due to the confined nature of a waveguide as well as the
physical principles of TIR and the boundary conditions on the electric and magnetic
fields, that light propagation in such structures takes the form of discrete eigenmodes.
The value and number of these modes depends on both the physical parameters and
geometry of the waveguide as well as the operating wavelength. For optical
communications, the wavelength is usually set at around 1550nm – the transparency
window for most optical fibers – and the remaining quantities are left to be specified by
the designer. For this problem however, all values will be specified and the primary
objective will be to solve for the modes of propagation. It is also important to note that
there are two mutually exclusive sets of modes comprised by the TE and TM
polarizations. The TE polarization will be used in this particular analysis.
The distribution of values for the index of refraction of each layer in the
waveguide must be known in most cases. The waveguide in Figure 2-3 has nf =1.50,
ns=1.45 and nc=1.40 where nf is the guiding layer, ns is the substrate layer and nc is the
cap layer. This type of structure is asymmetric since nf > ns >nc. In a symmetric structure
the substrate and cap layers have equal indices of refraction. The width of the guiding
layer h is 5µm and the operating wavelength λ is 1µm. The x=0 position is arbitrarily
chosen as the top interface since it is an asymmetric waveguide and this point cannot be
in the direct center.
nc = 1.40
x
λ = 1 µm
h = 5 µmnf = 1.50
ns = 1.45 z
Figure 2-3. A basic asymmetrical planar slab waveguide. Note the orientation of the axes.

42. 31
An important first step in this kind of problem is the choice of a proper
coordinate system. Because the individual components of each field will remain
independent of each other (no component coupling) throughout the structure during
propagation, the best coordinate system to use is a rectangular (Cartesian) one.
To completely specify the eigenmodes, the wave equation must be solved in
each dielectric layer and each solution must be connected at the layer boundaries using
the established boundary conditions. Using (2.07) and (2.09) the wave equation can be
rewritten as
, (2.17)0)( 2
0
2
=+∇ yiy EnkE
where ni = nf ,ns, or nc depending on the layer. The electric field points strictly in the y
direction (out of the page) because the polarization is TE as mentioned previously (refer
to Figure 2-2). The y dependence of the electric field can be ignored since each layer is
infinite in that direction. This means Ey will only be a function of x and z. Knowledge of
differential equations enables a simple trial solution for Ey:
)exp()(),( zjxEzxE yy
β+=
. (2.18)
In this equation β is the propagation coefficient in the z direction (aka the longitudinal
portion of the wave vector). The trial solution is plugged into (2.17) to yield
0))(( 22
02
2
=−+
∂
∂
yi
y
Enk
x
E
β . (2.19)
This equation completely describes the behavior of the E field along the x direction (the
transverse portion of the wave). The solution has the general form
)exp()( 0 xjExEy α±= , (2.20)
where E0 is the field amplitude at x=0 and 22
0 )( βα −= ink . The longitudinal part has
been left out since it is simply sinusoidal in z for all regions. There are two possible
scenarios here. When β < k0ni , the term under the radical is positive so the solution takes
an oscillatory form:
)exp()( 0 xjExEy κ= (2.21)

43. 32
where 22
0 )( βκα −±== ink and is known as the transverse wavevector. When β >
k0ni the term under the radical is negative and the solution becomes exponential:
)exp()( 0 xExEy γ−= (2.22)
where 2
0
2
)( ink−== βγα m and is known as the attenuation coefficient.
The diagram in Figure 2-4 helps to clarify the relationship between wavevector k,
transverse wavevector κ and longitudinal wavevector β. The only physical situation
occurs when β satisfies the condition:
fs nknk 00 << β . (2.23)
In this case, the wave is oscillatory in the guiding layer (nf) and decays
exponentially in the substrate (ns) and cladding (nc) layers. The above condition is
necessary to produce a guided wave in a structure where nf > ns >nc.
k
k2
= κ2
+ β2
x
κ
β
z
Figure 2-4. A diagram illustrating the relationship between wavevector k, transverse wavevector κ, and
longitudinal wavevector β.
To find the exact values for β which describe the allowed modes for the waveguide, the
generalized solutions in each layer must be connected via the boundary conditions. Each
solution has a transverse portion that describes the distribution in x and a longitudinal
portion describing the distribution along the z direction. As mentioned previously, the
longitudinal portions of all solutions have the same form so they can be left out for
simplicity. The transverse parts of the electric field amplitude are [30]:
⎪
⎩
⎪
⎨
⎧
+
+
−
=
))(exp(
)sin()cos(
)exp(
)(
hxD
xCxB
xA
xE
s
ff
c
y
γ
κκ
γ
(2.22)

44. 33
where A, B, C and D are the amplitude coefficients which will be determined by
application of the boundary conditions. γc and γs are the attenuation coefficients in the
cover and substrate respectively and κf is the transverse component of the wavevector k
within the guiding layer. In this case only two of the four boundary conditions from
(2.12) will be used. To reiterate, the tangential components of both E and H are
continuous across each boundary. The other two conditions will not be needed. This
results in an underspecified homogeneous system of four linear equations (2 boundaries
with 2 boundary conditions for each boundary) with five unknowns (A,B,C,D and the
value of κf for each mode) which, when fully solved, yields a transcendental equation in
the remaining unknown variable. Using Faraday’s law (2.01a) along with (2.02) to put
the tangential component of H in terms of E (the curl must be expanded into individual
components on the left hand side of the equation), three of the four boundary conditions
can be applied to find A in terms of B,C and D. The equation describing the amplitude of
the transverse E field across the structure is [30]:
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎪
⎨
⎧
+
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−
−
=
))(exp()sin()cos(
)sin()cos(
)exp(
)(
hxxxA
xxA
xA
xE
sf
f
c
f
f
f
c
f
c
y
γκ
κ
γ
κ
κ
κ
γ
κ
γ
(2.23)
The final boundary condition produces a transcendental equation in κf:
( )
scf
scf
f h
γγκ
γγκ
κ
−
+
= 2
)tan( (2.24)
Known as the characteristic equation, this equation can only be solved using numerical
or graphical methods. The result will show the discrete values for κf which can be used
in turn to solve for the values of β, the propagation constant for each mode. This type of
analysis can be performed for the TM polarized eigenmodes as well and will yield
similar equations [30]. Theoretically, any type of waveguide can be analyzed and
completely solved using this approach. However, in more complicated geometries or
multiple layer structures the analysis – particularly finding and solving the characteristic

45. 34
equation – becomes intractable. This necessitates the use of more powerful
computational packages such as RSoft.
Applying this approach to the example waveguide in Figure 2-3 and using the
graphical approach to solve for the modes of propagation produces nice pictorial
representations. A graph of each side of (2.24) is seen in Figure 2-5 while Figure 2-6
shows simple cross sectional pictures of the E field for each mode. Each intersection
point in Figure 2-5 represents a propagating mode. The total power in each mode is
normalized to be one.
0 0.5 1 1.5 2
x 10
4
−10
−8
−6
−4
−2
0
2
4
6
8
10
Graphical Solution for Modes
Propagating in a Simple 3−Layer Waveguide
Kappa
RHSandLHSofEquation2.24
Figure 2-5. Graphical solution of the transcendental equation (2.24) for an asymmetrical waveguide. Each
intersection of the red and blue graphs (aside from the vertical lines where the value of tangent is infinite)
represents a possible mode.

46. 35
(a)
−10 −8 −6 −4 −2 0 2 4
x 10
−4
0
1
2
3
4
5
6
7
Mode 0
β = 94087
x (cm)
Ey(x)(a.u.)
(b)
−10 −8 −6 −4 −2 0 2 4
x 10
−4
−4
−3
−2
−1
0
1
2
3
4
Mode 1
β = 93608
x (cm)
Ey(x)(a.u.)

47. 36
(c)
−10 −8 −6 −4 −2 0 2 4
x 10
−4
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
Mode 2
β = 92819
x (cm)
Ey(x)(a.u.)
(d)
−10 −8 −6 −4 −2 0 2 4
x 10
−4
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Mode 3
β = 91752
x(cm)
Ey(x)(a.u.)
Figure 2-6. Graphs (a) – (d) show pictures of the cross section of the E field for modes 0–3 respectively.
The value of β for each mode is also given on the graphs. The evanescent decay into the substrate and
cladding is shown by the blue and green sections of the plots respectively. It can be observed that higher
order modes penetrate further into the substrate and cladding.

48. 37
In communications it is common to design waveguides and fibers to only
support one eigenmode of propagation due to the fact that multiple modes will distort a
signal as a result of modal dispersion. A full explanation of different kinds of dispersion
in waveguides can be found in [30]. The formula describing the approximate number of
modes that will propagate in a waveguide is:
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡ −
=
π
22
Int
sf nnhk
m . (2.25)
In this case ‘Int’ means to round towards 0 (for example, if the number in parenthesis
came out to be 5.68395, m = 5). The number of modes is therefore dependent on three
parameters: the wavelength λ, the width h and the difference of the squares of the
indices of refraction of the guiding layer and the substrate or cladding layer (whichever
has the larger index of refraction). Although it is 1µm in the previous example, λ is
usually taken to be 1.55µm because at this particular wavelength the lowest possible
transmission losses are attained in most silica waveguides and optical fibers [30]. For
this reason it has been dubbed the communication wavelength. Unless otherwise
mentioned all simulations, calculations and results will assume this value for λ.
An interesting phenomenon to take note of is the evanescent decay of the mode
in both the cladding and substrate layers. It can be clearly observed that a finite portion
of each mode actually extends into these layers. This can be likened to a similar
occurrence in the energy levels of the finite potential well of quantum mechanics and
forms the basis for coupling between waveguides and other optical components – a very
important issue in the field of integrated optics. One other thing of note is the slight yet
noticeable asymmetry in the mode distributions which is not surprisingly a result of the
fact that the waveguide itself is asymmetric. Similar analysis can be performed on a
symmetric waveguide with similar results. A more complete discussion and analysis of
these topics can be found in [30, 31].

49. 38
2.7.2 Slot Confinement Waveguide
As it was noted before, the ideas from the earlier parts of this section and in
particular those of the previous section can be extended to more complicated structures.
Recently, some research has focused on guiding light in low-index materials by using
high index contrast waveguides. One of these structures is a slot confined waveguide or
simply the slot waveguide. This structure was proposed by a group at Cornell [32, 33]
and used as the basis for further analysis in another recent publication [34]. It is
essentially two planar Si slab waveguides placed in close proximity in a low index SiO2
background. A picture of this is seen in Figure 2-7 [32]. In this case nc is still the index
of refraction for the upper cladding but ns is now the index of refraction of the slot
material and nH is that of each guiding layer. It can be seen that the width of each guide
is |b – a| and the width of the slot is 2a.
Figure 2-7. Picture of the two dimensional slot confined waveguide proposed in [32]. It is essentially two
planar slab waveguides placed in close proximity in a low index SiO2 background [32, 33].
The idea is that by situating the two waveguides next to each other to form a slot
made of SiO2 with a lower index than the Si waveguides, the evanescent decay from
each waveguide will result in a highly concentrated flux density (D) in the slot as a
result of the boundary condition for D – namely, the continuity of its normal component.
The advantage of this over past methods of guiding and confining light in low index
materials is that the light is guided by TIR and not external reflections from interference
effects which are both lossy and wavelength dependent. The guided modes will
therefore be eigenmodes of the structure that are fundamentally lossless and not as
heavily dependent on wavelength as previous structures.

50. 39
The constitutive relations (2) show that the high index contrast at the interface
between the slot and the guiding layer is responsible for the large discontinuity in the E
field that results. Physically the best way to understand what is going on is to think of
the large discontinuity in E as a redistribution of photons resulting from the boundary
condition on D. This only occurs for the TM polarized modes since for TE modes the E
field (and hence the D field3
) has no component perpendicular to the interface. The x
(normal) component of the transverse E field of the fundamental TM eigenmode of the
slot waveguide can be found through analysis similar to that in the previous section to
be [32]
[ ] [ ]
[ ] [ ] [ ]⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎪
⎨
⎧
−−−⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+−⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+−⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
)(exp)(sin)sinh()(cos)cosh(
1
)(sin)sinh()(cos)cosh(
1
)cosh(
1
)(
2
2
2
22
2
bxaxa
n
n
aba
n
axa
n
axa
n
x
n
AxE
cHs
Hs
sH
Hs
c
Hs
Hs
s
Hs
H
s
s
y
γκγ
κ
γ
κγ
κγ
κ
γ
κγ
γ
(2.26)
where κH is the transverse wavevector, γc and γs are the attenuation coefficients in the
cladding and slot respectively. The value of A is written as [32]
( )
0
22
0
0
k
nk
AA HH κ−
= , (2.27)
where A0 is an arbitrary constant and k0 = 2π/λ0 is the wave number in vacuum. This
eigenmode can be thought of as an interaction between the fundamental TM eigenmodes
of each individual waveguide. The values of κH, γc, and γs can all be related to the
propagation constant (β) for each mode in a similar fashion to the simple planar
waveguide:
22
0
2
0
2
2
0
2
)(
)(
)(
βκ
βγ
βγ
−=
−=
−=
HH
ss
cc
nk
nk
nk
(2.28a-c)
3
This is not true in the most general case. Here, D is in fact parallel to E because it is assumed that this is
an isotropic and linear material. For further discussion see [27].

51. 40
The values of β for each eigenmode are calculated by solving the characteristic equation
for this structure [32]:
( )[ ] ( a
n
n
ab s
Hs
sH
H γ
κ
γ
κ tanhtan 2
2
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=Φ−− ). (2.29)
The continuity condition on D causes the E field to be enhanced by a factor of
(nH / ns)2
on the slot side of the boundary between the slot and the higher index
waveguides. This creates a higher overall power confinement within the thinner low
index slot layers. Power confinement will be discussed in the next section. If the two
waveguides are close enough together the enhancement can be quite significant but
deteriorates rapidly as the separation increases. A graph of the x component of the
transverse E field distribution of the fundamental TM mode is shown in Figure 2-8. In
this case the values used were: λ0=1.55µm, nH=3.48, ns= nc=1.44, a=72 nm, and b=152
nm and the slot width is 144nm [32]. Additional graphs and discussion of this type of
waveguide can be found in the next section on power confinement.
−6 −4 −2 0 2 4 6
x 10
−7
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Transverse E−field distribution of fundamental TM eigenmode
x (m)
Ex(x)(a.u.)
Figure 2-8. Graph of the x component of the transverse E field of the fundamental TM mode across the
regions of the slot confinement waveguide. The blue sections show the evanescent decay into the cladding
regions, the black sections show the distribution in the waveguides and the red section shows the behavior
within the slot. In this case the slot width is 144nm while the guiding layers are each 180nm.

52. 41
2.7.3 Power Confinement in Waveguides
One of the most important figures of merit for a waveguide is the power
confinement or confinement factor (CF) of each of its eigenmodes. It has already been
stated that the total power incident on a surface or cross sectional area of a waveguide
can be calculated in general by integrating the time averaged Poynting vector along the
direction of propagation (2.14) over the area comprising the surface. In the case of the
2D planar slab waveguide, the integral is instead performed across just the x direction
since it is considered to be essentially flat. The power confinement is obtained by
calculating the ratio of power flowing within the core or guiding layer to the total power
in all regions of the structure. For the planar slab waveguide this is written as
∫
∫
∫
∫
∞
∞−
−
==
dxxHxE
dxxHxE
dxxHxE
dxxHxE
P
P
xy
h
xy
regionsall
xy
core
xy
total
core
)()(
)()(
)()(
)()(
*
0
*
*
*
. (2.30)
Usually this quantity is expressed as a percentage by multiplying it by one
hundred. It is usually the case that higher order modes are less confined, mostly due to
the fact that they have lower attenuation factors and thus extend further into the low
index cladding and substrate layers.
The importance of the confinement factor is due to the fact that poorly confined
modes often exhibit higher power loss due to bending in the guiding structure (as in ring
resonators which will be discussed in a later section) and unintentional evanescent
coupling to nearby components. These phenomena are not directly relevant to this
discussion but can be explored further in [1, 30, 31].
The major advantage of the aforementioned slot waveguide is that it provides a
very high concentration of light in a relatively thin, low index layer (the slot) [32]. The
high power confinement of light that results is especially favorable in integrated optics
where power efficiency and dissipation are extremely important in device design. The
fact that light is concentrated in thin, nanometer-size layers could be considered even
more important however, since the diffraction limit has imposed limits on the size of
optical components and devices. The diffraction limit states that the lower limit of these

53. 42
sizes is on the order of the wavelength of the light within a particular material. Thus the
value of light intensity (power per unit area) becomes an important design parameter.
One of the primary objectives of this research is to determine optimum device
dimensions in terms of maximizing the power confinement within relatively small layers.
Through analysis it is found that the most crucial parameter in this endeavor is the ratio
of the slot width to that of the guiding layers. To provide a visualization of this, a few
graphs of the fundamental TM eigenmode of the slot waveguide are presented with
different values for the slot width (Figure 2-9) along with a plot of CF versus ratio
(Figure 2-10). In each plot the width of the surrounding guides is kept constant at a
specific value while the width of the slot is chosen to fit the chosen ratio. The first thing
to notice is that the advantage of confinement in a low index slot is only realized when
the slot is relatively thin since the evanescent decay becomes quite substantial at larger
thicknesses. It can also be seen that the relationship between slot to guide ratio and CF is
not linear and that there is instead a range of optimum values for the ratio. One might be
tempted to think a higher ratio and hence larger amount of low index material would
yield a higher confinement and that the best approach is to incorporate as much low
index material as possible. This trend is indeed true for ratios up to about 0.80. After this
point however, there is a steady tailing off of the confinement values. This is due to the
nature of the interaction between the eigenmodes of the two waveguides. The width of
the slot affects the ratio of the slot to guide width which in turn influences the combined
effective index of the two guiding layers and the slot layer, causing more light to be
redistributed into the outermost cladding and substrate layers. This results in a balancing
act between slot-to-guide ratio and CF, and is perhaps the most important idea to retain
from this section, since it provides valuable insight into the nature of light propagation
in such a device and also forms the basis of the major design concepts utilized in the
next chapter. A more enlightening explanation is given in the next chapter with the
presentation of simulation results and graphics for a more realistic device based on this
structure. The slot waveguide provides a workable theoretical basis for the design of a
three dimensional, multilayer, waveguide to be used as the resonant cavity in a Si-based
laser system which can achieve high light enhancement in low index Er-doped SiO2

54. 43
layers in order to maximize gain, minimize losses and potentially yield lower threshold
current densities.
(a)
−6 −4 −2 0 2 4 6
x 10
−7
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Transverse E−field distribution of fundamental TM eigenmode
Slot Width = 50nm; R = 0.28
x (m)
Ex(x)(a.u.)
(b)
−6 −4 −2 0 2 4 6
x 10
−7
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Transverse E−field distribution of fundamental TM eigenmode
Slot Width = 144nm; R = 0.80
x (m)
Ex(x)(a.u.)

55. 44
(c)
−6 −4 −2 0 2 4 6
x 10
−7
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
Transverse E−field distribution of fundamental TM eigenmode
Slot Width = 180nm; R = 1.00
x (m)
Ex(x)(a.u.)

56. 45
(d)
−6 −4 −2 0 2 4 6
x 10
−7
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Transverse E−field distribution of fundamental TM eigenmode
Slot Width = 270nm; R = 1.50
x (m)
Ex(x)(a.u.)
Figure 2-9. Figures (a) – (d) show graphs of the E field distribution of the transverse fundamental TM
mode for various slot widths and slot to guide layer thickness ratios. Notice how as the slot thickness
increases and consequently lowers the overall effective index, a larger evanescent tail leaking into the
cladding materials results. Thus the confinement factor begins to decrease as the total amount of the low
index material present in the structure increases beyond a certain point.
Confinement Factor vs Ratio of Slot Width to
Waveguide Width
55
56
57
58
59
60
61
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Ratio of Slot Width to Waveguide Width
ConfinementFactor(%)
Figure 2-10. Graph showing the relationship between confinement factor and ratio of slot width to
waveguide width. The width of the guiding layers is kept constant at 180nm while the slot width is chosen
to fit the desired ratio. The CF increases up to a ratio of about 0.80 because the added low index material
allows a larger amount of light in the desired region. The values saturate between 0.80 and about 1.1.

57. 46
2.8 Lasers
2.8.8 Introduction
Because this project deals with the design and optimization of a waveguide to be
used as the resonant cavity of a laser system, it is necessary to give an introduction to
the basic operating principles of lasers. Laser is an acronym that stands for Light
Amplification by Stimulated Emission of Radiation. All lasers are based on the
fundamental principles of absorption and emission of electromagnetic radiation. In order
to best understand the functionality of lasers it is critical to first understand these
processes. The next topic to cover is pumping and the different methods that are used to
achieve population inversion which is a necessity for lasing to occur. Next the topic of
amplification via resonant optical cavities will be covered. This is the most critical
aspect in any type of laser system since it is what distinguishes laser light from that of a
light emitting diode (LED) or any other arbitrary source for that matter. The three
distinguishing characteristics of laser light are: 1) directionality, 2) coherence, and 3) it
is monochromatic – that is, the range of wavelengths or bandwidth is relatively small
compared to arbitrary sources. It will be seen that particularly the last property of laser
light is the result of the resonant optical cavity in which the light emitting medium is
placed. This is the main reason that light from a laser is preferable over other light
sources in many systems and experiments. The cavity can be tuned to produce light of
very specific frequencies and wavelengths depending on what is needed in the particular
system or experiment. The next two ideas discussed are gain saturation and lasing
threshold which play a role in limiting the output power of a lasing cavity. The
minimum injection current needed to achieve lasing is called the threshold current and is
a very important parameter in the design of practical semiconductor lasers. Electrical
injection pumping, particularly with respect to silicon nanocrystals, is covered in the
next section. This is an important concept since the MURI project of which this study is
a part considers an electrically-pumped device. The concluding section deals with the
MURI laser design proposed by the principle investigators at each university and their
research groups. It also discusses the use of erbium as the extrinsic material which
provides light amplification.

58. 47
2.8.2 Three Processes: Absorption, Spontaneous Emission, Stimulated Emission
Stimulated absorption or simply absorption is the process by which an electron is
raised from a lower energy level (usually the ground level) to a higher one through
interaction with a photon. A diagram of this process for a simple two level system is
seen in Figure 2-11a. Absorption will only occur when the energy of the photon, given
by the quantity hν where h is Planck’s constant, is equal to or greater than the difference
in energy between the two levels.
A second critical process in a laser system is emission. Emission is just the exact
opposite of absorption – an electron makes a transition from a higher energy state to a
lower one and gives off a photon. There are two different types of emission:
spontaneous and stimulated. Spontaneous emission is the more common type of optical
emission and occurs when an excited electron randomly relaxes to a lower state emitting
a photon in the process. A diagram is shown in Figure 2-11b. By its nature, spontaneous
emission is completely unpredictable. It is impossible to know exactly where the emitted
photon will appear or which direction it will head in. It is possible however, to
determine a statistical average for the length of time between emissions for particular
transitions within a given material system. Often, spontaneous emission is treated as a
source of noise within a system since it creates additional light sources which are neither
predictable nor controllable [30].
Stimulated emission on the other hand is much more predictable and controllable.
It takes place in the same fashion as absorption except that the incoming photon now
interacts with an electron already in a higher energy state. The interaction of the photon
with the electron induces the transition and yields the release of another photon identical
in phase, direction, polarization and frequency to the original. A diagram of the process
is shown in Figure 2-11c. Thus, stimulated emission produces amplification of light
incident upon a material with electrons in excited states. It is one of the crucial elements
in producing laser light.

59. 48
(a)
Photon
(Spontaneous) Absorption
After:
Before: Electron in excited state after
absorbing the photon’s energy.Electron in ground state
(b)
Spontaneous Emission
Photon
After:
Before: Electron in ground state after
emitting a photon.Electron in excited state.
(c)
Stimulated Emission
2 Photons
Photon
Before: After:
Electron in excited state. Electron in ground state after emitting
photon.2 identical photons.Incoming photon interacts with it.
Figure 2-11. (a) Simplified representation of absorption, (b) spontaneous emission, and (c) stimulated
emission.

60. 49
2.8.3 Population Inversion and Pumping Schemes
Since stimulated emission acts as an amplification process and produces a
photon identical to one incident onto an electron in an excited energy state, it is really
the fundamental process in the production of laser light. In order to maximize the
number of identical photons and increase the intensity of the laser light, there must first
be a sufficient number of electrons in excited states with which incoming photons can
interact. For most systems in equilibrium, electrons will tend to relax to the lowest
energy state. Therefore energy must be introduced into the system in some way to raise
electrons from lower energy levels to higher ones by the process of absorption. This
results in the non-equilibrium condition of population inversion. The method of
achieving population inversion is called the pumping scheme.
The most common pumping schemes that are used in almost all laser systems are:
1) optical sources 2) electrical charge injection 3) chemical reactions 4) ion beams and 5)
electron discharges or beams [30]. Optical pumping involves the use of either a flash
lamp or another source of laser light to raise electrons to a higher energy state. Since it
requires an external source of light this type of pumping is not conducive to the laser
system that is the focus of this work. Instead the laser system studied here employs
electrical injection pumping which is commonly used in semiconductor lasers. These are
discussed further in a forthcoming section. The other types of pumping are not as
common and are not discussed in detail.
The simplest optical pumping scheme is a two level system. The idea is to raise
electrons from lower energy states to a specific higher energy state by providing them
the amount of energy unique to that specific absorption wavelength or frequency. In a
three level system the electrons are first excited to a higher energy level (L3) that is
unstable after which they undergo spontaneous emission or non-radiative decay and
relax to a lower, more stable level (L2) after a characteristic lifetime in the higher state.
This lower yet more stable level is still higher than the initial ground level (L1) of the
electrons. The lasing phase is between L2 and L1, where stimulated emission induces
transitions to lower energy states and the release of photons. The most efficient scheme
is a four level system, however. A diagram is shown in Figure 2-12. In this system the