44^ 



VII. — On the Theory of Optical Images, with Special Reference 



to the Microscope. 



By Lord Eayleigh.* 



(Reprinted by permission from the Philosophical Magazine, August 1S96.) 



The special subject of this paper has been treated from two distinct 

 points of view. In the work of Helmholtz f the method followed 

 is analogous to that which had long been used in the theory of the 

 telescope. It consists in tracing the image representative of a 

 mathematical point in the object, the point being regarded as self- 

 luminous. The limit to definition depends upon the fact that 

 owing to diffraction the image thrown even by a perfect lens is 

 not confined to a point, but distends itself over a patch or disk of 

 light of finite diameter. Two points in the object can appear fully 

 separated only when the representative disks are nearly clear of 

 one another. The application to the microscope was traced by 

 means of a somewhat extended form of Lagrange's general optical 

 theorem, and the conclusion was reached that the smallest resolv- 

 able distance e is given by 



e = i A /sin a, . . .. . . (1) 



A being the wave-length in the medium where the object is situated, 

 and a the divergence-angle of the extreme ray (the semi-angular 

 aperture) in the same medium. If A be the wave-length in 

 vacuum, 



* = *<,//*, (2) 



fju being the refractive index of the medium ; and thus 



e = -i A / jjl sin a (3) 



The denominator /x sin a is the quantity now well known (after 

 Abbe) as the " numerical aperture." 



The extreme value possible for a is a right angle, so that for 

 the microscopic limit we have 



e = ^ A / /x. (4) 



* The discussion of some further questions, now communicated to the Society, 

 depends upon principles laid down in this paper, which, though published several 

 years ago, does not seem to have attracted the attention of microscopists. It is 

 thought that its republication in connection with the new investigations will be 

 convenient to the reader. t Pogg. Ann., Jubelband, 1874. 



2 G 2 



