In this introductory chapter we focus on the interaction of optical fields with matter because it forms the basis of signal amplification in all optical amplifiers. According to our present understanding, optical fields are made of photons with properties precisely described by the laws of quantum field theory [1]. One consequence of this wave–particle duality is that optical fields can be described, in certain cases, as electromagnetic waves using Maxwell's equations and, in other cases, as a stream of massless particles (photons) such that each photon contains an energy hν, where h is the Planck constant and ν, is the frequency of the optical field. In the case of monochromatic light, it is easy to relate the number of photons contained in an electromagnetic field to its associated energy density. However, this becomes difficult for optical fields that have broad spectral features, unless full statistical features of the signal are known [2]. Fortunately, in most cases that we deal with, such a detailed knowledge of photon statistics is not necessary or even required [3, 4]. Both the linear and the nonlinear optical studies carried out during the last century have shown us convincingly that a theoretical understanding of experimental observations can be gained just by using wave features of the optical fields if they are intense enough to contain more than a few photons [5]. It is this semiclassical approach that we adopt in this book. In cases where such a description is not adequate, one could supplement the wave picture with a quantum description.

An optical fiber amplifier is a key component for enabling efficient transmission of wavelength-division multiplexed (WDM) signals over long distances. Even though many alternative technologies were available, erbium-doped fiber amplifiers won the race during the early 1990s and became a standard component for long-haul optical telecommunications systems. However, owing to the recent success in producing low-cost, high-power, semiconductor lasers operating near 1450 nm, the Raman amplifier technology has also gained prominence in the deployment of modern light-wave systems. Moreover, because of the push for integrated optoelectronic circuits, semiconductor optical amplifiers, rare-earth-doped planar waveguide amplifiers, and silicon optical amplifiers are also gaining much interest these days.

Interestingly, even though completely unrelated to the conventional optical communications technology, optical amplifiers are also finding applications in biomedical technology either as power boosters or signal-processing elements. Light is increasingly used as a tool for stretching, rotating, moving, or imaging cells in biological media. The so-called lab-on-chip devices are likely to integrate elements that are both acoustically and optically active, or use optical excitation for sensing and calibrating tasks. Most importantly, these new chips will have optical elements that can be broadly used for processing different forms of signals.

There are many excellent books that cover selective aspects of active optical devices including optical amplifiers. This book is not intended to replace these books but to complement them.

Introduction: diffraction gratings

When a polychromatic beam of light is shined upon a translucent planar material etched with a periodic pattern known as a diffraction grating, the different colors of the beam of light propagate in different directions. Such a diffraction grating can be used as a monochromator of light or as a spectrometer, as schematically illustrated in Fig. 8.1.

The physics of a diffraction grating can be understood by a simple interference argument: we suppose that light of wavelength λ is normally incident upon a thin opaque screen containing a large number of periodically arranged narrow apertures (slits) separated by a distance Δ. Because the light is normally incident upon the grating, the phase of the field is the same within each aperture. Each aperture produces a spherical wave, and we consider the overlap of the fields from two such elements in a direction θ from the normal to the surface; this is illustrated in Fig. 8.2.

As can be seen from the picture, the wave emanating from aperture 2 has to travel a distance Δsin θ farther than the wave emanating from aperture 1. This translates to a phase difference between the two waves of kΔsin θ, where k = 2π/λ is the wavenumber of the light.

Introduction: systems with different symmetries

It is obvious that the Cartesian coordinate system which we have been using up to this point is ideally suited for those situations in which one is trying to solve problems with a rectangular, or “box”-type geometry. Determining the electric potential inside of a cube, for instance, would be an ideal problem to use Cartesian coordinates, for one can choose coordinates such that faces of the cube each correspond to a coordinate being constant. However, it is also clear that there exist problems in which Cartesian coordinates are poorly suited. Two such situations are illustrated in Fig. 3.1. A monochromatic laser beam has a preferred direction of propagation z, but is often rotationally symmetric about this axis. Instead of using (x, y) to describe the transverse characteristics of the beam, it is natural to instead use polar coordinates (ρ,ϕ), which indicate the distance from the axis and the angle with respect to the x-axis, respectively. A point in three-dimensional space can be represented in such cylindrical coordinates by (ρ,ϕ,z). In the scattering of light from a localized scattering object, the field behaves far from the scatterer as a distorted spherical wave whose amplitude decays as 1/r, r being the distance from the origin of the scatterer.

Introduction: the Fabry–Perot interferometer

High precision interferometers typically employ reflecting surfaces to interfere an optical beam with itself multiple times. One of the earliest of these, developed by Charles Fabry and Alfred Perot, is also one of the most persistently useful and versatile interferometeric devices. First introduced in 1897 as a technique for measuring the optical thickness of a slab or air or glass [FP97], the device found its most successful application only two years later as a spectroscopic device [FP99]. In its simplest incarnation, the interferometer is a pair of parallel, partially reflecting and negligibly thin mirrors separated by a distance d; it is illustrated in Fig. 7.1. A plane wave incident from the left will be partially transmitted through the device, and partially reflected; the amount of light transmitted depends in a nontrivial way upon the properties of the interferometer, namely the mirror separation d, the mirror reflectivity r and transmissivity t, and the wavenumber k of the incident light. In this section we will study the transmission properties of the interferometer and show that a solution to the problem requires the summation of an infinite series.

The most natural way to analyze the effects of the Fabry–Perot is to follow the possible paths of the plane wave through the system and track all of its possible behaviors. We begin with a monochromatic plane wave incident from the left of the form U(z, t) = Aexp[ikz – iωt].

Why another textbook on Mathematical Methods for Scientists? Certainly there are quite a few good, indeed classic texts on the subject. What can another text add that these others have not already done?

I began to ponder these questions, and my answers to them, over the past several years while teaching a graduate course on Mathematical Methods for Physics and Optical Science at the University of North Carolina at Charlotte. Although every student has his or her own difficulties in learning mathematical techniques, a few problems amongst the students have remained common and constant. The foremost among these is the “wall” between the mathematics the students learn in math class and the applications they study in other classes. The Fourier transform learned in math class is internally treated differently than the Fourier transform used in, say, Fraunhofer diffraction. The end result is that the student effectively learns the same topic twice, and is unable to use the intuition learned in a physics class to help aid in mathematical understanding, or to use the techniques learned in math class to formulate and solve physical problems.

To try and correct for this, I began to devote special lectures to the consequences of the math the students were studying. Lectures on complex analysis would be followed by discussions of the analytic properties of wavefields and the Kramers–Kronig relations. Lectures on infinite series could be highlighted by the discussion of the Fabry–Perot interferometer.