AIRY-TYPE OBJECTIVES 



253 



Suppose that Eq. 5.3 is satisfied, that c(p, q) = 1, and that the object 

 point is located at Xq = y^ = 0. Then the primary diffraction integral 

 reduces to 



u{x, y) = rrf/^^'(/'^+92/) (^p f/^ 



(5.4) 



for points .r, y of the conjugate image plane. From Equations 3.5 and 

 5.4, with n = 1, 



Jo 



Jnpm 

 u 



■,^ Ji[27rp,J.r2 + /)-] 

 27rp,„(.r- + r) 



in which Jy and Ji are Bessel functions of zero and first order, respec- 

 tively. Introducing 



r = (.r^ + r)% 

 one finds that 



U(x, y) ^ U{r) = TTp, 



^2./i(27rrp„0 

 27rrp,^ 



(5.6) 



(5.7) 



Since the energy density E{r) is proportional to |t/(r)| 



E{r) 



2 4 



= 4 



J i(2irrp,n) 

 . 27rrp,„ . 



(5.8) 



Equation 5.8 describes the classical energy distribution produced 

 over the image plane by the coherent light emitted by an object point 

 which is located upon the optical axis. The discovery of this distribu- 

 tion law is attributed to Airy. We see therefore that \U{x, y)\- de- 

 scribes an Airy type of diffraction image when the objective is of the 

 Airy type. 



By introducing p,„ from Eq. 3.8 into Ecj. 5.8, we find that 



M^Eir) 

 7r-(N.A.)-^ 



= 4 



'■7i(27rrN.A./|M|)' 

 27rrN.A./|M| 



(5.9) 



Since the first positive root of ^1(2) = occurs at 2 = 3.8317, the first 

 zero value of E{r) occurs at 



r 3.8317 0.6098 



\M\ 



2tt N.A. 



N.A. 



wavelengths. 



(5.10) 



