OBJECTS OF PERIODIC STRUCTURE 269 



in which the pupil coefficients B^y, are numerically equal to the value of 

 the pupil function Pijp, q) at the points {p = Pq/M + v/2Ml, q = 

 qo/M + ii/2Mm). It follows from Eq. 3.24 that the values B^^ can 

 be altered at will by the choice of coating function cip, q). Hence the 

 Fourier coefficients F^_^ of the image function can be altered at will with 

 respect to the Fourier coefficients /^,^ of the object function. Since a 

 function described by a Fourier series depends on the Fourier coefficients, 

 it follows that the image function can be made quite different from the 

 composite object function. In particular, we should expect to be able 

 to increase or decrease and to reverse contrast in the image. In a 

 Zernike system of phase microscopy, the coating material of the dif- 

 fraction plate is distributed so as to modify Bq^q together with a highly 

 restricted set of the remaining pupil coefficients B^^^. In fact, it can 

 be shown that the points {p = po/M, q = qo/M) belong to the con- 

 jugate area of the diffraction plate. Consequently, P{po/M, qo/M) 

 is the value of the pupil function for points in the conjugate area. 

 Hence with respect to Eq. 10.16 only a highly restricted set of points 

 (p = po/M +v/2Ml, q = qo/M + v/2Mm) will fall in the conjugate 

 area. 



If the conjugate area is narrow, it can happen that all (p = po/^t + 

 v/2Ml, q = qo/M -\- n/2Mm) fall in the complementary area of the 

 diffraction plate when v 9^ Q ov ^x 9^ Q ox both. In this case 5o,o ^'^'ih 

 be said to be uniformly altered with respect to the pupil coefficients 

 5^,jx of the higher spectral orders. Since the zero-order coefficient Bq.o 

 is almost uniformly altered with respect to the higher ordered coeffi- 

 cients B^,ju in a Zernike system of phase microscopy, it will not be possible 

 to change the image function Fo{x, y, po, go) arbitrarily with respect to 

 the composite object function xo(-^o, Vo, Po, ?o)- Indeed, it can be shown 

 that the effect of altering Bo,o uniformly with respect to 5^,,, of the higher 

 spectral orders is mainly to alter contrast in the image. Consequently, 

 the Zernike system of phase microscopy is happily chosen so as to enable 

 one to alter contrast in the image without introducing artifact beyond 

 that already present in the ordinary system of microscopy. In making 

 this assertion, it is assumed that the conjugate and complementary areas 

 are coated uniformly and differently. 



With Fo(x, y, po, qo) determined by Eq. 10.21, the total energy density 

 G{x, y) in the plane of the image is given in accordance with Eq. 8.16 

 by the relation 



{x, y) = ff /^^°',^°^ ^ \Fo{x, y, po, 90) I' dpo dqo. (10.29) 

 J J no- — Po" - qo 



