Optical aberration

“ It is impossible to avoid a fringe of color when using only a single lens, but where we have a complete optical system, consisting of two condensers and an objective, it must be possible to adjust the combination so that practically no color is visible. Spherical aberration is best tested for by laying out very accurately, as in Figure 26, a set of small squares upon some material that will not be damaged by the heat of the lamp—mica for instance—and projecting this upon the screen. ”

“ 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. ”

“ By spherical aberration we mean that the light falling upon the centre of a lens is not brought to a focus at exactly the same spot as the marginal rays, and a general want of definition is the result, usually accompanied also by a want of 'flatness' in the image, that is to say the edges of the picture do not focus at the same time as the centre. ”

“ If the image of an object as seen in the microscope appears to be unusually distorted and indistinct toward the edge of the field, and satisfactory definition is limited to a small portion of the center, the cause is often attributed to the spherical aberration of the objective, while really this phenomenon has nothing to do with that optical defect of the objective, if any exists, but is caused by a lack of optical symmetry. ”

“ We have thus given a perfectly precise meaning to the application of the principle of relativity to stellar aberration. In doing so we have excluded everything which does not belong to the purely optical aspect of the problem. But stellar aberration is not a purely optical problem. For in practice we do not actually observe the apparent motion of the Sun and use the result to correct our observed positions of the stars. The motion which we do use is derived by calculation from the theory of gravitation. ”

“ If we suppose the focal length to be 66 ft., a single lens is practically perfect up to an aperture of 1·7 in.
Another obvious inference from the necessary imperfection of optical images is the uselessness of attempting anything like an absolute destruction of spherical aberration. An admissible error of phase of 1/4λ will correspond to an error of 1/8λ in a reflecting and 1/2λ in a (glass) refracting surface, the incidence in both cases being perpendicular. ”

“ 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. ”

“ Optical Power of Instruments.—Our investigations and estimates of resolving power have thus far proceeded upon the supposition that there are no optical imperfections, whether of the nature of a regular aberration or dependent upon irregularities of material and workmanship. In practice there will always be a certain aberration or error of phase, which we may also regard as the deviation of the actual wave-surface from its intended position. ”

“ Chromatic aberration is easily cured by using an achromatic or compound lens made by cementing together two lenses of crown and flint glass respectively, as in Fig. 40.
It will be seen that the flint glass component by itself is a concave lens and therefore neutralises in part, or in whole, the convex crown lens. Flint glass has both greater dispersive power and also greater refractive power than crown glass, but fortunately not to the same degree ”

“ The indistinctness of the image is partly due to the obliquity of the pencils which form parts of the image, and partly to what is termed spherical aberration. The first cause cannot be modified by the optician's skill, and is not important when the field of view is small. Spherical aberration causes those parts of a pencil which fall near the boundary of a convex lens to converge to a nearer (i.e. shorter) focus than those which fall near the centre. ”

“ But inasmuch as the telescope or line of vision is inclined at the angle of dispersion to the direction of the incident ray, ordinary aberration must come in, as it always does when an observer moves athwart his line of vision; and so there will be a spurious or apparent Doppler effect due to common aberration. That is to say a spectrum line will not be seen in its true place, but will appear to be shifted by an amount almost exactly imitative of a real Doppler effect—the imitation being correct up to the second order of aberration magnitude. ”

“ 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. ”

“ It will readily be conceived that the unequal refrangibility of light must give rise to a good deal of confusion in all images produced by refraction. For instance, a pencil of rays diverging from a point on the axis of a convex-lens will be collected, not to one single focus, but to a series of foci lying at different distances from the lens along its axis, and there will consequently be a confusion analogous to that arising from the aberration which we found to be produced by the spherical form of a refracting surface. ”

“ In optics, a spherical aberration is that wandering of the rays of light from the normal path which takes place when they are made to pass through curved lenses, or are reflected from curved mirrors, constituting portions of a sphere, instead of parts of a parabola. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ In most cases, two thin lenses are combined, one of which has just so strong a positive aberration (“under-correction,” vide supra) as the other a negative; the first must be a positive lens and the second a negative lens; the powers, however, may differ, so that the desired effect of the lens is maintained. It is generally an advantage to secure a great refractive effect by several weaker than by one high-power lens. By one, and likewise by several, and even by an infinite number of thin lenses in contact, no more than two axis points can be reproduced without aberration of the third order. ”

“ 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. ”

“ 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 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. ”

“ In general, the aberration is positive or negative, that is, Av is greater or less than the ultimate value, according as m/∆−1/∆‵ is positive or negative.
This is positive, if ∆ be less than (m+1) r, and negative when A is above that value; when ∆= (m+1) r, there is no aberration. See p. 54. Note.
When r is negative, or the surface convex, the aberration is always positive.
99. We will now pass on to the aberration in a lens, (Fig. 96.) ”

“ The spherical aberration of a diamond lens is extremely small, and when compared with that of a glass lens the difference is rendered strikingly apparent. This diminution of error in the diamond arises from the enormous refractive power possessed by this brilliant substance, and the consequent increase of amplification, with very shallow curves. The longitudinal aberration of a plano-convex diamond lens is only 0.955; while that of a glass one of the same figure is 1.166; both numbers being enumerated in terms of their thickness, and their convex surfaces exposed to parallel rays. ”

“ There is a very great difficulty introduced by the fact that a lens made out of a single piece of glass gives an image which is surrounded by fringes of color, so that some device has to be adopted in order to destroy, as far as possible, this enemy of good definition. In the early history of the telescope, this so-called chromatic aberration was considerably reduced by making small object-glasses of very great focal length. ”

“ An uncorrected residuum, rapidly increasing 163 with larger apertures, is then left, and this appears in the image amplified by the total power of the objective, so that with a non-homogeneous medium there is a maximum angular aperture which cannot be surpassed without undergoing a perceptible loss of definition, provided working distance is required. If we abolish the anterior aberration for all colours, by an immersion fluid which is equal to cover-glass in refractive and dispersive power, the difficulty is at once overcome. ”

“ 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. ”

“ Now it is plain that at the maximum state of , if it has one, all the refracted rays on the same same side with will pass through it, and passing from the section to the actual lens, the circle having for its radius will just contain all the rays, so that it will be the circle we seek.
In order to find we must know the extreme aberration let this be called
Since the aberration (longitudinal) varies as the square of the radius of the aperture,
and they are at a maximum together. ”