90 Dr Searle, Experiments with a plane diffraction grating 



Since m^^ + Wg- cannot exceed unity when the direction cosines are 

 real, the condition that a diffracted beam may exist is mg^ + n^^ 5 1, 

 or cos^ 62 < sin^T^g- I^ ^2 ^^^ V2 li® between and In, this requires 

 that ')72 + ^2 > i""- 



It is noteworthy that eg depends only upon e^ and iXJd, and 

 that 772 depends only upon tj^. 



We shall not further consider the case in which fi^ and /Xg are 

 unequal, but shall confine the work to the special case of /lij = /Xg- 

 The reader will find no difficulty in making the necessary modifi- 

 cations. 



§ 5. Diffraction of a plane wave; single tnediunfi. In practice 

 each medium is air, of refractive index n, relative to a vacuum. 

 If A is the wave length in air, Aq = /xA. We then obtain the simple 

 equations 



m2 = m^ T iXjd, or cos e^ = cos e^ T iXjd, (7) 



'^2^*^15 O^ cos 172 = cos T^j (8) 



Since -q may be restricted to lie between and tt, we have 



'n2 = Vi = 'n (9) 



The direction of the diffracted fay is easily constructed on a 



spherical diagram. Let the axes of the grating intersect a sphere 



about as centre in N , T, L (Fig. 2), and let NON' be a diameter. 



Let the continuation of the incident ray 



_L through meet the sphere in P^. The 



^\ great circle arc TPj measures e^. Calculate e, 



. .-^^--^^N by (7) and take TQ = €^ on TP^. About f 



2 y\ \ and L as poles draw small circles through 



N'f^ /IH"^ ^andPj. Theni>Pi=-7;. If the small circles 



T / / do not intersect, there will be no diffracted 



beam either by transmission or by reflexion. 



If the small circles intersect in the points Pgj 



Fig. 2. P,', then OP2, OP2 will be the directions of 



the two diffracted beams. Of the arcs NP2, 



NP2' of the great circle NP^P.^N' , one is greater and one less 



than Itt. If NP2 is less than \tt, it corresponds to the transmitted 



beam, and then iV^P2' corresponds to the reflected beam. 



When ej and i are given, there are two values of eg, and hence 

 there are two points Q_ and ^+ on TP^. Thus there will be two 

 directions {OP2)- and (0P2)+ for the transmitted diffracted beam, 

 and similarly for the reflected beam. 



It may happen that only one of the two beams {OP2)- and 

 (0P2)+ exists. Unless P^ is on the great circle LN, there will be 

 two distinct values of mg^, and the condition ^2^ + ^2^ ^ 1 may 

 be satisfied by the smaller value of m^^ but not by the larger. 



