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 ¼λ will correspond to an error of 1⁄8λ in a reflecting and ½λ 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 2½ 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 δf 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 ¼λ (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. |