A Dissertation Preliminary Report On - AJAY KUMAR...

A Dissertation Preliminary Report

On

A Design of High Temperature High Bandwidth Fiber Optic Pressure Sensors

Submitted in partial fulfillment of the requirements

For the degree of

MASTER OF TECHNOLOGY

IN

COMMUNICATION SYSTEMS

BY

AJAY KUMAR GAUTAM

(Roll No. P08EC901)

Under the guidance of

Prof. B. R. TAUNK & Prof. V. N. MISHRA

2008-2009

Electronics Engineering Department

Sardar Vallabhbhai National Institute of Technology Surat-395007, Gujarat, India.

Sardar Vallabhbhai National Institute of Technology Surat-395007, Gujarat, India.

Electronics Engineering Department

CERTIFICATE

This is to certify that AJAY KUMAR GAUTAM, Roll no. P08EC901 of M.Tech.-II (Communication System) has satisfactory completed a Project

Preliminary on “A DESIGN OF HIGH TEMPERATURE HIGH

BANDWIDTH FIBER OPTIC PRESSURE SENSORS” during the

year 2009-2010.

Signature of Guide Signature of HOD

Prof. B. R. Taunk Prof. B.R. Taunk

Head, ECED

Prof. V.N. Mishra

Signature of Internal Examiners:

(1)

(2) SEAL OF DEPARTMENT

Contents

List of figures i

Abstract ii

1. Introduction

1.1 Motivation

1

1

2. Fundamentals of pressure sensing 2-3

2.1 Pressure measurement sensors

2.2 Pressure measurement methods

2.3 Conventional electronic pressure sensors

2

2

2

3. Fundamentals of Optical Fiber

3.1 Basic structure of an optical fiber

4-13

5

3.2 Applications of Optical Fiber

3.3 Principle Of Operation

3.4 Mechanisms of attenuation

5

7

11

4. Fiber optic pressure sensors

4.1 Classification of FOPS

4.2 Fabry -Perot interferometer sensors

4.3 Fiber optic engine pressure sensors

4.4 Operating Principle of FOPS

4.5 Advantages of FOPS

4.6 Applications of FOPS

14-23

14

16

18

20

21

22

5. Analytical Treatment of Three Layer Optical Fiber

5.1 Fiber without coating

24-27

24

6. Result 28-31

7. Summary and Future Work

7.1 Summary

7.2 Future Work

32

32

32

Reference 33-34

List of Figures

Figure 1 Optical fiber types

4

Figure 2. Structure of Optical Fiber

5

Figure 3. The propagation of light through a multi-mode optical fiber

9

Figure 4. The structure of a typical single-mode fiber

10

Figure 5. Specular reflection

12

Figure 6. Illustration of an EFPI fiber optic sensor 18

Figure 7. Basic elements of an optical fiber sensing system

21

Figure 8. Classification of optical fiber sensors

21

Figure 9. Bessel function of first kind (integer value 0, 1)

28

Figure 10. Bessel function of first kind (non- integer value 0.5, 1.5)

29

Figure 11. Bessel function of second kind (integer value 0, 1) 30

Figure 12 Bessel function of second kind (non-integer value 0.5, 1.5) 31

i

Abstract

Pressure measurements are required in various industrial applications, including extremely harsh environments such as turbine engines, power plants and material- processing

systems. Conventional sensors are often difficult to apply due to the high temperatures, highly corrosive agents or electromagnetic interference (EMI) noise that may be present in

those environments. Fiber optic pressure sensors have been developed for years and proved themselves successfully in such harsh environments. Especially, diaphragm based fiber optic pressure sensors have been shown to possess advantages of high sensitivity, wide bandwidth,

high operation temperature, immunity to EMI, lightweight and long life.

Static and dynamic pressure measurements at various locations of a gas turbine engine are highly desirable to improve its operation and reliability. However, the operating environment, in which temperatures may exceed 600 °C and pressures may reach 100 psi (690

kPa) with about 1 psi (6.9kPa) variation, is a great challenge to currently available sensors. To meet these requirements, a novel type of fiber optic engine pressure sensor has been

developed. This pressure sensor functions as a diaphragm based extrinsic Fabry-Pérot interferometric sensor. One of the unique features of this sensor is the all silica structure, allowing a much higher operating temperature to be achieved with an extremely low

temperature dependence. In addition, the flexible nature of the sensor design such as wide sensitivity selection, and passive or adaptive temperature compensation, makes the sensor

suitable for a variety of applications.

ii


S V National Institute of Technology – Surat Page 1

Chapter 1

Introduction

1.1 Motivation

Pressure measurement is an essential technology in many industry applications, for

example, pressure monitoring in oil storage tanks and vacuum level control in chambers.

Some applications such as gas turbine engines and oil wells involve harsh environments.

Acquiring accurate pressure measurements in these harsh environments has always challenged

the available measurement technology. The motivation of this research is to meet the

recent increasing needs for optical fiber pressure sensors capable of operating accurately

and reliably in these harsh environments, especially in turbine engines.

Gas turbine engines employed in civilian and military aircraft consume large amounts of jet

fuel daily, and the energy consumption attributed to this industry is increasing. Under

increasing demand by engine users, manufacturers are extending operating envelopes of

gas turbine engines to their limits to achieve higher thrust, better efficiency, lower

emissions, improved reliability and longer engine life. The industry consensus is that these

goals can be realized by strategic measurements at various locations in an engine for

design optimization and real-time diagnosis during service . However, the operating

environment within the engine, characterized strong EMI and high temperature, pressure, and

turbulence, shortens the lifetimes of currently available sensors.

The widely used semiconductor pressure sensors have several major drawbacks. These include a

limited maximum operating temperature of 482ºC, poor reliability at high temperatures,

severe sensitivity to temperature changes, and susceptibility to electromagnetic interference.

Compared with conventional electronic sensors, fiber optic sensors have many advantages

including small size, light weight, high sensitivity, large bandwidth, high reliability, immunity

to electromagnetic interference and anti-corrosion and absence of a spark source hazard for

flammable environments. Fiber optic sensors can also survive at much higher temperatures than

conventional pressure sensors.

The basic operating principle of an extrinsic Fabry-Perot interferometric (EFPI) enables the

development of sensors that can operate in the harsh conditions associated with turbine engines

and other aerospace propulsion applications, where the flow environment is dominated by

high-frequency pressure caused by combustion instabilities, blade passing effects, and other

unsteady aerodynamic phenomena. Both static and dynamic pressures exist in turbine engines,

which must be measured by one sensor. Diaphragm-based Fabry-Perot Interferometric (DFPI)

fiber optic pressure sensors are capable of measuring static and dynamic pressure

simultaneously.



Chapter 2

Fundamentals of pressure sensing

Pressure is defined as the force per unit area, which is a derived quantity and as such has no

primary standard. Development of pressure standards is therefore based on the primary quantities

of mass and length.

1 psi = 51.714 mmHg = 6.8946 kPa

1 bar = 14.504 psi

1 atm = 14.696 psi

2.1 Pressure measurement sensors

Basically, there are two types of pressure measurement sensors, absolute and differential

pressure sensors, which are distinguished as follows:

2.1.1 Absolute pressure sensor

As the rear side of the sensing element is not accessible, pressure can only be applied on the

front side of the sensor. To achieve an absolute pressure signal, the reference pressure is set

to vacuum.

2.1.2 Differential pressure sensor

The rear side of the sensing element is accessible. Pressure can be applied to both sides

of the sensing element, and the difference in these pressures is measured. If atmospheric

pressure is taken as the reference pressure, the sensor works as a pressure gauge.

2.2 Pressure measurement methods

In general, there are two basic approaches to measuring pressure, either directly, by

determining the force applied to a known area, or indirectly, by determining some effect

of an applied pressure. The simplest direct method is balancing an unknown pressure

against the pressure produced by a column of liquid of known density (manometric

techniques). The second method uses an elastic member of known area as the sensing element

on which pressure acts and the resultant stress or strain is then measured to calculate the actual

pressure value.

2.3 Conventional electronic pressure sensors

In order to improve the sensitivity and resolution as well as to provide means for

compensating for nonlinear effects and the ability to transmit data over considerable distance,



electrical/electronic devices were later added for converting mechanical displacements into

an electrical signal thereby creating a whole family of electronic pressure transducers. Many

years of research and development of pressure measurement techniques have resulted in

various pressure transducers including:

Capacitive

Differential transformer

Inductive

Force balance

Piezoelectric

Piezoresistive

Potentiometric

Vibrating wire or tube

Strain gauges

In almost all these pressure transducers, the pressure signal is converted to the deflection

or movement of the pressure-sensing element, and thereafter measured by different electronic

sensing techniques. the transducers vary widely in performance and cost.



Chapter 3

Fundamentals of Optical Fiber

An optical fiber (or fibre) is a glass or plastic fiber that carries light along its length. Fiber

optics is the overlap of applied science and engineering concerned with the design and

application of optical fibers. Optical fibers are widely used in fiber-optic communications, which

permits transmission over longer distances and at higher bandwidths (data rates) than other forms

of communications. Fibers are used instead of metal wires because signals travel along them

with lessloss, and they are also immune to electromagnetic interference. Fibers are also used for

illumination, and are wrapped in bundles so they can be used to carry images, thus allowing

viewing in tight spaces. Specially designed fibers are used for a variety o f other applications,

including sensors and fiber lasers.

Light is kept in the core of the optical fiber by total internal reflection. This causes the fiber to

act as a waveguide. Fibers which support many propagation paths or transverse modes are

called multi-mode fibers (MMF), while those which can only support a single mode are

called single-mode fibers (SMF). Multi-mode fibers generally have a larger core diameter, and

are used for short-distance communication links and for applications where high power must be

transmitted. Single-mode fibers are used for most communication links longer than 550 metres

(1,800 ft).

Fig. 1 Optical fiber types.

Joining lengths of optical fiber is more complex than joining electrical wire or cable. The ends of

the fibers must be carefully cleaved, and then spliced together either mechanically or



by fusing them together with an electric arc. Special connectors are used to make removable

connections.

3.1 Basic structure of an Optical Fiber

The basic structure of an optical fiber consists of three parts; the core, the cladding, and the

coating or buffer. The basic structure of an optical fiber is shown in figure 2. The core is a

cylindrical rod of dielectric material. Dielectric material conducts no electricity. Light propagates

mainly along the core of the fiber. The core is generally made of glass. The core is described as

having a radius of (a) and an index of refraction n1. The core is surrounded by a layer of material

called the cladding. Even though light will propagate along the fiber core without the layer of

cladding material, the cladding does perform some necessary functions.

Fig. 2 Structure of Optical Fiber

3.2 Applications of Optical Fiber

3.2.1 Optical fiber communication

Optical fiber can be used as a medium for telecommunication and networking because it is

flexible and can be bundled as cables. It is especially advantageous for long-distance

communications, because light propagates through the fiber with little attenuation compared to

electrical cables. This allows long distances to be spanned with few repeaters. Additionally, the

per-channel light signals propagating in the fiber can be modulated at rates as high as

111 gigabits per second, although 10 or 40 Gb/s is typical in deployed systems. Each fiber can

carry many independent channels, each using a different wavelength of light (wavelength-

division multiplexing (WDM)). The net data rate (data rate without overhead bytes) per fiber is

the per-channel data rate reduced by the FEC overhead, multiplied by the number of channels

(usually up to eighty in commercial dense WDM systems as of 2008). The current laboratory

fiber optic data rate record, held by Bell Labs in Villarceaux, France, is multiplexing 155

channels, each carrying 100 Gbps over a 7000 km fiber.

For short distance applications, such as creating a network within an office building, fiber-optic

cabling can be used to save space in cable ducts. This is because a single fiber can often carry



much more data than many electrical cables, such as Cat-5 Ethernet cabling.[vague] Fiber is also

immune to electrical interference; there is no cross-talk between signals in different cables and

no pickup of environmental noise. Non-armored fiber cables do not conduct electricity, which

makes fiber a good solution for protecting communications equipment located inhigh

voltage environments such as power generation facilities, or metal communication structures

prone to lightning strikes. They can also be used in environments where explosive fumes are

present, without danger of ignition. Wiretapping is more difficult compared to electrical

connections, and there are concentric dual core fibers that are said to be tap-proof.

Although fibers can be made out of transparent plastic, glass, or a combination of the two, the

fibers used in long-distance telecommunications applications are always glass, because of the

lower optical attenuation. Both multi-mode and single-mode fibers are used in communications,

with multi-mode fiber used mostly for short distances, up to 550 m (600 yards), and single-mode

fiber used for longer distance links. Because of the tighter tolerances required to couple light into

and between single-mode fibers (core diameter about 10 micrometers), single-mode transmitters,

receivers, amplifiers and other components are generally more expensive than multi-mode

components.

3.2.2 Fiber optic sensors

Fibers have many uses in remote sensing. In some applications, the sensor is itself an optical

fiber. In other cases, fiber is used to connect a non-fiberoptic sensor to a measurement system.

Depending on the application, fiber may be used because of its small size, or the fact that

no electrical power is needed at the remote location, or because many sensors can

bemultiplexed along the length of a fiber by using different wavelengths of light for each sensor,

or by sensing the time delay as light passes along the fiber through each sensor. Time delay can

be determined using a device such as an optical time-domain reflectometer.

Optical fibers can be used as sensors to measure strain, temperature, pressure and other quantities

by modifying a fiber so that the quantity to be measured modulates

the intensity,phase, polarization, wavelength or transit time of light in the fiber. Sensors that vary

the intensity of light are the simplest, since only a simple source and detector are required. A

particularly useful feature of such fiber optic sensors is that they can, if required, provide

distributed sensing over distances of up to one meter.

Extrinsic fiber optic sensors use an optical fiber cable, normally a multi-mode one, to

transmit modulated light from either a non-fiber optical sensor, or an electronic sensor connected

to an optical transmitter. A major benefit of extrinsic sensors is their ability to reach places

which are otherwise inaccessible. An example is the measurement of temperature

inside aircraft jet engines by using a fiber to transmit radiation into a radiation pyrometer located

outside the engine. Extrinsic sensors can also be used in the same way to measure the internal

temperature of electrical transformers, where the extreme electromagnetic fields present make



other measurement techniques impossible. Extrinsic sensors are used to measure vibration,

rotation, displacement, velocity, acceleration, torque, and twisting.

3.2.3 Other uses of optical fibers

Fibers are widely used in illumination applications. They are used as light guides in medical and

other applications where bright light needs to be shone on a target without a clear line-of-sight

path. In some buildings, optical fibers are used to route sunlight from the roo f to other parts of

the building (see non- imaging optics). Optical fiber illumination is also used for decorative

applications, including signs, art, and artificial Christmas trees.Swarovski boutiques use optical

fibers to illuminate their crystal showcases from many different angles while only employing one

light source. Optical fiber is an intrinsic part of the light-transmitting concrete building

product, LiTraCon.

Optical fiber is also used in imaging optics. A coherent bundle of fibers is used, sometimes along

with lenses, for a long, thin imaging device called an endoscope, which is used to view objects

through a small hole. Medical endoscopes are used for minimally invasive exploratory or

surgical procedures (endoscopy). Industrial endoscopes (see fiberscope or borescope) are used

for inspecting anything hard to reach, such as jet engine interiors.

In spectroscopy, optical fiber bundles are used to transmit light from a spectrometer to a

substance which cannot be placed inside the spectrometer itself, in order to analyze its

composition. A spectrometer analyzes substances by bouncing light off of and through them. By

using fibers, a spectrometer can be used to study objects that are too large to fit inside, or gasses,

or reactions which occur in pressure vessels.

An optical fiber doped with certain rare earth elements such as erbium can be used as the gain

medium of a laser or optical amplifier. Rare-earth doped optical fibers can be used to provide

signal amplification by splicing a short section of doped fiber into a regular (undoped) optical

fiber line. The doped fiber is optically pumped with a second laser wavelength that is coupled

into the line in addition to the signal wave. Both wavelengths of light are transmitted through the

doped fiber, which transfers energy from the second pump wavelength to the signal wave. The

process that causes the amplification is stimulated emission.

Optical fibers doped with a wavelength shifter are used to collect scintillation light

in physics experiments.

Optical fiber can be used to supply a low level of power (around one watt) to electronics situated

in a difficult electrical environment. Examples of this are electronics in high-powered antenna

elements and measurement devices used in high voltage transmission equipment.

3.3 Principle Of Operation

An optical fiber is a cylindrical dielectric waveguide (nonconducting waveguide) that transmits

light along its axis, by the process of total internal reflection. The fiber consists of a core

surrounded by a cladding layer, both of which are made of dielectric materials. To confine the



optical signal in the core, the refractive index of the core must be greater than that of the

cladding. The boundary between the core and cladding may either be abrupt, in step-index fiber,

or gradual, in graded-index fiber

3.3.1 Index of refraction

The index of refraction is a way of measuring the speed of light in a material. Light travels

fastest in a vacuum, such as outer space. The actual speed of light in a vacuum is about 300

million meters (186 thousand miles) per second. Index of refraction is calculated by dividing the

speed of light in a vacuum by the speed of light in some other medium. The index of refraction

of a vacuum is therefore 1, by definition. The typical value for the cladding of an optical fiber is

1.46. The core value is typically 1.48. The larger the index of refraction, the slower light travels

in that medium. From this information, a good rule of thumb is that signal using optical fiber for

communication will travel at around 200 million meters per second. Or to put it another way, to

travel 1000 kilometres in fiber, the signal will take 5 milliseconds to propagate. Thus a phone

call carried by fiber between Sydney and New York, a 12000 kilometre distance, means that

there is an absolute minimum delay of 60 milliseconds (or around 1/16th of a second) between

when one caller speaks to when the other hears. (Of course the fiber in this case will probably

travel a longer route, and there will be additional delays due to communication equipment

switching and the process of encoding and decoding the voice onto the fiber).

3.3.2 Total internal reflection

When light traveling in a dense medium hits a boundary at a steep angle (larger than the "critical

angle" for the boundary), the light will be completely reflected. This effect is used in optical

fibers to confine light in the core. Light travels along the fiber bouncing back and forth off of the

boundary. Because the light must strike the boundary with an angle greater than the critical

angle, only light that enters the fiber within a certain range of angles can travel down the fiber

without leaking out. This range of angles is called the acceptance cone of the fiber. The size of

this acceptance cone is a function of the refractive index difference between the fiber's core and

cladding.

In simpler terms, there is a maximum angle from the fiber axis at which light may enter the fiber

so that it will propagate, or travel, in the core of the fiber. The sine of this maximum angle is

the numerical aperture (NA) of the fiber. Fiber with a larger NA requires less precision to splice

and work with than fiber with a smaller NA. Single-mode fiber has a small NA.

3.3.3 Multi-mode fiber

Fiber with large core diameter (greater than 10 micrometers) may be analyzed by geometrical

optics. Such fiber is called multi-mode fiber, from the electromagnetic analysis (see below). In a

step- index multi-mode fiber, rays of light are guided along the fiber core by total internal



reflection. Rays that meet the core-cladding boundary at a high angle (measured relative to a

line normal to the boundary), greater than the critical angle for this boundary, are completely

reflected. The critical angle (minimum angle for total internal reflection) is determined by the

difference in index of refraction between the core and cladding materials. Rays that meet the

boundary at a low angle are refracted from the core into the cladding, and do not convey light

and hence information along the fiber. The critical angle determines the acceptance angle of the

fiber, often reported as a numerical aperture. A high numerical aperture allows light to propagate

down the fiber in rays both close to the axis and at various angles, allowing efficient coupling of

light into the fiber. However, this high numerical aperture increases the amount of dispersion as

rays at different angles have different path lengths and therefore take different times to traverse

the fiber.

Fig. 3 The propagation of light through a multi-mode optical fiber

In graded- index fiber, the index of refraction in the core decreases continuously between the a xis

and the cladding. This causes light rays to bend smoothly as they approach the cladding, rather

than reflecting abruptly from the core-cladding boundary. The resulting curved paths reduce

multi-path dispersion because high angle rays pass more through the lower- index periphery of

the core, rather than the high- index center. The index profile is chosen to minimize the difference

in axial propagation speeds of the various rays in the fiber. This ideal index profile is very close

to a parabolic relationship between the index and the distance from the axis.

3.3.4 Single-mode fiber

Fiber with a core diameter less than about ten times the wavelength of the propagating light

cannot be modeled using geometric optics. Instead, it must be analyzed as

an electromagnetic structure, by solution of Maxwell's equations as reduced to

the electromagnetic wave equation. The electromagnetic analysis may also be required to

understand behaviors such as speckle that occur when coherent light propagates in multi-mode



fiber. As an optical waveguide, the fiber supports one or more confined transverse modes by

which light can propagate along the fiber. Fiber supporting only one mode is called single-

mode or mono-mode fiber. The behavior of larger-core multi-mode fiber can also be modeled

using the wave equation, which shows that such fiber supports more than one mode of

propagation (hence the name). The results of such modeling of multi-mode fiber approximately

agree with the predictions of geometric optics, if the fiber core is large enough to support more

than a few modes.

Fig. 4 The structure of a typical single-mode fiber.

1. Core: 8 µm diameter

2. Cladding: 125 µm dia.

3. Buffer: 250 µm dia.

4. Jacket: 400 µm dia.

The waveguide analysis shows that the light energy in the fiber is not complete ly confined in the

core. Instead, especially in single-mode fibers, a significant fraction of the energy in the bound

mode travels in the cladding as an evanescent wave.



The most common type of single-mode fiber has a core diameter of 8–10 micrometers and is

designed for use in the near infrared. The mode structure depends on the wavelength of the light

used, so that this fiber actually supports a small number of additional modes at visible

wavelengths. Multi-mode fiber, by comparison, is manufactured with core diameters as small as

50 micrometers and as large as hundreds of micrometers. The normalized frequency V for this

fiber should be less than the first zero of the Bessel function J0 (approximately 2.405).

3.3.5 Special-purpose fiber

Some special-purpose optical fiber is constructed with a non-cylindrical core and/or cladding

layer, usually with an elliptical or rectangular cross-section. These include polarization-

maintaining fiber and fiber designed to suppress whispering gallery mode propagation.

Photonic-crystal fiber is made with a regular pattern of index variation (often in the form of

cylindrical holes that run along the length of the fiber). Such fiber uses diffraction effects instead

of or in addition to total internal reflection, to confine light to the fiber's core. The properties of

the fiber can be tailored to a wide variety of applications.

3.4 Mechanisms of attenuation

Attenuation in fiber optics, also known as transmission loss, is the reduction in intensity of the

light beam (or signal) with respect to distance travelled through a transmission medium.

Attenuation coefficients in fiber optics usually use units of dB/km through the medium due to the

relatively high quality of transparency of modern optical transmission media. The medium is

typically usually a fiber of silica glass that confines the incident light beam to the inside.

Attenuation is an important factor limiting the transmission of a digital signal across large

distances. Thus, much research has gone into both limiting the attenuation and maximizing the

amplification of the optical signal. Empirical research has shown that attenuation in optical fiber

is caused primarily by both scattering and absorption.

3.4.1 Light scattering

The propagation of light through the core of an optical fiber is based on total internal reflection

of the lightwave. Rough and irregular surfaces, even at the molecular level, can cause light rays

to be reflected in random directions. This is called diffuse reflection or scattering, and it is

typically characterized by wide variety of reflection angles.

Light scattering depends on the wavelength of the light being scattered. Thus, limits to spatial

scales of visibility arise, depending on the frequency of the incident light-wave and the physical

dimension (or spatial scale) of the scattering center, which is typically in the form of some

specific micro-structural feature. Since visible light has a wavelength of the order of

one micron (one millionth of a meter) scattering centers will have dimensions on a similar spatial

scale.

Thus, attenuation results from the incoherent scattering of light at

internal surfaces and interfaces. In (poly) crystalline materials such as metals and ceramics, in



addition to pores, most of the internal surfaces or interfaces are in the form of grain

boundaries that separate tiny regions of crystalline order. It has recently been shown that when

the size of the scattering center (or grain boundary) is reduced below the size of the wavelength

of the light being scattered, the scattering no longer occurs to any significant extent. This

phenomenon has given rise to the production of transparent ceramic materials.

Fig.5 Specular reflection

Similarly, the scattering of light in optical quality glass fiber is caused by molecular level

irregularities (compositional fluctuations) in the glass structure. Indeed, one emerging school of

thought is that a glass is simply the limiting case of a polycrystalline solid. Within this

framework, "domains" exhibiting various degrees of short-range order become the building

blocks of both metals and alloys, as well as glasses and ceramics. Distributed both between and

within these domains are micro-structural defects which will provide the most ideal locations for

the occurrence of light scattering. This same phenomenon is seen as one of the limiting factors in

the transparency of IR missile domes. At high optical powers, scattering can also be caused by

nonlinear optical processes in the fiber.



3.4.2 UV-Vis-IR absorption

In addition to light scattering, attenuation or signal loss can also occur due to selective absorption

of specific wavelengths, in a manner similar to that responsible for the appearance of color.

Primary material considerations include both electrons and molecules as follows:

1) At the electronic level, it depends on whether the electron orbitals are spaced (or

"quantized") such that they can absorb a quantum of light (or photon) of a specific

wavelength or frequency in the ultraviolet (UV) or visible ranges. This is what gives rise

to color.

2) At the atomic or molecular level, it depends on the frequencies of atomic or molecular

vibrations or chemical bonds, how close-packed its atoms or molecules are, and whether

or not the atoms or molecules exhibit long-range order. These factors will determine the

capacity of the material transmitting longer wavelengths in the infrared (IR), far IR, radio

and microwave ranges.

The design of any optically transparent device requires the selection of materials based upon

knowledge of its properties and limitations. The lattice absorptioncharacteristics observed at the

lower frequency regions (mid IR to far- infrared wavelength range) define the long-wavelength

transparency limit of the material. They are the result of the interactive coupling between the

motions of thermally induced vibrations of the constituent atoms and molecules of the solid

lattice and the incident light wave radiation. Hence, all materials are bounded by limiting regions

of absorption caused by atomic and molecular vibrations (bond-stretching)in the far- infrared

(>10 µm).

Thus, multi-phonon absorption occurs when two or more phonons simultaneously interact to

produce electric dipole moments with which the incident radiation may couple. These dipoles

can absorb energy from the incident radiation, reaching a maximum coupling with the radiation

when the frequency is equal to the fundamental vibrational mode of the molecular dipole (e.g.

Si-O bond) in the far- infrared, or one of its harmonics.

The selective absorption of infrared (IR) light by a particular material occurs because the

selected frequency of the light wave matches the frequency (or an integral multiple of the

frequency) at which the particles of that material vibrate. Since different atoms and molecules

have different natural frequencies of vibration, they will selectively absorb different frequencies

(or portions of the spectrum) of infrared (IR) light.

Reflection and transmission of light waves occur because the frequencies of the light waves do

not match the natural resonant frequencies of vibration of the objects. When IR light of these

frequencies strike an object, the energy is either reflected or transmitted.



Chapter 4

Fiber optic pressure sensors

Fiber optic sensors can be used to measure pressure and possess a number of inherent

advantages including

(i) immunity to electromagnetic interference,

(ii) wide range of potential measurands,

(iii) high resolution,

(iv) remote sensing capability,

(v) high reliability and

A variety of fiber optic pressure sensors (FOPS) have been developed and proven themselves in

many applications.

The light transmitted through an optical fiber can be characterized by such parameters as

intensity, wavelength, phase, and polarization. By detecting the change of these parameters

resulting from the interaction between the optical fiber and the measurand, fiber optic sensors

can be designed to measure a wide variety of physical and chemical parameters.

4.1. Classification of FOPS

Accordingly, fiber optic sensors can be categorized into four major groups including: intensity

based fiber optic sensors, color modulated fiber optic sensors, phase modulated (or

interferometric) fiber optic sensors, and polarization modulated fiber optic sensors. More than

three decades of extensive research in fiber optic sensor technologies has greatly enhanced the

technical background of all the sensor categories, and the applications of each group of

the sensors are expanding very rapidly.

4.1.1 Polarization-modulated pressure sensor

The mainstream of developed polarization-modulated fiber optic sensors are based on two

different physical effects: the Faraday effect and the photoelastic effect. Sensors based on the

Faraday effect are mainly used to measure electrical or magnetic field with the typical

application of the measurement of the electrical current. On the other hand, photoelastic fiber

sensors are naturally suitable for developing into pressure sensors because the photoelastic

effect directly transfers the applied pressure into the change of the polarization property in

the optical medium. Although silica glass fiber itself exhibits a very weak photoelastic effect,

external optical crystals are often used as the sensing element for better control and more

accurate measurement.



The first fiber optic pressure sensor based on the photoelastic effect was introduced in 1982 by

Spillman. Since then, many photoelastic fiber sensors have been reported by different

authors with their emphases on the development of clever methods to compensate for the

optical power variation of the system. With a very good self- compensation mechanism, an

external photoelastic pressure sensor could achieve an accuracy of 0.2%. However, the self-

compensation had to be constructed at the same location as the external sensing element, which

made the sensor head very bulky and difficult to be protected in harsh environments.

4.1.2 Wavelength-modulated pressure sensor

The most popular wavelength-modulated fiber optic sensor has been the Fiber grating-based

sensor ever since the first fiber grating was manufactured in 1989 through transverse UV

exposure. Fiber sensors based on both Bragg gratings and long period gratings have been

developed for the measurement of temperature, stra in and pressure. By coating the grating

region with specially designed elastic material or encapsulating the grating into a glass bubble,

fiber grating sensors have been used to measure hydrostatic pressure with a typical resolution of

0.5% . Fiber grating sensors have the advantages of immunity to the optical power loss

variation of the optical network and the capability of multiplexing many sensors to share

the same signal processing unit. However, the long-term reliability of the fiber grating

sensors has been a concern due to the degradation of optical properties and mechanical

strength when the grating is exposed to high temperature and high pressure environments.

Moreover, when used for pressure measurements, fiber grating sensors exhibit relatively

large temperature dependence, which limits their scale of applications for harsh

environmental sensing.

In summary, although optical fiber-based pressure sensors have the potential opportunity to

replace the majority of conventional electronic pressure transducers in existence in today’s

sensor market because of their unique set of advantages that can’t be offered by other

technologies, technical difficulties still exist and delay this becoming a reality. The most

common concerns about the practical applications of fiber optic pressure sensors include the

stability issue and the cross-sensitivity among multiple environmental parameters. The

fluctuation of source power and the change in fiber loss can easily introduce errors to the

measurement results, which make most optical fiber-based sensors unstable. The fact that most

fiber sensors are cross sensitive to temperature changes also makes it difficult to use fiber

optic sensors to measure parameters other than temperature in many practical applications. In

order to be able to apply fiber optic sensors to real applications, research must be performed to

overcome these technical difficulties.

4.1.3 Intensity-based FOPS

In general, intensity-based FOPS are inherently simple and require only a modest signal

processing complexity through a direct detection of the change in optical power either in



transmission or reflection. A well-developed and successfully commercialized intensity-

based sensor is the multimode optical fiber microbend sensor, which bases its principle on

the physical phenomenon that mechanical periodic microbends can cause the energy of the

guided modes to be coupled to the radiation modes and consequently results in attenuation

of the transmitted light. Pressure sensors can thus be constructed by designing the mechanical

microbending device to transfer the applied pressure to the optical intensity change.

Although microbend pressure sensors have been reported with very high resolution

(typically better than 0.1%), the large hysteresis and the power fluctuation associated with the

optical source and fiber loss limit their accuracy within a few percent of the full scale. The large

size of the mechanical microbending mechanism also makes the microbend fiber optic pressure

sensor impractical in many sensing applications where the size of the sensor is restricted to a

very small dimension.

4.1.4 Interferometry based FOPS

To date, four types of interferometric FOPSs have been investigated for the

measurements of displacement, temperature, strain, pressure and acoustic signals. These

are the Mach-Zehnder, Michelson, Fabry-Perot, and Sagnac interferometers. Among them,

the first three interferometric sensors have been developed into pressure sensors while the

Sagnac interferometer has been primarily used for gyroscopes.

Mach-Zehnder and Michelson interferometers are the two intrinsic fiber sensors that were

investigated extensively for acoustic pressure detection in the early stage of fiber sensor

development. For example, underwater hydrophones based on these two interferometers

were reported to have very high resolution of 0.01% . However, due to the very low level of

photoelastic or stress-optic coefficients of the silica glass fibers, a very long length of sensing

fiber is necessary to obtain the desirable sensitivity, which unavoidably makes the sensor

thermally unstable. Another drawback associated with these two types of interferometric

sensors is the polarization-fading problem, which refers to the interference fringe visibility

as a function of the polarization status of the light transmitted inside the fibers. The

temperature instability and the polarization fading problem both render the Mach- Zehnder

and Michelson interferometric sensors unsuitable for the long-term measurement of DC

pressure signals where the sensor drift must be kept to a very small level.

4.2 Fabry-Perot interferometer sensors

The Fabry-Perot interferometer is a very useful tool for high precision measurement, optical

spectrum analysis, optical wavelength filtering, and construction of lasers. It is a high resolution,

high throughput optical spectrometer that works on the principle of constructive

interference. The Fabry-Perot interferometer is a very simple device that is based on the

interference of multiple beams. It consists of two partially transmitting mirrors that form a



reflective cavity. Incident light enters the Fabry-Perot cavity and experiences multiple reflections

between the mirrors so that the light can produce multiple interferences.

According to the different behaviors of the incident light, fiber optic Fabry-Perot sensors

can be classified into two types æ extrinsic F-P sensors and intrinsic F-P sensors. In

extrinsic sensors, the light can be allowed to exit the fiber and be modulated in a

separate zone before being relaunched into either the same or a different fiber. They form

an interferometric cavity outside the fiber, and the fiber just acts as a medium to transmit light

into and out of the Fabry-Perot cavity. In intrinsic sensors, the light can continue within the fiber

and be modulated. A Fabry-Perot cavity is formed by a section of fiber with its two end faces

cleaved or coated with reflective coatings.

4.2.1 Intrinsic Fabry-Perot Interferometer Sensor

In intrinsic sensors the fiber construction materials are deliberately chosen in order to give

sensitivity to one or more parameters. Often it is not cost effective to make highly specialized

fibers for sensing applications; therefore intrinsic sensors may utilize readily available

fiber in specialized configurations and in conjunction with sophisticated instrumentation.

Usually an Intrinsic Fabry-Perot Interferometer (IFPI) sensor is fabricated by splicing a section

of special fiber with its two endfaces coated with reflective films to regular fibers. The

interferometric superposition of multiple reflections at the two special fiber’s end faces

generates the output signal, which is a function of the F-P cavity length, the refractive

index of the special fiber, and the reflectance of the coating. The change of the F-P cavity

length or the refractive index of the special fiber can be detected by tracking the

interference output (either through the reflection or the transmission). Various physical or

chemical parameters such as temperature, pressure and strain can be measured with a high

resolution using an IFPI sensor.

4.2.2 Extrinsic Fabry-Perot Interferometer Sensor

In extrinsic sensors the performance of the device should be independent of the fiber and

depend only on the nature of the sensing element, hence it offers the flexibility to design the

Fabry-Perot cavity to accommodate different applications. A typical EFPI sensor configuration

is shown in Fig.7. It consists of a cavity that is formed between an input optical fiber and a

reflecting optical fiber. Although the two reflectors of forming the Fabry-Perot cavity can

be the surfaces of any optical components, a very simple way to form an EFPI will be using the

well-cleaved end faces of two fibers.



Fig. 6 Illustration of an EFPI fiber optic sensor

As shown in Fig. 6, the light from an optical source propagates along the input optical

fiber to the Fabry-Perot cavity that is formed by the input optical fiber and the reflecting optical

fiber. A fraction of this incident light R1, approximately 4%, is reflected at the end face of

the input optical fiber backward the input optical fiber. The light transmitting out of the input

optical fiber projects onto the fiber end face of the reflecting optical fiber. The reflected

light R2 from the reflecting optical fiber is partially recoupled into the input optical fiber.

Optical fiber Extrinsic Fabry-Perot interferometers (EFPI) have also been developed into

pressure sensors. Compared to the Mach-Zehnder and Michelson sensors, the EFPI sensor has

advantages such as high sensitivity, small size, simple structure, polarization independence,

and great design flexibility; EFPI fiber optic sensors are therefore attractive for many

sensing applications. Moreover, because the optical fibers are packed very closely together,

there is a potential advantage to minimize the temperature dependence of the sensor.

In summary, optical fiber interferometric sensors usually have the reputation of design flexibility

of the sensing element, large dynamic range, and extremely high resolutions. However, due

to the nonlinear periodic nature of the interference signal, the accurate detection of the

differential phase change of an interferometer becomes a real challenge. Very often, the

complexity of the phase demodulation part of the interferometric sensor contributes the most

to its high cost.

4.3 Fiber optic engine pressure sensors

Fiber optic pressure sensors are capable of working in hostile environments such as

turbine engines. Compared with hollow cylinder based pressure sensors for static pressure

measurement, diaphragm based configurations are more suitable for both static and dynamic

pressure measurements . However, these diaphragm based pressure sensors are still not

suitable for applications above 500ºC. Also, the large coefficient of thermal expansion

(CTE) mismatch will cause severe stress between different materials used in sensor



construction. This stress will degrade the sensor performance or lead to a failure. Even if the

same material is used to fabricate the sensor elements, the bonding adhesive used,

especially if epoxy-based, is still a major concern for the sensor’s performance. For example,

epoxy will exhibit a time-dependent viscoelastic dimensional change and will decompose at

high temperatures. Also, the bonding adhesive having a different CTE from the sensor

elements will cause a large temperature dependence in the pressure measurement or cause

the sensor to fail. Although anodic bonding is adhesive free bonding, it cannot be used for

bonding fused silica glass, which has a higher softening point and much lower CTE than other

glass and is the most compatible material to silica optical fiber.

The goal of this research was to develop a new diaphragm based fiber optic EFPI engine

pressure sensor, which has high sensitivity, high temperature capability, large bandwidth and

low thermal- induced measurement error. Also, the sensor must be reliable and anticorrosion.

In general, the fiber optic engine pressures have to satisfy several special requirements as

explained below.

4.3.1. High temperature capability

High temperature is very often involved in many harsh environments. For example,

temperatures in turbine engines can reach 500°C or much hotter depending upon which

region of the turbine. The high temperature is the main reason that renders most electronic

sensors inapplicable. Although optical fibers can sustain temperatures as high as 800°C

before the dopants start to thermally diffuse appreciably, extra attention must be paid to the

design and fabrication of the fiber sensor in order to maintain the desirable performance at

such high temperature.

4.3.2. High pressure capability

Pressures as high as 500 psi can be encountered in turbine engines. In order to be able to survive

in such high pressure environments, fiber optic pressure sensors must be designed and

fabricated with enough mechanical strength and with their optical paths hermetically sealed to

provide the necessary protection.

4.3.3. High Bandwidth

Dynamic pressures with frequency up to 50kHz exist in turbine engines. The pressure sensor

must have very high frequency response.

4.3.4. Good thermal stability

Fiber optic pressure sensors designed for high temperature applications must be thermally

stable or have the capability of compensating for temperature variations. Otherwise the

temperature fluctuation of the environment can easily introduce large errors in the pressure

measurement results.



4.3.5. Absolute measurement and self-calibration capability

Fiber optic pressure sensors with absolute readouts are much more attractive in

applications for harsh environments because of their exemption from initialization and/or

calibration when the power is switched on. In addition, the sensors are required to have self-

calibration capability so that the fiber loss changes and the source power fluctuations can be

fully compensated, or absolute measurement becomes meaningless.

4.3.6. Cost-effectiveness

As the market for fiber optic pressure sensors for harsh environment opens rapidly, the cost of

the sensors and instrumentation is becoming a concern of increasing Importance. In order

to achieve successful commercialization, fiber optic pressure sensor systems must be robust

as well as low cost. This requires that the complexity of the fiber sensor system must be kept to

the minimum and the technique and process of fabricating sensor probes must have the potential

of allowing mass production.

4.3.7. Installability

Fiber optic pressure sensors designed for harsh environment applications must be capable

of remote operation and flexible enough for easy installation. This requires the sensor size to

be small enough to fit in the limited space where the sensor will be located. Also, the

sensor packaging must be compatible with the standard installation ports.

4.4 Operating Principle of FOPS

Optical fibers are also attractive for applications in sensing, control and instrumentation. In these

areas, optical fibers have made a significant impact. For these app lications fibers are made more

susceptible and sensitive to the same external mechanisms against which fibers were made to be

immune for their effective operation in telecommunications.

An optical fiber sensing system is basically composed of a light source, optical fiber; a sensing

element or transducer and a detector (see Fig. 7). The principle of operation of a fiber sensor is

that the transducer modulates some parameter of the optical system (intensity, wavelength,

polarization, phase, etc.) which gives rise to a change in the characteristics of the optical signal

received at the detector.



Fig. 7 Basic elements of an optical fiber sensing system

The fiber sensor can be either an intrinsic one - if the modulation takes place directly in the fiber-

- or extrinsic, if the modulation is performed by some external transducer as depicted in Fig. 8

Fig. 8 Classification of optical fiber sensors



4.5 Advantages of fiber optic pressure sensors

Compared with conventional electronic sensors, fiber optic sensors have many advantages

including small size, light weight, high sensitivity, large bandwidth, high reliability, immunity

to electromagnetic interference and anti-corrosion and absence of a spark source hazard for

flammable environments. Fiber optic sensors can also survive at much higher temperatures

than conventional pressure sensors.

The inherent advantages of fiber optic sensors which include their ability to be lightweight, of

very small size, passive, low power, resistant to electromagnetic interference, high sensitivity,

wide bandwidth and environmental ruggedness were heavily used to offset their major

disadvantages of high cost and unfamiliarity to the end user.

Optical fiber sensors offer attractive characteristics that make them very suitable and, in some

cases, the only viable sensing solution.

Fiber optic pressure sensors have been shown to possess advantages of high sensitivity, wide

bandwidth, high operation temperature, immunity to EMI, lightweight and long life.

4.6 Applications Of FOPS

Fiber optic sensors are being developed and used in two major ways. The first is as a direct

replacement for existing sensors where the fiber sensor offers significantly improved

performance, reliability, safety and/or cost advantages to the end user. The second area is the

development and deployment of fiber optic sensors in new market areas.

For the case of direct replacement, the inherent value of the fiber sensor, to the customer, has to

be sufficiently high to displace older technology. Because this often involves replacing

technology the customer is familiar with, the improvements must be substantial.

The most obvious example of a fiber optic sensor succeeding in this arena is the fiber optic gyro,

which is displacing both mechanical and ring laser gyros for medium accuracy devices. As this

technology matures it can be expected that the fiber gyro will dominate large segments of this

market.

Significant development efforts are underway in the United States in the a rea of fly-by-light

where conventional electronic sensor technology are targeted to be replaced by equivalent fiber

optic sensor technology that offers sensors with relative immunity to electromagnetic

interference, significant weight savings and safety improvements.

In manufacturing, fiber sensors are being developed to support process control. Oftentimes the

selling points for these sensors are improvements in environmental ruggedness and safety,

especially in areas where electrical discharges could be hazardous.



One other area where fiber optic sensors are being mass-produced is the field of medicine,

where they are being used to measure blood gas parameters and dosage levels. Because these

sensors are completely passive they pose no electrical shock threat to the patient and their

inherent safety has lead to a relatively rapid introduction.

The automotive industry, construction industry and other traditional users of sensors remain

relatively untouched by fiber sensors, mainly because of cost considerat ions. This can be

expected to change as the improvements in optoelectronics and fiber optic communications

continue to expand along with the continuing emergence of new fiber optic sensors.

New market areas present opportunities where equivalent sensors do not exist. New sensors,

once developed, will most likely have a large impact in these areas. A prime example of this is

in the area of fiber optic smart structures. Fiber optic sensors are being embedded into or

attached to materials (1) during the manufacturing process to enhance process control systems,

(2) to augment nondestructive evaluation once parts have been made, (3) to form health and

damage assessment systems once parts have been assembled into structures and (4) to enhance

control systems.



Chapter 5

Analytical Treatment of Three Layer Optical Fiber

We consider the meridional cross-section of a three- layer fiber. The outermost layer is

considered as the infinitely extended free-space having an RI of n3 = 1. RIs of the other different

layers are represented as n1 and n2 with n1 > n2. The analysis of the fiber structure essentially

needs the use of the cylindrical polar coordinate system ( , , z ); z-axis being the optical axis

of the fiber along which the propagation takes place. There are two interfaces in the fiber

separating the different regions, and the parametric boundaries of the different layers are

considered to be = 1 and = 2 with 1 < 2.

Solutions of the wave equation with cylindrical symmetry for axial components of the

electric/magnetic fields Ez and Hz are sought for the three regions, and then matched at the

interfaces for continuity conditions. The wave equation is

(2/2) + (/)/ + (2/2)/2 + q2 = 0 (1)

Where stands for either Ez or Hz, as the case may be. Also

q2 = 2 - β2

Where

is the angular frequency of the wave in the unbound medium,

β is the axial component of the propagation constant, and

and , respectively, are the permeability and permittivity of the medium.

5.1 Fiber without coating

For three- layer dielectric fibers, the electric/magnetic fields in the central core section can be

taken in the form of Bessel function Jʋ (·) of the first kind, whereas those in the inner clad can

be represented by the linear combination of Bessel functions of the first and the second kinds,

i.e., Jʋ(·) and Yʋ(·). In the outer clad section, field essentially has decaying character with

increasing radial parameter, and therefore, the most suitable solution in this region can be

represented by the modified Bessel function Kʋ(·) of the second kind. The axial components of

the electric/magnetic fields (i.e., Ez and Hz ) for the different regions of the fiber can be written

on the basis of these considerations.

Region I: core (0 <= ρ <= 1)

Ez1 = C1Jʋ(1ρ)ejʋ (2a)



Hz1 = C2Jʋ(1ρ)ejʋ (2b)

Region II: inner cladding (1 <= ρ <= 2)

Ez2 = C3Jʋ(2ρ) + C4Yʋ(2ρ)}ejʋ (3a)

Hz2 = C5Jʋ(2ρ) + C6Yʋ(2ρ)}ejʋ (3b)

Region III: outer cladding (ρ >= 2)

Ez3 = C7Kʋ(3ρ)ejʋ (4a)

Hz3 = C8Kʋ(3ρ)ejʋ (4b)

In Eqs. (2a) – (4b), C1– C8 represent unknown constants to be determined by the boundary

conditions. Also 1, 2 and 3 are the quantities corresponding to the different regions of the

fiber, and may be given as

12 = k1

2 - β2 = 21 - β2 (5b)

22 = β2 - k2

2 = β2 - 22 (5c)

32 = β2 – k3

2 = β2 - 23 (5d)

Where 1, 2 and 3 are the dielectric constants, and is the relative permeability of the medium.

Also ni = ( i)1/2, where

i is the relative dielectric permittivity of medium i,

β is the longitudinal component of the propagation constant.

Those axial components can be used to determine the transverse field components (i.e., E , H

and E, H) corresponding to the different regions of the fiber. These field components are not

explicitly stated in the text, but used to develop the equations that can be obtained after

implementing the continuity conditions at the layer interfaces. As there are two interfaces in the

fiber, there can be eight equations generated altogether after implementing the boundary

conditions at the layer interfaces with the parametric coordinates = a and = b (with a < b).

In order to simplify the situation, the outermost region is considered to be infinitely extended.

The form of this set of eight equations is as

A1Jʋ(1a) – A3Jʋ(2a)- A4Yʋ(2a) = 0 (6)

A2Jʋ(1a) – A5Jʋ(2a)- A6 Yʋ(2a) = 0 (7)



A1(βʋ/q12a) Jʋ(1a) + A2(j /q1

2) 1Jʋ’(1a) - A3(βʋ/q2

2a) Jʋ(2a)

– A4(βʋ/q22a) Yʋ(2a) - A5(j /q2

2) 2Jʋ’(2a) - A6(j /q2

2a) 2Yʋ’(2a) = 0 (8)

- A1(j /q12) 1Jʋ

’(1a) + A2(βʋ/q12a) Jʋ(1a) +A3(j/q2

2) 2 Jʋ’ (2a)

+ A4(j /q22) 2Yʋ

’(2a) - A5(βʋ /q22a) Jʋ’(2a) – A6(βʋ/q2

2a) Yʋ(2a) = 0 (9)

A3Jʋ(2b) + A4Yʋ(2b) - A7Kʋ(3b) = 0 (10)

A5Jʋ(2b) + A6Yʋ(2b) - A8Kʋ(3b) = 0 (11)

A3(βʋ/q22b) Jʋ(2b) + A4(βʋ/q2

2b) Yʋ(2b) + A5(j/q22) 2Jʋ

’ (2a)

+ A6(j/q22) 2Yʋ

’ (2b) – A7(βʋ/q3

2b) Kʋ(3b) - A8(j/q32) 3Kʋ

’ (3b) = 0 (12)

– A3(j/q22) 2 Jʋ

’ (2a) – A4(j/q2

2) 2 Yʋ’ (2b) + A5(βʋ/q2

2b) Jʋ(2b)

+ A6(βʋ/q22b)Yʋ(2b) + A7(j/q3

2) 3Kʋ’ (3b) – A8(βʋ/q3

2b) Kʋ(3b) = 0 (13)

In Eqs. (6) – (13), A1 - A8 are unknown constants to be determined by the boundary conditions.

Primes (‘) are first differentiation of the Bessel function. Also, 1, 2 and 3 are the quantities as

defined in Eq. (5). For Eqs. (6) – (13) to be consistent, the determinant (2) formed by the

coefficients A1 - A8 must vanish, i.e.,

2 = 0 (14)

2 is essentially a 8 × 8 determinant, the explicit form of which is not incorporated into the text.

Eq.(14) determines the dispersion relation for the three- layer dielectric fiber without Au-

coating, the solutions of which will provide the actual values of modal propagation constants

satisfied by the fiber. Once again the form of 2 is complex, and therefore, one may rewrite Eq.

(14) as

21 + j22 = 0 (15)



where 21 and 22 are, respectively, the real and the imaginary parts of 2. Obviously, the valid

propagation constants of the sustained modes in the fiber will be only those for which both 21

and 22 simultaneously vanish.



Chapter 6

Result

As already stated that present project will be completed by using Bessel function of first kind,

second kind and Hankel function. Bessel function of first kind and second kind has been studied

and implemented by MATLAB. It is now possible to interpret the wave equation (1) in

numerical terms. This will give us an insight into the model properties of our waveguides. By

putting the particular values of n1, n2 and λ into equation (14) , one can obtain different values of

2 for a large number of equispaced values of β, in the range of n1k > β > n2k, where k is the free

space propagation constant.

6.1 Bessel function of first kind (integer value)

Fig. 9 Bessel function of first kind (integer value 0, 1)



6.2 Bessel function of first kind (non-integer value)

Fig. 10 Bessel function of first kind (non- integer value 0.5, 1.5)



6.3 Bessel function of Second kind (integer value)

Fig. 11 Bessel function of second kind (integer value 0, 1)



6.4 Bessel function of second kind (non-integer value)

Fig. 12 Bessel function of second kind (non- integer value 0.5, 1.5)



Chapter 7

Summary and Future Work

7.1 Summary

Bessel Function of first and second kind has been studied and implemented by using MATLAB.

Characteristics of Bessel Function of first and second kind have been shown. It is now possible

to interpret the wave equation (1) in numerical terms. This will give us an insight into the model

properties of our waveguides. By putting the particular values of n1, n2 and λ into equation (14) ,

one can obtain different values of 2 for a large number of equispaced values of β, in the range of

n1k > β > n2k, where k is the free space propagation constant.

7.2 Future Work

Future work requires more complex steps, in future we will study and implement Fiber with Au-

coating in helical fashion, and thus design our main pressure sensor. So brief introduction for

future work has been given here

Fiber with Au-coating (Four-layer Fiber)

In the case of such a fiber, in the central core section, the solution can be taken in the form of

Bessel function Jʋ(·) of the first kind; ʋ representing the azimuthal periodicity, which can take

only discrete values. Essentially the symbol ʋ represents the mode index. In the outermost clad

region, the field has a decaying character as one moves away from the fiber axis, and

therefore, the solution can be best represented by the modified Bessel function Kʋ(·) of the

second kind. In the remaining two intermediate regions, the solutions must be formed by

linear combinations in the region next to the fiber core, by Bessel function of the first and the

second kinds, i.e., Jʋ(·) and Yʋ(·), and in the remaining region before the outermost clad, by

the modified Bessel function of the second kind and Hankel function, i.e., Kʋ(·) and Hʋ(1)(.).



References

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