1166 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 



be taken simply as 



(since, in vacuo, 6„o = a„o sgn ^„ and sgn /?„ = +1 for the incident field), 

 and the amplitude a„o may be taken to be unity. Therefore equation (30) 

 may be written 



dttn 



dz 



j ^ ["/ (m - fJio)H,nHtn dS - j j KHtn*Htn dS 



- jj eo^en^n dS - j^nK , 

 = j -^ / (e - eo)EinEtn dS - j^nttn . 



We solve these equations by integrating through the small thickness of 

 the sample. Since we are interested only in results of first order in 8z, 

 a„ and 6„ on the right hand side of these equations may be replaced by 

 flno , bno ; that is by 1, 1. Writing 



Ia = -^ / (m - IJio)Hinfftn dS - j KHtn*fftn dS 



+ K'l -'^)eoE,J,^dS , 



Ib = -^ f (e - eo)E,Jtn dS, 



we have, upon integration from the incidence plane (1) to the exit plane 

 (2), 



a«2 - flnl = J(Ia - l3n)8Z, 

 hnl — bnl = J(Ib - I3n)5z. 



a„i , the amplitude at the entrance plane consists of a„o = 1, the incident 

 amplitude, plus a small reflected amphtude p. Thus a„i = 1 + p. Simi- 

 larly hn consists of a„o sgn /3„ = 1, and the small reflected amplitude 

 p sgn pn = —p (corresponding to a backward wave) . 6„2 is equal to a„2 

 (since the transmitted wave is a forward wave). Hence we have 



a„2 - (1 + p) = j(Ta - /3„) 5.', 



a.2 - (1 - p) = J{Ib - |3«) dz, 



