Introduction

The development of commercially available laser diodes (LDs) has revolutionised solid state lasers so much so that today the term solid state lasers normally implies laser diode pumping. The change started in the mid 1980s with the commercial release of GaAs LD arrays capable of generating between 50 mW and 250 mW around 800 nm and capable of optically pumping Nd doped laser hosts. Subsequently, the range of available wavelengths and the power levels from LDs has increased dramatically, particularly around 980 nm, 800 nm and 670 nm, so that almost every possible dopant and host combination is capable of being LD pumped. In parallel with the improving technology for the pump sources, a new understanding of techniques for modelocking solid state lasers, that allowed the pulse durations directly from the laser to approach the limit set by the gain bandwidth of the laser, has dramatically changed the pulse duration expectations from solid state lasers. Consequently the design of a modern modelocked solid state laser is quite different from previous designs. It is the purpose of this chapter to summarise the developments that have been made over the past five to ten years in developing compact modelocked sources based on LD pumped solid state lasers.

Laser diode pumping

Pumping using LDs has been a major factor in allowing the phrases compact or miniature to be applied to LD pumped solid state lasers (LDPSSLs). This claim will be justified by quoting typical values for various parameters that should be achievable.

Introduction

Single-mode fibers typically contain two eigenmodes with the same light intensity distribution, but with orthogonal polarization states. In perfectly isotropic fibers these modes are degenerate and have the same propagation constants. Hence isotropic fibers are polarization maintaining, i.e. the output polarization state is the same as the input polarization state. However, this statement is not generally valid in the presence of nonlinearities. Though an isotropic fiber is still polarization maintaining for linearly and circularly polarized light independent of the light intensity, an elliptical polarization state will rotate as a function of power.

The phenomenon of ellipse rotation was already well known in the 1960s and was commonly used to measure the refractive index of isotropic materials with Kerr-type nonlinearities (Maker and Terhune, 1965). By incorporating suitable polarization optics, ellipse rotation can also be used as an ultrafast saturable absorber for passive modelocking, as first realized by Dahlström (1972). However in this work stable CW modelocking was not obtained due to the low available power levels. The first application of nonlinear polarization evolution in optical fibers dates back to Stolen et al. (1982), who used a highly-birefringent fiber for shortening and cleaning the pulses from a separate external ultra-short pulse source. Nonlinear polarization evolution for the general case of low birefringence fibers was first analyzed by Winful (1985), who also predicted the phenomenon of polarization instability when high intensity light is launched close to the fast axis of the fiber.

Introduction

We will now study the properties of fluctuating electromagnetic fields, paying attention mainly to the optical region of the electromagnetic spectrum. It seems hardly necessary to stress that every electromagnetic field found in nature has some fluctuations associated with it. Even though these fluctuations are, as a rule, much too rapid to be observed directly, one can deduce their existence from suitable experiments that provide information about correlations between the fluctuations at two or more space-time points.

The simplest manifestations of correlations in optical fields are the well-known interference effects that arise when two light beams that originate from the same source are superposed. With the availability of modern light detectors and electronic circuitry of very short resolving time, other types of correlations in optical fields began to be studied in more recent times. These investigations, as well as the development of lasers and other novel types of light sources, led to a systematic classification of optical correlation phenomena and the complete statistical description of optical fields. The area of optics concerned with such questions is now generally known as optical coherence theory.

The first investigations of coherence phenomena are due to Verdet (1865, 1869) and von Laue (1907a, b). Some early investigations of Stokes (1852) and Michelson (1890, 1891a, b, c, 1892, 1920) although not explicitly mentioning coherence – because this concept is of a much later origin – have also contributed to the clarification and development of this subject.

Introduction

We have now prepared the way for a quantum treatment of problems that are of interest in optics. We have examined some basic properties of the quantized field and have explored the coherent-state representation and some of its virtues. Until now our discussions have been essentially formal; we have not concerned ourselves with what is actually measured in the laboratory, nor with questions concerning the interactions of the quantized field with other systems.

In the following sections we shall turn our attention to some of the quantum field theoretic quantities that are of particular interest in quantum optics and are of significance in the measurement of the field. Our discussions in this chapter will still be somewhat formal, in that we shall not yet come to grips with the problem of the interaction of the field with the apparatus. That problem will be tackled in Chapter 14. For the moment we shall continue to oversimplify and treat the field as a free field. However, within the idealized context of the free field we shall examine some of the questions encountered in measurements in quantum optics. We shall see that normally ordered correlations play a dominant role and examine some of their properties. Anti-normally ordered correlations will be seen to arise in much less common and less useful measurements of the field. We shall also examine the sense in which a photon may be said to be localized in a measurement in which a photoelectric emission is registered.

Introduction to statistical ensembles

The concept of a random or stochastic process or function represents a generalization of the idea of a set of random variables x1, x2, …, when the set is no longer countable and the variables form a continuum. We therefore introduce a continuous parameter t, such as time, that labels the variates. We call x(t) a random process or a random function of t if x does not depend on t in a deterministic way. Random processes are encountered in many fields of science, whenever fluctuations are present. Examples of a real random process x(t) are the fluctuating voltage across an electrical resistor, and the coordinates of a particle under-going Brownian motion. We shall see shortly that the optical field generated by any realistic light source must also be treated as a random function of position and time. Of course the parameter t may also stand for some quantity other than time, but for simplicity we shall take it to represent time. In our applications x(t) will frequently represent a Cartesian component of the electric or magnetic field vector in a light beam. To begin with we shall take x(t) to be real, but complex random processes will also be encountered.

The ensemble average

As x does not depend on t deterministically, we can only describe its values statistically, by some probability distribution or probability density.

Introduction

We have already encountered situations in which light of one frequency falling on an atomic system gives rise to light of different frequencies (cf. Section 15.6). The atom may here be considered to play the role of a (noisy) nonlinear transducer for the incident field. Even stronger nonlinear effects arise when one is dealing with a large number of atoms, or a nonlinear medium. Under these circumstances it is sometimes permissible to ignore the atomic structure and to treat the medium as a continuum, as in Maxwell's electromagnetic theory. The subject of the interaction of the incident field with the nonlinear medium is usually known as nonlinear optics.

The subject had its beginnings in an experiment in which a strong beam of red light (wavelength 6943 Å) from a ruby laser was allowed to fall on a quartz crystal, and a faint beam of blue light at a wavelength of 3472 Å, the first harmonic of the red, was produced (Franken, Hill, Peters and Weinreich, 1961). The development of the subject of nonlinear optics into a mature field is due largely to the work of Bloembergen and his collaborators. In the following we shall consider only a few simple illustrative examples of phenomena in nonlinear optics. More topics and more detail can be found in the books by Bloembergen and others (Bloembergen, 1965; Yariv, 1967, Chap. 21; Shen, 1984; Schubert and Wilhelmi, 1986; Butcher and Cotter, 1990; Boyd, 1992).

Introduction

The idea of making use of the phenomenon of stimulated emission from an atom or molecule for amplification of the electromagnetic field, and then combining the amplifier with a resonator to make an oscillator, is due to Townes and his co-workers (Gordon, Zeiger and Townes, 1954, 1955) and independently to Basov and Prokhorov (1954, 1955). The former constructed the first MASER, or microwave amplifier by stimulated emission of radiation. In 1958 Schawlow and Townes proposed an application of the same principle to the optical domain (Schawlow and Townes, 1958), in which a two-mirror Fabry–Perot interferometer serves as the optical resonator and an excited group of atoms as the gain medium. The first LASER, or light amplifier by stimulated emission of radiation, was constructed by Maiman (1960). The gain medium was ruby, which was excited by a bright flash of light from a gas discharge tube, whereupon the laser delivered a short optical pulse. The first continuously operating – or CW – laser was developed by Javan and his co-workers (Javan, Bennett and Herriott, 1961); it made use of a He:Ne gas mixture, that was continuously excited by a discharge, as the gain medium. This type of laser is still widely used today. Since that time many different kinds of laser have been developed, ranging in wavelength from the infrared to the ultraviolet. Some produce light at several different frequencies at once, and some are tunable over a wide range.

We have now derived a number of general properties of a quantized electromagnatic field, and have encountered some useful formalisms for treating certain problems in quantum optics. We have introduced the correlation functions of the field, and we have seen in a general way how they are related to measurements. In selecting examples for illustration we have tended to focus our attention largely on certain idealized quantum states of the field, such as Fock states and coherent states. However, there exists an important class of optical fields with simple properties, the so-called thermal fields, which includes most fields commonly encountered in practice, that has not yet been discussed. These fields are produced by sources in thermal equilibrium, and they exhibit many features that can be treated almost exactly in our formalism. We now turn our attention to such fields.

Blackbody radiation

The density operator

Blackbody radiation is the name given to an electromagnetic field in thermal equilibrium with a large thermal reservoir or heat bath at some temperature T. By definition, such a field is assumed to be coupled to the heat bath, and it is therefore not a strictly free field in the sense of the previous chapters. However, the coupling can be as weak as we wish, and it is well known from the general theory of statistical thermodynamics that the properties of a system with many degrees of freedom in thermal equilibrium (described by a canonical ensemble) are often similar to those of an equivalent isolated system (described by a microcanonical ensemble).

Light is both radiated and absorbed by atoms, and the interaction between the quantized electromagnetic field and an atom represents one of the most fundamental problems in quantum optics. However, real atoms are complicated systems, and even the simplest real atom, the hydrogen atom, has a non-trivial energy level structure. It is therefore often necessary or desirable to approximate the behavior of a real atom by that of a much simpler quantum system. For many purposes only two atomic energy levels play a significant role in the interaction with the electromagnetic field, so that it has become customary in many theoretical treatments to represent the atom by a quantum system with only two energy eigenstates. This is the most basic of all quantum systems, and it generally simplifies the treatment substantially.

In a real atom the selection rules limit the allowed transitions between states, so that, in some cases, a certain state may couple to only one other. Moreover, optical pumping techniques have been developed that allow such preferred states to be prepared in the laboratory, and they have been successfully used in experiments (Abate, 1974). The two-level atom approximation is therefore close to the truth and not merely a mathematical convenience in some experimental situations. In the following we begin by developing the algebra for such a two-level atom.

Dynamical variables for a two-level atom

We consider an atomic quantum system with the two energy levels shown in Fig. 15.1.

Prior to the development of the first lasers in the 1960s, optical coherence was not a subject with which many scientists had much acquaintance, even though early contributions to the field were made by several distinguished physicists, including Max von Laue, Erwin Schrödinger and Frits Zernike. However, the situation changed once it was realized that the remarkable properties of laser light depended on its coherence. An earlier development that also triggered interest in optical coherence was a series of important experiments by Hanbury Brown and Twiss in the 1950s, showing that correlations between the fluctuations of mutually coherent beams of thermal light could be measured by photoelectric correlation and two-photon coincidence counting experiments. The interpretation of these experiments was, however, surrounded by controversy, which emphasized the need for understanding the coherence properties of light and their effect on the interaction between light and matter.

Undoubtedly it was the realization that the subject of optical coherence was not well understood that prompted the late Dr E. U. Condon to invite us, more than three decades ago, to prepare a review article on the subject of coherence and fluctuations of light for publication in the Reviews of Modern Physics, which he then edited. The article was well received and frequently cited, and this encouraged us to expand it into a book. Little did we know then how rapidly the subject would develop and that it would become the cornerstone of an essentially new field, now known as quantum optics. Also the first experiments dealing with non-classical states of light were reported in the 1970s, and they provided the impetus for the new quantum mechanical developments.

Introduction

It has been known since the nineteenth century that when light falls on certain metallic surfaces, electrons are sometimes released from the metal. This is known as the photoelectric effect, and the emitted particles are called photoelectrons. If a positively charged electrode is placed near the photoemissive cathode so as to attract the photoelectrons, an electric current can be made to flow in response to the incident light. The device thereby becomes a photoelectric detector of the optical field, and it has proved to be one of the most important of all photometric instruments. Various means exist for amplifying the photoelectric current. In one important device, known as the photomultiplier and shown schematically in Fig. 9.1, the photoelectrons are accelerated sufficiently that on striking the positive electrode they cause the release of several secondary electrons for each incident primary electron, and these electrons are then accelerated in turn to strike other secondary emitting surfaces. After 10 or more similar stages of amplification, the emission of each photoelectron from the cathode results in a pulse of millions of electrons at the anode, which is large enough to be registered by an electronic counter. By counting these photoelectric pulses we have an extremely sensitive detector of light.

It has been found experimentally that photoelectric emission from a given surface occurs only if the frequency of the incident light is high enough to exceed a certain threshold value (see Fig. 9.2).

In Section 11.5 we showed that, although the complex amplitude of the electromagnetic field has a well-defined value in any coherent state, yet the real and imaginary (Hermitian and anti-Hermitian) parts of the field fluctuate with equal dispersions. The phenomenon of vacuum fluctuations is a manifestation of this effect, because the vacuum state is an example of a particular coherent state. This behavior is quite different from that of an ordinary, classical field. In a squeezed state, which is even more non-classical, as we shall see, one part of the field fluctuates less and another part fluctuates more than in the vacuum state. In general, a squeezed state is one in which the distribution of canonical variables over the phase space has been distorted or ‘squeezed’ in such a way that the dispersion of one variable is reduced at the cost of an increase in the dispersion of the other variable. In the following we shall examine the properties of squeezed states when the two canonical variables are two quadratures of the electromagnetic field. Although the squeezing terminology is sometimes applied to variables other than the two field quadratures, it is less meaningful in those cases. A number of review articles on squeezing have been published and can be consulted for more details (Walls, 1983; Schumaker, 1986; Loudon and Knight, 1987; Teich and Saleh, 1989, 1990; Kimble, 1992).

In the previous chapter we studied the interaction between a single two-level atom and the electromagnetic field, both when the field is treated classically and when it is quantized. We encountered some interesting phenomena such as Rabi oscillations, the a.c. Stark effect, and photon antibunching, all of which have been observed. These phenomena are essentially single-atom effects, in the sense that either they require a single atom for the effect to be observed, as in the last case, or at least they do not require more than one, although a group of atoms may be used in practice.

In this chapter we shall turn to a discussion of some effects that depend in an essential way on the presence of a group of atoms. In some cases we shall find that the group or collective behavior of the atomic system is relatively trivial, in the sense that we can account for the phenomenon by summing the contributions of the individual atoms to the total field, and treating each of them as if it acts almost independently of the others. This is the situation in free induction decay and in the photon echo. In other cases it is essential to include the effect of each atom on all the other atoms, because this modifies the behavior of each in a significant way. These phenomena, such as self-induced transparency and superradiance, are collective effects in a deeper sense. They are sometimes called cooperative effects.