The Theory of Microscopical Vision.
It has been said that no comparison can be instituted between microscopic vision and macroscopic; that the images formed by minute objects are not delineated microscopically under ordinary laws of diffraction, and that the results are dioptrical. This assertion, however, cannot be accepted unconditionally, as will be seen on more careful examination of the late Professor Abbe’s masterly exposition of “The Microscopical Theory of Vision,” and also his subsequent investigations on the estimation of aperture and the value of wide-angled immersion objectives, published in the “Journal of the Royal Microscopical Society.”
The essential point in Abbe’s theory of microscopical vision is that the images of minute objects in the microscope are not formed exclusively on the ordinary dioptric method (that is, in the same way in which they are formed in the camera or telescope), but that they are largely affected by the peculiar manner in which the minute construction of the object breaks up the incident rays, giving rise to diffraction.
The phenomena of diffraction in general may be observed experimentally by plates of glass ruled with fine lines. [Fig. 26] shows the appearance presented by a single candle-flame seen through such a plate, an uncoloured image of the flame occupying the centre, flanked on either side by a row of coloured spectra of the flame, which become dimmer as they recede from the centre. A similar phenomenon may be produced by dust scattered over a glass plate, and by other objects whose structure contains very minute particles, or the meshes of very fine gauze wire, the rays suffering a characteristic change in passing through such objects; that change consisting in the breaking up of a parallel beam of light into a group of rays, diverging with wide angle and forming a regular series of maxima and minima of intensity of light, due to difference of phase of vibration.[10]
Fig. 26.
In the same way, in the microscope, the diffraction pencil originating from a beam incident upon, for instance, a diatom, appears as a fan of isolated rays, decreasing in intensity as they are further removed from the direction of the incident beam transmitted through the structure, the interference of the primary waves giving a number of successive maxima of light with dark interspaces.
When a diaphragm opening is interposed between the mirror, and a plate of ruled lines placed upon the stage such as [Fig. 27], the appearance shown in [Fig. 27]a, will be observed at the back of the objective on removing the eye-piece and looking down the tube of the microscope. The centre circles are the images of the diaphragm opening produced by the direct rays, while those on the other side (always at right angles to the direction of the lines) are the diffraction images produced by the rays which are bent off from the incident pencil. In homogeneous light the central and lateral images agree in size and form, but in white light the diffraction images are radially drawn out, with the outer edges red and the inner blue (the reverse of the ordinary spectrum), forming, in fact, regular spectra the distance separating each of which varies inversely as the closeness of the lines, being for instance with the same objective twice as far apart when the lines are twice as close.
Fig. 27.
Fig. 27a.
The influence of these diffraction spectra may be demonstrated by some very striking experiments, which show that they are not by any means accidental phenomena, but are directly connected with the image which is seen by the eye.
The first experiment shows that with the central beam, or any one of the spectral beams alone, only the contour of the object is seen, the addition of at least one diffraction spectrum being essential to the visibility of the structure.
Fig. 28.
Fig. 28a.
When by a diaphragm placed at the back of the objective, as in [Fig. 28], we cover up all the diffraction spectra of [Fig. 27]a, and allow only the central rays to reach the image, the object will appear to be wholly deprived of fine details, the outline alone will remain, and every delineation of minute structure will disappear, just as if the microscope had suddenly lost its optical power, as in [Fig. 28]a.
This experiment illustrates a case of the obliteration of structure by obstructing the passage of the diffraction spectra to the eye-piece. The next experiment shows how the appearance of fine structure may be created by manipulating the spectra.
Fig. 29.
Fig. 29a.
When a diaphragm such as that shown in [Fig. 29] is placed at the back of the objective, so as to cut off each alternate one of the upper row of spectra in [Fig. 27]a, that row will obviously become identical with the lower one, and if the theory holds good, we should find the image of the upper lines identical with that of the lower. On replacing the eye-piece, we see that it is so, the upper set of lines are doubled in number, a new line appearing in the centre of the space between each of the old (upper) ones, and upper and lower set having become to all appearance identical, as seen in [Fig. 29]a.
Fig. 30.
Fig. 30a.
In the same way, if we stop off all but the outer spectra, as in [Fig. 30], the lines are apparently again doubled, as seen in [Fig. 30]a.
A case of apparent creation of structure, similar in principle to the foregoing, though more striking, is afforded by a network of squares, as in [Fig. 31], having sides parallel to this page, which gives the spectra shown in [Fig. 31]a, consisting of vertical rows for the horizontal lines and horizontal rows for the vertical ones. But it is readily seen that two diagonal rows of spectra exist at right angles to the diagonals of the squares, just as would arise from sets of lines in the direction of the diagonals, so that if the theory holds good we ought to find, on obstructing all the other spectra and allowing only the diagonal ones to pass to the eye-piece, that the vertical and horizontal lines have disappeared and are replaced by two new sets of lines at right angles to the diagonals.
Fig. 31.
Fig. 31a.
Fig. 32.
Fig. 32a.
On inserting the diaphragm, [Fig. 32], and replacing the eye-piece, we find in the place of the old network the one shown in [Fig. 32]a, the squares being, however, smaller in the proportion of 1 : √2, as they should be in accordance with the theory propounded.
An object such as Pleurosigma angulatum, which gives six diffraction spectra arranged as in [Fig. 33], should, according to this theory, show markings in a hexagonal arrangement. For there will be one set of lines at right angles to b, a, e, another set at right angles to c, a, f, and a third at right angles to g, a, d. These three sets of lines will obviously produce the appearance shown in [Fig. 33]a.
Fig. 33.
Fig. 33a.
Fig. 34.
A great variety of appearances may be produced with the same arrangement of spectra. Any two adjacent spectra with the central beam (as b, c, a) will form equilateral triangles and give hexagonal markings. Or by stopping off all but g, c, e (or b, d, f), we again have the spectra in the form of equilateral triangles; but as they are now further apart, the sides of the triangles in the two cases being as √3 : 1, the hexagons will be smaller and three times as numerous. Their sides will also be arranged at a different angle to those of the first set. The hexagons may be entirely obliterated by admitting only the spectra g, c, or g, f, or b, f, etc., when new lines will appear at right angles, or obliquely inclined, to the median line. By varying the combinations of the spectra, therefore, different figures of varying size and positions are produced, all of which cannot, of course, represent the true structure. Not only, however, may the appearance of particular structure be obliterated or created, but it may even be predicted before being seen under the microscope. If the position and relative intensity of the spectra in any particular case are given, the character of the resultant image, in some instances, may be worked out by mathematical calculations. A remarkable instance of such a prediction is to be found in the case recorded by Mr. Stephenson, where a mathematical student who had never seen a diatom, worked out the purely mathematical result of the interference of the six spectra b-g of [Fig. 33] (identical with P. angulatum), giving the drawing copied in [Fig. 34]. The special feature was the small markings between the hexagons, which had not, before this time, been noticed on P. angulatum. On more closely scrutinizing a valve, stopping out the central beam and allowing the six spectra only to pass, the small markings were found actually to exist, though they were so faint they had previously escaped observation until the result of the mathematical deduction had shown that they ought to be seen.
These experiments seem to show that diffraction plays a very essential part in the formation of microscopical images, since dissimilar structures give identical images when the differences of their diffractive effect is removed, and conversely similar structures may give dissimilar images when their diffractive images are made dissimilar. Whilst a purely dioptric image answers point for point to the object on the stage, and enables a safe inference to be drawn as to the actual nature of that object, the visible indications of minute structure in a microscopical image are not always or necessarily conformable to the real nature of the object examined, so that nothing more can safely be inferred from the image as presented to the eye, than the presence in the object of such structural peculiarities as will produce the particular diffraction phenomena on which these images depend.
Further investigations and experiments led Abbe to discard so much of his theoretical conclusions relating to superimposed images having a distinct character as well as a different origin, and as to their capability of being separated and examined apart from each other. In a later paper he writes: “I no longer maintain in principle the distinction between the absorption image or direct dioptrical image and the diffraction image, nor do I hold that the microscopical image of an object consists of two superimposed images of different origin or a different mode of production. Thus it appears that both the absorption image and the diffraction image he held to be equally of diffraction origin; but while a lens of small aperture would give the former with facility, it would be powerless to reveal the latter, because of its limited capacity to gather in the strongly-deflected rays due to the excessively minute bodies the microscopical objective has to deal with.”[11]
Abbe’s theory of vision has been questioned by mathematicians, and since his death Lord Rayleigh went more deeply into the question of “the theory of the formation of optical images,” with special reference to the microscope and telescope. He has shown that two lines cannot be fairly resolved unless their components subtend an angle exceeding that subtended by the wave-length of light at a distance equal to the aperture; also, that the measure of resolution is only possible with a square aperture, or one bounded by straight lines, parallel to the lines resolved.
Lord Rayleigh’s Theory of the Formation of Optical Images, with Special Reference to the Microscope.[12]
Of the two methods adopted, that of Helmholtz’s consists in tracing the image representative of a mathematical point in the object, the point being regarded as self-luminous; that of Abbe’s the typical object was not, as we have seen, a luminous point, but a grating illuminated by plane waves of light. In the latter method, Lord Rayleigh argues that the complete representation of the object requires the co-operation of all the spectra which are focussed in the principal focal plane of the objective; when only a few are present the representation is imperfect, and wholly fails when there is only one. He then proceeds to show, by the aid of diagrams and mathematical formula, how the resolving power can be adduced.
On further criticism of the Abbe spectrum theory, he observes “that although the image ultimately formed may be considered to be due to the spectra focussed to a given point, the degree of conformity of the image to the object is another question. The consideration of the case of a very fine grating, which might afford no lateral spectra at all, shows the incorrectness of the usually accepted idea that if all the spectra are utilised the image will still be incomplete, so that the theory (originally promulgated by Abbe) requires a good deal of supplementing; while it is inapplicable when the incident light is not parallel, and when the object is, for example, a double point and not a grating. Even in the case of a grating, the spectrum theory is inapplicable, if the grating is self-luminous; for in this case no spectra can be formed since the radiations from the different elements of the grating have no permanent phase-relations.” For these reasons Lord Rayleigh advises that the question should be reconsidered from the older point of view, according to which the typical object is a point and not a grating. Such treatment will show that the theory of resolving power is essentially the same for all instruments. The peculiarities of the microscope, arising from the divergence-angles not being limited to be small, and from the different character of the illumination, are theoretically only differences of detail. These investigations can be extended to gratings, and the results so obtained confirm for the most part the conclusions of the spectrum theory.
Furthermore, that the function of the condenser in microscopic practice in throwing upon the object the image of the lamp-flame is to cause the object to behave, at any rate in some degree, as if it were self-luminous, and thus to obviate the sharply-marked interference bands which arise when permanent and definite phase-relations are permitted to exist between the radiations which issue from various points of the object. This is capable of mathematical proof; and in the case where the illumination is such that each point of the row or of the grating radiates independently, the limit to resolution is seen to depend only on the width of the aperture, and thus to be the same for all forms of aperture as for those of the rectangular. That Abbe’s theory of microscopic vision is fairly open to the criticisms passed on it by Lord Rayleigh must be taken for granted.