A real lens has axial thickness, two radii of curvature, one for each surface (front and back), as well as some non-zero edge thickness. The line connecting the two centers of curvature is the optical axis. Thus far, we have disregarded the axial thickness of a lens by making it zero (t = 0). This produced a set of equations for a thin lens that relate the conjugate planes of the object and image. The t = 0 assumption gave the thin lens an optical power approximately equal to that of the thick lens. These equations were developed in order to solve paraxial optical relationships with analytical functions instead of ray tracing, resulting in a body of knowledge referred to as Gaussian optics. The real refractive lens has to have some axial thickness in the most common cases, except for the cases of spherical mirrors and single refracting surfaces (SRS), which mimic thin lenses.

Therefore, for real refractive lenses, the question becomes: from what surface or location does one measure the focal lengths, object distances, and image distances for a given setup? Chapter 6 showed that one can determine the optical power of a thick lens; however, the fiducial points from which to measure these distances were not determined. The cardinal points of a thick lens system will be explored in this chapter. Recall there are six cardinal points: two principal points, two focal points, and two nodal points, as was discussed for the thin lens.

Thus far, we have discussed images and objects qualitatively. Other than in the case of the pinhole camera, we have not examined the actual production of the image of an object, but only what we thought the image orientation and its handedness might be. In order to produce an optical image, the rays or optical radiation must converge or diverge upon refraction or reflection. Only surfaces of optical power, or curved optical surfaces, can have this effect on rays. This book focuses on geometrical optics; diffractive optical elements of power will not be discussed. Before discussing the effects of curved optical surfaces on rays, we will digress and cover two very important subjects: optical spaces and the sign convention used in this text.

Optical spaces

The concept of “optical spaces” plays a very important role in the understanding of optical systems as well as in the layout of their design. There are multiple optical spaces in most systems; however, the minimum is three: object space, image space, and lens space, as shown in Figure 5.1. One example of this is the idea of object space, which is defined as the space domain where the object lies in a homogeneous medium which has a given index of refraction. For many systems, this medium is typically air. One can visualize a region from negative infinity to the first glass surface of a lens as being the physical space where an object resides.

This book is an introduction to geometrical optics and is intended for use in courses on optical engineering. Although several excellent books already exist that cover both physical and geometrical optics, it has been our experience that the purpose of these books is not to teach the fundamentals of geometrical optics, but rather to introduce the subject in order to prepare students for more challenging physical optics courses to follow. In contrast, this book will teach the subject in such a way that it will provide future optical engineers with a solid background and the skills necessary to understand modern computer optical design programs used by lens designers. Furthermore, unlike previous textbooks, this book uses a right-handed coordinate system that lends itself more efficiently to lens design techniques, such as integrating multiple lenses in cascade.

This book is geared toward the professional engineer or student who wants to gain a broad understanding of geometrical optics with the details worked out. It uses a student's prior background in basic high school level algebra, trigonometry, and calculus to build a foundation for the concept of image formation, using linear equations to describe where the image is formed, its size and its classical third-order aberrations, and to teach all fundamental topics necessary to understand complex optics used in optical instruments.

The book has grown out of class notes developed for an introductory course in geometrical optics offered as part of the optical engineering curriculum at the University of Arizona College of Optical Sciences, Tucson, Arizona.

Background history

The history of optics is filled with examples of unique uses and situations that are beyond the scope of this book which is intended for the student on a first year optics course. However, a brief review is necessary to show how man has been trying to understand and describe light over the last 2500 years.

The word “optics” originated in a book on visual perception written by Euclid some 2000 years ago. Euclid developed geometrical theories to account for the observation of images by mirrors. Some names that come to mind in the history of optics are Ptolemy, Bacon, Brahe, Kepler, and more recently Newton, Huygens, Fermat, Young, and Einstein.

There is a story by Archimedes (212 bc) that the Greeks defended Syracuse (in modern-day Sicily) from the Roman fleet by reflecting sunlight with the soldiers' shields and burning the ships' sails by focusing the intense heat of the Sun's rays.

Muslims in the thirteenth century were purported to have the ability to create a burning mirror to use for burning cities (in the Holy Land). Roger Bacon, a monk under Pope Clement IV, was motivated by this threat to study optics as a weapon of war. He developed similar devices for the Christian crusaders battling the Muslims.

Ptolemy of Alexandria, a Greek from Egypt (about ad 190), knew that two transparent substances, glass and water, had indices of refraction of 3/2 and 4/3, respectively.

Although an optical design goal is to collect as much light as possible, the amount of light which actually enters optical systems is a small fraction of the light which radiates off the object being viewed. The only usable light in an optical system is the light which enters the system at appropriate angles to reach the image location. Optical apertures limit the amount of light that enters the system, and also limit the area of the recorded image. Stops determine both the field of view and the illumination within the image. The field stop determines the field of view, which in turn determines how much of the object can be seen through the optical system. The field of view is calculated as the maximum angle from the optical axis at which light can enter and pass through the system. Image illumination is limited by aperture stops that constrict the bundle of light rays entering the system. Rays far from the optical axis, or those entering the system at too steep an angle, will not reach the image plane. Image illumination determines whether the image can be detected by an eye, a recording medium, or a two-dimensional detector array.

A stop (sometimes called a diaphragm) can be visualized as an opaque plate with a circular hole, akin to a metal washer or window in a building. There are two types of stops in optical systems, and they are named according to their functions within the system.

Rays

In geometrical optics, light is assumed to travel in a definite direction from a source, exhibiting a behavior known as rectilinear propagation. One useful way to think of light is to imagine it traveling in a very narrow beam, which is to say that light can be modeled as a ray. The emergent ray can be thought of as being a very narrow line. A ray of light is like a mathematically infinite thin line, but a light ray has direction while a line does not. The rays of geometrical optics are perpendicular to the wavefronts, and indicate the most probable path of the photons of quantum optics. A wavefront is an undulation of energy which propagates from one point to another and, as it travels, it carries electromagnetic energy. The wavefront is conceptualized as a surface with a fixed phase across it, and periodic oscillations (sinusoid) in the direction of propagation, with each wavefront having its own wavelength, frequency, amplitude, phase, and polarization.

The point source in Figure 2.1 illustrates the relationship between wavefronts and rays. The source emits energy in a spherical shell in all directions (4π sr), propagating at the speed of light. A wavefront is the locus of points on rays that have the same optical path length from the point source (same number of periods). The wavefront is perpendicular to the rays. At an infinite distance, the spherical wavefront turns into a plane wavefront (at that point the radius equals infinity).

Paraxial ray tracing is a technique used in geometrical optics for predicting the paths light will take through an optical system. Its primary application is in the design of lens and mirror systems.

Rays may be viewed as comprising streams of photons emanating from a light source and propagating toward surfaces throughout the optical system. As discussed in Chapter 2, four things may happen as rays strike optical surfaces: they may be refracted (transmitted), reflected, scattered, or absorbed. These effects may occur singly or in combination, and all of the energy in the incident beam must be accounted for by these mechanisms.

To review, refracted light enters a transparent medium at an angle different from the incident angle. Some of the energy in the incident beam will also be reflected by the surface. The surface may scatter some of the incident light in one or more directions. Additionally, it may absorb some of the energy, resulting in a loss of intensity in the beam. Refracted and reflected rays may strike other surfaces, at which the same mechanisms will again take effect, and at which Snell's law may be applied.

Paraxial ray tracing is applied to systems in which diffraction and interference effects are insignificant. It is useful for modeling and optimizing the design of lens systems and instruments, (e.g. minimizing the effects of aberrations) before optical components are ordered or fabricated. The technique finds utility in many applications, a particular example being illumination engineering.

Many optical phenomena that appear in the sky may be explained using the geometrical ray tracing described in this book. A few that will be discussed here are rainbows, halos, sun dogs, and the optical illusion of mirages in the desert.

Rainbows

A rainbow is an optical phenomenon that causes a nearly continuous spectrum of light to appear in the sky. It is only present when the rain location is opposite the sun's direction from the observer – in fact, when they are 180° apart. As white light from the sun shines onto drops of water, a polychromatic bow is formed, with red in the outermost arc and blue/violet on the inside of the bow. The infrared color regions beyond the red are ignored; however, they do exist. This is the most traditional rainbow, in which sunlight is spread out into its spectrum of colors and diverted to the eye of the observer by spherical water drops. The “bow” part of the word comes from the fact that rainbows contain a group of nearly circular arcs of color (blue to red), all having a common center of curvature. A second rainbow sometimes appears as a fainter arc, with colors in the opposite order, i.e., with violet on the outside, and red on the inside.

In Chapter 3 we considered OPAs in which the parametric gain coefficient g is a constant along the fiber. This allowed us to obtain closed-form solutions for the gain in several important situations. In practice, however, fibers generally have properties that may cause g to vary as a function of z. Examples of these properties are: fiber loss, which causes the pump power to drop exponentially; non-uniform dispersion, which causes Δβ to fluctuate; birefringence (fixed or randomly varying), which introduces a complex evolution of SOPs. Under these circumstances the constant-g solutions are no longer applicable.

In several areas of physics that involve wave propagation in media with slowly varying properties, one often uses approximate solutions derived by making use of the Wentzel–Kramers–Brillouin (WKB) approximation, also referred to as the phase-integral method [1, 2]. This method was originally introduced in quantum mechanics. In optics, it has been used extensively to study propagation in multimode fibers [3]. It can lead to closed-form solutions if the properties vary in a simple fashion, such as linearly. If the variations are not simple, one may still be able to use the method to obtain some useful expressions involving the average of g along the fiber.

Karlsson first applied the WKB method to fiber OPAs, in the context of modulation instability (MI) [4]. Here we present a slightly different version, starting from the basic OPA equations.

Introduction

In the preceding chapters we have reviewed the state of the art in fiber OPAs, in terms of the current theoretical understanding and of their experimental performance and progress toward applications. Along the way we have identified a number of obstacles that stand in the way of further progress. In this chapter we review some areas of technology where advances would be desirable for the development of fiber OPAs or related devices. We also look at areas that have not been investigated much to date and may yet present interesting research and development opportunities.

Fibers

Conventional fiber structures, consisting of a cylindrical core surrounded by a cladding made from a lower-index material, offer limited opportunities for optimizing the fiber properties of fiber OPAs. In contrast, as mentioned in Section 2.1, holey fibers (HFs) have the potential for providing new generations of fibers suitable for fiber OPAs. In principle all HF properties, including loss, nonlinear index, mode effective area, and dispersion, can be optimized. To improve OPA performance significantly beyond the level reached with currently available fibers, all properties will eventually need to be simultaneously optimized in the same HF. This will pose an interesting challenge for the designers and manufacturers of HFs.

High-γ fibers

A large value of the fiber nonlinearity coefficient γ (see Eq. (2.24)) is beneficial in a number of ways. It makes it easier to obtain a large gain bandwidth at moderate pump power. It also reduces the product P0L required to achieve a given gain.

Introduction

In the preceding chapters we investigated in detail the gain spectra of fiber OPAs with CW pumps. This is an important case for several reasons. From a practical standpoint, CW pumps are a requirement in applications such as amplifiers for optical communication. Also, experiments with CW pumps are simple to set up and are thus widely used. From a theoretical standpoint, the assumption of a CW pump leads to the relatively simple sets of equations and analytic solutions that we have studied in previous chapters.

It is also often desirable, however, to use pumps that do not have constant intensities, for a variety of reasons. One reason is that pumps that are pulsed with a low duty cycle can generate high peak powers while having a modest average power; this is a common way of generating peak powers of several watts or tens of watts using relatively inexpensive EDFAs with sub-watt average output power. Another reason is to generate OPA gain only during specified periods of time, to select particular parts of a signal waveform in order to implement some signal processing function. Examples of this are optical sampling for displaying optical pulse shapes, optical sampling for generating multiple replicas of optical signals, and optical switching for demultiplexing optical pulse trains.

Here we present alternative techniques for calculating the output fields of pulsed-pump OPAs.

Introduction

In this chapter we introduce the basic types of fiber optical parametric amplifier (OPA) and simple models to study their characteristics. We will confine ourselves to situations in which all the waves are launched into the fiber with the same linear state of polarization (SOP) and remain in that state along the entire fiber. This allows us to consider a single component of the electric field and hence to write scalar equations for it. We first set up the basic OPA equations starting from Maxwell's equations for the case of nonlinear polarization. In the process we derive an expression for the fiber nonlinearity coefficient γ in terms of the waveguide properties and those of the interacting modes. We then proceed with the solution of the OPA equations in a variety of situations for which exact solutions are known. The solutions in the absence of loss and pump depletion are relatively simple, being expressible in terms of exponentials, or alternatively in terms of sinh and cosh functions. They are used extensively in practice as a first approximation to calculating the gain spectra of various fiber OPAs. Solutions in other regimes are more complicated; some involve Bessel functions, others involve Jacobian elliptic functions, etc. While it is more difficult to grasp their properties, they can be useful computational tools for obtaining accurate results when the exponential solutions are not applicable.

Introduction

If an optical gain medium is available, one can in principle convert it into an oscillator by providing optical feedback to form an optical resonator. If the gain medium is a fiber OPA, the resulting device is a fiber optical parametric oscillator (fiber OPO). There are many possibilities for designing the feedback system, and hence a variety of possible fiber OPOs. In the simplest versions the pump is supplied externally and is non-resonant, i.e. it passes through the OPO in one direction, interacting with the signal and idler within the nonlinear medium.

Let us assume that the optical cavity is resonant at the signal wavelength but not at the idler wavelength; we then have a singly resonant oscillator (SRO). If the cavity has a high quality factor Q, little or no signal escapes from it, and so the useful output of the SRO may have to be the idler. If the idler is not available, however, the output is a fraction of the signal, exiting through a partially transparent mirror or via a fiber coupler; in such a case the coupling introduces a signal loss, and so Q may not be very high. Most fiber OPOs that have been implemented to date are of the SRO type.

Introduction

In Chapter 3 we derived optical parametric amplifier (OPA) equations under the assumption that all four waves involved were in the same state of linear polarization along the fiber length. This led to a relatively simple set of four scalar equations. By solving these equations we then obtained a set of solutions that can provide a good preliminary understanding of the performance of fiber OPAs.

In this chapter we enlarge the class of OPAs by considering more general possibilities for the SOPs of the waves. We begin by developing a vector formalism for light propagating in arbitrary SOPs in isotropic fibers. We apply it to circularly polarized (CP) and linearly polarized (LP) waves; since some of the waves may have identical or orthogonal SOPs, this leads to a class of 12 basic types of OPA. We then turn to fibers with constant birefringence, such as polarization-maintaining fibers (PMFs), and explore the impact of the birefringence on the gain spectrum. Finally we consider the typical case of fibers that have weak birefringence with random longitudinal variations, and we investigate the effect of these variations.

Isotropic fibers

The treatment here follows the general approach introduced in [1].

Classification of basic OPA types

We make the following basic assumptions. (i) The fiber is lossless. (ii) The fiber does not exhibit birefringence, either linear or circular. (iii) The fiber operates in the fundamental mode at all frequencies.