410 WRIGHT— POLARIZED LIGHT IN THE 



For a beam of light entering a second medium the axes of reference may 

 be so chosen that the Z-axis is normal to the boundary surface while the 

 plane of incidence is the XZ plane. In this case ''i = sin i, i-, =r 0/3 = cos i, 

 if the positive Z-axis points into the second medium and i is the angle of 

 incidence. In adopting this convention and substituting the values of (9) in 

 (8 and 5a) we obtain, on eliminating /Wj, /x,, /x, the equation 



(q2 — fl_) (g2 — Q^^ cQs2 { — fl^^ sin2 i + 2031 sin i-cos i) 



,~ . . - .^, ) (11) 



^(OosSmi — ai2 cos 



0=.} 



In this equation 'q and the constants an- • • are complex quantities. In similar 

 manner we find, by observing that the wave normal and the polarization vec- 

 tor are at right angles, the expression 



t^ii'q^ — ojjo) ^ ^^(0^.., cos ! sin I — "a.nCOi-i). (12) 



If the polarization plane include an angle 5 with the plane of incidence 

 (i"i = — cos I -cos 5, Mi = sin 5, /x, = sin t cos 5) equation (12) can be written 



<.^„ ; g^ — fl2 2 012 cos i — 023 sin i . . 



tan o = _ ; ; — . = — z 7r~- ; . : . (12a) 



ai2 cos I — ao2 sm t ^ — on cos^ t — a^ sm^ 1 -{- 20,31 sm t cos i 



By means of this equation the azimuth of the wave Iji- can be computed. 



Boundary Conditions. — On passing from one medium to another, as from 

 air into a crystal plate, light waves encounter at the boundary surface of the 

 plate entirely new conditions. This passage from the system of forces opera- 

 tive in the first medium to that in the second is exceedingly rapid and is 

 accomplished within a very thin film ; but it is nevertheless a continuous 

 process since, physically speaking, there are no discontinuities in nature. The 

 boundary surface is an inhomogeneous film in which the dielectric constant 

 passes continuously, though rapidly, from that of the first to that of the 

 second medium. Now in order that the finite current be carried across the 

 boundary the components of the electric and magnetic forces parallel with 

 the boundary surface must be continuous through the boundary; for an 

 infinitely thin film the forces on either side of the film must therefore be 

 equal. If the boundary surface be the xy plane and the plane of incidence 

 the xs plane, the general boundary conditions are («)i=: (m),, (t/)i=: (v)2, 

 (X),= (X)., (Y)i=(Y)n. For periodic vibrations it is evident that {dX/dt)i 

 = {dX/dt)2 and (dY/dt)i = {dV/dt)2 may be used in place of (X),= (X)„ 

 (F)i= {¥).; the last equation of (5b) can be written {du>!dt)i = {dw!dt)2, 

 or (w)i^(zi;); for periodic vibrations. Only four of these conditions are 

 independent. 



In the case of light waves entering a crystal plate from air, there are 

 two components (magnetic vectors) on the air side of the boundary film, 

 Ue, that of the incident wave and ur that of the reflected wave; on the crystal 

 side there are the components M,, m, of the two refracted waves. Equations 

 (8) define the state of the light wave motion at any point and instant. For 

 the boundary film Z = o; in order that in the film the boundary conditions 

 hold for all instants of time, the relations must obtain 



