AMPLITUDE AND PHASE BY COMPLEX NUMBERS 37 



4. REPRESENTATION OF THE AMPLITUDE AND PHASE BY COMPLEX 

 NUMBERS 



The laws expressed verbally and graphically in the preceding section 

 can be formulated more elegantly and comprehensively by stating the 

 interference phenomena which take place between the undeviated and 

 deviated waves either in terms of the trigonometric functions, as in a 

 recent paper by Keck and Brice (1949), or in terms of the equivalent 

 complex numbers. Complex numbers are preferred because they are 

 more readily multiplied or added than the trigonometric functions. 



The undeviated and de\'iated waves are vectors whose complete 

 specifications include direction, phase, and amplitude of vibration. The 

 direction of vibration may be ignored because all portions of a wave train 

 which is emitted by the source of light continue to vibrate in the same 

 direction as the wave train traverses the object specimen and the optical 

 system of a phase microscope. Within the scope of an elementary 

 treatment of phase microscopy the direction of vibration of the wave 

 train has no influence upon the character of the image. It is, however, 

 necessary to bear in mind that the undeviated and deviated waves both 

 originate and recombine as two simple harmonic motions oscillating in 

 the same straight line. 



A number z of the form 



z = X -\- iy, (4.1) 



in which x and y are real numbers and i = ( — 1)', is called a complex 

 number, x and y are called the real and imaginary parts of the complex 

 number z. The absolute value of z is denoted by l^l and is defined as 



\z\ = (^2 + 7/2)*. ' (4.2) 



The square of the absolute value of a complex number is equal to the 

 sum of the squares of its real and imaginary parts. The complex 

 number z can be represented geometrically as the vector OP drawn 

 from to F in the rectangular coordinate system XY of Fig. 11.10. The 

 direction of the vector OP is specified by means of its angular rotation 6. 

 It will be noted that {x" -}- y~)' = \z\ is equal to the length or amplitude a 

 of the vector OP. The absolute value of a complex number is therefore 

 equal to the amplitude of the complex number. 



As indicated in Fig. 11.10, x = a cos d and y = a sin d. If these 

 values of x and y are substituted into Eq. 4.1, the complex number z 

 assumes the polar form 



z = a(cos d -\- i sin 6). (4.3) 



