In a book like this, the development of computer programs for various tasks and also execution of simulations for different processes and devices, plays an essential role. The fundamentals of many computer programs are supported by numerical methods. Therefore, in this Appendix we summarize main elements of numerical analysis with an emphasis on methods related to the development of programs used in this book, and also to understanding of operation of those programs.

There are many excellent textbooks devoted to numerical analysis. We found the books by Koonin [1], DeVries [2], Garcia [3], Gerald and Wheatley [4], Rao [5], Heath [6] and Recktenwald [7] of significant pedagogical value. The books by Press et al. [8] stand on their own as an excellent source of practical computer codes ready to use.

We concentrate on description and implementation of some practical numerical methods and not on the problems which those methods are typically used for. We start our discussion with a summary of methods of solving nonlinear equations.

There are many textbooks aimed to the introduction of numerical methods and their applications. Some of the most popular are: Applied Numerical Analysis Using MATLAB by Fausett [9], Numerical Methods for Physics by Garcia [3], Introduction to Scientific Computing by van Loan [10], Advanced Engineering Mathematics with MATLAB by Harman et al. [11], A Friendly Introduction to Numerical Analysis by Bradie [12].

In the present chapter we will combine some of the methods developed earlier to create the simple point-to-point optical simulator which represents the simplest photonic system. It involves the transmitter, optical fibre and receiver. Some issues of how to quantify the quality of transmission in such a system will also be reviewed. Performance evaluation and tradeoff analysis are the central issues in the design of any communication system. Using only analytical methods, it is practically impossible to evaluate realistic communication systems. One is therefore left with computer-aided techniques.

In the last 10–15 years the design of photonic systems has moved from the back-of-the-envelope calculations to the use of sophisticated commercial simulators, see for example, products advertised by Optiwave, like OptiSystem [1], by RSoft Design Group the Optical Communication Design Suite [2] and by VPI Photonics line of products [3], to just name a few important players. They contain sophisticated physical models and allow for rapid assessment of new component technologies in the system under design.

Around 1995 the design of optical communication systems (operating over medium distances) would involve only a balance of power losses and pulse spreading. Later on, the demand on billion-dollar systems required complex analysis during the design process. This in turn created the need for sophisticated simulators.

Computer simulations can quickly provide answers to several important questions essential to every engineer designing optical communication system, like: what repeater spacing is needed for a given bit rate, or what is the required power generated by a transmitter?

This chapter calls on trigonometry and calculus, both differentiation and integration.

In this chapter, we will see what happens when terahertz-frequency electromagnetic radiation encounters matter.

Imagine driving along a freeway. All the cars are travelling along at the same, high speed, 100 kmh. There is no stopping or turning on the freeway. (It is a little bit boring, really.) Then the freeway ends. The traffic slows, say to 60 kmh. Some cars pull over at the shops. Some cars turn off to the suburbs. Others continue through the town, and rejoin the freeway, and continue at 100 kmh.

No analogy is exact, but the cars on the freeway are something like light in a vacuum. In a vacuum, light travels at a steady speed and in a straight line. Entering the town is something like light encountering matter; the speed limit decreases. When light encounters matter, it slows down. The change in speed is related to the refractive index and the phenomenon of refraction. Cars stopping are something like the absorption of light by matter; those vehicles are lost from the traffic flow. Some cars turn aside, just as the scattering of light diverts it from its original trajectory. Some cars may even make a U-turn and head back down the freeway along the direction they just came. This is something like the reflection of light. You can't make a U-turn on the freeway.

Photonics (also known as optoelectronics) is the technology of creation, transmission, detection, control and applications of light. It has many applications in various areas of science and engineering fields. Fibre optic communication is an important part of photonics. It uses light particles (photons) to carry information over optical fibre.

In the last 20 years we have witnessed the significant (and increasing) presence of photonics in our everyday life. The creation of the Internet and World Wide Web was possible due to tremendous technical progress created by photonics, development of photonic devices, improvement of optical fibre, wavelength division multiplexing (WDM) techniques, etc. The phenomenal growth of the Internet owes a lot to the field of photonics and photonic devices in particular.

This book serves as an attempt to introduce graduate students and senior undergraduates to the issues of computational photonics. The main motivation for developing an approach described in the present book was to establish the foundations needed to understand principles and devices behind photonics.

In this book we advocate a simulation-type approach to teach fundamentals of photonics. We provide a self-contained development which includes theoretical foundations and also the MATLAB code aimed at detailed simulations of real-life devices.

We emphasize the following characteristics of our very practical book:

1. • learning through computer simulations

2. • writing and analysing computer code always gives good sense of the values of all parameters

3. • our aim was to provide complete theoretical background with only basic knowledge assumed

4. • the book is self-contained in a sense that it starts from a very basic knowledge and ends with the discussion of several hot topics.

In this chapter we summarize the beam propagation method (BPM). The method is widely used for the numerical solution of the Helmholtz equation and also for the numerical solution to the nonlinear Schroedinger equation (to be discussed in Chapter 15 dealing with solitons). It is the most powerful technique for studying the propagation of light in integrated optics. The method was originally introduced by Feit and Fleck in the late 1970s [1]. The BPM was initially based on FFT algorithm. Later on it has been extended to finite-difference based BPM schemes (FD-BPM) and finite-element BPM (FE-BPM) and many others. The main characteristics [2] of FD-BPM are the ability to simulate structures with large index discontinuity, less memory and time consumption in modelling complex structures, the possibility to incorporate wide-angle and full vector algorithms and the ability to incorporate transparent boundary conditions.

There is a large number of algorithms available in the literature and almost all of them are based on the concept of a propagator. Propagators are mathematical ‘objects’ which propagate fields from one space coordinate to another. An example of the system where propagation takes place, known as an optical waveguide, is shown in Fig. 12.1. It splits input optical signal into two arms, see also Fig. 12.2. The role of BPM is to determine field profile along the waveguide knowing the distribution of refractive index over the whole waveguide.

This chapter calls on maths, but the maths is relatively elementary. Section 4.1 requires only the basic operations of addition, subtraction, multiplication and division. Section 4.2 requires calculus. Section 4.3 requires trigonometry, but is a rather direct extension of the methods we have already met in Chapter 2. Section 4.5 assumes some geometrical facility in picturing planes, circles and ellipses. Section 4.6 assumes a background in integration and differentiation.

Let there be light! Light is a basic physical entity. One might argue that light is the most fundamental physical phenomenon of all.

Physics is about energy and matter and their interaction. This chapter is about energy in its purest form, energy in the form of light. Chapter 5 is about matter. Chapter 6 is about the interaction between light and matter. These three chapters constitute the physics core of the book.

We start with the photon (Section 4.1). A photon is a bunch, a packet, a particle of light. We will look at what it means to describe light in this way.

An alternative way to think about light is to think of it as a wave. Maxwell's equations lead inexorably to the conclusion that light is an electromagnetic wave (Section 4.2).

Waves are a generalisation of oscillations. Oscillations involve only one variable, time. Waves add to this an additional variable, displacement. So waves can be described mathematically by extending the ideas of Chapter 2 in a straightforward manner.

This chapter uses mathematics and the mathematics increases in sophistication as the chapter proceeds. Section 10.1 requires no mathematics. Section 10.2 assumes you know about sine and cosine and how to differentiate. Section 10.3 assumes you know about trigonometry and Fourier transforms. Section 10.4 assumes you can handle exponential notation and complex numbers.

Spectroscopy is the division of light into its separate frequencies. For example, sunlight is separated into its separate frequencies by raindrops when a rainbow forms. The separate frequencies make up the spectrum. The sun and the raindrops constitute a spectrometer, a device for producing a spectrum.

In the rainbow, the blue light is separated from the green light and the red light and so on. Breaking up light into its separate components is referred to as analysis. The opposite process is synthesis – making white light, for example, by combining red, green and blue, as is happening right now on the computer screen before me as I type.

Newton made a simple spectrometer by sending sunlight through a glass prism. The different colours emerged in different directions. This is the principle behind the dispersive spectrometer: the different colours are separated in space by an optical element such as a prism or a grating. The key characteristic of the core optical element is dispersion, that is, it acts on different frequencies differently. Section 10.2 deals with dispersive spectrometers.

This chapter uses a lot of trigonometry. And very little else.

In the last chapter we met oscillations. In this chapter we will see what happens when we combine oscillations.

We have already seen, in passing, two examples of combining oscillations. The time-bandwidth theorem concerns oscillations with a range of frequencies and says that the spread in time goes up as the spread in frequencies goes down. The Fourier theorem says that by combining oscillations of fixed frequencies – all multiples of a fundamental frequency – an oscillation of arbitrary profile is constructed. These two examples may involve many oscillations, even an infinite number. We will take a step back in this chapter and restrict ourselves to combining two oscillations.

Underpinning all our calculations is the assumption that to combine oscillations we simply add them together. This is called the principle of superposition.

We begin with the simple and proceed to the complex. We will start by combining oscillations that have a lot in common, then move on to oscillations that have less in common. We saw in Chapter 2 that three key properties characterise an oscillation: frequency, amplitude and initial phase. We will begin by combining oscillations that have the same frequency, amplitude and initial phase, then move on to oscillations that differ only in amplitude, or only in frequency, or only in initial phase.

This chapter employs trigonometry, differentiation, integration and vector algebra.

Electromagnetic radiation is produced

1. • by hot objects,

2. • by electric charges moving freely,

3. • by transitions between defined energy levels.

In Chapter 4, I discussed in detail the radiation given off by hot objects. Heat any object, and it will give off light. Thermal sources are perhaps the simplest and best-known sources of light. The sun radiates light because it is hot. Likewise the stars. The moon reflects the light of the sun. The embers of the fire glow because they are hot. Terahertz radiation is produced by thermal sources, but this will not be discussed in detail in this chapter; see Chapter 4 for further information.

An electromagnetic wave may be produced by ‘waving’, or appropriately moving, an electric charge, or charges. The principles behind the production of electromagnetic radiation by moving electric charges are set out in Section 7.2. The synchrotron and the free electron laser are two sources of terahertz-frequency electromagnetic radiation based on these principles.

Transitions between defined energy levels are the basis of the laser. The principles behind this are set out in Section 7.3. These principles inform the operation of the Gelaser, the quantum cascade laser and the molecular laser, which all operate at terahertz frequencies. Moreover, many additional schemes for terahertz generation are based on using visible or near-infrared lasers, either operating in the continuous mode, or operating in the pulsed mode.

In this chapter we summarize the operation of an optical receiver, which is an important part of an optical communication system. An overview of design principles for receivers used in optical communication systems is provided by Alexander [1]. The aim of a receiver is the recovery of the transmitted data. The process involves two steps [2]:

1. the recovery of the bit clock,

2. the recovery of the transmitted bit within each bit interval.

A block diagram of an optical receiver is shown in Fig. 10.1 [3], [4].

The receiver consists of a photodetector which converts the optical signal into electrical current. A good light detector should generate a large photocurrent at a given incident light power. They should also respond fast to the input changes and add minimal noise to the output signal. This last requirement is of crucial importance since the received signal is typically very weak. In digital optical communication systems the detection process is often conducted with a PIN photodiode.

There are generally two types of detection [4]: direct detection (also called incoherent detection) and coherent detection.

Direct detection detects only the intensity of the incident light. It is used mainly for intensity or amplitude modulation schemes. It can only detect an amplitude modulated (AM) signal.

Coherent detection can detect both the power and phase of the incident light. It is therefore used when phase modulation (PM) or frequency modulation (FM) is preferred. Coherent detection is also important in applications such as WDM.