Fiber-Optic Communications E-Book

Pierre Lecoy

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 384

Veröffentlichungsjahr: 2013

Beschreibung

This book describes in a comprehensive manner the components and systems of fiber optic communications and networks. The first section explains the theory of multimode and single-mode fibers, then the technological features, including manufacturing, cabling, and connecting. The second section describes the various components (passive and active optical components, integrated optics, opto-electronic transmitters and receivers, and optical amplifiers) used in fiber optic systems. Finally, the optical transmission system design is explained, and applications to optical networks and fiber optic sensors are detailed, including the most recent developments in switched networks, high bit-rate systems, and FTTH or radio over fiber.

Leseprobe

Table of Contents

Foreword

Introduction

Chapter 1. Multimode Optical Fibers

1.1. Overview of optics

1.2. Dielectric waveguide

1.3. Multimode optical fibers

1.4. Propagation in multimode optical fibers

1.5. Dispersion in multimode optical fibers

1.6. Appendix: detail of calculation in section 1.4.2

Chapter 2. Single-Mode Optical Fibers

2.1. Fiber optic field calculation

2.2. Single-mode fiber characteristics

2.3. Dispersion in single-mode fibers

2.4. Polarization effects in single-mode fibers

2.5. Non-linear effects in optical fibers

2.6. Microstructured (photonic) optical fibers

Chapter 3. Fiber Optics Technology and Implementation

3.1. Optical fiber materials and attenuation

3.2. Manufacturing of optical fibers

3.3. Optical fiber cables and connections

3.4. Extrinsic fiber optic losses

3.5. Optical fiber measurements

Chapter 4. Integrated Optics

4.1. Principles

4.2. Mode coupling and its applications

4.3. Diffraction gratings

Chapter 5. Optical Components

5.1. Passive non-selective optical components

5.2. Wavelength division multiplexers

5.3. Active optical components

5.4. Fiber optic switches

Chapter 6. Optoelectronic Transmitters

6.1. Principles of optoelectronic components

6.2. Light-emitting diodes (LED)

6.3. Laser diodes

6.4. Optical transmitter interface

6.5. Comparison between optoelectronic emitters

Chapter 7. Optoelectronic Receivers

7.1. Photodetectors

7.2. Optical receiving interface

7.3. Other photodetection schemas

Chapter 8. Optical Amplification

8.1. Optical amplification in doped fiber

8.2. Erbium-doped fiber amplifiers

8.3. Noise calculation in amplified links

8.4. Other types of optical amplifiers

Chapter 9. Fiber-Optic Transmission Systems

9.1. Structure of a fiber-optic digital link

9.2. Digital link design

9.3. Digital link categories

9.4. Fiber-optic analog transmissions

9.5 Microwave over fiber optics links

Chapter 10. Fiber-Optic Networks

10.1. Computer networks

10.2. Access networks

10.3. Wide area networks

10.4. Toward all optical networks

Chapter 11. Fiber-Optic Sensors and Instrumentation

11.1. Fiber optics in instrumentation

11.2. Non-coherent fiber-optic sensors

11.3. Interferometric sensors

11.4. Fiber optic sensor networks

Published in France in 2007 by Hermes Science/Lavoisier entitled “Télécoms sur fibres optiques” (3° Éd. revue et augmentée)

First published in Great Britain and the United States in 2008 by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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The rights of Pierre Lecoy to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Cataloging-in-Publication Data

Lecoy, Pierre.

  [Télécoms sur fibres optiques. English]

  Fiber-optic communications / Pierre Lecoy.

    p. cm.

  Includes bibliographical references and index.

  ISBN 978-1-84821-049-3

  1. Optical fiber communication. I. Title.

  TK5103.592.F52L4313 2008

  621.382'75--dc22

2008019705

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN: 978-1-84821-049-3

To Jean-Pierre NOBLANC, who was first my professor, then my colleague at Ecole Centrale Paris and my lab Director at France Telecom, who conveyed his passion for optical fibers and optoelectronics.

Foreword

At home, the quadruple play: digital television, radio, high speed Internet, landline and cell phones … At work, multiple and diverse computer applications combining voice, data and images … We conduct our business as if it is completely natural to exchange and receive all this information, but how does this transmission of information happen? This is what Pierre Lecoy, author of this book, skillfully and clearly explains, and reminds us of all the levels of creativity, mathematical calculations and technical magic that had to happen in order for a simple fiber glass cable, the optical fiber to be able to transport digital information at speeds that today exceed a dozen terabits per second or 10,000 billion bits per second on a single fiber!

Pierre offers his experience in the field of fiber optics telecommunications based on his double expertise acquired, on the one hand, within large manufacturing groups and, on the other hand, as a professor instructing generations of engineers. The combination of these two qualities is not exceptional because engineers frequently teach, but in essence, it is this combination which makes this book so interesting, especially as Pierre illustrates the importance of developing theoretical knowledge based on the development of applications and concrete accomplishments.

You will discover the professional competence of an industrialist as Pierre demonstrates a rich experience of several years in this field. He alternatively presents all the elements inherent to fiber-optic connections by focusing on the applications, their constraints and especially on choice criteria for solutions offered by technologies available on the market. Without going into too much detail, I will cite such examples as soliton propagation, micro-structured fiber optics, the principle of fiber bragg gratings in fibers, etc.

Pierre uses his competence as a professor and takes advantage of it by explaining the equations and different laws governing these fields in good company with Bessel, Laplace, Maxwell, Rayleigh and others (Fabry or Pérot). This is where Pierre’s experience comes in handy: when he describes and breaks down light propagation mechanisms though fibers and the different components of networks. The reader will thus learn or rediscover the different dispersions in an optical fiber, the calculation of pulse widening as well as non-linear effect processing such as Raman scattering, Brillouin scattering and the Kerr effect. Pierre also presents the principles controlling the production of fiber optics, the different types of cables, connections and associated measurements. In addition, he explains the different wavelength division multiplexing, modulation, switching technologies and so much more.

This is why I strongly invite you to delve into this book, the fruit of patient research and compilation work of all you can find on fiber optics. I greatly recommend it as, even though this book is rigorously scientific, it is an easy read. This book is certainly very meticulous but it is also practical with concrete examples, making it easy to understand.

In this way, the novice reader will be able to follow each chapter, reading about the construction of fiber-optic connections and understand all about multimode fibers, single-mode fibers, the different types of cables, optical components, optoelectronic transmitters and receivers, optical amplification, analog and digital transmissions over fiber optics, the different types of optics – from local area to operator – with an eye on the optical future, and finally, a complete chapter is dedicated to fiber optics in instrumentation as well as optical fiber sensors.

As for you, informed reader, the structure of this book in clearly delimited chapters will enable you to go directly to the heart of the subject most interesting to you.

Lastly, Pierre puts his professor hat back on and his long teaching practice by proposing 35 exercises, thankfully with their answers! A bibliography and website addresses successfully complete this project.

What more is there to say to you, dear reader, except maybe to insist on the triple application of fiber-optic telecommunications: at home, at work and in social relations. Currently at home there are over 25 million subscribers worldwide using fiber optics providing them with 100 Mbit/s and already in Japan, several thousand users can use throughputs of 1 Gbit/s enabling them to simultaneously receive several high definition television channels while having access to hyper-realistic online video. At work, high throughput digital data, combined with video telephony, ensure rapid circulation of large volumes of data, their storage and their query in relational database management systems simultaneously. As for the social aspect, much interest is based on the development of e-government, e-training, e-medicine and other type relations. Another significant advantage of these social evolutions resulting from fiber-optic telecommunications involves environmental protection by greatly decreasing the movement of people.

That is why I thank Pierre on your behalf for this body of work that he has successfully achieved in order to offer you these very interesting pages. Through his analysis, the fruit of deep thought processes sourced from his professional and diversified life, Pierre has shared his knowledge in this field with us. Finally, while remaining prudent and realistic, this book is part of the future with the evolution of SDH/Sonet networks toward all optical transport networks (OTN – Optical Transport Network) including a new optical layer, wavelength routing and multiprotocol label switching (MPLS) for packet transmission. As former director of training at Tyco Electronics and as current honorary president of Club optique, I can only applaud this methodology. Tomorrow is happening today.

Jean-Michel Mur Honorary President of Club optique

Introduction

Overview

Although the first optical information transmission experiments date back to late in the 19th century, a way has to be found to correctly direct or guide the light before thinking of its application to telecommunications. The emergence of lasers, around 1960, led to transmission experiments in the atmosphere. However, propagation instability problems (mainly caused by air index variations) have led researchers to abandon this solution. It is now reserved for short distance communications (infrared remote control, indoor communications, infrared laser link between two buildings), even though it is now being considered again for direct links between satellites.

Optical fibers are very thin angel hair type transparent glass and use the well-known principle of fountains of light. They were used in decorative applications before being used in a more useful way (lighting, endoscopy, remote optical measures). Their application in the field of telecommunications, although considered by theoreticians (Charles Kao) as soon as 1966, was made possible in the 1970s because of the progress in silica fiber-optic production technology, allowing a very light attenuation and an adequate mechanical resistance. It was also enabled by the availability of semiconductor laser diodes, which combine laser performance with electronic component ease of use, in particular because of the progress of III-V compound semiconductors. The development of efficient cables, connectors and passive components, and the availability of industrial connection processes, have also been essential for the development of the first commercial links around 1980.

The 1990s were significant in the development of optical amplification followed by wavelength division multiplexing, leading to an explosion of capabilities in response to new requirements caused by the growth of the Internet. The next revolution should be all-optical networks. Inaugurated by the emergence of the first all-optical switches in 2000, they were only prototypes and this concept remains to be clarified, particularly with standardization of protocols.

With approximately 100 million kilometers produced each year, and despite production activity fluctuations in this field which shows a strongly cyclic characteristic, optical fibers have become a mature technology present in increasingly wide application fields.

Fiber optics advantages

Transmission performance

Very low attenuation, very large bandwidth, possible multiplexing (in wavelength) of numerous signals and users on the same fiber, provide much higher range (over 100 km between transmitter and receiver) and capacity (bitrates of several Tbit/s are possible over a single fiber) systems than with copper wires or radio. However, depending on the considered use, other advantages can be decisive.

Implementation advantages

Including the light weight, very small size, high fiber flexibility, noticeable in telecommunications as well as for cabling in aeronautics, information technology, medicine, manufacturing, home automation, etc.

The most important practical advantage remains electrical (total insulation between terminals, possible use in explosive atmosphere, under high pressure, in medical applications) and electromagnetic safety (the fiber is not sensitive to parasites and does not create them either).

Conversely, the optical power used is weak and not dangerous. We can add total (or almost) invulnerability: it is not possible to hear the signal on an optical fiber without being spotted.

Economic advantage

Contrary to common belief, the global fiber-optic system cost (taking its installation and necessary equipment into account) is, in many cases, lower than the cost of a copper wire system (in particular with the recent increase in the price of copper), and its implementation, notably concerning connections, has become much easier and cheaper than with the first applications.

Fields of use

The main field of use is obviously telecommunications, but fiber optics easily overflow this field and work with a large number of manufacturing applications.

Telecommunications

The two main fields of use, related to network needs, have been urban links, with large capabilities and working without intermediate amplification or remote power-feeding, and submarine links such as trans-oceanic links, or coastal links without repeaters (exceeding 200 km, and over 300 km with optical amplification in terminals). Then, stimulated by the arrival of new operators, regional, national and international terrestrial connections have seen a very strong growth. They are at the base of the ATM network infrastructure.

In the early days of fiber optics, several experiments were carried out in the field of access networks for video communications and broadband service users. Direct fiber optics access or FTTH (fiber to the home) did not spread as expected in the 1980s because of economic constraints and the increase of possible throughputs with twisted pairs (available through ADSL). Intermediate solutions were then developed where the fiber was relayed, at the front end, by existing cables or radio contact. However, the need for increasingly high throughputs has reenergized this market; this movement started in Japan and Sweden and has spread to now represent a large part of the activity for manufacturers, operators and regulation authorities.

Data networks and links

Even for short distances, the use of fiber optics in the information technology field has rapidly progressed, particularly for electric insulation and insensitivity to electromagnetic disruptions. Fiber optics also enable development of multi-terminal networks and high bitrate networks, such as Fiber Channel or 1 Gbit/s (and now up to 10 Gbit/s) Ethernet, were designed for fiber optics from the start. Networks now reach “metropolitan” sizes and work together with railway networks or electricity transport without technical problems.

Industrial systems

There are various applications (telemetry, remote controls, video monitoring, field bus) where fiber insensitivity to parasites and its insulating character are essential advantages.

The massive parallelism of electronic and information technology architectures, the constant increase of frequencies on buses and the resulting electromagnetic accounting problems influence the increasing use of optical support (fibers or planar guides) to interconnect the different computer or embedded system cards (“optical backplane” concept), followed by the different chips from multiprocessor architecture, and even the different parts of a single chip in the future, in a SoC (system on chip) design.

Instrumentation and sensors

Fibers are more and more present in optical instrumentation, where they make it possible to carry out remote measurements in hard to access points. Sensors use fiber optics as a sensitive element and transmission support. However, their use remains limited, especially when material integration or total electromagnetic immunity is required. These applications are discussed in Chapter 11.

Finally, optical fibers still play a role in light transportation. Traditional applications (lighting, visualization, endoscopy) or more recent applications (laser beam transportation for industry, measurement, medicine) have seen their performance improve, and their cost decrease, thanks to the development of fiberoptic technologies.

Elements of a fiber-optic transmission system

Interfaces

In a point-to-point link as well as in a network, we find (see Figure 1):

– the optical transmitter interface which transforms the electric signal into optical signal. It mainly includes the optoelectronic transmission component, which can be a light-emitting diode (LED) or a laser diode (LD), components studied in Chapter 6. The interface also contains adaptation and protection circuits; it is connected to the cable by a connector or by an optical fiber pigtail that needs to be connected. Modulation is generally a light intensity modulation obtained by modulation of current going through the transmission diode or, at a very high bitrate, by an external modulation;

– the optical receiver interface containing a photodiode which converts the received optical signal into an electric signal. It is followed by a head amplifier, which must be carefully designed as its noise is generally the one limiting the minimum optical power that can be detected, and thus the range of the system (see Chapter 7). According to the application, we then find filtering or digital reshaping circuits.

Figure 1.Point-to-point fiber-optic link

Repeaters

When the link length requires it, one or several repeaters are inserted. Ancient link repeaters (installed prior to 1995) contained receiving and transmission interfaces, linked by amplification and regeneration circuits for digital transmissions, leading to signal interruption. Nowadays, terrestrial and submarine links use erbium-doped fiber amplifiers (see Chapter 8), and are entirely optical over distances exceeding 10,000 km.

Wavelength division multiplexing

Wavelength division multiplexing (WDM) enables the multiplexing of several signals in the same fiber-optic, even if they are from geographically different origins or in opposite direction. If wavelengths are close, they can be amplified by the same optical amplifier.

This was first used in local area networks or user links, as well as in sensor networks. It now increases even already installed optical cable capabilities significantly.

Dozens, and even hundreds of Gbit/s per fiber are reached in commercial links. Achieved in the laboratory ten years ago, the Tbit/s (1012 bit/s!) now corresponds to long distance infrastructure requirements; the last laboratory benchmarks have exceeded 10 Tbit/s and the theoretical limit has not yet been reached.

Fiber-optic networks

All optical networks can be developed which are not simple point-to-point links combinations interconnected by electric nodes. Optical network nodes can be passive (splitters, multi-branch couplers, wavelength division multiplexers) or active (switches, time-division multiplexers) components using integrated optics or micro-technologies (MEMS).

The development of large scale optical switching is one of the major issues today. In fact, the speed of optical transmissions is such that the bottleneck is now located in the network node electronics. However, specific architectures and protocols for optical routing must be developed.

Transmitted signals

The vast majority of applications (telecommunications, information technology) consist of digital transmissions, with bitrates from a few kbit/s to more than 10 Gbit/s. However, analog applications still subsist in video and telemetry fields.

There are more particular cases, such as retransmission over fiber optics of carrier microwave frequencies to 30 GHz, or even modulating a laser diode. This technique (already used for producing active antennae, notably for radars) is beginning to be used in satellite telecommunications stations or for retransmission of microwave signals in future access networks combining fiber and radio.

Fiberless infrared connections

Even though this book is mainly dedicated to fiber-optic telecommunications, we can mention the development of infrared wave use for line-of-sight communications at very short distance: remote control, hi-fi accessories, mobile robotics, which are traditional applications, as well as wireless local area networks. While benefiting from the practical advantages of “wireless”, infrared can transport high throughputs and resolve certain disruption and confidentiality problems raised by radio links. They are well adapted to indoor propagation (within a building, not going outside). Different protocols were defined, with the most well known being IrDA for interfacing PCs and various peripherals to 4 Mbit/s.

Direct links between buildings with 0.8 µm laser beams have recently been developed, used as microwave point-to-point beams. These free space optics (FSO) systems avoid the cost and delay of cabling. These beams are very directional because of a telescope system (lens), and work at low power (a few mW), but can be interrupted by strong fog. Very high (2.5 Gbit/s or more) bitrates can be transported over a few kilometers with no obstacles, interferences or a need for a license, contrary to radio. Important development in these systems is expected.

There are also link projects between satellites by laser beams in spatial vacuum over thousands of kilometers, obviously requiring great precision in the orientation of transmitters.

Chapter 1

Multimode Optical Fibers

1.1. Overview of optics

1.1.1. Introduction

We can begin studying light propagation in optical fibers and guides by light ray propagation in the sense of traditional geometric optics.

However, this hypothesis is only valid if the guide's transverse dimensions are much larger than the wavelength λ. The optical fiber is then considered multimode, which means that for a given wavelength, several light rays can propagate.

Transmission performances are not optimal, but this type of fiber has a certain economic advantage compared to single-mode fibers, which have a much smaller core.

1.1.2. Propagation of harmonic plane waves

As we know, a ray is not a real model because the light cannot remain concentrated on a very small line because of diffraction. It can, however, be approached using the plane wave model, which is a harmonic wave propagating without divergence in the z direction (see Figure 1.1):

Figure 1.1.Harmonic plane wave

This wave is a solution for the propagation equation (this topic will be addressed in the next chapter) for a sine wave of angular frequency ω:

as long as the wave width is infinite ( and have the same value, in amplitude and phase, in every point on a plane parallel to xOy, or phase plane).In practice, its width must be much larger than the wavelength. This is the case, for example, for beams radiated by gas lasers that have a width in the order of millimeters:

is the phase velocity, which is the phase front propagation velocity;

is the material impedance;

In the vacuum:

In dielectric material, we have:

We will only consider dielectrics which are perfectly transparent where index n is real. The previous relations then become:

the light velocity being divided by the refraction index:

In what follows, we will only use the vacuum wavelength, which we will note as λ.

The power transported by this wave is equal to the Poynting vector flow, noted as: , and is co-linear to in isotropic mediums.

1.1.3. Light rays

We can locally assimilate a light ray to a plane wave if the field variation on a distance similar to the wavelength is negligible; we can then note:

where is the wave vector with a direction that is tangent to the light ray and its module is equal to:

In the geometric optics hypothesis, and in the case of isotropic mediums, the “light ray” path (see Figure 1.2) can be calculated by ray equation:

with unitary vector tangent to the path and s being the curvilinear abscissa.

We can also note this as:

Light rays lean towards increasing indices. They can therefore be guided into a high index level, which is well known in atmospheric propagation of light or radio frequency (mirage effect) rays. This guidance mode is used in graded-index optical fibers.

Figure 1.2.Light ray propagation

1.1.4. Dielectric diopter

In Figure 1.3 we illustrate a plane diopter between two perfect dielectric mediums, i.e. with real n1 and n2 indices. These indices become complex in the case of a partially absorbing medium. This issue will not be addressed here.

An incident plane wave (i indexed) arriving in the diopter with an angle of incidence θi will split into a reflected wave (indexed r) with θr angle of reflection and a transmitted wave (t indexed) θt angle of refraction. We represent a path for the wave, traditionally called the “light ray”, but we must not forget that the wave spreads on both sides of this ray.

Since there are neither charges nor currents in the dielectric, there is phase continuity of the electromagnetic field, hence continuity of phase fronts on both sides of the interface, which is written as:

where:

Figure 1.3.Plane dielectric diopter

which means that they will propagate together in the direction of axis z while remaining in-phase between each other and that the resulting field will take the form:

1.1.5. Reflection of a plane wave on a diopter

The above relations give wave directions, but not the distribution of energy between them. We must conduct an electromagnetic analysis to determine it. To do this, we must split the incident wave into a component TE ( parallel to the diopter) and a component TM ( parallel to the diopter). These two cases are represented in Figure 1.4.

Due to field continuity at the interface between both mediums:

This continuity is a essential property of dielectric guides, and a fundamental difference from the metallic wave guide.

Figure 1.4.Reflection of a plane wave on a dielectric diopter

We can bring all cases down to the reflection of a wave perpendicular to the diopter by splitting field vectors and by noting E' and H' as their components parallel to the diopter.

For magnetic fields, sign conventions between incident and reflected waves are reversed because of the reversal of propagation direction. We then write:

1.1.6. Reflection coefficient

The reflection coefficient in complex amplitudes is defined by:

which is equivalent to the reflection, at the end of an electric line, over an unmatched impedance. This depends on polarization (TE or TM).

The reflection of any θi incidence wave then becomes equal to the reflection of a zero incidence wave between two mediums of impedance:

and:

There is phase reversal if n2 > n1.

Figure 1.5.Reflection coefficient versus the angle of incidence: a) case where n2 > n1; b) case where n2 < n1

For zero (or low) incidences, the reflection coefficient in power equals:

giving 4% for the air (n ≈ 1) – glass (n ≈ 1.5) interface.

This Fresnel reflection is one of the causes of fiber-optic access and connection losses, and especially of integrated and optoelectronic optical components, where the indices are much higher. This is remedied by antireflection layers (dielectric layer stacking for an index adaptation at the wavelength used).

1.1.7. Total reflection

If n2 < n1, there is total reflection for θi > θlim, angle of limit refraction given by:

The medium 2 impedance, which is in cos θt or 1/cos θt depending on its polarization, then becomes completely imaginary.

This is the equivalent of an electric line loaded by reactive impedance. The reflection coefficient becomes imaginary and is written as:

depending on incidence and polarization.

We then observe in medium 2 a non-homogenous wave; the wave vector's components are:

This wave's field is then written as:

This wave presents a wave profile that is progressive in direction z, in phase with the progressive wave in medium 1, and an exponential profile decreasing in direction x. It is called an evanescent wave.

Its Poynting vector is purely imaginary; the energy of its wave is therefore purely reactive and does not modify reflected wave energy, as long as medium 2 is perfectly transparent (otherwise there is partial wave absorption).

These interactions are used in guided optics for splitters, as well as in optical near field microscopy.

1.2. Dielectric waveguide

1.2.1. Planar dielectric waveguide model

The planar guide is a 2D model (infinite in lateral direction y) in which the 2a thickness and n1 index guide is surrounded by mediums of lower indices n2 and n'2 (see Figure 1.6).

These mediums must have an infinite thickness, or a thickness that is large compared to the penetration depth of the evanescent field.

We will study this guide in geometric optics. A light ray propagating with an angle θ with Oz axis is guided by total reflection if: cos θ > n2/n1.

θ is the angle complementary to the angle of incidence θi previously used.

By splitting all three wave vectors according to Ox and Oz, we note:

The sign of the quantity k02n2 – β2 determines the progressive (if positive) or evanescent (if negative) character of a wave in the n index medium. The ray therefore remains guided in the n1 index medium on the condition that:

The three waves have the same longitudinal β propagation constant. They will therefore progress in direction Oz by remaining phased between each other. The resulting field will take the form:

Figure 1.6.Plane dielectric wave guide

1.2.2. Notion of propagation modes

Distribution E(x) of field Ox is determined by conditions at limits. In fact, in order to observe a stationary wave on a guide section, which is called a propagation mode, the wave must be in phase after a round trip between both diopters. The phase shift: 4a. α + 2ϕ) must then be a multiple integer of 2π.

The first term of this phase shift is caused by propagation in Ox direction, and the second by phase shift ϕ at the total reflection which can be calculated according to section 1.1.7 since its expression is slightly different for TE and TM waves.

The phase adaptation equation can be graphically solved and deduce α, β and λ can be deduced by introducing the normalized variables:

Figure 1.7.Goos-Hanchen effect and field pattern in the guide

Modes are orthogonal between each other, i.e. the fields of both modes i and j are bound by the relation:

The number of solutions, or modes, is completely determined by the value of V. The m-order mode exists if V > m.π/2, its cut-off frequency being given by:

The cut-off wavelength of the m-order mode is λc/m. It means that this mode can only be guided at a lower wavelength (then at higher frequency), otherwise it is refracted. These results are the result of the geometric optics approximation, which is only valid if V >> 1; they are therefore not rigorous if there are few modes. The cut-off frequency of 0-order mode is zero, which means that it is guided regardless of the wavelength. This is obviously theoretical and does not consider infrared matter absorption.

1.2.3. Case of the asymmetric guide

This case is frequent in integrated optics where the guide, thin layer of n1 index, is inserted between a substrate of slightly lower index n'2, and a coating of index n2 lower than n'2. Again in geometric optics, the phase matching condition, therefore existence condition, of the m type mode of tilt θ on the axis, is written in this case:

In fact, when the mode leans towards its cut-off, its β propagation constant leans towards k0n2, and therefore becomes lower than k0n'2 at a wavelength λc where it will be able to propagate in the material of index n'2 (the substrate in the case of integrated optics). Geometrically, θ tends to become higher than 0'2, angle of limit refraction at substrate-guide interface, which causes a refraction in the material whose index is closest to the guide's index.

This phenomenon happens when the normalized V frequency decreases, i.e. when the wavelength increases or when the guide's thickness decreases.

In addition, internal reflection phase shifts do not have the same value for TE and TM modes, so the cut-off frequencies will not either. In an asymmetric guide, there is a range of wavelengths where mode TE0 only, and not mode TM0, is guided, i.e. a polarized light.

Figure 1.8.Search for a solution (curve intersection points) in both types of plane guides

1.2.4. Dispersion

Once the β constant is determined for each mode, we deduce:

– its phase velocity: phase front propagation velocity; and its inverse called the phase delay (by unit of length):

– its group velocity: pulse propagation velocity; and its inverse called the group delay (by unit of size):

These values depend on both the mode order and angular frequency, which leads to a double dispersion:

– intermodal dispersion: caused by group delay difference between modes;

– intramodal dispersion, or chromatic dispersion: caused by group delay variation for each mode with its angular frequency, therefore with its wavelength. In practice, this effect is combined with the variations of refraction indices with the wavelength (called material dispersion).

We can note the limit values of these propagation velocities:

The dispersion will be weaker as n1 and n2 are closer. Dispersion minimization is an essential point of optical fiber optimization.

1.3. Multimode optical fibers

1.3.1. Definition

When transverse dimensions of a guide are much larger than the wavelength, the equation in section 1.2.3 will have a solution even for high m, which means that a large number of modes can be guided.

We can then say that a mode corresponds to an authorized path, resulting from constructive interferences between multiple reflections on the diopter between both materials.

A fiber-optic is a circular dielectric waveguide which is very probably multimode if the core, or in other words the central part where light propagates, has a diameter much larger than the wavelength. This diameter is approximately between 50 and 200 µm for silica fibers, and 0.5 and 1 mm for plastic fibers. We can then simply, but correctly, study propagation by geometric optics.

In Chapter 2, modes will be more rigorously defined by the resolution of the propagation equation deduced from Maxwell equations.

We will see that a mode is characterized by its path and by the distribution of the electromagnetic field around it. We must insist on the fact that in a multimode guide, the different modes are on the same wavelength.

1.3.2. Multimode step-index fiber

The simplest type of multimode fiber is a step-index fiber (see Figure 1.9), directly emerging from optical applications. In this structure, the core of refraction index n1 is surrounded by a cladding of slightly lower index n2. These indices are close to 1.5 for silica fibers. This cladding plays an active role in guiding and is also surrounded by a coating.

Within the Oz cylinder axis fiber-optic, the ray is guided if angle θ that it produces with Oz remains lower than θ0, the limit refraction angle deduced from the Snell-Descartes laws:

If θ > θ0, the ray is refracted. Otherwise, it is guided by total reflection at the core cladding interface. This remains true if the fiber is not straight, as long as the bending is not too pronounced. This makes it possible to consider long haul transmission with low loss levels and without transmitted information leaking outside.

Figure 1.9.Step-index fiber

Because of this condition, the maximum angle of incidence at fiber input, i.e. the acceptance cone aperture, is given by the numerical aperture:

In return, it is the angle within which the light coming out of the fiber diverges.

Along with diameters, core at 2a, and cladding at 2b, the numerical aperture is the most important parameter of an optical fiber. A large numerical aperture makes it possible to couple a large quantity of light, even from a divergent source such as light-emitting diode (LED). On the other hand, it will lead to spreading of transmitted pulses over time, because there are large differences in the length, therefore in propagation time, of the different guided rays (intermodal dispersion effect).

These fiber optics are quite suitable for optical applications and for very short distance transmissions. There are a certain number of multimode fibers that are different in material (plastic, silica/silicone, or “all silica” – not widely used for step-index) and in their characteristics (see Chapter 2).

1.3.3. Multimode graded-index fiber

The graded-index fibers were designed specifically for telecommunications in order to minimize this intermodal dispersion effect without significantly reducing the numerical aperture or the coupled power. Their core index decreases according to a parabolic-like law from the axis to the core cladding interface (see Figure 1.10). In this way, the rays follow a sine type path, and those with the longest path go through lower index mediums, increasing their velocity and making it possible to approximately equalize propagation delays.

Figure 1.10.Graded-index fiber

The index profile law takes the form:

Exponent α is close to 2; its exact value is optimized to minimize intermodal dispersion and depends on the material and wavelength.

We again define the numerical aperture as with step-index fibers. However, n1 is the core index only on the axis. Since the numerical aperture decreases the further away from the axis we move, with equal conditions approximately half as much power is coupled as in a step-index fiber.

The cladding does not occur in guiding itself, but plays an important spatial filtering role by eliminating the most tilted rays.

Several “all silica” graded-index (GI) fiber standards have been normalized for short distance telecommunications applications (50/125 fiber, now used for high bandwidth local area networks), local computer networks (62.5/125) and first generation video distribution (85/125, still not widely in use). Graded-index plastic fibers appear for the Gigabit Ethernet.

1.4. Propagation in multimode optical fibers

1.4.1. Ray paths

Figure 1.9 and 1.10 diagrams only represent meridional rays, remaining in a plane containing the optical fiber axis. In order to calculate all paths, we must switch to 3 dimensions. Due to the symmetry of revolution of the optical fiber, we use cylindrical coordinates (see Figure 1.11), by noting as:

– r, z, Ψ coordinates from the path's current point, P;

– local trihedral.

We must solve the ray equation:

This vector has for components:

– over

– over

– over

The resolution of this equation will result in the following wave vector:

Figure 1.11.Calculation of ray paths in an optical fiber

1.4.2. Propagation equation resolution

Since index n(P) only depends on r, distance from point P to axis Oz, the gradient of n is along the direction. We can therefore solve the equation along and , directions in which n components are zero.

Along z, which has a constant direction, we immediately have:

Along which does not have a constant direction, the calculation is slightly longer (detailed in the appendix of this chapter) and leads to:

These two constants are deduced from initial conditions:

where rE, θE and (ϕE correspond to the point of incidence and to the tilt of the incident ray at the fiber-optic entrance (see Figure 1.11) where Θe is defined outside of the fiber.

The wave vector can then be written as:

The radial component f(r) can be deduced from constants and n(r) index profile through module of k which is given by:

1.4.3. Different ray types

Light rays will propagate where the wave vector is real, or where f(r)2 is positive; otherwise we have an evanescent field. This condition must not be met at any point in the cladding in order for them to remain guided in the core. We must therefore study the sign of:

1.4.3.1. Meridional rays

Meridional rays remain in the plane containing the axis Oz, so ϕ and v are zero and these rays propagate where: β < k0 n(r).

They are guided in the core if: β > k0 n2, which corresponds to the initial condition:

generalizing the definition of the numerical aperture at the point E.

1.4.3.2. Skew rays

ν is not zero and we thus observe 3 cases (see Figure 1.12):

– the ray is guided if f(r)2 < 0 in all the cladding, or as previously: β > k0 n2; this ray, which never cuts off the axis, is also called gallery mode;

– the ray is refracted if f(r)2 > 0 in all the cladding, or:

– the ray is leaking in the intermediate case: .

In the case of leaking rays, f(r) 2 < 0 in the part of the cladding neighboring the core (we have an evanescent field), but becomes positive again further away, thus defining a zone in the cladding where a ray can propagate once more. This ray will gradually attract the energy propagating in the core and thus leak in cladding through the evanescent field, based on a phenomenon similar to the tunnel effect in quantum mechanics.

This type of ray therefore cannot guide energy in the core over a long distance, but can propagate over short distances before elimination, and thus disrupt transmission and measures.

Figure 1.12.Different types of rays in a multimode fiber

1.4.4. Modes of propagation

Due to the wave vector expression, the electric field takes the following form:

This is a guided mode field if its distribution is stationary over a plane perpendicular to the fiber axis, which requires a phase matching:

– over the phase shift equals ; it must be an integer multiple of 2π, then ν, which is dimensionless, must be an integer, called azimuthal order. It is the number of field periods on a girth. v can be positive or negative, corresponding to both rotation directions of a skew ray around the fiber axis.

It has a maximum vM value that can be graphically determined (see Figure 1.12), with the knowledge that it is not possible to have:

In fact, there would no longer be a progressive field zone where the ray would propagate;

If m is large, we can neglect phase shifts on caustics before the first term (propagation based phase shift) and write, by using the expression of f(r):

– for a step-index fiber:

– for a parabolic graded-index fiber: N ≈ V2/4. These results are only valid if V is large.

1.5. Dispersion in multimode optical fibers

1.5.1 Intermodal dispersion

Due to the dispersion of propagation delay between modes, called intermodal dispersion, a light pulse injected at the fiber-optic entrance will arrive in the form of a large number of shifted pulses. Since the modes are numerous and the propagation delay difference between two modes is much lower than the photodetector response time, it will deliver the pulse envelope, or a pulse response with a standard deviation Δτim For convenience of measurement purposes, the half-maximum width of the received pulse is generally considered.

It is very important to note that the envelope and mid-height pulse width do not only depend on differences in propagation delay between modes, but also on the light power distribution between them. This depends on light injection at entry as well as on geometric conditions of the continuation of the fiber (bending, constraints) modifying this distribution. We can however calculate the maximum received pulse width, which is the propagation delay difference between the slowest and fastest modes.

1.5.2. Pulse broadening calculation

1.5.2.1. In step-index fibers

In multimode fibers, we can approximately consider rays propagating in locally homogenous mediums. In this approximation, group delay locally equals:

where:

s is the curvilinear abscissa along the ray. Propagation time over a fiber-optic L length is deduced by integration along this ray:

Global propagation delay of the meridional ray with a θa tilt on the axis then equals (see