Numerical Computation of Phase from Amplitude at Optical Frequenciesincluded on pages 667-679

The fact that nature ties the real and imaginary components of a complex variable function of frequency inextricably together, when the variable represents some physically real quantity or phenomena, has been recognized to varying degrees for nearly half a century. For example, Kramers 1 in 1927 noted the general relations between the refractive index and absorption resulting from the simple relationships to the real and imaginary parts of a complex dielectric constant. Eecause one of the relations was contained in an earlier paper of Kronig's, 2 this relationship is commonly known in the physical science world as the Ivramers-Kronig relation. The awareness of the relationship between 637 6.52 T H E BELL SYSTEM TECHNICAL JOURNAL, MAY 1963 the real and imaginary parts of the impedance of an electrical network emerged about the same time as the Kramers-Kronig work. 3 4 The usefulness of a quantitative solution to the general real and imaginary component relationship was soon recognized. Bode has provided us with a key to the solution of this problem (Ref. 5, Ch. XIV). He gives general integral equations relating the two components, but points out what many have since discovered, namely, that the general integrals can be readily evaluated for only the simplest of functions. Bode, however, presents a practical numerical integration technique for summing the imaginary component associated with a multiple straightline approximation to the real component as a function of frequency (Ref.