The mathematics used in this chapter is limited to the operations of addition, subtraction, multiplication, division and a little differentiation.

In this chapter, we look at sensors of terahertz-frequency electromagnetic radiation. Terahertz sensors detect invisible terahertz radiation and convert it to something perceptible to a human being – usually a number on a dial or on a screen.

Terahertz-frequency electromagnetic radiation is invisible. The human eye cannot see it. How can we tell it is there?

The other human senses are not much help. We cannot taste, or hear, or smell terahertz radiation. We can feel it – we perceive terahertz radiation as heat – but only if it is rather intense. The human body, on the whole, is rather insensitive to terahertz radiation.

Devices that are sensitive to terahertz radiation and convert it to a signal that humans understand are called sensors or detectors. The general term for a device that converts a signal from one form to another is a transducer. Most commonly, the transducer takes a physical property, such as temperature, acceleration or viscosity, and converts it to an electrical signal, since electrical signals are easily stored and manipulated. Of course, a terahertz-frequency electromagnetic wave is already an electrical signal, but it is at too high a frequency and often at too low an intensity for a human to sense easily. A terahertz sensor takes in the terahertz-frequency electromagnetic wave and gives out or records a reading at a lower frequency that is perceptible to you or me.

Wavelength division multiplexing (WDM) is a modern practical method of increasing transmission capacity in fibre communication systems. It uses the principle that optical beams with different wavelengths can propagate simultaneously over a single fibre without interfering with one another. In the wavelength range of 1280–1650 nm (like an All Wave fibre [1]) the useable bandwidth of a single mode fibre is about 53 THz. In recent years an improved (denser) WDM system known as DWDM is under development.

In this chapter we discuss some of the WDM devices and also provide some applications of BPM developed earlier to simulate those devices. We start by summarizing the basic WDM system.

Basics of WDM systems

WDM is the main technique used in the realization of all optical networks. WDM is the technology which combines a number of wavelengths onto the same fibre.

Key features include:

1. • capacity upgrade

2. • transparency (each optical channel can carry any transmission format)

3. • wavelength routing

4. • wavelength switching

Implementation of a typical WDM system employing N channels is shown in Fig. 13.1. In the shown system three wavelengths are multiplexed in one fibre to increase transmission capacity. The light of laser diodes with wavelengths recommended by the ITU is launched into the inputs of a wavelength multiplexer (MUX) where all wavelengths are combined and coupled into a single-mode fibre. When needed, propagating light can be amplified by an optical fibre amplifier and eventually imputed at the wavelength demultiplexer (DMUX) which separates all optical channels and sends them to different outputs.

In this chapter, we will give a basic introduction to optical sources with the main emphasis on semiconductor lasers. The bulk of our description is based on the rate equations approach. We start with a general overview of lasers.

Overview of lasers

Generic laser structure is shown in Fig. 7.1 [1], [2]. It consists of a resonator (cavity), here formed by two mirrors, and a gain medium where the amplification of electromagnetic radiation (light) takes place. A laser is an oscillator analogous to an oscillator in electronics. To form an oscillator, an amplifier (where gain is created) and feedback are needed. Feedback is provided by two mirrors which also confine light. One of the mirrors is partially transmitting which allows the light to escape from the device. There must be an external energy provided into the gain medium (a process known as pumping). Most popular (practical) pumping mechanisms are by optical or electrical means.

Gain medium can be created in several ways. Conceptually, the simplest one is the collection of gas molecules. Such systems are known as gas lasers. In a gas laser one can regard the active medium effectively as an ensemble of absorption or amplification centers (e.g. like atoms or molecules) with only some electronic energy levels which couple to the resonant optical field. Other electronic states are used to excite or pump the system. The pumping process excites these molecules into a higher energy level.

In this chapter we review the basic concept of metamaterials as those possessing simultaneously negative permittivity and permeability over the same frequency range. Theoretical principles and basic experimental results are reviewed. Possible applications including cloaking, slow light and optical black holes are described.

Introduction

Metamaterials are artificially created structures with predefined electromagnetic properties. They are fabricated from identical elements (atoms) which form one-, two- or three-dimensional structures. They resemble natural solid state structures. Metamaterials typically form a periodic arrangement of artificial elements designed to achieve new properties usually not seen in Nature [1]. In a sense, they are composed of elements in the same way as matter consists of atoms.

Metamaterials are characterized and defined by their response to electromagnetic wave. Optical properties of such materials are determined by an effective permittivity εeff and permeability μeff valid on a length scale greater than the size of the constituent units. In order to introduce such a description, one requires that the size of artificial inclusions characterized by d be much smaller than wavelength λ, i.e. d ≪ λ.

The name meta originates from Greek, μϵτα and means ‘beyond’. Main characteristics of metamaterials (MM) are:

1. • man-made,

2. • have properties not found in Nature,

3. • have rationally designed properties,

4. • are constructed by placing inclusions at desired locations.

With modern fabrication techniques it is possible to create structures which are much smaller than the wavelength of visible light.

This chapter calls on maths, but the maths is relatively elementary. The first four sections only require simple geometry. The final sections refer to Fourier transforms, but only at a descriptive level.

Sight is the most complete of our senses. Our eyes detect light. So our eyes are photon detectors. There's more: our eyes distinguish light of different colours. So our eyes are spectrometers. There's more: our eyes tell the direction the light is coming from. So our eyes are imaging devices. There's more: between them, our eyes let us build up a three-dimensional image of the scene we are viewing.

We extend our vision using instruments. For example, the telescope lets us see the distant; the microscope, the small.

We can extend our vision to other parts of the electromagnetic spectrum. To do this, we need an instrument sensitive to invisible radiation that converts it to something we can see. X-rays are an example. An x-ray viewer records x-rays arriving from different places and presents this in a way the eye can see. At first, photographic film was used to display x-rays. Now a computer monitor is standard. In principle, the x-rays could be separated according to frequency, but this is not usually done in practice. By taking multiple x-ray images from different angles, a three-dimensional x-ray image may be built up. This process is called computer-axial tomography or computer-aided tomography (CAT) or simply computer tomography (CT).

You don't know anything about trigonometry? That's OK, I'll teach you all you need to know as we go along. I will introduce summation notation, but explain it. If you know how to integrate, that might be an asset, but it is not strictly necessary as I will give you a visual description of what is involved. This should allow you to appreciate the meaning of the equations even if you do not have a full grasp of the apparatus of integration.

In this chapter we meet oscillations. We will look at the general way to describe any oscillation using mathematics.

To describe an oscillation in mathematical terms, we identify three key properties: how rapid it is, how large it is and when it starts. These three properties are more formally defined as frequency, amplitude and initial phase. We can express an oscillation mathematically by using a trigonometric function such as cosine or sine or by using a compact exponential notation involving complex numbers.

The time-bandwidth theorem appears over and over again in terahertz physics. It says that the product of the duration of a pulse (time) and the range of frequencies encompassed in the pulse (bandwidth) has a minimum value. Looked at in one way, if we have a short pulse, the pulse must involve a large range of frequencies. Looked at in another way, a well-defined frequency implies a very long pulse.

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 endswith the discussion of several hot topics.