1. Introduction

Geometrical optics is a theory of Light propagation based on the assumption that light travels along certain curves, called rays, which are determined by the laws of geometrical optics. Although experience has shown that this theory is essentially correct, there are still many cases in which light appears in places where there are no rays (i.e., in shadows). Such discrepancies between experience and geometrical optics are called diffraction effects. It is the purpose of this article to show that geometrical optics can be so modified as to include diffraction. The modification consists in introducing new rays, called diffracted rays, by extending the laws of geometrical optics. These new rays account for the appearance of light in shadows and also alter the light in lit regions.

It seems evident that diffracted rays should be produced when a ray hits an edge or a vertex or when a ray grazes an interface or a boundary. Geometrical optics does not describe what happens in any of these cases; hence we will extend it to do so. Then it will yield the diffracted rays.

Our extension of the laws of optics will be presented in two equivalent forms. The first is the explicit form, in which we enumerate the different situations in which diffracted rays are produced and describe the different kinds of diffracted rays which occur in each case. The second formulation is based upon an extension of Fermat’s principle. The equivalence of the two formulations follows from the usual considerations of the calculus of variations.

Once the diffracted rays have been introduced, we shall define diffracted wavefronts and the phase, or eiconal, function by means of them. In this way we shall obtain new solutions of the eiconal equation. Conversely, from appropriate solutions of this equation, diffracted wavefronts and rays can be determined.

A number of examples in which diffracted rays occur will be described to illustrate this part of our theory. In these examples the diffracted rays cover the shadows of ordinary geometrical optics. However, we shall also find certain cases in which shadows remain even after the introduction of diffracted rays. To obtain rays in such shadow regions, we shall further extend the concept of a ray by introducing imaginary rays. These rays can be used in much the same way as real rays. For example, a complex phase function can be defined in terms of them. It provides the analytic continuation, from a lit region into the shadow, of a solution of the eiconal equation.

The second part of our theory shows how the rays and wavefronts can be used for the quantitative description of the light distribution. This necessitates the introduction of an amplitude function and certain principles for its determination.

Finally we shall discuss the relation of our theory to previous work on diffraction. This may partially justify the introduction of the new rays by showing how some kinds of them have already appeared in special cases.

All our considerations can be applied to other single-integral variational problems and to other first-order partial differential equations in any number of variables. They lead to the introduction of diffracted extremals and diffracted characteristics and of complex characteristics. With the aid of these characteristics, additional branches of solutions of first-order equations can be constructed. Complex solutions, which are analytic continuations of certain real solutions, can also be obtained for analytic equations.

As an example, consider the Hamilton-Jacobi equation of Classical Mechanics. The characteristics of this equation are the classical-mechanical trajectories which satisfy Hamilton’s canonical equations or Newton’s equations of motion. The complex characteristics are complex-valued solutions of Newton’s or Hamilton’s equations. These complex trajectories enter the “forbidden regions” into which real trajectories cannot penetrate. They enable us to continue solutions of the Hamilton-Jacobi equation into these regions, and yield complex values for the solutions there. The appearance of trajectories in forbidden regions is related to the “tunnel effect” of quantum mechanics. In fact, the present considerations provide a new classical interpretation of this effect.

2. Diffracted rays

The first kind of diffracted ray is produced when an incident ray hits an edge (see Figs. 1 to 4). The incident ray produces infinitely many diffracted rays, traveling in directions determined by the law of diffraction. This law states that each diffracted ray which lies in the same medium as the incident ray makes the same angle with the edge as does the incident ray. Furthermore, the incident and diffracted rays lie on opposite sides of the plane normal to the edge at the point of diffraction. However, the diffracted ray need not lie in the same plane as the incident ray and the edge. Therefore the diffracted rays form the surface of a cone with its vertex at the point of diffraction.

If a diffracted ray and the incident ray lie in different media, the angle between the diffracted ray and the edge is related to the angle between the incident ray and the edge by Snell’s law (see Fig. 5a). But here again, the diffracted ray is not restricted to lie in the same plane as the incident ray and the edge. Therefore these diffracted rays also form the surface of a cone.

When an incident ray hits a vertex (e.g., a junction of two or more edges), it produces infinitely many diffracted rays which leave the vertex in all directions (see Fig. 6). Thus at a vertex a single incident ray produces a two-parameter family of diffracted rays.

When a ray grazes an interface or boundary surface (i.e., when it is tangent to the surface), the ray splits in two (see Fig. 7). One part continues, unaffected by the surface, as an ordinary ray. The other part travels along the surface. Its path on the surface is a surface ray, i.e., a curve which satisfies the differential equations for a ray, when these equations are specialized to a surface. A surface ray also makes Fermat’s integral stationary among all curves lying on the surface. At every point on its path this ray again splits in two, one part continuing along the surface and the other part leaving the surface along the tangent to the surface ray, provided that the tangent lies on the same side of the surface as the surface ray. In defining a surface ray, one must use the value of the index of refraction appropriate to that side of the surface from which the incident ray comes.

If the surface ray lies on that side of the surface having the lesser index of refraction, it also sheds another diffracted ray at each point on its path (see Fig. 8). This diffracted ray is a critically refracted ray, which leaves the surface at the critical angle on the side opposite the surface ray. Conversely, a surface ray is produced when a ray is incident at the critical angle on that side of the surface having the greater index of refraction (see Fig. 9). In this case the refracted ray is initially tangent to the surface and then proceeds along it as a surface ray. Surface rays are also produced by a ray incident at an edge or a vertex, since in these cases some of the diffracted rays leave the edge or vertex along the surfaces meeting there (see Fig. 5).

When a ray is incident on a surface of discontinuity of any derivative of the index of refraction, it is reflected and refracted just as at a surface of discontinuity of the index itself. Similarly, diffracted rays are produced when incident rays hit edges or vertices of such surfaces or when they graze these surfaces. In other words, such surfaces behave in all respects like discontinuity surfaces of the index of refraction itself.

Just as discontinuity surfaces of derivatives of the index of refraction must be included as surfaces, so must lines of discontinuity of any derivatives of surfaces be counted as edges. Thus an ordinary edge is a line along which first derivatives of a surface (i.e., the slopes) are discontinuous. Similarly, a line along which second derivatives of a surface (i.e., the curvatures) are discontinuous must also be counted as an edge. Discontinuities in higher derivatives of a surface also yield edges. In the same way, discontinuities in any derivatives of edges must be counted as vertices. In addition, isolated points (not on edges) at which derivatives of a surface are discontinuous also play the role of vertices. The apex of a cone is an example of such a point. Diffracted rays are produced when rays hit any of these edges or vertices, in the same way as they are produced at ordinary edges and vertices.

On the basis of the preceding descriptions, we may say that a ray is diffracted whenever it hits an edge or vertex or grazes a surface. In every such case the ray produces infinitely many diffracted rays. Thus the process of diffraction, which occurs in these cases, splits a single ray into infinitely many diffracted rays. It is natural to expect that the light intensity associated with these diffracted rays, as well as with rays reflected from discontinuity surfaces of derivatives of the refractive index, is much smaller than that associated with the incident ray. This is indeed the case, and is of particular importance in using our theory for the quantitative calculation of intensities.

Away from discontinuity or boundary surfaces, all rays—both ordinary and diffracted—are determined by the usual laws of geometrical optics. These laws, plus the laws of reflection and refraction and the foregoing laws governing diffraction, completely determine all real rays.

As a consequence of the theory of diffraction just described, diffracted rays will exist, in addition to ordinary rays, in any medium which is bounded or in which the refractive index or any of its derivatives is discontinuous. Of course, in specific examples the incident rays might be so arranged that no diffraction occurs (see Fig. 10). In such examples no incident ray can hit an edge or graze a surface; hence no shadows can be formed. Therefore the absence of diffracted rays in these examples is not unexpected.

3. Examples

Let us now consider some examples of diffracted rays. First, consider a plane wave (i.e., a set of parallel rays) incident obliquely upon a thin opaque screen in the form of a half plane (see Fig. 1). If the medium is homogeneous, then all the rays, incident, reflected, and diffracted, are straight lines. The only diffracted rays are produced by those incident rays which hit the edge of the screen. Since the incident rays are parallel to each other, the cones of diffracted rays will also be parallel to each other. Each point of the edge will be the vertex of a cone. One diffracted ray will reach each point in the medium, accounting for the appearance of light in the “shadow” and also providing additional light rays in the illuminated region.

If the incident rays are perpendicular to the edge instead of being oblique, the diffracted rays will also be perpendicular to the edge (see Fig. 2). Thus in this case each cone of diffracted rays is opened up to become a plane of diffracted rays.

As a third example, consider a plane wave incident obliquely upon a slit in a thin opaque screen (see Fig. 3). The rays which hit the edges of the screen give rise to diffracted rays. Some of the rays diffracted from one edge will hit the other edge and give rise to a new set of diffracted rays. Some of these new rays will in turn hit the opposite edge, producing still other diffracted rays, etc. Thus, in this case there is an infinite set of multiply diffracted rays. Some of these rays are shown in Fig. 3. These singly and multiply diffracted rays are the only diffracted rays which occur in this problem, and they account for the occurrence of light in the shadow behind the screen. In addition, the usual incident and reflected rays are present.

As a fourth example, let us consider a plane wave normally incident upon a plane screen which contains an aperture with a smooth rim (see Fig. 4). Each incident ray which hits the rim is perpendicular to it. Therefore each set of diffracted rays lies in the plane perpendicular to the edge at the point of diffraction. As in the case of the slit, multiply diffracted rays will also be produced. However, the cones of multiply diffracted rays will in general not be planes, since the diffracted rays which produce them will in general not be perpendicular to the edge.

In all the preceding examples, the diffracting edge is a caustic of the diffracted rays, i.e., a locus of points of intersection of neighboring rays. Obviously, this is always the case with a diffracting edge. However, in the last example above, the singly and multiply diffracted rays also possess other caustics in addition to the edge. The caustic of the singly diffracted rays can be determined very simply, because these rays lie in planes normal to the edge. Therefore, to locate the intersections of neighboring rays, it suffices to consider the intersection of the neighboring planes which contain the rays. Since all these planes are perpendicular to the plane of the screen, any two of them must intersect in a straight line which is also perpendicular to the screen. As the caustic is made up of such lines, it is a cylinder with generators perpendicular to the screen.

To determine its cross section, we consider the curve of intersection of the cylinder and the plane of the screen. In the plane of the screen, the diffracted rays are perpendicular to the rim, and therefore their envelope is just the envelope of the normals to the rim. But this envelope is called the evolute of the rim. Thus we see that the caustic of the singly diffracted rays is a cylinder with generators normal to the screen and with the evolute of the rim as its cross section.

If the screen is removed and replaced by a thin plate having the same rim as the aperture, the preceding considerations apply equally well. This fact is a geometrical form of Babinet’s principle. Furthermore, in the case of the plate, the caustic just described will lie (at least partly) in the shadow of the plate. A cross section of it would appear as a bright line in the shadow, since light intensity is greater at a caustic than elsewhere. If the plate has a circular rim, the evolute is a single point, the center, and the caustic is just the axis of the circular plate. The resulting bright spot on the axis is well known experimentally. The bright lines in the shadows of plates of other shapes have also been observed and found to be the evolutes of the rims [2], as the above considerations predict.

In the case of a plane wave obliquely incident upon a flat plate, a more detailed analysis shows that the caustic of the singly diffracted rays approaches a cylindrical shape far behind the plate. The generators of this limiting cylinder are parallel to the shadow boundary (i.e., to the incident-ray direction). The cross section of this cylinder is the evolute of the cross section of the shadow, i.e., of the curve obtained by cutting the shadow with a plane perpendicular to the incident-ray direction. This shadow cross section is just a projection of the rim of the plate. Thus, far behind a plate illuminated obliquely, there should be a bright line in any cross section of the shadow, and it should be the evolute of the boundary of the shadow. This prediction of the theory is also in agreement with the experimental observations of the bright lines in the shadows of various plates illuminated obliquely [3]. If the bright lines had also been observed closer to the plates, they should have been found to differ from the evolutes described above. However, such observations were not made.

Let us now examine another example, in which a plane wave in a homogeneous medium is incident upon a wedge of a different material. The incident rays corresponding to the plane wave will be reflected and refracted at the surfaces of the wedge in the usual manner. In addition, the rays which hit the edge will produce cones of diffracted rays, as in the preceding examples. However, now some diffracted rays will also be produced inside the wedge (see Fig. 5a). If i denotes the angle between an incident ray and the edge and r denotes the angle between one of the resulting diffracted rays and the edge, then r is determined by the equation

Here n is the relative refractive index of the two materials; i.e., n is the velocity in the surrounding medium divided by the velocity in the wedge. Thus if the light velocity is faster within the wedge material, the angle r is smaller than i.

Of the diffracted rays produced inside the wedge by a single incident ray, one proceeds along each wall of the wedge. These two rays are surface rays within the faster medium, if n < 1. Therefore they shed refracted rays back into the surrounding medium all along their paths. These refracted rays leave the surface at the critical angle i_c determined by

They also lie in a plane normal to the surface. The “first” critically refracted ray coincides with one of the diffracted rays produced by the incident ray in the outer medium, as can be shown by simple trigonometry. Thus the shed refracted rays due to a single incident ray lie in a plane sector bounded by the outer diffracted cone and a wall of the wedge. The refracted rays shed from one wall of the wedge due to all the incident rays are all parallel to each other and thus form a plane wave. In each medium the cones of diffracted rays from the different points on the edge are also parallel to each other and form a conical wave. Some sections of these wavefronts are shown in Fig. 5b and 5c. When n > 1, that is, when the faster medium is outside the wedge, the refracted rays are shed into the wedge, and then the resulting plane waves lie inside the wedge.

Next, suppose that a plane wave is incident upon an opaque cone, as in Fig. 6. Then, in addition to the usual reflected rays, diffracted rays will be produced by that incident ray which hits the vertex or tip of the cone. These rays will go in all directions from the tip, and the corresponding diffracted wavefronts will be spheres with the tip as center.

Let us now consider a plane wave normally incident upon an opaque cylinder of convex cross section in a homogeneous medium (see Fig. 7). In this case, in addition to the reflected rays, two surface rays will be produced by the two incident rays tangent to the cylinder. These rays will lie in a plane normal to the axis of the cylinder and will encircle the cylinder in opposite directions. At each point on its path each surface ray will shed a diffracted ray along the tangent to the cross section of the cylinder (see Fig. 7). Two diffracted rays will pass through each point in space, one coming from each of the surface rays. The two rays through a given point are the two tangents to the cross section which pass through that point. Actually, each of these rays represents an infinite number of diffracted rays, one being shed by the surface ray each time it encircles the cylinder. These rays account for the illumination in the shadow region and also provide additional light in the lit region. The cross sections of the diffracted wavefronts, i.e., the surfaces orthogonal to the diffracted rays, are just the involutes of the cross section of the cylinder.

Suppose that the cylinder of the previous example is composed of a homogeneous material with a lower light velocity and therefore a higher refractive index than the surrounding medium. Then the incident rays which hit the cylinder will produce refracted rays inside the cylinder in addition to the reflected rays in the surrounding medium (see Fig. 8). These refracted rays will hit the cylinder surface and again produce reflected and transmitted rays, and this process will be repeated ad infinitum. All of these multiply reflected and transmitted rays are ordinary rays. In addition to them, the grazing rays will produce diffracted rays in the outer medium as before. But now the surface rays will also shed refracted rays into the cylinder. These refracted rays will leave the surface at the critical angle of refraction. Once inside the cylinder, these diffracted rays will hit the opposite surface and be reflected and refracted, etc.

As a final example, let us consider a spherical wave incident upon the plane interface between two homogeneous media, as shown in Fig. 9. The corresponding incident rays are straight lines emanating from the center of the spherical wave. They are reflected and refracted at the interface in the usual way. However, if the second medium has the faster light velocity, there is a critical angle at which the refracted ray is parallel to the interface. Therefore this critically refracted ray is a surface ray in the faster medium. Consequently, it sheds refracted rays back into the slower medium. These shed rays leave the interface at the critical angle. The corresponding diffracted wavefronts are cones, one of which is shown in cross section in Fig. 9. It appears there as a straight-line segment.

4. A generalization of Fermat’s principle

In Sec. 2 we extended the laws of geometrical optics by introducing diffracted rays and giving an explicit characterization of them. Now we shall consider an alternative (but equivalent) extension of the laws of geometrical optics based upon a generalization of Fermat’s principle. This principle, which is the basis of ordinary optics, involves the index of refraction n(x). This is a real positive function which characterizes the optical behavior of the medium. In terms of it the optical length L of any curve x(s) connecting two points P and Q is defined as

The parameter s denotes arc length.

Fermat’s principle states that the optical rays connecting P and Q are those curves which make L stationary in the class C₀ of all smooth curves joining P and Q. This principle applies to an unbounded continuous medium (i.e., one in which n is continuous). It does not apply to bounded or discontinuous media [i.e., media in which n(x) is discontinuous]. We may try to apply it to such media by considering, instead of C₀, a class of curves with a finite number of corners. However this formulation turns out to be unsatisfactory, because it yields only some rays, “direct” ones, but does not include any reflected rays.

In order to obtain a principle valid for ordinary optics in discontinuous media, we introduce for each integer r ≥ 0 the class of curves C_r. This is the class of curves with exactly r points on the boundaries or discontinuity surfaces of the medium. These points are to be inner points of these surfaces; i.e., they may not lie on edges or vertices. Now we formulate Fermat’s principle as follows: The rays are those curves in each class C_r which make the optical length stationary in C_r. Upon examining the consequences of this formulation, we find that the class C₀ yields rays which do not touch the boundary or discontinuity surfaces; C₁ yields singly reflected or refracted rays; and C_r yields r-tuply reflected and/or refracted rays.

The preceding formulation of Fermat’s principle for discontinuous and/or bounded media is presumably implicit in older formulations of geometrical optics. Although it includes reflected rays, it still fails to take account of diffracted rays. Therefore we shall further modify Fermat’s principle by introducing additional classes of curves. For each triple of nonnegative integers r,s,t we shall define the class D_rst. This class consists of curves with r smooth arcs on the boundary or discontinuity surfaces, s points on edges of the boundary or discontinuity surfaces, and t points on vertices of these surfaces. Any number of the r arcs may be degenerate arcs, i.e., points. To each arc the value of n on one side of the surface is assigned. We now define the rays as those curves in each class D_rst which make the optical length stationary in D_rst.

The class D₀₀₀ is the previously considered class C₀, and thus it yields the direct rays. The class D_r00 contains the previously considered class C_r and thus yields r-tuply reflected and/or refracted rays. Each ray in any of the other classes has at least one point on an edge or vertex and is thus a new ray not included in ordinary optics. Some of the rays in D_r00 are also new rays, since they have arcs on the boundary or discontinuity surfaces.

From the above extension of Fermat’s principle a number of conclusions can be drawn which suffice to characterize the rays explicitly. First, let us consider any smooth arc of a ray not containing a boundary point in its interior. By applying the usual considerations of the calculus of variations, we conclude that each such arc must be an extremal, i.e., a solution of the Euler equations. Similarly, each boundary arc must be a surface extremal. Second, by applying the appropriate considerations of the calculus of variations to each corner at an inner point of a boundary surface, we find that the law of reflection or the law of refraction must be satisfied, according as the two parts of the ray lie on the same or on opposite sides of the boundary. Third, at each inner point of a boundary edge, we find that a law of diffraction must be satisfied. This law states that the two parts of the ray make equal angles with the edge, if they lie in the same region at the edge, and that the angles are related by Snell’s law if they lie in different regions. Fourth, at a vertex the two parts of a ray may make any angles. Finally, at an inner point of a boundary, a surface extremal and an extremal in space may join together smoothly.

The foregoing consequences of the extended Fermat principle are essentially the explicit rules given in Sec. 2 for the determination of the rays. Since these consequences also suffice to make the optical length stationary in each class, it follows that our two prescriptions for determining rays are equivalent.

All the preceding considerations are based on the assumption that the index of refraction n(x) is a piecewise smooth function of x. This means that space is divided into a finite number of regions in each of which n and its derivatives are continuous and have limits at the boundary. The boundary is also assumed to be piecewise smooth, i.e., to consist of a finite number of parts each having continuous derivatives which have limits at the edge. The edge is assumed to be piecewise smooth, i.e., to consist of a finite number of arcs each having continuous derivatives which have limits at each end point of each arc. The end points of the edge arcs are called vertices. Isolated points of the boundary at which the derivatives are discontinuous are also vertices.

The extended form of Fermat’s principle for discontinuous media is complicated, compared to the original form for continuous media. Therefore it is natural to inquire whether the complicated form can be deduced from the simple form by considering a discontinuous medium to be the limit of a family of continuous media. Then the rays of the discontinuous medium could be defined as the limits of families of corresponding rays in the family of continuous media.

The answer to this question is negative. It turns out that the indicated limit process yields only some, but not all, of the rays determined by the extended form of Fermat’s principle. In particular, many of the reflected rays are not obtained by the limit process, viz., those rays reflected at any angle from a slower medium or normally reflected from a faster medium. However, some diffracted rays are given by this limit process. This result may be summarized by stating that the geometrical optics of discontinuous media is not the limit of the geometrical optics of continuous media.

5. Diffracted wavefronts

In ordinary geometrical optics one always deals with normal congruences of rays. A normal congruence is a family of rays, all of which are normal to some surface. Such a surface is called a wavefront. The theorem of Malus guarantees that a normal congruence remains a normal congruence after reflection or refraction. Therefore reflected and refracted wavefronts can always be defined.

Now suppose some of the rays of a normal congruence undergo diffraction. If the resulting diffracted rays also form a normal congruence, then diffracted wavefronts can be defined as the surfaces normal to the family of diffracted rays. That this is indeed the case can be proved, providing an extension of Malus’s theorem to diffraction. The proof will not be given here. Some diffracted wavefronts have already been considered in the examples of Sec. 3.

In ordinary geometrical optics we define the eiconal, or phase, function Ψ(P) at a point P as the optical distance to P from some fixed wavefront, measured along an ordinary ray. We then show that Ψ satisfies the eiconal equation

and that the surfaces Ψ = constant are wavefronts. We also find that Ψ(P) is double-valued if an incident and a reflected ray pass through P. These two values of Ψ become equal as P tends toward the reflecting surface. Thus the reflecting surface is a branch surface of Ψ. If many rays—incident, reflected, and refracted—pass through P, then Ψ(P) is many-valued, and the reflecting and refracting surfaces are the branch surfaces on which two or three different branches are equal.

All the foregoing considerations can also be applied to diffracted rays. We first define the eiconal Ψ(P) as the optical distance to P from some fixed wavefront measured along any ray, ordinary or diffracted. We can then show that Ψ still satisfies the eiconal equation. With this new definition Ψ(P) is even more multiple-valued than before. Not only are the boundaries and discontinuity surfaces of n(x) branch surfaces of Ψ, but the discontinuity surfaces of derivatives of n(x) are also branch surfaces. Furthermore the edges and vertices are branch lines and branch points of Ψ.

In ordinary geometrical optics, in a continuous unbounded medium, it is possible to utilize the wavefronts and the eiconal equation as a basis for geometrical optics. We prescribe some smooth surface as an initial wavefront and consider a solution Ψ of the eiconal equation which has the value zero on the given surface. Because the eiconal equation is quadratic, there are two such solutions, but they differ from each other only in sign. For either solution we define the surfaces Ψ = constant as a family of wavefronts and the orthogonal trajectories of these wavefronts as rays. These rays are exactly the same as the rays given by Fermat’s principle. By choosing different initial surfaces, we obtain precisely all the rays given by Fermat’s principle. Thus we see that in this case the wavefront formulation is equivalent to the ray formulation.

No similar formulation in terms of wavefronts has been given for ordinary geometrical optics in bounded or discontinuous media. This is undoubtedly due to the occurrence of diffraction in such media. However, the present theory, which includes diffraction, can presumably be formulated in terms of wavefronts for any medium. To this end we proceed as above, by prescribing some smooth surface as an initial wavefront and by considering a solution Ψ of the eiconal equation which is zero on the given surface. Then we define the surfaces Ψ = constant to be wavefronts and their orthogonal trajectories to be rays, just as before.

The only difference is that now we must consider a multiple-valued solution which has as branch points, lines, and surfaces the vertices, edges, and surfaces determined by the boundaries and by the refractive index and its derivatives. This solution must be complete, in the sense that it must branch at every permissible branch point, line, or surface. Unfortunately, the foregoing requirements do not determine a unique solution Ψ. Other conditions, perhaps at the boundaries or discontinuity surfaces, must be imposed to obtain uniqueness. Consequently, the equivalence of the ray and wavefront formulations of our theory is not yet demonstrated, since the wavefront formulation is not complete.

In two dimensions, an interesting wavefront results from diffraction by an object bounded by a smooth convex curve. The diffracted wavefronts are the involutes of the curve. In three dimensions, toroidal wavefronts result from diffraction of a normally incident plane wave by a circular disk.

6. Imaginary rays

In an unbounded medium in which the refractive index and all its derivatives are continuous, no diffracted rays occur. This is clear from the above laws governing diffraction. Therefore in such a medium the theory so far presented coincides with ordinary geometrical optics. However, ordinary geometrical optics sometimes yields shadows in such media. An example is the region on that side of a caustic surface through which no rays pass (see Fig. 11). Experimentally some light is observed in these shadows. Since our theory fails to account for this light, the theory is incomplete.

To complete the theory, we introduce another new type of ray, which we call an imaginary ray. Such a ray is a complex-valued solution of the ray equations. Thus, an imaginary ray in a homogeneous medium is a complex straight line. The definition presupposes that n(x) is analytic or piecewise analytic. Now we may consider an analytic normal congruence of real rays. By analytic we mean that the rays of the congruence are analytic functions of two real parameters. Then complex values of these parameters determine imaginary rays of the same congruence. Therefore every analytic congruence contains imaginary rays. Some of them will enter the shadows of the type considered above and thus account for the light observed there.

To see this, consider a two-dimensional homogeneous medium. Suppose a given curve C is a caustic, i.e., an envelope of a normal congruence of rays. These rays are then straight lines tangent to C. Let t denote arc length along C, and let the parametric equations of C be x = x(t), y = y(t). Then the equations of a ray tangent to C at the point [x(t),y(t)] are

(1)

In (1) the parameter s is the signed distance from [x(t),y(t)] on C to (x,y) on the ray.

If the point (x,y) is given, the rays through it are determined by the solutions s,t of (1). Each solution yields one ray through (x,y). If C is convex, there will generally be two rays through each point on the convex side of C but no real rays through any point on the concave side. This is because there are no tangents from such points to C. However, if C is analytic, (1) may have complex solutions for s and t. Then for each solution s,t the point (x,y) lies on the complex ray (1) which is tangent to C at the complex point [x(t),y(t)]. The value of s is the complex distance from (x,y) to the point of tangency, and (x,y) is the only real point on the ray.

As an example, suppose that C is a circle of radius a. Let the polar coordinates (a,ϕ) denote a point on C and let (r,𝜃) denote a point off C. Then (1) becomes

(2)

Solving these equations for the point of tangency ϕ and the distance s, we obtain

(3)

(4)

From (3) we see that, for r ≥ a, there are two real values of ϕ, if cos ⁻¹ is restricted to the range 0 to π. There are likewise two real values of t, one corresponding to each value of ϕ. If r < a, (3) and (4) do not yield real values of ϕ and s, but they do give the complex values

(5)

(6)

Thus two complex straight lines through (r,𝜃) are tangent to the circle r = a at the two points with ϕ coordinates given by (5). These complex lines are the imaginary rays through (r,𝜃) which belong to the normal congruence having the circle r = a as caustic.

As a second example, consider as caustic the parabola x′ = ay′². If a line through (x,y) is tangent to the parabola at (x′,y′), then

(7)

Solving for the point of tangency, we obtain

(8)

Equation (8) shows that there are two, one, or no real points of tangency according as (x,y) is outside, on, or inside the parabola. In the latter case there are two complex points of tangency and thus two imaginary rays through (x,y). The distance s from (x,y) to the point of tangency is given in the original paper as

(9)

PhysicsLibrary conversion note. Equation (9) above preserves the first factor y² −x∕2 exactly as printed in the source. Equations (7)–(8) suggest two typographical anomalies in the printed formula: the first factor contains x∕2 rather than x∕a, and the last radical term is printed without a factor of y. Both have been preserved exactly as printed rather than silently altered in this conversion.

This distance is complex for points inside the parabola.

7. The complex eiconal, or phase, function

An analytic normal congruence of rays may be defined as the set of rays normal to a given analytic surface S. Such a congruence contains imaginary as well as real rays. By means of the real rays, we have already defined the real eiconal, or phase, function Ψ(P). This is the optical length to P from S along a ray of the congruence. Now we may define the complex eiconal Ψ(P) in exactly the same way by means of the complex rays. It is readily seen that Ψ(P) is complex and that it satisfies the eiconal equation. Furthermore, it is the analytic continuation of the real phase function, which is defined only at points P lying on real rays of the congruence.

In the same way, we can obtain the analytic continuation of the solution of the Cauchy problem for the eiconal equation. In this problem a surface S, not necessarily a wavefront, is given, and a function Ψ₀ of position on S is also given. We are to find a solution Ψ of the eiconal equation such that Ψ = Ψ₀ on S. This problem is usually solved by means of certain real rays through S. The analytic continuation can be obtained by applying the same considerations to the imaginary rays through S.

As an example, let us consider a two-dimensional homogeneous medium with n(x) = 1. We seek a solution Ψ of the eiconal equation (∇Ψ)² = 1 having the value Ψ = t on C. As before, t denotes arc length along C. Since the derivative of Ψ along C is unity, and since the length of ∇Ψ is unity, we see that ∇Ψ is tangent to C. Therefore the problem we have posed is a characteristic boundary-value problem, since C is everywhere characteristic (i.e., tangent to the rays). This problem has two solutions because the eiconal equation is quadratic. From the eiconal equation the derivative of Ψ along a ray is ±1. Therefore the solutions are given in terms of the parameters s and t by equations (1) and

(10)

At points (x,y) for which s and t are real, Ψ is also real. However, for points lying on imaginary rays, both s and t are complex; hence Ψ is complex.

In the case of the circle treated above, s is given by (4) or (6) and t = aϕ, where ϕ is given by (3) or (5). Then from (10) we have in this case

(11)

(12)

For the parabola previously considered, if t = 0 at x = y = 0, we find

(13)

When this value of t is used in (10), together with s, given by (9), two solutions Ψ result which are real on and outside the parabola but complex inside it.

Complex solutions of the eiconal equation can also be obtained without making use of imaginary rays. As an example, consider the above problem for any curve C. Let a(t) be the radius of curvature of C and ρ be the distance along the normal to C, measured positively toward the convex side of C. In terms of ρ and t the eiconal equation becomes

(14)

On the basis of the explicit solutions above, we assume that Ψ has the form

(15)

Upon inserting (15) into (14) and utilizing the boundary condition on C, we find

(16)

The remaining coefficients can be found from a recursion formula which we will omit. The result (16) shows that Ψ is real for ρ > 0 and complex for ρ < 0, and that the imaginary part of Ψ is proportional to |ρ|^3∕2 for ρ small.

Let us write Ψ = R + iI, where R and I are real. Then the eiconal equation yields

(17)

(18)

Equation (17) shows that the surfaces R = constant are orthogonal to the surfaces I = constant. In the next section these surfaces will be shown to be surfaces of constant phase and of constant amplitude, respectively, for a field associated with Ψ. Thus (17) shows that for this field these surfaces are mutually orthogonal.

8. Field and amplitude

To make our theory quantitative, we associate a field u(s) with each ray. It is composed of an amplitude A(s) and a phase Ψ(s) in the form

(19)

In (19), k = ω∕c is the propagation constant, determined by the angular frequency ω of the field and the propagation velocity c in empty space. Equivalently, k = 2π∕λ, where λ is the wavelength of the field in empty space. Thus our construction applies to a time-harmonic field. The time factor e^−iωt will be omitted. The total field at a point P is the sum of the fields (19) on all rays through P.

When we deal with light, u is either the electric or the magnetic field and therefore A is a vector. However, our theory also applies to other types of field (e.g., acoustic pressure). For simplicity we shall describe it for a scalar u and then indicate the modifications which occur for vector fields.

We first assume that the phase difference Ψ(P) − Ψ(Q) between two points on a ray is equal to the optical length L of the ray from Q to P. We also assume that a direction of propagation is associated with each ray and that Ψ increases in this direction. From these assumptions it follows that Ψ can be determined at any point P if it is known at some point Q on the same ray:

(20)

We further require that Ψ be constant on some wavefront of a normal congruence of rays. Then Ψ is just the eiconal, or phase, function previously introduced.

Next we assume that the principle of conservation of energy applies in its optical form. This states that the energy flux is the same at every cross section of a tube of rays. We assume that the energy flux per unit area is proportional to nA². Then the energy principle yields, for a narrow tube of rays,

(21)

Here n and A are evaluated at a point P on a ray in the tube, and dσ is the cross-sectional area of the tube at P. The quantities n₀, A₀, and dσ₀ are evaluated at some other point Q of the same ray. From (21) we obtain

(22)

Thus we can compute A at any point P on a ray, provided that we know the amplitude A₀ at some point Q on the same ray. The ratio dσ₀∕dσ in (22) is the ratio of the areas of the cross sections at P and Q. Since these cross sections are portions of wavefronts, this ratio is just the Jacobian of the mapping from a wavefront at P to that at Q by means of rays. When A is a vector, we assume that its amplitude satisfies (22). Its direction, if A is an electric or magnetic field, is obtained from A₀ by parallel transport along the ray with respect to the metric nds.

If ρ₁ and ρ₂ denote the principal radii of curvature of the wavefront at Q, then, in a homogeneous medium, the corresponding radii at P are ρ₁ + s and ρ₂ + s. Here s denotes the distance along the ray from Q to P. Since the area ratio is inversely proportional to the ratio of Gaussian curvatures, (22) becomes

(23)

From (23) and (20) we see that, in a homogeneous medium,

(24)

Here Ψ₀ = Ψ(Q).

The field (24) becomes infinite at two, one, or no points on a ray, according as both, one, or neither of the radii of curvature are finite. These points are on the caustics of the ray congruence. In these various cases, u decays for large s like s⁻¹, s^−1∕2, or s⁰, that is, as in a spherical, cylindrical, or plane wave. Later we shall indicate how to modify our theory in order to obtain a finite value for u on a caustic.

In homogeneous media it is often convenient to measure s from a point Q on the caustic C. To do this, we first rewrite (23) in the form

(25)

The left side of (25) has a limit as Q tends to C, and therefore the right side must also. This is understandable since A₀ becomes infinite and ρ₁ becomes zero as Q tends to C. Let us denote this limit by A′₀ = lim _Q→CA₀ρ₁^1∕2. Then (24) becomes

(26)

In a two-dimensional medium, or for cylindrical waves in three dimensions, ρ₂ is infinite, and (26) becomes

(27)

In two dimensions, as we have seen, Ψ₀ = nt, where t denotes arc length along C. Furthermore, since A′₀ varies from ray to ray, we may designate each ray by its point of tangency t and write A′₀ = A′₀(t). Then (27) becomes

(28)

If two rays pass through a point P, as is often the case near a caustic, then u(P) is a sum of two terms of the form (28).

Let us apply (28) to the congruence of rays tangent to a circular caustic of radius a. Making use of our previous results for s and t, we obtain for r > a,

u(r,𝜃) =	eikn[a𝜃−a cos −1(a∕r)+]
	+ eikn[a𝜃+a cos −1(a∕r)−]−iπ∕2.	(29)

For r < a we also obtain two terms, each corresponding to one of the imaginary rays through the point (r,𝜃). One of these terms increases with distance from the caustic, whereas the other decreases. We now assume that the increasing term must be omitted. Then we obtain for r < a,

(30)

For the result (30) we require that A′₀(t) be an analytic function of t.

Let us now consider the function v(r,𝜃) defined by

(31)

This function is an exact solution of the reduced wave equation in two dimensions if B is a constant. We now expand it asymptotically for large nka and nkr. This yields exactly (29) for r > a and (30) for r < a, provided that A′₀ is constant and that B = e^iπ∕4 A′ ₀. This agreement indicates that our construction yields the leading term in the asymptotic expansion with respect to k, for k large, of the exact solution of the wave equation. We believe that this is always the case.

So far, we have described how the amplitude varies along a ray. Now we shall explain how the initial value of the amplitude is to be determined. First, on rays which come from a source—even if it is at infinity—the amplitude must be prescribed. This prescription characterizes the source. Second, on a reflected or refracted ray at the point of reflection or refraction, we assume that the amplitude is proportional to that on the corresponding incident ray at this point. The proportionality factors are called reflection and transmission coefficients, R and T, respectively. For vector fields these coefficients are matrices. Third, on a ray diffracted from an edge or vertex we assume that the field is also proportional to that on the corresponding incident ray at the point of diffraction. The proportionality factor we call a diffraction coefficient (or matrix, in the vector case). Additional hypotheses must be made to treat the fields on diffracted rays which have arcs on boundaries, but we shall not consider them here.

We assume that the various coefficients just introduced are determined solely by local conditions at the point of reflection, refraction, or diffraction. Thus, for example, the reflection and transmission coefficients depend only upon the angle between the incident ray and the surface normal as well as upon the properties of the media at the point of reflection. Therefore they can be determined from the solution of a canonical problem, that of reflection and refraction of a plane wave at a plane interface. The diffraction coefficients can also be obtained from the solutions of appropriate canonical problems.

The various coefficients depend upon the type of field under consideration. Sound waves will have different coefficients from water waves, electromagnetic waves, or other waves. Consequently, these coefficients must be determined separately for different fields. Mathematically, this difference will be manifested by the differential equations and boundary conditions which occur in the canonical problems.

Canonical solutions can also be used to modify the results of our theory at and near caustics. Thus, for example, let us again consider a two-dimensional homogeneous medium in which a circular caustic occurs. Our results (29) and (30) for the field u become infinite on the caustic r = a. But the function v in (31), which is asymptotic to u for large k, remains finite on the caustic. Therefore we can use v instead of u on and near the caustic in order to obtain a finite value for the field. We can also assume that a finite value for the field at a point on any caustic can be obtained from the field u off the caustic by the same correction factor, involving the radius of curvature of the caustic at the point.

9. Relation to other work

Some types of diffracted rays and diffracted wavefronts have already occurred in the solutions of particular diffraction problems. Some others have been observed experimentally or have been introduced to explain particular experimental results. We will now describe some of this previous work.

First we recall Thomas Young’s proposal that diffraction through an aperture in a screen is an edge effect. This proposal is in agreement with the present theory, which even makes it precise. Next we note that Sommerfeld’s solution of Maxwell’s equations for two-dimensional diffraction of waves by a half plane contains a cylindrical wave emanating from the edge [4]. The cylindrical wavefronts of this wave are just the diffracted wavefronts, and the normals to these cylinders are the diffracted rays, of our theory. The solutions of Sommerfeld and Macdonald for two-dimensional diffraction by wedges also contain cylindrical waves emanating from the edge. Their solutions for the three-dimensional case contain the cone of diffracted rays from each edge point.

The bright lines in the shadows of plates, observed by G. G. Becknell and J. Coulson [2, 3] have already been mentioned and explained in terms of our theory. Later Nijboer observed similar bright lines in the diffraction patterns of apertures. He introduced diffracted rays emanating normally from the edge and found that the caustics of these rays were exactly the observed bright lines.

The present theory predicts the bright spot on the axis of a circular disk, as was noted above. This result is particularly interesting, because the observation of the bright spot was a strong argument for the wave theory of light. We now see that this result is also predicted by a ray theory. Therefore, if this ray theory had been available at the time of the controversy between ray and wave theory, it might have forestalled the acceptance of the latter.

The field diffracted through an aperture in a screen can be represented as an integral over the aperture and screen. Using Kirchhoff’s approximate values for the integrand, A. Rubinowicz [5] reduced this integral to a line integral along the aperture rim and evaluated it by the method of stationary phase. The stationary points which he obtained for a given field point P coincide exactly with the places on the edge at which the diffracted rays through P are produced. N. G. van Kampen [6] evaluated asymptotically the integrals given by the modified Kirchhoff method. His result also contains one stationary point corresponding to each edge-diffracted ray through P, and in addition one stationary point corresponding to each corner of the edge, accounting for the corner-diffracted rays. R. M. Lewis, B. D. Seckler, and the present author [7] have obtained similar results from W. Braunbek’s [8] modification of the Kirchhoff theory.

Surface rays appear in the asymptotic expansion for large ka of the field diffracted by a sphere or cylinder of radius a. This was originally shown by G. N. Watson and elaborated by B. van der Pol and H. Bremmer [9], B. Friedman [10], I. Imai [11], W. Franz [12], and others. The tangent rays shed by these surface rays are exhibited in the exact solution of W. Franz [12] and the approximate solution of W. Franz and K. Depperman [13]. The latter authors showed that calculations of radar reflection from cylinders, based on the idea of surface rays, agreed excellently with the measurements of Limbach. F. G. Friedlander [14] introduced surface rays and the associated wavefronts in studying diffraction by cylinders of convex cross section.

Surface rays produced by refraction at the critical angle occur in the work of E. Gerjuoy [15] and of L. Brekovskih [16]. These authors examined the field produced by a point source near a plane interface between two media, in the high-frequency limit. They found that each critically refracted ray gave rise to the appropriate diffracted rays. Such rays have been observed experimentally in acoustics.

Spherical waves emanating from the tip occur in the solution for the field diffracted by a circular or elliptic cone. The wavefronts and rays of these waves are just the diffracted wavefronts and rays predicted by the theory in this case. The rays leave the vertex in all directions.

Rays reflected from surfaces of discontinuity of derivatives of the index of refraction do not seem to have been considered before. However, the fact that such discontinuities do reflect at normal incidence was noticed by J. Feinstein [17] and S. A. Schelkunoff [18].

The possibility of using rays in a systematic way for the calculation of fields was investigated by R. K. Luneberg [19]. He suggested that the ray construction would yield the leading term in the asymptotic expansion of the field for large k. The procedure for obtaining further terms in this asymptotic expansion was given by M. Kline [20], both for Maxwell’s equations and for more general equations. Other authors have considered the same type of expansion for various equations. Thus F. G. Friedlander [21], H. Bremmer [22], and E. T. Copson [23] also considered Maxwell’s equations; S. C. Lowell [24] considered waves in shallow water; J. B. Keller [25] considered weak shock waves; G. D. Birkhoff [26], L. Brillouin [27], G. Wentzel [28], P. A. M. Dirac [29], and J. B. Keller [30] considered the Schroedinger equation of quantum mechanics; F. G. Friedlander and J. B. Keller [31] considered the reduced wave equation; and W. J. Trjitzinsky [32] considered a very general linear equation. All of these authors restricted their attention to the rays of ordinary geometrical optics.

Many diffraction problems have been solved with the ray method by C. Schensted [33], J. B. Keller, R. M. Lewis, and B. D. Seckler [34], J. B. Keller [25, 30, 35, 36], K. O. Friedrichs and J. B. Keller [37], B. R. Levy and J. B. Keller [38], S. N. Karp and J. B. Keller [39], B. D. Seckler and J. B. Keller [41], etc. Whenever possible, the fields constructed by the ray method were compared with asymptotic expansions (for large k) of exact solutions. In all such cases perfect agreement was obtained. In other cases numerical results were compared, and good agreement was obtained for ka ≥ 2, where a is a typical length in the problem.

All of these results suggest that the ray method does yield the leading terms in the asymptotic expansions of solutions of diffraction problems. However, a general proof of this statement has not yet been obtained. Partial results of this kind are given by R. K. Luneberg [19], M. Kline [20], W. J. Trjitzinsky [32], W. L. Miranker [40], and R. M. Lewis [42].

PhysicsLibrary source note. This PhysicsLibrary entry is a conversion of the public-domain material in Reference [1].

References

References

Institute of Mathematical Sciences, New York University, New York, N.Y.