671 



XVIII. — On the Resolution of Periodic Structures. 

 By Edward M. Nelson. 



(Read October 21, 1908.) 



It has been often noted that when periodic, or lined, structures 

 are resolved upon a bright field, they may become invisible with 

 dark-ground illumination. One may search microscopical text- 

 books in vain for an explanation of this phenomenon. The Abbe 

 " Spectrum Theory " may be wrung up to its breaking point, but 

 not a drop of enlightenment can be squeezed out of it ; Mr. Gordon's 

 " Antipoint Theory," however, at once supplies an answer. 



If one of Mr. Grayson's beautiful rulings be placed on the stage, 

 and a band, say of 45,000 lines to the inch — the lines diagram- 

 matically represented by the shaded portions in A, fig. 157 — be 

 examined with a xff~ m - object-glass, under a full cone of trans- 

 mitted light, B may be taken to represent the image as seen in the 

 Microscope. 



Now Mr. Gordon tells us that the bright field is made up of a 

 mosaic of antipoints, the diameter of the antipoint being inversely 

 as the W.A., that is, the larger the W.A., the smaller the antipoint. 

 In this supposed case the W.A. is equal to the N.A. of the object- 

 glass. A glance at C shows how it comes to pass that the broad 

 lines at A are imaged in the Microscope by the narrow lines at B : 

 for we see the half-antipoints (diagrammatically but not accurately 

 illustrated by semicircles) eating into each side of the broad lines, 

 leaving a narrow central part. If the antipoints were so large that 

 the semicircles met in the middle of the broad lines, there would 

 obviously be no resolution : the lines would remain invisible. 



Now let us see what happens when the lines are illuminated 

 upon a. dark ground. The lines will be bright, the interspaces 

 dark, and the antipoints will eat, not into the lines, but into the 

 spaces, and so broaden the lines. D, drawn to the same scale, 

 shows that as the half-antipoints now meet in the dark spaces, 

 there can be no resolution, and the 45,000-band will appear a 

 blank, as at E, the limit of resolution being lowered to the 30,000- 

 band. 



Next let us examine the case under different conditions. In 

 fig. 157 the assumption has been made that the breadth of the lines 

 is wider than that of the interspaces, which is probably the case 

 with the higher bands. Now a very little consideration will show 

 that when the spaces are wider than the lines, the above recorded 



