Then differentiating V in regard to x, and replacing ηi(n)by its value a1η(n−1)+ ... + anη, we can arrange dV/dx, and similarly each of d²/dx² ... dNV/dxN, where N = n², as a linear function of the N quantities η1, ... ηn, ... η1(n−1), ... ηn(n−1), andThe resolvent eqution.thence by elimination obtain a linear differential equation for V of order N with rational coefficients. This we denote by F = 0. Further, each of η1... ηnis expressible as a linear function of V, dV/dx, ... dN−1V / dxN−1, with rational coefficients not involving any of the n² coefficients Aij, since otherwise V would satisfy a linear equation of order less than N, which is impossible, as it involves (linearly) the n² arbitrary coefficients Aij, which would not enter into the coefficients of the supposed equation. In particular, y1,.. ynare expressible rationally as linear functions of ω, dω/dx, ... dN−1ω / dxN−1, where ω is the particular function φ(y). Any solution W of the equation F = 0 is derivable from functions ζ1, ... ζn, which are linear functions of y1, ... yn, just as V was derived from η1, ... ηn; but it does not follow that these functions ζi, ... ζnare obtained from y1, ... ynby a transformation of the linear group A, B, ... ; for it may happen that the determinant d(ζ1, ... ζn) / (dy1, ... yn) is zero. In that case ζ1, ... ζnmay be called a singular set, and W a singular solution; it satisfies an equation of lower than the N-th order. But every solution V, W, ordinary or singular, of the equation F = 0, is expressible rationally in terms of ω, dω / dx, ... dN−1ω / dxN−1; we shall write, simply, V = r(ω). Consider now the rational irreducible equation of lowest order, not necessarily a linear equation, which is satisfied by ω; as y1, ... ynare particular functions, it may quite well be of order less than N; we call it theresolvent equation, suppose it of order p, and denote it by γ(v). Upon it the whole theory turns. In the first place, as γ(v) = 0 is satisfied by the solution ω of F = 0, all the solutions of γ(v) are solutions F = 0, and are therefore rationally expressible by ω; any one may then be denoted by r(ω). If this solution of F = 0 be not singular, it corresponds to a transformation A of the linear group (A, B, ...), effected upon y1, ... yn. The coefficients Aijof this transformation follow from the expressions before mentioned for η1... ηnin terms of V, dV/dx, d²V/dx², ... by substituting V = r(ω); thus they depend on the p arbitrary parameters which enter into the general expression for the integral of the equation γ(v) = 0. Without going into further details, it is then clear enough that the resolvent equation, being irreducible and such that any solution is expressible rationally, with p parameters, in terms of the solution ω, enables us to define a linear homogeneous group of transformations of y1... yndepending on p parameters; and every operation of this (continuous) group corresponds to a rational transformation of the solution of the resolvent equation. This is the group called therationality group, or thegroup of transformationsof the original homogeneous linear differential equation.
The group must not be confounded with a subgroup of itself, themonodromy groupof the equation, often called simply the group of the equation, which is a set of transformations, not depending on arbitrary variable parameters, arising for one particular fundamental set of solutions of the linear equation (seeGroups, Theory of).
The importance of the rationality group consists in three propositions. (1) Any rational function of y1, ... ynwhich is unaltered in value by the transformations of the group can be written in rational form. (2) If any rational function be changedThe fundamental theorem in regard to the rationality group.in form, becoming a rational function of y1, ... yn, a transformation of the group applied to its new form will leave its value unaltered. (3) Any homogeneous linear transformation leaving unaltered the value of every rational function of y1, ... ynwhich has a rational value, belongs to the group. It follows from these that any group of linear homogeneous transformations having the properties (1) (2) is identical with the group in question. It is clear that with these properties the group must be of the greatest importance in attempting to discover what functions of x must be regarded as rational in order that the values of y1... ynmay be expressed. And this is the problem of solving the equation from another point of view.
Literature.—(α)Formal or Transformation Theories for Equations of the First Order:—E. Goursat,Leçons sur l’intégration des équations aux dérivées partielles du premier ordre(Paris, 1891); E. v. Weber,Vorlesungen über das Pfaff’sche Problem und die Theorie der partiellen Differentialgleichungen erster Ordnung(Leipzig, 1900); S. Lie und G. Scheffers,Geometrie der Berührungstransformationen, Bd. i. (Leipzig, 1896); Forsyth,Theory of Differential Equations, Part i., Exact Equations and Pfaff’s Problem(Cambridge, 1890); S. Lie, “Allgemeine Untersuchungen über Differentialgleichungen, die eine continuirliche endliche Gruppe gestatten” (Memoir),Mathem. Annal.xxv. (1885), pp. 71-151; S. Lie und G. Scheffers,Vorlesungen über Differentialgleichungen mit bekannten infinitesimalen Transformationen(Leipzig, 1891). A very full bibliography is given in the book of E. v. Weber referred to; those here named are perhaps sufficiently representative of modern works. Of classical works may be named: Jacobi,Vorlesungen über Dynamik(von A. Clebsch, Berlin, 1866);Werke, Supplementband; G Monge,Application de l’analyse à la géométrie(par M. Liouville, Paris, 1850); J. L. Lagrange,Leçons sur le calcul des fonctions(Paris, 1806), andThéorie des fonctions analytiques(Paris, Prairial, an V); G. Boole,A Treatise on Differential Equations(London, 1859); andSupplementary Volume(London, 1865); Darboux,Leçons sur la théorie générale des surfaces, tt. i.-iv. (Paris, 1887-1896); S. Lie,Théorie der transformationsgruppenii. (on Contact Transformations) (Leipzig, 1890).(β)Quantitative or Function Theories for Linear Equations:—C. Jordan,Cours d’analyse, t. iii. (Paris, 1896); E. Picard,Traité d’analyse, tt. ii. and iii. (Paris, 1893, 1896); Fuchs,Various Memoirs, beginning with that in Crelle’s Journal, Bd. lxvi. p. 121; Riemann,Werke, 2rAufl. (1892); Schlesinger,Handbuch der Theorie der linearen Differentialgleichungen, Bde. i.-ii. (Leipzig, 1895-1898); Heffter,Einleitung in die Theorie der linearen Differentialgleichungen mit einer unabhängigen Variablen(Leipzig, 1894); Klein,Vorlesungen über lineare Differentialgleichungen der zweiten Ordnung(Autographed, Göttingen, 1894); andVorlesungen über die hypergeometrische Function(Autographed, Göttingen, 1894); Forsyth,Theory of Differential Equations, Linear Equations.(γ)Rationality Group (of Linear Differential Equations):—Picard,Traité d’Analyse, as above, t. iii.; Vessiot,Annales de l’École Normale, série III. t. ix. p. 199 (Memoir); S. Lie,Transformationsgruppen, as above, iii. A connected account is given in Schlesinger, as above, Bd. ii., erstes Theil.(δ)Function Theories of Non-Linear Ordinary Equations:—Painlevé,Leçons sur la théorie analytique des équations différentielles(Paris, 1897, Autographed); Forsyth,Theory of Differential Equations, Part ii., Ordinary Equations not Linear(two volumes, ii. and iii.) (Cambridge, 1900); Königsberger,Lehrbuch der Theorie der Differentialgleichungen(Leipzig, 1889); Painlevé,Leçons sur l’intégration des équations differentielles de la mécanique et applications(Paris, 1895).(ε)Formal Theories of Partial Equations of the Second and Higher Orders:—E. Goursat,Leçons sur l’intégration des équations aux dérivées partielles du second ordre, tt. i. and ii. (Paris, 1896, 1898); Forsyth,Treatise on Differential Equations(London, 1889); andPhil. Trans. Roy. Soc.(A.), vol. cxci. (1898), pp. 1-86.(ζ) See also the six extensive articles in the second volume of the GermanEncyclopaedia of Mathematics.
Literature.—(α)Formal or Transformation Theories for Equations of the First Order:—E. Goursat,Leçons sur l’intégration des équations aux dérivées partielles du premier ordre(Paris, 1891); E. v. Weber,Vorlesungen über das Pfaff’sche Problem und die Theorie der partiellen Differentialgleichungen erster Ordnung(Leipzig, 1900); S. Lie und G. Scheffers,Geometrie der Berührungstransformationen, Bd. i. (Leipzig, 1896); Forsyth,Theory of Differential Equations, Part i., Exact Equations and Pfaff’s Problem(Cambridge, 1890); S. Lie, “Allgemeine Untersuchungen über Differentialgleichungen, die eine continuirliche endliche Gruppe gestatten” (Memoir),Mathem. Annal.xxv. (1885), pp. 71-151; S. Lie und G. Scheffers,Vorlesungen über Differentialgleichungen mit bekannten infinitesimalen Transformationen(Leipzig, 1891). A very full bibliography is given in the book of E. v. Weber referred to; those here named are perhaps sufficiently representative of modern works. Of classical works may be named: Jacobi,Vorlesungen über Dynamik(von A. Clebsch, Berlin, 1866);Werke, Supplementband; G Monge,Application de l’analyse à la géométrie(par M. Liouville, Paris, 1850); J. L. Lagrange,Leçons sur le calcul des fonctions(Paris, 1806), andThéorie des fonctions analytiques(Paris, Prairial, an V); G. Boole,A Treatise on Differential Equations(London, 1859); andSupplementary Volume(London, 1865); Darboux,Leçons sur la théorie générale des surfaces, tt. i.-iv. (Paris, 1887-1896); S. Lie,Théorie der transformationsgruppenii. (on Contact Transformations) (Leipzig, 1890).
(β)Quantitative or Function Theories for Linear Equations:—C. Jordan,Cours d’analyse, t. iii. (Paris, 1896); E. Picard,Traité d’analyse, tt. ii. and iii. (Paris, 1893, 1896); Fuchs,Various Memoirs, beginning with that in Crelle’s Journal, Bd. lxvi. p. 121; Riemann,Werke, 2rAufl. (1892); Schlesinger,Handbuch der Theorie der linearen Differentialgleichungen, Bde. i.-ii. (Leipzig, 1895-1898); Heffter,Einleitung in die Theorie der linearen Differentialgleichungen mit einer unabhängigen Variablen(Leipzig, 1894); Klein,Vorlesungen über lineare Differentialgleichungen der zweiten Ordnung(Autographed, Göttingen, 1894); andVorlesungen über die hypergeometrische Function(Autographed, Göttingen, 1894); Forsyth,Theory of Differential Equations, Linear Equations.
(γ)Rationality Group (of Linear Differential Equations):—Picard,Traité d’Analyse, as above, t. iii.; Vessiot,Annales de l’École Normale, série III. t. ix. p. 199 (Memoir); S. Lie,Transformationsgruppen, as above, iii. A connected account is given in Schlesinger, as above, Bd. ii., erstes Theil.
(δ)Function Theories of Non-Linear Ordinary Equations:—Painlevé,Leçons sur la théorie analytique des équations différentielles(Paris, 1897, Autographed); Forsyth,Theory of Differential Equations, Part ii., Ordinary Equations not Linear(two volumes, ii. and iii.) (Cambridge, 1900); Königsberger,Lehrbuch der Theorie der Differentialgleichungen(Leipzig, 1889); Painlevé,Leçons sur l’intégration des équations differentielles de la mécanique et applications(Paris, 1895).
(ε)Formal Theories of Partial Equations of the Second and Higher Orders:—E. Goursat,Leçons sur l’intégration des équations aux dérivées partielles du second ordre, tt. i. and ii. (Paris, 1896, 1898); Forsyth,Treatise on Differential Equations(London, 1889); andPhil. Trans. Roy. Soc.(A.), vol. cxci. (1898), pp. 1-86.
(ζ) See also the six extensive articles in the second volume of the GermanEncyclopaedia of Mathematics.
(H. F. Ba.)
DIFFLUGIA(L. Leclerc), a genus of lobose Rhizopoda, characterized by a shell formed of sand granules cemented together; these are swallowed by the animal, and during the process of bud-fission they pass to the surface of the daughter-bud and are cemented there.Centropyxis(Steia) andLecqueureuxia(Schlumberg) differ only in minor points.
DIFFRACTION OF LIGHT.—1. When light proceeding from a small source falls upon an opaque object, a shadow is cast upon a screen situated behind the obstacle, and this shadow is found to be bordered by alternations of brightness and darkness, known as “diffraction bands.” The phenomena thus presented were described by Grimaldi and by Newton. Subsequently T. Young showed that in their formation interference plays an important part, but the complete explanation was reserved for A. J. Fresnel. Later investigations by Fraunhofer, Airy and others have greatly widened the field, and under the head of “diffraction” are now usually treated all the effects dependent upon the limitation of a beam of light, as well as those which arise from irregularities of any kind at surfaces through which it is transmitted, or at which it is reflected.
2.Shadows.—In the infancy of the undulatory theory the objection most frequently urged against it was the difficulty of explaining the very existence of shadows. Thanks to Fresnel and his followers, this department of optics is now precisely the one in which the theory has gained its greatest triumphs. The principle employed in these investigations is due to C. Huygens, and may be thus formulated. If round the origin of waves an ideal closed surface be drawn, the whole action of the waves in the region beyond may be regarded as due to the motion continually propagated across the various elements of this surface. The wave motion due to any element of the surface is called asecondarywave, and in estimating the total effect regard must be paid to the phases as well as the amplitudes of the components. It is usually convenient to choose as the surface of resolution awave-front,i.e.a surface at which the primary vibrations are in one phase. Any obscurity that may hang over Huygens’s principle is due mainly to the indefiniteness of thought and expression which we must be content to put up with if we wish to avoid pledging ourselves as to the character of the vibrations. In the application to sound, where we know what we are dealing with, the matter is simple enough in principle, although mathematical difficulties would often stand in the way of the calculations we might wish to make.The ideal surface of resolution may be there regarded as a flexible lamina; and we know that, if by forces locally applied every element of the lamina be made to move normally to itself exactly as the air at that place does, the external aerial motion is fully determined. By the principle of superposition the whole effect may be found by integration of the partial effects due to each element of the surface, the other elements remaining at rest.
We will now consider in detail the important case in which uniform plane waves are resolved at a surface coincident with a wave-front (OQ). We imagine a wave-front divided into elementary rings or zones—often named after Huygens, but better after Fresnel—by spheres described round P (the point at which the aggregate effect is to be estimated), the first sphere, touching the plane at O, with a radius equal to PO, and the succeeding spheres with radii increasing at each step by ½λ. There are thus marked out a series of circles, whose radii x are given by x² + r² = (r + ½nλ)², or x² = nλr nearly; so that the rings are at first of nearly equal area. Now the effect upon P of each element of the plane is proportional to its area; but it depends also upon the distance from P, and possibly upon the inclination of the secondary ray to the direction of vibration and to the wave-front.
The latter question can only be treated in connexion with the dynamical theory (see below, § 11); but under all ordinary circumstances the result is independent of the precise answer that may be given. All that it is necessary to assume is that the effects of the successive zones gradually diminish, whether from the increasing obliquity of the secondary ray or because (on account of the limitation of the region of integration) the zones become at last more and more incomplete. The component vibrations at P due to the successive zones are thus nearly equal in amplitude and opposite in phase (the phase of each corresponding to that of the infinitesimal circle midway between the boundaries), and the series which we have to sum is one in which the terms are alternately opposite in sign and, while at first nearly constant in numerical magnitude, gradually diminish to zero. In such a series each term may be regarded as very nearly indeed destroyed by the halves of its immediate neighbours, and thus the sum of the whole series is represented by half the first term, which stands over uncompensated. The question is thus reduced to that of finding the effect of the first zone, or central circle, of which the area is πλr.
We have seen that the problem before us is independent of the law of the secondary wave as regards obliquity; but the result of the integration necessarily involves the law of the intensity and phase of a secondary wave as a function of r, the distance from the origin. And we may in fact, as was done by A. Smith (Camb. Math. Journ., 1843, 3, p. 46), determine the law of the secondary wave, by comparing the result of the integration with that obtained by supposing the primary wave to pass on to P without resolution.
Now as to the phase of the secondary wave, it might appear natural to suppose that it starts from any point Q with the phase of the primary wave, so that on arrival at P, it is retarded by the amount corresponding to QP. But a little consideration will prove that in that case the series of secondary waves could not reconstitute the primary wave. For the aggregate effect of the secondary waves is the half of that of the first Fresnel zone, and it is the central element only of that zone for which the distance to be travelled is equal to r. Let us conceive the zone in question to be divided into infinitesimal rings of equal area. The effects due to each of these rings are equal in amplitude and of phase ranging uniformly over half a complete period. The phase of the resultant is midway between those of the extreme elements, that is to say, a quarter of a period behind that due to the element at the centre of the circle. It is accordingly necessary to suppose that the secondary waves start with a phase one-quarter of a period in advance of that of the primary wave at the surface of resolution.
Further, it is evident that account must be taken of the variation of phase in estimating the magnitude of the effect at P of the first zone. The middle element alone contributes without deduction; the effect of every other must be found by introduction of a resolving factor, equal to cos θ, if θ represent the difference of phase between this element and the resultant. Accordingly, the amplitude of the resultant will be less than if all its components had the same phase, in the ratio
or 2 : π. Now 2 area /π = 2λr; so that, in order to reconcile the amplitude of the primary wave (taken as unity) with the half effect of the first zone, the amplitude, at distance r, of the secondary wave emitted from the element of area dS must be taken to be
dS/λr (1).
By this expression, in conjunction with the quarter-period acceleration of phase, the law of the secondary wave is determined.
That the amplitude of the secondary wave should vary as r-1was to be expected from considerations respecting energy; but the occurrence of the factor λ-1, and the acceleration of phase, have sometimes been regarded as mysterious. It may be well therefore to remember that precisely these laws apply to a secondary wave of sound, which can be investigated upon the strictest mechanical principles.
The recomposition of the secondary waves may also be treated analytically. If the primary wave at O be cos kat, the effect of the secondary wave proceeding from the element dS at Q is
If dS = 2πxdx, we have for the whole effect
or, since xdx = ρdρ, k = 2π/λ,
In order to obtain the effect of the primary wave, as retarded by traversing the distance r, viz. cos k(at − r), it is necessary to suppose that the integrated term vanishes at the upper limit. And it is important to notice that without some further understanding the integral is really ambiguous. According to the assumed law of the secondary wave, the result must actually depend upon the precise radius of the outer boundary of the region of integration, supposed to be exactly circular. This case is, however, at most very special and exceptional. We may usually suppose that a large number of the outer rings are incomplete, so that the integrated term at the upper limit may properly be taken to vanish. If a formal proof be desired, it may be obtained by introducing into the integral a factor such as e-hρ, in which h is ultimately made to diminish without limit.
When the primary wave is plane, the area of the first Fresnel zone is πλr, and, since the secondary waves vary as r-1, the intensity is independent of r, as of course it should be. If, however, the primary wave be spherical, and of radius a at the wave-front of resolution, then we know that at a distance r further on the amplitude of the primary wave will be diminished in the ratio a : (r + a). This may be regarded as a consequence of the altered area of the first Fresnel zone. For, if x be its radius, we have
{(r + ½λ)² − x²} + √ {a² − x²} = r + a,
so that
x² = λar/(a + r) nearly.
Since the distance to be travelled by the secondary waves is still r, we see how the effect of the first zone, and therefore of the whole series is proportional to a/(a + r). In like manner may be treated other cases, such as that of a primary wave-front of unequal principal curvatures.
The general explanation of the formation of shadows may also be conveniently based upon Fresnel’s zones. If the point under consideration be so far away from the geometrical shadow that a large number of the earlier zones are complete, then the illumination, determined sensibly by the first zone, is the same as if there were no obstruction at all. If, on the other hand, the point be well immersed in the geometrical shadow, the earlier zones are altogether missing, and, instead of a series of terms beginning with finite numerical magnitude and gradually diminishing to zero, we have now to deal with one of which the terms diminish to zeroat both ends. The sum of such a series is very approximately zero, each term being neutralized by the halves of its immediate neighbours, which are of the opposite sign. The question of light or darkness then depends upon whether the series begins or ends abruptly. With few exceptions, abruptness can occur only in the presence of the first term, viz. when the secondary wave of least retardation is unobstructed, or when araypasses through the point under consideration. According to the undulatory theory the light cannot be regarded strictly as travelling along a ray; but the existence of an unobstructed ray implies that the system of Fresnel’s zones can be commenced, and, if a large number of these zones are fully developed and do not terminate abruptly, the illumination is unaffected by the neighbourhood of obstacles. Intermediate cases in which a few zones only are formed belong especially to the province of diffraction.
An interesting exception to the general rule that full brightness requires the existence of the first zone occurs when the obstacle assumes the form of a small circular disk parallel to the plane of the incident waves. In the earlier half of the 18th century R. Delisle found that the centre of the circular shadow was occupied by a bright point of light, but the observation passed into oblivion until S. D. Poisson brought forward as an objection to Fresnel’s theory that it required at the centre of a circular shadow a point as bright as if no obstacle were intervening. If we conceive the primary wave to be broken up at the plane of the disk, a system of Fresnel’s zones can be constructed which begin from the circumference; and the first zone external to the disk plays the part ordinarily taken by the centre of the entire system. The whole effect is thehalf of that of the first existing zone, and this is sensibly the same as if there were no obstruction.
When light passes through a small circular or annular aperture, the illumination at any point along the axis depends upon the precise relation between the aperture and the distance from it at which the point is taken. If, as in the last paragraph, we imagine a system of zones to be drawn commencing from the inner circular boundary of the aperture, the question turns upon the manner in which the series terminates at the outer boundary. If the aperture be such as to fit exactly an integral number of zones, the aggregate effect may be regarded as the half of those due to the first and last zones. If the number of zones be even, the action of the first and last zones are antagonistic, and there is complete darkness at the point. If on the other hand the number of zones be odd, the effects conspire; and the illumination (proportional to the square of the amplitude) is four times as great as if there were no obstruction at all.
The process of augmenting the resultant illumination at a particular point by stopping some of the secondary rays may be carried much further (Soret,Pogg. Ann., 1875, 156, p. 99). By the aid of photography it is easy to prepare a plate, transparent where the zones of odd order fall, and opaque where those of even order fall. Such a plate has the power of a condensing lens, and gives an illumination out of all proportion to what could be obtained without it. An even greater effect (fourfold) can be attained by providing that the stoppage of the light from the alternate zones is replaced by a phase-reversal without loss of amplitude. R. W. Wood (Phil. Mag., 1898, 45, p 513) has succeeded in constructing zone plates upon this principle.
In such experiments the narrowness of the zones renders necessary a pretty close approximation to the geometrical conditions. Thus in the case of the circular disk, equidistant (r) from the source of light and from the screen upon which the shadow is observed, the width of the first exterior zone is given by
dx = λ(2r)/4(2x),
2x being the diameter of the disk. If 2r = 1000 cm., 2x = 1 cm., λ = 6 × 10-5cm., then dx = .0015 cm. Hence, in order that this zone may be perfectly formed, there should be no error in the circumference of the order of .001 cm. (It is easy to see that the radius of the bright spot is of the same order of magnitude.) The experiment succeeds in a dark room of the length above mentioned, with a threepenny bit (supported by three threads) as obstacle, the origin of light being a small needle hole in a plate of tin, through which the sun’s rays shine horizontally after reflection from an external mirror. In the absence of a heliostat it is more convenient to obtain a point of light with the aid of a lens of short focus.
The amplitude of the light at any point in the axis, when plane waves are incident perpendicularly upon an annular aperture, is, as above,
cos k(at − r1) − cos k(at − r2) = 2 sin kat sin k(r1− r2),
r2, r1being the distances of the outer and inner boundaries from the point in question. It is scarcely necessary to remark that in all such cases the calculation applies in the first instance to homogeneous light, and that, in accordance with Fourier’s theorem, each homogeneous component of a mixture may be treated separately. When the original light is white, the presence of some components and the absence of others will usually give rise to coloured effects, variable with the precise circumstances of the case.
Although the matter can be fully treated only upon the basis of a dynamical theory, it is proper to point out at once that there is an element of assumption in the application of Huygens’s principle to the calculation of the effects produced by opaque screens of limited extent. Properly applied, the principle could not fail; but, as may readily be proved in the case of sonorous waves, it is not in strictness sufficient to assume the expression for a secondary wave suitable when the primary wave is undisturbed, with mere limitation of the integration to the transparent parts of the screen. But, except perhaps in the case of very fine gratings, it is probable that the error thus caused is insignificant; for the incorrect estimation of the secondary waves will be limited to distances of a few wave-lengths only from the boundary of opaque and transparent parts.
3.Fraunhofer’s Diffraction Phenomena.—A very general problem in diffraction is the investigation of the distribution of light over a screen upon which impinge divergent or convergent spherical waves after passage through various diffracting apertures. When the waves are convergent and the recipient screen is placed so as to contain the centre of convergency—the image of the original radiant point, the calculation assumes a less complicated form. This class of phenomena was investigated by J. von Fraunhofer (upon principles laid down by Fresnel), and are sometimes called after his name. We may conveniently commence with them on account of their simplicity and great importance in respect to the theory of optical instruments.
If ƒ be the radius of the spherical wave at the place of resolution, where the vibration is represented by cos kat, then at any point M (fig. 2) in the recipient screen the vibration due to an element dS of the wave-front is (§ 2)
ρ being the distance between M and the element dS.
Taking co-ordinates in the plane of the screen with the centre of the wave as origin, let us represent M by ξ, η, and P (where dS is situated) by x, y, z.
Then
ρ² = (x − ξ)² + (y − η)² + z², ƒ² = x² + y² + z²;
so that
ρ² = ƒ² − 2xξ − 2yη + ξ² + η².
In the applications with which we are concerned, ξ, η are very small quantities; and we may take
At the same time dS may be identified with dxdy, and in the denominator ρ may be treated as constant and equal to ƒ. Thus the expression for the vibration at M becomes
and for the intensity, represented by the square of the amplitude,
This expression for the intensity becomes rigorously applicable when ƒ is indefinitely great, so that ordinary optical aberration disappears. The incident waves are thus plane, and are limited to a plane aperture coincident with a wave-front. The integrals are then properly functions of thedirectionin which the light is to be estimated.
In experiment under ordinary circumstances it makes no difference whether the collecting lens is in front of or behind the diffracting aperture. It is usually most convenient to employ a telescope focused upon the radiant point, and to place the diffracting apertures immediately in front of the object-glass. What is seen through the eye-piece in any case is the same as would be depicted upon a screen in the focal plane.
Before proceeding to special cases it may be well to call attention to some general properties of the solution expressed by (2) (see Bridge,Phil. Mag., 1858).
If when the aperture is given, the wave-length (proportional to k-1) varies, the composition of the integrals is unaltered, provided ξ and η are taken universely proportional to λ. A diminution of λ thus leads to a simple proportional shrinkage of the diffraction pattern, attended by an augmentation of brilliancy in proportion to λ-2.
If the wave-length remains unchanged, similar effects are produced by an increase in the scale of the aperture. The linear dimension of the diffraction pattern is inversely as that of the aperture, and the brightness at corresponding points is as thesquareof the area of aperture.
If the aperture and wave-length increase in the same proportion, the size and shape of the diffraction pattern undergo no change.
We will now apply the integrals (2) to the case of a rectangular aperture of width a parallel to x and of width b parallel to y. The limits of integration for x may thus be taken to be −½a and +½a, and for y to be −½b, +½b. We readily find (with substitution for k of 2π/λ)
as representing the distribution of light in the image of a mathematical point when the aperture is rectangular, as is often the case in spectroscopes.
The second and third factors of (3) being each of the form sin²u/u², we have to examine the character of this function. It vanishes when u = mπ, m being any whole number other than zero. When u = 0, it takes the value unity. The maxima occur when
u = tan u, (4),
and then
sin²u / u² = cos²u (5).
To calculate the roots of (5) we may assume
u = (m + ½)π − y = U − y,
where y is a positive quantity which is small when u is large. Substituting this, we find cot y = U − y, whence
This equation is to be solved by successive approximation. It will readily be found that
In the first quadrant there is no root after zero, since tan u > u, and in the second quadrant there is none because the signs of u and tan u are opposite. The first root after zero is thus in the third quadrant, corresponding to m = 1. Even in this case the series converges sufficiently to give the value of the root with considerable accuracy, while for higher values of m it is all that could be desired. The actual values of u/π (calculated in another manner by F. M. Schwerd) are 1.4303, 2.4590, 3.4709, 4.4747, 5.4818, 6.4844, &c.
Since the maxima occur when u = (m + ½)π nearly, the successive values are not very different from
The application of these results to (3) shows that the field is brightest at the centre ξ = 0, η = 0, viz. at the geometrical image of the radiant point. It is traversed by dark lines whose equations are
ξ = mfλ / a, η = mfλ / b.
Within the rectangle formed by pairs of consecutive dark lines, and not far from its centre, the brightness rises to a maximum; but these subsequent maxima are in all cases much inferior to the brightness at the centre of the entire pattern (ξ = 0, η = 0).
By the principle of energy the illumination over the entire focal plane must be equal to that over the diffracting area; and thus, in accordance with the suppositions by which (3) was obtained, its value when integrated from ξ = ∞ to ξ = +∞, and from η = −∞ to η = +∞ should be equal to ab. This integration, employed originally by P. Kelland (Edin. Trans.15, p. 315) to determine the absolute intensity of a secondary wave, may be at once effected by means of the known formula
It will be observed that, while the total intensity is proportional to ab, the intensity at the focal point is proportional to a²b². If the aperture be increased, not only is the total brightness over the focal plane increased with it, but there is also a concentration of the diffraction pattern. The form of (3) shows immediately that, if a and b be altered, the co-ordinates of any characteristic point in the pattern vary as a−1and b−1.
The contraction of the diffraction pattern with increase of aperture is of fundamental importance in connexion with the resolving power of optical instruments. According to common optics, where images are absolute, the diffraction pattern is supposed to be infinitely small, and two radiant points, however near together, form separated images. This is tantamount to an assumption that λ is infinitely small. The actual finiteness of λ imposes a limit upon the separating or resolving power of an optical instrument.
This indefiniteness of images is sometimes said to be due to diffraction by the edge of the aperture, and proposals have even been made for curing it by causing the transition between the interrupted and transmitted parts of the primary wave to be less abrupt. Such a view of the matter is altogether misleading. What requires explanation is not the imperfection of actual images so much as the possibility of their being as good as we find them.
At the focal point (ξ = 0, η = 0) all the secondary waves agree in phase, and the intensity is easily expressed, whatever be the form of the aperture. From the general formula (2), if A be theareaof aperture,
I0² = A² / 벃² (7).
The formation of a sharp image of the radiant point requires that the illumination become insignificant when ξ, η attain small values, and this insignificance can only arise as a consequence of discrepancies of phase among the secondary waves from various parts of the aperture. So long as there is no sensible discrepancy of phase there can be no sensible diminution of brightness as compared with that to be found at the focal point itself. We may go further, and lay it down that there can be no considerable loss of brightness until the difference of phase of the waves proceeding from the nearest and farthest parts of the aperture amounts to ¼λ.
When the difference of phase amounts to λ, we may expect the resultant illumination to be very much reduced. In the particular case of a rectangular aperture the course of things can be readily followed, especially if we conceive ƒ to be infinite. In the direction (suppose horizontal) for which η = 0, ξ/ƒ = sin θ, the phases of the secondary waves range over a complete period when sin θ = λ/a, and, since all parts of the horizontal aperture are equally effective, there is in this direction a complete compensation and consequent absence of illumination. When sin θ =3⁄2λ/a, the phases range one and a half periods, and there is revival of illumination. We may compare the brightness with that in the direction θ = 0. The phase of the resultant amplitude is the same as that due to the central secondary wave, and the discrepancies of phase among the components reduce the amplitude in the proportion
or -2⁄3π : 1; so that the brightness in this direction is4⁄9π² of the maximum at θ = 0. In like manner we may find the illumination in any other direction, and it is obvious that it vanishes when sin θ is any multiple of λ/a.
The reason of the augmentation of resolving power with aperture will now be evident. The larger the aperture the smaller are the angles through which it is necessary to deviate from the principal direction in order to bring in specified discrepancies of phase—the more concentrated is the image.
In many cases the subject of examination is a luminous line of uniform intensity, the various points of which are to be treated as independent sources of light. If the image of the line be ξ = 0, the intensity at any point ξ, η of the diffraction pattern may be represented by
the same law as obtains for a luminous point when horizontal directions are alone considered. The definition of a fine vertical line, and consequently the resolving power for contiguous vertical lines, is thusindependent of the vertical aperture of the instrument, a law of great importance in the theory of the spectroscope.
The distribution of illumination in the image of a luminous line is shown by the curve ABC (fig. 3), representing the value of the function sin²u/u² from u = 0 to u = 2π. The part corresponding to negative values of u is similar, OA being a line of symmetry.
Let us now consider the distribution of brightness in the image of a double line whose components are of equal strength, and at such an angular interval that the central line in the image of one coincides with the first zero of brightness in the image of the other. In fig. 3 the curve of brightness for one component is ABC, and for the other OA′C′; and the curve representing half the combined brightnesses is E′BE. The brightness (corresponding to B) midway between the two central points AA’ is .8106 of the brightness at the central points themselves. We may consider this to be about the limit of closeness at which there could be any decided appearance of resolution, though doubtless an observer accustomed to his instrument would recognize the duplicity with certainty. The obliquity, corresponding to u = π, is such that the phases of the secondary waves range over a complete period,i.e.such that the projection of the horizontal aperture upon this direction is one wave-length. We conclude that adouble line cannot be fairly resolved unless its components subtend an angle exceeding that subtended by the wave-length of light at a distance equal to the horizontal aperture. This rule is convenient on account of its simplicity; and it is sufficiently accurate in view of the necessary uncertainty as to what exactly is meant by resolution.
If the angular interval between the components of a double line be half as great again as that supposed in the figure, the brightness midway between is .1802 as against 1.0450 at the central lines of each image. Such a falling off in the middle must be more than sufficient for resolution. If the angle subtended by the components of a double line be twice that subtended by the wave-length at a distance equal to the horizontal aperture, the central bands are just clear of one another, and there is a line of absolute blackness in the middle of the combined images.
The resolving power of a telescope with circular or rectangular aperture is easily investigated experimentally. The best object for examination is a grating of fine wires, about fifty to the inch, backed by a sodium flame. The object-glass is provided with diaphragms pierced with round holes or slits. One of these, of width equal, say, to one-tenth of an inch, is inserted in front of the object-glass, and the telescope, carefully focused all the while, is drawn gradually back from the grating until the lines are no longer seen. From a measurement of the maximum distance the least angle between consecutive lines consistent with resolution may be deduced, and a comparison made with the rule stated above.
Merely to show the dependence of resolving power on aperture it is not necessary to use a telescope at all. It is sufficient to look at wire gauze backed by the sky or by a flame, through a piece of blackened cardboard, pierced by a needle and held close to the eye. By varying the distance the point is easily found at which resolution ceases; and the observation is as sharp as with a telescope. Thefunction of the telescope is in fact to allow the use of a wider, and therefore more easily measurable, aperture. An interesting modification of the experiment may be made by using light of various wave-lengths.
Since the limitation of the width of the central band in the image of a luminous line depends upon discrepancies of phase among the secondary waves, and since the discrepancy is greatest for the waves which come from the edges of the aperture, the question arises how far the operation of the central parts of the aperture is advantageous. If we imagine the aperture reduced to two equal narrow slits bordering its edges, compensation will evidently be complete when the projection on an oblique direction is equal to ½λ, instead of λ as for the complete aperture. By this procedure the width of the central band in the diffraction pattern is halved, and so far an advantage is attained. But, as will be evident, the bright bands bordering the central band are now not inferior to it in brightness; in fact, a band similar to the central band is reproduced an indefinite number of times, so long as there is no sensible discrepancy of phase in the secondary waves proceeding from the various parts of thesameslit. Under these circumstances the narrowing of the band is paid for at a ruinous price, and the arrangement must be condemned altogether.
A more moderate suppression of the central parts is, however, sometimes advantageous. Theory and experiment alike prove that a double line, of which the components are equally strong, is better resolved when, for example, one-sixth of the horizontal aperture is blocked off by a central screen; or the rays quite at the centre may be allowed to pass, while others a little farther removed are blocked off. Stops, each occupying one-eighth of the width, and with centres situated at the points of trisection, answer well the required purpose.
It has already been suggested that the principle of energy requires that the general expression for I² in (2) when integrated over the whole of the plane ξ, η should be equal to A, where A is the area of the aperture. A general analytical verification has been given by Sir G. G. Stokes (Edin. Trans., 1853, 20, p. 317). Analytically expressed—
We have seen that I0² (the intensity at the focal point) was equal to A²/λ²f². If A′ be the area over which the intensity must be I0² in order to give the actual total intensity in accordance with
the relation between A and A′ is AA′ = λ²f². Since A′ is in some sense the area of the diffraction pattern, it may be considered to be a rough criterion of the definition, and we infer that the definition of a point depends principally upon the area of the aperture, and only in a very secondary degree upon the shape when the area is maintained constant.
4.Theory of Circular Aperture.—We will now consider the important case where the form of the aperture is circular.
Writing for brevity
kξ/f = p, kη/f = q, (1),
we have for the general expression (§ 11) of the intensity
λ²f²I² = S² + C² (2),
where
S =∫∫sin(px + qy)dx dy, (3),
C =∫∫cos(px + qy)dx dy, (4).
When, as in the application to rectangular or circular apertures, the form is symmetrical with respect to the axes both of x and y, S = 0, and C reduces to
C =∫∫cos px cos qy dx dy, (5).
In the case of the circular aperture the distribution of light is of course symmetrical with respect to the focal point p = 0, q = 0; and C is a function of p and q only through √(p² + q²). It is thus sufficient to determine the intensity along the axis of p. Putting q = 0, we get
R being the radius of the aperture. This integral is the Bessel’s function of order unity, defined by
Thus, if x = R cos φ,
and the illumination at distance r from the focal point is
The ascending series for J1(z), used by Sir G. B. Airy (Camb. Trans., 1834) in his original investigation of the diffraction of a circular object-glass, and readily obtained from (6), is
When z is great, we may employ the semi-convergent series
A table of the values of 2z-1J1(z) has been given by E. C. J. Lommel (Schlömilch, 1870, 15, p. 166), to whom is due the first systematic application of Bessel’s functions to the diffraction integrals.
The illumination vanishes in correspondence with the roots of the equation J1(z) = 0. If these be called z1z2, z3, ... the radii of the dark rings in the diffraction pattern are
being thusinverselyproportional to R.
The integrations may also be effected by means of polar co-ordinates, taking first the integration with respect to φ so as to obtain the result for an infinitely thin annular aperture. Thus, if
x = ρ cos φ, y = ρ sin φ,
Now by definition
The value of C for an annular aperture of radius r and width dr is thus
dC = 2 π J0(pρ) ρ dρ, (12).
For the complete circle,
In these expressions we are to replace p by kξ/ƒ, or rather, since the diffraction pattern is symmetrical, by kr/ƒ, where r is the distance of any point in the focal plane from the centre of the system.
The roots of J0(z) after the first may be found from
and those of J1(z) from
formulae derived by Stokes (Camb. Trans., 1850, vol. ix.) from the descending series.1The following table gives the actual values:—
In both cases the image of a mathematical point is thus a symmetrical ring system. The greatest brightness is at the centre, where
dC = 2πρ dρ, C = π R².
For a certain distance outwards this remains sensibly unimpaired and then gradually diminishes to zero, as the secondary waves become discrepant in phase. The subsequent revivals of brightness forming the bright rings are necessarily of inferior brilliancy as compared with the central disk.
The first dark ring in the diffraction pattern of the complete circular aperture occurs when
r/ƒ = 1.2197 × λ/2R (15).
We may compare this with the corresponding result for a rectangular aperture of width a,
ξ/ƒ =λ/a;
and it appears that in consequence of the preponderance of the central parts, the compensation in the case of the circle does not set in at so small an obliquity as when the circle is replaced by a rectangular aperture, whose side is equal to the diameter of the circle.
Again, if we compare the complete circle with a narrow annular aperture of the same radius, we see that in the latter case the first dark ring occurs at a much smaller obliquity, viz.
r/ƒ = .7655 × λ/2R.
It has been found by Sir William Herschel and others that the definition of a telescope is often improved by stopping off a part of the central area of the object-glass; but the advantage to be obtained in this way is in no case great, and anything like a reduction of the aperture to a narrow annulus is attended by a development of the external luminous rings sufficient to outweigh any improvement due to the diminished diameter of the central area.2
The maximum brightnesses and the places at which they occur are easily determined with the aid of certain properties of the Bessel’s functions. It is known (seeSpherical Harmonics) that
J0′(z) = −J1(z), (16);
J2(z) = (1/z) J1(z) − J1′(z) (17);
J0(z) + J2(z) = (2/z) J1(z) (18).
The maxima of C occur when
or by 17 when J2(z) = 0. When z has one of the values thus determined,
The accompanying table is given by Lommel, in which the first column gives the roots of J2(z) = 0, and the second and third columns the corresponding values of the functions specified. If appears that the maximum brightness in the first ring is only about1⁄57of the brightness at the centre.
We will now investigate the total illumination distributed over the area of the circle of radius r. We have
where
z = 2πRr/λf (20).
Thus
Now by (17), (18)
z-1J1(z) = J0(z) − J1′(z);
so that
and
If r, or z, be infinite, J0(z), J1(z) vanish, and the whole illumination is expressed by πR², in accordance with the general principle. In any case the proportion of the whole illumination to be found outside the circle of radius r is given by
J0²(z) + J1²(z).
For the dark rings J1(z) = 0; so that the fraction of illumination outside any dark ring is simply J0²(z). Thus for the first, second, third and fourth dark rings we get respectively .161, .090, .062, .047, showing that more than9⁄10ths of the whole light is concentrated within the area of the second dark ring (Phil. Mag., 1881).
When z is great, the descending series (10) gives
so that the places of maxima and minima occur at equal intervals.
The mean brightness varies as z-3(or as r-3), and the integral found by multiplying it by zdz and integrating between 0 and ∞ converges.
It may be instructive to contrast this with the case of an infinitely narrow annular aperture, where the brightness is proportional to J0²(z). When z is great,
The mean brightness varies as z-1; and the integral ∫∞0J0²(z)z dz is not convergent.
5.Resolving Power of Telescopes.—The efficiency of a telescope is of course intimately connected with the size of the disk by which it represents a mathematical point. In estimating theoretically the resolving power on a double star we have to consider the illumination of the field due to the superposition of the two independent images. If the angular interval between the components of a double star were equal to twice that expressed in equation (15) above, the central disks of the diffraction patterns would be just in contact. Under these conditions there is no doubt that the star would appear to be fairly resolved, since the brightness of its external ring system is too small to produce any material confusion, unless indeed the components are of very unequal magnitude. The diminution of the star disks with increasing aperture was observed by Sir William Herschel, and in 1823 Fraunhofer formulated the law of inverse proportionality. In investigations extending over a long series of years, the advantage of a large aperture in separating the components of close double stars was fully examined by W. R. Dawes.