Spherical aberration

“ Spherical aberration is the inability to bring to a focus the rays of light that pass through the lens near the margins at the same time as those that pass through near the center. For these reasons—and, in lesser degree, some others—the eye cannot see sharp lines, and a lens that gives sharp definition to the edges of objects produces results that are esthetically unpleasing, because foreign to our experience. ”

“ This is called chromatic aberration, because the colored rays do not all converge to the same focus; thus the image seen through them is surrounded by a fringe or border of color.
Single lenses lack the power of producing a straight image of a straight object; the image will have the curve of the lens through which the light passes to form it; a double convex lens will give a greater curve than a plano-convex. This is called spherical aberration. ”

“ But if F still approaches the glass, the angle of incidence [Pg 26] continues to increase with the increasing divergence of the ray, till it will exceed that of emergence, which has in the meanwhile been diminishing, and at length the spherical error produced by them will recover its original proportion to the opposite error of the curve of correction. When F has reached this point F ́ ́ (at which the angle of incidence does not exceed that of emergence so much as it had at first come short of it) , the rays again pass the glass free from spherical aberration. ”

“ The Gaussian theory is only an approximation; monochromatic or spherical aberrations still occur, which will be different for different colours; and should they be compensated for one colour, the image of another colour would prove disturbing. The most important is the chromatic difference of aberration of the axis point, which is still present to disturb the image, after par-axial rays of different colours are united by an appropriate combination of glasses. ”

“ The final correction for achromatism is made, if necessary, by departing slightly from a plane in the curvature of the last surface of the flint lens, and the final correction for spherical aberration in the figuring of the surfaces. In a lecture delivered at the Royal Institution on 2d April 1886 Sir Howard Grubb, optician, of Dublin, said:
"A truly spherical curve is the exception, not the rule. ”

“ A lens having a spherical surface bends the rays so that they do not all come to a focus at the same point. What is the effect of this on our theoretically perfect image? Again it is slight blurring of the sharpness of outline. It is said the spherical aberration in a perfectly corrected optician’s lens is less than that in the lens of the human eye. This must be remembered in connection with our later remarks. In the lower animals, spherical aberration is nearly absent. Their vision therefore is more periscopic, and therefore more like that of an optician’s lens. ”

“ Hence, it is evident, that the first image Ff, is formed only by the union of those rays which pass very near the centre of the lens; but as the rays of light proceeding from every point of an object are very numerous, there is a succession of images formed, according to the parts of the lens where they penetrate, which necessarily produces indistinctness and confusion. This is the imperfection which is distinguished by the name of spherical aberration, or the error arising from the spherical form of lenses. ”

“ The specula ground and polished under Lord Rosse's method are almost entirely free from what is called spherical aberration,—that is to say, all rays proceeding from a single point of light, such as a star, are collected into a single point instead of being scattered in a round mass. ”

“ The diameter of a star image increases with the time of exposure. This increase is due in part to the diffraction of the telescope, to imperfect achromatism or spherical aberration of the objective, to irregular grinding of the glass, and especially to variations in the refraction of the air, which produce an oscillation of the image around a mean position. ”

“ A much graver fault is zonal aberration, where some intermediate zone of objective or mirror comes to a focus too long or too short, generally damaging the definition rather seriously, depending on the amount of variation in focus of the faulty zone. A typical case is shown in Fig. 162 taken within focus. Here two zones are abnormally strong showing, just as in the case of simple spherical aberration, too short focus. Outside of focus the intensities would change places, the outer and midway zones and center being heavy, and the strong zones of Fig. ”

“ It takes a rather sharp eye and considerable experience to detect small amounts of spherical aberration; perhaps the best way of judging is in quickly passing from just inside to just outside focus and back again, using a yellow screen and watching very closely for variations in brightness. Truth to tell a small amount of residual aberration, like that of Fig. ”

“ In a single lens having spherical surfaces, aberration may be caused by differences in the focal lengths of the various parts of the lens: rays passing through the outer part of the lens come to a focus nearer the lens than do rays passing through its central part. ”

“ Up to the present we have assumed that all rays falling on a convex lens are brought to a point or focus, but this is not strictly true, for the edges of a lens turn the rays rather too much out of their course, so that they will not come to a point; just as the rays reflected from a spherical mirror do not form a single focus. The marginal rays will be spread over a certain circular surface, just as the colour due to chromatic aberration covered a surface surrounding the focus. ”

“ ABERRATION IN REFLEXION AT SPHERICAL SURFACES.
15.We found in the beginning of the last Chapter, that a cone of rays incident on a spherical surface were not so reflected as to meet in a point, but that the point which we called the focus of the reflected rays was, in fact, the mathematical limit of the intersections with the axis of rays chosen more and more nearly coincident with it. Let us now examine how much the intersection of the axis with a ray at a small but sensible distance from it, differs from this.
Taking the centre of the surface as the point to measure from, (Fig. 12.) ”

“ Hence actual configurations on the sphere are only changed conformally by the effect of aberration, or in other words any small area is altered only in size and not in shape. We know that aberration merely changes the scale of a photograph of a small part of the sky, and the truth of this fact now becomes independent of the velocity (however large) of the observer. Also stars which appear to lie on a circle at any one time will continue to do so permanently. ”

“ In some cases we may fairly assume that λ is constant, but λ as well as φ is originally unknown and, if the principle of relativity be accepted in its widest extent, remains unknowable.
9. The geometrical significance of the new law of aberration is interesting. In the first place we may consider the stereographic projection of the celestial sphere on the tangent plane perpendicular to the direction of motion. The form of the law of aberration
shows that the effect of aberration is simply to alter the scale of the projection. ”

“ The ordinary aberration can be readily detected with modern astronomical instruments, because it goes through a yearly change in direction or amount; but systematic aberration is constant so long as the planetary system moves uniformly in a sensibly straight line. Only then in the course of ages, when the curvature of the sun’s path becomes apparent, can we hope to verify the existence of this kind of aberration. ”

“ The latter equations give
or
which express the apparent wave-velocity in a moving medium in terms of the absolute, or vice versa. For a vacuum we put W = W' = U, and we have
8. The laws of stellar aberration and of the Doppler effect are at once deducible, as Einstein [3] has shown. For a star situated in the direction making an angle φ with the direction of motion of the observer we have a = - cos φ and a' = - cos φ', where φ' is the apparent direction. Then we see that
These expressions can, however, be simplified. ”

“ If we have reason to be dissatisfied with the results of the efforts hitherto made to determine the constant of aberration, we have little ground for taking into account the second-order effects. But on the other hand an optical experiment has been performed which cannot be reconciled with the ordinary theory, and we have been forced to admit that the theory, if not actually erroneous, can be no more than an approximation to the truth. ”

“ The curvature at will have the same effect on the apparent position of the planet as it would have on that of a star in the same direction : as to the curvature at , if we draw perpendicular to produced, the curvature will have the effect of causing to be seen as if it were at . Now the angle between the tangents at and being that through which a star in the direction of is displaced by aberration to an observer at , and the distance being by hypothesis small (two or three radii of the planet suppose) , it follows that the angle is extremely small, and may be neglected. ”

“ If, on the other hand, we suppose the aperture given, we find that aberration begins to be distinctly mischievous when it amounts to about a quarter period, i.e. when the wave-surface deviates at each end by a quarter wave-length from the true plane. ”

“ When an object-glass has its aberrations balanced for viewing an opaque object, and it is required to examine that object by transmitted light, the correction will remain; but if it is necessary to immerse the object in a fluid, or to cover it with glass or talc, an aberration will arise from these circumstances, which will disturb the previous correction, and consequently deteriorate the definition; and this effect will be more obvious with the increase of the distance between the object and the object-glass. ”

“ But it failed to account for the fact proved by experiment that the aberration was unchanged when observations were made with a telescope filled with water. For if the tangent of the angle of aberration is the ratio of the velocity of the earth to the velocity of light, then, since the latter velocity in water is three-fourths its velocity in a vacuum, the aberration observed with a water telescope should be four-thirds of its true value. ”

“ We may simply say that experience with mirrors as nearly plane as those used by us has shown us that the method of observation supposed would suffice for angles of aberration at least twenty-five times that expected if the velocity-ratio is 10,000.
Since the experiment gives a null result, it is not worth the space to prove that what is true of this adjustment is true with sufficient approximation for an adjustment which differs from the assumed adjustment only by the rotation of mirror I by an angle of ten seconds around a perpendicular axis passing through its surface. ”

“ Practical Elimination of Aberrations.—The existence of an optical system, which reproduces absolutely a finite plane on another with pencils of finite aperture, is doubtful; but practical systems solve this problem with an accuracy which mostly suffices for the special purpose of each species of instrument. The problem of finding a system which reproduces a given object upon a given plane with given magnification (in so far as aberrations must be taken into account) could be dealt with by means of the approximation theory; in most cases, however, the analytical difficulties are too great. ”

“ This is, at the present day, the ordinary type, e.g., of telescope objective (fig. 10) ; the values of the four radii must satisfy the equations (2) and (4) . Two other conditions may also be postulated; one is always the elimination of the aberration on the axis; the second either the Herschel” or “Fraunhofer condition,” the latter being the best (vide supra, “Monochromatic Aberation”) . In practice, however, it is often more useful to avoid the second condition by making, the lenses have contact, i.e. equal radii. ”

“ In optical systems composed of lenses, the position, magnitude and errors of the image depend upon the refractive indices of the glass employed (see Lens, and above, “Monochromatic Aberration”) . Since the index of refraction varies with the colour or wave length of the light (see Dispersion) , it follows that a system of lenses (uncorrected) projects images of different colours in somewhat different places and sizes and with different aberrations; i.e. there are “chromatic differences” of the distances of intersection, of magnifications, and of monochromatic aberrations. ”

“ The minor axis, on the other hand, is not constant, but, as we have already seen, depends on the latitude, being the product of the major axis into the sine of the latitude. Assured that his explanation was true, Bradley corrected his observations for aberration, but he found that there still remained a residuum which was evidently not a parallax, for it did not exhibit an annual cycle. ”

“ In whatever part of the sky we observe, every star is affected by aberration. At the poles of the ecliptic, 231⁄2 degrees from the earth's poles, the annual aberration orbits of the stars are very small circles, 41" in diameter. Toward the ecliptic the aberration orbits become more and more oval, ellipses in fact of greater and greater eccentricity, but with their major axes all of the same length, until we reach the ecliptic itself; and then the ellipse is flattened into a straight line 41" in length, in which the star travels forth and back once a year. ”

“ The absence of this error is termed achromatism, and an optical system so corrected is termed achromatic. A system is said to be “chromatically under-corrected” when it shows the same kind of chromatic error as a thin positive lens, otherwise it is said to be “over-corrected.”
If, in the first place, monochromatic aberrations be neglected —in other words, the Gaussian theory be accepted—then every reproduction is determined by the positions of the focal planes, and the magnitude of the focal lengths, or if the focal lengths, as ordinarily happens, be equal, by three constants of reproduction. ”