598 



Appendix B 



Consider now the case shown in Figure 3 of a curved interface. A 

 section of a sphere AOA' with center at r separates two media of indices 

 of refraction n-^ and n 2 . A section of a spherical wave BOB' with center 

 at p reaches the spherical surface at time t = 0. A short time later, 

 BB' has reached AA' . The waves in the second medium move more 

 slowly. Consequently, the portion of the wavefront initially at has 



n 7 >n \ 



Figure 3. Refraction of a spherical wave at a spherical inter- 

 face. Light is traveling from left to right. A real object is 

 located in medium 1 at —p and a real image in medium 2 at q. 



only reached M during the time in which the wavefront in the first 

 medium has moved from B to A. Thus, a real image will form at q 

 due to the change in the curvature of the wavefront. 



To find a relationship between p, q, r, n l3 and n 2 , one proceeds just 

 as in SnelPs law to equate the time for light to travel from to M and 

 from B to A ; symbolically 



n x AB = n 2 0M 



(6) 



The entire problem (which is still far simpler than the refraction in the 

 eye) can be simplified mathematically by assuming only small sections 

 of spheres. This allows one to make the following approximations 



AA' = BB' 



">-*$■ 



MN = 



AA 



'2 



2? 



2r 



(7) 



The last three are based on the sagittal approximation illustrated in 

 Figure 4. Noting in Figure 3 that 



LN = LO + ON 



and 



OM = ON - MN 



(8) 



