CLASSICAL THEORY OF LIGHT 753 



The ratio A2IA1 of the ampHtudes of the first and second reflected 

 beams, and in general the ratios A n/A j of the amplitudes of any of the 

 reflected beams and the first, are determined altogether by ju and i. 

 An important consequence of this will presently appear. One can 

 however alter these ratios, e.g., by half-silvering the sides of the plate; 

 and the formulae which I am about to quote may be applied to the 

 case of two half-silvered mirrors facing each other in air, by setting 



/^= 1. 



Isolate then in mind a single incident wave-train. Denote by i 

 its angle of incidence upon the upper surface of the glass; by t the 

 thickness of the plate. A wave-front of the oncoming wave-train is 

 divided into two. During the time while the part which entered the 

 glass is advancing to the lower side, being reflected, returning to and 

 re-emerging from the upper side, the part which was first reflected goes 

 on to the level EE' of Fig. 3. The emerging wave coincides with the 

 first-reflected part of a new wave-front which was following along 

 after the old one at the interval E'D'. In general, there is a phase- 

 difference <p between these two. The condition for interference is 

 ideally satisfied; two wave-fronts coincide. The amplitude A of the 

 resultant light travelling away from the mirrors is obtained by the 

 prior formula from the amplitudes Ai and ^2 of the first and second 

 reflected beams and their phase-difference (p: 



A^' = Ai^ + A2- + 2AiA2Cos<p. (17) 



It is now our affair to deduce the value of this difference in phase. 

 This is a simple task, for the angle (p, expressed in radians or in degrees, 

 is Iw or 360 times the number of wave-lengths intervening between 

 the wave-front DD' and the wave-front EE' which, so to speak, has 

 gone on ahead leaving part of itself behind to combine with DD'. We 

 have therefore only to multiply It/X or 360/X into the distance D'E'; 

 which distance is found ^ by very easy manipulations of trigonometry, 

 aided by the equation (16), to be l^xt cos r\ so that for the phase- 

 difference in radians we have 



2iv 

 if = -r- 2/x/ COS r + Vo- (18) 



A 



The constant ^0 is inserted to leave room for the possibility that 

 abrupt changes of phase may occur at the instant of reflection, unequal 

 for the two reflecting surfaces. Experience shows that in this case of 



' For the distance D'E' is the difference between BD' and BE'. The latter is 

 evidently BD sin i, which is 2t tan r sin i, which is 2nt tan r sin r. The former is 

 the distance cT traversed by Hght in vacuo during the time T while the beam which 

 entered the glass is traversing its zigzag path BCD; this time is {fi/c)BCD, which is 

 2{fx/c)t sec r. 



