Mode of Formation of the Image in the Microscope

To obtain any indication of structure in the object at least two images of the source must be formed behind the object-glass, since two images are required if any interference is to be pro duced in the body-tube. The smallest numerical aperture that will admit two separate beams of light from the grating (the direct beam and one diffracted beam) is given by where s is the distance between adjacent similar details in the object, and is the wave-length of the light used to illuminate the object (as measured in air). In order that a lens of this numerical aperture may admit the two beams, the object must be illuminated obliquely, so that the direct beam is thrown into one side of the object glass aperture and one of the first diffracted beams is just admitted into the other side. The smallest struc ture-interval that could be resolved by this lens, in the sense that an interference pattern would be produced in the focal plane of the eyepiece, is thus obtained as It will be seen that this result is very nearly the same as that obtained earlier for the resolution of two luminous point sources.

The "Equivalence" Theory.

Many microscopists and physi cists did not accept Abbe's theory as being of general applicability, though it was clearly valid under the particular conditions which obtained in Abbe's experiments, i.e., with objects of regular pe riodic structure illuminated by light from a distant small source. Many experiments were carried out in order to disprove Abbe's theory. Altmann stated in 188o that the results Abbe had ob tained with illuminated gratings, using diaphragms behind the object-glass, could all be reproduced with self-luminous hot-wire gratings when similar diaphragms were used behind the object glass. Mandelstam carried out these experiments with hot-wire gratings in 1911 and confirmed Altmann's statement. The value of Abbe's experiments as evidence supporting his theory was thus seriously discounted. Evidence based on the experience of micro scopists was also put forward as contradicting Abbe's theory, not ably by R. Koch in Germany and by Lewis Wright and E. M. Nelson in England. Lewis Wright's views on the mode of forma tion of the image in the microscope were emphasized by E. M. Nelson's clear demonstrations of the greatly improved images to be obtained by the use of wide axial cones of illumination, as op posed to the narrow axial or oblique pencils more favoured by the adherents to Abbe's theory.

The views held by these people were that, with suitable illumi nation, any non-luminous body can be considered as "equivalent" to a self-luminous body, and that, under the conditions of illumin ation commonly used by microscopists, the image is formed in essentially the same way as the image of a luminous body. Ac cording to this "equivalence" theory the most perfect resolution should be obtained when the full aperture of the object-glass is being used, i.e., when the cone of illumination is of such an angle that direct rays would, except for such obstruction as the object produces, enter all parts of the object-glass aperture. It usually happens, however, that, as the angle of the cone of illumination is increased from some quite small initial value, resolution is steadily improved at first, but a condition is reached such that further in crease of angle of the cone causes contrast in the image to dimin ish very rapidly, so that resolution, in the sense of visibility of detail, is greatly impaired. In the absence of any satisfactory ex planation of this phenomenon, it seemed to be in direct conflict with the implications and consequences of the "equivalence" theory, and constituted a formidable stumbling-block to that theory until the causes of the loss of contrast were discovered and were shown to be quite distinct from, and independent of resolu tion and resolving power proper.

Lord Rayleigh (3rd baron) considered the formation of the microscope image on theoretical grounds, and showed in 1896 that definite interference of the type required by the Abbe theory cannot occur unless the angle of the cone of illumination is small compared with the ratio X/s, where s is the distance separating adjacent similar details in the object and X is the wave-length of the light used to illuminate the object (Lord Rayleigh, Scientific Papers IV., 1903). This means that, with a wide angle of illumina tion, the vibrations coming from adjacent points in the object are virtually independent of each other, provided they are separated by a distance which is not small compared with the wave-length of the light used, i.e., each of these points is "equivalent" to a self luminous point in so far as the light coming from it is concerned.

Theoretical investigations by Mandelstam 1), Wolfke (1912) and von Laue (1914) confirmed Lord Rayleigh's views. Recently the whole subject has been comprehensively investigated by Berek, and in a series of papers published in 1926 he gives a full discussion of the applicability and limitations of both the theories. The conclusions he arrives at are that, under the condi tions of illumination ordinarily used, the image in the microscope is formed, in the main, in the same way as is the image of a self luminous object. The image thus obtained is a "true" image in the sense that it bears a close resemblance to the object, and has a true "focus-plane." Under the conditions which Abbe used in his dif fraction experiments, i.e., with an object of regular periodic struc ture illuminated by a narrow pencil of almost parallel rays, the only "image" obtained is of the nature of an interference pat tern. This "image" has no true focus-plane, the appearance seen in the eyepiece shows very little variation if the eyepiece is moved through quite large distances along the axis of the microscope, and this appearance does not necessarily possess any true resemblance to the object.

The theory that under the conditions of illumination commonly used, the mode of formation of the image in the microscope is in the main, the same as the mode of formation of the image of a self-luminous object, can thus be taken as well founded. The experimental evidence supporting the theory has accumulated rapidly since the conditions for obtaining contrast have been properly understood. Even before these conditions had been thor oughly investigated, however, important evidence in support of the "equivalence" theory was available from observations made with so-called "dark-ground" illumination (see p. 44o). According to the Abbe theory, the resolving power of a lens when used with this form of illumination must necessarily be less than it would be if the object were illuminated with transmitted light. Moreover, no "dry" lens, when used with dark ground illumination provided by a "dry" condenser, could have a greater resolving power than that corresponding to a numerical aperture 0.5. According to the "equivalence" theory the resolving power of a lens should be the same for any method of illumination which ensures that the full aperture of the lens is used. Actual experiment shows that resolving power is not in any way reduced when dark-ground illumination is employed. In fact, the resolution obtainable with this method of illumination often appears to be better than that obtainable with ordinary illumination by transmitted light, since dark-ground illumination secures a high degree of contrast in the image obtained. Because of this marked gain in contrast resolu tion is therefore easily recognizable.