Submitted in partial fulfilment for the degree of Master of Science in Information Engineering.
City University, London Department of Electrical, Electronic & Information Engineering

Bragg Gratings in Semiconductor Waveguides

by Stephan Pachnicke
2001

TABLE OF CONTENTS

ABSTRACT 7

SYMBOLS AND ABBREVIATIONS 8

1 INTRODUCTION ... 10
1.1 Introduction ... 10
1.2 Optical communication systems ... 11
1.3 Aims and objectives ... 12
1.3.1 Previous works ... 12
1.3.2 Work in this project ... 13
1.4 Structure of this thesis ... 13

2 THEORY ... 15
2.1 Chapter overview ... 15
2.2 Maxwell’s equations ... 15
2.3 Boundary conditions ... 17
2.4 Reflection and refraction ... 18
2.5 Dispersion and loss ... 19

3 DIFFERENT TYPES OF BRAGG GRATINGS ... 21
3.1 Chapter overview ...21
3.2 Principle of operation ... 21
3.2.1 Uniform gratings ... 21
3.2.2 Chirped gratings ... 22
3.2.3 Apodised grating ... 23
3.2.4 Phase-shifted gratings ... 23
3.2.5 Multiple gratings ... 23
3.3 Applications of Bragg gratings ... 24
3.3.1 Demultiplexer in WDM systems ... 24
3.3.2 Dispersion compensation ... 24
3.3.3 Applications in lasers ... 25
3.3.4 Applications in sensing technology ... 25
3.4 Fabrication of Bragg gratings ... 26
3.4.1 Fabrication of fibre Bragg gratings ... 26
3.4.2 Fabrication of semiconductor Bragg gratings ... 27

4 SIMULATION METHODS ... 28
4.1 Chapter overview ... 28
4.1.1 Introduction ... 28
4.1.2 Theory of the FEM ... 28
4.2 Least squares boundary residual (LSBR) method ... 31
4.2.1 Introduction ... 31
4.2.2 Theory ... 31
4.3 Transfer matrix method (TMM) ... 33
4.3.1 Introduction ... 33
4.3.2 Theory for uniform gratings ... 33
4.3.3 Theory for apodised gratings ... 36
4.4 Overlap integral (OI) ... 37
4.5 Coupled mode theory (CMT) ... 37
4.5.1 Introduction ... 37
4.5.2 Theory ... 37

5 SIMULATION RESULTS ... 41
5.1 Chapter overview ... 41
5.2 Examined structures ... 41
5.3 Results obtained from FEM ... 43
5.4 Analysis of the discontinuity junction using LSBR ... 50
5.5 Simulation of a complete grating using TMM ... 58
5.5.1 Introduction ... 58
5.5.2 Input data ... 58
5.5.3 Uniform gratings ... 59
5.6 Results obtained from the CMT ... 61
5.7 Simulation of fabrication inaccuracy ... 65
5.8 Apodised gratings ... 76
5.8.1 Different apodisation profiles ... 76
5.8.2 Simulation results ... 77
5.9 Chirped gratings ... 83
5.10 Phase-shifted gratings ... 87
5.11 Multiple gratings ... 90

6 CONCLUSION ... 92
6.1 Discussion of the results ... 92
6.2 Conclusion ... 93
6.3 Further work ... 94

7 REFERENCES ... 95

ABSTRACT

Bragg gratings are important devices for both optical communications and sensing. These devices are used to design very narrow band optical filters, which can be used in wavelength division multiplexing (WDM). It is also perceived that Bragg gratings will be used to compensate the dispersion in modern fibre optic telecommunication networks. Semiconductor gratings are usually integrated into lasers to control the operating wavelength.
City University Photonic Modelling Group is a world leading research group on the use of rigorous numerical techniques to design and optimise advanced photonic devices for optical communications. The research group has already achieved results on hypothetical one-dimensional (1-D) and realistic two-dimensional (2-D) structures. In this project a combination of three numerical methods has been used, all of which are rigorous, to simulate realistic three-dimensional (3-D) structures in semi¬conductor waveguides. The combination of these three accurate methods, the finite element method (FEM), the least squares boundary residual (LSBR) method and the transfer matrix method (TMM) turned out to be superior to the widely used coupled mode theory (CMT).
The numerical study of different Bragg gratings shows interesting dependencies of the characteristics of the gratings on the different design parameters. The work was carried out for different mesh distributions, different numbers of mesh divisions and different computational parameters. Another focus of the work was on the stability of the transmission and reflection coefficients obtained from the LSBR program. Furthermore the effect of inaccuracy occurring during the fabrication process has been studied.
The results of this work have been compared to results found by other groups and fellows. We can say that this project is quite new in the field of reflection spectrum computation of realistic 3-D semiconductor Bragg gratings. Up to now only a few papers have been published on such 3-D semiconductor gratings.

1 INTRODUCTION

1.1 Introduction

The significance of fibre optics has grown rapidly in the last decades. In 1980 the first fibre networks were installed in the US and not earlier than 1988 the first transatlantic optical fibre cable was installed. Optical fibres are used in different fields of science and technology.
First of all fibre optic devices are used in the modern optical telecommunication. The main arguments for the use of fibre optics in this field are the extremely low loss and the very high bandwidth. But there are also other fields like medicine, which make use of fibres extensively. There the requirements are different and the small size and high flexibility of the fibre are the main arguments in favour.
Therefore it is understandable that this area of science is a field that needs a lot of research to satisfy not only the demands of future telecommunications but also of many other fields.
The main problem in today’s telecommunication network is the dispersion of light pulses. If the fibre is used in the very low loss area around 1.55 m wavelength it is necessary to compensate this dispersion. One way to do this is the usage of Bragg gratings. They consist either of a periodic perturbation of the refractive index in an optical fibre or a periodic change of the height of the rib in a semiconductor waveguide. Bragg gratings can be used for many different applications such as wavelength division multiplexing, dispersion compensation, sensing and in semiconductor lasers. Bragg gratings allow to implement very narrow-band optical filters. Semiconductor gratings are usually integrated directly in lasers where they act as tuneable mirrors.

1.2 Optical communication systems

The invention of the telephone by Graham Bell allowed global communication via telephone networks. At first twisted pair wires were used and were later replaced by coaxial cables with higher data rates and lower loss. As the amount of the transmitted data increased continuously, this technology also reached its limit very fast.
The invention of optical fibres was a revolution for the long distance communication, since it made it possible to transmit signals of very high bandwidths. At first the losses in the fibre were very high, but already in 1966 Kao predicted an attenuation of 3 dB/km and in 1968 a fibre with 20 dB/km was realised by Maurer/Corning. Another important step was the invention of the laser in 1960 by Maiman. With this laser it was now possible to use a coherent light source, which is needed to couple a light beam into the fibre. Further inventions like the GaAs laser, Bragg gratings, low loss fibres and sophisticated multiplexing techniques made it possible to transmit several data streams simultaneously over a single optical fibre. Due to the exponential growth of the world-wide-web the demand for very fast broadband transmission media to exchange large amounts of data all over the world has grown rapidly. The optical fibres, which are now commonly used, are the best choice for this application. Modern electrical or optical fibre amplifiers offer the possibility to compensate the loss occurring during the transmission. One of the main problems in the transmission of light signals is the occurring dispersion. This effect limits the maximal bandwidth or the length of the fibre.

Fig.1-1 is in the Downloadfile.

Fig. 1-1 Attenuation and dispersion of silica fibre [1]

Unfortunately the wavelengths for the minimal dispersion and the minimal loss are not the same and therefore a possibility needs to be found to compensate the dispersion, since one wants to make use of the minimal loss area. The relation between dispersion and loss depending on the wavelength is shown in Fig. 1-1.
Being able to compensate dispersion is becoming even more important since Erbium doped fibre amplifiers (EDFA) are available. The idea of the Erbium doped fibre was developed at Southampton University. EDFA makes it possible to amplify a weak optical signal without converting it into an electrical signal. A very new approach is the usage of solitons for the dispersion compensation in optical fibres. A soliton is a pulse excitation of a non-linear dispersive medium, which propagates without distortion. In this technique the very small non-linearity of the optical fibre is used to compensate the occurring dispersion by non-linear phase modulation of the pulse. The soliton technology has tremendous potential in the near future to long distance communications and optical switching.

1.3 Aims and objectives

The aim of this project is to characterise three-dimensional (3-D) semiconductor Bragg gratings accurately. For this a combination of three numerical methods will be employed. We believe that these methods are superior to the widely used coupled mode theory (CMT).
First the finite element method (FEM) will be used to calculate the modal solutions for the H-field components. These results are used as input data for the least squares boundary residual (LSBR) method. The LSBR method will compute the reflection and transmission coefficients at the discontinuity junctions of the grating. These coefficients are needed for the transfer matrix method (TMM), which will calculate the overall reflection spectrum of the Bragg grating.

1.3.1 Previous works

In the last years there have been several projects at City University dealing with Bragg gratings. In 1998 Markus Plura [1] wrote a thesis on “Accurate Characterisation of Bragg Gratings”. In this work one dimensional, planar Bragg gratings were analysed. These fibre gratings were simulated with the help of the FEM, LSBR and TMM methods.
In 1999 Jacek Gomoluch [2] extended this work and implemented the CMT method. The CMT was able to simulate fibre Bragg gratings accurately, but failed for semiconductor gratings with high refractive index variations.
In 2000 Jean-Alain Esclafer de La Rode [3] wrote a thesis on the “Assessment of the reflection spectra of two-dimensional uniform semiconductor Bragg gratings”. His work focused on assessing realistic asymmetrical rib-based semiconductor gratings.

1.3.2 Work in this project

In this project three-dimensional semiconductor gratings will be simulated numerically. First of all the FEM will be used to find the propagation constant for a given structure. The variation of the -values for different mesh sizes will be shown and Aitken’s extrapolation will be used to find the theoretically optimal value for an infinite mesh.
Then the LSBR will be employed to find the reflection and transmission coefficients at the discontinuity junction. The slight overshoot of the transmission coefficient  above 1.0, will be examined thoroughly and several approaches how to overcome this problem will be presented. Also the stability of the reflection coefficient  will be discussed and the results will be compared to the widely used impedance method.
Afterwards the TMM will be used to find the overall reflection spectra of semiconductor gratings. The effect of a random change in the grating length will be studied and reflection spectra for apodised, chirped, phase-shifted and multiple gratings will be obtained.
The CMT will be used to compare with our results and the limitations of this widely used method will be discussed.

1.4 Structure of this thesis

After a brief introduction into optical communications has been given in this chapter, essential parts of the theory needed for understanding the propagation of light will be given in the next chapter. The theory is based on Maxwell’s equations. At the end of Chapter 2 the two main problems of optical communications will be discussed namely dispersion and loss.
Chapter 3 will introduce the different types of Bragg gratings: fibre gratings and semiconductor gratings. The principle of operation will be explained and different applications of these devices will be discussed. Some fabrication techniques will be illustrated in this chapter.
The theory behind the different simulation methods will be given in Chapter 4. This includes first of all the three major simulation methods, the finite element method (FEM), the least squares boundary residual (LSBR) method and the transfer matrix method (TMM). But also a brief description of the coupled mode theory (CMT) will be given and some ways of calculating the coupling coefficient will be presented.
The simulation results will be shown in Chapter 5. First of all the simulated structure and the parameters we used will be introduced. There will also be an assessment of the different methods we employed during this project.
In Chapter 6 a conclusion will be drawn and the whole project will be critically reviewed. Some suggestions for further work in this area will also be made.

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