Excerpt

## Table of Contents

Acknowledgements

Declaration

Abstract

Symbols and Abbreviations

1 Introduction

1.1 Introduction

1.2 Optical communication systems

1.3 Aims and objectives

1.3.1 Previous works

1.3.2 Work in this project

1.4 Structure of this thesis

2 Theory

2.1 Chapter overview

2.2 Maxwell’s equations

2.3 Boundary conditions

2.4 Reflection and refraction

2.5 Dispersion and loss

3 Different types of Bragg gratings

3.1 Chapter overview

3.2 Principle of operation

3.2.1 Uniform gratings

3.2.2 Chirped gratings

3.2.3 Apodised grating

3.2.4 Phase-shifted gratings

3.2.5 Multiple gratings

3.3 Applications of Bragg gratings

3.3.1 Demultiplexer in WDM systems

3.3.2 Dispersion compensation

3.3.3 Applications in lasers

3.3.4 Applications in sensing technology

3.4 Fabrication of Bragg gratings

3.4.1 Fabrication of fibre Bragg gratings

3.4.2 Fabrication of semiconductor Bragg gratings

4 Simulation Methods

4.1 Chapter overview

4.1.1 Introduction

4.1.2 Theory of the FEM

4.2 Least squares boundary residual (LSBR) method

4.2.1 Introduction

4.2.2 Theory

4.3 Transfer matrix method (TMM)

4.3.1 Introduction

4.3.2 Theory for uniform gratings

4.3.3 Theory for apodised gratings

4.4 Overlap integral (OI)

4.5 Coupled mode theory (CMT)

4.5.1 Introduction

4.5.2 Theory

5 Simulation Results

5.1 Chapter overview

5.2 Examined structures

5.3 Results obtained from FEM

5.4 Analysis of the discontinuity junction using LSBR

5.5 Simulation of a complete grating using TMM

5.5.1 Introduction

5.5.2 Input data

5.5.3 Uniform gratings

5.6 Results obtained from the CMT

5.7 Simulation of fabrication inaccuracy

5.8 Apodised gratings

5.8.1 Different apodisation profiles

5.8.2 Simulation results

5.9 Chirped gratings

5.10 Phase-shifted gratings

5.11 Multiple gratings

6 Conclusion

6.1 Discussion of the results

6.2 Conclusion

6.3 Further work

7 References

## Acknowledgements

First and foremost, I wish to acknowledge sincerely my project supervisor Professor B.M.A. Rahman for his help and guidance throughout the whole project. His excellent supervision was essential for the project work to be carried out.

Also, I would like to thank all members of the Photonics Device Modelling Group for their help and support. Without their experience many problems could not have been sorted out that quickly.

Moreover, I would like to take the opportunity to thank the German Academic Exchange Service (DAAD) for their financial support of my studies at City University.

Also, I am grateful to the Cusanuswerk for granting me a scholarship, which was a great help for my studies, and I gratefully appreciate the support by e-fellows.net and by NOKIA.

Finally, I would like to express my deepest gratitude to my parents for their encouragement and support throughout my studies in London.

## Declaration

The author hereby grants powers of discretion to the City University librarian to allow this thesis to be copied in whole or in part without further reference to the author. This permission covers only single copies made for study purposes, subject to normal conditions of acknowledgement.

## 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.

## Symbols and Abbreviations

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## 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 mm 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.

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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 *b* -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 *t* 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 *r* 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.

## 2 Theory

### 2.1 Chapter overview

This chapter will introduce the basic theory, which is required to understand the principles of light propagation. First of all Maxwell’s equations will be explained and the boundary conditions at discontinuity junctions, will be given. The principle of reflection and refraction will be mentioned briefly. Afterwards the major problems in optical communication namely dispersion and loss will be discussed.

### 2.2 Maxwell’s equations

The theory explaining the propagation of light is split into two different parts. It is said that light has a dual nature. There is on the one hand the wave theory by Hooke and Huygens and on the other hand the corpuscular theory of Newton [4]. In 1864 Maxwell combined the equations of electromagnetism and showed that they suggest the existence of transverse electromagnetic waves. These equations can either be written in integral or in differential form. For the analysis of light propagation, it turns out that the differential form is more convenient. In (2.1) - (2.4) these equations are given in differential form [5]:

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(2.1)

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(2.2)

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(2.3)

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(2.4)

where **H** is the magnetic field intensity [A/m], **J** the current density [A/m²], **D** the electric flux density [C/m²], **E** the electric field intensity [V/m], **B** the magnetic flux density [Wb/m²] and *r* the charge density [C/m³]. The current density **J** and the charge density *r* can be neglected in this project because only dielectric materials will be utilised.

The electric flux density **D** and the electric field intensity **E** are related by the following equation:

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(2.5)

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(2.6)

Because all materials are considered isotropic the permittivity *e* and the permeability *µ* are scalars and not tensors. Since the materials are linear, *e* and *µ* are not functions of the field intensity, and all calculations can be carried out using phasors. Using equations (2.2) and (2.5) the three-dimensional wave equations can be derived:

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(2.7)

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(2.8)

Using the relations illustration not visible in this excerpt and illustration not visible in this excerpt, equations (2.7) and (2.8) reduce in phasor notation to the three-dimensional Helmholtz equations:

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(2.9)

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(2.10)

### 2.3 Boundary conditions

In general the solutions of Maxwell’s equations have to satisfy certain boundary conditions [6]. It is a good approximation in many practical problems to treat good conductors as of infinite conductivity when finding the form of fields outside the conductor. In case of finite conductivity, we will find that all fields and currents concentrate in a thin region near the surface and this region reaches zero thickness as the conductivity approaches infinity. If the field is zero within the perfect conductor, continuity of tangential electric field at the boundaries requires that the surface tangential electric field is zero outside the boundary:

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(2.11)

In vector notation this can be written as:

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(2.12)

For optical devices these boundary conditions become important at discontinuities. Consider a discontinuity junction with two different mediums and a unity vector **n** pointing from one medium to the other. In absence of any surface currents (**J=0**) and surface charges (*r* =0) the following boundary conditions apply:

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(2.13)

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(2.14)

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(2.15)

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(2.16)

(2.13) is called the continuity of the tangential magnetic field, (2.14) the continuity of the tangential electric field, (2.15) the continuity of the normal magnetic flux density and (2.16) the continuity of the normal electric flux.

### 2.4 Reflection and refraction

Reflection is an important physical effect for the propagation of light in an optical fibre. The light, which is fed into a fibre, will follow the fibre with few losses as long as the bend is not too sharp. The first condition for total internal reflection is that the refractive index of the cladding is lower than that of the core. The amount of light that is reflected depends on the nature of the surface of the interface. To minimise the loss the surface has to be very regular and smooth. According to the law of reflection the angle of incidence equals the angle of reflection and the incident ray, the reflected ray as well as the normal ray always lie in the same plane. Total-reflection can be regarded as a special case of refraction with ideal materials.

But not all rays will be reflected. There may also be a ray passing from the higher refractive index side to the lower refractive index side. In this case the refracted ray increases its speed and changes its direction such that the angle between the ray and the normal gets larger. When the ray comes from the other direction, passing from the lower refractive index side to the higher one, the speed of the ray, as well as its angle to the normal, will decrease.

This behaviour can be calculated using Snell’s law:

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(2.17)

where *n1* and *n2* are the refractive indices of the two materials. The refractive index is defined as the ratio of the speed of light in vacuum *c0* to the speed of light in the medium *v*.

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(2.18)

At some point *q2* will reach 90°. Then *q1* will be called the critical angle, *qcrit*.

If *q1* is increased any further, then total internal reflection will occur as described above. In this case *q2* equals 90° and the critical angle can be calculated as follows:

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(2.19)

### 2.5 Dispersion and loss

As already mentioned earlier there are two major problems occurring in the transmission of signals in optical fibres: dispersion and loss. Dispersion describes the spreading of a signal in time. At the input, a series of pulses – representing binary information – are launched onto an optical waveguide. Dispersion causes each of the pulses to spread in time. When they arrive at the output, the pulses have broadened to the point where they begin to overlap adjacent pulses. This is called inter-symbol-interference (ISI). The spreading limits the maximum data rate of a communication link. There are three types of dispersion, which can occur in optical waveguides.

Material dispersion is caused by the material itself, which has different refractive indices for different wavelengths. Or in other words *n* is a function of the frequency. Therefore each wavelength will travel at a different speed and the signal pulse will smear out. This effect is even stronger in semiconductor materials.

Waveguide dispersion occurs due to different time delays or velocities at different wavelengths due to different power-distribution in core and cladding. It can be reduced by careful design.

Modal dispersion only occurs in multimode fibres. Each allowed mode in the waveguide travels with a different group velocity. In step-index (SI) fibres pulse spreading occurs due to different geometrical paths. The paths taken by particular modes may be longer than the paths taken by other modes and the pulse tends to spread out. This problem can be overcome by using graded-index (GI) fibres. In these fibres the refractive index is maximal in the centre of the core and gets smaller to the cladding. Thus the light will travel faster in the outer regions of the core and compensate the longer distance it has to travel compared to an axial ray directly through the centre of the core.

In multi-mode fibres all three types of dispersion may occur, whereas in single-mode fibres only material dispersion and waveguide dispersion exist. There are several ways to overcome dispersion. The first one is to use a single mode fibre, which has no modal dispersion. Another possibility is to utilise dispersion-shifted fibres. These fibres are designed in such a way that the waveguide dispersion is enhanced to compensate material dispersion. However additional loss occurs, because an additional fibre of a certain length has to be attached to the end of the transmission fibre. A better way to compensate dispersion is to use a chirped Bragg grating. In this type of Bragg grating the grating period is uniformly changed so that different wavelengths reflect from different position of the grating. This allows the high and the low spectrum of the optical signal to be shifted back to their original position.

Loss is the other great problem in optical communication. The transmission losses in optical fibres arise from three different mechanisms: absorption in the glass, scattering by imperfections in the glass and radiation from the core due to bending or geometric non-uniformities. The lowest losses have been found in silica glass. The absorption is due to inherent electronic and molecular vibrations and to the presence of contaminants. The electronic absorption occurs at ultraviolet wavelengths and sets a basic limit to the minimum attenuation to be achieved. The molecular absorption occurs at infrared wavelengths. Impurity ions would also cause absorption, but these can be removed by careful manufacturing. The minimum absorption is wavelength dependent with a minimum in the infrared region of 1.1 µm to 1.8 µm. Scattering losses for fused silica are caused by microscopic inhomogenities, which modify the refractive index. They are caused by the process of cooling the glass from being a pure liquid to its normal super cooled liquid state. The scattering losses can be reduced by cooling the glass over a longer interval of time.

Unfortunately the minimum loss area and the zero dispersion area are not the same. The dispersion is zero at the wavelength of 1.33µm, while the loss is considerable high at this wavelength. For the lowest attenuation at 1.55µm dispersion on the other hand is higher. However, since the loss is at its minimum, this wavelength is used as the main transmission wavelength. Today the minimal loss is approx. 0.2 dB/km.

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Fig. 2-1 Attenuation and dispersion of a silica fibre[1]

**[...]**

- Quote paper
- Dr.-Ing. Stephan Pachnicke (Author), 2001, Bragg gratings in semiconductor waveguides, Munich, GRIN Verlag, https://www.grin.com/document/50382

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