470 



NATURE 



[December 9, 1920 



depend on functions of the form (assuming Huvgens's 

 prmciple) 



Intensity- constxJ,«(?|^ . j*), 



where J, is the Bessel function of the first order, 

 R the radius of the wave-surface at the lens, o 

 the semi-aperture, and fc the distance of the point 

 from the axis. At present it has not been found 

 possible to give an expression for the light distribu- 

 tion in the presence of spherical aberration, cr away 

 from the focus, which can be physically interpreted 

 in such a simple way. Prof. Conrady has been able, 

 however, to determine the distribution numerically 

 (Monthly Notices, R.A.S., vol. l.xxix.. No. 8) in a 

 series of simple cases by mechanical quadratures 

 which give the value of the integral 



I = const 

 where 



X {° '"{y(art)cos2,„ + J(,2(aa)sin»,„H'''), 



iTck 



i\ representing the phase which is supposed to vary 

 in a spherical reference-surface by an amount depend- 

 ing on the spherical aberration. Prof. Conrady 

 assumes a series of likely cases for phase distribution. 

 No analytical expression for »; can be obtained for any 

 but the simplest optical systems, but empirical expres- 

 sions can easily be derived from the result of trigono- 

 metrical ray tracing or Hartmann tests in the most 

 complex systems, thus enabling the truth of the 

 numerical results for light distribution to be checked 

 experimentally. The value of 



ria = -^l 6da (verv nearly), 

 A J 



where 6 is the angular aberration derived from the 

 calculated or observed lateral aberration or lateral 

 intercept due to displacement from the centre of the 

 spherical reference surface. 



It is not too much to say that the thorough solution 

 of the problem is of the greatest importance in the 

 study of the performance of optical instruments. In 

 order to search for the actual phenomena predicted 

 by Prof. Conrady in theoretical cases, and further to 

 explore the subject, I have recently carried out a 

 critical examination of the image of a very small 

 source of light (a fine "pinhole" in a silver film) pro- 

 duced by a microscope objective having excellent 

 spherical correction, and for which the curves, glasses, 

 etc., were known. The spherical aberration intro- 

 duced by varying the tube-length can thus be cal- 

 culated for any conditions. A nearly linear relation 

 was found between the phase difference of the 

 paraxial and marginal rays at the marginal focus rmd 

 the reciprocal of the tube-length. 



As a check on the calculation, I was able to devise 

 a method of performing a test on the microscope 

 objective very similar to the well-known Hartmann 

 test employed for telescope object-glasses. By this 

 means the properties of the objective became well 

 known. ^ 



The changes in the distribution of light at the best 

 focus in the presence of varying amounts of spherical 

 aberration have been examined quantitatively, both 

 visuallv and photographically. A perceptible loss of 

 light from the central disc, estimated at 20 per cent., 

 occurs when the residual aberration at the best visual 

 focus amounts to o-a^A. Such light is scattered into 

 the surrounding field ; it does not appear in the first 

 bright rincr at this focus. The sizes of the rings in 

 the diffraction pattern at the best visual focus do not 

 depart measurably from the theoretical values in the 



NO. 2667, VOL. 106] 



presence of residual aberration amounting to 0-6A. at 

 the focus where there is least confusion of phase. 



The "out of focus" appearances present nianv 

 points of great interest. It appears that the succes- 

 sive bright rings retain a marked individuality, but 

 suffer periodic variations in brightness and " thick- 

 ness. " This causes the dark rings between them to 

 suffer corresponding variations in "darkness " and in 

 radius. Under a high magnification the familiar 

 broad, dark diffraction rings which appear to grow 

 in the expanding "out of focus" disc are found to 

 grow as the result of this periodic motion of the 

 smaller dark interference rings, tTie whole action 

 resembling that of a model to illustrate the propaga- 

 tion of the compression waves of sound. 



If, when the aberration is a minimum, we go 

 sufficiently far out of focus to introduce a path- 

 difference between marginal and paraxial rays, 

 '^P = o5^, on either side of the focus, we find that 

 the first dark ring has nearly filled %vith light. .-\t 

 about dp=i\ the central disc has lost all its light, and 

 the first bright ring is at a maximum. At about 

 dp=i-^X the central disc has again reached a maxi- 

 mum, the first ring is at its minimum, the second 

 at a maximum, and the third nearly equal to 

 the second. We thus see the first broad, dark diffrac- 

 tion ring between the central disc and the annulus of 

 light formed by the second and third bright rings. 

 So the various changes progress, the location of the 

 successive bright rings being given fairly nearly bv 

 the ordinary theory. When, however, a definite 

 amount of aberration is introduced, sufficient to cause 

 a residual variation of phase of q:,\ at the best focus, 

 the changes are violently dissimilar on the two sides 

 of the focus — a fact which is fairlv well known. On 

 one side there is a quick dissolution of the central 

 concentration into a mere haze, while on the other 

 a bright and well-formed ring svstem is found 

 in which the broad diffraction rings spread out with 

 much the same action as before, except that the 

 periods of the variations are altered from those in 

 the "no aberration" adjustment. Further, on this 

 side of the focus, as was suggested by Prof. Conrady 's 

 numerical results, a central concentration persists 

 which diminishes considerably in size as compared 

 with the " best focus " disc, but remains brighter 

 than the rest of the ring system up to a path- 

 difference dp = T,X — a displacement in the actual 

 focussing point of 7 mm. in a total tube-length of 

 23 cm. It can easily be seen that this effect is quite 

 capable of rendering possible instrumental perform- 

 ances, so far as resolving power alone is concerned, 

 far in excess of anv value possible in the absence of 

 spherical aberration, although this would be a per- 

 fectly legitimate conclusion only for such cases as that 

 of a double-star resolution by a telescope objective. 

 \ fairly complete set of photographs to illustrate the 

 various appearances has been made. These have 

 been measured and examined for the purpose of 

 intensity determinations. 



The importance of these matters lies in the 

 determination of the effects of aberration, ex- 

 pressed in ray intercepts, on the distribution of 

 light in the image, as the distribution suggested 

 by ray concentration is often nothing approach- 

 ing the truth. The "out of focus" appearances, 

 too, are of great ' importance in dealinsr with 

 "roundness of field." It is possible, in the light of 

 such results, to form ideas as to legitimate tolerances 

 in design and manufacture. 



The investifation, to become complete, must be 

 extended to other types of aberration, but it is hoped 

 shortlv to publish a complete account of the experi- 

 mental work, of which the foregoing risumi may 



