1911 Encyclopædia Britannica/Diffraction of Light/7

7. Influence of Aberration. 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. In general, we may say that aberration is unimportant when it nowhere (or at any rate over a relatively small area only) exceeds a small fraction of the wave-length (λ). Thus in estimating the intensity at a focal point, where, in the absence of aberration, all the secondary waves would have exactly the same phase, we see that an aberration nowhere exceeding 1/4λ can have but little effect.

The only case in which the influence of small aberration upon the entire image has been calculated (Phil. Mag., 1879) is that of a rectangular aperture, traversed by a cylindrical wave with aberration equal to cx³. The aberration is here unsymmetrical, the wave being in advance of its proper place in one half of the aperture, but behind in the other half. No terms in x or x² need be considered. The first would correspond to a general turning of the beam; and the second would imply imperfect focusing of the central parts. The effect of aberration may be considered in two ways. We may suppose the aperture (a) constant, and inquire into the operation of an increasing aberration; or we may take a given value of c (i.e. a given wave-surface) and examine the effect of a varying aperture. The results in the second case show that an increase of aperture up to that corresponding to an extreme aberration of half a period has no ill effect upon the central band (§ 3), but it increases unduly the intensity of one of the neighbouring lateral bands; and the practical conclusion is that the best results will be obtained from an aperture giving an extreme aberration of from a quarter to half a period, and that with an increased aperture aberration is not so much a direct cause of deterioration as an obstacle to the attainment of that improved definition which should accompany the increase of aperture.

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.

As an application of this result, let us investigate what amount of temperature disturbance in the tube of a telescope may be expected to impair definition. According to J. B. Biot and F. J. D. Arago, the index μ for air at t° C. and at atmospheric pressure is given by

$\mu -1={\tfrac {\cdot 00029}{1+\cdot 0037t}}.$

If we take 0° C. as standard temperature,

δμ＝−1·1 × 10⁻⁶.

Thus, on the supposition that the irregularity of temperature t extends through a length l, and produces an acceleration of a quarter of a wave-length,

1/4λ＝1·1 lt × 10⁻⁶;

or, if we take λ = 5·3 × 10⁻⁵,

lt＝12,

the unit of length being the centimetre.

We may infer that, in the case of a telescope tube 12 cm. long, a stratum of air heated 1° C. lying along the top of the tube, and occupying a moderate fraction of the whole volume, would produce a not insensible effect. If the change of temperature progressed uniformly from one side to the other, the result would be a lateral displacement of the image without loss of definition; but in general both effects would be observable. In longer tubes a similar disturbance would be caused by a proportionally less difference of temperature. S. P. Langley has proposed to obviate such ill-effects by stirring the air included within a telescope tube. It has long been known that the definition of a carbon bisulphide prism may be much improved by a vigorous shaking.

We will now consider the application of the principle to the formation of images, unassisted by reflection or refraction (Phil. Mag., 1881). The function of a lens in forming an image is to compensate by its variable thickness the differences of phase which would otherwise exist between secondary waves arriving at the focal point from various parts of the aperture. If we suppose the diameter of the lens to be given (2R), and its focal length gradually to increase, the original differences of phase at the image of an infinitely distant luminous point diminish without limit. When ƒ attains a certain value, say ƒ₁, the extreme error of phase to be compensated falls to 1/4λ. But, as we have seen, such an error of phase causes no sensible deterioration in the definition; so that from this point onwards the lens is useless, as only improving an image already sensibly as perfect as the aperture admits of. Throughout the operation of increasing the focal length, the resolving power of the instrument, which depends only upon the aperture, remains unchanged; and we thus arrive at the rather startling conclusion that a telescope of any degree of resolving power might be constructed without an object-glass, if only there were no limit to the admissible focal length. This last proviso, however, as we shall see, takes away almost all practical importance from the proposition.

To get an idea of the magnitudes of the quantities involved, let us take the case of an aperture of 1/5 in., about that of the pupil of the eye. The distance ƒ₁, which the actual focal length must exceed, is given by

√ (ƒ₁² + R²) − ƒ₁ ＝1/4λ;

so that

ƒ1 ＝2R2/λ

(1).

Thus, if λ = 1/40000, R = 1/10, we find

ƒ₁＝800 inches.

The image of the sun thrown upon a screen at a distance exceeding 66 ft., through a hole 1/5 in. in diameter, is therefore at least as well defined as that seen direct.

As the minimum focal length increases with the square of the aperture, a quite impracticable distance would be required to rival the resolving power of a modern telescope. Even for an aperture of 4 in., ƒ₁ would have to be 5 miles.

A similar argument may be applied to find at what point an achromatic lens becomes sensibly superior to a single one. The question is whether, when the adjustment of focus is correct for the central rays of the spectrum, the error of phase for the most extreme rays (which it is necessary to consider) amounts to a quarter of a wave-length. If not, the substitution of an achromatic lens will be of no advantage. Calculation shows that, if the aperture be 1/5 in., an achromatic lens has no sensible advantage if the focal length be greater than about 11 in. 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 inquire what is the greatest admissible longitudinal aberration (δƒ) in an object-glass according to the above rule, we find

δƒ＝λα−2

(2)

α being the angular semi-aperture.

In the case of a single lens of glass with the most favourable curvatures, δƒ is about equal to ᲃ, so that α⁴ must not exceed λ/ƒ. For a lens of 3 ft. focus this condition is satisfied if the aperture does not exceed 2 in.

When parallel rays fall directly upon a spherical mirror the longitudinal aberration is only about one-eighth as great as for the most favourably shaped single lens of equal focal length and aperture. Hence a spherical mirror of 3 ft. focus might have an aperture of 21/2 in., and the image would not suffer materially from aberration.

On the same principle we may estimate the least visible displacement of the eye-piece of a telescope focused upon a distant object, a question of interest in connexion with range-finders. It appears (Phil. Mag., 1885, 20, p. 354) that a displacement δƒ from the true focus will not sensibly impair definition, provided

δƒ < ƒ2λ/R2

(3),

2R being the diameter of aperture. The linear accuracy required is thus a function of the ratio of aperture to focal length. The formula agrees well with experiment.

The principle gives an instantaneous solution of the question of the ultimate optical efficiency in the method of “mirror-reading,” as commonly practised in various physical observations. A rotation by which one edge of the mirror advances 1/4λ (while the other edge retreats to a like amount) introduces a phase-discrepancy of a whole period where before the rotation there was complete agreement. A rotation of this amount should therefore be easily visible, but the limits of resolving power are being approached; and the conclusion is independent of the focal length of the mirror, and of the employment of a telescope, provided of course that the reflected image is seen in focus, and that the full width of the mirror is utilized.

Fig. 5.

A comparison with the method of a material pointer, attached to the parts whose rotation is under observation, and viewed through a microscope, is of interest. The limiting efficiency of the microscope is attained when the angular aperture amounts to 180°; and it is evident that a lateral displacement of the point under observation through 1/2λ entails (at the old image) a phase-discrepancy of a whole period, one extreme ray being accelerated and the other retarded by half that amount. We may infer that the limits of efficiency in the two methods are the same when the length of the pointer is equal to the width of the mirror.

We have seen that in perpendicular reflection a surface error not exceeding 1/8λ may be admissible. In the case of oblique reflection at an angle φ, the error of retardation due to an elevation BD (fig. 5) is

QQ′ − QS＝BD sec φ(1 − cos SQQ′)＝BD sec φ (1 + cos 2φ)＝2BD cos φ;

from which it follows that an error of given magnitude in the figure of a surface is less important in oblique than in perpendicular reflection. It must, however, be borne in mind that errors can sometimes be compensated by altering adjustments. If a surface intended to be flat is affected with a slight general curvature, a remedy may be found in an alteration of focus, and the remedy is the less complete as the reflection is more oblique.

The formula expressing the optical power of prismatic spectroscopes may readily be investigated upon the principles of the wave theory. Let A₀B₀ be a plane wave-surface of the light before it falls upon the prisms, AB the corresponding wave-surface for a particular part of the spectrum after the light has passed the prisms, or after it has passed the eye-piece of the observing telescope. The path of a ray from the wave-surface A₀B₀ to A or B is determined by the condition that the optical distance, ∫μ ds, is a minimum; and, as AB is by supposition a wave-surface, this optical distance is the same for both points. Thus

∫μ ds (for A)＝ ∫μ ds (for B)

(4).

We have now to consider the behaviour of light belonging to a neighbouring part of the spectrum. The path of a ray from the wave-surface A₀B₀ to the point A is changed; but in virtue of the minimum property the change may be neglected in calculating the optical distance, as it influences the result by quantities of the second order only in the changes of refrangibility. Accordingly, the optical distance from A₀B₀ to A is represented by ∫(μ + δμ)ds, the integration being along the original path A₀ . . . A; and similarly the optical distance between A₀B₀ and B is represented by ∫(μ + δμ)ds, the integration being along B₀ . . . B. In virtue of (4) the difference of the optical distances to A and B is

∫δμ ds (along B0 . . . B) − ∫δμ ds (along A0 . . . A)

(5).

The new wave-surface is formed in such a position that the optical distance is constant; and therefore the dispersion, or the angle through which the wave-surface is turned by the change of refrangibility, is found simply by dividing (5) by the distance AB. If, as in common flint-glass spectroscopes, there is only one dispersing substance, ∫δμ ds＝δμ.s, where s is simply the thickness traversed by the ray. If t₂ and t₁ be the thicknesses traversed by the extreme rays, and a denote the width of the emergent beam, the dispersion θ is given by

θ＝δμ(t₂ − t₁)/a,

or, if t₁ be negligible,

θ＝δμt/a

(6).

The condition of resolution of a double line whose components subtend an angle θ is that θ must exceed λ/a. Hence, in order that a double line may be resolved whose components have indices μ and μ + δμ, it is necessary that t should exceed the value given by the following equation:—

t＝λ/δμ

(7).