268 THE DIFFRACTION THEORY OF MICROSCOPY 



Equations 10.22 and 10.23 are nothing but Lummer's theorem. The 

 image is similar to the object, provided that all B^^ are equal to unity 

 or any other constant. We have seen in Section 5 that Pip, q) = I for 

 an uncoated Airy-type objective. It follows from Eqs. 10.16-10.18 

 that By,j, = 1 for an idealized Airy-type objective of infinite numerical 

 aperture. Fourier series have the property 



/..M-^0 (10.24) 



as V and /x become large. Hence 



Xo -^ M^Fo (10.'25) 



as V and n become large with Airy-type objectives. 



Equation 10.25 expresses Lummer's theorem in its more useful form. 

 It states that the object and image approach similarity as the number of 

 spectral orders reaching the image is increased. 



A study of Eqs. 10.16-10.18 shows also that the resolution formula 

 for the (j^, /x)th spectral order is 



4:1m 2 , , . . 



T^^T"^]^ - m1^ I M A wavelengths. (10.26) 



{V m -j- H l )' JN .A. objective + JN . A. condenser 



The resolution formula for simply periodic structure whose spacing is 

 21 wavelengths along the A'o direction is obtained by setting m — oo in 

 Eq. 10.26. Thus 



21 ^ — '^' wavelengths. (10.27) 



-'■^ •■^■objective I -^^ -A. condenser 



The choice of the greater-than sign in Eqs. 10.26 and 10.27 gives the 

 necessary condition for resolution of the [v, ju)th spectral harmonic. 

 The choice of the equality sign gives the physical limit of resolution. 

 The physical limit may be approached but never attained. Abbe's 

 resolution formula for the first spectral order is obtained by setting 

 V = ±1 in Eq. 10.27. 



The above phenomena are consistent with well-known experimental 

 and theoretical facts about the image formation with periodic structure 

 in the ordinary microscope. Hence the formulation of Section 8 yields 

 predictions consistent with known facts. 



That the formulation is capable of explaining phase microscopy with 

 periodic structures will now be demonstrated in a general manner from 

 Eqs. 10.20 and 10.21. Note that the composite object function xo is 

 determined by the Fourier coefficients/^,^ of the object function whereas 

 the image function Fq is determined by the Fourier coefficients 



F.,, ^ B,J,,, (10.28) 



