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.

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.

The field of terahertz physics does not boast many elementary books. (I hope you have enjoyed this one.) Why not pursue the subject further through some of the recent books that take the subject to a higher level?

Sensing with Terahertz Radiation

Daniel Mittleman (Editor)

(Springer, 2003)

An edited collection of articles focussing on the detection of terahertz radiation.

Terahertz Optoelectronics

Kiyomi Sakai (Editor)

(Springer, 2005)

A book with much detailed technical information, featuring Japanese contributions to the field.

Terahertz Spectroscopy: Principles and Applications

Susan L. Dexheimer (Editor)

(CRC Press, 2008)

A book with much technical information, especially concerning spectroscopy.

Intense Terahertz Excitation of Semiconductors

S. D. Ganichev and W. Prettl

(Oxford University Press, 2006)

The subject matter is limited in both method (high intensity radiation) and the materials under investigation (semiconductors), but the treatment is definitive, even magisterial.

Principles of Terahertz Science and Technology

Yun-Shik Lee

(Springer, 2009)

A comprehensive overview citing much recent research literature.

Introduction to THz Wave Photonics

X.-C. Zhang and Jingzhou Xu

(Springer, 2010)

A volume summing up the recent research and with a strong emphasis on imaging.

Terahertz Techniques

Erik Brundermann, Heinz-Wilhelm Hübers and Maurice FitzGerald Kimmitt

(Springer, 2012)

A modernisation and amplification of Kimmitt's 1970 classic, ‘Far-Infrared Techniques’.

This chapter is largely descriptive. It only makes use of rather simple mathematics: algebra and trigonometry (including the exponential function).

The last chapter was about sources of terahertz radiation. The next chapter will be about sensors of terahertz radiation. In this chapter we will look at what happens between the source and the sensor. We will look at how terahertz radiation is manipulated: bent, focussed, collimated. We will use optics as a general term for manipulating light and optical elements as a general term for specific devices that manipulate light.

Following the progression of an electromagnetic field through a collection of optical elements can be carried out at various levels of sophistication.

1. Geometrical optics. At the simplest level, light rays can be traced through the optical system using the approximation that they travel in straight lines and that the effects of wave properties, namely diffraction, are ignored. The assumption that wave properties are neglected means this is referred to as geometrical optics or ray optics. The assumption that the optical elements, in particular lenses, are thin, and that the angle of divergence of light from the optical axis is small – in other words, that only paraxial rays are considered – means this is referred to as Gaussian optics, in memory of the one who developed this method. Associated with Gaussian optics are fundamental geometrical formulas you might have met in previous studies that go by names such as the thin lens formula, the lens makers’ formula, the mirror formula, the focal ratio or the f -number, formulas for magnifiers, microscopes and telescopes.

2. […]

This book aims to teach. It is directed to students. It is not directed to researchers.

Research literature is typically written in the third person. That is appropriate. In this book I will often write in the first person – there, I just did! – and address you – yes, you, the reader – in the second person. When I employ ‘we’ it is not as a royal plural, but as you and me together, author and reader, engaged in the common pursuit of applying ourselves to terahertz physics. I hope you are comfortable with, or will become comfortable with, this use of personal pronouns.

This book is a text book. So it contains text. But not unrelieved text. To give pedagogical structure, the text is divided into parts, into chapters, into sections and into subsections. But there is more, to accommodate different learning styles. Perhaps you are a visual learner; there are diagrams for you. Perhaps you are a native speaker of the language of mathematics; equations succinctly express much of the material. Perhaps you learn by example; there are examples sprinkled throughout. Perhaps you learn by doing; there are exercises aplenty. If you like reading dictionaries or telephone directories, you might like the glossary.

The previous chapters described the unusual electronic phenomena brought about by disorder and interaction and how they can be understood at least qualitatively. To conclude this book, a number of open questions and possible future research directions are addressed. These demonstrate that the field of electron glass is far from being closed and that some of its major issues are not understood or are under debate. First and foremost, there exists no statistical or thermodynamic theory of nonergodic systems, and for that reason there is no full understanding of any glasses – less so of quantum glasses such as is the electron glass. Indeed, the purpose of this book is to present a comprehensive review of the current state of the field, with the anticipation that further experimental, theoretical, and numerical work will follow and help elucidating the issues that remain unclear today.

• Processes responsible for glassy properties At the present, there is no consensus as to the electronic processes that lead to long relaxation and memory phenomena. Does the glassy behavior require many-electron transitions or are single hops enough? Is it only very low energy transitions that linger at very long times? The proposed suggestions for the slow relaxation of conductivity are slow formation of the Coulomb gap, slow formation of quasi-particles, slow manyelectron transitions off the percolation path affecting the paths, many-electron transitions traversing the current carrying path, exchange of particles between the percolation path and clusters off the path, all as discussed in Chapter 7.

Chapter 3 serves as an introduction to glasses and in doing so focuses on what is characteristic of glasses in general. To briefly summarize, the phenomenological common features are an extremely long or perpetual state of nonequilibrium, disobeying time homogeneity (aging), and memory effects. Structurally, the common features are interaction and disorder, the combination of which leads to frustration. As far as experiments are concerned, one can excite the glass with some generalized force – thermodynamically an intensive quantity – and measure a responding dynamical variable (an extensive quantity) or a generalized susceptibility. For example, in structural glasses, one measures deformation or viscosity responding to stress, in spin glasses one measures magnetization or magnetic susceptibility responding to applied magnetic fields, and in electron glasses one measures conductivity or electrical susceptibility in response to applied electric fields. Alternatively, one can measure slow relaxation of this variable after cooling from high temperature.

In relation to Chapter 3, this chapter is primarily intended to point out the more subtle differences between the electron glass and other glasses, with some particular attention to the spin glass, which is a close relative of the electron glass and is the most widely studied glass.

Structural glasses

Before all else, it is useful to start with structural glasses, which, in more than one sense, is the mother of other glasses. The silica glass (“window glass”) is by far the oldest glass that humans took interest in, mainly as an object of fashion.