Analytical and Numerical Analysis of Low Optical Overlap Mode Evanescent Wave

Chemical Sensors

A thesis presented to

the faculty of

the Russ College of Engineering and Technology of Ohio University

In partial fulfillment

of the requirements for the degree

Master of Science

Anupama Solam

August 2009

This thesis titled

Analytical and Numerical Analysis of Low Optical Overlap Mode Evanescent Wave

Chemical Sensors

ANUPAMA SOLAM

has been approved for

the School of Electrical Engineering and Computer Science

and the Russ College of Engineering and Technology by

Ralph D. Whaley, Jr.

Assistant Professor of Electrical Engineering and Computer Science

Dennis Irwin

Dean, Russ College of Engineering and Technology 3

ABSTRACT

SOLAM, ANUPAMA, M.S., August 2009, Electrical Engineering

Analytical and Numerical Analysis of Low Optical Overlap Mode Evanescent wave

Chemical Sensors (106 pp.)

Director ofThesis: Ralph D. Whaley, Jr

Demands for integrated optical (IO) sensors have tremendously increased over the years due to issues concerning environmental pollution and other biohazards. Thus, an integrated optical sensor with good detection scheme, sensitivity and low cost is needed.

This thesis proposes a novel evanescent wave chemical sensing (EWCS) technique for ammonia (gaseous) and nitrite (aqueous) detection utilizing a low optical overlap mode

(LOOM) waveguide structure. This design has the advantage of low modal fill factor and more field interaction with the sensing region compared to fiber sensors; hence eliminating the difficulty faced by traditional EWCS designs. Effective refractive index is a crucial parameter for analyzing the sensitivity, which is compared using the analytical and numerical methods such as finite element method (FEM) and semi-vectorial beam propagation method (BPM). The sensitivity of LOOM structures are calculated and compared analytically and numerically. Finally, these waveguide structures are analysed when integrated with a Mach-Zehnder Interferometer.

Approved: ______

Ralph D. Whaley, Jr.

Assistant Professor of Electrical Engineering and Computer Science

ACKNOWLEDGMENTS

It’s a wonderful opportunity to thank my advisor, Dr. Ralph Whaley, Jr. PhD. for his consistent support and guidance in every step for the completion of my thesis. I highly respect and appreciate his ideas and inputs, and learned a lot of things besides academics such as being systematic, polite, humble and enthusiastic. Secondly, I would like to thank my committee members Dr.Savas Kaya, Dr.Avinash Kodi and Dr.Karen Coschigano for their extended support. I am grateful to Dr.Gines Lifante, who had provided me literature help and replies to various questions through email and also helping me with the results for Semi-vectorial BPM simulations. I also thank the Rsoft research group, mainly Mr.

Matthew Frank, for troubleshooting various problems while working with the FEM software. My father Mr. Sharath Chander, my mother Mrs. Jhansi Rani and my sister

Anusha have always been encouraging and showed tremendous interest towards my study and research, which contributed to my thesis to a great extent. I thank my friend

Bhargav Kota, who has been motivating and always there for any kind of help and support. I would also like to thank my friends from India, Priyanka Kamath, Deepthi

Chandu, Lakshmi Iyer and Kriti Reddy who stood by me always. I thank my seniors at

Ohio University: Chandana Venkatayogi, Jyothsna Jakka, Vijaya Hari, Tanu Sharma,

Sulalita Chaki and Mohor Chatterjee for their encouragement and understanding. Krishna

Manoharan, one of our research members, helped a lot with his inputs and suggestions throughout the process. I thank him and Aarthi Srinivasan, for her warmth and help. I also thank my relatives Mr. P.Sri Kumar and Mrs. Shailaja for following me up all the while. Last, but not the least, I would like to thank our janitor, Mrs. Rose Mary.

TABLE OF CONTENTS

Page

ABSTRACT ...... 3

ACKNOWLEDGMENTS ...... 4

LIST OF TABLES ...... 9

LIST OF FIGURES ...... 10

CHAPTER 1: INTRODUCTION ...... 12

1.1 Overview of Optical Sensors ...... 14

1.2 Bio-Chemical Sensor ...... 16

1.3 Advantages of Planar Structures over Fibers for Optical Bio-sensing ...... 17

1.4 Importance and Phenomenon of Evanescent Wave Sensing ...... 18

1.5 Types of Evanescent Wave Sensors ...... 21

1.6 Working Principle of Integrated Optic (IO) Sensors ...... 23

1.7 Single Mode Operation ...... 25

1.8 LOOM Structure ...... 26

1.9 Analytes ...... 28

1.9.1 Gaseous Detection of Ammonia ...... 28

1.9.2 Aqueous Detection of Nitrites...... 28

1.10 Material Used ...... 29

1.10.1 Indium Phosphide (InP) ...... 29

1.10.2 Amorphous Zinc Oxide (a-ZnO) ...... 30 6

CHAPTER 2: ANALYTICAL AND NUMERICAL METHODS TO OBTAIN

EFFECTIVE INDEX ...... 31

2.1 Characteristics of Optical Waveguides ...... 31

2.2 Channel Waveguides ...... 33

2.3 Analytical Solution: Effective Index Method (EIM) ...... 36

2.3.1 Theoretical Calculation of Effective Index for a Symmetric Planar

Waveguide Structure ...... 40

2.3.2 Wave Equations of a planar waveguide ...... 42

2.3.3 Other Important Waveguide Parameters ...... 43

2.3.4 Mode Parameters ...... 44

2.3.5 Eigen Value Equation ...... 45

2.4 Effective Index Calculations using Finite Element Method (FEM) ...... 49

2.5 Effective Index Calculations Using Semi-Vectorial Beam Propagation

Method (BPM) ...... 53

2.6 Vertical Cut Profile: Electric Field ...... 53

2.7 Confinement Factor ...... 55

2.8 Modal Fill Factor ...... 56

2.9 Gaseous and Aqueous Detection ...... 57

2.10 Modal Analysis of the Buried Structure ...... 58

CHAPTER 3: SENSITIVITY ANALYSIS ...... 60

3.1 Analytical Method ...... 60

3.2 Numerical Method: Sensitivity Analysis using FEM Approach ...... 62 7

3.3 Sensitivity Analysis of Buried Waveguide Structure ...... 64

3.4 Surface Sensitivity of an Air LOOM Structure ...... 66

CHAPTER 4: MACH ZEHNDER INTERFEROMETER ANALYSIS ...... 67

4.1 Principle of Operation of an MZI ...... 68

4.2 Analysis of MZI ...... 69

CHAPTER 5: RESULTS AND DISCUSSION ...... 70

5.1 Analytical and Numerical Results for Effective Index Calculations ...... 70

5.2 Comparison of the Modal Fill Factor and Confinement Factor ...... 72

5.3 Sensitivity Analysis ...... 73

5.4 Mach-Zehnder Interferometer Output ...... 75

CHAPTER 6: CONCLUSION ...... 77

REFERENCES ...... 79

APPENDIX A.0 EIM CALCULATIONS FOR STRUCTURE 1(TE) ...... 86

APPENDIX A.1 EIM CALCULATIONS FOR FINAL STRCUTURE 1(TE) ...... 87

APPENDIX A.2 EIM CALCULATIONS FOR STRUCTURE 1 (TM) ...... 88

APPENDIX B.0 FEM SETTINGS ...... 89

APPENDIX C.0 CALCULATION OF ELECTRIC FIELD PLOTS ...... 90

APPENDIX D.0 SENSITIVITY CALCULATIONS FOR STRUCTURE 1(TE) ...... 91

APPENDIX D.1 SENSITIVITY CALCULATIONS FOR STRUCTURE 1(TM) ...... 92

APPENDIX D.2 MOST SETTINGS ...... 93

APPENDIX E.0 MZI OUTPUT FOR STRUCTURE 1 ...... 94

APPENDIX E.1 MZI OUTPUT FOR STRUCTURE 2 ...... 98 8

APPENDIX F.0 MFF AND CF CALCULATIONS FOR STRUCTURE 1 ...... 102

APPENDIX F.1 MFF AND CF CALCULATIONS FOR STRUCTURE 2 ...... 104

APPENDIX G.0 ABSORPTION SPECTRA OF GASEOUS AMMONIA ...... 106

LIST OF TABLES

Page

Table 1: Optical and geometrical properties of the LOOM based waveguide structures.47

Table 2: List of effective index values obtained...... 48

Table 3: Rf values for different cladding thicknesses...... 65

Table 4: Comparison of Neff between analytical and numerical methods...... 70

Table 5: Percentage of error calculated for the three methods...... 71

Table 6: Comparison of the MFF and CF...... 72

LIST OF FIGURES

Page

Figure 1: Layout of the Evanescent wave chemical sensors used...... 14

Figure 2: Schematic Figure of IO EWS...... 16

Figure 3: Operating principle of a biochemical sensor ...... 17

Figure 4: Classification of biochemical sensors ...... 17

Figure 5: Classification of optical sensors ...... 18

Figure 6: Single mode waveguide with evanescent wave penetration ...... 19

Figure 7: Penetration depth of an evanescent wave ...... 20

Figure 8: Three cases when the incidence angle of the light ...... 21

Figure 9: Shows types of EWS ...... 23

Figure 10: An integrated optic sensor with an adlayer ...... 24

Figure 11: Basic integrated sensor effects ...... 25

Figure 12: Dispersion curve ...... 26

Figure 13: The structure is a nanoscale membrane waveguide...... 27

Figure 14: The SEM shows an InP-based LOOM guide ...... 27

Figure 15: A CCD image of a guided LOOM mode...... 28

Figure 16: Fundamental structure of a dielectric waveguide ...... 32

Figure 17: Index profile distributions of step index and graded index waveguides ...... 33

Figure 18: Shows the 3 kinds of channel waveguides ...... 35

Figure 19: Rib waveguide and diffused waveguides structures ...... 36

Figure 20: Effective index approximation method...... 37 11

Figure 21: Flowchart of EIM ...... 38

Figure 22: Two LOOM based structures...... 40

Figure 23: Matlab output ...... 46

Figure 24: Figure showing mesh plot of structure 1...... 50

Figure 25: Figure showing the mode profile of structure 1 ...... 51

Figure 26: Vertical Electric field mode profile of structure 1...... 51

Figure 27: Horizontal electric filed mode profile of structure 1 ...... 52

Figure 28: Figure showing the index profiles for structure 1...... 52

Figure 29: showing the Hx and Ey field amplitude of an TM mode...... 55

Figure 30: shows the active region of a rib waveguide structure...... 56

Figure 31: Waveguide structures for gaseous and aqueous detection ...... 58

Figure 32: Mode plot and the vertical cut profile of the buried waveguide structure ...... 58

Figure 33: Figure showing the steps to obtain effective index ...... 62

Figure 34: MOST outputs for (a) TE and (b) TM, for structure 1 ...... 63

Figure 35: Absorbance versus Concentration plot for buried waveguide structure ...... 65

Figure 36: Absorbance versus Concentration for Air LOOM waveguide structure ...... 66

Figure 37: Schematic diagram of an IO MZI...... 68

Figure 38: Comparisons of the analytical and numerical methods for structure 1 ...... 74

Figure 39: Comparison of analytical and numerical methods for structure 2 ...... 75

Figure 40: Plot of power outputs P1 and P2 for structure1 at 1550nm...... 76

Figure 41: Plot of power outputs P1 and P2 for structure2 at 550nm ...... 76

CHAPTER 1: INTRODUCTION

In the recent years, optical sensors have attracted considerable attention, especially in the application of biochemical species detection. They have excellent advantages such as good compactness and robustness, immunity to electromagnetic interference, high sensitivity, shorter response time, low cost, and high compatibility with fiber optic networks [0]. However, optical fiber-based systems do not seem promising with respect to fabrication, efficiency and miniaturization. However, planar waveguide-based platforms employing evanescent wave sensing techniques have shown tremendous improvement [2] and evanescent wave sensors have proven to be highly sensitive [3-5].

Over a decade, biochemical sensors have shown profitable improvement, in various sensing applications and in industry, with a revenue report from high-tech market research firms indicating an increase in compound annual growth rate of about 11.5%, with the next generation biochemical sensors revenue rising from $2.3 billion in 2002 to nearly 4 billion in 2007 [6-7]. The applications are broad-reaching, including the areas such as environmental, chemical, biomedical, industrial and other process control fields

[8]. Added to this, it has been suggested that “modern optical sensors provide high- quality applications for monitoring environmental processes, process control, food analysis, medical diagnostics, investigation of the biosphere, drug detection and even extra-terrestrial research” [6, 9]. Recent rise in terrorist activities has also demanded for the need to detect biochemical explosives [6]. Irrespective of the application, realizing a highly sensitive, miniaturized, reduced cost biochemical sensor is desirable. With the

growing interest in the integration of waveguide sensors with existing electronic and 13

photonic platforms, comes the demand for high-performance sensing architectures that

can be directly incorporated into the standard CMOS process sequence.

Goals of this Thesis

The main aim of this thesis is to introduce the concept of Low Optical Overlap Mode

(LOOM) structure into integrated optical sensors incorporating an evanescent wave

chemical sensing technique. LOOM structures incorporating amorphous zinc oxide (a-

ZnO) material are analyzed using Finite Element Method (FEM). The structure utilizing

InP as the core material is a non-resonant structure as the wavelength of operation does not match the absorbance peak of the analyte, while the resonant structure with a-ZnO as

the core material works at the wavelength where the analytes show peak absorbance. The

critical parameter in assessing the sensitivity of the sensor is effective refractive index,

which is analyzed and compared using both analytical and numerical methods, FEM and

semi-vectorial BPM as previously mentioned. Modal analysis of LOOM structures using

FEM software are used to solve for the field components using a non-uniform mesh for

both guided and leaky modes. This thesis has two areas of sensing as shown in Figure 1:

gaseous and aqueous. An air LOOM and buried waveguide structures are designed for

detecting ammonia (gaseous) in air and nitrites (aqueous) in water. Sensitivity analysis is

performed and compared using the above mentioned methods. The other parameters such

as confinement factors and modal fill factors are also compared. Choice of the materials

used is important, and this is discussed in detail in the future. These LOOM structures

when integrated with Mach-Zehnder Interferometer architectures are also realized. 14

Evanescent wave sensor using LOOM mechanism

Gaseous ammonia Nitrite detection in detection water

Rib waveguide Rib waveguide structure structure Buried waveguide structure (InP) (a‐ZnO) (a‐SiC:H) non‐resonant resonant

Figi ure 1: Layout of the Evanescent wave chemical sensors used.

1.1 Overview of Optical Sensors

The current trends used for the development of optical sensors include miniaturization, low cost, compatibility with mass production processes, disposability besides multiple functionalities on a single platform and multiple analyte detection [2]. Interest and progress in the integrated optics sensors over the last two decades have tremendously increased due to the shortcomings of the conventional Electromagnetic Field (EMF) sensors, utilizing active metallic probes, thus producing noise. Some of the broad 15 scientific applications of EMF sensors are as process control, microwave-integrated circuit testing, electromagnetic compatibility measurements and electric field monitoring in medical apparatuses. Comparatively, photonic EMF sensors have very good galvanic insulation, high sensitivity and a very wide bandwidth. [10]. Optical sensors are widely divided based on the two segments, sensing mechanism and sensing architecture. The various sensing mechanisms are fluorescence, surface plasmon resonance, raman scattering, absorption change, photon migration spectroscopy and effective index change in the guiding structures which are discussed in detail in [1]. Based on the second category, they are further classified in terms of architecture as interferometer, anti resonant reflecting optical waveguides (ARROW), hollow waveguides, surface plasmon resonance, bragg gratings, silicon slot waveguides, integrated optical micro cavities and low optical overlap mode (LOOM) structures, which are also explained in detail in [1,

11]. Figure 2 shows a schematic overview of an integrated optical evanescent wave sensor (IO EWS).

Figure 2: Schematic Figure of IO EWS.

1.2 Bio-Chemical Sensor

Optical Bio-chemical sensing is an extremely wide field incorporating various devices

such as contact-less monitors, fiber-optic based devices, planar waveguides, evanescent

wave interrogation and interferometers. Some of the diverse applications in current

analytical chemistry are environmental monitoring, chemical process analysis, food

storage and manufacturing, clinical monitoring, space and aeronautics [12]. The bio-

sensing applications in the life sciences, such as binding and dissociation kinetics of antibodies and receptor-ligand pairs, protein-DNA and DNA-DNA interactions, epitome

mapping, phage display libraries and whole cell and virus protein interactions [13]. As

quoted in [1] “A biosensor is a device which consists of a biologically or biophysically-

derived sensing element integrated with a physical transducer that transforms a

measurand into an output signal.” Figure 3 shows the operating principle of a

biochemical sensor. In optical biosensors, there are two kinds of sensing mechanisms,

homogeneous sensing and surface sensing. The former relates to a change in the

refractive index of the cover medium due to the presence of analytes, and the latter is due

to an increase in the thickness of the adlayer deposited [1]. The adlayer refers to the

sensing layer which is deposited on top of the core layer to specifically bind the analytes.

It is used mainly to improve the specificity of the sensor. The two sensing mechanisms

are discussed in detail in later chapters.

Figure 3: Operating principle of a biochemical sensor [1].

The structure of a biosensor is shown in the Figure 4, which is based on [14]. They are classified based on different parameters such as nature of the sensor, monitored parameters, type of a transducer, operational mode, number of analytes, etc.

Figure 4: Classification of biochemical sensors.

1.3 Advantages of Planar Structures over Fibers for Optical Bio-sensing

In spite of the attractive advantages of optical fiber (OF) bio-chemical sensors, such as sensing in the inaccessible areas, transmitting optical signals over great distances with 18 low power loss, high immunity to electromagnetic disturbances, flexibility in their geometry, resistance to corrosion , electrically less passive, and better compatibility with telemetry; [15-17] they have a few disadvantages when compared to planar devices in terms of fabrication and mass production. In planar optical devices, deposition and patterning of reagents is easier, also different materials can be used. They have high precision as multiple devices can be fabricated on a single sheet. The devices are robust and hence have better flexibility. Planar devices are easily incorporated into optically associated instrumentation. Lastly, manufacturing equipment for optical planar devices is easily available [15]. Figure 5 shows the classification of optical sensors.

Figure 5: Classification of optical sensors.

1.4 Importance and Phenomenon of Evanescent Wave Sensing

In comparison to various other sensing methods, evanescent wave sensors (EWS) have various advantages, one of them being miniaturization, as the mode of operation is light, 19

and the sensor does not incorporate any bulky components. The interaction of the

evanescent field with the analyte forms the main concept of the sensing mechanism

which is explained in more detail in the forthcoming sections; and because the power in

the sensing region is high, these sensors have high sensitivity. They have immunity from

scattering effects and have better control in design besides reduction in cost [18].

Evanescent waves are those waves which penetrate into the cladding or the region where

light is reflected off the surface. In optical terminology, when total internal reflection

occurs at an angle greater than the critical angle, the sinusoidal waves reflect off from an

interface, and the waves which penetrate into, are called the evanescent waves [19].

Figure 6 shows the evanescent field decaying exponentially into the cover region.

Figure 6: Single mode waveguide with Evanescent wave Penetration.

The above mentioned mechanism is used in sensing, when the analytes or species to be detected are in contact or are made to interact with the evanescent field (which decay exponentially from the surface of the waveguide), forms the basis for evanescent wave 20 sensing. The operating wavelength is selected such that the analytes show peak absorption, so that there is a change at the output either in the intensity or power or any other parameter, when measured using a detector. In this thesis, the operating wavelength is chosen to be 1550nm and 550nm; while 1550nm is the non resonant wavelength which is purely simulated to compare and check with the previous literature; while for the

550nm; where many analytes show peak absorption as shown in Appendix G.0 is the resonant structure as mentioned earlier [20].

Figure 7 shows the penetration depth, when reflected off from a surface of higher index incident onto a lower index. The penetration depth (δ) is the exponential decay of the electric field, where N is the effective refractive index of the medium, and nc is the index of the cladding region, containing the analyte and is calculated using [21-22]

2 2 −1/ 2 (Equation 1.1) δ = ()λ / 2π [N − nC ]

Figure 7: Penetration depth of an Evanescent wave. 21

Figure 8 which is based on [22] shows the three cases when the incident light travelling in medium 1 with higher refractive index, strikes medium 2 having lower index. The three cases show the incident angle less than the critical angle, equal to critical angle and greater than critical angle; when the concept of evanescent wave arises, due to total internal reflection.

Figure 8: Three cases when the incidence angle of the light (a) less than critical angle (b) equal to critical angle and (c) greater than critical angle.

1.5 Types of Evanescent Wave Sensors

Evanescent wave sensors are those, in which the light of the evanescent wave interacts with the analyte, and this is further classified, based on the kind of interaction between the two [23]; and is as follows:

1. Open-Clad-Type Evanescent Wave Sensor

This type of a sensor typically has a three layer structure with the substrate,

guiding film; as well as the cladding, as shown in the Figure 9(a). It is called

Open clad type as the cladding acts as the absorptive medium.

2. Buffered-Clad-Type Evanescent Wave Sensor

This is a four layer type of a sensor as shown in the Figure 9(b). The cladding still

acts as the absorptive medium with the buffer layer as an addition to the previous

type of a sensor. Buffer layer is used to increase or decrease the sensitivity or

sometimes protects the guiding film from harmful substances [23].

3. Sensing-Layer-Type Evanescent Wave Sensor

This is also a four layer type of a sensor, where the cladding is no longer acting as

the absorptive medium, while the sensing layer does. Here, the analytes do not

absorb the working wavelength light directly and these sensors are the most

common type of Evanescent wave sensors [23].

Figure 9: (a) shows open clad type EWS, (b) shows buffered clad typpe EWS and (c) shows sensing layer type EWS [22]

1.6 Working Principle of Integrated Optic (IO) Sensors

Generally, efff ective refractive index (Neff) of a guided mode is considered to be the most crucial physical parameter under measurement. It is because for an IO sensor, a chemically selective layer or adlayer (as shown in Figure 10) based on [1], is deposited on the waveguide surface to bind the analyte (liquid or gaseous). Subsequently the evanescent field senses the changes in the index distribution caused by the layer, thus innducing an effective refractive index change [21, 24]. Figure 11, based on [21], depict a basic integrated sensor effect. Neff depends on the polarization (Transverse Electric, TE; 24 or Transverse Magnetic TM), mode number (m), wavelength (λ), other properties of the film, such as its refractive index, thickness and index of the substrate and cover.

The following effects cause effective refractive-index changes

1. Changes in the cover index containing the sample or anlayte.

2. Changes formed due to the formation (thickness) of an adlayer due to

adsorption or bound molecules formed from the bulk gaseous or liquid

samples.

3. Only in case of a micro porous waveguide, an effect called adsorption

or desorption of molecules causes changes in its refractive index, thus

causing changes to the effective refractive index. In this case, the

evanescent field is not responsible but only the field in the waveguide

[21].

When the above three effects together result in the effective index change, then

⎛ ∂N ⎞ ⎛ ∂N ⎞ ⎛ ∂N ⎞ (Equation 1.2) ΔN = ⎜ ⎟d + ⎜ ⎟Δn + ⎜ ⎟Δn ⎜ ∂d ⎟ F′ ⎜ ∂n ⎟ C ⎜ ∂n ⎟ F ⎝ F′ ⎠ ⎝ C ⎠ ⎝ F ⎠

Figure 10: An integrated optic sensor with an adlayer 25

Figure 11: Basic integrated sensor effects.

1.7 Single Mode Operation

The main reason for choosing the fundamental mode or single mode of operation is to avoid the modal dispersion caused in multimode waveguide structures [25]. The mode number m determines the number of modes, with m=0, corresponding to the fundamental mode and m=1, 2, 3…. corresponding to first, second, third and higher order modes. The figure below shows the modal dispersion curves taken from [26] for both transverse electric (TE) and transverse magnetic (TM) modes with core index being 3.38, cladding and substrate index being 3.17. It is a plot of the normalized parameters. As it can be seen for the figure that the first curve represents the fundamental mode (TE and TM), while the others correspond to the higher order modes. 26

Figure 12: Dispersion curve [24].

1.8 LOOM Structure

The difficulty faced by most evanescent wave sensor architectures lies in the lack of strong interaction between the analytes (target species) and the evanescent field. This is primarily due to the use of strongly confined waveguide designs [11]. The concept of

Low Optical Overlap Mode (LOOM) structures arises, where a nanoscale core region is used, which results in a large and distended mode [27].These structures have a very small percentage (usually less than 1%) of the optical mode confined in the guiding material.

This is discussed in detail in the future chapters. These large modes are sufficient to provide a strong interaction with the analytes, as the field is pushed out of the core region, enhancing the field interaction with the analytes. All the upper and lower cladding 27 layers are air, while the core is a dielectric or semiconductor depending on the application. Figures 13, 14 and 15 show the LOOM structure, SEM of the structure and

CCD image of the LOOM taken from [11].

Figure 13: The structure is a nanoscale membrane waveguide where the guided optical mode is represented by the dashed circle lines [11].

Figure 14: The SEM shows an InP-based LOOM guide indicating (a) membrane, (b) rib, (c) effluent vias, and (d) MEMS release V-groove [11].

Figure 15: A CCD imaage of a guided LOOM mode with dashed lines indicating V- groove position [11].

1.9 Analytes

1.9.1 Gaseous Detection of Ammonia

Ammonia is one of the most harmful gaseous pollutants present in the atmosphere. It is corrosive in nature, colorless and has a strong odor. Ammonia is present in air, wateer and soil naturally. The main source of ammonia entering the human body is via breathing.

The Environmental Protection Agency (EPA) reports show that exposure to excess amounts of ammonia cause irritation and burns to eyes, mouth, skin and lungs and sometimes may cause permanent blindness, lung disease or even death. The maximum permissible level of ammonia in air is 50ppm, above which it causes irritation and bad odor. The absorption spectra of gaseous ammonia are shown in Appendix G.0. Thus, detection of ammonia in air is important for human safety [28].

1.9.2 Aqueous Detection of Nitrites

One of the harmful agents present in water is nitrogen compounds, which are present in the form of nitrites, nitrates and ammonia [29]. Nitrites and nitrates are compounds of 29 nitrogen atom that contain two oxygen atoms attached to the nitrogen, while for nitrates; they are joined to three oxygen atoms. Nitrites and nitrates are readily converted into each other. Runoff from the golf courses, home lawns, gardens, farms, and septic systems, excess usage of fertilizers and bad management of animal feedlots are some of the sources for nitrate content in water. Division of Environmental and Occupational

Health Services Consumer and Environmental Health Services report that the maximum contaminant levels of nitrates can be 10 ppm, nitrites can be 1 ppm, while both combined can be 10 ppm in water. Excess nitrogen compounds in human body react with hemoglobin in the blood and this reduces its ability to carry oxygen to various parts of the body. It even causes a disease called blue baby syndrome in infants. Also, it is found that they cause cancer in animals [30]. Thus determining the nitrite/ nitrate content in water is important to monitor the pollution levels.

1.10 Material Used

1.10.1 Indium Phosphide (InP)

Indium Phosphide is applicable only at a wavelength of 1.55µm and not below, as this is very lossy at lower wavelengths. Since, the waveguide absorption loss is the product of the modal fill factor and the material absorption, a lower waveguide loss is possible, for a

LOOM structure with high quality core material [11]. Compared to most frequently used semiconductors such as silicon and gallium arsenide, for high power and high frequency electronic applications, InP has an edge due to its high electron velocity. It is used widely in the integration of active devices (lasers), as it has a direct band gap. 30

1.10.2 Amorphous Zinc Oxide (a-ZnO)

Amorphous Zinc Oxide (a-ZnO) has received popularity in the recent past due to its physical and chemical properties and wide range of applications. It is a wide band gap material of 3.37 eV at room temperature; which makes it a promising candidate for short wavelength emitters [31-32] besides having higher breakdown voltages, ability to sustain high electric fields, high-power operation ability and reduced electronic noise [32]. It has excellent chemical and thermal stability, has electrical and optoelectronic properties for solar cells [33], and has high exciton binding energy which enables semiconductor lasers with reduced thresholds [34]. It is observed that this material is lossless at a wavelength around 550 nm [35], because of the wide band gap, while most other materials are lossy around that wavelength and thus it is used as a waveguide at lower wavelengths. Since most of the analytes as discussed in the above section have high absorption at lower wavelengths, using this material proves to improve the sensitivity of the sensor device.

Since the absorption of light by the anlaytes determines the sensor sensitivity, i.e by reducing the output at the detector, which detects the amount of anlaytes present, thus, it is demanded that the material used is lossless at that wavelength. Thus, choosing a-ZnO for EWS waveguide technique proves to be advantageous.

CHAPTER 2: ANALYTICAL AND NUMERICAL METHODS TO OBTAIN

EFFECTIVE INDEX

Effective indices of optical waveguides are often calculated using a wide variety of methods, both analytical and numerical. Analytical methods include the Effective Index method and Marcitili method which are simple and often yield an accurate result; while numerical methods (such as Beam Propagation methods (BPM) and Finite Element

Method (FEM)) solve for much more complex waveguide structures, which incorporates a numerical technique such as Fast Fourier Transform, Finite Difference and Finite

Element Methods [37].

The main aim of this chapter is to calculate the effective index using the analytical method: “Effective Index Method” and the numerical methods: “Finite Element Method” and “Semi-vectorial Beam propagation Method”. Using the values obtained from the effective index method, the vertical cut profile for the electric field is plotted analytically and compared with the other methods. Modal analyses of the structures are performed analytically and critical parameters such as modal fill factor and confinement factor are determined in this chapter.

2.1 Characteristics of Optical Waveguides

In general the fundamental structure of a dielectric waveguide consists of a high index core region, also the film, surrounded by a lower index cladding region, the bottom cladding, (sometimes called the substrate) and the upper cladding (cover) as shown in

Figure 16. The propagation of a guided optical wave is in the longitudinal direction, parallel to the planar structure. There are two kinds of waveguides, planar optical 32 waveguides and non planar optical waveguides. They are divided based on the feature of optical confinement, which is the interaction between the cladding region and the core. In the planar waveguides, the optical confinement is restricted to one direction, while for the non planar waveguides; it is surrounded by all directions as shown in the Figure 16(a) and 16(b) [38].

(a) 1D (b) 2D

Figure 16: Fundamental structure of a dielectric waveeguide and (a) 1D planar Optical waveguide and (b) 2D non planar optical waveguuide.

Further, the planar waveguides are divided based on the refractive index distribution, as step index and graded index waveguides. Step Index waveguides have constant refractive index in the core, sandwiched between the substrate and the cover. They have an abrupt index difference between the core and the cladding regions. Frequently, the cover is air, 33 with an index of 1. When the cover index and the substrate index are the same, the waveguide is said to be Symmetric, if different, the term Asymmetric is used [38].

Graded index waveguides have the refractive index dependent upon the deposition profile

[39]. Figure 17 shows the index profiles of a step and graded index waveguide.

Figure 17: Index profile distributions of step index and graded index waveguides.

2.2 Channel Waveguides

Channel waveguides are an important group of a non planar waveguide. The three main types are the Buried waveguides, Stripe waveguides and the Rib waveguides [38].

2.2.1 Buried Channel Waveguides

This type of waveguide is formed by inducing a high index core region, which is buried in the lower index substrate region. The typical cross section geometry of the core is rectangular, though it can have any shape. The Figure 18(a) shows a buried channel waveguide with a rectangular core, having a width ‘w’ and a thickness ‘d’ [38]. It is 34 difficult to grow this type of a waveguide structure and requires re-growth, the advantage of using the buried channel waveguides is its high confinement factor.

2.2.2 Stripe-Loaded Waveguides

This type of a waveguide includes a strip on top of a core with an index of n1 greater than n3, that of the strip. The waveguide core beneath the loading strip has a thickness ‘d’, and the width ‘w’, described by the loading strip as shown in Figure 18(b) [39]

2.2.3 Ridge Waveguides

A ridge waveguide is similar to the stripe waveguide, except that the ridge on top is the core, with its index higher than the surrounding regions, thus has better optical confinement. Here, the thickness of the core is‘d’, that of the ridge and the width is ‘w’ as shown in the Figure 18(c). [38]

18(a) 18(b)

18(c)

Figure 18: showing the 3 kinds of channel waveguides, 18(a) buried waveguide, 18(b) Stripe waveguide and 18(c) ridge waveguide

The other types of the Channel waveguides are described as below

2.2.4 Rib Waveguides

This has a structure similar to strip or ridge waveguide, but the only difference is that the strip/ridge together with its lower surface constitutes the core region. The thickness and width being ‘d’ and ‘w’, defined by the rib, as shown in the Figure 19(a) [38]. These kinds of waveguides are modeled in this thesis.

2.2.5 Diffused Waveguides

This kind of waveguide is fabricated when a high index core region is grown in the substrate with a process of diffusion. Due to the diffusion process, the boundaries are not sharp, but curved as shown in the Figure 19(b). It has a thickness‘d’ and a width ‘w’ [38].

(a) (b)

Figure 19: Rib waveguide and diffused waveguides structures as in (a) and (b).

2.3 Analytical Solution: Effective Index Method (EIM)

EIM is one of the simplest approximate methods to find the effective indices in channel waveguides with arbitrary index profiles and waveguide geometries [38]. Since it is difficult to analyze the channel waveguides (rib waveguides) as the optical confinement is dependent on two directions, it is hence necessary to understand the physics associated with the structures to analyze the index distribution and hence, the analytical approach becomes critical. The basic concept is to convert a channel waveguide into two planar waveguides, treated independently to find the effective index of the propagation mode for each, as shown in the Figure 20.

Figure 20: Effective index approximation method, showing two planar waveguides in two different directions.

The EIM solves for two planar waveguides, described in two directions: x and y, while planar waveguide 1 has the light confined in x direction, while the planar waveguide 2 has it in the y direction. EIM is a good approximation, if it satisfies the following conditions [38].

1. The width of the waveguide is larger than its thickness, w>d

2. The wave guiding along its width is not stronger than its perpendicular or in its

thickness direction.

EIM can be applied to Step index and Graded index waveguides, and also to all of the non planar waveguides described before (if the two conditions mentioned above are satisfied) [38].

The technique of EIM is to solve for the propagation constants of the planar waveguides, and then obtain the effective indices; Figure 21 shows the flowchart to obtain the effective index, using EIM. Since effective index is mode dependent, we must specify the 38 mode (number of the mode) we are looking for. In this technique we assume that when the electric field is in the y direction or parallel to the planar surface, it is Transverse

Electric (TE)-like mode, while when it is in x direction or perpendicular to the planar surface, then it is Transverse Magnetic (TM)-like mode [38].

Figure 21: Flowchart of EIM.

A LOOM based rib waveguide structure is considered, with its core being InP, (n1=3.17) at a wavelength of 1550nm, substrate and cover being air, (n2,3=1) thickness of the membrane is 50nm, and the rib thickness is 50nm, and the width ‘a’ of the rib is 3µm, as shown in Figure 15. This structure is divided into three regions, Region 1, Region 2 and

Region 3 as in Figure 20. Each of these regions are treated as a symmetric slab planar waveguide structure with a propagation constant β and hence the effective index Neff, 39

(Neff = λβ/2π) are calculated. Since the effective index is mode dependent as mentioned before, the x-dependence of the y component for the TE like mode, or the electric field,

Êm, y (x), and the magnetic field Ĥm, y (x) for TM like mode is calculated [38]. Here, ‘m’ corresponds to a specific mode number. The above mentioned procedure is used to calculate β1, β2 and β3, for the three regions and hence the corresponding Neff1, Neff2 and

Neff3 are calculated as shown in Table 2 for the resonant and non-resonant structures.

Region 1 consists of a core index n1 = 3.17 and n2 = n3 = 1, with a width of 100nm, while regions 2 and 3, have the similar index distribution, but the width is 50nm each. For the structure proposed here, β2 = β3. After obtaining the three effective indices, we treat the structure with n1=Neff1 and n2 = n3 = Neff2 or Neff3. Now, we apply the above procedure again to calculate the propagation constant for the final structure, having a y-dependence on the y-component for the TE like mode or the electric field Ên, y(y) and the magnetic field Ĥn, y(y) for TM like mode. Firstly, the three regions are analyzed in the x-direction

(vertical), the corresponding effective indices are obtained and those effective indices are considered to get a final structure in the y-direction (horizontal), with a width of 3µm.

This final structure is again a three slab symmetric waveguide as mentioned before, but for this final structure, to obtain a TE- like mode, we consider the TM field of the vertical structure; and for TM-like mode, it’s the TE field of the initial vertical structure [38]. The two LOOM waveguide structures, with structure 1 having a core of InP and structure 2 having a-ZnO are as shown in Figure 22.

Figure 22: Two LOOM based structures, with structure 1 having InP as the core and structure 2 having a-ZnO as core material and the other dimensions are as shown above.

2.3.1 Theoretical Calculation of Effective Index for a Symmetric Planar Waveguide

Structure

From [38], we understand that “a waveguide mode is a transverse field pattern whose amplitude and polarization profile remains constant along the longitudinal direction”.

The electric and magnetic fields of a mode can be written as

ˆ (Equation 2.1) Eυ (r,t) = Eυ (x, y)exp(iβυ z −iωt)

ˆ (Equation 2.2) Hυ (r,t) = Hυ (x, y)exp(iβυ z −iωt)

ˆ ˆ Where, ‘υ ’ is the mode index, Eυ (x, y) and Hυ (x, y) are the mode field patterns and βυ is the propagation constant of the mode [37].

Using the basic Maxwell Equations, we obtain for a linearly, isotropic dielectric waveguides [38-39]. 41

δH (Equation 2.3) ∇× E = −μ 0 δt

δE (Equation 2.4) ∇ × H = ε δt

Since, the optical fields in the waveguides are in the form of equations (2.1) and (2.2), the above Maxwell equations (2.3) and (2.4) are simplified as following

∂Eˆ (Equation 2.5) z − iβEˆ = iωμ Hˆ ∂y y 0 x

∂Eˆ (Equation 2.6) iβEˆ − z = iωμ Hˆ x ∂x 0 y

∂Eˆ ∂Eˆ (Equation 2.7) y − x = iωμ Hˆ ∂x ∂y 0 z

∂Hˆ (Equation 2.8) z − iβHˆ = iωεEˆ ∂y y x

∂Hˆ (Equation 2.9) iβHˆ − z = iωεEˆ x ∂x y

∂Hˆ ∂Hˆ (Equation 2.10) y − x = iωεEˆ ∂x ∂y z

From the above equations, the transverse components of the electric field and magnetic fields are expressed in terms of the longitudinal components [38] as:

∂Eˆ ∂Hˆ (Equation 2.11) ()k 2 − β 2 Eˆ = iβ z + iωμ z x ∂x 0 ∂y

∂Eˆ ∂Hˆ (Equation 2.12) ()k 2 − β 2 Eˆ = iβ z + iωμ z y ∂y 0 ∂x 42

∂Hˆ ∂Eˆ (Equation 2.13) ()k 2 − β 2 Hˆ = iβ z + iωε z x ∂x ∂y

∂Hˆ ∂Eˆ (Equation 2.14) ()k 2 − β 2 Hˆ = iβ z + iωε z y ∂y ∂x

Where, k 2 = ω 2μ ε (x, y) (Equation 2.15) 0

Thus, once the longitudinal components Ez and Hz are known, the other field components

Ex, Ey, Hx, and Hy can be calculated.

2.3.2 Wave Equations of a planar waveguide

The two modes of a planar, 3-slab waveguide are either TE or TM. The homogeneous wave equations of a planar waveguide with an index distribution n(x) and is independent of the y-axis [36] is as follows:

TE Mode:

Using equations (2.11)–(2.14), for any TE mode we have Ez=Ex=Hy=0 and

∂Hz/∂y=0.Thus, the only non zero components left are Hx, Ey and Hz [36] .The wave equation for Ey can be derived using other field components as,

∂ 2 Eˆ (Equation 2.16) y + ()k 2 − β 2 Eˆ = 0 ∂x2 y

ω 2 (Equation 2.17) where, k 2 = ω 2μ ε (x) = n2 (x) 0 c2

And, the other components are obtained in terms of Ey as follows

ˆ β ˆ (Equation 2.18) H x = − E y ωμ 0 43

1 ∂Eˆ (Equation 2.19) Hˆ = y z iωμ ∂x 0

TM Mode:

ˆ In case of a TM mode, Hz=Hx=Ey=0 as ∂ Ez / ∂y =0. Thus, the non zero components are

Ex, Hy, Ez The wave equation for Hy can be derived using other field components [38] as

∂ 2 Hˆ 1 dε ∂Hˆ (Equation 2.20) y + ()k 2 − β 2 Hˆ = y ∂x 2 y ε dx ∂x

Where, k 2 is the same as defined in equation (2.17).

β (Equation 2.21) Eˆ = − Hˆ x ωε y

1 ∂Hˆ (Equation 2.22) Eˆ = y z iωε ∂x

As mentioned previously, the structure is broken down into 3 regions, with each region being a symmetric waveguide structure. In the analysis below, we can see how the effective index of region 1 is solved.

2.3.3 Other Important Waveguide Parameters

Asymmetry Factor: This is the measure of the asymmetry of the waveguide meaning that the cladding regions are different, which depends on the polarization of the mode [39].

2 2 n2 − n3 (Equation 2.23a) aE = 2 2 For TE modes n1 − n2

4 2 2 n1 n2 − n3 (Equation 2.23b) aM = 4 2 2 For TM modes n3 n1 − n2 44

Where, a = 0 for symmetric waveguides as, n2 = n3.

Normalized Waveguide Thickness or the V-number: It is a dimensionless normalized parameter of a waveguide and determines the number of modes supported and is defined as

2π (Equation 2.24) V = d n2 − n2 λ 1 2

Where k is the propagation constant,

2π (Equation 2.25) k = λ

2.3.4 Mode Parameters

A guided mode is one in which the power is retained inside the core as the wave, is perfectly reflected off both the interfaces. This mode can exist only if it satisfies a transverse resonance condition, such that the reflected wave has constructive interference with itself. The transverse component (x-component) of the wave vector inside the core is h1 = k1cosθ, where θ is the angle of incidence and the longitudinal component (z- component) is β = k1sinθ. Similarly, the transverse components for the substrate and cover regions can be defined as h2 = k2cosθ, and h3=k3sinθ, where k1, k2 and k3 are the propagation constants in the respective regions [38], defined as

2πni (Equation 2.26) ki = ,i = 1,2,3 λ

Applying boundary conditions at the n1/n2 and n1/n3 interfaces, the above can be rewritten as, 45

2 2 2 (Equation 2.27) k − β = h1

2 2 2 (Equation 2.28) β − k2 = γ 2

2 2 2 (Equation 2.29) β − k3 = γ 3

But, equations 2.28 and 2.29 are imaginary as β>k2>k3, and the guided mode fields decay exponentially in the transverse directions of the substrate and the cover regions, thus introducing new parameters called decay parameters as γ2=|h2| and γ3=|h3| [38].

The only parameter to be determined is β, which specifies the waveguide mode as the transverse field parameters h1, γ2 or h2 and γ3 or h3 are characterized by k1, k2 and k3, which are well determined. Another interesting feature relating mode number and the evanescent field is that, as the mode number increases, the penetration of the wave corresponding to that mode deepens. This phenomenon occurs because as there is an increase in the mode number, the propagation constant decreases, thus lowering γ3 and hence, enhances the penetration depth [39].

2.3.5 Eigen Value Equation

Equation for TE [38]

⎛ h d mπ ⎞ V 2 − h2d 2 (Equation 2.30) tan⎜ 1 − ⎟ = 1 , m= 0, 1, 2.. ⎝ 2 2 ⎠ h1d

Equation for TM [38]

2 2 2 2 ⎛ h1d mπ ⎞ n1 V − h1 d (Equation 2.31) tan⎜ − ⎟ = 2 , m= 0, 1, 2,.. ⎝ 2 2 ⎠ n2 h1d

Where, ‘m’ determines the mode number. 46

The equations above mentioned are solved using Matlab, graphically by plotting left hand side and right hand side, as a function of (h1d). The solutions give the value of β, and hence the effective index, Neff, could be calculated from the relation Neff = λβ/2π,

as discussed before.

Figure 23 shows the graphical solution obtained by plotting the left and right hand side of the eigen value equation, for a TM mode, which determines h1d for structure1.

Figure 23: Matlab output showing a plot of left and right hand side of the Eigen value equation for structure1(TM).

After obtaining the neff for structure1 as shown in Appendix A.0, which is for region 1 in

Figure 20, a similar analysis for region 2 and region 3 is performed as in Appendix A.0, but the value of d is changed to 50nm from 100nm. Also, for this structure region 2 and region 3 are identical to each other. After obtaining the effective indices for the 3 regions, 47 as previously mentioned, the structure is analyzed as a 3 slab waveguide, with a width of

3µm, as shown in Figure 20.

Again, a similar analysis is done to obtain the final effective index of a quasi-TE mode, for the entire structure as in Appendix A.1. Similarly, calculations are performed for a

TM mode as in Appendix A.2, in the similar way as done for the TE modal analysis.

Table 1 shows the optical and geometrical properties of the LOOM based waveguide structures

Table 1: Optical and geometrical properties of the LOOM based waveguide structures.

Structure 1 Structure 2 Structure 3

Operating wavelen nm) 1550 1550 550

Substrate index ns 1 1 1

Core index nf 3.17 1.79 1.79

Cover index nc 1 1 1 thickness h1(nm) 100 100 100

Thickness h2 (nm) 50 50 50

Width (nm) 3000 2000 2000

Table 2 shows the effective indices obtained for two structures, using InP and a-ZnO, both at 1550nm and a-ZnO at 550nm when solved in the x-direction. 48

Table 2: List of effective index values obtained.

Structure Effective index Region II Region I Region III Region I (100 nm Regions II, III (50 nm

TE mode TM mode TE mode TM mode

1.8114 1.005 1.3204 1

1.071 1.009 1.0746 1.005

1.357 1.103 1.152 1.0234

The results obtained for the final structures, in the y-direction are discussed in the chapter

5 “Results and Discussion”. The numerical methods are discussed as below 49

2.4 Effective Index Calculations using Finite Element Method (FEM)

This is the second method based on Computer Aided Design (CAD) employed for the effective index calculations. FemSIM is a product or simulation tool of Rsoft Design

Group based on the Finite Element Method. It is a generalized mode solver that calculates a transverse or a cavity mode on a non-uniform mesh, for any arbitrary structure. Some of the other waveguide related applications of finite element method are plasmonics, surface plasmon resonance, microwave crystal waveguides, photonic crystal waveguides, optical fibers and waveguides [40]. FemSIM divides the entire waveguide into small rectangular blocks or mesh, where boundary conditions are applied and

Maxwell’s equations are solved for each block [35]. It calculates for the effective index for each mode specified, and filters the TE and TM modes accordingly.

For TE modes, the symmetry options in the mesh options are made to be symmetric, and the modes are sorted with respect to highest effective index in the mode options, while for TM modes, it is set to anti-symmetric in mesh options and the modes are sorted from lowest losses. By incorporating these settings in the software, we can filter both TE and

TM modes, for the structures under examination.

The simulation results obtained from the FEM are shown in the Figures 24-28. Figure 24 represents the mesh plot of the rib waveguide structure, which is divided into rectangular blocks and Maxwell’s equations are solved across each box along the boundaries. Figure

25 shows the mode output plot, in terms of the intensity, where scale on the right side of the Figure shows the color having maximum and minimum intensity. This gives the 50 effective index of the waveguide structure. Figures 26 and 27 show the vertical and horizontal electric field mode profiles, cut at x= -0.0036 and y= -0.013 respectively.

Figure 28 shows the contour map of transverse index profile with the core having an index of 3.17 and the other region being air [41].

Figure 24: Figure showing mesh plot of structure 1. 51

Figure 25: Figure showing the mode profile of structure 1.

Figure 26: Vertical Electric field mode profile of structure 1. 52

Figure 27: Horizontal electric filed mode profile of structure 1.

Figure 28: Figure showing the index profiles for structure 1. 53

Appendix B.0 gives the details of the FEM settings. The results obtained are compared to the effective index method in the chapter 5 Results and Discussion.

2.5 Effective Index Calculations Using Semi-Vectorial Beam Propagation Method

(BPM)

Beam Propagation Method (BPM) is the most common tool used for the study of optical devices, due to its simplicity and high speed of performance [42]. It solves for the transverse electric fields considering the boundary conditions, and also takes the polarization property of the propagating electromagnetic waves into account and is based on finite difference scheme [43]. Conventional BPM uses fast fourier transform (FFT) technique which has a limitation as the boundary conditions are ignored, while the recently developed finite difference (FD) techniques incorporate the vector boundary conditions. Neglecting the vector boundary conditions has an adverse affect, as the discontinuity at the index interfaces having large differences, affects the transverse component of the electric field [42]. The simulations and results using this method are performed and calculated by Dr. Gines Lifante, professor at Universidad Autonoma de

Madrid, Spain using finite difference BPM algorithms. The result obtained using this method is compared with the above methods in the results and discussion chapter.

2.6 Vertical Cut Profile: Electric Field

Modal Analysis of the Air LOOM structures is performed using the analytical method.

Firstly, the electric field is plotted analytically as shown in Appendix C.0. From the effective index method, the effective indices values are obtained, and plugging in these 54 values, the electric field equations are plotted. The propagation constant β can be obtained as Neff is known, and then γ2, γ3 and k are calculated using equations 2.29-2.31, and these are substituted in the field equations as shown below [39]:

-γ x Ae2 x 0

(Equation 2.32) Ey(x) = A( – ) 0 < x < -d

γ (x+d) A( + )e 3 x

Where A is a constant parameter which is related to the energy carried by the mode [39], and is taken as 1 for convenience here. For a confined mode, the electric field decays exponentially in the cover and the substrate as predicted and is sinusoidal in the core region. The above equations are defined for the guided TE modes, while for the guided

TM modes, it is as below

0 cos sin 0 Equation 2.33 cos sin

The above equations are the solutions of magnetic fields associated with TM polarized modes [39]. Similar to the TE modes, the solution to the TM modes also decays exponentially in the cladding and substrate regions, while it varies sinusoidally in the core region. Solving for the electric field, using the above field equations, we obtain the following results as shown in Figure 29.

Figure 29: showing the Hx and Ey field amplitude of an TM mode, with n= 1.00155.

The plots above show a maximum amplitude of 1, while the Ey plot shows a different behavior at the centre (x=0), it is discontinuous across the index interface and drops to a value close to 0, while the Hx field is continuous across the same index interface, but its derivative would be discontinuous, which is in accordance to the Maxwell’s equations.

From the plots obtained, the modal fill factor is calculated graphically which is discussed in the chapter 5 “Results and Discussion”.

2.7 Confinement Factor

The confinement factor is defined to be the ratio of the power confined in the core region to the total power. This factor determines the amount of power present in the core region

[38].

d / 2 d / 2 E(y)H (x)dydx Pcore ∫∫−−d / 2 d / 2 (Equation 2.34) Γ = = ∞ ∞ Ptotal E(y)H (x)dydx ∫∫−∞ −∞

When more power is confined in the core region, then there is less power in the sensing regions and hence, interaction between the modes and the analytes is reduced. Thus we 56 aim for less confined structures for our requirement, i.e for an evanescent wave IO sensor. Figure 30 shows the active region where, the ratio of the power in the active region to the total power determines the confinement factor. The confinement factor is calculated analytically as well as using Semi-Vectorial Imaginary BPM. The results are as discussed later in chapter 5 “Results and discussion” and the Matlab files associated with the calculation of the confinement factor and modal fill factor are in Appendix F.0,

F.1 for the respective structures.

Figure 30: shows the active region of a rib waveguide structure.

2.8 Modal Fill Factor

The modal fill factor (MFF) is defined as the ratio of the electric field in the core region to the electric field in the entire region. Modal fill factor is crucial in determining the amount of field present for the interaction between the guided mode and the analytes. As for the requirement, we need lower value of the modal fill factor in the core; since when more of the mode is in the cover region, there is a better scope for the specie to interact with the mode and hence increase in the sensitivity as mentioned in the confinement factor concept. Modal fill factor is a crucial parameter in evanescent wave sensing 57 mechanism as it is the evanescent field which couples or interacts with the analytes, when compared to the confinement factor, which is associated with the power.

d / 2 ∫ E(y)dy −d / 2 (Equation 2.35) MFF = ∞ ∫ E(y)dy −∞

The MFF for a LOOM structure is almost less than 1%, and thus has better interaction with the analytes compared to other sensing types. As mentioned MFF is very important and is different to the CF, for the LOOM structures. As the electric field component (Ey) obtained is a smaller value (as more field is pushed outside the core) than the other component Hx, while the CF which is the cross product of the two components Ey and Hx, is thus different. This is discussed further in Chapter 5.

2.9 Gaseous and Aqueous Detection

Figure 31 shows gaseous (ammonia) sensing and aqueous (nitrites) sensing waveguide structures with core being a-ZnO for the former and a-Si:C:H buried in a-Zno for the latter. The reason for choosing a-Si:C:H material is to have an ideal index contrast between the a-ZnO and the core material, which is responsible for the propagation of the mode [27]. It has low loss at the operating wavelength which is 550nm and the index is greater than the index of a-ZnO, thus has good index contrast.

Figuure 31: Waveguide structures for gaseous and aqueous detection.

2.10 Modal Analysis of the Buried Structure

Figure 32: Mode plot and the vertical cut profile of the buried waveguide structure.

Figure 32 depicts the mode profile of a buried structure obtained using FEM, with an effective index of 1.789 for mode 0. The scale shows the intensity variations, with the color pink representing least intensity and red for maximum intensity. The Figure on the right shows the vertical cut profile, which shows the field discontinuity across the index 59 interfaces, which is in accordance to the Maxwell’s equations [27]. This buried structure with a-ZnO is a novel waveguide structure proposed to detect nitrites in water. The sensitivity (surface) analysis of the buried structure is discussed in chapter 3, for varying the top clad thickness.

CHAPTER 3: SENSITIVITY ANALYSIS

In this chapter, sensitivity (homogeneous) analysis is performed and compared for the previously discussed LOOM based waveguide structures using both analytical method and numerical methods (FEM). Also, the sensitivity (surface) analysis for a buried structure, for aqueous sensing, is performed. For gaseous sensing, InP (structure 1) and a-

ZnO (structure 3) are used as the core materials. Below is the explanation of the analytical method approach.

3.1 Analytical Method

This method is used to obtain the sensitivity for the LOOM based waveguides which is based on the EIM calculations [44]. As already mentioned, a change in the chemical/physical quantity either due to homogeneous sensing or surface sensing, affects the total effective index of the guided mode, which is used in the EIM analysis.

From [45], sensitivity is defined “as the rate of change of the modal effective permittivity, relative to the cover permittivity,” where ε=n2. However, in case of optical detection, we are more interested in the phase of a light wave than dielectric constant.

Hence the sensor sensitivity is redefined as the rate of change of effective index relative to its cover index change, in case of homogeneous sensing (as shown in equation 3.1) and in case of surface sensing, it’s relative to the change in the film thickness. The homogeneous sensing is discussed below:

∂N (Equation 3.1) S = nc = nc0 ∂nc 61 where N is the effective refractive index, obtained from the EIM, as shown in the Figure

33, nc is the index of the cover region and nc0 is nc, unchanged/unperturbed value [46].

Using the variational theorem as mentioned in [46], for dielectric waveguides, we obtain the equation of sensitivity for quasi TE and quasi TM modes as the following equations

3 TE 1 Q + k0 wn2 A1 (n1a2 a2 + n2a1 a2 ) (Equation 3.2) S TE = w 3 2 2 N k0 wn2 a2 ()()n1a2 + n2a1 + 2a1 a2 a1 + a2

⎡ TM TM ⎤ TM 1 TM 2a (A − A ) (Equation 3.3) S = ⎢A + 1 2 1 ⎥ w N 1 ⎣⎢ ()a1 + a2 ()2 + k0w a2 ⎦⎥ where k0 is the vacuum wave number or propagation constant, n1 is the effective index obtained for the structure with height h1 and n2 is the effective index obtained for the structure with height h2, (as shown in the Figure 33).

The other parameters are as follows: [44, 46]

TE 2 TE 2 TE 2 2 TE 2 2 (Equation 3.4) Q = 4a1a2 A2 n1 − 4a1a2 A1 n2 + 2a2 A1 a1 a2 + 2a1A2 a1 a2

2 2 2 2 (Equation 3.5) a1 = n1 − N and a2 = N − n2

TE nc (Equation 3.6) Ai = i = 1,2 ⎛ c ⎞⎛ 1 1 ⎞ c ⎜1+ i ⎟⎜h k + + ⎟ i ⎜ f ⎟⎜ i 0 ⎟ ⎝ i ⎠⎝ ci si ⎠

−1 ⎛ c n4 + f n4 ⎞ 1 ()2c + n2 ⎜ i f i c ⎟ i c ⎜ f n4 ⎟ 3 TM ⎝ i c ⎠ nc ci (Equation 3.7) Ai = i = 1,2 ⎡h k ()c + f n2 ()s + f n2 ⎤ ⎢ i 0 + i i c + i i s ⎥ n2 4 4 4 4 ⎣⎢ f ()ci n f + fi nc ci ()si n f + fi ns si ⎦⎥

2 2 ci = ni − nc i =1,2 2 2 (Equation 3.8) si = ni − ns i =1,2 2 2 fi = n f − ni i =1,2 62

Figure 33: Figure showing the steps to obtain effective index.

[46] discusses the derivation of the sensitivity equations for quasi-TE and quasi-TM in detail. For different etch depths (h1-h2), and varying the cover index from 1 to 1.1, we obtain the sensitivity. Appendix D.0 and D.1 shows the sensitivity calculations for quasi-

TE and quasi-TM modes respectively for structure 1 and similarly the calculations are repeated for structure 2, by changing the concerned values .The results obtained using this method are reviewed in detail in Results and Discussion.

3.2 Numerical Method: Sensitivity Analysis using FEM Approach

Using the MOST option in the FEM software, we can calculate the sensitivity as discussed above in the analytical method. By setting the cover index values to vary from

1 to 1.1 in steps of 0.025, we get the sensitivity for different etch depths, as considered

Mode 0 63 before. Appendix D.2 shows the calculations of the sensitivity using FEM. The Figure 34 shows the MOST output, effective index versus the cover index for both TE and TM modes.

Figure 34: MOST outputs for (a) TE and (b) TM, for structure 1.

From the MOST output plots, we see that the output is as expected, with an increase in the cover index value, the effective index increases for both TE and TM modes. For an increase in the cladding index from 1 to 1.1, for a TE mode, the effective index increases by 0.84%, while for a TM mode, it increases by almost 7.92% approximately. This indicates that the TM modes are far more sensitive than the TE modes, as observed from the plots. Similarly, the same analysis is performed for structure 3 and the final results and comparisons with the analytical approach are discussed in results and discussion.

3.3 Sensitivity Analysis of Buried Waveguide Structure

For the buried waveguide structure as described in chapter 2, we calculate the surface sensitivity of the waveguide structure by varying the cladding thickness.

[47] States that the evanescent wave absorbance of the analytes is given by

p(0) (Equation 3.9) A = ln( ) p

Where p(0) is the power transmitted in the absence of the anlaytes, while p is the power transmitted in presence of the analytes.

The relation between p and p(0) [45] is given as

(Equation 3.10) P = P(0)exp(−γCL) where (L) is the length of interaction between the guided wave and the anlaytes in cm,

(C) is the concentration of the analytes in ppb and (γ) is the Evanescent wave absorption coefficient in cm-1(which is the product of the bulk absorption coefficient of the analytes

-1 (αm ) in cm and the effective fraction of the total guided power in the sensing region( rf ))

(Equation 3.11) γ = rfα m

Thus, using the above equations, we obtain an expression for absorbance, [47]

(Equation 3.12) A = rfαmCL

Using FEM, the effective fraction of the guided power in the sensing region is calculated for different cladding thicknesses as tabulated in the Table 3. As the thicknesses of the cladding region increases, the fraction of the guided power in the sensing region or in other words the confinement factor also increases, due to an elongated evanescent wave tail.

Table 3: Rf values for different cladding thicknesses.

Thickness of the clad (nmEffective guided power

sensing region Rf

15 0.83%

25 0.92%

50 1.08%

Absorbance is calculated for a sensing length of about 1cm (which is typically used), and for the above three cladding thicknesses. The values of the bulk absorption coefficient are obtained for different nitrite concentrations in water using [47]. Figure 35 below shows the absorbance versus concentration plot, for a buried structure, and is in accordance to

Beer Lambert’s Law. As the concentration of the analytes increase, the absorbance also increases, and the maximum absorbance is for a cladding thickness of 50nm, as predicted.

Figure 35: Absorbance versus Concentration plot for buried waveguide structure. 66

3.4 Surface Sensitivity of an Air LOOM Structure

Surface sensitivity analysis is performed for an air LOOM waveguide structure and the output is as shown in the Figure 36. Since there is no upper clad, as in the aqueous sensor, there is a single plot, which shows high absorbance values due to significant overlap of the optical mode with the anlaytes or target species [27]. As the concentration of the analytes increase to 600ppb, the absorbance almost increases linearly approximately to 4*103.

Figure 36: Absorbance versus Concentration for Air LOOM waveguide structure.

CHAPTER 4: MACH ZEHNDER INTERFEROMETER ANALYSIS

As mentioned in Chapter 1, interferometers are one of the classifications of Optical sensors when divided based on their sensing architecture. There are many kinds of

Interferometers, the most popular being michelson interferometer, mach-zehnder interferometer (MZI), fabry perot interferometer and sagnac interferometer. Among these

MZI is most commonly used because of its high sensitivity and easy integration [1]. Also, among the other interferometers, it is only the mach-zehnder interferometer which facilitates for a simple and an easily accessible reference arm due to which extrinsic disturbances can be avoided, thus improving the resolution of the sensor [48]. MZI has attracted several promising applications such as in chemical and biochemical sensing by finding optical parameters, which determines the concentration of the chemical, or biochemical species [49], sensing pressure, electric fields, antigen-antibody interaction, optical parameters such as effective indices and absorption coefficients [48]. In spite of increase in research in this area, integrated optical (IO) MZI are never used in commercial sensor applications due to the problems associated with the extrinsic disturbances such as light sources, temperature fluctuations, and other associated technological faults; and the intrinsic factors such as fringe order ambiguity, directional ambiguity and sensitivity fading [48]. The various fabrication technologies involved for these MZI based sensors are typically CMOS-compatible and even glasses and III-V semiconductors have been suggested [1]. A graded index waveguide is generally used for glass based MZI, and because such type of guiding structures have weak evanescent 68 fields, the sensitivity is low. To improve the glass waveguide sensitivity, channel-planar composite optical waveguide (COWG) has been suggested [1, 50].

LOOM waveguide structures based IO MZI have been studied in this thesis, which has the advantage of increased evanescent fields, which facilitates the interaction between the analytes and the fields. Figure 37 shows the schematic diagram of an IO MZI.

Figuure 37: Schematic diagrram of an IO MZI.

4.1 Principle of Operation of an MZI

As we can see from the Figure 36, the incoming optical signal is divided into a Y- junction by two separate arms, where one is the sensing arm and the other, the reference arm. In the sensing arm, the interaction between the analyte and the evanescent field takes place, by removing a portion of the cladding region to facilitate the interaction.

After propagating in the two arms, due to the interaction of the anlaytes in the sensing arm, there is a change in the optical effective index, which induces a phase shift, and it is this phase shift which determines the analyte concentration or parameter under 69 measurement such as refractive index of the media under investigation etc [1, 50]. These phase shifted optical signals again interfere at the output by another Y-junction.

4.2 Analysis of MZI

LOOM based structures are analyzed using an MZI technique. The reference arm consists of a rib waveguide structure with InP as the core material (structure 1). While in the sensing arm, the cladding index is changed from 1 to 1.1 in intervals of 0.01, to monitor the effective index change with varying cladding index. Effective Index Method (EIM) approximation is used to analytically calculate the effective index change with respect to change in the cladding index. This induces a change in the effective indices between the two arms, thus introducing a phase shift ΔΦ. Hence, the output power of the two arms, is calculated based on [50, 26] as below

(Equation 4.1) Δneff = neff (sensing arm) – neff (reference arm)

(Equation 4.2) ΔΦ = where L is the length of interaction and is typically 1cm [2]

λ is the working wavelength.

2 ΔΦ (Equation 4.3) P1 = P0 sin ( )

2 ΔΦ (Equation 4.4) P2 = P0 cos ( )

Thus, using the above mathematical equations and also the EIM approximations, the output power from the two arms of an MZI is calculated as in Appendix E.0 and E.1 for the two LOOM waveguide structures incorporating InP as the core material at 1550nm and a-ZnO at 550nm. The results obtained are shown in chapter 5 Results and

Discussions. 70

CHAPTER 5: RESULTS AND DISCUSSION

5.1 Analytical and Numerical Results for Effective Index Calculations

The results obtained from the analytical and numerical calculations for effective index for

the three waveguide structures are compared in Table 4. The two numerical techniques

show good accordance in the values.

Table 4: Comparison of Neff between analytical and numerical methods.

Analytical Method Numerical Methods

(EIM) FEM Semi‐Vectorial BPM

Structure TE mode TM mode TE mode TM mode TE mode TM mode

1.79748 1.00155 1.79445 1.01271 1.79525 1.01314

1.0514 1.00589 1.05868 1.00367 1.0608 1.00393

Table 4 (continued)

1.35179 1.0973 1.35044 1.09504 1.35149 1.09418

Percentage of Error with respect to the numerical technique, FEM method is tabulated in table 5

Table 5: Percentage of error calculated for the three methods.

EIM Semi-Vectorial BPM

TE (%) TM (%) TE (%) TM (%)

Structure1 -0.168 1.1019 -0.0445 -0.0424

Structure2 0.687 -0.2211 -0.200 -0.0259

Structure3 -0.094 -0.206 -0.077 0.07853

The effective indices calculated using the above three methods, including the results for semi-vectorial BPM [1] are compared and good agreement is noted. It shows the relative error with respect to FEM. The maximum error is approximately 1.11% for the TM mode for structure 1, while it is less than a percent for all the other modes and structures. 72

5.2 Comparison of the Modal Fill Factor and Confinement Factor

Table 6: Comparison of the MFF and CF.

Structure Analytical Method Numerical Method

(Semi‐Vectorial BPM)

Confinement f Confinement f Ey Hx Ey Hx (CF) (CF)

0.002 0.024 0.0045 0.003 0.003 0.0084

0.006 0.033 0.0127 0.09 0.24 0.21

The values obtained for a TM mode using both the analytical and numerical methods, semi-vectorial BPM are shown in the Table above. As expected, LOOM structures prove to have less MFF, as more of the mode is pushed out of the active region and thus facilitates for a better interaction with the anlaytes, in turn increasing the sensitivity of the sensor. For the structure 1, the confinement factor for a TM mode using analytical methods show 0.45% and using the semi-vectorial BPM, it is 0.84% which are both less 73 than 1% as predicted. There is some kind of discrepancy in the results obtained for structure 2, as the analytical method show a confinement factor value of about 1.27% while the semi-vectorial BPM claims to be about 21%. This discrepancy could be because the fundamental LOOM might have bled into a less powerful multimode at some point in the semi-vectorial BPM, whose MFF is predicted to be higher than the fundamental LOOM. Also, there is a unique point to be noted here, for these LOOM structures the MFF and the CF values do not have the same order of magnitude, unlike for most other structures. This is because for the MFF, the Ey component of the electric field has an order of magnitude change, due to an extended mode (out of the core region into the clad) compared to the Hx component. Thus, there is a change in an order of magnitude in CF, which is the cross product of Ey and Hx components. In this thesis, the crucial parameter is the MFF, and not CF, as the electric field is the concern and not the power.

5.3 Sensitivity Analysis

A quick recap of the sensitivity analysis, in case of homogeneous sensing, for different etch depths (h1=100nm, and h1-h2=40-80 nm), we obtain the sensitivity with varying cladding index (1-1.1). Figure 38 shows the comparison between the analytical and the numerical (FEM) sensitivity analysis for waveguide structure 1. The analytical method shows sensitivity as about 87% comparing to its numerical method, which is about 83%

(approximately) for TM modes. The relative error between the two methods is calculated to be less than 5%. 74

Figure 38: Comparisons of the analytical and numerical methods for the sensitivity versus etch depth for structure 1.

Similarly, the sensitivity analysis is performed for structure 2, at a wavelength of 550nm, where the core material is a-ZnO. The Figure 39 shows the comparison between the analytical method and the FEM technique. The analytical method shows that the sensitivity of a TM mode is about 45% while FEM predicts it to be around 43%. The TE

Modes have lesser sensitivity and both the methods show less than 21%.

Figure 39: Comparison of analytical and numerical methods for sensitivity versus etch depths for structure 2.

5.4 Mach-Zehnder Interferometer Output

The output from the MZI is shown in the Figures 40-41. When the LOOM structures are integrated onto a sensing architecture such as an MZI, the output powers from the two arms of an MZI are plotted with varying cladding index. With an increase in the cladding index from 1 to 1.01, the change in the power outputs at the MZI output ports is very high, thus the sensitivity is high. Comparing the outputs from the structure 1 and structure

2, the sensitivity using the InP waveguide structure is high compared to the a-ZnO. The plots below show the output power plots for structure 1 and 2, with structure 1 working at

1550nm, and structure 2 at 550nm, as mentioned earlier.

Figure 40: Plot of power outputs P1 and P2 with varying cladding index, for structure1 at 1550nm.

Figure 41: Plot of power outputs P1 and P2, with varying cladding index for structure2 at 550nm. 77

CHAPTER 6: CONCLUSION

The purpose of this thesis is to introduce Low Optical Overlap Mode (LOOM) structures based on Evanescent wave sensing in Integrated Optics (IO) sensors. LOOM based structures as predicted have extended mode which could interact with the surrounding anlaytes in a much better way than the commonly used EWS. The shortcomings of EWS are overcome by the introduction of the LOOM technique, as this has a modal fill factor which is less than 1% and has nanoscale core regions with less confined power compared to the fiber optics. The modal analysis of LOOM structures used in both gaseous sensing and aqueous sensing are performed and studied. The analytical and numerical techniques performed to calculate the effective index show good agreement. The confinement factor and modal fill factor are calculated using analytical and semi-vectorial BPM, which proves the presence of the extended evanescent field. Using the effective index values obtained from the analytical methods, the electric field profile is plotted, which is as expected assuming the boundary conditions and in accordance with maxwell’s equations.

Sensitivity analysis gives good results from both the analytical methods and FEM. The

LOOM structures prove to have good sensitivity compared to other EWS architectures.

This thesis aims at the Modal analysis of LOOM structures and comparison of analytical and numerical techniques for various critical parameters.

When integrated onto a MZI sensor architecture, these LOOM waveguides prove to have high sensitivity, as a small change of the cladding index, of about 0.01, induces a large percentage change in the output power sensed.

FUTURE WORK

Integrating LOOM structures in the detection of nitrites and ammonia is one of the prime concerns. Also fabricating amorphous materials incorporating the LOOM waveguide structures is challenging. Finally, integrating these waveguide structures, with the MZI architecture is difficult with respect to fabrication stand point, and this needs to be realized.

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APPENDIX A.0 EIM CALCULATIONS FOR STRUCTURE 1(TE)

% Appendix A.0 %Effective Index Method Calculations,InP,TE0 clc clear all n1=3.17; %index of the core region n2=1; %index of the substrate n3=1:0.01:1.1; %array of index of the cover region d= 100*10^-9; %thickness of the core region lmd=1550*10^-9; %working wavelength k=2*3.14/lmd; %propagation constant V=k*d*((n1^2-n2^2)^0.5); %Normalized parameter or V-number a=0; %parameter determining asymmetry m=0; %mode number k1=2*3.14*n1/lmd; %propagation constant in the core region %%A Number of modes, it supports M = V/3.14; h1=[10^4:10^6:10^9] A=h1.*d; fn1=tan(A/2- (m*3.14)/2); f1=abs((V^2-A.^2).^0.5); f2=A; fn2=f1./f2; grid on; Figure(1); plot(A,fn1,A,fn2); A1=0.5797; h11=A1/d k2=2*3.14*n2/lmd; beta=abs(k1^2-h11^2)^0.5 gama2=abs(beta^2-k2^2)^0.5 gama3=gama2; q2=beta^2/k1^2+beta^2/k2^2-1; q3=q2; dm=d+((gama2*q2)^-1)+((gama3*q3)^-1) N=gama2*d*(V^2+2*gama2*d); D=V^2*(2+gama2*d); Cf=N/D neff=lmd*beta/(2*3.14) 87

APPENDIX A.1 EIM CALCULATIONS FOR FINAL STRCUTURE 1(TE)

%Effective Index Method Calculations, InP, TE final structure clc clear all n1=1.81147;%effective index obtained from region 1 n2=1.32046;%effective index obtained from region 2 n3=1.32046;%effective index btained from region 3 d= 3*10^-6;%width of the core, i.e width of the rib lmd=1550*10^-9;%working wavelength k=2*3.14/lmd;%propagation constant V=k*d*((n1^2-n2^2)^0.5);%normalized parameter a=0;%assymetry factor m=0;%mode number k1=2*3.14*n1/lmd;%propagation constant of the core %Number of modes, it supports %M = V/3.14; h1=[10^5:10^5:10^8] A=h1.*d; %Eigen value equation fn1=tan(A/2- (m*3.14)/2);%left hand side of the eigen value equation f1=abs((V^2-A.^2).^0.5); f2=A; fn2=f1./f2;%right hand side of the eigen value equation grid on; Figure(1); plot(A,fn1,A,fn2); %from the point of intersection, A1=2.724; h11=A1/d k2=2*3.14*n2/lmd; beta=abs(k1^2-h11^2)^0.5 gama2=abs(beta^2-k2^2)^0.5 gama3=gama2; % %Effective guide thickness is calculated q2=beta^2/k1^2+beta^2/k2^2-1; q3=q2; dm=d+((gama2*q2)^-1)+((gama3*q3)^-1) N=gama2*d*(V^2+2*gama2*d); D=V^2*(2+gama2*d); Cf=N/D neff=lmd*beta/(2*3.14) 88

APPENDIX A.2 EIM CALCULATIONS FOR STRUCTURE 1 (TM)

%Appendix A.2 %% Effective Index Method Calculations, InP,TM0 clc clear all n1=3.17; %index of the core region n2=1; %index of the substrate n3=1; %index of the cover region d= 100*10^-9; %thickness of the core region lmd=1550*10^-9; %working wavelength k=2*3.14/lmd; %propagation constant V=k*d*((n1^2-n2^2)^0.5); %normalized parameter or V-number a=0; %asymmetry factor m=0; %mode number k1=2*3.14*n1/lmd; %propagation constant of the core %%%Number of modes, it supports M = V/3.14; h1=[10^5:10^6:10^9] A=h1.*d; %Eigen value equation fn1=tan(A/2- (m*3.14)/2); f1=abs((n1^2).*((V^2-A.^2).^0.5)); f2=(n2^2).*A; fn2=f1./f2; grid on; Figure(1); plot(A,fn1,A,fn2); A1=0.609; h11=A1/d k2=2*3.14*n2/lmd; beta=abs(k1^2-h11^2)^0.5 gama2=abs(beta^2-k2^2)^0.5 gama3=gama2; % %Effective guide thickness is calculated q2=beta^2/k1^2+beta^2/k2^2-1; q3=q2; dm=d+((gama2*q2)^-1)+((gama3*q3)^-1) N=gama2*d*(q2*V^2+2*gama2*d); D=V^2*(2+q2*gama2*d); Cf=N/D neff=lmd*beta/(2*3.14) 89

APPENDIX B.0 FEM SETTINGS

APPENDIX C.0 CALCULATION OF ELECTRIC FIELD PLOTS

%analytical calculation of the electric field plots clc clear all nf=3.17; ns=1; nc=1; d= .1*10^-6; lmd=1.55*10^-6; k=2*3.14/lmd; m=0; k1=2*3.14*nf/lmd; k2=2*3.14*nc/lmd; k3=2*3.14*ns/lmd; n=1.81147;%effective index beta=2*3.14*n/lmd; kf=((k1^2-beta^2)^0.5); gamac=((beta^2-k2^2)^0.5); gamas=((beta^2-k3^2)^0.5); a=1; x1= 0:1e-11:3e-6; Hy1=(exp(-gamac.*x1)); Figure plot(x1,Hy1); hold on clad_totey=sum(Hy1(1:300001)) x2=-d:1e-11:0; Hy2=(cos(kf.*x2)-(a.*(((gamac)/(kf)).*sin(kf.*x2)))); plot(x2,Hy2); hold on core_totey=sum(Hy2(1:10001)) x3=-3e-6:1e-11:-d; Hy3=(cos(kf*d)+(a*((gamac)/(kf))*(sin(kf*d)))).*(exp(gamas.*(x3+d))); plot(x3,Hy3); hold on sub_totey=sum(Hy3(1:290001)) elec_totey=clad_totey+sub_totey+core_totey modfillfactorey=core_totey/elec_totey

APPENDIX D.0 SENSITIVITY CALCULATIONS FOR STRUCTURE 1(TE)

%Analytical calculation of sensitivity for TE mode of structure 1 clc clear all nc=1;%Index of the cover nf=3.17;%index of the film ns=1;%index of the substrate n1=1.81147;%effective index obtained for height h1 n2=1.32046;%effective index obtained for height h2 N=1.00086;%total effective index considered w=3*10^-6;%width of the core h1=100*10^-9;%height of region 1 h2=57*10^-9;%height of region 2 lmd=1550*10^-9;%working wavelength k0=2*3.14/lmd;%propagation constant c1=n1^2-nc^2; s1=n1^2-ns^2; f1=nf^2-n1^2;

A1n=(nc); A1d=((c1^0.5)*(1+(c1/f1))*(h1*k0+(1/(c1^0.5))+(1/(s1^0.5)))); A1=A1n/A1d; c2=n2^2-nc^2; s2=n2^2-ns^2; f2=nf^2-n2^2;

A2n=(nc); A2d=((c2^0.5)*(1+(c2/f2))*(h2*k0+(1/(c2^0.5))+(1/(s2^0.5)))); A2=A2n/A2d; a1=n1^2-N^2; a2=N^2-n2^2;

Q=((4*a1*a2*A2*(n1^2))- (4*a1*a2*A1*(n2^2))+(2*a2*A1*(a1^2)*((a2^2)))+(2*a1*A2*(a1^2)*(a2^2))); % sensitivity of TE, modes Sn=abs((Q+(k0*w*n2^3*A1*((n1*a2*(a2^0.5))+(n2*a1*(a1^0.5)))))); Sd=abs(((k0*w*(n2^3)*(a2^0.5)*(n2*a1+n1*a2))+(2*(a1^2)*(a2^2)*(a1+a2)))); S=Sn/Sd

APPENDIX D.1 SENSITIVITY CALCULATIONS FOR STRUCTURE 1(TM)

%Analytical calculation of sensitivity for TM modes of structure 1 clc clear all nc=1;%index of the cover region nf=3.17;%index of the film ns=1;%index of the substrate n1=1.0057;%effective index obtained in region 1 n2=1.00001;%effective index obtained in region 2 N=1.78;%total effective index considered w=3*10^-6;%width of the core h1=100*10^-9;%height h1 of region 1 %i=1; h2=57*10^-9;%height h2 of region 2 lmd=1550*10^-9;%working wavelength k0=2*3.14/lmd;%propagation constant c1=n1^2-nc^2; s1=n1^2-ns^2; f1=nf^2-n1^2;

A1n=(2*c1+nc^2)*(((c1*nf^4+f1*nc^4)/(f1*nc^4))^-1)*(1/(nc^3*(c1^0.5))); A1d= ((h1*k0/nf^2)+(((c1+f1)*nc^2)/((c1*nf^4+f1*nc^4)*(c1^0.5)))+(((s1+f1)*ns^2)/((s1*nf^ 4+f1*ns^4)*s1^0.5))); A1=A1n/A1d;

c2=n2^2-nc^2; s2=n2^2-ns^2; f2=nf^2-n2^2; A2n=abs((2*c2+nc^2)*(((c2*nf^4+f2*nc^4)/(f2*nc^4))^-1)*(1/(nc^3*(c2^0.5)))); A2d=abs(((h2*k0/nf^2)+(((c2+f2)*nc^2)/((c2*nf^4+f2*nc^4)*(c2^0.5)))+(((s2+f2)*ns^2 )/((s2*nf^4+f2*ns^4)*s2^0.5)))); A2=A2n/A2d; a1=n1^2-N^2; a2=N^2-n2^2; %sensitivity calculations S= (N^-1)*(A1+((2*a1*(A2-A1))/((a1+a2)*(2+k0*w*(a2^0.5))))) %i = i+1;

APPENDIX D.2 MOST SETTINGS

MOST settings

APPENDIX E.0 MZI OUTPUT FOR STRUCTURE 1

%% Calculation of the output power from the two arms of an MZI, with %% varying cladding index, using the effective index approximation, for %% structure 1, InP with a wavelength of 1550nm. clc clear all

%effective index calculations in the sensing arm for a thickness of 100nm n1=3.17; n2=1; %n3=1.01; d= 100*10^-9; lmd=1550*10^-9; k=2*3.14/lmd; V=k*d*((n1^2-n2^2)^0.5); i = 1; for n3=1.0:0.01:1.1

a=((n1^4)*(n2^2-n3^2))/((n3^4)*(n1^2-n2^2)); m=0; M = V/3.14; k1=2*3.14*n1/lmd; k2=2*3.14*n2/lmd; k3=2*3.14*n3/lmd; % % Number of modes, it supports

h1=[10^6:5*10^4:3*10^7]; A=h1.*d; %%Eigen value equation fn1=tan(A); f1=abs(((V^2-A.^2).^0.5)); f2=abs((((1+a)*V^2-A.^2).^0.5)); fn2n=A.*(f1+f2); f3=(A.^2); f4=abs(((V^2-A.^2).^0.5).*((((1+a)*V^2)-A.^2).^0.5)); fn2d=abs(f3-f4); fn2=(fn2n./fn2d); % grid on; % Figure; % plot(h1,fn1,h1,fn2); % hold on % plot(h1,fn1-fn2,'-r'); 95

x = find((fn1-fn2)>0.001); h11 = h1(x(1))

beta=((k1^2-h11^2)^0.5); neff1(i)=abs(lmd*beta/(2*3.14)); i = i+1; end; neff1 % grid on; % Figure(1) % plot(h1,fn1,h1,fn2); % hold on % plot(h1,fn1-fn2,'-r');

%effective index calculations for the sensing arm with a thickness of 50nm n1=3.17; n2=1; n3=1.01; d= 50*10^-9; lmd=1550*10^-9; k=2*3.14/lmd; V=k*d*((n1^2-n2^2)^0.5); for n3=1.0:0.01:1.1

a=((n1^4)*(n2^2-n3^2))/((n3^4)*(n1^2-n2^2)); m=0; M = V/3.14; k1=2*3.14*n1/lmd; k2=2*3.14*n2/lmd; k3=2*3.14*n3/lmd; % % Number of modes, it supports

h1=[10^6:10^4:5*10^7]; A=h1.*d; %%Eigen value equation fn1=tan(A); f1=abs(((V^2-A.^2).^0.5)); f2=abs((((1+a)*V^2-A.^2).^0.5)); fn2n=A.*(f1+f2); f3=(A.^2); f4=abs(((V^2-A.^2).^0.5).*((((1+a)*V^2)-A.^2).^0.5)); 96

fn2d=abs(f3-f4); fn2=(fn2n./fn2d); % grid on; Figure(1); plot(h1,fn1,h1,fn2); hold on plot(h1,fn2-fn1,'-r'); hold off % x1 = (fn2-fn1)<0.7; % y1 = (fn2-fn1).*x1; % x = find((y1)>0.6); %[val,x] = min(fn2-fn1); %h11 = h1(x(1))

beta=((k1^2-h11^2)^0.5); neff2=abs(lmd*beta/(2*3.14)); end neff2

% final 3D effective index calculations using the above results for i=1:11 n10=neff1(i); n20=neff2(i); n30=neff2(i); d0= 3*10^-6; lmd=1550*10^-9; k=2*3.14/lmd; V0=k*d0*((n10^2-n20^2)^0.5); a=0; m=0; k10=2*3.14*n10/lmd; %Number of modes, it supports h10=[10^6:10^4:10^7]; A0=h10.*d0; %Eigen value equation fn10=tan(A0/2- (m*3.14)/2); %left hand side of the eigen value equation f10=abs((V0^2-A0.^2).^0.5); f20=A0; fn20=f10./f20; %right hand side of the eigen value equation

% Figure(1); 97

% plot(h10,fn10,h10,fn20); % hold on % plot(h10,fn10-fn20,'-r') %from the point of intersection, A=1.22 x1 = (fn20-fn10)<0.1; y1 = (fn20-fn10).*x1;

x0 = find((y1)>0.001); h110 = h1(x0(1)) beta0=abs(k10^2-h110^2)^0.5; neff3(i)=lmd*beta0/(2*3.14); i=i+1; end neff3 %MZI calculations, after obtaining the effective indices for varying %cladding indices nefftot1=1.569444583834116; nefftot2=neff3; delneff=nefftot2-nefftot1; L=1*10^-2; delphi=(2*3.14*L.*delneff)/lmd; P0=1; P1= P0*(sin(delphi./2).^2); P2=P0*(cos(delphi./2).^2); % Figure % plot(P1) % Figure % plot(P2) % n3; % Figure % plot(P1,P2) Figure plot(n3,delphi)

APPENDIX E.1 MZI OUTPUT FOR STRUCTURE 2

%% Calculation of the output power from the two arms of an MZI, with %% varying cladding index,using effective index approximation for structure %% 2, with a-Zno as core, with a wavelength of 550nm clc clear all

%Effective index calculations in the sensing arm, with a thickness of 100nm n1=1.79; n2=1; %n3=1.01; d= 100*10^-9; lmd=550*10^-9; k=2*3.14/lmd; V=k*d*((n1^2-n2^2)^0.5); i = 1; for n3=1.0:0.01:1.1

a=((n1^4)*(n2^2-n3^2))/((n3^4)*(n1^2-n2^2)); m=0; M = V/3.14; k1=2*3.14*n1/lmd; k2=2*3.14*n2/lmd; k3=2*3.14*n3/lmd; % % Number of modes, it supports

% plot(h1,fn1-fn2,'-r'); x = find((fn1-fn2)>0.001); h11 = h1(x(1))

beta=((k1^2-h11^2)^0.5); neff1(i)=abs(lmd*beta/(2*3.14)); i = i+1; end; neff1 grid on; Figure(1) plot(h1,fn1,h1,fn2); hold on plot(h1,fn1-fn2,'-r');

%effective index calculations for the sensing arm, for a thickness of 50nm n1=1.79; n2=1; d= 50*10^-9; lmd=550*10^-9; k=2*3.14/lmd; V=k*d*((n1^2-n2^2)^0.5);

for i= 1:0.01:1.1 a=((n1^4)*(n2^2-n3^2))/((n3^4)*(n1^2-n2^2)); m=0; M = V/3.14; k1=2*3.14*n1/lmd; k2=2*3.14*n2/lmd; k3=2*3.14*n3/lmd; % % Number of modes, it supports

h1=[10^5:10^4:3*10^6]; A=h1.*d; %%Eigen value equation fn1=tan(A); f1=abs(((V^2-A.^2).^0.5)); f2=abs((((1+a)*V^2-A.^2).^0.5)); fn2n=A.*(f1+f2); f3=(A.^2); f4=abs(((V^2-A.^2).^0.5).*((((1+a)*V^2)-A.^2).^0.5)); fn2d=abs(f3-f4); fn2=(fn2n./fn2d); % grid on; 100

Figure(1); plot(h1,fn1,h1,fn2); hold on plot(h1,fn2-fn1,'-r'); hold off % x1 = (fn2-fn1)<0.7; % y1 = (fn2-fn1).*x1; % x = find((y1)>0.6); %[val,x] = min(fn2-fn1); %h11 = h1(x(1))

beta=((k1^2-h11^2)^0.5); neff2=abs(lmd*beta/(2*3.14)); end neff2

% Final 3D effective indices calculations using the above obtained results for i=1:11 n10=neff1(i); n20=neff2(i); n30=neff2(i); d0= 2*10^-6; lmd=550*10^-9; k=2*3.14/lmd; V0=k*d0*((n10^2-n20^2)^0.5); a=0; m=0; k10=2*3.14*n10/lmd; %Number of modes, it supports h10=[10^6:10^4:10^7]; A0=h10.*d0; %Eigen value equation fn10=tan(A0/2- (m*3.14)/2); %left hand side of the eigen value equation f10=abs((V0^2-A0.^2).^0.5); f20=A0; fn20=f10./f20; %right hand side of the eigen value equation

% Figure(1); % plot(h10,fn10,h10,fn20); % hold on % plot(h10,fn10-fn20,'-r') %from the point of intersection, A=1.22 x1 = (fn20-fn10)<0.1; 101

y1 = (fn20-fn10).*x1;

x0 = find((y1)>0.001); h110 = h1(x0(1)) beta0=abs(k10^2-h110^2)^0.5; neff3(i)=lmd*beta0/(2*3.14); i=i+1; end neff3

%MZI calculations nefftot1=1.32968716389294; nefftot2=neff3; delneff=nefftot1-nefftot2; L=1*10^-2; delphi=(2*3.14*L.*delneff)/lmd;

P0=1; P1= P0*(sin(delphi./2).^2); P2=P0*(cos(delphi./2).^2); % Figure % plot(P1) % Figure % plot(P2) % n3; % Figure % plot(P1,P2) Figure plot(n3,nefftot1,n3,nefftot2)

102

APPENDIX F.0 MFF AND CF CALCULATIONS FOR STRUCTURE 1 modal fill factor and confinement factor calculations using EIM for structure 1 clc clear all nf=3.17; ns=1; nc=1; d= .1*10^-6; lmd=1.55*10^-6; k=2*3.14/lmd; m=0; k1=2*3.14*nf/lmd; k2=2*3.14*nc/lmd; k3=2*3.14*ns/lmd; n=1.81147;%effective index beta=2*3.14*n/lmd; kf=((k1^2-beta^2)^0.5); gamac=((beta^2-k2^2)^0.5); gamas=((beta^2-k3^2)^0.5); a=1; x1= 0:1e-10:12e-6; Ey1=(exp(-gamac.*x1)); % figure % plot(x1,Ey1); hold on clad_totey=sum(Ey1) x2=-d:1e-8:0; Ey2=(cos(kf.*x2)-(a.*(((gamac)/(kf)).*sin(kf.*x2)))); % plot(x2,Ey2); hold on core_totey=sum(Ey2) x3=-12e-6:1e-10:-d; Ey3=(cos(kf*d)+(a*((gamac)/(kf))*(sin(kf*d)))).*(exp(gamas.*(x3+d))); % plot(x3,Ey3); hold on sub_totey=sum(Ey3) elec_totey=clad_totey+sub_totey+core_totey; modfillfactorey=core_totey/elec_totey

nf=3.17; ns=1; nc=1; d= .1*10^-6; lmd=1.55*10^-6; k=2*3.14/lmd; m=0; k1=2*3.14*nf/lmd; k2=2*3.14*nc/lmd; 103 k3=2*3.14*ns/lmd; n=1.0057;%effective index beta=2*3.14*n/lmd; kf=((k1^2-beta^2)^0.5); gamac=((beta^2-k2^2)^0.5); gamas=((beta^2-k3^2)^0.5); a=((nf/nc)^2); x1= 0:1e-10:12e-6; Hy1=(exp(-gamac.*x1)); % figure % plot(x1,Hy1); hold on clad_tothy=sum(Hy1) x2=-d:1e-8:0; Hy2=(cos(kf.*x2)-(a.*(((gamac)/(kf)).*sin(kf.*x2)))); % plot(x2,Hy2); hold on core_tothy=sum(Hy2) x3=-12e-6:1e-10:-d; Hy3=(cos(kf*d)+(a*((gamac)/(kf))*(sin(kf*d)))).*(exp(gamas.*(x3+d))); % plot(x3,Hy3); hold on sub_tothy=sum(Hy3) elec_tothy=clad_tothy+sub_tothy+core_tothy; modfillfactorhy=core_tothy/elec_tothy core_cf= (Ey2.*Hy2); result2= sum(core_cf); cladding_cf= (Ey1.*Hy1); result1=sum(cladding_cf); sub_cf= (Ey3.*Hy3) result3=sum(sub_cf); result4=result2+result1+result3; %result4= sum(total_cf) cf=result2/result4

104

APPENDIX F.1 MFF AND CF CALCULATIONS FOR STRUCTURE 2

%%modal fill factor and confinement factor calculations using EIM for structure 2 clc clear all nf=1.79; ns=1; nc=1; d= .1*10^-6; lmd=0.55*10^-6; k=2*3.14/lmd; m=0; k1=2*3.14*nf/lmd; k2=2*3.14*nc/lmd; k3=2*3.14*ns/lmd; n=1.3573;%effective index beta=2*3.14*n/lmd; kf=((k1^2-beta^2)^0.5); gamac=((beta^2-k2^2)^0.5); gamas=((beta^2-k3^2)^0.5); a=1; x1= 0:1e-10:12e-6; Ey1=(exp(-gamac.*x1)); figure plot(x1,Ey1); hold on clad_totey=sum(Ey1) x2=-d:1e-8:0; Ey2=(cos(kf.*x2)-(a.*(((gamac)/(kf)).*sin(kf.*x2)))); plot(x2,Ey2); hold on core_totey=sum(Ey2) x3=-12e-6:1e-10:-d; Ey3=(cos(kf*d)+(a*((gamac)/(kf))*(sin(kf*d)))).*(exp(gamas.*(x3+d))); plot(x3,Ey3); hold on sub_totey=sum(Ey3) elec_totey=clad_totey+sub_totey+core_totey modfillfactorey=core_totey/elec_totey

nf=1.79; ns=1; nc=1; 105 d= .1*10^-6; lmd=0.55*10^-6; k=2*3.14/lmd; m=0; k1=2*3.14*nf/lmd; k2=2*3.14*nc/lmd; k3=2*3.14*ns/lmd; n=1.1035;%effective index beta=2*3.14*n/lmd; kf=((k1^2-beta^2)^0.5); gamac=((beta^2-k2^2)^0.5); gamas=((beta^2-k3^2)^0.5); a=((nf/nc)^2); x1= 0:1e-10:12e-6; Hy1=(exp(-gamac.*x1)); figure plot(x1,Hy1); hold on clad_tothy=sum(Hy1) x2=-d:1e-8:0; Hy2=(cos(kf.*x2)-(a.*(((gamac)/(kf)).*sin(kf.*x2)))); plot(x2,Hy2); hold on core_tothy=sum(Hy2) x3=-12e-6:1e-10:-d; Hy3=(cos(kf*d)+(a*((gamac)/(kf))*(sin(kf*d)))).*(exp(gamas.*(x3+d))); plot(x3,Hy3); hold on sub_tothy=sum(Hy3) elec_tothy=clad_tothy+sub_tothy+core_tothy; modfillfactorhy=core_tothy/elec_tothy core_cf= (Ey2.*Hy2); result2= sum(core_cf); cladding_cf= (Ey1.*Hy1); result1=sum(cladding_cf); sub_cf= (Ey3.*Hy3) result3=sum(sub_cf); result4=result2+result1+result3; %result4= sum(total_cf) cf=result2/result4

106

APPENDIX G.0 ABSORPTION SPECTRA OF GASEOUS AMMONIA

Absorption spectra of gaseous ammonia [20].