40 



AN ELEMENTARY THEORY OF PHASE MICROSCOPY 



parallelogram method for finding the resultant of two different simple 

 harmonic motions oscillating in the same straight line. If, therefore, the 

 amplitudes and phases of two interfering waves are represented by two 

 complex numbers Zi and Z2, the amplitude and phase of the resultant 

 wave is given by the complex number which is equal to the sum of the 

 complex numbers Zi and Z2. With respect to the third analogy, let the 

 amplitude and phase of the wave which is incident upon a medium be 

 represented by the complex number zi = aie'^^, and let the amplitude 



Fig. 11.11. Geometrical representation of the sum ze^^ of two complex numbers 



zi = aie*^i and 22 



(126 



162 



and phase transmission of the medium be represented by the complex 

 number 22 = 02^'^"- ^2 is then the optical path of the medium expressed 

 in radians instead of wavelengths. Now it is a well-known experi- 

 mental fact that, when an incident wave of amplitude Ui and phase di 

 has passed through a medium of amplitude transmission 02 and phase 

 transmission 62, the amplitude of the emergent wave is aia2 and the 

 phase of the emergent wave is di + 62. If we now represent the 

 emergent wave by the complex number z = ae'^, experiment requires 

 that 



z = ae'^ = aia2e''^^i+^2) = aie'^'-a2e'^\ (4.11) 



The complex number describing the amplitude and phase of the emergent 

 wave is therefore given by the product of the complex numbers rep- 

 resenting the amplitude and phase of the incident wave and the ampli- 

 tude and phase transmission of the medium. 



5. VISUAL EFFECT PRODUCED BY A COMPLEX WAVE 



Light waves vibrate so rapidly that the eye is not able to follow the 

 instantaneous values of the electric vector but can detect only the time 



