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 ƒ 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 ¼λ. 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 = ¼λ;
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.