Light Scattering by Small Particles

PREFACE

The scattering of electromagnetic waves by a homogeneous sphere is a problem with a known solution. I first met this problem when I needed some numbers and curves in an astrophysical investigation. I soon learned that it is a long way from the formulae containing the solution to reliable numbers and curves. Subsequent conversations and correspondence with other research workers, notably in chemistry, showed that the same difficulty was felt in other fields.

The studies on which the present book is based were started in 1945 in an attempt to compile the data available in the literature and to fill in the gaps, where needed. Several related problems, such as the scattering by cylinders, were added to the original topic.

For clearer presentation, the problems of mathematical physics dealing with the scattering properties of single particles (part II) have been separated from the problems arising in specific fields of application (part III). The properties of the particles that should be known in order to describe the optical properties of a medium consisting of such particles have been defined in general terms (part I).

New formulae or numerical results are contained in almost all chapters. They are noted in the references at the end of each chapter. The reference lists have steadily grown in the course of the years; they probably are fairly complete, but no systematic bibliographical study has been made.

Although the book has a mathematical character, requirements of mathematical rigor do not dominate the presentation. Arguments based on physical intuition are given wherever they illuminate the subject more clearly than a mathematical derivation. Simple results that arise under special sets of assumptions are often derived both ways. In view of the wishes expressed by several colleagues, I have not shrunk from a certain inconsistency in the level of presentation. For instance, chapter 17, which comes closest to an actual research report, contains less explanation of elementary detail than some earlier chapters, which may be consulted by research workers without special mathematical training.

Personal acknowledgment of the support received from numerous friends and colleagues in writing this book is impossible. I wish to express my thanks both for the information they have contributed and for their inspiring questions.

H. C. VAN DE HULST

Leiden, The Netherlands

March, 1957

PART I

Basic Scattering Theory

1.INTRODUCTION

1.1.Scattering, Absorption, Extinction

This book is a treatise on the scattering of light. Hardly ever is light observed directly from its source. Most of the light we see reaches our eyes in an indirect way. Looking at a tree, or a house, we see diffusely reflected sunlight. Looking at a cloud, or at the sky, we see scattered sunlight. Even an electric lamp does not send us light directly from the luminous filament but usually shows only the light that has been scattered by a bulb of ground glass. Everyone engaged in the study of light or its industrial applications meets the problem of scattering.

Scattering is often accompanied by absorption. A leaf of a tree looks green because it scatters green light more effectively than red light. The red light incident on the leaf is absorbed; this means that its energy is converted into some other form (what form of energy is irrelevant for our purpose) and is no longer present as red light. Absorption is preponderant in materials such as coal and black smoke; it is nearly absent (at visual wavelengths) in clouds.

Both scattering and absorption remove energy from a beam of light traversing the medium: the beam is attenuated. This attenuation, which is called extinction, is seen when we look directly at the light source. The sun, for instance, is fainter and redder at sunset than at noon. This indicates an extinction in the long air path, which is strong in all colors but even stronger in blue light than in red light. Whether scattering or absorption is mainly responsible for this extinction cannot be judged from this observation alone. Looking sideways at the air, through which the sun shines, we see that actually blue light is scattered more strongly. Measurements show that all light taken away from the original beam reappears as scattered light. Therefore, scattering, and not absorption, causes the extinction in this example.

Other terminology is sometimes used but is not recommended. Here the word absorption is used in the sense of extinction as defined above.¹ Actual absorption is then designated as pure absorption or true absorption. Throughout this book terms will be used as defined above, so that

Extinction = scattering + absorption.

1.2.Subject Limitations

Only a few of the multitude of scattering phenomena are treated in this book.

A first restriction is that we shall always assume that the scattered light has the same frequency (i.e., the same wavelength) as the incident light². Effects like the Raman effect, or generally any quantum transitions, are excluded.

1.21Independent Scattering

A second, most important limitation is that independent particles are considered. The distinction is roughly this: the scattering by well-defined separate particles, such as occur in a fog, is within the province of this book, whereas the scattering by a diffuse medium, as for instance a solution of a high polymer, is not discussed.

A more precise distinction may be made. If light traverses a perfectly homogeneous medium, it is not scattered. Only inhomogeneities cause scattering. Now, in fact, any material medium has inhomogeneities as it consists of molecules, each of which acts as a scattering center, but it depends on the arrangement of these molecules whether the scattering will be very effective. In a perfect crystal at zero absolute temperature the molecules are arranged in a very regular way, and the waves scattered by each molecule interfere in such a way as to cause no scattering at all but just a change in the overall velocity of propagation. In a gas, or fluid, on the other hand, statistical fluctuations in the arrangement of the molecules cause a real scattering, which sometimes may be appreciable. In these examples, whether or not the molecules are arranged in a regular way, the final result is a cooperative effect of all molecules. The scattering theory then has to investigate in detail the phase relations between the waves scattered by neighboring molecules. Any such problem, in which the major difficulty is in the precise description of the cooperation between the particles, is called a problem of dependent scattering and is not treated in this book.³

Frequently, however, the inhomogeneities are alien bodies immersed in the medium. Obvious examples are water drops and dust grains in atmospheric air and bubbles in water or in opal glass. If such particles are sufficiently far from each other, it is possible to study the scattering by one particle without reference to the other ones. This will be called independent scattering; it is the exclusive subject of this book.

It may be noted that waves scattered by different particles from the same incident beam in the same direction still have a certain phase relation and may still interfere. The fact that the wavelength remains the same means that the scattered waves must be either in phase and enhance each other or out of phase and destroy each other, or any intermediate possibility. The assumption of independent scattering implies that there is no systematic relation between these phases. A slight displacement of one particle or a small change in the scattering angle may change the phase differences entirely. The net effect is that for all practical purposes the intensities scattered by the various particles must be added without regard to phase. It thus seems that the scattering by different particles is incoherent, although in the strict sense this is not true. An exception must be made for virtually zero scattering angles. In these directions no scattering in the ordinary sense can be observed. (See chap. 4.)

What distance between particles is sufficiently large to ensure independent scattering? Early estimates have shown that a mutual distance of 3 times the radius is a sufficient condition for independence. This may not be a general rule, but a more precise discussion is beyond the scope of this book. In most practical problems the particles are separated by much larger distances. Even a very dense fog consisting of droplets 1 mm in diameter and through which light can penetrate only 10 meters has about 1 droplet in 1 cm³, which means that the mutual distances are some 20 times the radii of the drops. The same is true for many colloidal solutions.

1.22.Single Scattering

A third limitation is that the effects of multiple scattering will be neglected. Practical experiments most often employ a multitude of similar particles in a cloud or a solution. The obvious relations for a thin and tenuous cloud containing M scattering particles are that the intensity scattered by the cloud is M times that scattered by a single particle, and the energy removed from the original beam (extinction) is also M times that removed by a single particle. This simple proportionality to the number of particles holds only if the radiation to which each particle is exposed is essentially the light of the original beam.

Each particle is also exposed to light scattered by the other particles, whereas the light of the original beam may have suffered extinction by the other particles. If these effects are strong, we speak of multiple scattering and a simple proportionality does not exist. This situation may be illustrated by a white cloud in the sky. Such a cloud is like a dense fog; its droplets may be considered as independent scatterers. Yet the total intensity scattered by the cloud is not proportional to the number of droplets contained in it, for not each droplet is illuminated by full sunlight. Drops within the cloud may receive no direct sunlight at all but only diffuse light which has been scattered by other drops. Most of the light that emerges from a cloud has been scatttered by two or more droplets successively. It is estimated (for a very thick cloud) that about 10 per cent emerges after a single scattering.

Multiple scattering does not involve new physical problems, for the assumption of independence, which states that each droplet may be thought to be in free space, exposed to light from a distant source, holds true whether this source is the sun or another droplet. Yet the problem of finding the intensities inside and outside the cloud is an extremely difficult mathematical problem. This problem has been studied extensively in many ramifications. It is usually called the problem of radiative transfer. Common applications are the transfer of radiation in a stellar atmosphere and the scattering of neutrons in an atomic pile. The cases treated so far refer to rather simple forms both of the single scattering pattern (isotropic scattering, Rayleigh scattering) and of the entire cloud (infinite or finite slab with plane boundaries, sphere). The reader is referred to the literature for further details.

A simple and conclusive test for the absence of multiple scattering is to double the concentrations of particles in the investigated sample. If the scattered intensity is doubled, only single scattering is important. Another criterion may be the extinction. The intensity of a beam passing through the sample is reduced by extinction to e−τ of its original value. Here τ is the optical depth of the sample along this line. If τ < 0.1 single scattering prevails; for 0.1 < τ < 0.3 a correction for double scattering may be necessary. For still larger values of the optical depth the full complexities of multiple scattering become a factor. They may not prevent a determination of the scattering properties of a single particle, but they certainly make the interpretation much less clear. Caution is invariably required when the optical depth is not small in all directions through the sample.

Concluding this section it may be noted that this book treats only the very simplest case occurring in the theory of many particles. This leaves room for a thorough treatment of the scattering theory for one particle.

1.3.Historical Review

A proper understanding of the subject will be helped greatly by a review of its history, even though this has to be brief and can only show some of the highlights.

The nature of light was the subject of speculation and research by nearly all the great scientists of the seventeenth century. Snell's law, Newton's rings, Huygens' principle, and Fermat's principle date from this era. The general feeling was that light was something in the ether as sound is in air, but the phenomenon of polarization seemed to present insurmountable difficulties in such a concept, so that the century closed with the problem of the nature of light unsolved. Nor did the eighteenth century add much to the solution of this problem.

The decisive steps were taken by Young and Fresnel at the beginning of the nineteenth century. Young studied diffraction phenomena and showed that the pattern of maxima and minima in the shadow space behind a hair was caused by interference of waves coming from both sides of the hair. The nature of these waves remained obscure to him. Fresnel showed that these waves originate from the undisturbed wave front at either side of the obstacle. In this explanation Fresnel drew upon the old principle of Huygens that each point of a wave front may be considered as a center of secondary waves. By combining this principle with Young's principle of interference, Huygens' rule that the envelope of the secondary waves forms a new wave front had a natural explanation. If part of the original wave front is blocked by an obstacle, the system of secondary waves is incomplete so that diffraction phenomena occur. The exact agreement obtained between theory and experiments in a number of difficult problems left no doubt that Fresnel's explanation was correct. It will also form the basis of many problems discussed in this book.

The final explanation of polarization was given by Young's suggestion that the ether must exhibit transverse vibrations, like a rigid solid. It was a happy circumstance that in the same period Malus had discovered that polarization occurs at reflection and Brewster had measured the intensities of the polarized components at any angle of incidence. Then Fresnel, taking up Young's idea of transverse vibration, was able to derive these intensity rules theoretically on the basis of the simple boundary condition that the tangential component of the amplitude of vibration must be continuous.

One of the outstanding achievements of the later nineteenth century was Maxwell's electromagnetic theory of light, by which electric and optical phenomena were linked together. The modern way of expressing the boundary condition is, therefore, to say that the tangential component of the electric field must be continuous. Yet this improvement is not always essential to our problem. Many scattering problems involving polarization might as well be formulated in Fresnel's terminology as in modern terms by means of electric and magnetic fields.

The nineteenth century, particularly its second half, was the era of the great mathematical physicists: Poisson, Cauchy, Green, Kirehhoff, and the paragons Stokes and Rayleigh, if a very incomplete enumeration may suffice. With the exception of Stokes's discussion of the nature of natural and partly polarized light as a superposition of many polarized waves (sec. 5.13. of this book), no fundamental problems in optics were solved. The quest was for new skill in the mathematical formulation of complex phenomena rather than for added physical insight into simple phenomena. Coordinate systems in which the wave equation is separable were found. Fresnel's version of Huygens' principle was given a mathematical basis by Kirehhoff; Bessel functions and related functions were made into a powerful tool. A problem typical for this era was the scattering of light by a homogeneous sphere, one of the main topics of this book. It was among the more difficult problems and, though many special cases had been solved before, its full solution was formulated by Mie only in 1908.

This period came to an end with the rise of quantum mechanics. Debye was possibly the last one to study scattering problems of this kind with the devotion, insight, and mathematical technique displayed by the masters of the nineteenth century. Soon afterwards most of the mathematical physicists of top rank began devoting their time to studies of quantum mechanics or other fields of current interest. The scattering problems discussed in this book became a subject for applied scientists, interested in numerical results, or for students writing doctor's theses, of whom the writer was one. Formulae and numerical results were gradually collected, but few important ideas were added during this period.

The final stage of this brief and oversimplified history of the subject under review is its rather curious comeback in more recent years. The new interest devoted to it springs from very diverse sources, ranging from unemployment, which helped to start the New York Mathematical Tables Project, to the invention of radar and the development of quantum mechanics. Also new research in astronomy and chemistry prompted more extensive calculations than had been made before. It is important to see what quantum mechanics has to do with it. The analogy between traveling electrons and waves of light, or sound, had been an important help in the early development of quantum mechanics. So it was evident that scattering of an electron by an atom should have points of analogy to scattering of light, or sound, by a solid particle. In the late thirties quantum mechanics had developed so far that the need for accurate computations of scattering cross sections was felt. To this end new methods were devised, which partly were variants of the methods established in the domain of optics thirty or more years before and partly were of a new character. This has stimulated new studies of optical scattering problems. The method of phase shifts and the variational methods were new and have since found their application also in optical problems.

1.4.Sketch of the Book

The book has one theme: single scattering by independent particles. This means that only those experimental conditions are considered in which the particles are so far from each other that each of them is exposed to a parallel beam of light (i.e., light from a distant source) and has sufficient room to form its own scattering pattern, undisturbed by the presence of other particles. The book consists of three parts, which treat three distinct phases of the subject.

Part I. This part gives general theorems for particles that may have arbitrary size, shape, and composition. It is shown that the scattering by any finite particle is fully characterized by its four amplitude functions, S1 S2, S3, and S4, which are complex functions of the directions of incidence and of scattering. Knowledge of these functions suffices for computing the intensity and polarization of scattered light, the total cross sections of the particle for scattering, absorption, and extinction, and the radiation pressure exerted on the particle. For homogeneous spheres only two such functions S1(ϕ) and S2(ϕ) are needed, where ϕ is the scattering angle.

Chapter 2 presents all that can be expressed in terms of intensities without the introduction of phases and complex numbers. Chapter 3 introduces phases and complex amplitudes. Chapter 4 gives the main theorems derived for a single particle as well as a medium of independent particles; sec. 4.42 is the one most often referred to, as it gives the formulae for homogeneous spheres. The simplifications resulting if arbitrary particles are distributed in orientations with certain symmetry relations are summarized in chap. 5.

Part II. This is the main part of the book. It specifies the amplitude functions for a great variety of special particles. These chapters contain many cross references, as one special case and another special case often have a common limiting case. The final aim is, in each case, to derive the amplitude functions and the cross sections. Roughly, we can distinguish three groups of chapters.

Chapters 6 to 8 discuss particles that do not have a special form. They are, respectively, very small, very soft, and very large.

Chapters 9 to 14 treat homogeneous spheres of arbitrary size. The two parameters in all formulae are x = 2πa/λ (a = radius, λ = wavelength) and m, the refractive index. The rigorous solution (Mie) is derived, its limiting forms discussed, and many numerical results reproduced in tables and graphs. A survey of these chapters can best be gained from the Contents and from sec. 10.1.

Chapters 15 to 17 are devoted to particles of other regular forms, namely long circular cylinders, some miscellaneous geometrical forms, and large bodies with a smoothly curved surface.

Part III. This part gives selected applications in many domains of chemistry, physics, meteorology, and astronomy. Whereas the earlier parts are reasonably complete, this part is meant to give only typical examples of practical problems in which the preceding theories have proved important. Some common features of such applications are discussed in chap. 18, and chaps. 19 to 21 give examples from different fields. A scientist who is not directly concerned with the mathematical complexities of the subject may find it useful to turn to the chapter on his own field first in order to find his way into the formulae, graphs, and tables of the preceding parts.

References

The dependent scattering by polymers, etc., not treated in this book, is surveyed by

G. Oster, Chem. Revs., 43, 319 (1946).

B. H. Zimm, P. Doty, and R. Stein, Theory and Application of Light Scattering, New York, John Wiley & Sons, in preparation.

The multiple scattering, also excluded, is discussed by

S. Chandrasekhar, Radiative Transfer, Oxford, Oxford Univ. Press, 1950.

V. Kourganoff, Basic Methods in Transfer Problems, Oxford, Oxford Univ. Press, 1952.

H. C. van de Hulst, The Atmospheres of the Earth and Planets (2nd ed.), chap. 3, G. P. Kuiper, ed., Chicago, Univ. of Chicago Press, 1952.

An excellent book on the history is

E. T. Whittaker, A History of the Theories of Aether and Electricity (2nd ed.), part I, London, Longmans, Green & Co., 1952.

¹ E.g., in the term interstellar absorption.

² This may tecdhnically be ccalled coherent scattering. However, this term is often used with a different connotation: and assembly of particles is said to scatter incoherently if the positions of the individual particles vary sufficiently (sec. 1.21).

³ See the references at the end of this chapter. References appear throughout at the end of each chapter.

2. CONSERVATION OF ENERGY AND MOMENTUM

2.1. Scattering Diagram and Phase Function

In accordance with the limitations imposed upon our subject in the first chapter (sec. 1.2) we shall consider a single particle of arbitrary size and form, illuminated by a very distant light source. We shall inquire into the properties of the scattered light at a large distance from the particle. This implies the assumption that other particles leave sufficient room about this particle for the distant scattered field to be established (assumption of independence, sec. 1.21).

The most important property of the scattered wave is its intensity. By intensity I we shall understand the energy flux per unit area; its c.g.s. units are erg per cm² per sec. In optics this is called the irradiance. The incident wave and the scattered wave at any point in the distant field are unidirectional, i.e., each confined to one direction or to a very small solid angle around this direction. The term intensity as used in this book refers to the total energy flux in this solid angle. The waves are also assumed to be monochromatic, i.e., confined to one frequency or to a small frequency interval. The intensity refers to the total energy flux in this interval. Changing to other units (m.k.s. units: watt per m²) makes no difference in the formulae, except where I is expressed directly in terms of electric and magnetic fields. With the same exception and with the exception of the formulae giving the radiation pressure we can read everywhere for I the illuminance, i.e., luminous flux per unit area (units: lumen per m² = lux).

Neither the incident nor the scattered light is completely characterized by its intensity; the additional properties are polarization and phase. The phases cannot be measured directly, but they are of importance in the correct formulation of the scattering of polarized light. So, throughout the second and main part of this book we shall work with scattering functions S1(θ, φ) and S2(θ, φ), which are complex numbers and describe amplitude and phase of the scattered waves. In the present chapter the relations are derived that can be formulated in terms of intensities without reference to phase. The phases are introduced in chap. 3 and applied to scattering theory in chaps. 4 and 5.

The scattered wave at any point in the distant field has the character of a spherical wave, in which energy flows outward from the particle. The direction of scattering, i.e., the direction from the particle to this point, is characterized by the angle θ which it makes with the direction of propagation of the incident light and an azimuth angle φ (Fig. 1).

Let I0 be the intensity of the incident light, I the intensity of the scattered light in a point at a large distance r from the particle, and k the wave number defined by k = 2π/ψ, where ψ is the wavelength in the surrounding medium. Since I must be proportional to I0 and r−2 we may write

Here F(θ,φ) is a dimensionless function (F/k² is an area) of the direction but not of r. It also depends on the orientation of the particle with respect to the incident wave and on the state of polarization of the incident wave.

Fig. 1. Definition of scattering angle. Incident light is from below in this and in other figures.

The relative Values i, or of F, may be plotted in a polar diagram, as a function of θ in a fixed plane through the direction of incidence. This diagram is called a scattering diagram of the particle. When F(θ, φ) is divided by k²Csca where Csca is the area defined below, another function of direction, the phase function¹, is obtained. The phase function has no physical dimension, and its integral over all directions is 1.

2.2. Conservation of Energy

Let the total energy scattered in all directions be equal to the energy of the incident wave falling on the area Csca. By this definition and by the preceding equation we have

where dw = sin θ dθ dφ is the element of solid angle and the integral is taken over all directions. Likewise, the energy absorbed inside the particle may by definition be put equal to the energy incident on the area Cabs, and the energy removed from the original beam may by definition be put equal to the energy incident on the area Cext. The law of conservation of energy then requires that

Cext = Csca + Cabs

The quantities Cext Csca, Cabs, are called the cross sections of the particle for extinction, scattering, and absorption, respectively. They have the dimension of area. Generally, they are functions of the orientation of the particle and the state of polarization of the incident light.

Non-absorbing particles have Cext = Csca. This cross section will sometimes be denoted by C without suffix.

2.3. Conservation of Momentum; Radiation Pressure

According to Maxwell's theory, light carries momentum as well as energy. The direction is that of propagation and the amount is determined by energy

where c is the velocity of light. We shall consider the component of momentum in the direction of propagation of the incident wave (which will further be called the forward direction). The momentum removed from the original beam is proportional to Cext Of this, the part Cabs is not replaced, but the part Csca is partially replaced by the forward component of the momentum of the scattered light. This component in any direction is proportional to I cos θ The total forward momentum carried by the scattered radiation is, therefore, proportional to

This equation defines the weighted mean of cos θ with the scattering function as weighting function. Examples are worked out in secs. 10.62 and 12.5. It follows that the part of the forward momentum that is removed from the incident beam and not replaced by the forward momentum of the scattered light is proportional to

For non-absorbing particles

This momentum is given to the scattering particle. A certain force is, therefore, exerted on the scattering particle in the direction of propagation of the incident wave. This is the well-known phenomenon of radiation pressure The force is equal to the force that would be exerted by the incident light on the area Cpr of a black wall. Its magnitude is

Force = I0Cpr/c.

Generally, the particle will also suffer a component of force perpendicular to the direction of propagation of the incident light. Its magnitude may be derived in a similar way. The influence of this component cancels out in a cloud of particles which are oriented at random. The particle is generally also subject to a torque. Its calculation requires not only the distant field but also the field approximated to a higher power of I/r.

2.4. Efficiency Factors

Most particles have an obvious geometrical cross section G. A sphere of radius a has, for instance, G = πra². The dimensionless constants

will be called the efficiency factors for extinction, scattering, absorption, and radiation pressure, respectively. For quite general particles these factors depend on the orientation of the particle and on the state of polarization of the incident light. For spheres they are independent of both. In all cases we have

Qext = Qsca + Qabs.

2.5. Scattering Diagram for Polarized Light

The way in which F(θ, φ) depends on the form and size of the scattering particle is the main subject of this book and will be treated in chaps. 6 to 17. The way in which it depends on the state of polarization of the incident light will be treated here inasmuch as it can be formulated without referring to phase effects. The derivation is found in secs. 4.41, 5.13, and 5.14.

The full relations indicating how the intensity and state of polarization of the scattered light depend on the intensity and state of polarization of the incident light are contained in the matrix equation

Here I, Q, U, and V are the Stokes parameters of the scattered light, the meaning of which is explained in sec. 5.13, and I0, Q0, U0, and V0 are the corresponding parameters of the incident light. The matrix F consists of 16 components, each of them a real function of the directions of incidence and scattering. The first of the four equations contained in this matrix equation reads

On comparing this equation with the one in sec. 2.1, we find that the value of F is

which specifies the manner in which F depends on the state of polarization of the incident light, defined by the quantities Q0/I0, U0/I0, and F0/I0. For incident natural light the latter quantities are zero, so that

F = F11.

In the most general case the matrix F is asymmetric. For a single particle the number of independent constants is reduced to 7 because 9 relations exist between the 16 elements (sec. 5.14). For a cloud consisting of many particles the number of constants actually is 16. However, symmetry relations reduce the number of independent constants in most practical circumstances (secs. 5.2 and 5.3). For instance, the scattering matrix of a homogeneous spherical particle is characterized by 3 independent constants, i1, i2, and δ, which are functions of the angle θ (secs. 4.42 and 9.31). In this case 10 of the 16 constants are zero, and the other 6 are quadratic functions of the complex amplitude functions S1 and S2(θ). Full formulae are found in sec. 4.42.

2.6. Scattering and Extinction by a Cloud Containing Many Particles

Let a cloud contain many scattering particles and be optically thin so that the incident intensity I0 for each particle is the same (cf. sec. 1.22). It is then possible to write the equation

for each particle, each particle being denoted by an index i. The particles need not be similar. By summation we find that a formula of the same form as the first equation of sec. 2.1 holds for the entire cloud and that

This addition is based on the assumption that phase effects may be neglected (sec. 1.22).

This formula may be applied to a volume element V of an extended medium that contains N identical particles per unit volume, each characterized by the same function F(θ, δ). The number of particles in the element then is NV, and the scattered intensity at a distance r is given by

If the projected area of the volume in this direction is A, the radiation is contained in the solid angle A/r² so that the average brightness (radiance) of the scattering element is

In terms of luminous units I0 is the illuminance, B is the luminance and is measured in lumen per steradian or candle per m².

Many evident variations on these results may be made. Among them may be mentioned the one that, if polarizations are involved, the summation formula holds for each of the 16 elements Fik separately.

A similar addition formula holds for the cross sections Cext, Csca, and Cabs. The somewhat hidden physical ground of this rather obvious assumption is explained in secs. 4.22 and 4.3.

One very common application is the extinction by a cloud of spherical particles of the same composition but of different sizes. Here the efficiency factor Qext(a) and the extinction cross section Cext(a) = πa²Qext(a) are functions of the radius a. Let there be N(a) da particles per cm³ with radii in the interval da so that N(a) da = N is the total number per cm³. The extinction coefficient of the medium, which equals the total cross section per cm³ (sec. 4.3), is then

Frequently it is desirable to change from the integration variable a to the variable x = 2πa/ψ.

¹ The word phase has come into this expression via astronomy (lunar phases) and has nothing to do with the phase of a wave.

3. WAVE PROPAGATION IN VACUUM

The preceding chapters dealt with intensities only. Throughout this book we shall describe a wave not only by its intensity but also by its phase. This phase plays an important role in chap. 4 (the general extinction formula) and in chap. 5 (the general formulation of polarized light).

As preparation for these and further chapters we shall review the very simplest problem: the phase relations in a plane wave traveling in vacuum. This problem was first successfully discussed by Fresnel.

3.1.Fresnel's Formulation of Huygens' Principle

3.11. Complex Numbers

Complex amplitudes are an indispensable tool in the mathematical physics of wave phenomena. We briefly recall their definition and simple properties. The amplitude a and the phase α of a periodic wave are combined into a complex amplitude:

A = aeiα = a cos α + ia sin α.

Here i is the imaginary unit . A complex number may be graphically displayed as a point in a plane (the complex domain) by plotting the real part (a cos α) horizontally along the real axis and the imaginary part a sin α) vertically along the imaginary axis. The factors −1 and i may always be written in an exponential form by means of

−1 = eiπ, i = eiπ/2.

The physical quantity represented by a complex expression containing A as a factor is always assumed to be equal to the real part of that expression. The intensity of the wave is proportional to the square of the amplitude. This square may be written in various ways as

|A|² = A.A* = (a cos α + ia sin α)(a cos α − ia sin α) = a².

The vertical bars denote the modulus or absolute value, and the asterisk denotes the conjugate complex value ((i replaced by −i wherever it occurs explicitly or implicitly).

3.12. Derivation of Fresnel's Formula

Let a light source be at infinite distance so that we have a plane wave of constant intensity. Let the wave propagate in the positive z-direction. Further, t = time, ψ = wavelength, k = wave number = 2π/ψ, c = velocity of light, ω = circular frequency = kc. The field of the light wave may then be represented by the complex expression¹

u = e−ikz + iω

Here u may represent any component of the electric or magnetic fields;

Fig. 2. Explanation of rectilinear propagation by Fresnel.

in the early nineteenth century, when light was still considered an elastic vibration of the ether, it was called the disturbance.

Polarized light is characterized by two such amplitudes. All arguments in this chapter are valid for these two amplitudes separately to the degree of approximation to which they are valid at all. This means that the state of polarization is preserved. For simplicity the theory is presented with one amplitude only.

The planes z = constant are the planes of constant phase and therefore are called wave fronts. Figure 2 shows two such wave fronts, I and II, at a mutual distance , which we suppose to be much larger than the wavelength:

The disturbance at front I may be considered as the cause of the disturbance at front II at a time that is /c seconds later. Roughly, the disturbance at A‘ is caused by that at A, the disturbance at B’ is caused by that at B. This corresponds to rectilinear propagation. However, this rule holds only approximately. If for instance all light at the left side of A is screened off, there is not a sharp shadow edge at A′. Apparently the disturbance at A′ is caused to a certain extent by the disturbances at all points close to A. Huygens visualized this idea by assuming that all points of I were centers of secondary spherical waves and that the envelope of these waves determines the new wave front II (Huygens' principle). Huygens could explain the laws of reflection and refraction in this way, but the question how large a surrounding of A cooperates in determining the disturbance at A′ remained open. Therefore, a quantitative theory could not be derived.

A very plausible solution of this problem was found by Fresnel, and its essential correctness was proved by the successful solution of many diffraction problems. Later it proved to be an approximation (valid whenever kl 1) of the rigorous formula. Fresnel assumed that the secondary waves from all points of I should cooperate at A′ (or indeed at any point beyond I) according to the principle of interference, just discovered by Young. This means that the disturbances due to these waves have to be added, each with its proper phase.

Let dS be a surface element of plane I at a distance r from A′. Then a spherical wave emitted by dS causes the disturbance

at A′. Here uI is the disturbance at any point of plane I, and q is a constant to be fixed later. With rectangular coordinates in plane I, centered at A, we have

dS = dx dy

and if we anticipate that only points of I for which

x and y

effectively influence the disturbance at A′, we also have

The r in the denominator may be replaced by . The analytic representation of Huygens' principle thus becomes

Here uI has been retained under the integrals because in diffraction problems in general uI is a function of x and y, e.g., zero where the light is screened and 1 where it is not screened.

In the propagation treated in this section uI is independent of x and y. The remaining integrals are of the type

(If the limits are not ∞ this is called Fresnel's integral.) The relation reduces then to

The factors cancel out, as they should, for we know from the first formula in this section that the result should be

This finally fixes the constant q:

The full result is: The disturbance caused by an area dS of a wave front with disturbance uI at a point at the distance r, which is in a direction not too far from the direction of propagation, is

This formula suffices for a quite accurate solution of most diffraction problems. The derivation presented here closely resembles the one given by Fresnel in 1818.

3.13. Quantitative Examples

The applications of this formula are so well known that we may refer to textbooks on physical optics for further details. Straightforward applications in this book are found, e.g., in sec. 8.2 (diffraction by opaque bodies) and in sec. 11.3 (diffraction by transparent spheres).

Most textbooks emphasize the relative contributions of the various surface elements dS to the final result. For instance, if, as in Fig. 2, the disturbance at a distance from the wave front I is sought, this wave front may be subdivided into successive zones, for which the phase of exp (—ikr) is such that surface elements in those zones give contributions of alternating sign to the final amplitude. These are the Fresnel zones. The central zones contribute most effectively. The outer zones are less effective because their phases change so rapidly that their contributions tend to cancel out. The contribution of all points outside A lags behind in phase, because BA′ > AA′. The average phase lag is π / 2; it is compensated by the factor i in Fresnel's formula at the end of sec. 3.12.

It may be stressed that the same formula admits of quite simple quantitative applications. The quantitative definition of the effective area, useful for precise computations of intensity, is the following: The amplitude at a distance l beyond a plane wave front is such as if an area lλ of the wave front contributes with equal phase and the remaining part of the wave front not at all.

Some examples may illustrate the application of this rule.

(a) A lens without aberration is placed in a parallel beam of light. By what factor does the intensity at its focus exceed the intensity in the undisturbed beam? Answer: The light at the focus comes with the same phase from the entire area S of the lens at a distance f, where f is the focal distance. If the lens were absent the light would come effectively from the area λf at the same distance. So the amplitude is increased by the factor S / λf, the intensity by the factor S² / λ²f². This formula is correct for any form of the lens and can be found in a much more elaborate manner from the theory of Fraunhofer diffraction near the focus.

(b) Rectilinear propagation. A pencil of light of length can exist only if its width at its base is large compared to √(λl). Since has to be λ for the preceding theory to have any sense at all, the required width is certainly > λ but in cases of long pencils considerably greater. More precisely; a pencil of width of the order of pλ can lead an independent existence over a length of the order of p²λ. This implies quite clearly that it is impossible to trace rays by geometrical optics through a particle with a size of the order of λ or smaller. Only for very much larger particles is this method permitted. For applications see secs. 8.1 and 13.24.

3.2. Converging and Diverging Beams

3.21. Amplitude Outside a Focal Line

The preceding derivation was made for the propagation in a plane wave. The writer does not know whether Fresnel himself has considered converging and diverging beams. The extension of his reasoning to this case is so simple and yet so important that it will be given here. It will be applied in sec. 12.22.

A