Introduction

In many visual optical systems, prisms play an important role in the formation of the final image. They are used to control or change the direction of the image forming beam and are often used to change the orientation of the image. In these operations, the prism may only refract the beam, may refract and reflect it or may only reflect it but one or more times. Therefore in discussing the structure and properties of prisms, it is convenient to classify prisms according to whether they (a) are purely refracting, (b) combine refraction with internal reflections or (c) are purely reflecting.

The optical construction of most prisms is relatively simple but the ray paths inside a prism, which reflect the beam one or more times, can be complex and some of the properties of the prism are not readily obvious from looking at the ray paths. However these properties are sometimes made clearer by examining the unfolded version. Unfolding a prism involves reflecting each reflecting surface in the next reflecting surface along the ray path. Ideally, this process will become clearer when examples are discussed later in this chapter.

Refracting-only prisms

Refracting prisms occur frequently in visual optical and ophthalmic instruments. They can be used to deviate beams or images, as aids to focussing to form double or split images and often to compensate for eye problems such as phorias and squint.

Just as lenses may be examined using finite (or real) ray tracing on one hand or paraxial theory on the other, so can refracting prisms, but before we examine the paraxial properties of prisms, we will examine their effects on finite rays.

Introduction

The visual performance of a user viewing an image through an optical instrument depends upon a complex interaction between the eye and the instrument. Thus one cannot fully assess an optical instrument without a general understanding of visual optics and occasionally a knowledge of the visual capabilities of the individual user, which must include any anomalies of his or her visual system.

Many visual optical instruments are monocular and some people suffer visual discomfort when using monocular visual instruments for any length of time. Most of this probably stems from the tendency to keep one eye closed while viewing through the instrument. A binocular equivalent instrument will eliminate this problem and offers the possibility of a stereoscopic image. However, binocular instruments may lead to other problems. In this chapter, we will only discuss the ergonomics of monocular instruments and leave the ergonomics of binocular instruments and potential problems until the next chapter.

Instrument focussing

Image vergence

If an instrument is to be correctly focussed, the image must be within the accommodation range of the user. Classically, the ideal image vergence of visual optical instruments is taken as zero, that is the image is at infinity. The reasoning justifying this situation is that it was traditionally believed that the eye prefers viewing with relaxed accommodation. Unfortunately this approach neglects instrument accommodation and any refractive error of the user. The nature of instrument accommodation will be discussed later in Section 36.1.2. The different types of refractive errors and their distribution among the adult population has already been discussed in Chapter 13 and some representative values are shown in Figure 13.9.

Introduction

This chapter deals with the properties of mirrors or reflecting surfaces. Two types of mirrors will be looked at: (a) plane mirrors and (b) curved mirrors. So far, we have tended to concentrate on the paraxial properties of optical systems and therefore looked at such phenomena as image formation in terms of paraxial rays, because these rays are aberration free. When we discuss the properties of plane mirrors, we need not restrict ourselves to paraxial rays because plane mirrors are aberration free and therefore we can use either paraxial or finite rays. However, when we discuss the properties of curved mirrors, these are not free of aberration and therefore we must return to paraxial optics.

Optical systems can be constructed solely of mirrors or reflecting elements and such systems are called catoptric systems. Systems that consist of refracting and reflecting elements are called catadioptric systems. In Chapter 4, we showed how to use paraxial ray tracing to study the properties and image formation in such systems. In this chapter, we will look at the properties of single mirrors and simple mirror systems, starting with plane mirrors.

Plane mirrors

In this section, we will investigate the properties of plane mirrors. These properties are useful in their own right, but are also very useful in understanding the properties of systems of plane mirrors or reflecting surfaces, such as occur in some reflecting prisms, which are discussed in Chapter 8.

The optics of plane mirrors can be analysed using Snell's law. For reflection, Snell's law reduces to the statement that the angle of reflection is equal to the angle of incidence.

Introduction

So far, all the visual optical instruments that have been discussed have been essentially describable in terms of geometrical or ray optics. In this and the following chapter, we will describe some important visual optical instruments and devices based upon physical or wave optics. The essential difference between geometrical and physical optics is the difference between the geometrical description of the propagation of light by rays and the physical optical description using waves. The wave nature of light gives rise to two very important phenomena, interference and diffraction. This chapter is concerned with interference and the next will look at diffraction.

The phenomenon of interference allows us to produce visual stimuli in which the light level varies sinusoidally across the pattern. Since early work by Campbell and Green (1965), the visual system is now regularly analysed in terms of its response to sinusoidally varying luminance profiles. This is often done by measuring the threshold contrast of sinusoidal patterns at a range of spatial frequencies. The resulting function is known as the contrast sensitivity function (see Chapter 35). Sinusoidal patterns can be produced in several ways. They may be produced in printed form, for example as photographs, or can be produced on television screens or by optical interference techniques. This chapter is concerned with the last mode of production.

There are two fundamental differences between the production of sinusoidal patterns by interference techniques and those in printed or television screen forms.

Introduction

Camera lenses are optical systems designed to give a real image of distant objects. Therefore they must have a positive power with their aberrations corrected to produce reasonably good image quality for an object plane at infinity, that is for the image formed in the back focal plane of the lens. Some camera lenses known as “macro-lenses” are designed to give optimum image quality at intermediate or close object distances rather than for the object at infinity. Examples of macro-type lenses are those used in photocopiers and for photographing documents.

Camera lenses usually have a variable aperture stop, known as the diaphragm, whose diameter is varied within limits, for control of the image illuminance and hence film exposure (Section 21.4) and the depth-of-field (Section 21.6).

While a camera lens may consist of a single positive power lens, such a lens is not suitable when the focal length is very long or very short. Also a single lens would give less than adequate image quality for many applications, particularly those requiring large apertures or wide fields-of-view. Thus most camera lenses are complex optical systems, often consisting of two or more separate lenses. Designs suitable for very long and very short focal lengths are discussed in Section 21.2 and aberration considerations for wider apertures and field angles are discussed in Section 21.10.

Field-of-view, focal length and image sizes

The field-of-view of a camera system is usually limited by a field stop placed in the image or film plane. In 35 mm cameras, the field stop is 35 × 24 mm; thus the image or frame size on the film has these dimensions.

Introduction

A number of photometric and colorimetric instruments are visual optical systems as these devices have a viewing system to observe a scene either for alignment or to make a subjective judgement of brightness (luminance) or colour.

Photometers

Photometers are either subjective or objective instruments and are designed to measure the absolute or comparative brightness of sources or scenes. In the subjective type, the observer has to make a match between the brightness of a standard scene and that of a second scene. In the objective instruments, an observer uses the viewing system only for alignment.

Subjective photometers

The Lummer–Brodhun photometer

The Lummer–Brodhun photometer is designed to compare the luminous intensities of two light sources. Often one is a secondary standard source of known luminous intensity, and in this case, the luminous intensity of the second or unknown source can be found.

The instrument is shown in Figure 24.1. The light from the two sources illuminates both sides of a white diffuse block as shown in the diagram. Both sides of the block are viewed simultaneously through an optical system which contains a prism assembly and an eyepiece. One side of the block (the left side in the diagram) is seen through the central part of the prism and the other side of the white block is seen through the peripheral part. The two beams are brought together but separated by a sharp boundary by the prisms. The eyepiece is focussed on the sharp boundary. The field-of-view seen through the eyepiece is shown in the diagram.

In a recent series of experimental papers, it has appeared that lasers that are well described by the simplified TSD rate equations (7.42)–(7.44) display what is called, in laser physics, antiphased dynamics. Its simplest manifestation is that when N modes oscillate, the total intensity displays many properties that are those of a single-mode laser. In the simplest experiment, the laser is initially in a steady state. A control parameter is suddenly changed and the relaxation toward the new steady state is recorded. This transient evolution is then Fourier analyzed to evaluate its frequency content. The result of this experiment is that each mode is characterized by as many frequencies as there are modes, whereas the total intensity, which is simply the sum of all modal intensities, is characterized by only one frequency. This frequency is the single-mode relaxation oscillation frequency (7.51) and is also the highest of the N frequencies. This property has been observed with a Nd-doped optical fiber laser [1]. Using a LiNdP4O12 (abbreviated as LNP) laser oscillating on two or three modes, it was shown that the noise spectrum of the laser displayed the same antiphase dynamics [2]. With the same laser, antiphase dynamics was also reported in the case where a feedback loop induces a chaotic output [3]. The feedback loop has the effect of injecting part of the output beam in the laser after each modal intensity has been subjected to a modulation. Antiphase dynamics also plays a significant role when a multimode laser undergoes a Feigenbaum cascade toward chaos. It has been reported that though the total intensity displays this usual route to chaos, the modal intensities do not [4].

A rigorous description of the light–matter interaction requires the use of quantized Maxwell equations to describe the light field and the Schrödinger equation to describe the material medium. This book does not attempt to deal with quantum optical problems that necessitate a field quantization. Instead we focus on properties that are generally grouped under the umbrella of nonlinear optics. Although many definitions of this expression exist, we use it in the sense that the properties we deal with do not depend in a crucial way on the field quantization. For practical purposes, this means in general that the average photon number is large and that we can neglect spontaneous emission as a dynamical process. Rather, we include the consequence of spontaneous emission, that is, the instability of the atomic levels, in a purely phenomenological way. When this is done, we call the material equations the Bloch equations instead of the Schrödinger equation. References [1] through [4] are the classical textbooks that contain a tentative justification for the transition from the Schrödinger to the Bloch equations. None of these attempts is satisfactory from a fundamental viewpoint because the real difficulty is to incorporate the finite lifetime (or natural linewidth) of the atomic energy levels in the Schrödinger equation. This remains an open problem as of now.

The Maxwell–Schrödinger equations

We consider atoms interacting with an intense electromagnetic field. By intense we mean a field for which the quantization is not necessary.

Laser theory has attracted a large number of studies centered on the stability problem [1]–[4]. There are at least three motivations for these studies. First, the laser equations are rather simple equations that can be derived from first principles with a minimal admixture of phenomenology. The complexity of their solutions was not fully appreciated until Haken showed the equivalence between the three single-mode laser equations on resonance [equations (1.58)–(1.60) with Δ = 0] and the Lorenz equations derived in hydrodynamics [5]. However, the domains of parameters that are relevant for optics and hydrodynamics are not the same. New asymptotic studies were suggested for the laser equations. Second, infrared lasers have been built that can be modeled quite accurately by the two-level equations, at least in some domains of parameters. A good analysis of this topic is found in [6]. A marked advantage of optics over hydrodynamics is that in general the time scales are much shorter. Hence experimental data can be accumulated, and averaging procedures can be used to separate the effect of noise from deterministic properties. Third, the laser stability becomes an essential question when the laser is used as a tool in scientific or industrial applications.

Stability means that a perturbation applied to the laser decreases in time. This requires the solution of an initial value problem that in general is too difficult to be solved analytically.

Introduction

Usage has reserved the expression optical bistability mostly for coherently driven passive systems. An atomic system is called passive when there is no population inversion. This is the case, for example, of a system at thermal equilibrium. On the contrary, the laser that we have described in Chapter 1 is a driven active system, because a population inversion is created. The laser equations we have studied so far describe an incoherently driven laser. However, nothing prevents driving a lasing cavity with a coherent field emitted by another laser. This is realized in a whole class of lasers that are used mainly as frequency converters. In optical bistability (OB), the driving field is usually a coherent field. Thus we have to account for two differences between the laser and the optically bistable system: (1) In OB, there is no inversion of population: 〈|A2|〉>〈|B2|〉, in the notation of Chapter 1. Hence, in the absence of interaction with a field, the population difference D relaxes toward a negative value. (2) The pumping is coherent, meaning that an external laser field is added to the cavity field.

Using the single-mode equations (1.48)–(1.50), we have to change the sign of D and Da (which amounts to keeping D defined in (1.34) as it is but changing P into –P and A into –A) and to add a source term in the equation for the complex field amplitude.

Nonlinear optics is a fairly young science, having taken off with the advent of the laser in 1960. Nonlinear optics (NLO) deals with the interaction of electromagnetic waves and matter in the infrared, visible, and ultraviolet domains. The frontiers of NLO are somewhat blurred, but microwaves and γ-rays are clearly outside its domain. This book is entirely devoted to a study of NLO in a resonant cavity and when the quantum nature of the electromagnetic field is not of prime importance. More precisely, we study those aspects of cavity NLO in which fluctuations in the number of photons and atoms are not relevant. This particular area of optics is dominated by the Maxwell–Bloch equations, which constitute its paradigm in the sense of T. S. Kuhn. The status of the Maxwell–Bloch equations is quite peculiar. From a fundamental viewpoint, they describe the laws of evolution of the first moments of a density operator, which verifies the von Neumann equation. However, to account for the finite lifetime of the atoms and of the field in the necessarily lossy cavity, some legerdemains have to be introduced to obtain the Maxwell–Bloch equations. Stated more explicitly, the von Neumann equation for a large but finite system does not explain irreversibility, whereas the Maxwell–Bloch equations fully include the irreversible decay of the atoms and of the cavity field. This problem is not specifically related to optics but reflects the general failure of statistical mechanics to explain convincingly the irreversible evolution of macroscopic systems.

Introduction

Up to now, we have described in detail many properties of steady bifurcations and limit points, that is, critical points where a stable steady state solution loses its stability and coincides with another steady state solution. At a few places, we have also met the so-called Hopf bifurcation where a steady solution loses its stability and a time-periodic solution emerges. However, we have not yet studied in any detail a Hopf bifurcation for lack of a suitable example. Even the simple-looking trio of laser equations on resonance [equations (1.58)–(1.60) with Δ = 0, E and P real] yield such complex expressions that it is hard to separate conceptual difficulties from mere computational problems. In this chapter, we make an intrusion upon a domain that has not yet been considered in this book. The motivation is both to cover an important topic of cavity nonlinear optics and to provide a pedagogical example of a Hopf bifurcation.

In the preceding chapters, this book has dealt exclusively with processes in which only one photon is either absorbed or emitted. Other phenomena, however, rely on multiphoton transitions [1, 2]. In the original Bohr formulation of atomic transitions and in much of the ensuing quantum mechanical formulation, resonance conditions on atomic transitions express conservation laws but give no constraint on the number of photons needed to achieve the transition.

In this chapter, we remove one assumption that has been implicit since the beginning of this book, namely the unidimensional aspect of the cavity. We now take into account the transverse variation of the field in the resonant cavity. In dealing with transverse effects, there are two possible approaches, very much like in the multimode optical bistability (OB) studied in Chapter 6. One possibility is to project the Maxwell–Bloch equations on a suitable basis. In the introduction of Chapter 6, the difficulty of selecting this suitable basis was explained for 1-D cavities. This difficulty is amplified by the transverse dimensions, especially because of the lateral boundaries. The other approach is to derive global equations (that are still partial differential equations) for slowly varying amplitudes. They are generally variants of well-known nonlinear partial differential equations of mathematical physics, and therefore a large number of results are directly available. However, when transferring results from another domain to optics, some care must be exercised because the relevant domains of parameters are not always compatible. The classic example is the canonical set of parameters for the Lorenz equations that includes b = γ║/γ⊥ = 8/3 whereas for atomic transitions the upper bound of b is 2. The reader will find a wealth of results, mostly for the modal expansions, in the special issues and reviews [1]–[4]. A review more specifically oriented toward the global amplitude equations is found in [5] and a mathematical study of these equations is presented in [6].

Introduction

Until now, we have always assumed that the cavity losses are linear, that is, field-independent. This need not always be true, and in this chapter we investigate how nonlinear losses affect the operation of a nonlinear optical device. We consider two examples. The first one is the laser with a saturable absorber, the second is parametric amplification in the presence of a saturable absorber.

The modeling of lasers with a saturable absorber (LSA) has a history that is practically as long as that of the laser. Problems related to the LSA are still a subject of debate. An LSA is a laser that contains both an active (or amplifying) medium and a passive (or absorbing) medium. For instance, if population excitation is produced in the laser cavity but inversion is not achieved in the whole cavity, some domains of the laser amplify the radiation and others absorb it. The salient feature of this situation is that both amplification and absorption result from the resonant interaction of light with atoms. Hence, both processes contribute to the nonlinear response. One way to look at an LSA is to consider it as a generalization of a laser in which the losses are as nonlinear (i.e., intensity-dependent) as the gain. From the viewpoint of dynamical systems, the LSA is a prototype of competition between nonlinear gain and nonlinear losses. As a result, there has been over the years an irresistible temptation to attribute to saturable absorption properties that are not explained by the standard laser equations, derived in Chapter 1.

In Chapters 1 to 5, we have dealt with single-mode ring cavities, either for lasers or for optical bistability. In this chapter, we come back to laser theory to consider the properties of multimode cavities. This subject is immense and our goal can only be modest.

The single-mode unidirectional ring laser is the model of choice for theoreticians who want to study fundamental aspects of laser theory. The simplicity of its evolution equations, equations (1.58)–(1.60), makes the model attractive. Its equivalence with the Lorenz equations [1], which have become the generic model to study chaos in ordinary differential equations, increases the relevance of the ring laser model. Of importance is the fact that the laser model lends itself quite naturally to a complexification of the variables. It suffices that the detuning be nonzero to have a coupling between the phase and the amplitude of the electric field and of the atomic polarization. This opens the door to an even richer phenomenology of complex behaviors.

The ring configuration for a laser is not simply an idealization intended for theoreticians. A number of lasers operate in this configuration. Dye lasers and some coherently pumped lasers are built with ring cavities. Laser gyroscopes are essentially ring lasers. If the ring cavity is perfectly symmetric with respect to the two directions of propagation, there is no preferential direction of oscillation and both directions must be taken into account. This is the simplest example of a multimode laser and we analyze some of its properties in Section 9.3.