Introduction

While atomic magnetometers can measure magnetic fields with exceptional sensitivity and without cryogenics, spin-altering collisions limit the sensitivity of sub-millimeter-scale sensors [1]. In order to probe magnetic fields with nanometer spatial resolution, magnetic measurements using superconducting quantum interference devices (SQUIDs) [2–4] as well as magnetic resonance force microscopes (MRFMs) [5–8] have been performed. However, the spatial resolution of the best SQUID sensors is still not better than a few hundred nanometers [9] and both sensors require cryogenic cooling to achieve high sensitivity, which limits the range of possible applications. A related challenge that cannot be met with existing technology is imaging weak magnetic fields over a wide field of view (millimeter scale and beyond) combined with sub-micron resolution and proximity to the signal source under ambient conditions.

Recently, a new technique has emerged for measuring magnetic fields at the nanometer scale, as well as for wide-field-of-view magnetic field imaging, based on optical detection of electron spin resonances of nitrogen-vacancy (NV) centers in diamond [10–12]. This system offers the possibility to detect magnetic fields with an unprecedented combination of spatial resolution and magnetic sensitivity [8, 12–15] in a wide range of temperatures (from 0 K to well above 300 K), opening up new frontiers in biological [10, 16, 17] and condensed-matter [10, 18, 19] research. Over the last few years, researchers have developed techniques for nanoscale magnetic imaging in bulk diamond [11, 12, 20] and in nanodiamonds [21–23] along with scanning probe techniques [10, 24].

Introduction

Soon after the development of optical magnetometers based on the radio-optical double resonance method (see Chapter 4), it was realized by Bell and Bloom [1] that an alternative method for optical magnetometry was to modulate the light used for optical pumping at a frequency resonant with the Larmor precession of atomic spins. In a Bell-Bloom optical magnetometer, circularly polarized light resonant with an atomic transition propagates through an atomic vapor along a direction transverse to a magnetic field B. Atomic spins immersed in B precess at the Larmor frequency ΩL, and when the light intensity is modulated at Ωm = ΩL, a resonance in the transmitted light intensity is observed. The essential ideas of the Bell–Bloom optical magnetometer are reviewed in Chapter 1 (Section 1.2), and can be summarized in terms of what Bell and Bloom termed optically driven spin precession: in analogy with a driven harmonic oscillator, in a magnetic field B atomic spins precess at a natural frequency equal to ΩL and the light acts as a driving force oscillating at the modulation frequency Ωm. From another point of view, the Bell-Bloom optical magnetometer can be described in terms of synchronous optical pumping: when Ωm = ΩL, there is a “stroboscopic” resonance in which atoms are optically pumped into a spin state stationary in the frame rotating with ΩL. Depending on the details of the atomic structure, the spin state stationary in the rotating frame can be either a dark state that does not interact with the modulated light or a bright state for which the strength of the light–atom interaction is increased.

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

This book is all about things that oscillate at terahertz frequencies, as introduced in Chapter 1. In Chapters 2 and 3 we saw how oscillations can be described using mathematics. In Chapter 4 we moved from the purely mathematical to physical reality, in treating the oscillatory nature of light. Now, in this chapter, we look at oscillations in matter.

The oscillations that can be sustained in matter are described by quantum mechanics. In this chapter, the wave mechanics approach to quantum mechanics is introduced via the Schrödinger equation. The equation is then solved for many cases of general and practical interest: the free particle, confined states such as in quantum wells and multiple quantum wells, and the oscillations of molecules, both rotational and vibrational. In all cases I deal with here, the emphasis is on the terahertz frequency range.

Learning goals

By the time you finish this chapter you should be able to discuss terahertz phenomena related to

1. • free particles,

2. • particles confined in structures such as semiconductor energy bands and quantum wells,

3. • molecules that are free to rotate and vibrate.

Wave mechanics

There is no escaping it. We can't really discuss terahertz phenomena in matter without discussing quantum mechanics.

As the name implies, quantum mechanics is about quanta, or distinct amounts. Applied to oscillations, quantum mechanics says that oscillations cannot occur at any arbitrary frequency, but only at particular, distinct frequencies.

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.