Optics, the science which treats of the nature of light, and of the laws of the phenomena of light and vision. For the theories of light, and other branches of the subject, see the articles Aberration, Chromatics, Fluorescence, Light, Spectrum, Spectrum Analysis, Spectacles, Stereoscope, and Vision. The present article will be devoted chiefly to the laws of reflection (catoptrics) and those of simple refraction (dioptrics). These form a large portion of geometrical or formal optics, in which, without regard to any theory, the actual phenomena of light are observed and generalized, and the laws of the changes effected in the rays by surfaces and media are ascertained. In connection with the transmission of light one other general fact may be noticed. It is that, with the exception that some degree of dimness will arise when the interposed body of air is of great extent, a given surface, as that of the side of a house, illuminated in the same degree, appears equally luminous, at whatever distance it may be regarded.

This equal brightness at different distances is readily explained when we remember that the actual intensity of light from a point or unit of surface diminishes in inverse ratio as the square of the distance increases; and that, since any linear magnitude diminishes in the inverse ratio of the simple distance, so a surface must also appear lessened in the ratio of the square of distance; less light comes to the eye from a given surface at increased distance, but the actual surface becomes contracted into an apparent surface less in the same proportion; and thus one effect balances the other, and the actual illumination is reduced by the effect of the aerial perspective only. - The ancient Greeks and the Arabians made considerable progress in formal optics, but chiefly in the discovery of the law of reflection, and of consequences flowing from it. They had attained the idea of rays of light, the fact of their ordinary straight-lined transmission, and the law of equality of the angles of reflection and incidence, and deduced with much completeness the properties of shadows, perspective, and the convergence of rays by concave mirrors. Euclid and the followers of Plato, however, taught that these rays proceed from the eye, not from the visible object.

Aristotle reasoned that an interposed medium was necessary to vision; this he considered to be light, and defined as " the transparent in action." Of special treatises on light, the earliest known are the "Optics" of Euclid, Heron's " Catoptrics," and Ptolemy's " Optics." In the last of these occurs an elaborate collection of measurements of the refraction at different angles, from air to glass, and from glass to water - tables of much interest, as furnishing the oldest extant example of an accurately conducted physical investigation by experiment. Tycho Brahe introduced a correction for atmospheric refraction into astronomical calculations; the telescope appears to have been invented separately by Metius and Jansen about the year 1608; and Kepler, with his usual fertility of mathematical elements and of hypotheses, and incited by these advances, strove earnestly to find the true law of relation of the angle of refraction to that of incidence, but reached only a near approximation. The actual relation, known as the "law of the sines," was discovered by Willebrord Snell, about 1621. Descartes, who unjustly claimed this discovery, has really the merit of having applied it so as to explain the general formation and the angles of the rainbow.

Newton in 1672 published his remarkable discoveries in connection with the decomposition of light by aid of the prism, with the doctrine and measure of the refrangibilities of the different colors, and the agreement of the phenomena witli those of the rainbow. His discoveries resulted in improvements in the telescope, and also in explaining a prominent defect in the refracting telescope, that of the colored borders of images, due to chromatic aberration. Dollond about 1757 discovered the possibility of achromatic combinations of lenses, and produced the first of these. The first notice of double refraction is that of Bartholin, 1660; but Huygens first satisfactorily explained the phenomena,, by means of his since renowned undulatory theory of light, his treatise upon which was written in 1678, and first published in 1690. He also first observed the fact of polarization; though the distinct discovery of this phenomenon was not made until more than a century later, namely, by Mains in 1808, who commenced a thorough study of the subject; and this was much extended by Young, Fresnel, Arago, Brewster, Biot, and Seebeck. Hooke appears first to have studied the colors of thin plates, which he described in 1664; and these colors Newton and Young afterward turned to very important use.

Diffraction and the fringes of shadows were discovered by Grimaldi in 1665; depolarization, with the production of periodical colors in polarized light, by Arago in 1811; the relation of optical properties to the symmetry and axes of crystals, by Brewster in 1818. The general explanation of most of these phenomena by the undulatory theory is due to the labors of Young and Fresnel, from 1802 to 1829; and these have since been carried forward and corrected by the labors of Airy, Hamilton, Lloyd, Cauchy, and many others. Still other discoveries in optics, especially the more recent, as those made in connection with color, the velocity and physical modifications of light, the various optical instruments, and photography, will be found mentioned under the proper heads.

I. Catoptrics

When rays of light fall on a surface of an opaque, and in some degree smooth or polished body, a portion of those rays, greater or less, but never the whole, is thrown off again from such surface, and this light is said to be reflected. Opaque surfaces reflecting in a high degree are termed specula, or mirrors. Suppose a ray or minute beam incident on a polished plane surface in any direction whatever, and let fall at the point of incidence a perpendicular to the surface; then, first, it is universally true that the reflected ray will be situated in the same plane in space in which this perpendicular and the line of the incident ray are situated. Thus we may always determine the plane, vertical to the reflecting surface, in which to look for the reflected ray. The angle I O P, fig. 1, included between the perpendicular and incident ray, is termed the angle of incidence; that between the same perpendicular and the reflected ray, POR, the angle of reflection. These angles are always equal. Thus, the fundamental and universal law of reflection from plane surfaces is simply this: the paths of the incident and reflected rays always lie in the same plane with the perpendicular to the reflecting surface drawn to the point of incidence; and in that plane the angle of reflection is always equal to the angle of incidence.

This law is strictly verified by experiment and measurement. Necessary consequences of its truth are, that beams or rays parallel before incidence on a plane mirror will remain parallel after reflection, and that divergent rays will after reflection continue to diverge, and convergent rays to converge, at the same rates as before impinging on the reflecting surface. All the facts relating to images in plane mirrors follow from the same law. But a very important truth in relation to images, and one too often lost sight of, must be premised. Parallel rays or beams of light, or a single beam, may show us the existence of the object emit-ting them, but they do not enable us to determine its place or distance. We can do this in regard to an object or image, or any point in it, only by means of pencils of light, divergent in themselves, proceeding from the points or point to the eye. We necessarily judge of the size of this object chiefly by the angle subtended at the eye by a line joining its extreme points (the visual angle); and of its distance by the amount of reconvergent action the eye must exert upon the pencils painting its several points, in order to focus them upon the retina, as well as by the convergency of the axes of the two eyes upon the place of the object, if near. (See Stereoscope, and Vision. ) The pencils of light from the various points of an object before a plane mirror, being divergent at the same rate after as before reflection, and the eye of necessity seeing the object in the direction in which the rays of light finally come to it, the determination of the position and size of images resolves itself into investigating the images of a series of points.

And first, the case of a single point, A, fig. 2, placed before a plane mirror, M N, will he considered. Any ray, A 13, incident from this point on the mirror, is reflected in the direction B O, making the angle of reflection D B O equal to the angle of incidence D B A. If now a perpendicular, A 1ST, he let fall from the point A on the mirror, and if the ray O B be prolonged below the mirror until it meets this perpendicular in the point a, two triangles are formed, A B N and N B a, which are equal, for they have the side B N common to both, and the angles A N B, A B N, equal to the angles a N B, a B N; for the angles A N B and a N B are right angles, and the angles A B N and a B 1ST are equal to the angle O B M. From the equality of these triangles, it follows that a N is equal to A N.; that is, that any ray, A B, takes such a direction after being reflected, that its prolongation below the mirror cuts the perpendicular A a in the point a, which is at the same distance from the mirror as the point A. This applies also to the case of any other ray from the point A, A C for example. From this the important consequence follows, that all rays from the point A, reflected from the mirror, follow after reflection the same direction as if they had proceeded from the point a.

The eye is deceived, and sees the point A at a, as if it were really situate at a. Hence in plane mirrors the image of any point is formed behind the mirror at a distance equal to that of the given point from its front surface, and on the perpendicular let fall from this point on the mirror. It is manifest that the image of any object will be obtained by constructing according to this rule the image of each of its points, or at least of those which are sufficient to determine its form. Fig. 3 shows how the image a b of any object, A B, is formed. It follows from this construction that in plane mirrors the image is of the same size as the object; for if the trapezium A B C D be applied to the trapezium D C a 5, they are seen to coincide, and the object A B agrees with its image. A further consequence of the above construction is, that in plane mirrors the image is symmetrical in reference to the object, and not inverted. When an object is between two plane mirrors nearly parallel, the primary images seen in each of these are reflected as if at a greater distance in the other, and so on, forming in each mirror a long succession of images, growing more and more remote.

As the mirrors are turned, approaching a right angle with each other, the number of repetitions grows less, and the whole take a circular arrangement. At a right angle, the object and three images are visible, arranged as represented in fig. 4. The rays O C and O D from the point O, after a single reflection, give, the one an image O', and the other an image O", while the ray O A, which has undergone two reflections at A and B, gives a third image O"'. When the angle of the mirrors is 600, five images are produced, and seven when it is 45°. The number of images continues to increase in proportion as the angle diminishes, and when it is zero, that is, when the mirrors are par-rallel, the number of images is theoretically infinite. (See Kaleidoscope.) The amount of light reflected from a surface of given size and polish is different with mirrors of different material; and it increases in all cases with increase of the angle of incidence, though not in all cases regularly. We observe the image of the sun in water near midday without difficulty; but when near the horizon the brightness of the reflected light is usually intolerable.

Remembering that the surface impinged on by any single ray of light is extremely small, it will be seen that any curved reflector is in effect simply a collection of a great number of such minute planes; and that, if we consider the rays falling on such a surface as reflected from the same points in as many different planes tangent to the surface at the points of incidence, we at once extend the law for plane surfaces to all curved surfaces whatever. To the points of incidence of rays on any curved surface, K A B, fig. 5, let fall lines C K, C I, C A, etc, perpendicular (normal) to the surface at those points; each reflected ray will be in the plane containing its incident ray and its proper normal; and the angles of reflection, C K l, C I l, etc, and of incidence, L K C, L I 0, etc, will be equal for each ray on the two sides of its normal. Ordinary concave and convex mirrors are parts of spherical surfaces. The former must reflect parallel rays convergent, convergent rays more rapidly so, etc. The latter must reflect parallel rays divergent, divergent rays more so, etc.

Parallel rays falling on a concave mirror are reflected to a focus distant from the surface half the radius of curvature of such surface, i. e., at one fourth the diameter of the sphere, as shown in fig. 6, where C D being the normal at the point of incidence D, the angle of reflection C D F is equal to the angle of incidence G D C, and is in the same plane. It follows from this that the point F, where the reflected ray cuts the principal axis, divides the radius of curvature A C very nearly into two equal parts. For in the triangle D F C, the angle D C F is equal to the angle C D G, since they are alternate and opposite angles; likewise the angle C D F is equal to the angle C D G, from the laws of reflection; therefore the angle F D C is equal to the angle F C D, and the sides F C and F D are equal as being opposite to equal angles. The smaller the arc A I), the more nearly does D F equal A F; and when the arc is only a small number of degrees, the right lines A F and F C may be taken as approximately equal, and the point F may be taken as the middle of A C. So long as the aperture of the mirror does not exceed 8° or 10°, any other ray, H B, will after reflection pass very nearly through the point F. Hence, when a pencil of rays parallel to the axis falls on a concave mirror, the rays intersect after reflection in the same point, which is at an equal distance from the centre of curvature and from the mirror.

This point is called the principal focus of the mirror, and the distance A F is the principal focal distance. If the angle of aperture of the mirror exceeds 10°, not all of the reflected rays will meet in one and the same focal point, but, by reason of the various angles of incidence made by the incident rays on the curved surface, the further the point of incidence of a ray is from the centre M of the mirror A M B, fig.

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Fig. 1.

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Fig. 2.

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Fig. 3.

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Fig. 4.

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Fig. 5.

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Fig 6.

7, the nearer to that centre will the ray be reflected; but incident rays included in an angle of aperture of 10° will approximately be reflected to one focus F. Fig. 7 is an accurate representation of the paths of the reflected ray of an incident beam of parallel rays. M is the centre of figure of the spherical mirror A M B, C is the centre of curvature, and F is the focus. This departure from a true focus of rays reflected from spherical mirrors is called "spherical aberration." The curved line A L F formed by the intersections of the reflected rays is called a "caustic." This caustic can be easily seen by placing a piece of paper in the same plane with the axis of the mirror, or by observing the reflection from a curved polished clock spring placed on a piece of white paper in the sunshine. Spherical aberration can only be avoided by using mirrors of small angles of aperture, or by the use of mirrors having paraboloid surfaces, as shown in section in fig. 8. It is a well known property of the parabola that a normal bisects the angle made by a diameter at the point of contact, with the line drawn from that point to the focus; hence all rays, R M, O B, in fig. 8, parallel to the principal axis A X, will be reflected to one point F, the focus of the mirror; and conversely, if F be a luminous point, all rays emanating from it which fall on the mirror will proceed outward in parallel lines.

This last mentioned property of paraboloid mirrors is applied in their use as reflectors on locomotives, and in lighthouses. But a large fraction of the rays emanating from the light at the focus of the paraboloid do not strike the mirror, and therefore diverge and are not useful in illuminating distant objects. To render these diverging rays parallel to the axis of the mirror, Thomas Stevenson, engineer of the English board of northern lighthouses, devised in 1834 the ingenious plan of placing a lens, L, fig. 9, before the mirror, to intercept the cone of rays, MfN, which is usually lost by divergence. Opposite this lens is a portion of the mirror, a b, which is not paraboloidal but spherical, and the principal focus of the spherical mirror and of the lens is at f. By this simple device the cones of rays af b and M f N are brought into a beam of parallel rays, R S, which proceed in the same direction as the rays reflected by the paraboloid. Thus all of the rays are available, and from this property of these instruments they have been termed holophotal reflectors (Gr. bliog, entire, and, light). The object before a common concave mirror being anywhere without the centre of curvature, the image is between such centre and the focus, inverted, real, and reduced in size; and the places of object and image are interchangeable - the foci are "conjugate," i. e., mutual.

When the object is brought within the principal focus, the image is erect, virtual (behind the mirror), and magnified. The image with convex mirrors is always virtual, diminished, nearer the mirror than the object, and erect.

II. Diopteics

When a ray or a minute beam of light passes through any surface of division, separating vacuum from any medium, or any one medium from another of different density, a portion of the light is reflected at such surface, and another portion, never the whole, is transmitted. This transmitted light is always bent out of its course at the surface of division, never within the medium, if this be homogeneous; and the light is then said to be refracted. If the medium be one of varying density, like the atmosphere, the ray is bent continually within it; but this case is equivalent to its passing through a succession of surfaces, dividing media more and more or less and less dense. Suppose a ray or minute beam of light transmitted at a point through a plane dividing surface, M N, fig. 10, between space and a medium, or any two media, and coming to such point in any direction whatever; let fall to this point of transmission, O, a perpendicular to the surface, O P, and passing through it, so as to lie in both the media; then, first, it is universally true that the ray, after refraction, will be situated in the same plane in space in which this perpendicular and the line of the ray before refraction are situated.

Thus we may always determine within what plane, vertical to the refracting surface, to look for the ray after refraction. The angle I O P, included between the perpendicular line and the ray before refraction, is termed the angle of incidence, and may be represented by I; that between the same perpendicular on the other side of the surface and the line of the ray after refraction, R O D, is the angle of refraction, R. These angles, the media being of different density, are never equal; nor have the angles themselves any direct ratio to each other. But if in the course of the ray before, and also after refraction, equal radii, O B, O R, measured from the point where the ray penetrates the surface, be taken, and from the extremities of these radii perpendiculars, B A and R S, be let fall on the perpendicular line already drawn, these latter perpendiculars, B A (or I' S') and R S, will be the sines of the angles in which they are respectively, i. e., the sines of the angles of incidence and refraction. For any two given media, no matter what the angle of incidence, the corresponding angle of refraction is such that the ratio of the sines is always the same - is a constant value.

Thus, the fundamental and universal law of refraction at plane surfaces is also simple, though the conditions to be kept in view are much more complex than in the case of reflection ; it is this : The paths of the ray before and after refraction always lie in the same plane with the perpendicular to the refracting surface drawn to the point of transmission, and on opposite sides of that perpendicular; and in that plane the sines of the angle of incidence and of refraction have in all cases the same ratio for any two given media. This is " Snells law;" and it also is rigidly verified by measurements. Suppose the refraction be that of a ray passing from air into ordinary crown glass; then, for all angles of incidence, the ratio Sine I / sine R = 3/2, very nearly. The anigle of incidence is the greater, and the refraction is therefore toward the perpendicular. This is the case whenever the ray passes from a less to a more dense medium. And as, in all such cases, we have sine I / Sine R > 1, this fact of a ratio greater than unity expresses a refraction toward the perpendicular. The value which the ratio sine I / sineR may have, being constant for any two media, is called for such media the " index" or " co-efficient of refraction," c.

From air to water, c=4/3; from air to diamond, c-=5/2; from water to crown glass, 9/8; from crown glass to diamond, 5/3. When light passes successively from air through water, crown glass, and diamond, these refractions are not added; but the ray has in any one of the media precisely the course it would have had if passed from vacuum or from air directly into the given medium. Tims, in the case supposed, the successive refractions would be 4/3×9/8×5/3=c=5/2, the same as if the light had passed at once from air to diamond; and so in all cases. When the ray passes, on the other hand, from a denser medium to a rarer, we always find the ratio sine I/sine R=c < 1; and this signifies that the ray is then bent from the perpendicular. Thus, from crown glass to air, c=2/3; from water to air, c=¾; and so on. That is, in all these cases, sine I must be less than sine R, or sine R > sine I. But the angle of incidence may vary from 0° up to 90° ; and the angle of refraction cannot exceed 90°, because this is the whole space between any surface and a perpendicular to it. Hence, for light going toward the rarer medium, there will be a limit of the angle of incidence beyond which no angle of refraction can be found sufficiently large.

Rays meeting the surface at an angle greater than this limit cannot pass tin- surface. There is a mathematical impossibility, and hence a physical; and the light is wholly thrown back into the medium, i. e., totally reflected. Fig. 11 gives a correct view of the paths of the rays proceeding from a radiant point R in the interior of a mass of water whose surface S S' is contiguous to air. R P is the path of the ray which is perpendicular to the surface S S'. The rays which diverge are bent away from the perpendicular when they pass the surface S S' into the air, and their directions are shown by the lines 1, 2, 3, etc.; but when the divergence has become so great that the sine of the angle of refraction (in air) must be greater than the radius in order that the law of the constancy of the ratio of the sines shall hold, the rays do not pass through the surface S S', but are reflected from that surface, as shown by the lines a, b, c. This total reflection is readily observed on looking in certain directions into a prism; its highly transparent surfaces serve as mirrors for objects situated so that their light falls without a certain angle; for crown glass, 41° 48'. Any small transparent body of a density unlike that' of the medium it is in, and bounded by a curved and a plane or by two curved surfaces, is termed a lens.

The combination of spherical surfaces, either with each other or with plane surfaces, gives rise to six kinds of lenses, sections of which are represented in fig. 12; four are formed by two spherical surfaces, and two by a plane and a spherical surface. A is a double convex lens, B is a plano-convex, C is a converging concavo-convex, D is a double concave, E is a plano-concave, and F is a diverging concavo-convex. The lens C is also called the converging meniscus, and the lens F the diverging meniscus. The first three, which are thicker at the centre than at the borders, are converging; the others, which are thinner at the centre than at the borders, are diverging. Lenses are most conveniently made of glass, and with spherical surfaces. As with mirrors, so with lenses, by considering any curved surface as composed of a multitude of minute plane surfaces, Ave at once extend to them the law of refraction; and it is then only necessary to know the angles of incidence and the value of c, in order to trace the course of the rays. The refraction toward a perpendicular at the first surface of a lens will conspire with that from the perpendicular at the second surface, both occurring in the same actual direction in space.

A ray passing through the centres of curvature, C and F, fig. 13, of the surfaces, passes also through the middle point of the lens, and is not refracted. This line M F is the axis of the lens. Rays parallel to this axis are, when the lens is convex, brought to meet in a real focus F lying at some point in the axis; they are made to diverge as from a virtual focus somewhere in this line, whenever the lens is concave. The aperture of a lens is the total arc or number of degrees of curvature of surface on the two sides of the axis through which light is allowed to pass. Hence, it does not depend on size alone; and the minute lens which is merely a bead of glass has almost necessarily a much greater aperture than a lens of some inches or feet focus. The principal focus F of a double convex or double concave lens, of crown glass, of equal curvatures, is at the centre F of the sphere of which the lens surface BND forms part; the focal distance is equal to radius; for the plano-convex and plano-concave, it is equal to twice the radius. The general rule for finding the focal distance is: For the meniscus and concavo-convex lens, divide twice the product of the radii of curvature by their difference; for the double convex and concave, by their sum.

When, for the double convex lens, the object is at any distance greater than twice the radius, on one side, the image is always somewhere between the focus and the other side of the sphere or the distance of twice the radius, on the other; and here, again, the places of object and image are interchangeable; the foci are conjugate. Fig. 14 shows the manner in which the image I of the candle C is funned by the lens L S. Cones of rays, having for their basis the surface of the lens and for their apices every point on the surface of the candle facing the lens, are refracted by the lens to points in the image corresponding to the points in the candle from which the rays emanated. When the object is brought within the principal focus on either side, the image is then on the same side, or virtual, erect, beyond the focal distance, and magnified. So, in the former case, the real image is magnified by bringing the object nearer the focus. The simple act of bringing an object at less than the ordinary distance of distinct vision from the eye, as when we look at small objects close to the eye through a pin hole, increases the visual angle, and so proportionally magnifies them.

Hence it is that, for objects viewed as placed within the principal focus, the magnifying power increases with diminution of focal distance of the lens, being determined conveniently by the quotient of the ordinary limit of vision, say 8 inches, divided by the focal distance of the lens. Thus a lens, focal distance fa of an inch, has a linear magnifying power of 8-1/50=400 times; and of course a superficial magnifying power of 4002=160,000 times. Thus are explained the very high powers obtained by the use of minute spherical lenses in form of beads, of perfect glass. But it is only for a small aperture, say 6° or at most 8°, that the rays are brought rigidly to one focus. Enlarging the aperture, the successive rings lying with-out bring their light to foci successively nearer the lens; passing their foci, these rays diverge, and form an indistinct border of light about the image. This is spherical aberration of lenses. It is to some extent corrected by peculiar forms of lens, hence called aplanatic; the least spherical aberration thus obtained is with a double convex lens, the radii of whose curvatures are as 1:0; this, with the surface whose radius is 1 toward the object, gives an aberration of 1.07 times its own thickness.

III. Dispersion

The dispersion of light is the separation of the colors existing, actually or potentially, in white or solar light. It may occur by refraction, by diffraction, or by interference. (See Color.) The total length of spectrum obtained by prisms, i. e., the total dispersion, and also the amount of spreading out of the different colors, differ with the nature of the medium or prism employed. Calling the refrangibility of the violet ray V, and of the red R', for a given prism, and the coefficient of refraction c, the dispersive power is =V'-R'/c-1 This ratio, for oil of cassia, is 139; for flint glass, 052; Canada balsam, 045; diamond, 038; crown glass, 036; water, 035; rock crystal, 026. Thus, for example, the total dispersion raid length of spectrum for a hollow glass prism filled with oil of cassia, are about four times those of crown glass; and of Hint glass, 1½ time those of crown glass. Now, lenses, like prisms, must disperse or decompose light. The different colors are really brought to foci that, in the case of convex lenses, lie in the following order: The focus of the least refrangible or red ray corresponds with the true place of the principal focus; and the more refrangible rays are brought to foci within this, as the orange, yellow, green, blue, indigo, and violet, lying nearer and nearer to the lens.

These colored rays cross at their foci, and again diverge; the effect is a colored border or fringe, mainly blue or red, as the case may be, surrounding the image, and more marked as the aperture of the lens is greater, and in objects toward the margin of the field of view. This is chromatic aberration of lenses. It is almost perfectly corrected by combining lenses in various ways, thus forming achromatic combinations. The principal of these is usually that of correcting, for example, the less dispersion of crown glass by the greater dispersive power of flint glass. To do this, a concave of flint of less entire curvature is combined with a crown glass, convex, and of the greater entire curvature. The dispersion is corrected; hut part of the refractive or lens effects remains undestroyed, and the focal distance becomes greater. (See Achromatic Lens).

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Fig. 7.

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Fig. 8.

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Fig. 9.

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Fig. 10.

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Fig. 11.

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Fig. 12.

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Fig. 13.

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Fig. 14.