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Table Of Contents
TABLE OF CONTENTS
1
CHAPTER ONE - INTRODUCTION
4
1.1Introduction
4
1.2Types of Fibers for the
1.2.1Solid core fibers
IR
1.2.2Liquid core fibers
1.2.3Hollow waveguides
1.2.4Why
1.3The
hollow waveguides
Research Subject and Main Goal
1.4References
for chapter one
6
6
16
16
24
25
27
CHAPTER TWO - ATTENUATION MECHANISMS IN HOLLOW
WAVEGUIDES
32
List of Symbols For Chapter 2
32
2.1Reflection
34
from a thin film
2.2Scattering from rough surfaces
2.2.1The Rayleigh criterion
2.2.2The
general solution for scattering from rough surface
surface
2.2.3Scattering from a randomly rough
2.2.4The normally distributed surface
2.3Measurements
of surface roughness
43
43
45
50
55
58
2.4Summary
of chapter 2
62
2.5Refrences
for chapter 2
63
CHAPTER THREE - RADIATION PROPAGATION THROUGH
HOLLOW WAVEGUIDES
65
List of Symbols For Chapter 3
65
3.1Theory
3.1.1The
3.1.2The
67
67
71
74
mode approach
ray model
3.1.3Ray propagation through straight waveguides
2
3.1.4Ray
propagation through bent waveguide
3.2Experimental results and
3.2.1Experimental setups
theoretical calculations
3.2.2The
3.2.3The
attenuation of straight hollow waveguides
attenuation of bent hollow waveguides
3.2.4The dependence of the attenuation on the waveguide’s radius
3.2.5The dependence of the attenuation on the coupling conditions
3.3Discussion
3.3.1Straight hollow waveguides
3.3.2Bent hollow waveguides
3.3.3The
3.3.4The
dependence on the waveguide’s inner diameter
influence of coupling conditions
82
87
87
90
95
102
104
110
110
111
112
113
3.4Conclusion
114
3.5Summary
114
of chapter 3
3.6References
for chapter 3
116
CHAPTER FOUR - PULSE DISPERSION IN HOLLOW WAVEGUIDES
119
List of Symbols For Chapter 4
119
4.1Theory
4.1.1Pulse
4.1.2Pulse
120
120
121
dispersion in straight and smooth multi-mode fibers
dispersion in real hollow waveguides
4.2Experimental results and discussion
4.2.1Experimental setup
4.2.2Experimental results and discussion
122
122
123
4.3Summary
127
4.4References
for chapter four
129
CHAPTER FIVE - IMPROVING THE HOLLOW WAVEGUIDES
130
List of Symbols For Chapter 5
130
5.1Theory
5.1.1Bloch theory
5.1.2Maxwell equations
a multilayer mirror for the infrared
135
135
143
147
results and discussion– multilayer mirror
149
5.1.3Designing
5.2Experimental
3
5.2.1Dielectric
mirror
5.2.2Metal dielectric mirror
5.3Hollow
waveguides made of multilayer films
150
152
160
5.4Conclusion
163
5.5Summary
163
5.6Refrences
for chapter five
165
CHAPTER SIX – SUMMARY
167
6.1Summary
167
6.2Future
168
work
4
Chapter One - Introduction
1.1 Introduction
In 1966 Kao and Hockham [1.1] described a new concept for a transmission
medium. They suggested the possibility of information transmission by optical
fibers. In 1970 scientists at Corning Inc. [1.2], fabricated silica optical fibers
with a loss of 20 dB/km. This relatively low attenuation (at the time)
encouraged
scientists
from around
the
world
that
perhaps
optical
communication could become a reality. A concentrated effort followed, and by
the mid 1980s there were reports of low loss silica fibers that were close to the
theoretical limit.
Today, silica based fiber optics is a mature technology with major impacts in
telecommunications, laser power transmission, sensors for medicine, industry,
military, as well as other optical and electro optical systems. While silica based
fibers exhibit excellent optical properties out to about 2m, other materials are
required for transmission of longer wavelengths in the infrared (IR). These
materials can be glassy, single crystalline and polycrystalline. Examples of
such materials include fluoride and chalcogenide glasses, single crystalline
sapphire, and polycrystalline silver halide. Depending on their composition
these materials can transmit to beyond 20m. Consequently, optical fibers
made from these materials enables numerous practical applications in the IR.
For example, IR transmitting fibers can be used in medical applications such as
for laser surgery and in industrial application such as metal cutting and
machining using high power IR laser sources (e.g. Er:YAG, CO, CO2 lasers).
5
More recently, there has been considerable interest in using IR transmitting
fibers in fiber optic chemical sensors systems for environmental pollution
monitoring using absorption, evanescent, or diffused reflectance spectroscopy
since practically all molecules posses characteristic vibration bands in the IR.
Aside from chemical sensors, IR fibers can be used for magnetic field, current
and acoustic sensing, thermal pyrometry, medical applications, IR imaging, IR
countermeasures and laser threat warning systems. While low loss silica fibers
are highly developed for transmission lines in telecommunications applications,
the IR transmitting materials still have large attenuation and need to be
improved.
The materials described above are used to fabricate solid core fibers; however
there is another class of fibers based on hollow waveguides, which has been
investigated, primary for CO2 laser power transmission. These waveguides
posses hollow core and are based on hollow tubes with internal metallic coating
with or without a dielectric coating. These waveguides may be good candidates
for transmitting infrared radiation.
This chapter has two main sections. The first section of this chapter is a survey
of the current research of IR fibers and waveguides. It describes each type of
fiber shortly and at the end of the section compares the different types. The
second section outlines the objective of this thesis. It describes the major
contribution of my work to the field of hollow waveguides to the IR region.
6
1.2 Types of Fibers for the IR
IR fibers can be used for many applications. In the last two decades IR
materials and fibers were investigated intensively in order to produce
commercial IR fibers for different applications. These fibers are divided into
three categories; solid core fibers, liquid core fibers and hollow waveguides.
Each category can be divided to subcategories see table 1.1.
Subcategory
Solid Core
Glass
Crystalline
Liquid Core
Hollow Waveguides
Type
Silica based
Fluoride based
Chalcogenide
Single crystal
Polycrystalline
Fused silica
Examples
Na2O-CaO-SiO2
ZABLAN
Sapphire
AgBrCl, KRS-5
C2Cl4
Metal/ Dielectric
Refractive index <1
Table 1.1 – Categories of IR Fibers
1.2.1 Solid core fibers
Solid core fibers guide the laser radiation through total internal reflection.
These fibers are made of different kinds of glasses, single crystalline materials
and polycrystalline materials.
1.2.1.1 Glass fibers
Silica and silica based fibers
Silica based glass fibers [1.3] can be optically transparent from the near
ultraviolet (NUV) to the mid-infrared (MIR) range of the electromagnetic
spectrum. Optical fibers made from these glasses are widely used in the near
infrared (NIR) at wavelengths close to the zero material dispersion (1310nm)
7
and minimum loss (1550nm) wavelengths of silica. Such fibers provide the
backbone of modern optical telecommunication networks. Since the late 1970s
these fibers has been manufactured routinely. It is possible to manufacture very
long fibers with very low attenuation (0.2dB/km).
Multicomponent glasses, specifically soda-lime silicate (Na2O-CaO-SiO2) and
sodium borosilicate (Na2O-B2O3-SiO2) and related compositions, in which
silica comprises less then 75% mol of the glass, were early candidates for
optical communication fibers. Core cladding index differences were typically
achieved by varying the concentration or type of the alkali in the respective
glasses or by adding GeO2 to the core glass. Graded index profiles in the fiber
could be tailored by using crucible designs, which permitted more, or less,
interfacial contact and interdiffusion between the core and cladding glasses
during the fiber draw.
Up to the mid 1970’s significant efforts were made to fabricate low loss
multicomponents telecommunication fibers. It was recognized that Rayleigh
scattering in many multicomponent silicate glasses could be lower than in high
silica content glasses and that the achievable losses were largely determined by
extrinsic impurities. Many innovative approaches were tried to minimize the
concentrations of these impurities. The efforts yielded fibers with losses as low
as 3.4 dB/km at 840nm. Further loss reduction was dependent on reducing –OH
contamination to sub parts per million and transition metals to low ppm levels.
The intrinsically lower strength, reliability, and radiation hardness of these
fibers also present significant obstacles for their practical utilization.
8
High silica content fibers are compositionally simple. In most instances, the
cladding glass is 100% SiO2 while the core glass is 90 to 95% SiO2 with a few
percent of dopants to increase the refractive index in order to achieve a guiding
structure. The cladding glass in the vicinity of the core may also be doped to
achieve specific refractive profiles.
These fibers are sensitive to moisture. Given the opportunity, moisture can
diffuse from the surface to the core of the fiber with an attendant increase in
attenuation at communication wavelengths due to overtone and combination
absorptions of OH vibration.
Exposure to ionizing radiation can produce defect centers in fibers, which also
contributes to optical loss. Natural radiation, which is approximately 0.1 to 1
rad/year can be sufficient to produce significant degradation over system
lifetime.
For practical purposes the strength of a fiber should be sufficient to withstand
initial handling stresses including those generated in cabling and deployment.
Furthermore this strength should not degrade during the system lifetime.
Fluoride glass based fibers
Fluoride glasses based on ZrF4 [1.3] are predicted to have minimum optical
loss of less than 0.01 dB/km, which is more than an order of magnitude lower
than the 0.12 dB/km predicted and practically realized for silica fibers. This
phenomenon is related to fact that these are low phonon frequency glasses and,
hence, the multiphonon energy is shifted to longer wavelengths. In addition,
fluoride glasses possess low non-linear refractive indices and, in some cases, a
9
negative thermo optic coefficient (dn/dT). Furthermore, these glasses are
excellent hosts for rare earth elements. As a result, there are many applications
for optical fibers, such as low loss repeater-less links for long distance
telecommunications, fiber lasers and amplifiers, as well as infrared laser power
delivery. More recently, there has been interest in using fluoride fibers in
remote fiber optic chemical sensor systems for environmental monitoring,
using diffuse reflectance and absorption spectroscopy.
Historically, the first fluoride glasses were based on beryllium and have been
known since the 1920’s. These glasses are very stable and have many unique
properties including good UV transparency, low index of refraction, low
optical dispersion and low non-linear refractive index. However, the
combination of high toxicity, volatility, and hygroscopic nature of BeF2 poses
serious problems in melting, forming, handling, and disposal of these glasses.
These glasses were investigated primarily because of their high resistance to
damage.
A second kind of HMF glasses is fluorozirconate glasses. They are so named
since the major component is ZrF4. The glass formed is ZrF4-BaF2 (ZB). The
main disadvantage of these glasses is their crystallization instability. It is
possible to add more metal fluoride to facilitate stable glasses. For example the
addition of ThF4 and LaF3 form ZBT and ZBL. Unfortunately these glasses are
not stable enough for practical uses. In addition Th is radioactive.
It has been demonstrated that the addition of a few percent of AlF3 to
fluorzirconate glasses greatly enhances the glass stability. The glasses ZrF4-
10
BaF2-LaF3-AlF3 (ZBLA) and ZrF4-BaF2-LaF3-AlF3-NaF (ZBLAN) are very
stable and may be used to fabricate fibers.
All fluoride glasses have excellent optical properties and transmit more than
50% between 0.25m and 7m. While fluoroaluminate glasses do not transmit
as far as fluorozirconate glasses, non-ZrF4 based on heavy metal can transmit
to even longer wavelengths. As a result, fluoride glasses are candidate
materials for both bulk windows and fibers for the IR.
The ultra low loss application for fluoride glass fibers have not been realized
because
of
problems
associated
with
microcrystallization,
crucible
contamination and bubble formation during fiber fabrication. Nevertheless,
fibers with losses greater then than 10dB/km are routinely obtained and can be
used for chemical sensor application as well as high power UV, visible and IR
laser transmission.
Chalcogenide glass based fibers
Chalcogenide compounds [1.3] of some elements belonging to groups 4B and
5B in the periodic table exhibit excellent glass forming ability. Based on the
wide infrared transmission range of the As2S3 glass, various glass compositions
have been developed as optical component materials for the 3 to 5m and 8 to
12m bands.
Chalcogenide glasses can be classified into three groups: sulfide, selenide and
telluride. Glass formation can be achieved when chalcogen elements are melted
and quenched in evacuated silica glass ampoules with one or more elements,
such as As, Ge, P, Sb, Ga, Al, Si, etc. The properties of the glasses change
11
drastically with glass composition. For example while some sulfide glasses are
transparent in the visible wavelength region, the transmission range of selenide
and telluride glasses shift to the IR region with increasing the contents of
selenium and tellurium. The mechanical strength and thermal and chemical
stabilities of chalcogenide glasses, which are typically lower than oxide
glasses, are sufficient for practical fiber applications.
The attenuation of chalcogenide fibers depends on the glass compound and the
fiber’s drawing technique. Typical attenuation of sulfide glass fibers is 0.3
dB/m at 2.4m. The attenuation of selenide fibers is 10dB/m at 10.6m and
0.5dB/m at 5m. The attenuation of telluride fibers is 1.5dB/m at 10.6m and
4dB/m at 7m.
Chalcogenide glass fibers may be used for temperature monitoring, thermal
imaging, chemical sensing and laser power delivery. The main disadvantage of
these fibers is the poisonous nature of some of the materials they are made of, a
thing that makes them hard to handle and unsuitable for medical application.
1.2.1.2 Crystalline Fibers
About 80 crystals are listed as IR optical materials [1.3]. Many of these
materials have similar IR transmission characteristics, but may possess very
different physical properties. Table 1.2 lists the optical properties of several
crystals. While low transmission loss is usually a prime consideration,
mechanical strength cannot be overlooked. Fibers that tend to be brittle will be
difficult to bend and therefore lose much of their attractiveness as optical
fibers.
12
Crystal
Si
Ge
Al2O3
BaF2
CaF2
CsBr
CsI
CuCl
MgF2
KCl
AgBr
KRS-5
TlCl
ZrO2-Y2O3
Transmission
Range (m)
1.2-7
1.8-23
0.15-6.5
0.14-15
0.13-10
0.21-50
0.25-60
0.19-30
0.11-9
0.2-24
0.45-35
0.6-40
0.4-30
0.35-7
Refractive
Index
3.426
4
1.7
1.45
1.4
1.662
1.739
1.88
1.337
1.457
2
2.36
2.193
2.009
Table 1.2 -IR Materials
Single crystal fibers
Unlike glass fibers, which are pulled at high speed from a heated preform,
single crystal fibers [1.3] have to be grown at much slower rate from a melt.
Long distance transmission using crystalline fibers is therefore not practical.
Instead, the early development of crystalline IR transmitting fibers was driven
primarily by the interest in fibers with good transmission at the 10.6m
wavelength of the CO2 laser. Such fibers could deliver laser power to targets
for surgery, machining, welding and heat treatment. Excellent IR optical
materials, such as the halides of alkali metals, silver and thallium, were
considered as promising candidates for fiber development.
More recently, solid-state lasers with output near 3m have emerged as
excellent medical lasers because of the strong water absorption at that
wavelength in human tissues. Currently silica based fibers do not transmit at
that wavelength. Fluoride glass fibers, on the other hand, have excellent
13
transmission in the 2-3m region, but their chemical stability in wet
environment is a problem. Therefore, single crystal fibers that are free from the
above constraints and that can handle high laser power are sought for new
medical lasers, and sapphire fiber is a prime candidate for fiber delivery of
Er:YAG laser.
Besides applications in optical fiber beam delivery, single crystal fibers also
find potential use in fiber-based sensors. In applications where sensor must
operate in harsh environment, the optical property of fiber materials is not the
only consideration. High melting temperature, chemical inertness, and
mechanical strength often dictate the choice for fiber materials. Sapphire is one
example of a single crystal that possesses an unusual combination of these
properties.
The main advantage of single crystal fiber is the purity of the material. Fibers
made from very pure crystals have low transmission loss due to low absorption
and scattering losses. These fibers have some disadvantages. They are hard to
manufacture, it is hard to fabricate long fibers, and some of them are brittle or
made of toxic materials.
 Sapphire fibers
Sapphire fibers have good physical properties. Sapphire is a strong
material, it is chemically stable, it is not soluble in water, it has high
melting temperature and it is biocompatible. Sapphire fibers are
transparent up to 3.5m [1.4,1.5].
14
Sapphire fibers are grown from a melt at rates up to 1mm/min [1.6].
They are made as core only fibers (without cladding) with diameters in
the range of 0.18mm to 0.55mm. Their bending radius depends on their
diameter and ranges between 2.8cm to 5.2cm. Sapphire fibers have
attenuation of 2dB/m at 2.93m and maximum power delivery of
600mJ.
 AgBr Fibers
Bridges et al. [1.7] fabricated AgBr fibers using a pulling method. In
this method a reservoir of single crystal material is heated and the
material is then pulled through a nozzle to form the fiber. Using this
method they fabricated fibers up to 2m with diameters between 0.35mm
to 0.75mm at a rate 2cm/min. These fibers are able to transmit up to 4W
of CO2 laser at 10.6m without any damage and attenuation of about
1dB/m.
1.2.1.3 Polycrystalline fibers
The first to propose the fabrication of infrared fibers from crystalline materials
(AgCl) was Kapany [1.8] in the mid 1960s. However it took another decade
until the fabrication of polycrystalline infrared fibers was reported by Pinnow
et al.[1.9]. These authors fabricated fibers made of thallium halide crystals,
TlBr-TlI (KRS-5). The motivation for fabricating fibers from heavy metal
halides was to realize predictions of ultra low attenuation of these materials in
the IR.
15
The first attempts to fabricate IR fibers from KRS-5 , KCl, NaCl, CsI, KBr, and
AgCl [1.3] resulted in optical losses of two orders of magnitude higher than
those of the original crystals. Much of the work over the years on these fibers
was concentrated on finding the origin of the fibers loss and improving the
fabrication process.
Besides the low theoretical optical loss of polycrystalline materials, there are
some practical requirements from these materials. First, the crystal must be
deformed plastically in a typical temperature range with speeds higher than
1cm/min. This requirement is needed in order to manufacture a long fiber in a
reasonable amount of time. Second, the crystal must be optically isotropic, due
crystallographic reorientation. Therefore it must posses a cubic crystal
structure. Third, the composition of the crystal must be of solid solution.
Finally, the recrystallization process of the materials must be a one that does
not cause the degradation of the optical material. From this point of view the
suitable materials are thallium halides, silver halides and alkali halides.
 Thallium Halides
Thallium halides fibers made of TlBr-TlI (KRS-5) are fabricated
using the extrusion method [1.10]. These fibers have a low
theoretical attenuation about 6.5dB/km. However due to material
impurities and scattering the achieved attenuation is much lower,
in the range of 120-350dB/km.
The scattering effects [1.11] in these fibers are caused by three
factors. The first is scattering by surface imperfections, the
16
second is due to residual strains and the last is due to the grain
boundaries and dislocation lines.
Furthermore these fiber have additional drawbacks [1.12] such as
aging effects, sensitivity to UV light and solubility in water.
 Silver Halides
Polycrystalline silver halides fibers are made of AgClxBr1-x by
Kaztir et al. in Tel-Aviv University [1.13, 1.14]. These
waveguides are manufactured using the extrusion method. The
optical losses of these waveguides are very low, 0.15dB/m at
10.6m. The attenuation is caused by bulk scattering, absorptions
in material defects and extrinsic absorption.
1.2.2 Liquid core fibers
Liquid core fibers are hollow silica tubes filled with liquid which is transparent
in the infrared [1.17]. The measured attenuation of a liquid core fiber filled
with C2Cl4 is very high (about 100dB/m) for 3.39m. fibers with liquid core
made of CCl4 have attenuation of about 4dB/m at 2.94m.
The only potential advantage of these fibers over solid core fibers is that it is
easier to manufacture a clear liquid. Hence there are less losses due to
scattering.
1.2.3 Hollow waveguides
The concept of using hollow pipes to guide electromagnetic waves was first
described by Rayleigh in 1897 [1.3]. Further understanding of hollow
waveguides was delayed until the 1930s when microwaves generating
17
equipment was first developed and hollow waveguides for these wavelengths
were constructed.
The success of these waveguides inspired researchers to develop hollow
waveguides to the IR region. Initially these waveguides were developed for
medical uses especially high power laser delivery. But more recently they have
been used to transmit incoherent light for broad band spectroscopic and
radiometric applications [1.15, 1.16].
Hollow waveguides present an attractive alternative to other types of IR fibers.
They can transmit wavelength at a large interval (well beyond 20m); their
inherent advantage of having an air core enable them to transmit high laser
power (2.7kW [1.16]) without any damage to the waveguide. Moreover they
have a relatively simple structure and low cost.
However, these waveguides also have some disadvantages. They have a large
attenuation when bent; their NA (numerical aperture) is small and they are very
sensitive to the coupling conditions of the laser beam.
Hollow waveguides may be divided into two categories. The first include
waveguides whose inner wall materials have refractive indices greater than
one. These waveguides are also known as leaky waveguides. Thus the guidance
is done via total reflection. The second includes waveguides whose inner wall
has a refractive index smaller than one. These are known as attenuated total
reflection (ATR) waveguides and the guidance mechanism is similar to solid
core fibers.
18
1.2.3.1 Leaky Waveguides
Leaky waveguides are made of hollow tubes that are internally coated by thin
metal and dielectric films. The choice of the hollow tube and the thin layers
materials depends on the application and coating technique. Table 1.3
summarizes the different kinds of materials used to manufacture hollow leaky
waveguides. These waveguides may have a circular or rectangular cross
section.
Tube
Teflon, Poly-Imide,
fused silica, glass
Copper, Stainless
steel
Nickel, Plastic,
Glass
Plastic, Glass
Metal Layer
Ag
Dielectric Layer
AgI
Cu
CuO ,ZnS, ZnSe
Ag, Ni, Al
Si, ZnS, Ge
Ag
AgI
Silver
Ag
AgBr
Stainless steel
Plastic
Ag
Ag, Al, Au
PbF2
ZnSe, PbF4
Table 1.3 – Types of hollow waveguide
Group
Croitoru et al.
[1.18]
Croitoru et al.
[1.20]
Miyagi et al. [1.25]
Harrington et al.
[1.28]
Morrow et al.
[1.29]
Luxar et al. [1.31]
Laakman et al.
[1.31]
19
Croitoru et al.
This group from Tel-Aviv University [1.18-1.22] develops hollow waveguides
for
a
large
wavelengths.
waveguides
range
The
are
of
Figure 1.1
Cross Section of a Hollow Waveguide
hollow
made
of
Black – tube
Gray – metal layer
Light gray – dielectric layer
different kind of tubes, mainly
fused silica, Teflon and polyimide.
The
deposited
thin
using
films
a
are
patented
electro less method in which solutions containing Ag are passed through the
tubes and coat it. The dielectric layer is built by iodination of some of the Ag
layer to create a AgI layer. Figure 1.1 shows a cross section of the waveguide.
The choice of the tube depends on the application and is an optimization
among the desirable characteristics such as flexibility, roughness and chemical
inertness. Fused silica tubes are very smooth [1.23] and flexible (at small bore
radius). However their thermal conductivity is very poor. Plastic tubes are very
flexible. But they also have a large surface roughness and are damaged easily at
high temperatures. In the past few years there has been an attempt to develop
hollow waveguides using metal tubes.
Croitoru et al. have made hollow waveguides for different wavelengths.
However mainly waveguides for CO2 lasers at 10.6m and Ee:YAG lasers at
2.94m were manufactured, since these wavelengths are used in many medical
applications. Table 1.4 summarizes the characteristics of these waveguides.
20
Inner Diameter [mm]
Attenuation [dB/m]
Maximum input power
Mode of operation
NA
=10.6m
Fused Silica
Teflon
0.7
1
0.2-0.5
0.2-0.75
50 W
80 W
CW, Pulse
CW, Pulse
0.0338
0.1563
=2.94m
Fused Silica Teflon
0.7
1
0.8
1.1
900mJ
1650mJ
Pulse
Pulse
0.0475
-
Table 1.4 – Characteristics of Hollow Waveguides Made by Croitoru et al.
The group also investigated hollow waveguides for Ho:YAG laser at 2.1m
that transmit up to 1200mJ and have attenuation of 1.5dB/m, waveguides for
CO2 laser at 9.6m that have attenuation of 2.8dB/m and waveguides for free
electron lasers at 6-7m that have attenuation of 1.8-4 dB/m [1.24].
Miyagi et al.
Miyagi and his co-workers [1.25-1.26] have pioneered the development and
theoretical characterization of hollow waveguides. They used a three steps
method to fabricate the waveguides. In the first step a metal rod, usually made
of aluminum, was placed in a sputtering chamber where it was coated by
dielectric and metal thin films. Next, the coated pipe was placed in an
electroplating tank where a thick metal layer was deposited on top of the
sputtered layers. Finally, the metal rod was etched leaving a hollow waveguide
structure similar to the one in figure 1.1.
Miyagi et al. fabricated hollow waveguides for different wavelengths. They
measured about 0.3dB/m for hollow waveguides optimized for CO 2 lasers at
10.6m and about 0.5dB/m for Er:YAG lasers at 2.94m. Recently this group
has made hollow waveguides with dielectric thin films made of plastic
materials [1.27].
21
Harrington et al.
Harrington’s group [1.28] has made hollow waveguides similar to those made
by Croitoru et al. They deposit thin Ag and AgI layers using wet chemistry
method similar to that of Croitoru and co-workers. Using this method
waveguides of bore size between 0.25mm to 0.75mm and up to 13m long have
been made. The measured attenuation of a 0.5mm bore size waveguides is
about 0.5dB/m for CO2 lasers at 10.6m and about 1dB/m for Er:YAG lasers at
2.94m.
Morrow et al.
The waveguides developed by Morrow and coworkers [1.29] are constructed
from silver tubes. The tube is made of silver and there is no need to deposit a
metal layer inside the tube. The first step is to etch the bore of the tube in order
to smoothen it. Then a dielectric layer (AgBr) is deposited using a wet
chemistry method. The attenuation of a 1mm bore hollow waveguide for CO2
lasers at 10.6m is less than 0.2dB. However the beam shape quality of this
waveguide is very poor due to mode mixing which is caused by the surface
roughness of the silver tube.
Laakman et al. and Luxar et al.
Laakman et al. [1.30] have developed first a hollow waveguide with a
rectangular cross section. They used Ag, Au or Al as a metal layer and ZnSe or
ThF4 as a dielectric layer. Such waveguides had attenuation of about 1.7dB/m.
Later, the group begun to fabricate hollow waveguides with circular cross
section. Their fabrication technique involved depositing a Ag film on a metal
22
sheet and then over coating it with a dielectric layer, PbF2 [1.31]. The same
method was used by Luxar et. al as well. The attenuation of such waveguides
with bore size of 0.75mm was about 0.5dB/m for CO2 lasers at 10.6m.
Garmire et al. and Kubo et al.
Garmire et al. [1.32] developed hollow waveguides with rectangular cross
sections. They use polished metal strips made of Au or Al separated by brass
spacers 0.25mm to 0.55mm thick. The distance between the spacers was
several millimeters.
The measured attenuation of such a hollow waveguide with cross section of
0.5mm by 7mm was about 0.2dB/m. The maximum power that was transmitted
through it was 940W.
Kubo et al. [1.33] fabricated similar hollow waveguides. They used Al strips
and Teflon spacers. Such a waveguide with a cross section of 0.5mm by 8mm
transmitted laser power of 30W and the measured attenuation was about
1dB/m.
These waveguides had several advantages. They had small attenuation, they
were made of cheap and common materials and they were able to transmit high
laser power for a long time without damaging the waveguides. However they
also had some drawbacks. Their size made them inapplicable for medical
applications and for several industrial ones. There was a need for two focusing
lenses in order to couple the laser beam into the waveguide, and they had a
large attenuation when bent.
23
1.2.3.2 Attenuated Total Reflection (ATR) Hollow Waveguides
Hollow waveguides with n<1 were suggested by Hidaka et al. [1.34] in 1981.
In these waveguides the air core (n=1) has a refractive index greater than the
inner walls, therefore the guiding mechanism is the same as in regular core clad
fibers. To be useful the ATR must have an anomalous dispersion in the region
of the laser wavelength.
Haikida et al. made ATR hollow waveguides for CO2 lasers. These waveguides
were made of glass tubes with bore size of 0.6mm to 1mm, which are made of
lead and germanium doped silicates. By adding heavy ions to the silica glass, it
was possible to create the anomalous needed for CO2 laser guiding. Such
waveguides were able to transmit laser power of up to 20W and attenuation of
5.5dB/m [1.35].
Another kind of ATR hollow waveguides was made of chalchogenide glass
[1.36]. These waveguides had a bore size of 0.1mm to 0.3mm and were several
meters in length. The attenuation of these waveguides was very high (5dB/m).
Gregory et al. [1.37] have developed ATR waveguides made of sapphire. The
attenuation of these waveguides was much lower than that of the previous one,
1.5dB/m and they were able to transmit about 2KW of laser power [1.38].
24
1.2.4 Why hollow waveguides
The following table summarizes the advantages and drawbacks of each type of
fiber.
Fiber type
Glass


Advantages
Known and
established
manufacturi
ng process
Can be made
with very
small
diameter






Single crystal


High
melting
point
(sapphire).
Low
theoretical
attenuation.






Polycrystalline

Low
theoretical
attenuation.






Drawbacks
Brittle
Some of the
materials are
toxic.
Impurities
cause
scattering
High
attenuation
Moisture
sensitive
Ionization
radiation
sensitive
Some of the
materials are
sensitive to
UV radiation.
Some of the
materials are
toxic.
Hard to
manufacture.
Brittle
Soluble in
water.
Impurities
cause
scattering
Some of the
materials are
sensitive to
UV radiation.
Some of the
materials are
poisonous.
Hard to
manufacture.
Brittle
Soluble in
water.
Impurities
25
Liquid core

Non toxic


Hollow waveguides





Able to
transmit
high power
Low
attenuation
Non toxic
materials
Easy to
manufacture
Support a
wide range
of
wavelengths




cause
scattering
The fibers are
brittle
Large
attenuation
Sensitive to
temperature
beyond a
certain value.
Some of the
tube have
large surface
roughness.
Hard to
manufacture
long fibers
Surface
roughness
causes
scattering
Table 1.5 – Advantages and Drawbacks of Infrared Fibers
As can be seen from the table, hollow waveguides are good candidates for
transmitting infrared radiation. They support a wide range of wavelengths from
the x ray region [1.39] through the visible and to the IR. This characteristic
makes them able to transmit a wide selection of lasers for many applications.
Hollow waveguides may deliver very high power. This characteristic makes
them usable for industrial applications such as cutting metals. In addition they
are non-toxic which makes them very suitable for medical applications.
Moreover they may also be used for military applications such counter measure
detection, and for civilian applications such as imaging, spectrometry,
thermometry and signal delivering.
1.3 The Research Subject and Main Goal
26
The purpose of this research is to develop a new type of hollow waveguides
and to improve the hollow waveguides that are made by Croitoru et al. These
goals will be achieved by a theoretical and experimental study of laser
propagation through hollow waveguides. The development of a new type of
hollow waveguides will be based on the notion of photonic crystals and relating
it to hollow waveguides.
The theoretical study will be based on the development of an improved ray
model. Although many ray models have been suggested over the years, none of
them could accurately predict the transmission of laser radiation through
hollow waveguides. I have been successful in introducing the effect of surface
roughness into the ray model. This has enabled me to determine the influence
of the waveguide’s physical parameters (length, inner diameter, roughness), the
coupling conditions (focal length of the coupling lens, off center coupling) and
the shape of the laser beam at the entrance to the waveguide on its transmission
and energy distribution at the distal end. This ray model can also determine the
pulse dispersion caused by the waveguide. In the future, the introduction of
different coupling devices and tips can be included as well.
The experimental study is based on the measurement of the waveguide’s
transmission and laser beam shape as a function of the above parameters. The
experimental setup consists of a light source (laser, monochromator), a detector
(power/ energy meter, beam analyzer) and several types of hollow waveguides.
The experimental setup for the pulse dispersion measurement consists of a Qswitched Er:YAG laser, fast detector, oscilloscope and a hollow waveguide.
27
One of the ways to improve the current hollow waveguides and to develop new
types of hollow waveguides is by using photonic crystals structures. Photonic
crystals are periodic structure of dielectric materials that introduce energy
bandgaps for photons. These bandgaps are similar to those introduced to
electrons in crystals with periodic potentials. Such bandgaps cause radiation
with specific wavelengths and at any incidence angle to fully reflect (R=1)
from the photonic crystal structure. Introducing such a structure to hollow
waveguides will enable us to improve their performance and enlarge their
usage possibility. Although periodic dielectric structures have been
investigated for many years, I am suggesting to use a metal and dielectric
structure that may consist of fewer number of layers.
1.4 References for chapter one
1.1 C. K. Kao and G. A. Hockham, Proc. IEE, 133, 1158, (1966).
1.2 F. P. Karpon, D. B. Keck and R. D. Maurer, Appl. Phys. Lett., 17, 423,
(1970).
1.3 J. S. Sanghera and I.D. Aggarwal “Infrared Fibers Optics”, CRC Press
1998.
1.4 G. N. Merberg, J. A. Harrington, “Optical and Mechanical Properties of
Single-Crystal Sapphire Optical Fibers”, Applied Optics 1993, v. 32, p. 3201.]
1.5 G. N. Merberg, “Current Status of Infrared Fiber Optics for Medical Laser
Power Delivery”, Lasers in Surgery and Medicine 1993, p. 572.
28
1.6 R. W. Waynant, S. Oshry, M. Fink, “Infrared Measurements of Sapphire
Fibers for Medical Applications”, Applied Optics 1993, v. 32, p. 390.
1.7 T. J. Bridges, J.S. Hasiak and A. R. Strand, “Single Crystal AgBr Infrared
Optical Fibers”, Optics Letters 1980, v. 5, p. 85-86.
1.8 N. S. Kapany, “Fiber Optics: Principles and Applications”, Academic
Press, 1962.
1.9 D. A. Pinnow, A. L. Gentile, A. G. Standlee and A. J. Timper,
“Polycrystalline Fiber Optical Waveguides for Infrared”, Applied Physics
Letters 1978, v. 33, p. 28-29.
1.10 M. Ikedo, M. Watari, F. Tateshi and H. Ishiwatwri, “Preparation and
Characterization of the TLBr-TlI Fiber for a High Power CO2 Laser Beam”, J.
of Applied Physics 1986, v. 60, p. 3035-3036.
1.11 V. G. Artjushenko, L. N. Butvina, V. V. Vojteskhosky, E. M. Dianov and
J. G. Kolesnikov, “Mechanism of Optical Losses in Polycrystalline KRS-%
Fibers”, J. of Lightwave Technology 1986, v. 4, p. 461-464.
1.12 M. Saito, M. Takizawa and M. Miyagi, “Optical and Mechanical
Properties of Infrared Fibers”, J. of Lightwave Technology 1988, v. 6, p. 233239.
1.13 A. Saar, N. Barkay, F. Moser, I. Schnitzer, A. Levite and A. Katzir,
“Optical and Mechanical Properties of Silver Halide Fibers”, Proc. SPIE 843
1987, p. 98-104.
1.14 A. Saar and A. Katzir, “Intristic Losses in Mixed Silver Halide Fibers”,
Proc. SPIE 1048 1989, p. 24-32.
29
1.15 M. Saito, Y. Matsuura, M. Kawamura and M. Miyagi, “Bending Losses of
Incoherent Light in Circular Hollow Waveguides”, J. Opt. Soc. Am. A 1990, v.
7. p. 2063-2068.
1.16 A. Hongo, K. Morosawa, K. Matsumoto, T. Shiota and T. Hashimoto,
“Transmission of Kilowatt-Class CO2 Laser Light Through Dielectric Coated
Metallic Hollow Waveguides for Material Processing”, Applied Optics 1992, v.
31, p. 6441-6445.
1.17 H. Takahashi, I. Sugimoto, T. Takabayashi, S. Yoshida, “Optical
Transmission Loss of Liquid-Core Silica Fibers in the Infrared Region”, Optics
Communications 1985, v.53, p.164.
1.18 I. Gannot, J. Dror, A. Inberg, N. Croitoru, “Flexible Plastic Waveguides
Suitable For Large Interval of The Infrared Radiatio Spectrum”, SPIE 1994.
1.19 J. Dror, A. Inberg, R. Dahan, A. Elboim, N. Croitoru, “Influence of
Heating on Prefornence of Flexible Hollow Waveguides For the Mid-Infrared”
1.20 I. Gannot, S. Schruner, J. Dror, A. Inberg, T. Ertl, J. Tschepe, G. J.
Muller, N. Croitoru, “Flexible Waveguides For Er:YAG Laser Radiation
Delivery”, IEEE Transaction On Biomedical Engineering, 1995, v. 42, p.967.
1.21 A. Inberg, M. Oksman, M. Ben-David and N. Croitoru, “Hollow
Waveguide for Mid and Thermal Infrared Radiation”, J. of Clinical Laser
Medicine & Surgery 1998, v. 16, p.125-131.
1.22 A. Inberg, M. Oksman, M. Ben-David, A. Shefer and N. Croitoru,
“Hollow Silica, Metal and Plastic Waveguides for Hard Tissue Medical
Applications”, Proc. SPIE 2977 1997, p.30-35.
30
1.23 M. Ben-David, A. Inberg, I. Gannot and N. Croitoru, “The Effect of
Scattering on the Transmission of IR Radiation Through Hollow Waveguides”,
J. of Optoelectronics and Advanced Materials, 1999, No. 3, p-23-30.
1.24 I. Gannot, A. Inberg, N. Croitoru and R. W. Waynant, "Flexible
Waveguides for Free Electron Laser Radiation Transmission", Applied Optics,
Vol. 36, No. 25, pp 6289-6293, September 1997.
1.25 M. Miagi, K. Harada, Y. Aizawa, S. Kawakami, “Transmission Properties
of Dielectric-Coated Metallic Waveguides For Infrared Transmission”, SPIE
484 Infrared Optical Materials and Fibers III 1984.
1.26 Y. Matsuura, M. Miagi, A. Hongo, “Dielectric-Coated Metallic Holllow
Waveguide For 3m Er:YAG, 5m CO, and 10.6m CO2 Laser Light
Tranamission”, Applied Optics 1990, v. 29, p. 2213.
1.27 Y. Wang, A. Hongo, Y. Kato, T. Shimomura, D. Miura and M. Myagi,
“Thickness and Uniformity of Fluorocabon Polymer Film Dynamically Coated
Inside Silver Hollow Glass Waveguide"
1.28 J. Harrington, “A Review of IR Transmitting Hollow Waveguides”, Fibers
and Integrated Optics 2000, v. 19, p. 211-217.
1.29 P. Bhardwaj, O. J. Gregory, C. Morrow, G. Gu and K. Burbank,
“Preformance of a Dielectric Coated Monolithic Hollow Metallic Waveguide”,
Material Letters 1993, v. 16, p. 150-156.
1.30 K. D. Laakman, Hollow Waveguides, 1985, U.S. Patent no. 4652083.
1.31 K. D. Laakman and M. B. Levy 1991, U.S. Patent no. 5,500,944.
31
1.32 E. Garmire “Hollow Metall Waveguides With Rectangular Cross Section
For High Power Transmission”, SPIE 484 Infrared Optical Materials and
Fibers III 1984.
1.33 U. Kubo, Y. Hashishin, “Flexible Holloe Metal Light Guide For Medical
CO2
Laser”, SPIE 494 Novel Optical Fiber Techniques For Medical
Application 1984.
1.34 T. Haidaka, T. Morikawa and J. Shimada, “Hollow Core Oxide Glass
Cladding Optical Fibers For Middle Infrared Region”, J. Applied Physics 1981,
V. 52, p. 4467-4471.
1.35 R. Falciai, G. Gireni, A. M. Scheggi, “Oxide Glass Hollow Fibers For
CO2
Laser Radiation Transmission”, SPIE 494 Novel Optical Fiber
Techniques For Medical Application 1984
1.36 A. Bornstein, N. Croitoru, “Chalcognide Hollow Fibers”, Journal Of NonCrystaline Materials 1985, p. 1277.
1.37 C. C. Gregory and J. A. Harrington, “Attenuation, Modal, Polarization
Properties of n<1 Hollow Dielectric Waveguides”, Applied Optics 1993, v. 32,
p. 5302-5309.
1.38 R. Nubling and J. A. Harrington, “Hollow Waveguide Delivery Systems
for High Power Industrial CO2 Lasers”, Applied Optics 1996, v. 34, p. 372380.
1.39 F. Pfeiffer, “X-Ray Waveguides”, Diploma Thesis, July 1999.
32
Chapter Two - Attenuation Mechanisms In Hollow
Waveguides
List of Symbols For Chapter 2
i,  ..... Angle of incidence
t ............. Transmittance angle
r ............. Reflectance angle
rp ............. Reflection coefficient for the amplitude of p state polarization
rs ............. Reflection coefficient for the amplitude of s state polarization
tp ............. Transmission coefficient for the amplitude of p state polarization
ts.............. Transmission coefficient for the amplitude of s state polarization
Rp ............ Reflection coefficient for the power of p state polarization
Rs ............ Reflection coefficient for the power of s state polarization
n .............. Index of refraction
Wavelength
 ........ Phase shift
d .............. Dielectric layer thickness
 ............. Surface roughness
r ............ Path difference
k.............. Wave vector
R,r .......... Radius vector
 ............. Angular frequency
E ............. Electric field vector
H............. Magnetic field vector
.............. Scattering coefficient
 .............. Separation parameter
T ............. Correlation distance
33
As was seen in the previous chapter, hollow waveguides are composed of two
thin films (metal and dielectric), which are deposited on the inner wall of a
hollow tube. In order to understand the transmission of the laser beam through
such a waveguide we should examine how the characteristics of the dielectric
layer (thickness, index of refraction and roughness) influence the transmission
of the waveguide. The thickness of the dielectric layer and its index of
refraction influence the reflection coefficient of the thin layer. The roughness
of the surface influences the scattering of the incident laser beam.
This chapter deals with the two main attenuation mechanisms, which influence
the transmission of the laser through the hollow waveguide. The first is the
reflection of the laser beam from a thin layer, and the second is the scattering of
the laser beam from a rough surface. Understanding these mechanisms will
enable us to analyze in the next chapter the dependence of the waveguide’s
attenuation on the waveguide’s characteristics, its radius of bending and the
coupling conditions.
Although these two mechanisms had been discussed in many papers, I chose to
include them for two reasons. The first is that they are the most important
building blocks of the ray model that I will develop in the next chapter. The
second is the scattering theory that will be described here has not yet been
implemented to hollow waveguides. Furthermore these two mechanisms may
be used as a tool for designing hollow waveguides.
The chapter ends with the description of roughness measurements made by an
atomic force microscope. These measurements show the roughness of the inner
34
layers and the height distribution of the scattering centers. These measurements
will be used as parameters for the theoretical calculations that will be presented
in the next chapter.
2.1 Reflection from a thin film
First we will examine the properties of the reflection coefficient of a thin film,
especially how the characteristics of the dielectric film (thickness and index of
refraction)
influence
the
Figure 2.1
Reflection and refraction at a boundary
reflection coefficient. When
electromagnetic
impinges
on
radiation
a
boundary
i
r
n1
n2
t
between two materials with
different index of refraction it
undergoes two processes, reflection and refraction (figure 2.1). According to
the law of reflection and Snell’s law the relations between the angle of
incidence, i, the angle of reflection, r, and the angle of refraction, t, are [2.1]
i=r
(2.1.1)
n1sin(i)=n2sin(t)
(2.1.2)
The reflection (r) and transmission (t) coefficients are given by Fresnel’s
coefficients for TE (s) and TM (p) polarization [2.2]
rp 
n1 cos t   n2 cos i 
n1 cos t   n2 cos i 
rs 
n1 cos i   n2 cos t 
n1 cos i   n2 cos t 
(2.1.3)
(2.1.4)
35
tp 
2n1 cos i 
n1 cos t   n2 cos i 
(2.1.5)
ts 
2n1 cos i 
n1 cos i   n2 cos t 
(2.1.6)
and the reflection coefficient for the power are given by1
R p  R s  rp2  rs2
(2.1.7)
When the material, which the radiation (at a certain wavelength) is advancing
through, is completely transparent, its index of refraction is real. If the material
is only partially transparent such as in the case of metals and semiconductors,
its index of refraction is complex and can be expressed as
n  n  ik
where k is the extinction
coefficient, and n
(2.1.8)
Figure 2.2
Reflection from a thin layer
is the
I
r2
index of refraction.
2
Since we are dealing with
1
t2t’2r2
n2
t2r1 t2r1r’2
n1
reflection from a thin layer
0
t2t1
t2r1r’2t1
n0
the reflection coefficient is
influenced not only by the
change in the index of reflection but also by the multiple reflections inside the
thin layer. Figure 2.2 shows the path of a ray through a thin layer with a
thickness d and index of refraction n1.
1
The reflected electric, Er, field is related to the incidence electric field, Ei, by Er=rEi.The electric power is the
power of the electric field (Er)2=(rEi)2=r2Ei2=REi2
36
The ray that is reflected from the lower boundary undergoes a phase shift of e -i
where

2

n1d cos1 
(2.1.9)
As we can see from figure 2.2, the reflectance from the thin layer is the sum of
all the internal reflections plus the first reflection. The reflection is then given
by
rtot 
r2  r1e 2i
1  r1r2e  2i
(2.1.10)
and the reflection coefficient for the power is
R
r12  r22  2r1r2 cos( 2 )
1  r12 r22  2r1r2 cos( 2 )
(2.1.11)
This expression is correct for any polarization (TE and TM). For an
unpolarized radiation we have to take the average of the two polarizations. In
the case of unpolarized radiation the reflection coefficient is given by
Rtot 
Rs  Rp
2
(2.1.12)
As was described in the first chapter, hollow waveguides are made of two thin
layers (Ag and AgI), that are deposited on the inner wall of a hollow tube, and
an air core. The air and the AgI are dielectric materials, which absorb little of
the radiation. The Ag layer is a metal layer with a complex index of refraction,
which absorbs some of the radiation. In order to get maximum reflectance we
need to find which state of polarization is reflected better, and how the
dielectric layer properties (thickness, index of refraction) influence the
radiation’s reflection.
37
In order to calculate the reflection coefficients of the different layers one has to
know the index of refraction of Ag and AgI. The index of refraction of Ag
depends on the radiation wavelength [2.3-2.5]. For infrared radiation Beatie
[2.4] found the following relations for the index of refraction and absorption of
Ag
n  a2  b4
(2.1.13)
k  c  d3
(2.1.14)
where  is measured in m and a=0.12, b=1.3e10-4, c=7.2, d=3.6e10-3. The
index of refraction of AgI is 2.2 for all wavelengths [2.5].
38
The
first
hollow
waveguides,
which
Figure 2.3
Reflection Coefficient Vs. Angle of Incidence
were developed, had
only
a
metal
film.
Using equations 2.1.32.1.6 one can calculate
the
reflection
coefficients
(Rs,
Rp,
Rtot) of the metal layer
for the different types of polarization. Figure 2.3 shows the reflection
coefficients
silver
from
layer
=10.6m
(i.e.
a
for
Figure 2.4
Reflection Coefficient Vs. Angle of Incidence
CO2
laser). We can see from
the figure that the layer
does not reflect well
TM
polarization
at
large
angles
of
incidence. To overcome this drawback Miyagi [2.6] suggested adding a
dielectric layer over the metal layer. The dielectric layer could be made of
several materials Si, Ge or as in our case AgI. As can be seen from figure 2.4,
the dielectric layer improves the reflection of TM polarization at large angles.
The dielectric layer decreases slightly the reflection coefficients of the TE
39
polarization but the total reflection coefficient increases. Therefore the
introduction of a dielectric layer improves the hollow waveguide.
If we examine equations 2.1.9 and 2.1.10 closely we can see that the
reflectance of the waveguide depends on the laser’s wavelength, the materials
deposited on the inner wall, the thickness of the dielectric layer and the angle
of incidence. Since in most applications the laser’s wavelength is constant, we
need to adapt the materials and the thickness of the dielectric layer to it. If we
need to transmit several wavelengths, the parameters of the materials need to
be adapted to all of them.
We can see from equation 2.1.11 that the index of refraction of the inner layers
plays a major role in determining the waveguide’s reflectance. In order to
achieve maximum transmission we need to select materials with indices of
refraction, which are suitable to the wavelengths we want to transmit.
The angle of incidence and the thickness of the dielectric layer determine the
nature of interference (constructive or destructive) of the incident radiation. We
need to build the hollow waveguide with the appropriate thickness of the
dielectric layer [2.7].
In order to build waveguides suitable for transmitting a certain laser
wavelength we wrote a computer program which calculates the reflection
coefficient as a function of the thin layer parameters (thickness, index of
refraction), the laser wavelength and angle of incidence.
Since we know the wavelength of the laser, which we want to transmit, we first
have to choose the material for the dielectric layer. The dielectric layer needs to
40
have two characteristics: it has to be transparent at the laser wavelength, and it
has to have the appropriate thickness and index of refraction, which enable
maximum reflectance for various angle of incidence. Figures 2.5 to 2.7 show
the reflection coefficient as a function of the index of refraction, the layer
thickness and the angle of incidence. For the calculation we took the laser
wavelength to be =10.6m, which is the wavelength of CO2 laser and
assumed that the metal layer is silver.
[m]
Figure 2.5
Reflection coefficient vs. thickness and index of refraction
of the dielectric layer
=850, =10.6m, metal layer – Ag
41
[m]
[degrees]
Figure 2.6
Reflection coefficient vs. thickness of the dielectric layer
and the laser’s angle of incidence
n=2.2, =10.6m, metal layer – Ag
[degrees]
Figure 2.7
Reflection coefficient vs. index of refraction of the dielectric layer
and the laser’s angle of incidence
d=0.8m, =10.6m, metal layer – Ag
42
Figure 2.8 shows the reflection coefficient as a function of the laser wavelength
and thickness of the dielectric layer. For the calculation the laser’s angle of
incidence is 85o, the index of refraction of the dielectric layer (n) is 2.2, which
is the index of refraction of AgI, and the thickness of the dielectric layer is
0.8m. The metal layer is silver.
[m]
[m]
Figure 2.8
Reflection coefficient vs. thickness of the dielectric layer and the laser’s wavelength
nd=2.2, metal layer – Ag
We can use the above graphs in order to choose the thickness and index of
refraction of the dielectric layer suitable for the laser wavelength and its angle
of incidence. Notice that for d=0 the index of refraction has no influence and
the reflection coefficient is the one for a metal layer.
As an example let us design a waveguide for CO2 laser (=10.6m). According
to the above graphs we will get maximum reflectance (0.992R1) for d1m
and n2.
43
2.2 Scattering from rough surfaces
The second attenuation mechanism that affects the transmission of the hollow
waveguide is the scattering of the laser beam from a rough surface. The tube in
which the thin layers are deposited is rough and as we shall see later the
deposition of the thin layers increases the roughness of the waveguide’s wall.
2.2.1 The Rayleigh criterion
Before attempting to determine the scattering coefficient quantitatively, we
shall first consider a
more
Figure 2.9
Specular reflection and difused scattering
elementary
question. When does a
smooth surface become a




rough one, or for what
values of wavelength,
Specular Reflection
Diffuse Scattering
surface roughness and angle of incidence does specular reflection change to
diffused scattering (figure 2.9)?
Rayleigh suggested [2.8]
Figure 2.10
Derivation of the Rayleigh criterion
a way of formulating the
2
1
relation involving these





parameters: Consider two
rays 1 and 2 (figure 2.10) incident on a surface with irregularities of heights 
at a grazing angle . The path difference between the two rays is
44
r  2 sin  (2.2.1)
and hence the phase difference is
 
2

r 
4

sin 
(2.2.2)
If the phase difference is small, the two rays will be almost in phase as they are
in the case of perfectly smooth surface. If the phase difference increases, the
two rays will interfere until for    where they will be in phase opposition
and cancel each other. If there is no energy flow in this direction, then it must
have been redistributed in other directions, for it can not have been lost. Thus
for    the surface scatters and hence is rough, while for   0 it reflects
specularly and is smooth. We may now establish a value of phase difference
between theses two extremes to distinguish between “rough” from “smooth.
We may arbitrarily choose the value of half way between the two cases, i.e.
   . By by substituting this value in equation 2.2.2 we obtain the relation
2
known as the “Rayleigh criterion”, namely that a surface is considered smooth
for


8 sin 
(2.2.3)
However there seems to be a little point in assessing an exact dividing line,
which must in essence be more or less conventional. A safer way of expressing
the basic idea of the Rayleigh criterion is to use the right hand side of equation
2.2.2 as a measure of effective surface roughness. It may then be stated that a
surface will be effectively smooth only under two conditions:
45

 0 or   0

(2.2.4)
i.e. the surface roughness is small in comparison to the radiation wavelength or
when the grazing angle is small. If either one tends to zero then according to
equation 2.2.2 the phase difference will tend to zero, thus implying that the
surface is smooth.
2.2.2 The general solution for scattering from rough surface
In this section we will derive the general solution for scattering from a rough
surface [2.8]. The rough surface will be given by the function
   x, y 
(2.2.5)
The mean level of the surface is the plane z=0 (figure 2.11).
All quantities associated with the incident wave will be denoted by the
subscript 1 and those associated with the scattered field - by the subscript 2.
Thus the incident wave is E1 and the scattered wave is E2. The medium in the
space z   is assumed to be free space. We shall assume E1 to be a harmonic
plane wave of unit amplitude:
E1  exp ik1  r  i t 
(2.2.6)
where
k1 
is the propagation vector,
(2.2.7)
Figure 2.11
Basic notion
which will always lie in the
xz plane (this plane will
2 k 1
 k1
z
r
k1
r
k2
1
2
x
46
denote the plane of propagation of the rays in the ray model in the next chapter,
thus I will discuss only the influence of scattering on the propagation vector in
that plane) (figure 2.11) and r is the radius vector
r  xx0  yy 0  zz 0
(2.2.8)
In particular, for points on the surface S, we have
r  xx0  yy 0   x, y z 0
(2.2.9)
The angle of incidence, included between the direction of propagation of E1
and the z axis, will be denoted by 1; the scattering angle, included between z0
and k2, where
k 2  k1 
2
(2.2.10)

will be denoted by 2 (figure 2.11).
Let P be an observation point and let R’ be the distance from P to a point x, y,
 x, y  on the surface S. Then the scattered field is given by the Helmholtz
integral
E1 P  
1
4


E 
  E n  n dS
(2.2.11)
where


exp ik 2 R '
R'

(2.2.12)
In order to deal with plane waves, rather then spherical ones, we let R '   ,
i.e. we move the point P to the Fraunhofer zone of diffraction and then as seen
in figure 2.12
47
to P
R0=OP
(x,y
)
 
k2  r
OC 
k
R’=BP

k2
C

r
B
O
X
Figure 2.12
k 2 R '  k 2 R0  k 2  r
(2.2.13)
where R0 is the distance of P from the origin, so that

exp k 2 R0  k 2  r 
R'
(2.2.14)
Within this approximation the field on surface S will be
E S  1  R E1
(2.2.15)
and
 E 
   1  R E1k1  n
 n  S
(2.2.16)
48
where n is the normal to the surface at the considered point and R is the
reflection coefficient of a smooth plane given in section 2.1. The second
relation follows from the first by differentiating the incident and reflected
waves or by taking the magnetic field as
H S  1  RH1
(2.2.17)
and using Maxwell equation
  E  
H
t
(2.2.18)
To simplify the calculations we will limit ourselves to the case of one
dimensionally rough surface i.e.
 x, y    x
(2.2.19)
which is constant along the y coordinate, so that n always lies in the plane of
incidence xz in figure (2.13). Then
k1
1

n$n

x
  x
Figure 2.13
   1     1  arctan  ' x
Substituting 2.2.14, 2.2.15 and 2.2.16 in 2.2.13 we find
(2.2.20)
49
E2 
i exp ikR0 
Rv  p n exp iv  r dS
4 R0 S
(2.2.21)
where
v  k sin  1  sin  2 x 0  k cos 1  cos 2 z 0
(2.2.22)
p  k sin  1  sin  2 x 0  k cos  2  cos  2 z 0
(2.2.23)
n  x0 sin   z 0 cos 
(2.2.24)
r  xx0   x z 0
(2.2.25)
dS  sec  dx
(2.2.25)
tan    '  x 
(2.2.26)
For a surface extending from x=-L to x=L we may thus write 2.2.20 in the
scalar form
ik exp ikR0 
'
 L a  b exp iv x x  iv z dx
4 R0
L
E2 


(2.2.27)
where
a  1  r  sin  1  1  R  sin  2
(2.2.28)
b  1  r  cos  2  1  R  cos  1
(2.2.29)
To get rid of the factor in front of integral 2.2.27, we normalize the expression
2.2.27 by introducing a scattering coefficient

E2
E20
(2.2.30)
50
E20 is the field reflected in the direction of specular reflection (1=2) by a
smooth, perfectly conducting plane of the same dimensions under the same
angle of incidence at the same distance. E20 is given by
E20 
k exp ikR0 L cos 1
 R0
(2.2.31)
and the scattering coefficient is given by
  1 , 2  
L
F2
exp iv  r 
2 L L
(2.2.32)
with
F2  1 , 2   sec  1
1  cos 1   2 
cos 1  cos 2
(2.2.33)
2.2.3 Scattering from a randomly rough surface
The rough surfaces met in nature are best described by the statistical
distribution of their deviation from a certain mean level. This does not yet,
however, describe the surface completely, since this distribution does not tell
us whether the hills and valleys of the surface are crowded, close together, or
whether they are far apart. A second function, the correlation function or the
autocorrelation coefficient, describes this aspect of the surface. It turns out that
the statistical distribution suffices for determining the mean value of the field,
but to calculate its variance (or power) the general two dimensional distribution
is needed.
In this section we shall apply the general solution derived in the previous
section to the case where  x, y  is a random function. Again we will limit
ourselves to the case of one dimensional roughness,  x  .
51
Let  x  be a random variable assuming values z with a probability density
w(z): Let the mean value be
 0
(2.2.34)
and consider the mean value of the integral
L
 exp iv  r dx
L
L
  exp iv x x  exp iv z z  dx  exp iv z 
L
L
 exp iv x dx (2.2.35)
x
L
We notice that
exp iv z  

 wz  exp iv z dz   v  2
z
(2.2.36)
z

is the definition of the characteristic function  vz  , associated with the
distribution w(z). We therfore have
L
 exp iv  r dx
L
L
  v z   exp iv x x dx
(2.2.37)
L
Substituting 2.2.36 and 2.2.37 in 2.2.32, we find the important relation
   vz  0
(2.2.38)
where
 0  sin cvx L 
(2.2.39)
is the scattering coefficient of a smooth surface.
Note that as L>>,  0 will be equal to unity for the specular direction (1=2,
vx=0), but it will rapidly tend to zero as 2 leaves the specular direction. Hence
 will equal zero for any direction of scattering except for a narrow wedge
2
The characteristic function
 v
of a distribution p(z) is
 v  

 pz  exp ivzdz

52
about the specular direction; if  vz  equals zero in the specular direction 
will equal to zero in all directions. However, it should be remembered that  is
a complex quantity and that we may not infer   0 from   0 (unless  is
real and non negative).
Since  is a complex quantity, its mean value is of little use except as a
stepping stone to determine the mean value of
   *
(2.2.40)
Note that the mean square of 2.2.40 is
 *   2 
E22
2
E20
(2.2.41)
whitch is proportional to the mean scattered power: It is also related to the
variance of , denoted by D, and the variance of the scattered field DE2 
by
 *    *  D     * 
1
DE2  3 (2.2.42)
2
E20
The root mean square value of  is
 RMS   *
(2.2.43)
and the mean scattered power is given by
2
P2  12 Y0 E2 E2*  12 Y E20
 *
where Y0=1/120 is the admittance of free space.
3
D    *    *
(2.2.44)
53
Thus we can find all the quantities of interest by determining the value  * .
To find  * , we shall assume that the surface S is large enough, to get rid of
edge effect terms. From 2.2.32 we have

*
F2
 22
4L
L L
  exp iv x
x
1
 x2  exp iv z  1  2  dx1dx2
(2.2.45)
 L L
where
 1   x1 
 2    x2 
(2.2.46)
From the theory of statistics 4
exp iv z  1  2  
 
  W z , z  exp iv 
1
2
z
1
  2 dz1dz2  2 v z  v z  (2.2.47)
  
is the two dimensional characteristic function of the distribution W(z1,z2),
which will be equal to the product of the distribution w(z1) and w(z2) only if  1
and  2 are independent. Now we can assume that  x  is a “purely random”
process, that does not contain any non random periodic components.  1 and  2
will obviously be independent if they are far apart, i.e. for large values of the
“separation parameter”, defined by
  x1  x2
(2.2.48)
But when  is small, i.e. when x1 and x2 are taken near each other,  1 and  2
will be correlated, and when =0, they will be identical. Thus the distribution of
, w(z), is insufficient to determine the two dimensional characteristic function
4
 2 v1 , v2  
 
  p z, y  exp iv z  iv y dzdy
2
  
1
2
54
2.2.47. In addition to the statistical distribution of  we must know the
correlation function
L
B   lim
1
  x   x   dx
L  2 L 
L
(2.2.49)
or its autocorrelation coefficient, which is related to the correlation function by
C   
B   
D 
2

 1 2   1  2
 12   1
2
(2.2.50)
Since   0 we get
C   
 1 2
 12
(2.2.51)
It follows directly from equation 2.2.49 that
lim C   1
(2.2.52)
 0
i.e. full correlation (linear dipendence) and
lim C    0
(2.2.53)
 
i.e. independence.
For a purely random surface C  will decrease monotonously from its
maximum value C 0  1 to C  0 . Let the distance in which C  drops to
the value e-1 be T. This distance, which will be called the “correlation
distance”, is smaller then L.
From 2.2.42, 2.2.45, 2.2.38 and 2.2.36 we find that
D  

*
  
*
F22
 2
2L
L L
  exp iv  x
x
1


 x 2   2 v z ,v z    2 v z  *2 v z  dx1 dx 2
 L L
(2.2.54)
when  1 and  2 are independent
55
 2 v z ,v z    2 v z  *2 v z 
(2.2.55)
but  1 and  2 are independent for all but small ; hence the square bracket in
2.2.54 will vanish for all but small . Substituting 2.2.48 in 2.2.54 we get
F22
D   2
2L
 exp iv   v ,v    v  v d
L
x
2
z
z
2
*
2
z
z
(2.2.56)
L
with the only significant contribution to the integral coming from the region
near =0.
Equation 2.2.54 shows that to determine the variance we must know the
dimensional distribution W(z1, z2; ) of the surface, i.e. know the dimensional
distribution of  at two points x1 and x2 separated by any distance .
2.2.4 The normally distributed surface
We take the normal distribution as the most important and typical of a rough
surface for substitution in the formulae of the preceding section.
Let  be normally distributed with the mean value
 0
(2.2.57)
and standard deviation . The distribution of  is given by
 z2 
1
wz  
exp   2 
 2
 2 
(2.2.58)
The standard deviation , which in this case is due to 2.2.57 and is also the root
mean square of , describes the roughness of the surface.
The characteristic distribution of 2.2.36 is
 v   exp  12  2vz2 
(2.2.59)
56
The rough surface is not uniquely described by the statistical distribution of ,
as it tells us nothing about the distances between the hills and valleys of the
surface; i.e. about the density of irregularities. This is described by the
autocorrelation coefficient given by 2.2.50. The autocorrelation coefficient is
an even function of . We take as a sufficiently general autocorrelation
coefficient the function
 2 
C    exp   2 
 T 
(2.2.60)
where T is the “correlation distance”.
Substituting 2.2.59 in 2.2.38 and expressing vz explicitly, we find that
 2 2
  0 exp 


2

cos 1 cos 2 2 

(2.2.61)
since 0 vanishes everywhere except near the direction of specular reflection
(vx=0)
 spec
 1  4  cos 1  2 
 exp   2 
 
 

 

(2.2.62)
This result was also obtained experimentally by several others such as Bennett
et al. [2.9]
The two dimensional normal distribution of two random variables 1, 2, with
mean values zero and variances 2, and whitch is correlated by a correlation
coefficient C, is
wz1 , z2  
 z 2  2Cz z  z 2 
exp   1 2 1 2 2 2 
2 1  C
2 1  C 2


1


The characteristic function of this distribution is given by
(2.2.63)
57
 2 vz ,vz   exp  vz2 21  C 
(2.2.64)
Substituting 2.2.60 into this expression and expanding it in an exponential
series we have
 2 vz ,vz   exp  vz2 2  vz2 m m 2exp   m



m 0
2

 (2.2.65)
T 
2
We shall work with the quantity vz2 2 . For briefness we therefore introduce a
new symbol g, where
g  v z  2

cos 1 cos 2 

(2.2.65)
Using 2.2.59 we then find
 v ,v    v  v   e  gm! exp  m

2
z
z
z

g
*
z
m
m 0
2


T 
2
(2.2.66)
Substituting 2.2.66 into 2.2.56 we obtain
D  


F2
gm
  m 2 
ivx  g
e
exp

d

T2
2 L 

m  0 m!
(2.2.67)
where we have replaced the integration limits by ; this is permitted since the
integral receives significant contributions only from the region near =0.
Using the integral

e
 at
cos btdt 


a
e
b2
4a
a  0
(2.2.68)
we then find
D 
 F 2T
2L

gm
  v 2T 2

exp  x

4
m


m0 m! m
eg 
Using 2.2.38 and 2.2.41 we can also write
(2.2.69)
58

*
  sin v L  2
 F 2T
x
 
 e  
  vx L 
2L

g
In order to find the scattering
coefficient we should compute

gm
  v 2T 2
 
exp  x
4m  
m


 m!
m 0
(2.2.70)
Figure 2.14
Normalized scattering coefficient vs. scattering
angle
(/=0.0001,T/=1000, L=100)
the series in equation 2.2.70. The
series is calculated numerically
dotted line - /=0.0001
solid line - /=0.01
dashed line - /=1
until the last term is less then
0.01%. The scattering coefficient
was calculated using a MATLAB
program. Figures 2.14 and 2.15
show the scattering coefficient for
various angles of incidence and
Figure 2.15
Normalized scattering coefficient vs. scattering
angle
(=80,T/=1000, L=100)
surface roughness respectively.
We can see that as the angle of
incidence decreases, or as the
surface roughness increases, the
normalized scattering coefficient
increases. This is also predicted
from equation 2.2.62.
2.3 Measurements of surface roughness
In order to apply the above theory to real hollow waveguides (as will be
explained in the next chapter) we first have to measure the surface roughness
and its distribution [2.10].
59
The surface roughness was measured using an atomic force microscope
(AFM). We measured the surface roughness of fused silica, Teflon and glass
waveguides coated with Ag and AgI thin filmes. We cut several the
waveguides open and measured the roughness of little pieces. The distribution
of the surface height was calculated using the AFM software.
Table 2.1 summarizes the surface heights measurements. As can be seen from
the table, the smoothest surface is that of fused silica waveguides and the
roughest surface is that of the Teflon waveguides. It is clear from the table that
except for the fused silica tube, the addition of the AgI layer smoothen the
surface roughness hence decreasing the surface scattering.
Waveguide type
Teflon
Fused silica
Glass
Ag layer roughness
(nm)
298
10
42
AgI layer roughness
(nm)
100
24
20
Table 2.1
Surface roughness of different waveguides
Figures 2.16 to 2.17 a, b and c show the surface roughness of Ag layer, AgI
layer and the height distribution of the AgI layer for glass and Teflon
waveguides respectively. Similar results were obtained for fused silica
waveguides. We can see from the figures that the AgI layers looks smoother
than Ag layers.
60
Figure 2.16
AFM measurements for glass waveguide’s Ag and AgI layer
(a) Ag layer
(b) AgI layer
61
(c) AgI height distribution
Figure 2.17
AFM measurements for Teflon waveguide’s Ag and AgI layer
(a) Ag layer
62
(b) AgI layer
(c) AgI height distribution
2.4 Summary of chapter 2
This chapter laid the foundation for the ray model that I will discuss in the next
chapters. It described the main attenuation mechanisms in hollow waveguides.
These mechanisms determine the performance of hollow waveguides. The
reflection from the thin film determines the wavelength that will be transmitted
by the waveguide and its minimal attenuation, while the surface roughness
63
determines the amount of scattering the laser beam encounters while
propagating through the waveguide. Surface scattering has not been applied to
hollow waveguides yet and I will use the above theoretical analysis as one of
building blocks of my ray model.
I also measured the roughness of the different layers within different types of
tubes. As was shown in the above table, different tubes have different
roughness. Hence they impose different roughness on the deposited layers. The
differences in the roughness among the tubes are the cause for the variation in
attenuation among them. This will be shown in the next chapter, where I will
use the above measurements results as parameters for the ray model and the
theoretical calculations.
2.5 Refrences for chapter 2
2.1 D. Marcuse, “Theory of Dielectric Optical Waveguides”, ch. 1,
Accademic Press, 1991.
2.2 Chopra, “Thin Film Phenomena”, ch. 2, McGraw Hill 1969.
2.3 E. D. Palik: “Handbook of Optical Constants of Solids”, D.W. Lynch
and W.R. Hunter, “Comments on optical constants of metals and an
introduction to the data of several metals - Silver”, p. 275, Accademic press.
2.4 J. R. Beatie, “The annomalous skin effect and the IR properties of silver
and aluminum”, Physica 1957, v. 23, p. 898.
2.5 W. G. Drisco;l and W. Vaughan, “Handbook of Optics”, McGraw Hill
1978.
64
2.6 M. Miyagi, A. Hongo and S. Kawakami, “Transmission characteristics of
dielectric coated metalic waveguides for infrared transmission: slab waveguide
model”, IEEE J. Quantun Electronics, 1983, v. 9, p. 136.
2.7 M. Alaluf, J. Dror, R. Dahan, N. Croitoru, “Plastic hollow fibers as a
selective mid-IR radiation transmission medium”, J. Appl. Phys., 1992, v. 72,
p. 3878.
2.8 P. Beckman and A. Spizzichino “The Scattering of Electromagnetic
Waves from Rough Surfaces”, Pergamon Press, 1963.
2.9 H. E. Bennet “Specular reflectance of aluminized ground glass and height
distribution of surface irregularities”, J. Opt. Soc. Am. 1963, v. 53, p. 1389.
2.10 M. Ben-David, A. Inberg, I. Gannot and N. Croitoru, “The Effect of
Scattering on the Transmission of IR Radiation Through Hollow Waveguides”,
J. of Optoelectronics and Advanced Materials, 1999, No. 3, p-23-30.
65
Chapter Three - Radiation Propagation Through Hollow
Waveguides
List of Symbols For Chapter 3
.............. Propagation constant
 ............. Attenuation constant
a, T, r ...... Waveguide’s inner radius
 .............. Polar distribution of the ray model
r .............. Radial distribution of the ray model
R ............. Reflection coefficient
n .............. Index of refraction
Wavelength
 ............. Angle of entrance
Angle of propagation
d,  .......... Dielectric layer thickness
 ............. Surface roughness
k.............. Wave vector
 ............. Angular frequency
E ............. Electric field vector
S.............. Scattering coefficient
I .............. Laser beam energy
z .............. The distance between two refractions
p .............. Number of times a ray impinges on the waveguide’s wall
l............... waveguide’s length
A ............. Attenuation
T ............. Transmission
0 ............ Laser spot size
f .............. focal length
D ............. lens diameter
66
y .............. Deviation from the waveguide’s center
P.............. laser power
R ............. Bending radius
 .............. The ratio of the experimental M2 factor to the theoretical one
In the previous chapter I introduced the two main attenuation mechanisms in
hollow waveguides, Fresnel reflection related properties (thin film thickness,
index of refraction etc.) and the surface roughness. These mechanisms are
structural mechanisms. In this chapter I will relate the attenuation mechanisms
to the transmission/ attenuation of the hollow waveguide.
There are two methods to analyze the propagation of infrared radiation through
hollow waveguides. The first is known as the mode model. The mode model
solves Maxwell equations in cylindrical coordinates and derives from the
solution the attenuation of the hollow waveguide as a function of the
waveguide parameters.
The second method is the ray model. This model is applicable as long as the
laser wavelength is small compared to the waveguide’s inner diameter. This
condition is satisfied in our case, where the inner diameter is larger than 0.2mm
and the wavelength is smaller than 12m.
This chapter is composed of two main parts. The first part is the theoretical
part, in which I will describe briefly the mode approach and develop the
improved ray model. The improved ray model will be the base of the
theoretical study of hollow waveguides. The main improvements, that I have
introduced in this model, are: introduction of surface roughness to the
67
attenuation calculations and geometrical calculations for bent hollow
waveguides and off center coupling. Although the ray model is not new. The
approach I have developed enables to take into account many of the hollow
waveguide’s parameters as well as the laser beam and coupling conditions
parameters. This ray model, as will be demonstrated, gives very accurate
results when compared to the experimental results.
The second part of this chapter describes the experimental results, which were
obtained using different hollow waveguides. The experimental results are
compared to the theoretical calculations and are found to be in good agreement
with them. This study enables me to identify the drawbacks of the current
generation of hollow waveguides and to suggest a way to improve and develop
the next generation of hollow waveguides. This will be done in chapter five.
Note: This chapter deals only with the spatial aspect of laser propagation
through hollow waveguides. The temporal aspect will be dealt in the next
chapter.
3.1 Theory
3.1.1 The mode approach
In the mode approach, we solve Maxwell equations in cylindrical coordinates
with the appropriate boundary conditions. The solutions to this problem give
the propagation conditions of each mode along the hollow waveguide. Many
researchers used this method to derive the attenuation of hollow waveguides.
Among the first who analyzed the propagation of each mode in hollow
waveguides were Marcatili and Schmeltzer [3.1]. They analyzed the
68
attenuation coefficients of each mode in hollow metallic and dielectric
waveguides, assuming the waveguide’s inner diameter is much larger then the
laser wavelength. According to Marcatili and Schmeltzer the propagation
constants of the different modes is given by
 nm 
2
2 
 1  unm  

  n   
1

1

Im

 




  2  2a  
 a  

(3.1.1)
and the attenuation coefficients is given by
3
u  
  nm  3 Re  n 
 2a  a
2
 nm
(3.1.2)
where a is the waveguide’s inner diameter, unm is the mth root of the equation
J n1 unm   0
(3.1.3)
and  n is a function of the propagation mode and the index of refraction. For the
HE11 mode, which is the main mode of a CO2 laser,
1 2
  1
2
n 
 2 1
(3.1.4)
Equation 3.1.2 indicates that the waveguide’s attenuation is proportional to a-3.
Hence as the waveguide diameter decreases the attenuation increases. This
theoretical result is not accurate. Experiments done by Harrington [3.2] have
shown that the waveguide’s attenuation is proportional to a-2 and nor a-3.
While Marcatili and Schmeltzer analyzed the mode propagation in hollow
metallic waveguides, Miyagi et al. analyzed the mode propagation in hollow
metallic-dielectric waveguides [3.3, 3.4].
69
Miyagi et al. used the following argument. Let us look at a hollow cylindrical
waveguide with inner radius T and index of refraction n0  1 . The waveguide is
coated by a metallic layer with index of refraction n-ik, and a number of
dielectric layers made of two materials of width  iT and index of refraction ain0
(i=1,2) and a layer of width  near the air core. Figure 3.1 shows the index of
refraction profile for such a waveguide. Miyagi assumes that there are no
energy losses in the dielectric layers in the dielectric layers and that a1<a2.
n(x )
n0
1 2

a1
n  ik
a2
Dielectric layers
metal
1
1
Figure 3.1
The index of refraction profile of Miyagi’s waveguide
The layers widths satisfy the condition
 i ai2  1 2 n0 k0T 
1

2
(3.1.5)
The attenuation constant is given by
  n0 k0
u02
F
n0 k0T 3
(3.1.6)
where F is a constant that is a function of  and the polarization of the laser
radiation.
70
When there is an odd number of dielectric layers
m  2m p  1
(3.1.7)
the minimum of the attenuation coefficients is achieved when
 a  1
2
1
1
2
 a
1
n0 k0T   tan 
1
2
 a1  1 4
1
mp
m p
 a1 
  C 2
 a2 

  s

(3.1.8)
where s is an integer and C is given by
C
a12  1
1
a22  1
(3.1.9)
For the hybrid modes
Fmin
2
1 mp 
a1
 Fmetal C 1 
1
2
 a12  1 2
 a1 
 
 a2 
2m p
2
m p
C
2

  Fmetal  Fdiel

(3.1.10)
where
Fmetal 
n
n  k2
2
(3.1.11)
For a waveguide without a dielectric layer the attenuation is given by
  n0 k0
u02
F 0 
3
n0 k0T 
(3.1.12)
where F 0  for the hybrid mode is equal to n 2 .
The above equations show that the bigger the metal index of refraction the
smaller the waveguides attenuation. This is true for the infra red region where
k>>n. The addition of the dielectric layers causes the attenuation to decrease
since Fdiel  1 . As an example let us take a waveguide made of Al (n-ik=20.558.6i) and Ge (n=4). The calculated attenuation is 0.08dB/m while the
calculated attenuation without a dielectric layer is 12dB/m.
71
3.1.2 The ray model
According to the ray model [3.5, 3.6] one can decompose the laser beam into
separate rays, and use the laws of geometrical optics to calculate the ray
propagation through the waveguide. This model may be used since in our case
<<ID, where  is the wavelength of the coupled laser beam and ID is the inner
diameter of the waveguide’s cross section.
We assume that multiple incidences on the metal and dielectric layers guide the
rays, by refraction and reflection. The dielectric layer has a normal distribution
of heights (equation 2.2.58) as was measured experimentally.
The “traditional” ray model used the following conditions to describe the laser
beam propagation through hollow waveguides:
1. Fresnel’s equations (equations 2.1.3-2.16) give the reflection
coefficient each time the ray impinges the waveguide’s wall.
2. The rays propagate only frontally and not rotationally (there are
no skewed rays). According to Miyagi [3.7] the contribution of
skew rays is of the second order and can be neglected.
3. Two coordinates represent the laser beam cross-section and the
point the ray enters into the waveguide; r (with Gaussian
distribution) and  (with uniform distribution) (figure 3.2a),
where 0  r  R (R is the laser spot size) and 0    2 . The
angle  also determines the plane in which the ray propagates.
72
4. The angle of entrance, , (figure 3.2b) has a Gaussian
distribution. This angle determines the angle of propagation, ,
by =90-.
r


Figure 3.2
The ray parameters
5. The rays have random polarization, TE or TM.
6. The total energy of the laser beam, I, is the sum of the energy of
all rays, and is given by
I   I i  r ,   , ri ,  i 
(3.2.1)
i
where r, and  are the standard deviations of the Gaussian beam size
and the angle respectively.
The traditional ray model and mode approach did not give accurate results
when trying to compare their results with the experimental ones. One of the
reasons is that these models did not take into account the surface roughness.
The surface roughness causes scattering ,which (as shall be seen in the next
73
chapter) increases the attenuation and changes the energy distribution outside
the waveguide. Miyagi et al. [3.8] tried to use equation 3.3.14 in order to take
into account the influence of the surface roughness. They assumed that the ray
continues at the same angle after it impinges on the waveguide’s wall and is
attenuated according to equation 3.3.14. This is of course not accurate since
the surface scatters the ray’s energy in a range of directions and changes the
angle of incidence.
To improve the ray model we introduced the influence of surface roughness as
described in chapter 2. The new ray model adds the following assumptions:
1. The surface of the dielectric layer is rough.
2. The roughness centers are distributed randomly.
3. The scattering is produced only on the surface of the incident
layer and not inside the AgI layer, since the AgI layer is more
granular than the Ag layer, hence roughness is greater.
4. The scattered energy is taken only in the positive direction;
scattered energy in the negative direction is assumed lost. The
scattering coefficient (S) is given by
90
S
 S  

0
90
 S  
(3.2.2)
  90
5. The scattering of the ray changes the ray’s angle of propagation.
The new angle is the average angle of the scattered energy.
6. The laser beam is decomposed to a minimum of 105 rays.
74
Using these assumptions one can calculate the attenuation and the beam shape
outside the waveguide as a function of the waveguide’s parameters (length,
inner diameter), and the coupling conditions (focal length of the coupling lens,
off center movement of the ray).
When the laser beam hits the waveguide’s inner wall it undergoes two
processes: reflection from a thin layer and scattering. According to the
assumptions mentioned above, the energy of each ray after one incident with
the inner wall is
I i  I i 0 R S  
(3.2.3)
where R() is the reflection coefficient, given by Fresnel law, and S() is the
scattering coefficient (equation 3.2.2).
Using the assumptions mentioned above one can write a computer program,
which can simulate the propagation of a laser beam through hollow
waveguides. The simulation uses ray tracing to find how different parameters
(length, inner diameter, bending etc.) affect the attenuation of different
waveguides and energy distribution outside them.
3.1.3 Ray propagation through straight waveguides
First let us look at a straight waveguide. In this section I will analyze the
influence of the waveguide’s geometrical parameters (length, inner diameter)
and the coupling conditions (focal length of the coupling lens and off center
coupling) on the waveguide’s attenuation and energy distribution of the laser
beam outside the waveguide.
3.1.3.1 The influence of the waveguide’s geometrical parameters
75
In order to analyze the influence of the waveguide’s geometrical parameters let
us first assume that the waveguide is perfectly smooth i.e. S()=1 [3.10]. Using
this assumption, the distance a ray passes between two refractions from the
waveguide’s wall, zi, can be calculated using figure 3.3 and is given by
zi  2  2r tan  i 
(3.3.1)
z
i
Figure 3.3
Ray propagation in a smooth waveguide
where r is the waveguide’s radius. The number of times a ray impinges on the
waveguide’s wall of length l, pi, is given by
 l
pi  int 
 zi



l

  int 
 2r tan   
i 


(3.3.2)
The total reflection coefficient of the ray is the multiplication of all the
reflections it passed on the way. It is given by
Rtotal  i   R i  i
p
(3.3.3)
The transmission, T, and the attenuation, A, are given by
Ti 
I i ,out
I i ,in
(3.3.4)
Ai  10 log( T )
Using equations 3.3.2 and 3.3.3 we get
 


1
 log R i 
Ai  10int 
  2r tan  i  

(3.3.5)
76
We can conclude from equation 3.3.5 that the attenuation is proportional to the
waveguide’s length.
Equation 3.3.5 also helps us to find the dependence of the attenuation on the
waveguide’s radius. The waveguide’s radius appears in the equation explicitly
and indirectly through the propagation angle  i . The angle of propagation is
determined by the coupling lens focal length and by the laser beam spot size.
The laser spot size at the entrance to the waveguide is given by
0 
1. 9 f
D
(3.3.6)
and the maximum angle at the entrance is
tan   
D
f
(3.3.7)
The angle of incidence and the angle at the entrance are related by   90   .
The maximum spot size is limited by the waveguide’s radius r
r
 max
2

0.95 f
D
(3.3.8)
Using equations 3.3.5 – 3.3.8 it is possible to find the dependence of the
attenuation on the waveguide’s radius




l
l
  int 
int 
r

 2r tan   
 2r
 0.95


  int  0.95 l   1
2
2

 4r  r


(3.3.9)
The reflection coefficient does not depend on the waveguide’s radius. Hence
the attenuation is inversely proportional to r2. This result is in accordance with
Harrington’s [3.2] experimental results.
77
More recently Harrington et al. [3.21] showed experimentally that the
attenuation varies as 1/r3. It is possible to show that for small angles of
incidence the ray model gives a 1/r3 dependence for the attenuation. Let us look
at equations 3.3.5 through 3.3.9. In the derivation of the attenuation
dependence I did not take into account the influence of the waveguide’s radius
on the reflection coefficient. According to Saito et al. [3.22] for small angles of
incidence
R( )  1  C (3.3.10)
and

1
(3.3.11)
r
using equations 3.3.10 and 3.3.11, expanding the log(R) into a power series and
taking the second term in the series (the first one equals to zero) gives us
log R   log 1  C   C 
1
r
(3.3.12)
equation 3 along with equation 3.3.9 in the dissertation gives us
A
1
r3
(3.3.13)
which agrees with the calculations of Marcatilli et al. and Miyagi et al.
theoretical calculations and with Harrington et al. experimental results.
The result mentioned above may become more complicated when we introduce
the surface roughness into the equations. Miyagi [3.11] tried to introduce the
effect of surface roughness by multiplying the reflection coefficient by
 1  4n cos   2 
S    exp   
 
 2

 

(3.3.14)
78
This equation assumes that the ray continues to propagate at the same angle
after it impinges the wall. The results obtained by Miyagi’s model were not in
agreement with the experimental results.
Using the theory developed in chapter 2 we can calculate the effect of the
surface roughness and insert it into the attenuation. In using this theory one has
to remember that each time the ray impinges at the waveguide’s wall it changes
its angle of propagation. Thus the next time it impinges at the wall, the
reflection coefficient and the scattering coefficient will be different. This
calculation could be done numerically by a ray-tracing program, which was
developed especially for this purpose.
The theoretical results of the ray model for different waveguide’s parameters
are shown in the experimental section along with the experimental results.
3.1.3.2 The influence of the coupling conditions on the waveguide’s
attenuation
3.1.3.2.1 The coupling lens
In the previous section we saw that the waveguide’s attenuation depends on the
angle of propagation of the laser beam. The angles of propagation of the rays,
which constitute the laser beam, are determined by the coupling lens (equations
3.3.7-3.3.8). The smaller the focal length the smaller the angles of incidence,
thus the increase in the reflection coefficient and scattering, which increase the
attenuation.
The focal length also determines the size of the laser’s spot at the waveguide’s
entrance (equation 3.3.7). The smaller the focal length the smaller the laser’s
79
spot size. Clearly we want a spot size smaller then the waveguide’s radius. This
would help us to couple the laser beam to the waveguide and it does not
damage the waveguide’s wall.
As one can see there is a conflict between the need for large and small focal
lengths. On one hand we need a large focal length, in order to get large angles
of incidence. On the other hand a small focal length causes a small laser’s spot
that helps us to couple the laser beam to the waveguide. We need to find an
optimum focal length. Harrington [3.12] found the desired ratio between the
laser’s spot size and the waveguide’s inner diameter should be 0.6.
3.1.3.2.2 Off center coupling
Till now we assumed that the laser beam enters the waveguide at its
geometrical center. Practically it is very difficult to couple the laser beam right
at the geometrical center of the waveguide. Let us now examine what happens
when the laser beam does not enter at the geometrical center of the waveguide.
This is known as off center coupling.
Let us look at a laser beam that enters the waveguide with inner diameter r, at
distance y from the center of the waveguide (see figure 3.4).
r

y
O
R
80
Figure 3.4
One of the assumptions of the ray model is that there are no skewed rays.
Hence when the laser beam does not enter at the center of the waveguide’s
cross section the rays will propagate in a waveguide with an effective radius
which is smaller or equal to the waveguide’s true radius. In order to find the
effective radius each ray sees, let us look at figure 3.5.

O’
90+
reff
y
O
Figure 3.5
The geometry of an off center coupling
Using the sine theorem
sin   
y cos 
r
(3.3.15)
and the effective radius is given by
reff  r cos 
(3.3.16)
Using equation 3.3.15 and 3.3.16 one can see that the effective radius depends
on the initial deviation of the ray from the center of the waveguide, the
waveguide’s radius and the angle  . Figure 3.6 shows the effective radius as a
function of  for different deviations and r=0.5mm.
81
Effective Radius Vs. Teta
Waveguide's IR= . mm
.
.
.
.
.
(mm
)
Effective Radius
.
.
Teta (degrees)
Y= . m m
Y= . m m
Y= . m m
Y= . m m
Figure 3.6
Effective radius vs. teta
As can be seen from figure 3.6 the longer the deviation the smaller the effective
radius (except for  =90o,270o ). As was explained earlier, as the radius of the
waveguide decreases the attenuation increases. In the case of off center
coupling, the effective radius is smaller than the waveguide’s radius and hence
the attenuation increases.
Figure 3.7 shows the effective radius vs.  for different waveguides radii. As
can be seen from the figure as the waveguide’s radius increases the effect of off
center coupling decreases since the effective radius is longer.
82
Effective Radius Vs. Teta
0.6000
0.4000
0.3000
0.1000
]
0.2000
[mm
Effective Radius
0.5000
0.0000
0
100
200
300
Teta [deg]
Reff (ID=0.5mm)
Reff (ID=0.7mm)
Reff (ID=1mm)
Figure 3.7
Effective radius vs. teta
Combining the above arguments with the assumptions of the ray model enable
us to determine the effect of off center coupling on the waveguide’s attenuation
and the energy distribution of the laser beam outside it. The theoretical results
are shown in the experimental section with the experimental ones.
3.1.4 Ray propagation through bent waveguide
The main purpose of optical fibers is to deliver the laser radiation through bent
trajectories. However bending the fiber causes the attenuation to increase due
to the change in the angle of propagation.
When dealing with bending one has to remember that there are two ways a ray
can propagate. The first is the normal way in which the ray impinges on the
inner and the outer walls of the waveguide (figure 3.8a). The second is known
as the whispering gallery mode (WGM), in which the ray impinges only on the
outer wall (figure 3.8b).
83
(a)
(b)
Figure 3.8
Ray propagation in a bent waveguide, (a) the normal way, (b) whispering gallery mode
(WGM)
When a ray impinges on a bent waveguide’s wall it changes its angle of
propagation. Since this change usually decreases the angle, the attenuation
increases. Figure 3.9 shows the geometry of a bent waveguide. From the
geometry one can calculate the change in the angle of incidence.
 
  R Rr 2sin
R
sin  ' 
out
(3.4.1)
0
where R is the bending radius, R0 – is the waveguide radius and rout is the place
the ray enters the curvature.
84
’
R+2R0

R+rout
O

Figure 3.9
The geometry for a bent waveguide
Figure 3.10 shows  ' as a function of  for various radii of curvature. As can
be seen from the figure the change in the angle of incidence increases as the
radius of the curvature decreases. That change causes the attenuation to
increase.
85
100
90
70
60
50
40
20
]
30
[deg
angle of incidence after bending
80
10
0
0
10
20
30
40
50
60
70
80
90
angle of incidence before bending [deg]
R=40cm
R=10cm
R=5cm
R=1cm
Figure 3.10
’ as a function of 

As we mentioned earlier there are two modes of propagation in a bent
waveguide: the normal mode and whispering gallery mode (WGM). Not each
ray can propagate in WGM. The condition for such mode of propagation is
derived using the following geometry.
100
86
x

R
R+d
d
Figure 3.11
The geometry of WGM propagation
The condition for WGM propagation is then given by [3.13]
0  x  2d
(3.4.2)
x  R  d 1  sin  
Combining the above arguments with the assumptions of the ray model enable
us to determine the effect of bending on the waveguide’s attenuation and the
energy distribution of the laser beam outside it.
87
3.2 Experimental results and theoretical calculations
3.2.1 Experimental setups
In order to characterize hollow waveguides under different conditions, I had to
use different experimental setups. This section describes the experimental
setups that were used.
3.2.1.1 Experimental method for measuring the attenuation of straight
waveguides
One of the most important parameters that characterize a hollow waveguide is
its attenuation. There are two methods to measure the waveguide’s attenuation:
a destructive one and a non-destructive one. The first method, which is also the
most common one, is known as the cutback method [3.14]. In this method we
measure the output power from a hollow waveguide with a known length.
Afterwards we cut a piece of the waveguide and measure the output power and
the waveguide’s length again. The waveguide’s attenuation is calculated as
follows: for a given waveguide the attenuation per unit length, α, is measured
in units of dB/m. The relation between the output power of a waveguide with
different lengths, l, is
Pl1   Pl 2 10
A
10
(3.5.1)
where A is the total attenuation of the waveguide.
Using equation 3.5.1 we get
A  10 log
   
P l1
P l2
and the attenuation per unit length is given by
(3.5.2)
88

A
l1  l2
(3.5.3)
The following experimental setup was used to measure the waveguide’s
attenuation. The laser beam was coupled to the hollow waveguide using a
focusing lens, and its power at the waveguide’s output was measured using a
detector. During the experiment the laser output power was kept constant.
Surgical knife was to cut the waveguides.
laser
focusing
lens
hollow waveguide
detector
Figure 3.12 – Experimental Setup For Measuring The Waveguide’s Attenuation
Although the cutback method is the most common one to measure the hollow
waveguides attenuation, it has one major drawback. During the measurement
process the waveguide is being destroyed. An alternative method that
overcomes this drawback is the non-destructive method [3.15]. In this method
the laser beam enters the waveguide at different points using another hollow
waveguide. Usually a fused silica waveguide with inner diameter of 0.7mm is
used to couple the laser beam into a waveguide with inner diameter of 1mm, a
0.5mm one is used to couple the beam into a 0.7mm waveguide etc. The non
destructive method is a repetitive process and the attenuation is calculated in
the same manner as in the first method. The two methods produce similar
results.
89
3.2.1.2 Experimental method for measuring the attenuation of bent
waveguides
Hollow waveguide are used to deliver infrared radiation under straight and bent
trajectories. Therefore it is very important to measure the waveguide’s
attenuation as a function of the bending radius. Knowing this dependence will
enable us to learn more about the hollow waveguides limitations and to design
more suitable waveguides for medical and industrial applications.
The following experimental setup (figure 3.13) was used to measure the
waveguide’s attenuation as a function of the radius of curvature. The laser
beam was coupled to the hollow waveguide using a focusing lens, the
waveguide was bent using a bending rail with different radii of curvature, and
the output power was measured using a detector. During the experiment the
laser output power was again kept constant. The result of the experiment is a
graph of the waveguide’s attenuation as a function of 1/R (R is the radius of
curvature).
waveguide
bent rail
laser
focusing
lens
detector
Figure 3.13 – Experimental Setup For Measuring A Bent Waveguide’s Attenuation
90
It is possible to measure the energy distribution outside the waveguide if we
use a beam profiler instead of a regular detector. In this case we need to work
with a pulsed laser or to use a chopper before the beam profiler.
3.2.2 The attenuation of straight hollow waveguides
3.2.2.1 Experimental results and theoretical calculations
The attenuation of the different types of straight waveguides (fused silica,
glass, Teflon and polyimide) was measured using one of the two methods that
were described in section 3.5.1. I used a Synrad CO2 laser, a 50mm lens a an
Ophir power meter or a Spiricon beam profiler. Figures 3.14 to 3.20 show the
attenuation as a function of the waveguide’s length for different type of
waveguides. Table 3.1 shows the measured attenuation, the calculated one, and
the correlation between the experimental results and the theoretical model
calculations for different type of hollow waveguides.
1.6
1.4
1.2
1
0.8
0.6
[dB
]
Attenuation
[dB]
0.4
0.2
0
60
70
80
90
100
110
Waveguide's Length [cm]
experimental
theoretical
Figure 3.14 – Theoretical And Experimental Attenuation of a Straight Waveguide:
Fused Silica Waveguide, ID=0.5mm, L=1m, Laser Wavelength=10.6m
91
[dB
]
Attenuation
2
1.8
1.6
[dB] 1.4
1.2
1
0.8
0.6
0.4
0.2
0
0.5
0.7
0.9
1.1
1.3
1.5
1.7
1.9
Waveguide's Length [m]
experimental
theoretical
Figure 3.15 – – Theoretical And Experimental Attenuation of a Straight Waveguide:
Fused Silica Waveguide, ID=0.7mm, L=1.9m, Laser Wavelength=10.6m
0.4
0.35
0.25
0.2
0.15
[dB
]
Attenuation
[dB] 0.3
0.1
0.05
0
50
55
60
65
70
75
80
85
90
Waveguide's length [cm]
experimental
theoretical
Figure 3.16 – – Theoretical And Experimental Attenuation of a Straight Waveguide:
Fused Silica Waveguide, ID=1mm, L=1m, Laser Wavelength=10.6m
92
[dB
]
Attenuation
2
1.8
1.6
[dB] 1.4
1.2
1
0.8
0.6
0.4
0.2
0
40
50
60
70
80
90
100
110
Waveguide's Length [cm]
experimental
theoretical
Figure 3.17 – Theoretical And Experimental Attenuation of a Straight Waveguide:
Teflon Waveguide, ID=1mm, L=1m, Laser Wavelength=10.6m
1.4
1.2
0.8
0.6
0.4
[dB
]
Attenuation
[dB] 1
0.2
0
50
60
70
80
90
100
110
Waveguides's Length [cm]
experimental
theoretical
Figure 3.18 Theoretical And Experimental Attenuation of a Straight Waveguide:
Teflon Waveguide, ID=2mm, L=1.1m, Laser Wavelength=10.6m
93
[dB
]
Attenuation
0.90
0.80
0.70
[dB] 0.60
0.50
0.40
0.30
0.20
0.10
0.00
0
10
20
30
40
50
Waveguide's Length [cm]
experimental
theoretical
Figure 3.19 – Theoretical And Experimental Attenuation of a Straight Waveguide:
Glass Waveguide, ID=1.6mm, L=0.9m, Laser Wavelength=10.6m
1.2
1
0.6
0.4
[dB
]
Attenuation
[dB] 0.8
0.2
0
50
60
70
80
90
Waveguide's Length [cm]
experimental
theoretical
Figure 3.20 – Theoretical And Experimental Attenuation of a Straight Waveguide:
Polyimide Waveguide, ID=1mm, L=1m, Laser Wavelength=10.6m
94
Waveguide’s
Inner
Experimental Theoretical Correlation Experimental
Type
Diameter Attenuation Attenuation
Method
[mm]
[dB/m]
[dB/m]
Fused Silica
0.5
0.78
0.81
0.7208
Cut back
0.7
0.54
0.67
0.9628
Cut back
1.0
0.43
0.35
0.9767
Nondestructive
Teflon
1.0
1.86
1.64
0.9312
Cut back
2.0
1.59
1.56
0.9348
Glass
1.6
1.43
1.46
0.9699
Nondestructive
Polyimide
1.0
2.30
2.46
0.9834
Nondestructive
Table 3.1 – Theoretical And Experimental Attenuation
For Different Types Of Hollow Waveguides
3.2.2.1 Experimental and theoretical beam shapes
Knowing the attenuation of each type of hollow waveguide is not enough. For
many applications, especially medical ones, it is also important to know what is
the energy distribution at the output of the waveguide.
Figures 3.21 and 3.22 show the experimental beam shape (a) and theoretical
one (b) of a fused silica hollow waveguide (ID=1mm) and Teflon hollow
waveguide (ID=1mm). One can see the similarity between the experimental
beam shapes and the theoretical ones.
95
(a)
(b)
Figure 3.21 – Experimental (a) And Theoretical (b) Beam Shapes:
Straight Fused Silica Waveguide, ID=1mm, L=1m
(a)
(b)
Figure 3.22 – Experimental (a) And Theoretical (b) Beam Shapes:
Straight Teflon Waveguide, ID=1mm, L=1m
3.2.3 The attenuation of bent hollow waveguides
3.2.3.1 Experimental results and theoretical calculations
The attenuation of the different types of waveguides (fused silica, Teflon and
polyimide) as a function of the bending radius was measured using the
experimental method that was described in section 3.2.1. I used the same
equipment as in the measurement of straight waveguides. Figures 3.23 to 3.28
show the attenuation as a function of the radius of curvature for different type
of waveguides. Table 3.2 shows the dependence of the attenuation on the
bending radius for the experimental results, the theoretical calculations ones,
96
and the correlation between the experimental results and the theoretical
method.
The dependence of the attenuation on the bending radius is given by
A  Rx
(3.3.1)
where x is given in table 3.2.
3.00
2.50
2.00
1.50
1.00
[dB
]
Attenuation
[dB]
0.50
0.00
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
1/R [1/cm]
Experimental
Theoretical
Figure 3.23 – Theoretical And Experimental Attenuation of Bent Waveguides:
Fused Silica Waveguide, ID=0.5mm L=1m, Laser Wavelength 10.6m
3.5
3
2
1.5
1
[dB
]
Attenuation
[dB] 2.5
0.5
0
0
0.02
0.04
0.06
0.08
0.1
1/R [1/cm]
experimental
theoretical
Figure 3.24 – Theoretical And Experimental Attenuation of Bent Waveguides:
Fused Silica Waveguide, ID=0.7mm L=1m, Laser Wavelength 10.6m
97
2.5
2
1.5
1
[dB
]
Attenuation
[dB]
0.5
0
0
0.01
0.02
0.03
0.04
0.05
0.06
1/R [1/cm]
Experimental
Theoretical
Figure 3.25 Theoretical And Experimental Attenuation of Bent Waveguides:
Fused Silica Waveguide, ID=1mm L=1m, Laser Wavelength 10.6m
3.5
3
2
1.5
1
[dB
]
Attenuation
[dB] 2.5
0.5
0
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
1/R [1/cm]
Experimental
Theoretical
Figure 3.26 – Theoretical And Experimental Attenuation of Bent Waveguides:
Teflon Waveguide, ID=1mm L=1m, Laser Wavelength 10.6m
98
3
2.5
2
1.5
1
[dB
]
Attenuation
[dB]
0.5
0
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
1/R [1/cm]
Experimental
Theoretical
Figure 3.27 Theoretical And Experimental Attenuation of Bent Waveguides:
Teflon Waveguide, ID=2mm L=1m, Laser Wavelength 10.6m
2.5
2
1.5
1
[dB
]
Attenuation
[dB]
0.5
0
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
1/R [1/cm]
Experimental
Theoretical
Figure 3.28 – Theoretical And Experimental Attenuation of Bent Waveguides:
Polyimide Waveguide, ID=1mm L=1m, Laser Wavelength 10.6m
99
Waveguide’s
Type
Fused Silica
Teflon
Polyimide
Inner
Diameter
[mm]
0.5
0.7
1.0
1.0
2.0
1.0
Experimental Theoretical Correlation
Dependence Dependence
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
-1
0.9475
0.9891
0.7441
0.9549
0.9561
0.9742
Table 3.2 – Theoretical And Experimental Dependence Of The Attenuation On The
Bending Radius For Different Types Of Hollow Waveguides
3.2.3.2 Energy distribution at the end of the waveguide [3.16]
Knowing the attenuation as a function of length and radius of curvature is not
sufficient. For medical purposes, knowing the beam shape of the laser beam
outside the hollow waveguide is very important. Simple Gaussian beam shape
will enable us to make clean cuts, while more complex beam shapes, will cause
damage to the surrounding area.
The new ray model also enables us to calculate the energy distribution outside
the hollow waveguide. In order to compare between the two beam shapes let us
define the parameter  which is the ratio of the M2 factor of the experimental
beam shape to the M2 [3.17] of the calculated beam shape. The beam quality
M2 is the ratio of the laser beam's multimode diameter, Dm, to the diffractionlimited beam diameter, d0 [3.25].
D 
M   m 
 d0 
M x2,exp
x  2
M x ,cal
2
2
y 
M y2,exp
M y2,cal
(3.3.2)
100
A good correlation is achieved when  is close to one. Figures 3.29 to 3.33
show the experimental beam shapes (a) and the theoretical ones (b) for a bent
fused silica hollow waveguide (ID=1mm), for radii of curvatures of 20cm,
28cm, 36cm, 40cm, and 44cm respectively. The beam shapes were obtained
using a pyroelectric camera (made by Spiricon). As can be seen from the
images, the experimental beam shapes and theoretical ones are quite similar.
.
(a)
(b)
Figure 3.29 – Experimental (a) And Theoretical (b) Beam Shapes of Bent Waveguides:
Fused Silica Waveguide, ID=1mm, R=20cm
(a)
(b)
Figure 3.30 – Experimental (a) And Theoretical (b) Beam Shapes of Bent Waveguides:
Fused Silica Waveguide, ID=1mm, R=28cm
101
(a)
(b)
Figure 3.31 – Experimental (a) And Theoretical (b) Beam Shapes of Bent Waveguides:
Fused Silica Waveguide, ID=1mm, R=36cm
(a)
(b)
Figure 3.32 – Experimental (a) And Theoretical (b) Beam Shapes of Bent Waveguides:
Fused Silica Waveguide, ID=1mm, R=40cm
(a)
(b)
Figure 3.33 – Experimental (a) And Theoretical (b) Beam Shapes of Bent Waveguides:
Fused Silica Waveguide, ID=1mm, R=44cm
In order to get a quantitative result that describes the similarity of the beam
shapes we calculated x and y. Table 3.3 summarize x and y for various radii
of curvatures.
102
Radius of Curvature
(cm)
20
28
36
40
44
x
y
0.81
1.15
1.25
1.08
0.87
0.9
1.2
1.02
1.05
1.03
Table 3.3
3.2.4 The dependence of the attenuation on the waveguide’s radius
3.2.4.1 Experimental results and theoretical calculations
I measured the attenuation of fused silica hollow waveguides as a function of
the waveguide’s radius. Figure 3.34 shows the attenuation as a function of the
waveguide’s radius. The correlation between the experimental results and the
theoretical ones is practically one (0.9999).
It is possible to find the dependence of the attenuation on the waveguide’s
radius. The experimental results show that the attenuation is proportional to
r-n where n=2.12±0.07 and the theoretical results show the attenuation is
proportional to r-n where n=2.15±0.05.
1.6
1.4
0.8
0.6
[dB
]
Attenuation
[dB] 1.2
1
0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Waveguide's radius [mm]
experimental
theoretical
Figure 3.34 – Experimental And Theoretical Attenuation, Straight Fused Silica
Waveguides
103
3.2.4.2 Energy distribution at the end of the waveguide [3.20]
As was explained in the previous section it is important to measure the beam
shape of the laser beam outside the hollow waveguide. Figures 3.34 to 3.36
show the experimental beam shapes (a) and the theoretical ones (b) for fused
silica hollow waveguides with different radius. Again I used the test in order
to get a quantitative result (table 3.4).
(a)
(b)
Figure 3.24 – Experimental (a) And Theoretical (b) Beam Shapes:
Fused Silica Waveguide, ID=0.5mm, L=1m
(a)
(b)
Figure 3.25 – Experimental (a) And Theoretical (b) Beam Shapes:
Fused Silica Waveguide, ID=0.7mm, L=1m
104
(a)
(b)
Figure 3.26 – Experimental (a) And Theoretical (b) Beam Shapes:
Fused Silica Waveguide, ID=1mm, L=1m
Waveguide’s Radius
(mm)
0.25
0.35
0.5
x
y
1.02
1.09
0.85
0.97
0.99
0.77
Table 3.4
3.2.5 The dependence of the attenuation on the coupling conditions
3.2.5.1 The influence of the focal length of the coupling lens
The laser beam has to be coupled to the waveguide. The most common way to
couple it is by using a simple lens. As I showed in section 3.1 the focal length
of the coupling lens determines the laser spot size at the entrance to the
waveguide and the laser beam divergence.
When trying to couple a laser beam, one has to consider two elements: the laser
spot size and the laser beam divergence. Smaller spot sizes enable us to couple
the laser beam easily and prevent wall damage. However it also mean a large
divergence angle and thus a larger attenuation.
Figures 3.37 through 3.40 show the attenuation of a fused silica hollow
waveguides (figures 3.37-3.39) and a glass hollow waveguide (figure 3.40).
105
Table 3.5 shows the correlation between the theoretical results and the
experimental ones.
6.00
5.00
4.00
3.00
2.00
[dB
]
Attenuation
[dB]
1.00
0.00
0
20
40
60
80
100
Focal Length [mm]
experimental
theoretical
Figure 3.37 – Theoretical And Experimental Attenuation for Different Focal Lengths:
Fused Silica Waveguide, ID=0.5mm L=1m, Laser Wavelength 10.6m
6.00
5.00
3.00
2.00
[dB
]
Attenuation
[dB] 4.00
1.00
0.00
0
50
100
150
200
250
Focal Length [mm]
experimental
theoretical
Figure 3.38 – Theoretical And Experimental Attenuation for Different Focal Lengths:
Fused Silica Waveguide, ID=0.7mm L=1m, Laser Wavelength 10.6m
106
[dB
]
Attenuation
5.00
4.50
4.00
[dB] 3.50
3.00
2.50
2.00
1.50
1.00
0.50
0.00
0
50
100
150
200
250
Focal Length [mm]
experimental
theoretical
Figure 3.39 – Theoretical And Experimental Attenuation for Different Focal Lengths:
Fused Silica Waveguide, ID=1mm L=1m, Laser Wavelength 10.6m
6.00
5.00
3.00
2.00
[dB
]
Attenuation
[dB]4.00
1.00
0.00
0
50
100
150
200
250
Focal Length [mm]
experimental
theoretical
Figure 3.40 – Theoretical And Experimental Attenuation for Different Focal Lengths:
Glass Waveguide, ID=1.6mm L=1m, Laser Wavelength 10.6m
Waveguide’s
Type
Fused Silica
Glass
Inner
Diameter
[mm]
0.5
0.7
1.0
1.6
Table 3.5
Correlation
0.9999
0.9891
0.9715
0.9918
107
3.2.5.2 Off center coupling
Coupling the laser beam into the hollow waveguide is not easy. If the laser
beam does not enter the waveguide at the center of its cross section it traverses
the waveguide near the waveguide’s wall. This increases the laser beam
attenuation due to increased absorption in the waveguide’s wall.
3.2.5.2.1 Experimental results and theoretical calculations
The coupling position of the laser beam into the waveguide was changed by
changing the position of the hollow waveguide by using a micrometer XYZ
positioner. Figures 3.41 to 3.43 show the attenuation as a function of the
distance from the waveguide’s cross section center, for different waveguides.
As can be seen from the graph and from table 3.6, the experimental results and
theoretical ones are very close.
4.50
4.00
3.50
3.00
2.50
2.00
1.50
1.00
0.50
0.00
(dB
)
Attenuation
[dB]
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
Distance From The Waveguide's Center (mm)
experimental
theoretical
Figure 3.41 – Theoretical And Experimental Attenuation for Off Center Coupling:
Fused Silica Waveguide, ID=0.7mm L=1m, Laser Wavelength 10.6m
108
8.00
7.00
6.00
[dB]
Attenuation
5.00
4.00
(dB
)
3.00
2.00
1.00
0.00
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Distance From The Waveguide's Center (mm)
experimental
theoretical
Figure 3.42 – Theoretical And Experimental Attenuation for Off Center Coupling:
Fused Silica Waveguide, ID=1mm L=1m, Laser Wavelength 10.6m
6.00
5.00
4.00
Attenuation
[dB]
3.00
(dB
)
2.00
1.00
0.00
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Distance From The Waveguide's Center (mm)
experimental
theoretical
Figure 3.43 – Theoretical And Experimental Attenuation for Off Center Coupling:
Glass Waveguide, ID=1.6mm L=1m, Laser Wavelength 10.6m
109
Waveguide’s
Type
Fused Silica
Glass
Inner
Diameter
[mm]
0.7
1.0
1.6
Correlation
0.9897
0.9693
0.9834
Table 3.6
3.2.5.2.2 Energy distribution at the end of the waveguide [3.20]
It is important to measure the beam shape of the laser beam outside the hollow
waveguide. Figure 3.44 to 3.46 show the experimental beam shapes (a) and the
theoretical ones (b) for different deviations from the center of a fused silica
hollow waveguide.
(a)
(b)
Figure 3.44 – Experimental (a) And Theoretical (b) Beam Shapes for Off Center
Coupling: Fused Silica Waveguide, ID=1mm, L=1m, Beam deviation 0.15mm from the
center
(a)
(b)
Figure 3.45 – Experimental (a) And Theoretical (b) Beam Shapes for Off Center
Coupling: Fused Silica Waveguide, ID=1mm, L=1m, Beam deviation 0.3mm from the
center
110
(a)
(b)
Figure 3.46 – Experimental (a) And Theoretical (b) Beam Shapes for Off Center
Coupling: Fused Silica Waveguide, ID=1mm, L=1m, Beam deviation 0.4mm from the
center
It can be seen that the theoretical beam shapes are similar to the experimental
ones.
3.3 Discussion
3.3.1 Straight hollow waveguides
As can be seen from the graphs (figures 3.14 to 3.20) and from table 3.1, the
theoretical calculations are similar to the experimental results. Moreover the
correlation between the experimental results and the theoretical calculations are
more than 0.9 in most cases.
It is noticed that there is a difference between the waveguides attenuation. The
difference between waveguides within one type is due to the differences in the
waveguide’s inner diameter and the surface roughness (the surface roughness
varies for different tubes). The larger the waveguide’s inner diameter the
smaller its attenuation. This was shown theoretically in section 3.1.
The difference in attenuation among waveguides with the same inner diameter
that are made of different types of tubes is due to the differences in the surface
scattering. In chapter 2 I showed the measurements of the surface roughness for
111
different types of waveguides. The larger the surface roughness the larger the
attenuation. Since Teflon waveguides have larger surface roughness their
attenuation is larger as well.
The surface roughness influences the beam shape of the transmitted laser beam
as well. We can see from Figures 3.21 and 3.22 that the beam shape of the
Teflon waveguide is more complex and it contains higher modes than the beam
shape of the fused silica waveguide. The difference in the beam shapes is due
to the scattering from the surface. The larger the surface roughness the larger
the scattering and as a consequence there is more mode coupling between
lower modes and higher modes. This coupling increases the attenuation and
gives rise to a more complex beam shapes. The scattering from the rough
surface changes the angle of incidence of the rays and thus mode coupling
occur.
Although our experimental results fits the theoretical calculations of the ray
model they are different from the ones obtained by Harrington et al. [3.21]. Our
experimental results differs from the ones achieved by Harrington et al. since
we did not used optimal coupling conditions and took into account the surface
roughness.
3.3.2 Bent hollow waveguides
As can be seen from figures 3.23 to 3.28 and table 3.2 the results of the
theoretical calculations are similar to the experimental results. Moreover the
correlation between the experimental results and the theoretical calculations is
more than 0.95 for most cases.
112
As expected the attenuation increases with the decrease in the bending radius.
This is due to the change in the angle of incidence caused by bending the
waveguide. Bending the waveguide causes the angle of incidence to decrease,
thus increasing the attenuation. The change in the angle of incidence also
causes mode coupling. Usually lower modes are coupled to higher ones. It is
worthwhile to note that for most cases the attenuation increases when the
bending radius decreases. However for special bending radius the attenuation
decreases due to the presence of whispering gallery modes.
As can be seen from table 3.2 the attenuation changes as 1/R. This result is
with accordance to other works that has been done in the past [3.18, 3.19].
These works also show that the attenuation dependence on the bending radius
is -1.
The correlation between the experimental beam shapes and the theoretical
beam shapes is very good. The shapes differ by no more than 20%. The
similarity between the beam shapes can also be seen through figures 3.29 to
3.33. Although the original shape of the laser beam at the waveguide’s entrance
is Gaussian, the beam shape the beam shape at the distal end is more complex.
This is due to mode coupling caused by bending.
3.3.3 The dependence on the waveguide’s inner diameter
In the theoretical section (section 3.1) I showed that the waveguide’s
attenuation is proportional to r-n where n=2, where r is the waveguide’s radius.
In this section I showed that n=2.12 and 2.15 for the experimental and
theoretical attenuation respectively. This result is very close to the theoretical
113
prediction. The deviation is due to the influence of scattering, which is taken
into account in the simulations, but not in the analytical derivation of the
attenuation dependence (equation 3.3.9).
The experimental beam shapes and the theoretical ones are similar, as can be
seen from figures 3.34 to 3.36. The deviation of the  factor is less than 25%,
which shows a good similarity between the beam shapes.
As can be seen from figures 3.34 trough 3.36 the beam shapes of different
waveguides differ from one another. The beam shapes of the waveguides that
have larger inner diameter have more secondary modes. This difference is due
to the fact that higher modes have smaller angles of incidence with the
waveguides wall and thus are highly attenuated. Since the attenuation is a
function of the waveguides radius, higher modes will be more attenuated in
smaller core waveguides than in larger core ones. This is clearly shown in the
beam shapes figures.
3.3.4 The influence of coupling conditions
The coupling conditions have a great influence on the attenuation and beam
shapes of hollow waveguides. The coupling lens determines the spot size of the
laser beam and its numerical aperture (NA). The smaller the focal length of the
coupling lens the larger the NA and thus the attenuation increases. This is
clearly shown in figures 3.37 to 3.40. One has to use the appropriate lens when
coupling the laser beam to the hollow waveguides. It is well known that the
desired ratio between the spot size and the waveguide’s inner diameter is 0.6.
114
Coupling the laser beam off center with regards to the waveguide’s crosssection causes the attenuation to increase and the beam shape to be spoiled. It is
obviously seen from figures 3.41 to 3.46 that when the beam is further out of
the center, the beam is “spoiled”, symmetry is gone, the main peak is reduced
and higher modes appear. We can also see that our model predicts well the
experimental results. The loss of symmetry is due to the large attenuation of the
lower modes, which, propagate now near the waveguide’s wall. When the
lower modes are highly attenuated, the higher modes are more emphasized.
3.4 Conclusion
The comparison between the experimental result and the theoretical calculation
that was made in this chapter shows that the theoretical model gives the correct
results. The model predicts accurately the waveguide’s attenuation and the
laser beam shape outside the hollow waveguide for complicated situations such
as bending and off center coupling. This is useful for designing new kinds of
hollow waveguides and improving the current ones.
The model is also useful in finding new applications for hollow waveguides.
As was mentioned earlier, infrared radiation may be used for medical
applications. The model may predict for example how much the laser beam
will be attenuated when it is transmitted through endoscopes and how the beam
shape would look like at its end.
3.5 Summary of chapter 3
In this chapter I described the improved ray model. Although ray models have
been used in the past, they have not taken into account the surface roughness in
115
such a manner that they could predict correctly the performance of hollow
waveguides. Moreover, the improved ray model has more features that enables
analyze complex situations correctly. Whispering gallery modes enable to
analyze bent hollow waveguides correctly and off center coupling may predict
the complexity in the beam shape.
The second section of the chapter is a characterization of hollow waveguides,
their performance and the parameters that influence their attenuation and the
shape of the output beam. The experimental results are also used to examine
the validity of the improved ray model.
As can be seen from the above graphs and beam shapes, the improved ray
model predicts the performance of hollow waveguides very well. The
theoretical calculations and the experimental results enable us to better
understand the influence of different parameters on the performance of hollow
waveguides and to design hollow waveguides that suit a desired application.
116
3.6 References for chapter 3
3.1 E. J. Marcatili and R. A. Schmeltzer, “Hollow Metallic and Dielectric
Waveguides for Long Distance Optical Transmission and Lasers”, The Bell
System Technical Journal, July 1964, p. 1783.
3.2 J. A. Harrington, “Laser Poewr Delivery in Infrared Fiber Optics”, SPIE
1649, Optical Fibers in Medicne 1992.
3.3 M. Miyagi and S. Kawakami, “Design Theory of Dielectric-Coated
Circular Metalic Wavwguides for Infrared Transmission”, Journal of
Lightwave Technology, 1984, p.116.
3.4 miyagi
3.5 D. Mendelovic, E. Goldenberg, S. Rushin, J. Dror and N. Croitoru, “Ray
Model for Transmission of Metalic-Dielectric Hollow Bent Cylindrical
Waveguides”, Applied Optics, 1989, v. 28, p. 708.
3.6 O. Morhhaim, D. Mendelovic, I. Gannot, J. Dror and N. Croitoru, “Ray
Model for Transmission of Infrared Radiation Through Multibent Cylindrical
Waveguides”, Optical Engineering, 1991, v. 30, p. 1886.
3.7 M. Miyagi, Y. Matsuura, M. Saito and A. Hongo, “Spectral attenuation of
incoherent IR light in circular hollow waveguides”, Applied Optics v. 27, n. 20,
p. 4169-4170.
3.8 K. Takatani, Y. Matsuura and M. Miyagi, “Theoretical and experimental
investigations of loss behavior in the infrared quartz hollow waveguides with
rough inner surfaces”, Applied Optics v. 34, n. 21, p. 4352-4357.
117
3.9 A. Inberg , M. Ben-David, , M. Oksman, A. Katzir and N. Croitoru,
“Theoretical and Experimental Studies of Infrared Radiation Propagation in
Hollow Waveguides”, Optical Engineering, 2000, 39, 5, 1384-1390.
3.10 M. Ben-David, A. Inberg, A. Katzir and N. Croitoru, “Hollow
Waveguides for Infrared Radiation”, Chemistry and Chemical Engineering,
Journal of the Chemical Engineers and Chemist, The Association of Engineers
and Architects in Israel, No. 34-35, 12-18, 1999.
3.11 M. Miyagi, Y. Matsuura, M. Saito and A. Hongo, “Loss characteristics of
circular hollow waveguides for incoherent infrared light”, J. Opt. Soc. Am. A,
v. 6, n. 3, p. 423-427
3.12 R. L. Kozodoy, A. T. Pagkalinawan, J. A. Harrington, “Small bore hollow
waveguides for delivery of 3m laser radiation”, Applied Optics v. 35, n. 7, p.
1077-1082.
3.13 M. Ben-David, I. Gannot, A. Inberg and ,N. Croitoru, “Bending Effect
on IR Hollow Waveguides Transmission”, EBIOS 2000, 4-8 July, Amsterdam
Netherland.
3.14 M.Bass “Handbook of Optics”, second edition, McGraw Hill 1995.
3.15 R. Dahan, J. Dror, A. Inberg and N. Croitotu, “Nondestructive Method for
Attenuation Measurement in Optical Hollow Waveguides”, Optics Letters,
1995, v. 20.
3.16 M. Ben-David, I. Gannot, A. Inberg and ,N. Croitoru, “Bending Effect
on IR Hollow Waveguides Transmission”, EBIOS 2000, 4-8 July, Amsterdam
Netherland
118
3.17 O. Svelto, “Principles of Lasers”,1998, 4th ed.:481-482, Plenum Press
3.18 M. Miyagi, Y. Matsuura, M. Saito and A. Hongo, “Loss characteristics of
circular hollow waveguides for incoherent infrared light”, J. Opt. Soc. Am. A,
v. 6, n. 3, p. 423-427
3.19 R. L. Kozodoy, A. T. Pagkalinawan, J. A. Harrington, “Small bore hollow
waveguides for delivery of 3m laser radiation”, Applied Optics v. 35, n. 7, p.
1077-1082.
3.20 I. Gannot, M. Ben-David, A. Inberg, N. Croitoru, and R. Waynant, “Beam
Shape analysis of waveguide delivered IR lasers”, Optical Engineering,
Accepted for publication 2001.
3.21 Y. Matsuura, T. Abel, and J. A. Harrington, "Optical properties of smallbore hollow glass waveguides", Applied Optics, 34, 6842-6847, (1995).
3.22 Saito, Sato, and Miyagi, JOSA A vol. 10, pg. 277-282 (1993).
3.23 Abe et al., IEEE Transactions on Microwave Theory and Techniques, v.
39 n. 2, February 1991.
3.24 Matsuura et al., Applied Optics, v. 31, n. 30 October (1992).
3.25 O. Svelto, “Principles of Lasers”.
119
Chapter Four - Pulse Dispersion In Hollow Waveguides
List of Symbols For Chapter 4
c .............. Speed of light
fast .......... The travel time for the fastest mode
slow ......... The travel time for the slowest mode
............ Pulse dispersion
Angle of propagation
L ............. waveguide’s length
The previous chapter dealt with the influence of hollow waveguides on the
spatial distribution of the laser beam. However the laser beam is influenced by
the hollow waveguide in the time domain as well. For medical purposes such as
laser tissue interaction and diagnostics there is a need to understand the
delivery of short laser pulses through hollow optical waveguides.
In this chapter, I describe a theoretical model for calculating the pulse
dispersion within the hollow optical waveguide and the experimental results of
pulse dispersion, which correspond to these calculations [4.5]. Although pulse
dispersion has been discussed in many papers, such as in [4.6], it is always with
regard to a single mode of propagation and the calculations are done using
mode theory. However in many cases the laser beam is a multi-mode which
makes the calculation very difficult, and previous calculations did not take into
account difficult situations such as surface roughness, bending, off center
coupling etc.
120
The ray model that I have developed enables to calculate the pulse dispersion
for many complicated situations. Moreover it is very straightforward, simple to
code into a computer and it does not require a long running time. These
advantages along with good agreement between the theoretical calculations and
the experimental results (as will be shown), make the ray model a very useful
tool in evaluating the pulse dispersion.
4.1 Theory
Different light components (modes or wavelengths) travel through the
waveguide with different velocities or through different optical paths.
Therefore, laser pulses that travel along the fiber may become longer by a
certain amount of time that we will call . This phenomenon is well known
and is called pulse dispersion.
In a multi-mode fiber, the dominant dispersion mechanism is the modal or
inter-modal dispersion (i.e., different modes travel in different optical paths
because of the different incidence and reflectance angles at and from the
reflecting/refracting layers). In a single-mode fiber, the dominant dispersion
mechanism is the chromatic dispersion (i.e., different wavelengths travel with
different velocities).
4.1.1 Pulse dispersion in straight and smooth multi-mode fibers
Different modes travel at different optical paths through the fiber and the result
is a modal dispersion that can be evaluated using figure 4.1 [4.1].
121
2
0
Figure 4.1
The shortest path traveling mode in figure 1 corresponds to ray 0, which travels
directly to the destination along the fiber axis. The travel time for that mode is
 fast 
Ln1
c
(4.1.1)
where L is the fiber length, and n is the refractive index of the core.
1
The longest path of a traveling mode corresponds to ray 2, which travels with
the maximum angle with respect to the fiber axis. (This is limited by NA of a
fiber). In the hollow waveguide case there is no real NA but a limiting input
angle, which is 25o full angle or “NA” of 0.22). The travel time for this mode is
 slow 
Ln1
c cos
(4.1.2)
The modal dispersion is obtained by subtracting the two equations [4.8]
 
Ln1  1

 1

c  cos

(4.1.3)
As can be seen from equation 4.1.3 the pulse dispersion depends on the fiber
material properties, its length and the incidence angle of the input ray.
4.1.2 Pulse dispersion in real hollow waveguides
The calculation mentioned above is useful only for straight and smooth
waveguides. However in a practical real life use, hollow waveguides are being
bent and they have a certain amount of surface roughness. Bending and
roughness cause a change of the incidence angle that leads to increasing pulse
122
dispersion. In order to calculate the real pulse dispersion under these
conditions, we can use the ray model, which we described in chapter 3 [4.2].
Our ray model calculates the new angle of incidence after each time a ray
impinges on the waveguide’s wall taking into account the roughness of the
surface, which causes scattering of the impinging ray. Knowing the correct
angle of incidence enables us to calculate the ray trajectory along the
waveguide and from it, the time it takes the ray to exit at the distal tip. The
pulse dispersion is therefore given by:
   slow   fast
(4.1.4)
4.2 Experimental results and discussion
4.2.1 Experimental setup
There are several methods to measure the pulse width of a laser beam. One of
them is to use an autocorelator setup [4.4]. We measured the pulse dispersion
directly, i.e. we measured the pulse width at the waveguide entrance and the
pulse width at the output of the waveguide. The pulse dispersion is the
difference between them. We chose this method since the theoretical
calculations have shown that the pulse dispersion would be in the order of
nano-seconds, i.e. the magnitude of the pulse itself.
Figure 4.2 shows the experimental setup that was used to measure the pulse
width of the laser beam after propagating through the waveguide. We used a Qswitched Er-YAG laser (MSQ, Israel); the laser wavelength was 2.m and
pulse width 70nm. The waveguide was manufactured by our group. We used a
3m long fused silica waveguide with inner diameter of 0.7mm. The waveguide
123
was bent at a constant radius (8cm) at different bending angles i.e. several turns
for angles greater than 360o. For each bending we measured the pulse width
using a fast IR detector (VIGO by Boston Electronics, MA). The received
signal was captured by HP 500 MHz digital oscilloscope and transferred to a
Microsoft EXCEL program on the PC through HP BenchLink XL Software.
There are several ways to couple the laser beam to the waveguide, using
conventional lens or other coupling devices. We coupled the laser beam to the
waveguide using a glass funnel [4.3]. The funnel has two functions. The first is
to smoothen the laser beam energy. The second is to decrease the laser’s N.A.,
because the funnel has a low N.A.. The funnel we used has NA of 0.033, which
corresponds, to a very small input angle of 3.8o full angle.
Oscilloscope
Computer
Q switched
Funnel
waveguide Fast IR detector
Er-YAG
Figure 4.2 – Experimental Setup
4.2.2 Experimental results and discussion
We measured the pulse dispersion at various bending angles. Figure 4.3 (a and
b) shows the experimental results (points), their best fit curve (dotted line) and
the theoretical calculations for a waveguide with (solid line) and without
roughness (squares). As can be seen from figure 4.3 there is a good correlation
124
between the experimental results and the theoretical ones (the calculated
correlation between the curve fit and the theoretical calculations is 0.92).
Moreover it can be seen that the major contribution to the pulse dispersion is
due to the surface roughness.
The figure also shows that the pulse dispersion increases as the angle of
bending increases. This is due to mode coupling between lower modes of
propagation and higher ones. The mode coupling is caused by two
mechanisms. The first is scattering from a rough surface. Each time a ray
impinges on the waveguide’s wall it scatters, which means that its angle of
reflection is different from the angle of incidence. Since each angle
corresponds to a different mode of propagation, mode coupling occurs.
The second mechanism is bending. Here changing the radius of the curvature
changes the angle of incidence, and therefore the mode of propagation. The two
mechanisms cause lower modes to couple to higher ones. The coupling cause a
longer optical path and thus to a significant pulse dispersion even in a straight
waveguide.
Pulse Dispersion [nsec]
125
50
40
30
20
10
0
0
200
400
600
800
Bending [degrees]
experimental
best fit
theoretical - with roughness
theoretical - without roughness
(a)
Pulse dispersion [nsec]
0.25
0.2
0.15
0.1
0.05
0
0
200
400
600
800
Bending [degrees]
theoretical - without roughness
(b)
Figure 4.3 – Experimental and Theoretical Results (a – all results, b – without
roughness)
The measurements and calculations correspond to the optical funnel used.
However, calculation can be made for each input angle, which is generated by
a focusing lens. Usually, we use lenses with focal lengths between 50 and 150
mm. We repeated those calculations to even a broader range of 25 to 200 mm.
126
We obtained pulse broadening as a function of focal length. This can be seen in
the graph in Fig 4.4. As can be seen from the figure a smaller focal length leads
to a larger pulse dispersion. The theoretical pulse dispersion is about 10% of
the original for a 25mm focal length, and almost negligible compared to the
original pulse at focal lengths higher than 100mm. These theoretical
calculations are in agreement with the experimental results, which were
obtained by Prastisto et. Al [4.4]. They measured the pulse dispersion of micro
pulses from FEL laser. They found an increase of about 50% in the pulse width
for a small focal length and a negligible at longer focal lengths.
Figure 4.4
Although the relative pulse dispersion of Paratisto et al., Matsuura et al. [4.7]
and ours are similar (about 50%). The absolute value of the pulse dispersion in
each experiment is different. The difference among the different pulse
dispersion measurements (Partisto et al. Matsuura et. al and ours) is due to
different experimental conditions, laser type, pulse width, mode distribution,
127
fiber length, bending radius and coupling conditions. While Matsuura et al. use
almost a single mode laser beam and grazing angle coupling conditions
(f=300mm) our beam structure is more complex and is composed of a lot of
modes. This leads to larger pulse dispersion.
I used the ray model to calculate the pulse dispersion of Matsuura et al. laser
pulse using the same setup they used (f=300mm, l=1m, ID=1mm, pulse width
196fs and =775nm) for a waveguide without roughness and with roughness.
The results are summarized in table 4.1.
Pulse dispersion (fs)
Ben-David et al. ray model
MatSuura et al.
without roughness
with roughness
Straight waveguide
21
25
17
Bending waveguide
46
127
Max: 178
(R=50cm)
Average: 133
Table 4.1
As can be seen from the table the results are similar.
4.3 Summary
I studied the temporal behavior of a laser pulse after it was transmitted through
an IR hollow core optical waveguide. Using the ray model enables to analyze
the pulse dispersion in hollow waveguides. The model takes into account the
waveguide’s rough surface and enables to estimate the pulse dispersion in
complicated situations such as bending and multi-mode laser beams. We
measured experimentally pulse broadening and compared it to our theoretical
calculations. The experimental results were in agreement with the calculations.
128
These results help us to understand the laser pulse behavior in hollow
waveguides. It will help us to characterize laser pulses to transmitted tissue and
may assist in designing surgical or diagnostic procedures.
129
4.4 References for chapter four
4.1 L. Kazovsky, S. Benedetto, A. Willner, “Optical Fiber Communication
Systems”, Artech House, 1996.
4.2 I. Gannot, M. Ben-David, A. Inberg, N. Croitoru, and R. Waynant, “Beam
Shape analysis of waveguide delivered IR lasers”, Optical Engineering,
January 2002.
4.3 I. Ilev and R. Waynant, "Grazing-incidence-based hollow taper for infrared
laser-to-fiber coupling", Applied Physics Letters 74, 2921 (1999).
4.4 H.S. Pratisto, S. R. Uhlhorn and E. Duco Jansen, “Beam Delivery of the
Vanderbilt Free Electron Laaser with Hollow Wave Guides: Effect on
Temporal and Spatial Pulse Propagation”, Fiber and Integrated Optics; 20, p.
83-94, 2001.
4.5 I. Gannot, M. Ben-David and Ilko K. Ilev, “Pulse Dispersion in Hollow
Optical Waveguides”, submitted for publication.
4.6 Steven G. Johnson, Mihai Ibanescu, M. Skorobogatiy, Ori Weisberg,
Torkel D. Engeness, Marin Soljacic, Steven A. Jacobs, J. D. Joannopoulos, and
Yoel Fink, “Low-loss asymptotically single-mode propagation in large-core
OmniGuide fibers”, Optics Express; v. 9, n. 13, p. 748-778, December 2001.
4.7 Y. Matsuura and M. Miyagi, “Delivery of Fentosecond Pulses by Flexible
Hollow Fibers”, J. of Applied Physics, v.91 n.2, 2002.
4.8 D. Sadot, Notes from the course “Optical Communication” Ben-Gurion
University”.
130
Chapter Five - Improving The Hollow Waveguides
List of Symbols For Chapter 5
c .............. Speed of light
B ............. Magnetic field
E ............. Electric field
.............. Charge density
J .............. current density
D ............. Displacement field
 .............. Dielectric constant
r .............. radius vector
t............... time
 ............. angular frequency
 ............. Operator
a .............. Lattice constant
u .............. Wave function
iAngle of incidence
tAngle of transmittance
rAngle of reflection
k.............. Wave vector
d .............. Layer thickness
r .............. Reflection coefficient
t............... Transmission coefficient
Wavelength
131
As we saw in chapter 3, hollow waveguides have two drawbacks: high
attenuation when bend and high sensitivity to the laser beam’s angle of
incidence. Moreover, each waveguide is “tuned” for delivering only a specific
wavelength [5.13].
One of the ways to overcome these drawbacks is to
introduce the notion of photonic crystals.
Photonic crystals are materials patterned with a periodicity in dielectric
constant, which can generate (as will be shown later) a range of “forbidden”
frequencies called photonic bandgaps. Photons with energies lying in the
bandgap cannot propagate through the medium. This provides a way to
construct materials with high reflectance (R=1), such as a “perfect mirror”,
laser cavities etc.
The theoretical study of photonic crystal began in the early sixties. Yariv et al.
[5.1] showed that a periodic multilayer structure of dielectric materials might
reflect photons completely, thus generating a “perfect mirror”. Later on
Joannopoulos et al. [5.2] showed that the phenomena could be expanded to two
and three dimensions. Perfect mirrors made of dielectric materials have been
made by many companies and research groups [5.6, 5.8]. Most recently Fink et
al. [5.7] reported of a “perfect” mirror made of polystyrene and tellurium.
Although the idea of photonic crystals has been known for many years, only
one dimensional structures i.e. multi-layer mirrors have been built. Only
recently there have been successful attempts to manufacture photonic crystal
fibers. These fibers could be categorized into two groups. The first one is multi
layer hollow waveguides [5.14] and the second one is “holey fibers” [5.15].
132
Building long multi-layer hollow waveguide is not an easy task and was not
accomplished yet due to manufacturing problems.
One of the first attempts at achieving a PBG structure was the “holey” fiber; an
ordinary, pure silica fiber with a triangular array of axial air holes running the
length of it; see Figure 5.0. A single, missing air hole in the center, the
“defect”, was predicted to exhibit wave guidance by the PBG effect [5.16]. It
did exhibit wave guidance, but the structure was not precise enough to produce
the required band gaps. In 1999, Crystal Fibre A/S was founded in Denmark to
mass-produce and commercially sell these fibers. Although they do not operate
on the photonic bandgap principle, these fibers are commonly called Photonic
Crystal Fibers (PCFs).
pitch
cladding
a
Air
holes
Figure 5.0
General structure of a holey waveguide
Triangular lattice fibers have some advantages over the more exotic designs
that have been proposed for true PCFs. They can be produced with ordinary
silica fiber drawing technologies [5.17]. Hollow silica tubes are bundled
together in the desired configuration. Solid glass rods are added to the center to
133
form the defect, and the whole structure is heated and drawn out on an ordinary
fiber-drawing tower [5.18].
While the holes do not retain their relative sizes and locations perfectly, the
general triangular lattice structure survives. The resulting fibers are not regular
enough to exhibit photonic bandgaps, but they are regular enough to display
wave guidance based on the “effective index difference” between the defect
and the lattice.
Crystal fibers have some very desirable optical properties. They can be made
single mode over a large bandwidth, while maintaining large core diameter,
and still have reasonable performance at low wavelengths. They are also
comparatively simple to manufacture, and the critical parameter a/(this
parameter determines the bandwidth) can be adjusted with relative ease.
While crystal fibers have certain desirable optical properties, they also have
some undesirable mechanical properties. The bridges” between air holes are
thin and prone to breakage, especially given the brittle nature of silica. Crystal
fibers tend to crack around the holes and lose chunks of glass as the fiber
pieces are separated. Finally, it is difficult to use fiber-bonding epoxies on a
crystal fiber, as the epoxy could plug the air holes, increasing their refractive
index and killing wave guidance at the entrance of the fiber.
In this chapter I will propose a way to improve the current hollow waveguides
by using a multi-layer structure also known as a photonic crystal. The first part
of the chapter describes the theory behind photonic crystals. It includes a
134
description of photonic crystals, using Bloch theory and the more conventional
Maxwell equation approach.
The second part describes our efforts to design and manufacture a multi-layer
mirror as a step in fabricating a multi-layer hollow waveguide. Many
researches around the world have been designing and manufacturing multilayer mirrors. However none of the methods used are applicable to the
manufacturing of long hollow waveguides. We propose a new approach, using
metal and dielectric layers instead of two dielectric layers. This approach has
two advantages: the first is that the method of deposition may be used for long
tubes and the second one is that using metal and dielectric layers enables us to
use a smaller number of layers (in fact two pairs is enough) as opposed to the
large number of layers needed for dielectric structures.
The last section shows a theoretical calculation that compares the performance
of multi-layer hollow waveguide to a conventional one.
135
5.1 Theory
There are two ways for analyzing photonic crystals. The first is to look at the
material as a lattice and to solve Maxwell equations using the symmetry
principles, which lead to Bloch theory [5.1, 5.2]. The second is to solve
Maxwell equations with the appropriate boundary conditions [5.3].
5.1.1 Bloch theory
5.1.1.1 The macroscopic Maxwell equations
All macroscopic electromagnetic phenomena, including the propagation of
light in a photonic crystal, is governed by the four macroscopic Maxwell
equations. In cgs units, they are
B  0
1 B
0
c t
  D  4
E 
H 
(5.1.1)
1 D 4

J
c t
c
where E and H are the macroscopic electric and dielectric and magnetic fields,
D and B are the displacement and magnetic induction fields, and  and J are
the free charges and currents.
We will restrict ourselves to propagation within mixed dielectric medium with
no free charges or currents   J  0 . Next we relate D to E and B to H with
the constitutive relations appropriate to our problem. Quite generally, the
components Di of the displacement field D are related to the components Ei via
the power series,
Di    ij E j   k ij Ei E j  OE 3 
j
j
(5.1.2)
136
However, for many dielectric materials, it is good approximation to employ the
following assumptions. First, we assume that the fields’ strengths are small, so
that we are in the linear region, and we can use only the first term in 5.1.2.
Second, we assume that the material is macroscopic and isotropic, so that
Er,  , Dr,  are related by a scalar dielectric constant  r,   . Third, we
ignore any explicit frequency dependence of the dielectric constant. We simply
choose the value of the dielectric constant appropriate to the frequency range of
the physical system we are considering. Fourth, we focus only on low loss
dielectrics, which means we can treat  r  as purely real.
When all is said and done, we have Dr    r Er  . An equation similar to
(5.1.2) relates B and H. However, for most dielectric materials the magnetic
permeability is very close to unity and we may assume B=H.
With all these assumptions in place, the Maxwell equations (5.1.1) become
  H r, t   0
1 H r, t 
0
c t
   r Er   0
  Er, t  
  H r, t  
(5.1.3)
1 Er, t 
0
c t
In general both E and H are complicated functions of time and space. But since
the Maxwell equations are linear, we can separate out the time dependence by
expanding the fields into a set of harmonic modes. Maxwell equations impose
restrictions on a field pattern that happens to vary sinusoidally (harmonically)
with time. The solutions for the Maxwell equations are
Hr, t   Hr ei t
Er, t   Er ei t
(5.1.4)
137
To find the equations for the mode profiles of a given frequency, we insert the
above equations into (5.1.3). The two divergence equations give the simple
conditions:
  Hr     Er   0
(5.1.5)
These equations have a simple physical interpretation. There are no point
sources or sinks of displacements and magnetic fields in the medium. The two
curls relate E(r) and H(r):
i
H r   0
c
i
  H r    r Er   0
c
  Er  
(5.1.6)
We can decouple these equations in the following way. Divide the bottom,
equation of (5.1.6) by  r  , and then take the curl. Then use the first equation
to eliminate E(r). The result is an equation entirely in H(r):
 1
  
  
  Hr     Hr 
  r 
 c
2
(5.1.7)
This is the master equation. In addition to the divergence equation (5.1.5), it
determines H(r). Now we can find the magnetic field for a given photonic
crystal, and using the first equation (5.1.6) we can derive the electric field
  ic 
  Hr 
Er   
  r  
(5.1.8)
From the master equation we can derive the allowable electromagnetic modes.
If H(r) is an allowable mode, the other results will just be constants times the
original function H(r). This situation arises often in mathematical physics, and
is called an eigenvalue problem. If the result of an operation on a function is
138
just the function itself, multiplied by some constant, then the function is called
an eigenfunction or eigenvector of the operator, and the multiplicative constant
is called the eigenvalue.
In this case we identify the left side of the master equation as an operator
 acting on H(r) to make it look like an eigenvalue problem:
 
H r     H r 
c
(5.1.9)
 1

Hr     
  Hr 
  r 

(5.1.10)
2
where
The eigenvectors H(r) are the field patterns of the harmonic modes and the
eigenvalues  / c 2 are proportional to the squared frequencies of those modes.
Notice that the operator  is a linear operator.
Our eigenvalue problem resembles the eigenvalue problems in quantum
mechanics, where a Hermitian operator (the Hamiltonian) is operated on the
wave function and the eigenvalues can be derived by the variation principle and
symmetry properties. These useful similarities can be useful in the
electromagnetic case as well.
5.1.1.2 Translational symmetry and solid-state electromagnetism
In both classical mechanics and quantum mechanics, we learned the lesson that
symmetries of a system allow one to make general statements about that
system’s behavior.
139
There are several types of symmetries: inversion, continuous translation and
discrete translation. Since a photonic crystal material has a periodic structure of
the dielectric constant we will consider translation symmetry.
Photonic crystals, like the familiar crystals of atoms, have discrete translation
symmetry. That is, they are not invariant under translations of any distance –
only under distances that are a multiple sum of a fixed step length. The
simplest example of such a system is a structure that is repetitive in one
direction (figure 5.1).
z
a
y
x
Figure 5.1
For this system we have continuous translational symmetry in the x direction,
but we have also discrete translational symmetry in the y direction. The basic
step length is the lattice constant a, and the basic step vector is called the
primitive lattice vector, which in this case is a  ayˆ . Because of the symmetry,
 r    r  a . By repeating this translation, we see that  r    r  R for any R
that is an integral multiple of a.
Because of the translational symmetries,  must commute with all the
translation operators in the x direction, as well as the translation operators for
lattice vectors R  layˆ (l is an integer) in the y direction. With this knowledge
140
we can identify the modes of  as simultaneous eigenfunctions of both
translational operators. These eigenfunctions are plane waves:
 
Tdxˆ eikx x  eikx  x  d   eikx d eikx x
TR e
ik y y
e
ik y  y  la 
 e
 e
ik y la
(5.1.11)
ik y y
Where Tdx and TR are the translation operators in x and y directions
respectively.
We can begin to classify the modes by specifying k x and k y . However not all
values of k y yield different eigenvalues. Consider two modes, one with wave
vector k y and the other k y 
2
. A quick insertion into the above equations
a
shows that they have the same TR eigenvalues. In fact, all of the modes with
wave vectors of the form k y  m
2
, where m is an integer, form a degenerate
a
set: they all have the same TR eigenvalues of e
integral multiple of b 
ik y la
. Augmenting k y by an
2
leaves the state unchanged. We call b the primitive
a
reciprocal lattice vector.
Since any linear combination of these degenerate eigenfunctions is itself an
eigenfunctions with the same eigenvalues, we can take linear combination of
our original modes to put them in the form
H k x ,k y r   eikx x  ck y ,m z e
m


i k y  mb y
 eikx x e
ik y y
 c z e
k y ,m
imby
 eikx x e
ik y y
uk y  y, z  (5.1.12)
m
where the c’s are expansion coefficients to be determined by the explicit
solution, and uk  y, z  is a periodic function of y.
y
141
The discrete periodicity in the y direction leads to a y dependence for H that is
simply the product of a plane wave with a y periodic function. We can think of
it as a plane wave, as it would be in free space, but modulated by a periodic
function because of the periodic lattice:
H ... y....e
ik y y
uk y  y...
(5.1.13)
This result is commonly known as Bloch’s theorem. In solid-state physics the
above equation is known as Bloch state.
One key fact about Bloch state is that the Bloch state with wave vector k y and
the Bloch state with the wave vector k y  mb are identical. The k y ’s that differ
by integral multiple of b are identical in the physical point of view. Thus the
mode frequencies must also be periodic in k y :  k y    k y  mb . In fact we
only need to consider k y in the region   a  k y   a which is called the
Brillouin zone. This result can be expanded to three dimensional photonic
crystal [5.2].
5.1.1.3 Photonic band structure
We saw that it is possible to derive the solutions for the wave equations from
symmetry principles. All the information about the modes is given by the wave
vector k and the periodic function uk r  . To solve for uk r  we insert the Bloch
state into the master equation:
142
  k  
H k  
 Hk
 c 
2
 1
   k   ikr
  
  e ikr u k r   
 e u k r 
  r 
  c 
2


ik    1 ik    uk r     k   uk r 
  r 
  c 
2
(5.1.14)
  k  
 k u k r   
 u k r 
 c 
2
Here we have defined  k as a new Hermitian differential operator that depends
on k:
 1
ik   
 k  ik   
  r 

(5.1.15)
The function u, and therefore the mode profiles, are determined by the
eigenvalue problem in the forth equation in 5.1.14, subjected to the condition
uk r   uk r  R 
(5.1.16)
Because of the periodic boundary condition, we can regard the eigenvalue
problem as being restricted to a unit cell of the photonic crystal. This leads to a
discrete spectrum of eigenvalues. We can find for each value of k an infinite
set of modes with discretely spaced frequencies, which we can label with a
band index n.
For a multilayer film, which is composed of two materials, Yablonovich [5.4]
showed that there could exist a band gap in the frequency band structure. In
this band gap light with frequency that lies within the band gap is not allowed
to propagate and is reflected completely.
143
5.1.2 Maxwell equations
Consider the linearly polarized wave shown in figure 5.2 [5.3], impinging on a
thin dielectric film between two semi-infinite transparent media. Each wave ErI,
E’rII, EtII, and so forth, represents the resultant of all possible waves traveling in
that direction, at that point in the medium. The summation process is therefore
built in. The boundary conditions require that the tangential components of
both the electric field, E, and the magnetic field, H, be continuous across the
boundaries.
HiI
EiI
ErI
kiI
iI
I
n0
EtI
iII
d
ErII
EiII
n1
II
ns
tII
HtII
EtII
KtII
Figure 5.2
Fields at the boundaries
144
At boundary I
'
EI  EiI  ErI  EtI  ErII
(5.2.1)
and
HI 
0
EiI  ErI n0 cos iI   0 EtI  ErII' n1 cos iII
0
0
(5.2.2)
where use is made at the fact that E and H in non-magnetic media are related
through the index of refraction and the unit propagation vector:
H
0 ˆ
nk  E
0
(5.2.3)
At boundary II
EII  EiII  ErII  EtII (5.2.4)
and
H II 
0
EiII  ErII n1 cos iII   0 EtII ns cos tII (5.2.5)
0
0
the substrate having an index ns. A wave that transverses the film once
undergoes a phase shift of k0 h  k0 2n1d cos iII  2 , so that
EiII  EtI exp  k0 h
(5.2.6)
and
ErII  ErI exp  k0 h  (5.2.7)
so that equations 5.2.4 and 5.2.5 can now be written as
EII  EtI exp  k0 h  ErI exp  k0 h (5.2.8)
and
145
H II  EtI exp  k0 h   ErI exp  k0 h 
0
n1 cos iII (5.2.9)
0
These last two equations can be solved for EtI and E ' rII , which when
substituted into 5.2.1 and 5.2.2, yield
EI  EII cosk0 h   H II
i sin k0 h
Y1
(5.2.10)
and
H I  EII Y1i sin k0 hcosk0 h  H II cosk0 h
(5.2.11)
where
Y1 
0
n1 cos iII
0
(5.2.12)
When E is in the plane of incidence, the above calculations result in similar
equations, provided that now
Y1 
 0 n1
0 cos iII
(5.2.13)
In matrix notation, the above linear relations take the form
 EI   cosk0 h 
H   
 I  Y1i sin k0 h 
i sin k0 h 
 E 
Y1   II 
cosk0 h    H II 
(5.2.14)
or
 EI 
 EII 
H   M I H 
 I
 II 
(5.2.15)
The characteristic matrix M I relates the fields at the two adjacent boundaries.
It follows, therefore, that if two overlaying films are deposited on the substrate,
there will be three boundaries or interfaces, and now
146
 EII 
 EIII 
 H   M II  H 
 II 
 III 
(5.2.16)
Multiplying both sides of this expression by M I we obtain
 EI 
 EIII 
 H   M I M II  H  (5.2.17)
 I
 III 
In general, if p is the number of layers, each with a particular value of n and h,
then the first and the last boundaries are related by
 E p 1 
 EI 
 H   M I M II .....M p  H  (5.2.18)
 I
 p 1 
The characteristic matrix of the entire system is the resultant of the product (in
the proper sequence) of the individual 2x2 matrices, that is
m12 
m
M  M I M II .....M p   11

m21 m22 
(5.2.19)
From the above equations we can derive the reflection and transmission
coefficients. By reformulating equation 5.2.14 and setting
Y0 
0
n0 cos  iI
0
0
Ys 
ns cos  tII
0
(5.2.20)
we obtain
 EiI  ErI  
 EtII 
E  E Y   M I  E Y  (5.2.21)
rI
0
 iI
 tII s 
When the matrices are expanded the last relation becomes
1  r  m11t  m12Ys t
and
(5.2.22)
147
1  r Y0  m21t  m22Yst
(5.2.23)
where
r
E
ErI
, t  tII
EiI
EiI
(5.2.24)
Consequently,
r
Y0 m11  Y0Ys m12  m21  Ys m22
Y0 m11  Y0Ys m12  m21  Ys m22
(5.2.25)
t
2Y0
Y0 m11  Y0Ys m12  m21  Ys m22
(5.2.26)
and
To find r or t for any configuration of films, we need only to compute the
characteristic matrices for each film, multiply them, and then substitute the
resulting matrix elements into the above equations.
5.1.3 Designing a multilayer mirror for the infrared
As stated before, one of the solutions to the attenuation problems in hollow
waveguides may be the use of a multilayer film inside the hollow tube. In order
to get “perfect” reflection we will design a multilayer film, which totally
reflects radiation around a given wavelength 0 . The simplest form of a
photonic crystal is the quarter wave stack, which is a multilayer film where
each layer has a width of a quarter wavelength. We will use equation 5.2.25 for
the design.
As an example let us design a multilayer mirror for a laser with a wavelength
6m. We will use two dielectric materials that have indices of refraction, which
are far apart. Such materials could be Germanium (n=4) and Zinc Selenide
148
(n=2.4) [5.5]. A mirror made of Germanium and Zinc Selenide has layers that
are 0.375m and 0.625m thick respectively.
Figure 5.3 shows the calculated reflectance of a multilayer film made of
different number of pairs of Ge and ZnSe, as a function of wavelength, for 0 o
angle of incidence. As can be seen from the figure the more pairs there are the
sharper is the region where the reflectance is “perfect”.
Figure 5.3
Reflectance Vs. Wavelength
(dotted line – 2 pairs, dashed line – 4 pairs, solid line – 6 pairs)
Figure 5.4 shows the reflectance of a multilayer mirror with 6 pairs of layers as
a function of wavelength for different angles of incidence. As can be seen from
the figure the reflectance pattern shifts to shorter wavelengths but its shape
stays the same.
149
Figure 5.4
Reflectance Vs. Wavelength
(dotted line – 0o, dashed line – 30o, solid line – 60o)
These theoretical results may help us understand the dependence of a
multilayer film on the laser wavelength and the laser beam angle of incidence,
thus helping us in designing a multilayer film that reflects a certain wavelength
range.
5.2 Experimental results and discussion– multilayer mirror
Commercial companies and research teams have been building onedimensional photonic crystals for many years. They reported the use of films
made of GaAS - AlAs [5.6], polystyrene – tellurium (PS – Te) [5.7] and
Na2AlF – ZnSe [5.8]. These mirrors usually have a large number of layers and
exhibits a 100% reflection for a large interval of wavelengths.
150
Before applying the above-mentioned theory to hollow waveguides we tried to
build two kinds of multilayer mirror. The first was a dielectric mirror made of
Si and SiO2. The second was a metal dielectric mirror made of silver and silver
iodine.
5.2.1 Dielectric mirror
Since most of the teams and companies have made dielectric multilayer mirrors
we tried to build a mirror made of Si and SiO2. The refractive index of Si is
3.415 and the one of SiO2 is 1.54 [5.5]. Since their refractive indices are far
apart they may be a good candidates for a “perfect” mirror.
5.2.1.1 Deposition of the layers
The layers were deposited using the sputtering technique on a glass plate. We
used two targets, one made of Si and one made of SiO2.The mirror was build
for the reflection of near IR radiation around 0=700nm. The thickness of each
layer was determined by the time material is sputtered on the glass. We built
mirrors made of one pair of Si-SiO2, 2 pairs, 4 pairs and 6 pairs.
5.2.1.2 Reflection measurements
The reflection measurements were made using a FTIR (Brucker Germany),
which emits radiation in the visible and near IR (400nm<  < 1200nm). Figures
5.5 and 5.6 show the experimental results (fig. 5.5) and the theoretical
calculations (fig. 5.6) of the reflectance as a function of the wavelength for the
different number of pairs respectively.
151
Reflectivity Vs. Wavelength
1
0.9
0.8
Reflectivity
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
400
500
600
700
800
900
1000
1100
Wavelength [nm]
1 pair
2 pairs
4 pairs
6 pairs
Figure 5.5
Experimental reflectance as a function of  for a multilayer mirror made of Si and SiO2
Figure 5.6
Experimental reflectance as a function of  for a multilayer mirror made of 4 layers of
Si and SiO2
Blue – Ag (Si-SiO2), Green – Ag 2(Si-SiO2) , Red - Ag 4(Si-SiO2), Light green - Ag 6(SiSiO2)
As may be seen from the figures there is a good agreement between the
experimental results and theoretical ones only for the mirrors made of one pair
and two pairs.
152
The deviation between the experimental results and the theoretical ones is due
to the inability to exactly control the thickness of the layers and a poor
adherence between the pairs. Since the structure is very sensitive to the changes
in the layers thickness, any deviation will cause the reflectance to drop from
one.
5.2.2 Metal dielectric mirror
Manufacturing multilayer mirrors is not an easy task. It is necessary to control
the thickness of the deposited layers very accurately otherwise the reflection
would decrease. Moreover it is very difficult to use the current coating methods
in order to deposit multilayer thin films inside a hollow tube.
Another possibility is to make multilayer mirrors made of metal and dielectric
materials [5.12]. Metals reflect radiation very well and if made thin enough (of
the order of nm) they also transmit some of the radiation and almost do not
absorb it. We tried to manufacture a multilayer mirror made of silver (Ag) and
silver iodine (AgI). We are very familiar with these materials and know how to
deposit them inside a hollow tube.
5.2.2.1 Deposition of the layers
The deposition of the layers was done using a chemical electro less method.
The main advantage of this method over other methods, such as CVD, is the
ability to deposit thin layers inside hollow tubes and not just coat simple plates.
This will enable us to develop in the future “lossless” hollow waveguides.
We used Ag for the metal layer and AgI for the dielectric layer. The Ag and
AgI layers were deposited on a sapphire slide. The sapphire slides were
153
cleaned, and their surface was activated in PdCl2 solution before silver electro
less plating. An Alpha-step 500 was used to measure the thickness of the
deposited Ag and AgI layers.
The deposition of Ag was made from AgNO3 solution (0.05 M), buffered by
ammonia and citric acid to pH = 10-10.6, at room temperature. Ammonia and
citric acid were also used to complex metal-ions. The minute quantities of
additives were introduced for brightness and softness of the deposit and for
stabilization of the electrolyte. Hydrazine hydrate was used as a reducer. To
obtain the pair Ag/AgI layer, an Ag layer of thickness dAg >70 nm was
deposited and by reaction with iodine a part dAg was transformed into AgI
layer. Since the density of AgI is smaller than that of Ag, it was easy to obtain
values of thickness of AgI of dAgI 50 nm, without reducing drastically dAg.
5.2.2.2 Properties of thin Ag layers
In order to use thin silver layers in a multilayer structure we must first
determine their optical properties, especially for which thickness they are
transparent and what is their index of refraction.
In order to determine the transmission of a thin silver film we deposited thin
silver films of different dAg on a sapphire plate with known transmission
properties and measured their transmission. The measurement was done using a
FTIR (Brucker, Germany). Figure 5.7 shows the transmission of the sapphire
plate, a sapphire plate coated with a 30nm thick Ag film and another with a
50nm thick AgI film, as a function of wavelength. As can be seen from figure
5.7, the values of transmission of the Ag+sapphire and AgI+sapphire layers
154
were identical to the transmission of the sapphire itself, in the region where the
transmission of sapphire is higher than 40%. This proves that the thin films are
transparent in the region where the substrate in not absorbing.
100
Sapphire 1 mm
Ag 30 nm
90
80
70
Transmission, %
60
50
40
30
20
10
0
100 4
5
6
7
6
7
90
8
9
8
9
Sapphire 1 mm
AgI 50 nm
80
70
60
50
40
30
20
10
0
4
5
Wavelength, m
Figure 5.7 – Transmission of Ag+Sapphire and AgI+Sapphire as a function of
wavelength
Figure 5.8 shows the transmission of different silver layers as a function of
wavelength, for various values if dAg. As can be seen from the figure, thin Ag
films are transparent for dAg<75nm. Using thin Ag films of thickness less than
75nm will enable us to build a multilayer mirror.
155
T ransmission
1
0 .8
0 .6
0 .4
0 .2
0
3
4
5
6
Wa v e le n g th [m ic ro n ]
36 nm
50 nm
75nm
120 nm
Figure 5.8 – Transmission of Ag as a function of wavelength for various dAg
The advantage of using metallic materials such as silver is that for a certain
interval of wavelengths range the ratio of the index of refraction of the metal to
that of the dielectric material (nm/nd) is large enough to obtain the behavior of a
photonic crystal. This enables us to use a small number of pairs in order to
manufacture an omnidirectional mirror .The index of refraction of Ag is a
function of wavelength and changes between 3.74 at 6m to 14 at 14m. Such
index of refraction yields the large ratio needed to obtain a photonic crystal
1.7 
n Ag
n AgI
 6.36
6m    14m 
(5.4.1)
The following graph (figure 5.9) shows the calculated reflectance of our
structure (Ag/AgI multilayer 4 pairs) as a function of wavelength for three
angles of incidence (blue – 80o, green – 45o and red – 25o). As can be seen
from the figure perfect reflectance and omni-directionality are achieved for
156
the two wavelengths range. The boarder one is between 10.5m to 14m
and the narrower one is between 6.5m to 7.5m
Figure 5.9 – The reflectance of a Ag/AgI multilayer mirror (4 pairs)
as a function of wavelength for various angles of incidence
blue – 80o, green – 45o and red – 25o
5.2.2.3 Reflectance measurements
Using the above electro less method we manufactured multilayer mirrors made
of pairs of Ag-AgI. We made mirrors with 1 to 4 pairs of Ag-AgI layers on a
thick silver substrate. Such a substrate reflects back the part of the radiation,
which might be transmitted by the multilayer structure. This makes it possible
to avoid losses due to partial photonic mirror effects. The main difference
between a good mirror made of a perfect thick Ag and the multilayer maybe
demonstrated
by
(omnidirectional).
the
angle
dependence
of
the
reflected
radiation
157
The reflectance of the mirrors as a function of wavelength was measured using
a FTIR (Brucker – Equinox 55). Figure 5.9 shows the reflectance of a thick Ag
layer as a function of wavelength for various angles of incidence. As can be
seen, the reflectance (R) decreases as the angle of incidence () increases. The
Reflectance
change is from R=95% at =100 to R=70% at =600.
1
0.5
0
3
4
5
6
7
8
9
10
Wavelength (m)
10 degrees
20 degrees
40 degrees
50 degrees
30 degrees
Figure 5.9 – The reflectance of a thick Ag layer for various angles of incidence
Figure 5.10 a and b show the reflectance of a 4 pairs mirror as a function of
wavelengths for various angles of incidence: (a) experimental results and (b)
theoretical calculations. As can be seen from the figures there is a good
agreement between the theoretical calculations and the experimental results.
There is a negligible dependence of the reflectance on the angle of incidence.
The difference in the reflectance is less than 2%. This result shows that the
reflectance is insensitive to the angle of incidence (omnidirectional) as opposed
to the strong dependence of the reflectance of the thick Ag layer.
R eflectan ce
158
1
0 .9
0 .8
0 .7
0 .6
0 .5
6
8
10
12
14
W a v e le n g th [  m ]
1 0 d e g re e s
2 0 d e g re e s
4 0 d e g re e s
5 0 d e g re e s
3 0 d e g re e s
(a)
1
0 .9
Reflectance
0 .8
0 .7
0 .6
0 .5
0 .4
0 .3
0 .2
0 .1
0
6
7
8
9
10
11
12
13
14
Wav eleng th [m]
1 0de gre es
2 0d egre es
30d eg rees
40 deg ree s
50 de gree s
(b)
Figure 5.10 – The reflectance of 4 pairs multilayer mirror for various angles of
incidence. a – experimental results, b – theoretical calculation
Figures 5.11 a and b show the reflectance as a function of wavelength for
mirrors made of different numbers of pairs of Ag and AgI: (a) experimental
results and (b) theoretical calculations for the angle of incidence of 600. The
theoretical calculations were made using the method described in [5.10]. As
159
can be seen from these figures, there is a good agreement between the
theoretical calculation and the experimental results. As the number of layers
increases the reflectance increases. A mirror that is made of 4 pairs has a
reflectance in the range of 98% to 100%. However for practical applications it
can be seen that the increase in the reflectance is getting smaller with the
Refle ctance
number of pairs and for certain wavelength range two pairs might be sufficient.
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
6
8
10
12
14
Wavelength [ m]
2 pairs
3 pairs
4 pairs
Reflectance
(a)
1.005
1
0.995
0.99
0.985
0.98
0.975
0.97
0.965
0.96
0.955
6
7
8
9
10
11
12
13
Wavelength [micron]
1pair
2pairs
3pairs
4pairs
(b)
Figure 5.11 – The reflectance of multilayers mirror for various number of Ag-AgI
pairs, angle of incidence 60o; a – experimental results, b – theoretical calculation
14
160
5.3 Hollow waveguides made of multilayer films
In order to examine the influence of multilayer structure on the performance of
hollow waveguides the above theoretical calculation was inserted into our ray
model (presented in chapter 3) [5.11], which predicts the attenuation and beam
shape of a hollow waveguide under different conditions. Figure 5.12 shows the
calculated transmission of a bent hollow waveguide (ID=1mm, l=1m) made of
the traditional structure (tube, metal layer, dielectric layer) as a function of the
bending radius. Figure 5.13 shows the calculated transmission of a bent hollow
waveguides (ID=1mm, l=1m) with a multilayer structure as a function of the
bending radius.
Transmission Vs. Radius of Curvature
1
0.95
0.9
Transmission
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0.5
0
50
100
150
200
250
300
350
Radius of Curvature [mm]
400
Figure 5.12 – Transmission vs. Radius Of Curvature
Traditional Hollow Waveguide (theoretical calculation)
450
500
161
Transmission Vs. Radius of Curvature
0.999
Transmission
0.9985
0.998
0.9975
0
50
100
150
200
250
300
350
Radius of Curvature [mm]
400
450
500
Figure 5.13 – Transmission vs. Radius Of Curvature
Multilayer Hollow Waveguide (theoretical calculation)
Figure 5.14 and 5.15 show the calculated transmission of hollow waveguide as
a function of the focal length of the coupling lens for a traditional and
multilayer waveguides respectively.
162
1
0.98
transmission
0.96
0.94
0.92
0.9
0.88
0.86
20
40
60
80
100
focal length [mm]
120
140
160
140
160
Figure 5.14 – Transmission vs. Focal Length
Traditional Waveguide (theoretical
calculation)
0.9994
0.9992
transmission
0.999
0.9988
0.9986
0.9984
0.9982
20
40
60
80
100
focal length [mm]
120
Figure 5.15– Transmission vs. Focal Length
Multilayer Hollow Waveguide (theoretical calculation)
163
As can be seen from the figures the transmission of the multilayer waveguide is
much better then the transmission of the traditional waveguide. This due to the
fact that the omnidirectional mirror’s reflectance is less influenced by the angle
of incidence then a thin film deposited on a metal layer.
5.4 Conclusion
Hollow waveguides are suitable for delivering infrared radiation even though
they have some drawbacks. As we showed the performance of the hollow
waveguides can be improved as we introduce the notion of multilayer mirror to
the calculation.
5.5 Summary
In this chapter I proposed a way to improve the performance of hollow
waveguides by using the concept of photonic crystals. I surveyed two methods
for dealing with multilayer mirrors: Bloch theory and Maxwell equations. I
used these theories in order to design a multilayer mirror for the IR.
We tried to make experimentally two types of multilayer mirrors. The first one
was made of Si and SiO2. This experiment was not successful due to
insufficient control over the layers thickness.
The second one was made from silver and silver iodine. I showed that thin
silver layers are transparent in the IR and that a multilayer mirror made from
these materials has the desired properties of a photonic crystal. Using these
materials we will be able to manufacture multilayers inside a hollow tube to
fabricate a multilayer hollow waveguide.
164
I introduced the concept of photonic crystals to the improved ray model and
showed that hollow waveguides made of multilayers have better properties than
the conventional hollow waveguides. To date we have not been successful in
manufacturing a multilayer hollow waveguides due to technical problems
relating to the control of the layer’s thickness.
165
5.6 Refrences for chapter five
5.1 P. Yeh, A. Yariv and C. S. Hong, “Electromagnetic propagation in periodic
media I. General Theory”, J. Opt. Soc. Am., V. 67, No. 4, p. 423-438, April
1977.
5.2
J. D. Joannopoulos, R. D. Meade, J. N. Winn, “Photonic Crystals –
Molding the Flow of Light”, p. 3-53, Princeton University Press 1995.
5.3 E. Hecht, “Optics”, 2nd edition, p. 363-368, Addison – Wesley Publishing
1987.
5.4 E. Yablonovich, “Inhibited spontaneous emission in solid-state physics and
electronics”, Physical Review Letters, V.58, No. 20, p. 2059-2062, May 1987.
5.5 M. Bass, “Handbook of Optics”, V. 2, McGraw-Hill, 1995.
5.6 M. H. MacDougal, H. Zhao, P. D. Dapkus, M. Ziari and W. H. Steier,
“Wide bandwidth distributed bragg reflectors using oxide/GaAs multilayers”,
Electronic Letters, V.30, No. 14, p. 1147-1148, July 1994.
5.7 Y. Fink, J. N. Winn,, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos and E.
L. Thomas, “A dielectric omnidirectional reflector”, Science, V. 282, p. 16791682, November 1998.
5.8 S. Bains, “Dielectric stack can be used as omnidirectional mirror”, OE
Reports, p. 3, June 1999.
5.9 J. R. Beattie, “The anomalous skin effect and infrared properties of silver
and aluminum”, Physica XXIII, p.898-902, 1957.
166
5.10 M. Ben-David, I. Gannot, A. Inberg and N. N. Croitoru, “Optimizing
Hollow Waveguides Performance Through Omnidirectional Mirrors”, BIOS
2001.
5.11 A. Inberg , M. Ben-David, , M. Oksman, A. Katzir and N. Croitoru,
“Theoretical and Experimental Studies of Infrared Radiation Propagation in
Hollow Waveguides”, Optical Engineering, 2000, 39, 5, 1384-1390.
5.12 M. Ben-David, I. Gannot, A. Inberg, G. Rezevin and N. Croitoru,
“Electroless deposited broadband omnidirectional multilayer reflectors for midinfrared (MIR) lasers”, BIOS 2002.
5.13 J. Harrington, “A Review of IR Transmitting Hollow Waveguides”, Fibers
and Integrated Optics 2000, v. 19, p. 211-217.
5.14 www.omni-guide.com
5.15 J.C. Knight, T.A. Birks, P.St.J. Russell and D.M. Atkin, “All-silica singlemode fiber with photonic crystal cladding,” Opt. Lett. 21 (1547-1549) 1996;
Errata, Opt.Lett. 22 (484-485) 1997
5.16 Jes Broeng et al. Waveguidance by the photonic bandgap effect in optical
fibres. Journal of Optics A, pages 477–482, July 1999
5.17 Jes Broeng. Photonic crystal fibers: A new class of optical waveguides.
Optical Fiber Technology, pages 305–330, July 1999
5.18 Jes Broeng et al. Review paper: Crystal fibre technology. DOPS, 2000.
167
Chapter Six – Summary
6.1 Summary
Hollow waveguides are very good candidates for the delivery of infrared
radiation. They have relatively low attenuation; they are non-toxic and may be
suitable for many applications such as industrial and medical ones.
In order to develop better hollow waveguides it is important to understand the
attenuation mechanism, and the physical parameters that influence the delivery
of the radiation through hollow waveguides. I developed a tool (a computer
simulation program) that uses an improved ray model. The improvements made
in this model include surface roughness, whispering gallery modes and
different coupling conditions. Using the ray model enabled me to determine the
attenuation/ transmission of different kinds of hollow waveguides and the
energy distribution of the laser radiation after propagating through them. The
improved ray model calculations were found to be in good agreement with the
experimental measurements.
The ray model was also used to predict the pulse dispersion caused by a hollow
waveguide. It was shown that bending the hollow waveguide significantly
increases the pulse dispersion. This is because of coupling between lower
modes of propagation and higher ones.
The experimental measurements and the theoretical calculations revealed the
hollow waveguide’s drawbacks; i.e. high attenuation when bent, high
sensitivity to surface roughness and to the coupling conditions. Introducing the
notion of photonic crystals and designing a multilayer waveguide might
168
overcome these drawbacks. Theoretical calculations for such a waveguide
show that it is the right approach for solving the hollow waveguides problems.
Proving theoretically that photonic crystals may improve the performance of
hollow waveguides is very exciting. However, building a multilayer hollow
waveguide is very difficult. Since we have lots of experience with flow
chemistry we have chosen to use silver and silver iodine as the layers. This
method although unorthodox in the field of photonic crystals decreases the
number of pairs needed and thus make the manufacturing less difficult.
We have shown that thin silver layers, of the order of tens of nano-meters,
behave as a dielectric layer and may be used as a building block for a
multilayer mirror or hollow waveguide. We have built multilayer mirrors made
of silver and silver iodine. The mirrors show the properties of photonic crystals
and the two materials may serve as the building blocks for a multilayer hollow
waveguide.
In conclusion an improved ray model helps us understand the attenuation
mechanisms in hollow waveguides and is used in order to improve them. One
of the ways to improve the current hollow waveguides is by introducing
photonic crystals and manufacturing a multilayer hollow waveguide,
6.2 Future work
Further research of the issues that were investigated in this thesis includes
further development of the ray model. The model can be expanded to include
heat dissipation along the waveguide. This will help us understand the factors
that may cause damage to the waveguides and to find constraints under which
169
these waveguides could be utilized.. Furthermore it can be expand to
applications such as thermal imaging and tissue welding.
There is a lot of work to be done in the field of multilayer hollow waveguides
in particular designing, manufacturing and investigating multi-layer hollow
waveguides and adopting them as a mean for radiation delivery in many
applications.