Introduction

In this chapter we shall begin our discussion of nonlinear phenomena that are important in optics. When one or more electromagnetic waves propagate through any medium they produce polarizations in the medium that, in principle, oscillate at all the possible sum and difference frequencies that can be generated from the incoming waves. These polarizations, which oscillate at these new frequencies, give rise to corresponding electromagnetic waves. Thus, we get phenomena such as harmonic generation, for example, when infrared light is converted into visible or ultraviolet light, and various other frequency-mixing processes. These nonlinear processes can be described by a series of nonlinear susceptibilities or mixing coefficients. These coefficients will be defined and their origin traced to the anharmonic character of the potential that describes the interaction of particles in the medium.

Anharmonic potentials and nonlinear polarization

When an electromagnetic wave propagates through a medium a total electric field acts on each particle of the medium. This total field contains components at all the frequencies contained in the input wave or waves. Each particle of the medium will be displaced from its equilibrium position by the action of this field. Positive ions and nuclei will be displaced in the direction of the field, while negative ions and electrons will be displaced in the opposite direction to that of the field. The resultant separation of centers of positive and negative charge creates dipoles in the medium.

Introduction

When an electromagnetic wave propagates through amedium stimulated emissions increase the intensity of the wave, while absorptions diminish it. The overall intensity will increase if the number of stimulated emissions can be made larger than the number of absorptions. If we can create such a situation then we have built an amplifier that operates through the mechanism of stimulated emission. This laser amplifier, in common with electronic amplifiers, has useful gain only over a particular frequency bandwidth. Its operating frequency range will be determined by the lineshape of the transition, and we expect the frequency width of its useful operating range to be of the same order as the width of the lineshape. It is very important to consider how this frequencywidth is related to the various mechanisms by which transitions between different energy states of a particle are smeared out over a range of frequencies. This line broadening affects in a fundamental way not only the frequency bandwidth of the amplifier, but also its gain. A laser amplifier can be turned into an oscillator by supplying an appropriate amount of positive feedback. The level of oscillation will stabilize because the amplifier saturates. Laser amplifiers fall into two categories, which saturate in different ways. A homogeneously broadened amplifier consists of a number of amplifying particles that are essentially equivalent, whereas an inhomogeneously broadened amplifier contains amplifying particles with a distribution of amplification characteristics.

Homogeneous line broadening

All energy states, except the lowest energy state of a particle (the ground state) cover a range of possible energies.

Introduction

In this chapter we will put the concept of coherence on a more mathematical basis. This will involve the formal definition of a number of functions that are used to describe the coherence properties of optical fields. These include the analytic signal, various correlation functions, and the degree of coherence for describing both temporal and spatial coherence. We shall see that the degree of temporal coherence is quantitatively related to the lineshape function and that the degree of spatial coherence between two points is determined by the size, intensity distribution, and location of an illuminating source. We will use the wave equation, and special solutions to the wave equation called Green's functions, to show how spatial coherence varies from point to point.

The chapter will conclude with a discussion of how the intensity fluctuations of a source depend on its coherence properties, and we will examine a specific scheme, namely the Hanbury Brown–Twiss experiment, by means of which this relationship is studied. This discussion will involve a discussion of “photon statistics,” namely the time variation of the “detection” of photons from a source. In a quantum-mechanical context, square-law detectors respond to these quantized excitations of the optical field, which we call photons.

In classical coherence theory it is advantageous to represent the real electromagnetic field by a complex quantity, both for its mathematical simplicity and because it serves to emphasize that coherence theory deals with phenomena that are sensitive to the “envelope” or average intensity of the field.

The author of a text generally feels obligated to explain the reasons for his or her writing. This is a matter of tradition as it provides an opportunity for explaining the development and philosophy of the text, its subject matter and intended audience, and acknowledges the help that the author has received. In the case of a second edition of a text, as is the case here, a new preface provides an opportunity for the author to explain the revisions of the second edition and to further acknowledge help from colleagues. I hope to accomplish these tasks briefly here.

The first edition of this text grew over many years out of notes that I had developed for courses at the senior undergraduate and beginning graduate student level at the University of Manchester, Cornell University, and the University of Maryland, College Park. These courses covered many aspects of laser physics and engineering, the practical aspects of optics that pertain to an understanding of these subjects, and a discussion of related phenomena and devices whose importance has grown from the invention of the laser in 1960. These include nonlinear optics, electro-optics, acousto-optics, and the devices that take practical advantage of these phenomena. The names given to the fields that encompass such subject matter have included laser physics, optical electronics, optoelectronics, photonics, and quantum electronics. The fundamentals of these subjects have not changed significantly in the years that have intervened since the publication of the first edition.

Introduction

In this chapter we shall discuss in some detail the operating principles, characteristics, and design features of solid-state lasers in which the laser medium is an insulating or glassy solid. In many of these lasers the active particles are impurity ions doped into a host matrix. These lasers are pumped optically, with a pulsed or continuous lamp, and most commonly by another laser. Our discussion will build on the brief introduction to one laser of this class, namely the ruby laser, given in Chapter 3. The chapter will conclude with a discussion of the characteristics of the radiation emitted by such lasers and how this radiation can be modified and controlled in time.

Optical pumping in three- and four-level lasers

Optical pumping in an insulating solid-state laser is illustrated schematically in Fig. 7.1. Light from the pumping lamp(s) excites ground-state particles into an absorption band, labeled 3 in the figure. Ideally, particles that reach this state should transfer rapidly into the upper laser level, level 2. If transfer occurs preferentially to level 2 rather than to level 1, a population inversion will result between levels 2 and 1, and laser action can be obtained. The drain transition from level 1 back to the ground state should be fast, in order to keep level 1 from becoming a “bottleneck.” The performance of the laser will be influenced by several factors.

In this chapter we shall discuss both from a ray and from a wave standpoint how light can be guided along by planar and cylindrical dielectric waveguides. We shall explain why optical fibers are important and useful in optical communication systems and discuss briefly how these fibers are manufactured. Some practical details of how fibers are used and how they are integrated with other important optical components will conclude the chapter.

Introduction

We saw in the previous chapter that a Gaussian beam can propagate without beam expansion in an optical medium whose refractive index varies in an appropriate manner in the radial direction. This is a rather specific example of how an optical medium can guide light energy. However, we can discuss this phenomenon in more general terms. By specifying the spatial variation in the refractive index, and through the use of the wave equation with appropriate boundary conditions, we can show that dielectric waveguides will support certain “modes” of propagation. However, it is helpful initially to see what can be learned about this phenomenon from ray optics.

Introduction

All lasers are to some extent tunable. Their output frequency can be varied continuously without discontinuous changes in output power by moving the position of the oscillating modes under the gain profile. However, if the gain profile is not very wide then this range of tunability is limited. For any atomic gas laser, for example, where gain profiles typically have Doppler widths on the order of 1 GHz, tunability of a single axial mode over about a 1 GHz range can be accomplished by changing the optical length nl of the cavity. This can be done by moving one of the mirrors with a piezoelectric transducer and thereby varying the geometric length l, or by adjusting the laser pressure so as to adjust the index n. Although a tunable frequency range of 1 GHz might seem large in absolute terms, it represents a very small fraction of the operating frequency of the laser, 1014−1015 Hz, say. Discontinuous tunability in the infrared can be obtained by using a molecular gas laser, for which several vibrational–rotational transitions have gain. Systems, such as the CO2 laser, can offer many lines over a relatively broad wavelength region, but a graph of output power vs. frequency is not continuous.

To achieve continuous tunable laser operation over a broad wavelength region we must use an amplifying medium with a broad gain profile.

When the laser was first invented, it was described as “a solution looking for a problem.” This comment did not long survive scrutiny, and nowadays lasers are ubiquitous in many aspects of daily life, with many technological, artistic, and educational applications. This chapter highlights some of the important application areas where lasers have become essential.

Optical communication systems

Introduction

Optical communications systems have a long history. Ancient man signaled with smoke and fire, often relaying messages from mountain top to mountain top. However, this optical communication scheme had limited transmission capacity. Such messages could serve as a warning, as Queen Elizabeth the First of England planned when she had a network of bonfires erected to be set in the event of a seaborne invasion from Spain. The smoke signals transmitted by native Americans had the capacity to transmit various messages. Since the end of the eighteenth century,messages have been passed by semaphore – the use of flags to indicate the transmission of one letter at a time. This form of communication could transmit information at a rate of about one letter per second over a direct line of sight, although messages could be relayed over long distances. Such means of communication were not very secure: anyone in the line of sight to the message sender could read the information (if he or she knew the code).

Introduction

Gas lasers operate using a large number of different atomic and molecular gases and gas mixtures. On a macroscopic scale gases are intrinsically uniform, so the properties of the laser medium are immune to the defects and structural issues that affect lasers using solid materials. In addition, continuous variations of the composition and pressure of a gas medium offer a degree of flexibility in laser design. On the other hand, gases must be held in a container and lack the robustness offered by solid materials. Just as vacuum and gas-filled electronic tubes have been largely supplanted by solid-state devices, except in niche applications, gas lasers, especially atomic-gas lasers, have been largely replaced by lasers using solid active media. Even so, some gas lasers remain important in various applications, and the renewed interest in optically pumped gas lasers, which until relatively recently had seemed only of historical interest, suggests that no laser should necessarily be written off as “out-dated” prematurely. In this chapter we shall consider some of the fundamental processes which are used to produce, and maintain, population inversion in atomic gases. We shall see that the technological features of gas lasers, and the efficiency with which they can be made to operate, are intimately connected with the particular mechanism used to excite the upper laser level. Our attention will be concentrated on a consideration of gas lasers in which the laser action involves energy levels of a neutral or ionized atom.

Introduction

To give a little more practical emphasis to some of the ideas we have dealt with so far, let us consider some of the details of the two laser systems in which population inversion and laser oscillation were first demonstrated. One of these lasers uses an amplifying medium that is a crystalline solid – the ruby laser; in the other the amplifying medium is a gas – a mixture of helium and neon. In each case, the amplifying medium is pumped into a state of population inversion by feeding energy into it in an appropriate way. Laser oscillation occurs when the amplifying medium is placed between a pair of suitable aligned mirrors that provide the necessary optical feedback to cause oscillation to occur. The ruby laser was the first operational laser, being demonstrated on May 16, 1960 by Theodore Maiman of the Hughes Aircraft Company in Malibu, California [1].

That the ruby laser was the first laser to be demonstrated surprised many in the scientific community. This is because the ruby laser is a three-level laser, which was expected to be much more difficult to operate than a four-level laser. This is an important distinction, which we will examine before describing the first two lasers in detail.

Three- and four-level lasers

The distinction between three- and four-level lasers can be illustrtated with the aid of Fig. 3.2. Energy is fed into the system to move particles from the ground state, level 0, to a pumping energy level, level 3.