General Optical Principles
GENERAL OPTICAL PRINCIPLESKATOPTRICS, DIOPTRICS, PHYSIOLOGICAL OPTICS.
By WILLIAM S. DENNETT, M. D., AND COLMAN WARD CUTLER, M. D.
OF NEW YORK CITY.
Light from its source spreads from center to circumference not as the arrow flies, but as the wave passes. The continually repeated cycle at the origin is imitated in all its essentials at each surrounding particle, which, being thus made luminous, transmits in turn what it has received to others next removed.
This is not the place to discuss at length the wave theory of light, but let it he remembered that the image on the retina is the result of purely mechanical processes into which the time element necessarily enters. Whatever the nature of the cycle at the origin, it has to do with a mass of matter controlled by elastic forces; hence its period is constant. The conditions at half cycle periods are such as may be represented by algebraic equals and opposites, compounding into zero if both are impressed on the body at the same time.
The passage of light through space is the transference of motion from one body to another, or to many others whose reactions bring or tend to bring the first to rest, and which are brought to rest in turn by those on whom they act.
The time element in this process of light propagation is also determined strictly in accordance with mechanical laws, and hence the spherical shell of a wave surface is deformed or distorted by any change in the density or structure of the medium through which it passes.
At the outset, in a homogeneous medium, the wave surfaces are spherical, and the light received by any body to which the wave has reached is measured by the area of wave surface which it intercepts. This means that the body is, as it were, a buffer to the moving masses of which the medium is composed.
If the recipient is at an equal distance front two such sources of light whose phases and cycles are similar, it will of course receive twice the light that it would from one. Now, the whole theory of transmission by waves implies that every separate point of a wave front is itself, while the wave is passing, nothing other than an instantaneous source of light, and may be treated as such, and that the results traceable to any one lumi element fig. 57, 1~ are the same as may be obtained by the summation of results due to similar conditions as they exist at some later period in every separate element, a, be c. do etc., along the whole wave surface. Thus it happens that any point, p, equally distant from the points, b and a, receives double the amount of light or energy from both these points that it does from either.
A change in the form of the wave front so that, as at a', d, it curves in a circle about the point p toward which it is advancing, makes that point the recipient of all the energy which was distributed along its are.
Image forming optical instruments are devices by which each light wave that comes from one of a configuration of points, the object, is made to curve around the corresponding one of another configuration of points, the image.
Fig. 58 delineates this process in its simplicity,where a lens, is made of such medium as will delay by its density the progress of the wave, and is so shaped that it will give to the wave front a circular section. The ray, a , indicative of the direction toward which the wave at any point is moving, is identical with the radius of the curved wavesurface at that point, and the radius of a circle measures its straightness of are, just as the reciprocal of the radius measures its curvature.
Thus it will be seen that the study of the propagation and distribution of light is very much, at bottom, the study of curves, and, as curves are determined by the properties of their normals or radii, it is possible for Geometrical Optics to be cultivated as a degenerate form of Physical Optics, dealing principally with the positions of points and the lengths of line segments.
The accessibility of certain truths when sought by geometrical methods, and the accessibility of the methods themselves as instruments of research, are their all sufficient but not their only recommendation. In the pages that follow only occasional reference will be made to the physical aspects of the case, but attention is here invited to the fact that not only as a figure of speech, but in the accurate mathematical sense, rarity is the reciprocal of density, straightness of curvature, and slowness of velocity. From these hints it will be found that the formula used in the study of refracting and reflecting surfaces and centered lens systems give abundant evidence of their physical origin, and recognition of this relationship will be an easy and legitimate mnemonic device.
Thus in Equation 13, page 108, one may read each term as the value in droplets of a lens or a pencil. One recognizes the f's as typical of focal distances, and the r as a radius, but f and f,' are also radii, and their magnitudes measure the flatness of the incident and refracted waves; I/f’ is the curvature of a wave surface, and u’' is the coefficient of slowness for wave travel in the medium thus indexed, while u"', u" is the lag of the wave as it passes from one medium to another; and so on until the whole physical theory is read from the necessary geometrical relations.
Refraction and Reflection. With Snell"s law for a stepping stone we now pass to the geometrical consideration of refraction and reflection. This law for nearly a hundred years was the expression merely of the results of experience in the observation of refracted light. It is now generalized and applied to both reflection and refraction. Its consistency with the wave theory of light may be seen as follows:
When a wave surface whose section may be represented by a b (Fig. 59) passes through d, the surface separating one medium from another in which for any reason whatever light makes its way at a different rate of speed, if the wave surface immediately before its passage is parallel to the surface separating the two media, it will be parallel to it immediately after its passage, because at no time have the circumstances governing its speed differed along the whole line of the wave front, the change having taken place everywhere at the same instant. The length of section is immaterial so that it be straight. Its straightness as a measurable quantity is the are divided by the radius so whatever the curve for a section as small as you please, the above other and statement is practically true, neither end of the wave gains on the other and it continues to advance in a straight line.
If the wave enters a retarding medium whose surface, do, is not parallel to its own, instead of making its way as it otherwise would to the position a", be, the spread of the light disturbance from particle to particle has covered, say, a smaller area in the new medium than in the old, and the limit, of its advance is along the common tangent of the circles whose radii are proportional to the time since they began to form in the new medium. Since V" in the line v' represents the velocity of propagation in the medium,' and n the medium the desired relations are easily established. Each is perpendicular to its wave front and is consequently a radius or ray; a", b, shows the place to which the wave would have advanced bad the character of the medium Dot changed at do, and a", b" shows the place to which it really has advanced during the same interval of time. Each forms the side of a right angled triangle whose hypothenuse is the separating surface, and who respective bases are corresponding sections of wave surface, and form wit the surface of separation the angles i' and i". One of these angles is the angle of incidence; the other is the angle of refraction. Hence the sine of the angle of incidence is to the sine of the angle of refraction as the velocity at incidence is to the velocity after refraction, or, as usually stated,
In practice it is easy to locate the centers from which the waves come and which they go, and easy to locate the center of the optical surface; connecting these centers, p', p", or p"' and n with the point of incidence a (Fig.60) gives us the three radii, each of course perpendicular to the surface to which it belongs, and consequently mutually inclined to each other as are surfaces.
Through the relations of these radii the law was discovered, through them it is most easily proved, and through them it is most frequently stated, angles of incidence, reflection, and refraction being defined as angles made by the incident, reflected, or refracted ray (perpendicular) with the radius of the surface.
The ability to transfer the attention from surfaces to rays, and to replace velocities by their reciprocals, is a great geometrical advantage, though it gives a show of artificiality to the whole theory of optical instruments as far as we have occasion to pursue it.
If u” however accented, is taken to represent 1/v, Equation I may be written sin I’u’ = sin i “ u”. Equation 2 is Snell's law.
As here used, u”,u”, etc. represent the time needed for light to travel unit cc in the medium with which each is connected; they might be called cients of slowness or coefficients of sine magnitude; they are, in fact, called indices of refraction.
The time needed for light to spread unit distance in etber or in air, which is very nearly the same is the standard of measurement, and is assumed to be 1. The actual value in seconds for ether, for air, or for other media is of no special import to us here; we need only the relative magnitudes, which are known or easily obtained, and are represented by u appropriately accented. When u is equal to 1, it is often omitted from a product as a matter of brevity and convenience. In all the formula here used it will be written for the sake of symmetry and clearness.
With this much of physical explanation and the law of sines as the rule the road, we may proceed to speak of rays and foci as of pencils and points, hoping that their true significance will not be forgotten, and believing that the little effort that is necessary to identify physical with geometrical relationship will more than pay for itself as a guard against error and as a mnemonic aid.
We shall use the word refraction in its most general sense, including refraction and reflection. If exceptions to this usage occur, they will be noted.
The first general problem that presents itself in the study of image form Optical instruments is this: Given waves of circular section, what will be their curve in either medium after incidence on the spherical surface which separates it from another of different index?
The problem may be solved by the aid of Fig. 60, A, in which waves at h would converge upon the point p' except that the optical surface changes their curvature, giving them a center at p" In this particular case n, h, p', Ulf are known, and p"' is sought, but the solution desired should enable us to determine the position of any one of the quantities when the others are given, h being the point where the optical surface meets the line connecting its center with that of the incident wave.
At h the incident wave and the optical surface have a common tangent, and there is no change in the direction of the wave or of its radius; consequently, the center of the two waves will be on a line with the center of the optical surface. At any other point of incidence the law of sines applied to the two known radii will indicate the third, and its cross with the axis at pl, will be approximately the center of wave curvature. The solution is as follows:
FIG. 60 Typical cases of refraction and reflection, showing the relative positions as expressed by Snell's law to be the same for rays and normals as for the surfaces to which they belong. At h the waves are parallel to the optical surface; a is any p6int common to optical and wave surfaces )), is the center of the incident wave, p" of the refracted, and p111 of the reflected wave. The values ofr~dii, curvatures. and focal distances are ordinarily considered positive when the centers to which they appertain lie to the right of h; in (A) they are all positive.
For the convenience of a one letter notation draw Fig. 61 identical with Fig. 60, but represent the radius of the refracting surface by r, the distance of any point _p from the center of the refracting surface by g appropriately accented, the distance of any point p from a by e, also appropriately accented, and distances from h by f. Then in Fig. 61 will be seen one triangle whose sides are r, e', and g, and whose vertex measures the angle of incidence, and another triangle whose sides are r, e", and g", and whose vertex is the angle of refraction. The angle between r and y may be called o.
From the well known property of triangles come these two equations:
It should be noticed here that when the point a (Fig. 61) is placed very near to h the pole of the optical surface, e is nearly equal in value to f, and at the limit, when a and h become identical, any e is exactly equal to the corresponding f. The value off at the instant when a and h coincide is the value that gives accurately the curvature of the wave at h. If the wave is circular in section, p" determined for one point on its surface is determined for all. When the refracted wave has not a circular section, it is usual in practice either to shut off that portion of Its surface which departs appreciably from a uniform curve, and assumes that all the rays cross at the limiting position of p", or to name for the focal point that position of p" which is Dearest to the greatest number of rays at once. Some information may be obtained concerning the curve of the wave by substituting for e' and el' in Eq. 8 the value which each pos virtue of its being opposite to the angle 0 in the triangle to which it belongs; thus:
Squaring 7 and substituting the value e from 9,
It is not necessary to ask here the full significance of this formula, but onlv to remark that when g' is equal to (u”/u”) g”, o disappears from the equation, and consequently the refracted wave has a circular section. One such position may be found for pl on either side of r. The distance from any position of p" to the limiting position when a h ~ 0 is the aberration for the angle o (longitudinal spherical aberration), and there is no aberration for such values of 9' or g" as cause 0' to disappear.
As will be readily appreciated, any irregularity in the curvature of the refracted wave interferes with the point to point correspondence of the image to its object. The optical surfaces of most instruments are spherical, and many circumstances conspire to limit our use of these surfaces to that part which is so near the axis as to be practically without aberration, or to have only so much aberration as may be ignored or eliminated by compensatory errors; so in all first approximations p" in its limiting position is taken as the focus conjugate to p' ; and since the e's and the f's are in this position identical, Eq. 7 may be written thus:
Designating these segments by their terminal points, as in Fig. 60, the nature of the relation sought becomes apparent:
In (h n pp") we have an anharmonic range in which the two foci are con to the center and the pole of the optical surface, and the cross ratio is the ratio of wave velocity in the two media. It is worth while to study into this a little if necessary, for, besides furnishing the easiest possible method of remembering the relations of the foci to their surface, it shows that the relations are reciprocal, and that the two foci, being given a surface of any curve, may be placed, or a curve corresponding to any place may be determined in precisely the same way.
Any combination of lenses and mirrors may be replaced by an equivalent surface : this is of' very general utility, and, moreover, in the theory of thin pencils the circle of least confusion is located between the first and second focus of the pencil by the harmonic variety of this relation, the ratio being, as in the case of the mirror, equal to 1. (See p. 127.)
Again, when g in Eq. 11 is replaced by its equal (f r), we have the following:
which, when reduced, as it easily can be, gives the most important formula in this part of the book :
In as brief a treatise on geometrical optics as this must be, Eq. 13 may be consider an epitome of all that has gone before and a key to all that follows. It should be co mitted to memory and associated with Fig. 60, A, until each is a " word sign " for the other. It should never be written in any other form until it has become so familiar the eye that from any side an error of transcription would be discovered at a glance It is general in its application for the focal distances of axial pencils for a surface any circular curvature, plus or minus, between any media of whatever index. It might just as well have been deduced from any of the special cases pictured in Fig. and the preceding applies and may be read equally well in connection with any one assume of these cases. p" is used in this' figure to indicate the position which P" a a when IL""= IL"; that is, in all cases of reflection. Fig. 60, A, was chosen as the t by which all may be classed and remembered, because in it all the curvatures, all the focal distances, and other magnitudes are positive quantities; and if Eq. 13 is remem bered as belonging to the case where all the quantities are plus, no confusion need arise in interpreting apparent anomalies of sign when a numerical equation of this form presents itself.
The discussion of Eq. 13 is much more simple than its derivation. If the optical surface is a plane, r becomes infinite and the last member vanishes, and consequently or te, which must be construed to mean that the conjugate foci of a plane refracting surface are on the same side of the surface and at distances whose ratio is the same as the indices for the two media. It' any value represented by f ', f ", or r has a minus sign, it of course represents a distance to the left of h. If f7 or.f " represents an infinite value, the inference is that the wave surface is perfectly flat, that the rays are parallel.
Only in one case can y' and y"' be replaced by quantities having different signs. Thaty"should equal ,u,,' would indicate a position of the wave that physical conditions can only account for by the supposition that it is a reflected wave that is, turned back into the medium. whence it came and consequently travelling with the same velocity as before. Therefore the numerical value of y"I must be the same as pl. And it can be stated in this connection that when the indices differ in sign their numerical values do not differ, and u 1. This only happens in cases of' reflection.
It is not only unnecessary, but it is confusing, to make any distinction between problems of reflection and refraction other than what is indicated by the signs of the refractive indices.
The simplicit and generality of the conditions is such that the laws, the methods, the formula and their interpretations are the same for katoptrics as for dioptries.
Katoptries is that part of the science of optics that deals with the phenomena of reflection, especially from regular surfaces like mirrors.
Dioptries treats of the phenomena of refraction, and with the definitions we dismiss the distinction, except in such degree as it is shown by the signs of the indices. Eq. 13 is the open sesame to all of Optics that we require. When the quantities that are represented by and It" are of unlike sign, they are equal and we are dealing with reflection. All other cases are refractive.
The inverse situation is covered by the rule which tells us to treat all mirrors as optical surfaces between media whose indices are 1 and 1.
Cardinal Points four in number, may be named in' connection with a !single optical surface (Fig. 62). They are n, the center of the surface, h, the principal point, f' the first principal focus, and f" the second principal focus.
The Center. Since concentric circles are parallel, the wave whose center of curvature before incidence is it will have it for a center after incidence the ray that passes through n is unrefracted.
It will be seen hereafter that the relative size of object and image is the ratio of respective distances from n; that they approach n together; that each is inverted in ing through n; and that when they meet at ,if the size of one, in terms of the I is numerically equal to the ratio of the velocities of the light waves by which the respective images are formed. It will be seen also that the center n is to the optical surface what the two nodal points n' and n" are to the lens or the optical system.
The principal point h is the point where the optical surface is pierced the line connecting its center with the radiant.
Object and image approach h together. At h they are equal and congruent (see page 112), and to h of the optical surface correspond the two principal points, h" a h"', of the system.
The principal foci, F' and F"', are the same for the surface as for the system.
The first principal focus, F', is the center of those waves which after incidence become plane. In other words, F' is the cross of rays that are made parallel by incidence on the optical surface.
The second principal focus is the center of those waves that before incidence on the surface were parallel ; or it may be stated thus : Rays previously parallel cross after incidence at the last principal focus.
These foci are found by giving to the variables of Eq. 13 such values a will impose the required conditions.
To find f" substitute for f I in Eq. 13 and solve for f. This because the center of a plane wave or the focus of a parallel pencil is at infin¬ity. If f”= , I/f” = 0, and so disappears from the expression, and we have f’=- u’r/u”-u’=F’, (14) the necessary result of the condition imposed.
The second principal focus, F", is found in the same way, for when F’=oo, 1/f’=0, and f / = "Ilr . = Ill. if a To apply this, suppose light from air is incident on a convex glass surface whose radius is one fifth meter (.20 a). Replacing by 1, the index for air, by 1.54, the index for glass, and r by .20, Eq. 15 gives F11 = 1.54 x.20 . .308 . 57. 1.54 1 .54 If the surface had been concave, as in Fig. 60 (0), r would have been equal to . and F"" would have had the same value, with a contrary sign to indicate that it was the left of A. If the surface is to be a mirror, the same equations are used, and u is put equal to thus from Eq. 14: Fly ,ul/.20 _y".20~.20=.10.
21U// 2 For F" one obtains the same result, showing that the principal focus for either A of a reflecting surface is halfway between the center and the surface.
When F1 and F", the principal foci, are known, a very simple formula may be obtained for placing the conjugate of any other given focal point thus, multiplying Eq. 13 by r and then dividing each numerator by It" it becomes P11'r F/r P f Replacing each numerator by the values obtained from Eqs. 14 and I we have F11 _ F1 _ 1. P f free from fractions and subtract F"F" from each side :Fllfl+ F1JY1 flll FvF11 ~ FIF11 (FIT f 1) (f FI) FIF11. (I'' f") is u// (Fig. 63) and (to FI) is ul.
III Changing the sign convention so that one accented quantity measures dis the left and two accented quantities are measured distance toward the right,
We get a very convenient symmetrical notation for the relation of conjugate to principal foci: for the relation given in Eq. 16 there is a very simple graphic solution. As the line k (Fig. 64) is turned on the point p whose rectangular coordinates are F' and F, parts cut off from the axes are respectively equal to f and in', due regard being had sense.
Conjugate Images: Object and image are corresponding configurations of points. By this is meant that to each point in one configuration there cornds a point in the other configuration whose relation to it and to some optical surface is that by which in the preceding paragraphs p' has been conted with oil. The path of the light wave being reversible, either config¬may in theory play the part of object to the other as image. Their distances from each other and from the cardinal points of the surface are. ermined by previous considerations. Their relative magnitudes are to be determined.
The magnification of an object by its image is ordinarily of two kinds, longitudinal and transverse. With the longitudinal, which may be obtained, for example, by comparing (Fig. 65) q' s' with q11 s", we will not here concern ourselves. The following is an easy geometrical determination of the transverse dimensions of object and image: Let the line p" q' perpendicular to the axis be represented by J", its conjugate by j", minus because it is the opposite side of the axis, and it is important to distinguish an is on from an upright image. From the point p' let two lines be drawn, one allel to the axis and one through F', the first principal focus, and let be continued till they meet the optical surface. As these lines are rays, their course after meeting the surface is determinate. That parallel to the axis pass through the second principal focus Ell, and that from the first principal focus will be made parallel to the axis. Where these two refracted rays in will be the focus conjugate to p', and p" q" will in this case be j".
The three horizontal lines of the figure are parallel. The two j's are within required limits perpendicular to them, hence the triangles on the I are all similar, and the triangles on the right are all similar; so we have these two equations from a comparison of the sides of similar triangles From these two equations we may learn where an object must be placed in order that object and image may be equal and cosensal. For such a condition it must be equal to 1. This can only be the case in (18), where f and f " are both equal nothing; therefore the only place is at the surface itself, and there object and image meet and are of the same size. To find where object and image are equal in size opposite in sense we put 1. This condition is imposed upon (18), when f 2 F1 and when J'/ = 2 I''
By replacing I'' and I'' by their equals from Equations 14 and 15, and letting f and f' each equal to r, Equation 18 reduces to u’’/u’’ = j’/j’’.This may be construed to mean, that when the two images meet, as they must, in the center of the optical surface, their dimensions are proportional to the velocity of light in the media to which they respectively correspond.
For refraction it will be seen, e. y., that image and object are cosensal, but when,as in reflection, u' ~ ,u 1, and therefore image and object are of opposite sense and equal in size.
In practice the center of a concave mirror may be found by placing a needle in A vicinity and moving it until its point is coincident with the POiDt Of its image. the cross ratio (see page 108) by which the cardinal points of the mirror are connected with the conjugate foci being 1, (nhffl) is an harmonic range, and, any three points being given, the fourth may be determined by the well known formula: 2 1 + 1 n h fh f lh'
The graphic solution is convenient, as it may be done with a pencil and arraign edge only. If three consecutive elements are given, asked, her'' (Fig. 66), connect th three points by straight lines with any other point, a, not in a line with them. Throu any point on the middle line draw two diagonals, as in the figure, and complete the quadrilateral. Its fourth side will cut the axis at n, the point required. If one of the middle points of the range is sought, as It, connect the two contiguous elements with any point, a, as before. Cross the triangle thus formed by any line n c, put in the two
FiG. 66. Graphic construction by which the following questions are answered: Given the surface ofa mirror, what must its curvature be, or where must its center be in order to produce a picture off at i oroff'atf? Given the center, where must the surface be? Given the mirror and the object, where iwill the image be? or the mirror and the image, where must the object be? diagonals, and draw through their intersection the line a h; h is the fourth harmonic sought.
An analogous construction serves for surface, lens, or system. Take three points, c, d, and e (Fig. 67), equally distant from the line a n, and so placed that the distances c d and c e are proportional to the indices of the first, and last media. From a through each of the other points draw a line. The axis of any optical surface may be placed across cil of four lines, so that three of the lines cross it at any three cardinal points. ~tel.lrth point is determined by the cross of the axis a and the fourth line. This dnwing will answer too for all systems whose first and last media are in this ratio.
Before proceeding to show that other systems of more surfaces than one may, if their centers are colinear, be treated much in the same way as single it is necessary to prove Helmholtz's formula connecting the size of each image with the inclination to the axis of any ray common to them all. Let f ' a f 11 (Fig. 68) be the ray between two images. Assuming the figure. to be made up of two right angle triangles,
Substituting in Eq. I I the values obtained from Eqs. 20 and 21, we have the relation sought
Here we begin the study of centered optical systems by calling attention to the fact that the geometrical relations of object and image are such that dis tinction is often unnecessary; that an object and) i images are frequently spoken of as (n + 1) images; and that any image may be considered object or image, at convenience.
The position and size of any image may of course be determined for an number of surfaces by proceeding step by step from the object to the fin image through as many refractions and reflections as are necessary to attain it. This laborious method is avoided by the localization of cardinal points, which fulfil the same function for the system as do those previously described for the single surface.
Of focal points for the system this must be said : They are Measured not from the first and last surface (Fig. 69), but from two principal points the “first " and the 11 last," whose functions are described below, and whose tions and distances from their respective sit surfaces are designated by h' and h" The, "second" principal point, principal focus, principal plane, nodal point, and on, are properly so named for a single surface, but for a system of surfaces to use ordinal adjective thus is sometimes' misleading. We shall use the term last principal point or (In+ 1)th principal point, and soon, giving it the ordinal adjective and the number of primes that corresponds to the medium to which it appertains This is not so In ,in innovation as a conscientious adhesion to the spirit and method of the notation nomenclature in detail. Something is gained if the accents on letters serve to I the phenomena to which their existence is due. The ability to locate other cardinal points a set, in fact, for each medium reached by waves that were parallel at incident on the system may not be of any special importance, but it is of advantage to has characters systematically named and accented. It enables us to read our records a and to locate easily the processes to which the characters refer.
The removal of the origin for the estimation of focal distances accounts for appearance of h" and h"" in the denominators of Eqs. 23 and 24 (infra). The obscurity, any, vanishes when it is remembered that h and h"" as distances are, by convention, counted plus when measured into the system from the first and last surface respectively. and the surface foci, are measured both right or both left, each from its surface, while e the foci for the system, are measured both in the same direction as the f's anti Vs. This is the reason whv in Eq. 23 F' has been replaced by (W h"), while Eq. 24, V" has been replaced by (W11 h111)
We may now proceed to the consideration of three media separated by two surfaces. In this system are three images (Fig. 70),J'J", andj"', each worresponding to light distribution in the similarly accented medium, J" serving as image to j' by the first surface, and as object to the image j"' by the ,econd surface.
The first 1)rincipal focus of the system is the focus conjugate by the F surface (Fig. 71) to the first principal'foctis of the 0 surface. Changing the surface of reference, thus from one surface to another, demands, of course, that (/) be replaced by (6 it), do, being the distance between the two surfaces A' If'), is obtained from Eq. 13 by the following substitution, and, being the only unknown quantity, its value is immediately forthcoming:
Principal Planes. There are definite reasons for replacing the one prin cipal point oil the pole of the single surface by the two points, h' and h' not necessarily on any surface. We may imagine a plane through each cardinal point perpendicular to the axis and designated by the name of point. On the principal plane, which is tangent to and within required limits is coincident with the single surface (Fig. 72), the end points of incident rays are arranged in a configuration that is identical with the beginning points of refracted or reflected rays ; and it will be remembered that conjugate images approach this plane together until their corresponding points are united ea to each and the two images become identical. No such single plane can placed in any system of optical surfaces, but two planes perpendicular to t axis may always be found such that the configuration of end points a' b' h (Fig. 73) of' incident rays on one surface is congruent with a' ', b", an C11+1 the beginning points of' reflected or refracted rays in the last or (it + I)th medium, such also that when the first image moves toward one of these plan and disappears in it, the final image moves also toward the second plane and disappears in it. A little consideration will convince the student that. if J, the middle image of the three index system (Fig. 74), be so placed that it as an object produces two images (one by each surface) equal in size and cosensal, these two images will lie in planes which answer the above description. We shall call this middle image J. From these planes along the axis conjugate foci of' the system are measured.
Whatever transformations take place within the system are comparatively unimpor ant if only we may receive light emergent from one plane apparently unc ange since it., entrance at the other. If also these planes are so related that the a * act a proaches, ,ne as its image approaches the other, until in size and sense alike a a disappears in its plane, then the two principal planes are quite fit to replace the single plane of the Single surface, and Fig. 73, which we use here to illustrate the system, becomes exactly what Fig. 72 would become if pulled apart and separated by the distance between he' and no''.
We now proceed to find the position of this middle image, indicating principal foci as, usual by capital letters, other focal distances by small letters. Of course the distance of the middle image from the F surface will be indicated by f 11, its distance from the (P surface by o Thus it is seen that the middle image will have in the two surfaces conJugates that are equal and cosensal if it divides the middle medium into parts proportional to the principal foci appertaining thereto. If d represents the distance between the surfaces and J the place of the middle image, Jh will be equal to f"=df’’/f’’+o’’. The conjugate focal distance Jhl may be found by substituting this value for f" in Eq. 13.
In like manner Eq. 13 applied to the (P surface will give the value of for hill from that of (/)" ~ d (P11__ Pit + (Pit
It is hardly necessary to repeat that hl and hill used as magnitudes define the distances of the principal points from their surfaces: they are usually considered positive when in the middle medium. It is not uncommon to give to an optical system a symmetrical notation, so that the direction F, Fn h'h" are considered positive when each is measured from its own principal plane away from the other.
Optical Center. It remains for us to determine what point or points, if any, may be found along the axis of the system having properties like those of the centers of single surfaces. There is, generally speaking, Do point through which as through a center light will pass without change of direction. Only in the special case where the centers of the surfaces are coincident can this happen. One may assume, however, that somewhere is a point so situated that light passing through it will be equally and oppositely refracted at the two surfaces. In this case the first and final paths, though not necessarily identical, must be parallel.
The optical center is the name by which this point is known, and to determine its place we make use of Equation 22. By it the linear dimensions of 0 are connected with those of its first and last image; thus, We may drop out the middle term of this equation, and as the condition imposed is that a' is equal to a"', the other tangents also disappear, giving the condition to which we must conform in locating the three points. By usual notation we use q to measure distances from the first center, and y those from the second, and remember that which may be easily proved.
Referring to Fig. 70, where the distribution along the axis of the cardinal points of the two surfaces is shown in its relation to 0 and its two images, have two expressions for the relative size of each pair:
Dropping out the middle terms and multiplying Eq. 30 by u' and Eq.
The two right hand members are equal by Eq. 28. Expressing the equality of the two left hand members after substituting the numerators from and dividing by u", By composition and alternation, Calling 8 the distance between I and p, we find here again, for the optical center as for the middle image, we must divide a distance into parts propor¬tional to the principal foci of the two surfaces, but this time we must use principal foci for rays that are parallel in the middle medium, whereas bee we used the principal foci for rays that were parallel outside the middle medium.
In a three index system the optical center, and in the lens, where the first last media are the same, both middle image and, optical center can be geometrically, as in Fig. 75. The surface ends of any two parallel are connected by a straight line; its cross with the axis is the optical r of the lens.
The image of the optical center in each surface gives the nodal point cor to that surface; it may be found by Eq. 13 as above, remembering that
These two points are called nodal points, and transformations of waves and incident to the passage of light from. one of these points to the other may in many cases be ignored, for we know that what goes into the system if directed to n' will come out unchanged in direction as if from n'''. So where, again, we have, as in the case of the principal points, lost space, and geometrical constructions which give graphic solutions with single surfaces may be used equally well for systems; but every picture thus formed will be broken in two, some of its lines parting at the principal plane, some at the center. The two halves being separated by translation parallel to the axis, there will result a similar construction, except that the surface h takes on a finite thickness equal to h' h"', and the center n instead of being a point is stretched out into a line, reaching from n' to n"', and equal in length to h, h"'.
It is hardly possible in an article as short as this must be to include rigid demonstrations of everything necessary to its usefulness. Little, so fir, has been omitted which was necessary to show both geometrical and functional relations existing between the cardinal points of the optical system and the center, the pole, and the two foci of the single surface.
The student who desires to pursue the matter in the same thorough manner must be referred to Helmholtz for whatever of proof' is omitted from the remaining pages.
By a continuation of the methods used above it can be proved that when the principal points are located for any system of two surfaces, and when the principal foci of the system are measured from these points; that when the nodal points are placed, and g', g''' 9 and g''' be used as above to indicate distances from them; and when, as in Eq. 17, it' and u'' are used to measure distances away from the center, with the principal foci as origins, then not only Eq. 13, but also Eqs. 17 and 18, apply equally well to the system as to the surface, if only allowance be made for the lost space between the principal planes and between nodal points.
This fact is of great practical utility, as it gives no restriction at all in cases where the thickness of the lens is small as compared with its focal length. In most of the cases where spectacles are used the thickness of the glass may be ignored. When we add to this statement the extension which is warranted by fact that not only may surfaces be compounded into systems without change of properties, but these systems still further compounded, the one with the other, it will appear that for every set of surfaces, howevI many in number, an equivalent set of eight points may be determined as follows: The optical center, the middle image, the two principal points, the two nodal points, and the two foci.
The following formula give the places of the cardinal points where three media are concerned. They are applicable to media separated either by surfaces or systems, if only it be remembered to measure d from the last principal point of the first medium to the first principal point of the last medium, and to measure the distance between nodal points of the component systems in like manner.
In Fig. 74, where d is h n and 8 is it v we may let x and x'' represent t sections of d by j and y' and y'' the sections of 3 by 0. (See also Fig. 70.)
The middle image j divides the distance d into x' and x'' :
The optical center divides the distance 8 into y' and y''
From Fig. 75 it is easily proved that 8 and d are similarly divided by 0. We may therefore substitute d for o h for n and n for V in Equations 37, and so obtain the formula for the position of the optical center as measured from the two surfaces.
Principal Points. h! and h''' as linear magnitudes are positive when measured from h and n the extremes of d toward the middle medium:
Principal Foci. F' and F''' are considered positive when each principal comes between its focus and the other principal point:
Nodal points, )I' and no'', are measured inward from the extremities of it: n/ = h1+ F5/// a/ Rd P14)"' F'4)'1
From these last equations, by the substitution of the values of the I's and the O's as obtained by Eggs. 14 and 15, are deduced the simplest expres the cardinal points of any system. They flow fro m the above Without complication or difficulty, and are obtained by the ordinary of elimination. Expressed in terms of u'u it u '''r and p they reduce to vulgar fractions having common denominator. This term; being constant for the system, may calculated once for all, and so is abbreviated to N, there being no physical significance here intended. It is merely an abbreviation borrowed from Helmholtz.
Eqs. 39 to 4.5 may be used without restriction.
these general formula may be much simplified by the imposing of certain condition which often occur in practice. Thus, if the middle medium is very thin, d may be considered equal to o. In that case H is also equal to o, and h, h', J,h '', and all coincide; so the last term in N disappears, and our system is practically described by the two values of f' and f''' The two terms only of their denominators being left, we write in full, as follows:
If both radii are now supposed alike, the middle drops out of the account, and we have a single optical surface between the first and third medium a condition realized in the passage of light through the cornea and aqueous.
A still more important condition that may be imposed on a system of two surfaces is that the first and last media shall have the same index. This gives the lens proper.
Lens. It would seem the part of wisdom to confine the term "lens to such combinations, and to use the word 11 system " for others. In this way a distinction is made which is in keeping with the derivation of the word and with ordinary mechanical constructions, and which is continually in evidence through the simplicity of the resulting formula , while a lens that is used as a window between two different media is such only in name, and the name so used is definitive only of a triviality. We shall use the word " lens " only for two index systems. The crystalline lens of the eye is not excluded from this category, is the aqueous and vitreous are of the same refractive power.
Eq. 41 to 45 by letting It"' = IV, we have the formula characteristic of lenses :
Fig. 77, illustrative of the preceding paragraphs, show , the disposition along the axis of the cardinal points of several optical systems. a is a single optical surface, and to it corresponds the aphakic eye and the schematic eye of Listing. b is the general case of two surfaces separating three media, all of different indices. In this the nodal points and the principal points are not identical. c is a true lens as described above, in form resembling the crystalline. In it, as in d ef g, other lenses, principal points ' and nodal points coincide, and it maybe noted that, assuming u"> u' and d less than r+p, positive lenses are thicker in the middle.
Double convex and double concave lenses have their principal points between the curved surfaces. In plano convex and plano concave lenses h', ' h''' and J all come together on the curved surface. In the meniscus they pass out of the substance of the lens and arrange themselves in the medium farthest from the centers of curvature. j corresponds to the human eye, it to the eye with a spectacle lens before it.
The continuity of a series of systems is seen by looking, for example, at system b, and noting that the point f''' in the relevant formulae is such a function of It"' and o that one may be increased as the other is decreased without altering the place of f''' ; so that wherever in a system of three media f''' happens to be placed by making the compensatory changes in o and It"', u''' may be brought to be equal to It" without altering the places of the principal foci. In this way, without changing the disposition of the foci, h''' may be varied until it is equal to r in which case h' will be equal to nothing. In other words, the single surface may be treated as a system in which the third index of refraction is equal to the second, and whose second surface has an infinite curvature, and whose center an(] surface are both coincident with the center of the first surface. Such a substitution of values may always be made in the use of Eqs. 47 to 50, where one of the component systems is a Single surface.
The Diopter. Consistent with any scheme that measures the direction of light propagation as positive, the curvature of the wave is considered positive when its center is in front of it, for its radius must be then positive, and so, whether mirror, refracting surface, or system, its strength as an optical factor is estimated by the curve of the wave, the con¬vergence of the rays that may be produced by it. The unit which is now universally and almost exclusively used in the estimation of the strength of lenses is the diopter, suggested by Nagel and named by Monoyer. It is to the credit of ophthalmologist that in their optical work inches are being fast forgotten. Lenses are thus described by giving to each the reciprocal of its focal length in meters, and placing before this number the sign + or to denote whether it has a real or virtual focus for parallel rays. The convenience this method is its chief recommendation, as combinations of lenses are subject to computation by simple addition in an all but universal standard of measurement, instead of requiring pencil and paper computation oils in terms that are none too rapidly becoming archaic.
The focal length of a lens whose dioptric number is given is of course the reciprocal of that number in meters, or one hundred times that reciprocal in centimeters.
In comparing the two systems it may be said of one that it designates the lenses by their focal lenghts in inches, the other gives to a lens its additive value in droplets. To reduce accurately from either system to the other, one divides 39.37 by the number of the lens. A sufficient approx¬imation for all test case examples is to use 40 as the dividend. Thus a glass of 8 ineb focus is equal to 5 droplets. A three diopter lens has a focaI distance of one third of a meter that is, 33 c.m.-or, if its old number in inches is desired, divide 40 by 3. It is approximately No. 13' accurately it is 13.123, unless the method of calculation has proved superior to the method of its original manufacture and measurement, which for ordinary spectacle lenses is quite likely to be the case.
For all thin lenses the distance between the principal planes may be ignored, and the equa¬tions that have been used for surfaces may be used without restriction; and in their use they admit of such simplification as comes from putting a/ = a/// ~ 9/ ~ 9///. There are but three cardinal points to such a lens. The middle image, the optical center, the two nodal points, and the two principal points are all united in a single point halfway between the two principal foci.
The strength or power of a lens is the convergence that it can produce in parallel rays. It is also the curvature it can give to a plane wave that passes through it; it is also the reciprocal of its focal distance. Either one of these definitions implies the other. Whichever way it is defined, 1/t is its measure. This definition must be modified for a single surface or a system other than a lens. The dioptric strength of such a system is consistently considered to be the measure of the curvature in air or vacuum which it will impress on a wave that was flat before reaching it. Some such convention must be adopted, as the convergence produced is greater on the side of the lesser index, though the system is the same (Fig. 78). With this limitation we can evaluate systems as well as lenses in diopters, and the value will be .the index of the last medium divided by the length of the principal focus in that medium. With this convention the dioptric value of a system is the same for light travelling in either direction.
It is hardly necessary to define further the word "focus," or the word 44 conjugate," which has been used so often to signify that two points or two configurations of points are associated as object and image through the agency of some surface or system.
Virtual and Real Images. But the distinction of virtual and real has not been mentioned thus far in relation to foci and images. A focus or image is real when it is a place from which light really emanates or to which it actually attains. It is virtual when the physical conditions that it represents, though having no real existence, are such that they would account for the reactions taking place at some other point if there were no break in the homogeneity of the intervening medium.
Thus we see in Fig. 79 light from any point of j" falls on the screen k" as if coming from j,", though no light waves or rays enter the medium behind the reflecting surface.
Again, were the surface a refracting surface, the light would fall on the screen k'' as if coming from j'' j'' the virtual image of j' though none of the waves that are disposed as if coming from j'' are in the medium in which j" is placed. We may say, consistently with the notation of this article, that when the image j finds itself in a medium whose accents are different from its own, the image is virtual. Examples of real images are seen in Figs. 72, 73, and 78. Fig. 80 shows a concave lens with its virtual focus at F’’’.
We take note here that the general forms of the lenses given in Fig. 77 may be described by the following terms: Double convex, double concave, plano convex, plano concave, concavo convex, or convexo concave; the last variety when thinnest on the edges is called a meniscus.
Applying Eqs. 48 to 50 to obtain the characteristic properties of this group, one easily proves that the principal points of the double convex and the double concave variety are between the two surfaces; that in the plano they are both united on the curved surface; that for the coneavo convex type they pass out of the substance of' the lens on the side of the greater curvature. It will be found also that when radii, surfaces, and indices are so arranged that the strength of the lens is negative that is, when the lens has a virtual focus f''' falling on the left in the figure and f' on the right then h' and h''' are also transposed, each being found between the other and its own principal focus. With one exception the lenses that are thickest in the middle are of positive focal length, and all positive lenses whose index is greater than that of the surrounding medium are thicker in the middle than at the edges. The one exception of a minus lens that is thinner at the edges occurs when r is greater than p when d is greater than the distance between the centers, and when u'' (p r) is algebraically less than (It" u')d. Equation 49 will under such conditions give a minus value for f'''.
The human eye, as has been said ' is a centered system of optical stirfaces like that given in Fig. 77 (j). We copy here from Czapski's table of dimensions and constants, given for reference in his book on optical instruments, where figures collected from various sources by Helmholtz furnish what might be called a composite reproduction of the type and where also are tabulated the results of carefill measurements and calculations in a single case by Tscherning. Along the vertical line of Fig. 81 are the cardinal points and other points of interest as arranged on the axis of file eye. Between cornea and retina the spaces are correctly given on an enlarged scale of 2.5 to 1. All distances are in meters, so that when applied to use in the above formulae the strength of a lens or system will be expressed in the diopter, the familiar unit of the testcase.
The cases in which practice suggests or renders useful the application of the above formula are not infrequent. We mention only two: One a case of axial myopia in which a supposition that the dioptric system of the eye has remained the same, but the retina has been displaced backward an amount which is easily calculated from the strength of the glass needed to give distinct distant vision. Suppose the size of the retinal image is required for the corrected eye. the correcting lens is usually made as thin as possible; hence its optical center and all the cardinal points except the two foci are at its geometrical center. f'' is minus, and measured along the axis, f' is plus; d is the distance of the correcting lens from the cornea added to 0.0017532, the distance of the cornea in front of h' of the eye. Both foci of the emmetropic eye may be obtained from the table, and thus the figures are all obtainable for getting principal points and nodal points for the complete system through the application of formulae 36 to 41.
Another interesting case occurs where the lens has been removed and a strong plus glass is worn. The nodal points of the glass may be calculated without difficulty, or, if used for reading, a plano convex, with the flat side in front, will be acceptable to the patient, and its nodal points are on the convex surface.
The surface of the cornea is the principal point of the eye, and its, curvature read from the ophthalmometer locates its center, which is the nodal point for the aphakic eye, or this center may be assumed to be like the average and supplied from Fig. 81.
It can be hardly thought necessary to guide the student farther, as lie has now all the points of the component systems which are required to give the cardinal points of the equivalent or resultant system, and these being found, the magnification is forthcoming by Eq. 18 or Eq. 20.
Astigmatic Surfaces and Pencils. We pass now to a very brief consideration of astigmatic surfaces and pencils. We have thus far assumed that the optical surfaces were spherical that is to say, surfaces of revolution about their common axis, and whose principal sections were circular.
It happens that such is not always the case. Imperfections of the cornea or lens give for the surfaces of the eye itself imperfect approaches to sphericity ; and even if that were not so, a displacement of any center or radiant focus from the axis of the system produces the same change in the transmitted or reflected pencil that would result from imperfect curvature of the surface.
For the small pencils with which we deal there is only one form of astigmatism. It is that which would be given to a pencil of' light by the optical action of a toric surface. A sphere is the surface developed by die revolution of a circle about one of its diameters. A torus is developed by the revolution of a circle about any line that is in the same plane, but not a diameter. Roughly speaking, when the axial line is a cord the torus is shaped like an apple with a dimple in its blossom end equal to that in its stem end. When the axial line is not a cord, the torus is like an anchor ring. When the line is at an infinite distance from the circle the toric surface is a cylinder.
The toric lenses in use are supposed to be such as might be sliced from a toric surface by a plane parallel to its axis of development. Such a lens is centered optic~lly when both its centers, the center of the circle and the center about which in its development the circle revolves, are on the axis of' the system.
It will need but little consideration to convince the reader that in two different sections of such a surface the problems relative to the transmission of light will be exactly similar to those which we have just considered as true for any plane whatever of the spherical surface.
plane section of the toric surface may be taken perpendicular to the circumference of the developing circle, or coincident with that circumference, and in either case it will be a circular section. In one case it will be the section of least, and in the other the section of greatest, curvature, with foci correspondingly shorter and longer than in other sections; and in each case may the optical conditions be described and determined by the same laws and for as those previously considered for a spherical surface,, which is a surface of circular section.
The section of the toric surface through its two centers, both of which we suppose to be on the axis of our optical system, may take place through a meridian not coincident with the section of greatest or least curvature, and then consecutive rays from any axial point will not be reunited by this surface on the axis, but near it. The result is that an axial pencil directly incident on such a surface has the characteri sties that are portrayed by Fig. 82, showing the general form of the pencil from Aubert, and the distribution of its component rays as in a diagram by Edward Jackson from. Norris and Oliver's System of Diseases of the Eye.
The point along the axis that can be most satisfactorily utilized as a focal point is at F in the figure. It is the place where the rays are collected into the smallest bundle. It is called "the circle of least confusion," and its place between F1 and I', divides that distance in such ratio that it is a fourth harmonic to F, and F,. Consequently, it is determined by the same formula and constructions that are used to locate the conjugate foci in a spherical mirror (see pp. 108 and 113).
Astigmatism is usually an anomaly and not a desideratum. It is measared and discussed in terms of the diopter, which have proved equally useful whether applied to pencils or lenses.
The amount of astigmatism is the strength of a lens which under ordinary optical conditions would change the convergence of the meridian of least to that of greatest curvature, or vice versa. The correcting lens must be essenlly a toric, and one also whose focal anomaly is exactly equal and opposite that of the pencil to be corrected. For simplicity the cylinder is usually ksen, and, having only one finite focus, it is designated by the dioptric value the correction required.
In correcting the anomalies of refraction and accommodation it is not in general possible to use a simple lens, either cylinder or sphere. One gives cylinder necessary to make either of the extreme foci coincident with the ther. and then adds whatever of spherical correction is required. The par combination of cylinder and sphere that is used is more a matter of commerecial than of physiological interest.
The astigmatism that has been described as produced by a toric lens is the only kind that has been successfully and systematically corrected. It is .. thin pencils " the only kind that exists, and for pencils as large as may r the pupil it is the only kind that merits attention, aberration being so known by its own name as to be considered, if at all, under a separate head.
The classification of astigmatism into, “simple," " compound," " myopic", hyperopic," and so on may have its clinical advantages, but it seems to writer to be of very doubtful propriety. We deal only with one kind of astigmatism. It may have its existence in a myopic eye, a paper weight, or, lass door of a Gothic house, but a nomenclature that takes cognizance of such facts is confusing to the novice unless he clearly understands that the astigmatism and its method of correction is the same in every case.
For those who find it convenient to classify astigmatism by its associated anomalies it may be stated that when the retina of the eye at rest falls behind the posterior focal line, the condition is what is called “compound myopic astigmatism ;" when it falls on the posterior focal line, it is called ''Simple myopic astigmatism;" when it falls between the two focal lines, it is called “mixed astigmatism."
When the retina passes through the first focal line it is called "simple hyperopic astigmatism'' and when in front of both focal lines the anomaly is said to be " compound hyperopic astigmatism " (see also p. 227). This cumbrous and useless attempt at precision, as it is usually taught, merely serves to conceal the fact that there is a point on the axis between tile first and second focal lines through which the retina must pass to obtain the best image compatible with that particular degree of astigmatism.
The construction for finding this point has been given above. The distribution along the axis of the four letters in Fig. 82 is h f2 : h f1 fof2 : fof1 or, briefly, (h F, F2F1)
1. When the retina of the eye at rest passes through this point (fo), the case should be considered simply as one of astigmatism. If the retina passes behind this point, there is myopia as well ; if in front of it, hypermetropia.
The glasses found in most trial cases for the correction of astigmatism are cylinders in pairs, both plus and minus, quarter numbers to 2.50, and half numbers to 6. The spherical lenses are usually in quarter numbers to 2.50, half numbers to 7, whole numbers to 14, and then increasing two diopers at a step to 20 or 9 2, a pair each of both kinds, plus and minus, the cylinders usually plano cylinders, the sphericals double convex or double concave.
Optic Axis; Line of Vision; Line of Fixation; Line of Sight.We have spoken of the eve as a centered system, and such it is in type. Its principal points, its nodal points, its center of motion, as well as the cardinal points of the lens, are usually all on one line or nearly so. This line is called the optic axis. It is approximately the axis of symmetry for the whole organ. It is sometimes the case that the macula, the center of the most acute perceptive power, is directly in this line, but oftener it is not. When tile optic axis passes through the macula it is the line of vision as well, meaning by the line of vision or the line o sight the line on which the object must be placed in order that the visual act should be most advantageously, performed. Under these circumstances also the optic axis is the line of fixation, for it is the line passing through the center of motion and indicative of the eye's position or aim.
An excentric position of the macula lutea is so common as to be the rule rather than the exception. It is usually toward the outer side of the optic axis. Consequently, the line of vision is no longer coincident with that axis, but crosses it with a slight "fault" at the nodal points, and the line of fixation connecting the center of motion of the eye with the object on which it is trained has now a position which differs from the optic axis almost as much as the line of vision.
The angle 0MA (Fig. 83) is taken as the measure of this lack of symmetry due to the excentricity of the macula. It is called the angle gamma, y. It is reckoned as plus when the optic axis falls outside of the visual axis.
Another peculiarity of construction must be considered in connection with the form and position of the cornea.
It is convenient, and in some measure consistent with existing conditions, to look upon the cornea as ellipsoidal rather than spherical in its contour. Its horizontal section if the curve were completed would occupy a position in the average emmetropic eve something like that pictured in Fig. 84. Here it is seen that the corneal major axis does not coincide either with the visual axis or the optic axis. The lack of symmetry thus pictured is usually measured by the angle which the major axis of the cornea makes with the visual axis. This angle is known as a, the angle alpha, and is reckoned plus when the visual axis pierces the cornea on its nasal side. In high myopia the angle a negative (see also p. 96).
Mirrors. In the eye itself are no plane surfaces, and no surfaces whose function is comparable to that of the mirror; but such surfaces must be considered as being intrinsic parts of many instruments. The mirror like of the dioptric surfaces of the eye is made use of in various methods investigation.
A mirror being only a special case Of single optical surface where u'= u", may be most, satisfactorily discussed in connection with previous studies by o the substitution of u' for p" in the general formulae 13 and 18.
Substituting and reducing, we have 1/f” + 1/f’ = 2/r
As has been previously mentioned, this formula is suggestive of the harmonic relation for which a construction has already been given (Figs. 66 and 67). Whichever side of the surface is used, the principal focus is halfway between center and the surface. It is found from Eq. 13 in the usual way.
It is evident from the formula or from the graphic constructing that image and object are always on the same side of the principal focus; also that they are always separated by the surface or the center, never by both; also, whenever the object, the image that is on the same side of the principal focus as the reflecting surface is a virtual image.
The relation between the size of object and image is precisely the same as for dioptric surface, and may be determined either by Eq. 18 or 21.
We have but one more present application for Eq. 13, and that is for the special case where the surface, either dioptric or katoptric, is plane. In such case r=oo, the second member disappears, and which may be construed as saying that the foci conjugate to a plane optical surface vary as their respective indices. Make this a reflecting surface again by putting u” = ,u’, and we find that foci conjugate to a plane mirror are of equal length and of opposite sense; thus :
1 I f" f' (53) Substituting cc for f' or f" in Eq. 18 we find that r plane surfaces j' = 1, j'' (54) showing that in reflection or refraction the image is equal in size to the object The ambiguity of sign enters the equation on account of the double interpretation which may be given to the expression for infinity.
It must be remembered that the conditions to which these formula have been applied, and to which alone they are considered applicable, are such as exist for centered surfaces and pencils of light whose rays make very small angles with each other and with the axis of the system
The Prism. The prism enters a system optically through the decentering of one or more of its surfaces. The prismatic lens in its simplicity differs from the ordinary lens in no other way, and the prismatic element in the lens is measured by the angle between the two lines that contain the cardinal points of the two surfaces. To qualities which the prismatic glass possesses by virtue of its curved surfaces must be added those that are due to the noncoincidence of the two axes, and these are best studied in the case of the plane prism. The action of a prismatic lens as used in ophthalmology is the added action of the simple lens and the plane prism. The plane prism is made up of two plane optical surfaces inclined to each other at an angle less than 180'. The first and third media are usually alike. These conditions cannot be considered analogous to any previously discussed, as on one or both planes the pencil is oblique; neither is it possible to look upon both planes as centered on any finite axis. Consequently, we have to begin again with the law of Snell, and we confine ourselves to refraction in a principal section.
The apex or edge of the prism is the intersection of the two planes forming its sides or faces. A principal section is a section of the prism by a plane perpendicular to the edge. A base apex line is the line of intersection of either with a principal section. From Snell's law we know that a ray of light which before incidence is confined to a plane of principal section will pass through the prism without passing out of that plane. Such plane is pictured . 85, where angles made with the normal to the first surface are designated by 0, those made with the second surface by 0, and where the primes shows in what medium the light ray making the angle is situated.
If R is the refracting angle of the prism and D the total angular deviation caused in any ray passing through the prism, the following relations are easily established:
D + R=0"+u".
D=o+ip R.
Applying Eq. 2 to the angles in question, gives
sin 0/ =u" sin 0",
u=R+D m',
u"=R o",
and u' sin I (R + D) o' sin R
Hence, by easy trigonometry,
sin (R + D) cos o cos (R + D) sin sin R cos 0 1, cos R sin 0"
(55) (56) (57)
(58) (59) (60) (61)
(62)
When the prisms are thin, as in most spectacle lenses, the angles R and .R D may be substituted for their sines, and 1 for their cosines, giving
D R (u" cos 0" u' Cos 01 and this is still further simplified in D = (u" ,u')R, by limiting the angle of incidence to one so small that COS 0/ cos u
(63)
(64)
When the light ray passes symmetrically through the prism, as in Fig.
$6. R may be substituted for 0" and 0", giving 2
D = sin ' sin R/2 R/2
2 (11 2 2
(65)
which is useful, because it expresses the action of the prism on light which passes through in its position of minimum deviation a term which defines itself.
The deviation at position of perpendicular incidence or perpendicular exit is given by A simple transposition of 65 gives the formula for getting the index of refraction from the deviation and refracting angle.
Total Reflection. There is one special condition that comes to our notice generally in connection with reflection and refraction at plane surfaces. We may take as illustration Fig. 87, and ask guidance of Snell's law when the wave whose normal incident from the denser medium (11"), makes with the limiting surface an angle whose sine, multiplied by u"/u" is greater than 1. The path that Snell's law would seem to indicate for the refracted wave would be an impossible path, for there is no angle whose sine is greater than 1. Under such conditions refraction does not take place.
There is no break in the continuity of the phenomena, for when the angle o is so great that u’’ sin I ; then sin 1, and the refracted ray, r' is parallel to the surface. The wave front, in other words, is perpendicular to the optical surface, and neither recedes nor approaches it.
A still greater increase of the angle o" would so increase o' that its general direction would be into, instead of out from, (u") The angle would have a minus sine, but its numerical value could be nothing other than u" since the medium is (u") and this is the relation characteristic of reflection. Under such conditions all the light that is not destroyed is reflected, and the phenomenon is known as total reflection.
The prism is of use in ophthalmology chiefly on account of its causing a deviation in the path of light, and thus furnishing an instrument which may be used either as cause of, or compensation for, slight anomalies of the position of the eye itself. The practical application to such purposes is given elsewhere. In that application it is necessary to take cognizance of its value as used to cause deviation of light, and thus an apparent displacement of any object through it. The relation between the refracting angle and the deviation produced being such, prisms have until recently been described by their refracting angles as Pr. 1', Pr. 2', and so on. By Eq. 65 it will be seen that the deviation produced by any prism of ordinary glass, D = (1.54 1) R, is very nearly one half the refracting angle of the prism; and since one half a degree is about the smallest increment which ophthalmologists have found useful, the scale is a very convenient one, and in spite of criticisms is still much in use. Its only fault is that the numbers on the glasses do not correspond to the values for which they are used. To remedy this defect it has been proposed to number prisms by the angular deviation in degrees, replacing the degree mark by a small d to avoid confusion, thus Pr. 1d, Pr. 2'. This is the Deviation angle System of Jackson. The unit in this system is about double the value of the unit of the Refracting angle System.
To obviate the necessity of making any material change in the size of the working unit, it was proposed to give to each prism the value of its angular deviation in terms of the radian, the only unit of angle that is recognized in works on analysis and mathematical philosophy. One one hundredth of t his, the radian angle, which, in accordance with "C. G. S." (Centimeter(Irramme Second) nomenclature, is a centrad, is so near the unit of the Refracting angle System as to be practically indistinguishable from it. This is the Centrad System of Dennett. The Refracting angle System and the Centrad System so nearly coincide that for glass of any ordinary index some number between 0 and 35 will be identical for the two systems, and the others of the scale will be so near as to admit of interchange under ordinary circumstances. Centrads are prescribed thus: Pr. I'D', Pr. 2 .
The Prism diopter Scale of Prentice does not differ much from the Centrad Scale and does not differ appreciably from it in the numbers that are most used. It gives to every prism the value of the tangent of the deviation in hundredths of the radius. Centrads and the prism diopters are compared in Fig. 88.
The same fault may be found with the Prism diopter Scale as with the Refracting angle Scale namely, the number on the glass is a transcendental function of the value toe which the glass is used. Within the limits of common use the three scales are alike, and the choice is one of symbol and sentiment only. Prism diopters are described thus: Pr. 1, Pr. 2, and so on.
To Prentice is due also the suggested change of the ' to d for the degree deviation, and to z~, for the tangent deviation. The author has extended the symbolism to the centrad system by inverting the triangle for it.
There remains only the Meter angle System, it having been suggested that the “Meter Angle “of Nagel be adopted as a unit for prism nomenclature.
The Meter Angle. The. meter angle is the angle made by the visual and the median plane when the eye is directed to a point in that plane meter's distance from the center of rotation. The value of this angle depends, of course, on the interocular distance, which must needs be conventionalized if it is used for purposes of prism notation. An interocular dis .06 iiakes the meter angle equal to 3V. Though a little narrow for an adult, it is perhaps as good a distance as any to assume. The advantage of this unit is supposed to consist in this, that for any point of fixation con and accommodation are expressed in the same terms, the inclination of the axis to the median line being the same in meter angles as the accom in diopters. The writer is not aware that the meter angle is in actual use as a prism unit. Its relation to convergence may be seen in Fig. 89, and the following notation has been suggested: Pr. 1", Pr. 2".
Accommodation is that fLMCti0n of the eye that makes clear vision possible at varying distances.
This adjustment for all distances between the far point, punctum remotum, and the near point, punctum proximum, is accomplished by the action of the ciliary muscle in changing the form of the lens.
The theory of this process, which has been generally accepted, is that of Helmholtz. The ciliarv muscle may be considered as made up of two partsan outer, formed of longitudinal fiber , which arise at the junction of the cornea and sclera, and pass backward to a diffuse attachment in the outer layers of the choroid, called the tensor choroidea or muscle of Brucke; and an inner portion, formed of fibers which have an approximately circular course, called compressor lentis or Muller's muscle. When the ciliary muscle contracts, the choroid and ciliary processes are drawn forward, and by the contraction of circular fibers the circumference of the ciliary processes is narrowed, the la or suspensory ligament of the lens relaxed, and the lens, being released from the tension which this has exerted on its capsule, tends to assume a more convex shape. This hypothesis has not been seriously disputed until Tscherning, following in the footsteps of Thomas Young, developed a theory which, as it becomes more generally understood, may in part prove a danger rival to that of Helmholtz.
Briefly, Tscherning asserts that the accommodation does not depend on a relaxation of the zonula of Zinn but on its tension through the agency of the ciliary muscle, whereby the peripheral portion of the lens is flattened and the curve of the anterior surface from an approximately spherical approaches a hyperboloid form. The theories of Helmholtz and Tscherning are illustrated by Fig. 90.
FIG. 90. A, accommodation according to Helmholtz. The dotted line represents the thicker form assumed by the lens when the traction of the zonula is diminished by the contraction of the cities muscle Be accommodation according to Tscherning. The unbroken lines show the lens at rest. I invited lines show the change occurring during accommodation, supposed to be due to the traction of the zonula being increased by the contraction of the ciliary muscle. It will be seen that the increased dioptric power of the lens maybe obtained either by relaxation of the zonula or by contraction. Tscherning believes that the changes which he has observed in the lens during accommodation prove that the latter theory is correct, while Hess (Graefe's Arch., xIii. 1, S. 288; Jbid, x1ift. 3, S. 477) opposes it strongly.
As regards the change in the lens itself, Tscherning's view seems abundantly proven by numerous experiments.' The action of the ciliary muscle is still undetermined. The older description, its given above, is supported by the diagrams according to 1wanoff,' but these results have not been corroborated in recent times, although they appear in some of the best text books. Tscherning believes that the inner portion of' the muscle retracts, having its more fixed attachment posteriorly in the choroid, which is steadied by the tension of the vitreous, this being increased during accommodation by the backward traction of the lens. This retraction of the oblique fibers of Muller's muscle, which is probably not as purely a circular muscle as has heretofore been described, makes trac¬tion on the zonula and produces the changes in the lens. The iris as a diaphragm cuts off the peripheral parts of the lens, so that whichever view is taken of the mechan¬ism of accommodation the optical conditions remain practically the same.
By accommodation is meant the muscular effort, the" change in the shape of the lens, and the effect produced on vision. The muscular effort is selfevident. The change in the pupillary portion of the lens is seen from the changes which the reflexes called the images of Purkinje undergo during accommodation. These images are catoptric that is, formed by reflection from the cornea, the anterior and the posterior surfaces of the lens. In the pupillary space pictured in Fig. 91 are seen the reflections of two bright squares, one above another: a is reflected from the surface of the cornea, 6 from anterior surface of lens, c from posterior surface of lens. They are best seen in a dark room when a bright light is thrown on the eye from the side opposite the observer.
During accommodation the reflex of the anterior surface of the lens becomes smaller, which indicates an increase in convexity. In some eyes the image changes its position in a manner to indicate a slight advancement of the surface (Helmholtz), but this is Dot constant (TscherniDg). The posterior surface of the lens becomes slightly more convex, but does not change its position. The pupil contracts during accommodation. According to Tscher the portion of the iris between the pupillary border and the periphery retires a little, corresponding to the flattening, of the peripheral portion of the lens which he has proven takes place. It has been stated that the tension of the anterior chamber diminishes during accommodation. Foerster (1864) observed that in patients with small keratoceles the protrusion diminished or disappeared during accommodation, to reappear when this was relaxed.
When the accommodation is relaxed the eye is adjusted for a far point. When the greatest accommodative effort compatible with clear vision is made, the adjustment is for the near point.
Range of Accommodation. Accommodation is measured by its effect on the vision, and the effect may be described either in terms of distance traversed between the far and near points, as measured from the eye (range of accommodation), or in diopters, expressing the increase of the refractive power of the lens (amplitude or power of accommodation). The additional strength which the lens gains may be considered as a separate lens placed in front of the crystalline.
The focal distance of such a lens being A, the distance of the far point from the eye R, and of the near point P, the range of accommodation would be A = P R, and, as the refractive power of a lens is the inverse of its focal distance, the refractive power of the lens which we assume to represent accommodation would be
A P R
The application of this to emmetropia is
CC = P
the far point being at infinity.
The power of accommodation is measured by the strength of a lens sufficient to give the rays leaving the near point the direction in the vitreous which they would have if without it they came from infinity, or in emmetropia the accommodation is measured by the dioptric value of the near point.
For example, an emmetrope whose near point was at 10 cm. would have 10 diopters accommodation; thus:
1 = 1 1 ~ I= 10 D. A .10 .10
A myope with a far point at 50 cm. (2 diopters of myopia) would have an accommodative ability of 8 diopters; thus:
IOD 2D=8D. ~._10 .50
In hyperopia only convergent rays are focussed on the retina, and the far point is a virtual focus behind the eye. It has therefore a negative value.
We may best not alter the formula, but remember that a negative sign in its last denominator makes that fraction additive, as seen in the following example, where a person whose hypermetropia is 2 D, and whose near point is 10. cm., is shown to have an accommodative power of 12 D:
A .10 .50) = .10 + 50
Practically, the accommodation in hyperopia equals the sum of the lens required to bring vision to infinity with that representing the dioptric value of the near distance. It will be seen from what has preceded that the measurement of the far point is equivalent to the determination of the static refraction of the eye. The near point is the nearest point at which very small type can be seen most distinctly, and is usually measured by Jaeger's test type.
Relative Accommodation. Ordinarily, accommodation and convergence are exerted together, the eyes being directed to the point for which vision is adjusted, but a considerable latitude or independence of these functions in their relations to each other is possible. If, for instance, an emmetrope fixes at a point 33 cm. from the eye, the corresponding accommodation would be 3 10. but a certain amount of relaxation of accommodation and of additional pDwer is possible with the same convergence. This relative accommodation varies for each point of fixation. The normal relations have been tabulated by Donders.'
The practical applications are numerous. A lack of unity between accommodation convergence is seen in the normal eye at the near point. The function of convergence being stronger than that of accommodation, the absolute near point is attained at sacrifice of binocular vision, convergence over acting, and thus reinforcing accommodation. In hyperopia the accommodation required is greater than the convergence, and the same tendency of the two functions to reinforce each other offers a stimulus to latter which may result in convergent strabismus. In myopia less accommodation is required; accordingly there is less incentive to converge, and insufficiency of convergence or even divergence may occur.
Presbyopia. The power of accommodation diminishes progressively from the earliest youth. As a result, the near point recedes from the eye, until at about the age of forty in emmetropia it reaches the distance of 22 cm., and the strength of accommodation has become about 4.5 D. Near vision then is rendered difficult, and from this time on convex glasses must be used to bring the near point nearer and to compensate for the diminishing power of accommodation. The cause of this change is a physiological sclerosis of the crystalline lens, which renders it less elastic in response to the force of the ciliary muscle. The table (Table 11.) and accompanying curve, devised by Donders, shows the decrease in the amplitude of accommodation as well as the change in the static refraction, beginning at about the age of fifty five, by which an acquired hyperopia takes place; the curve, p p, represents the changes in the near point ; the curve r r, the far point.
As has been said, presbyopia begins at the time when near vision becomes difficult. This period varies with the refraction of the eye, for the reason that the strength of the accommodation required to brin the near point to a comfortable distance depends on the position of the Far point. Thus in myopia the far point is nearer the eye than in hyperopia, and the same strength of accommodation will continue the range of useful vision for Dear work at its proper distance later in life ; that is to say, presbyopia is postponed in myopia and anticipated in hyperopia as compared with emmetropia.
It will be seen that a myope of 3 _D will reach the age of sixty without discomfort, while a hyperope of the same degree would be able to overcome his hyperopia and to bring the near point to the reading distance, at the latest, up to twenty ive years.
It is important to remember that the accommodation cannot be sustained at its maximum. There must always be a reserved power, as in any other continuous work, and that is why the near point is said to be at 22 cm., allowing 0.5 D 1 D reserve above the accommodation required for the average reading distance.
The working distance is decidedly arbitrary, depending on the kind of work done or the habit of the individual as regards the distance the work is held from the eyes and on the visual acuity, for if this is diminished, the work must be brought nearer in order to obtain larger images, and the accommodation must be aided accordingly.
Visual Acuity. Vision is measured by the size of the smallest object which can be recognized at a fixed distance in the most favorable light with the best optical adjustment. The size of the object is expressed by the visual angle formed by lines that pass through its extremities, through the nodal points of the eye, to the inverted image on the retina. The size of the image on the retina varies as the distance. of the posterior nodal point from the which distance is greatest in myopia and least in hypermetropia. Axial ametropia is referred to, as that is the commonest form.
When the ametropia is corrected by a lens placed at the anterior focus of eye. the retinal image is the same size as if the eye were emmetropic.
A stronger lens is needed for the correction of myopia the farther the lens is placed from the eye, and a weaker lens suffices for hypermetropia if removed from the eye. Differently stated, this is: a concave lens loses strength and a convex lens gains strength if removed from the eye, which explains the tendency of presbyopes to slide their glasses down the nose as presbyopia increases. It is obvious that to attain the highest visual for a great distance the eye must be placed in a condition to see to the best advantage ; that is to say, the ametropia must be corrected for infinity, consequently the glass that gives the highest visual acuity is the measure of static refraction.
The distance usually chosen for the examination of vision is 6 m. So a distance is taken because it is desirable to measure acuity uninfluenced by the effect of accommodation and rays of light that enter the eye from any on an object 6 m. away, however wide the pupil, are practically parallel and meet on the retina.
Snellen's type are so devised that each letter subtends an angle of five minutes, each part of' a letter and each space being One fifth of the whole in measurement. A visual angle of five minutes has been assumed as representing the average of a great manv measurements of the eyes of individual Of all ages, and Snellen acknowledges that a great many young person have a greater visual acuity.
It has been said above that visual acuity is measured by the ability to recognize an object at a given distance. This means that the parts of which it is composed can be differentiated: each part of one of Snellen's letters subtends an angle of one minute (Fig. 93).
The perception of a single object, however, would be a reliable test of vision, as its visibility would depend on the intensity of the light by which it was seen and would be, in some measure, independent of its size and the distance for instance, a fixed star is visible, although its apparent size is almost infinitely small and its image smaller than one of the perceptive elements of retina. Two stars, however, cannot be distinguished as separate unless they are about sixty seconds apart; that is, unless the distance between their images on the retina equals at least the breadth of a perceptive element. If &stance were smaller, both images would fall upon the same or upon adjacent elements. In the first case both would produce a single sensation, and in the second case there would be two sensations, but upon adjacent elements,so that it could not be told whether there were two points of light or one which fell upon both elements.
the fact that the diameter of the cones in the macula corresponds quite closely to the smallest distances between the images of two objects that can be recognized as two I the conclusion has been drawn that the cones are elements.
Suellen's letters are arranged in lines, over each of which are Roman numerals indicating the distance, D, at which the letters of that line appear under an angle of five minutes or the distance at which they can be read by an eye of normal vision. The distance at which they can be read by the eye that is being tested is d. The formula, then, for visual acuity is V= d . As examinations are ordinarily made at a fixed distance of about six meters, 11 d is constant, the value of the fractional expression being varied with the value of the 11 D " which designates the smallest legible letters, thus V ~ d = 6 is D 6 normal vision. V= 6 indicates that what the patient ought to see at sixty meters he can see at only six meters, all acuity of 0.1. It is best, however, to leave the fraction unreduced, thus recording the exact distance at which the test was made. If vision is inferior to 6 , the test types may be brought 60 nearer, and the distance recorded at which the largest is read, as 3 . If this 60 is not enough, the distance may be noted at which the fingers of the outstretched hand can be counted against a dark background, or, still farther, onl ' v the movements of the hand may be seen, and finally light perception only, at varying distances, or, simply, the differentiation of light from darkness (L. P.) may be all there is to record.
A better system than that of Snellen is one devised by Monoyer, in which the lines progress in tenths from 1. to 0.1. The regularity of the interval is a decided advantage, and has been utilized by Dennett, with the modification that the size of every letter in each line has been so chosen as to ensure its uniform visibility.
For the illiterate, characters may be used which can be described without being named, or Burchardt's series of dots may be used. The most common are the E's in different posi¬tions (Fign 94). Guillery proposed to measure the visual acuity simply by the use of a black dot on a white ground. By comparison with the letters of Suellen he found that such a dot seen at an angle of 50,"' would correspond to the normal visual acuity; at 5 m. it would have a diameter of 1.2 mm. An acuity of one half would be shown by the ability to see a dot of double the area at the same distance. The dots are placed in various parts of squares and are to be localized by the patient.'
Entroptic Phenomena. Objects in the eye in front of the sensitive layer of the retina intercept light that passes through the pupil and throw shadows which under certain conditions can be perceived. Since Listing' the examination of objects in our own eyes has been called entoptic observation.
If a clear sky is looked at through a pin hole in a dark card placed near the anterior focus of the eye the rays thus reaching the retina parallel or if a flame at a distance of 5 m. is seen through a strong convex glass held two or three inches from the eye, a bright disk of light will be seen formed by circles of diffusion, upon which various objects are visible: (1) The traces of the lids on the cornea formed by half closing the eyes. These horizontal Iines remain an instant after the pressure has ceased, and in some cases show a more lasting effect of constriction, leading to an irritable condition called “tarsal asthenopia." ‘The tears and drops of mucus are seen following the movements of the lids. (2) The lens or some of its parts may become visible if a very small opening is used, the light being homocentric. Physiologically, the radiating star shaped figure of the lens and numerous small round objects like hyaline globules may be seen. These increase with age until the senile the beginning of cataract, may also become apparent to the possessor in this manner (Donders). (3) In the vitreous there are always floating bodies, fibers, which as musce votitantes cause alarm to the nervous observer assured of their insignificance. (4) A very interesting application of the entopic method is the observation of the retinal vessels (Purkinje). may be seen in three ways:
(a) in a darkened room a candle is held at a short distance from the eye regards the distance. The vessels come into view as dark lines on a background. They seem to move when the candle is moved.
(b) On looking through a stenopaic opening at the sky, if the opening is motion the vessels a distinctly seen even to the smallest around the macula.
If a strong light is focussed on the sclera as far as possible from the and moved slightly from side to side, the same phenomena occur. planation given by Heinrich Miller (1855) is that the shadow of the vessels falls on the sensitive layer of the retina.
In the last experiment Muller measured the movement of a vessel projected on a at a known distance, and the movement of the focus on the sclera which prothis excursion, and calculated the distance behind the retinal vessel at which the layer must lie, his result coinciding very closely with the actual distance the vessels and the layer of rods and cones.
Konig and Zumft' have recently attempted to apply this principle to the of color vision, and have claimed that different colors are seen at levels, violet being perceived by the most anterior portion of the layer, red by the most posterior. Considerable doubt has been raised, however, by Koster 2 as to the accuracy of these statements.
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