398 Prof. Henry A. Rowland on Gratings 



we can write this 



\/A 2 + B 2 sm[e + b(p-Vt)]. 



The energy or intensity is proportional to (A 2 + B 2 ) . 

 Taking the expression 



(A + iB)e- ib( P~ Vi \ 



when i= s/ — 1, its real part will be the previous expression 

 for the displacement. Should we use the exponential expres- 

 sion instead of the circular function in our summation, we 

 see that we can always obtain the intensity of the light by 

 multiplying the final result by itself with —i in place of +i, 

 because we have 



( A + iB) £-^-™> x (A - 1 B) e»(p-v*> = A 2 + B 2 . 



In cases where a ray of light falls on a surface where it is 

 broken up, it is not necessary to take account of the change 

 of phase at the surface, but only to sum up the displacement 

 as given above. 



In all our problems let the grating be rather small com- 

 pared with the distance of the screen receiving the light, so 

 that the displacements need not be divided into their com- 

 ponents before summation. 



Let the point x\ y 1 , z ! be the source of light, and at the 

 point w, y, z let it be broken up and at the same time pass 

 from a medium of index of refraction I' to one of I. Consider 

 the disturbance at a point x", y 11 , z n in the new medium. It 

 will be 



e -ib(lp+Vp-Vt) 



where 



p 2 = x" 2 + y" 2 + z" 2 + x 2 +y 2 + z 2 -2(xx J! +yy" + zz"), 



p* = x' 2 +y' 2 +z 12 + x 2 +y 2 + z 2 -2(xx' +yy' + zz'). 



Let the point x, y, z be near the origin of coordinates as 

 compared with x f , y ! , z f or x", y", z", and let a, ft, y and a', ft', y> 

 be the direction-cosines of p and p. Then, writing 



R= I' VaP+y» + z'* + I vV' 3 +y' 2 + *" 3 , 



\=ia+r«', 



f.=lft + l'ft>, 



v=i7 +iy, 



we have, for the elementary displacement, 



p — ib[R— Vt-\x— fiy— vz+icr2] . 



