A New Calculus for the Treatment of Optical Systems VII Properties of the N-Matrices

The preceding papers of this series have examined the properties of an optical calculus which represented each of the separate elements of an optical system by means of a single matrix M. This paper is concerned with the properties of matrices, denoted by N, which refer not to the complete element, but only to a given infinitesimal path length within the element.If M is the matrix of the optical element up to the point z, where z is measured along the light path, then the N-matrix at the point z is defined by (A)N≡(dM/dz)M-1.Thus one may write symbolically, (B)N=dlogM/dz,and (C)M=M0 exp(∫Ndz).A general introduction is contained in Part I. The definition and general properties of the N-matrices are treated in Part II. Part III contains a detailed discussion of the important special case in which the optical medium is homogeneous, so that N is independent of z; Part III contains in Eq. (3.26) the explicit relation which corresponds to the symbolic relation (C). Part IV describes a systematic method, based on the N-matrices, by which the optical properties of the system at each point may be described uniquely and quantitatively as a combination of a certain amount of linear birefringence, a certain amount of circular dichroism, etc.; the method of resolution is indicated in Table I. Part V treats the properties of the inhomogeneous crystal which is obtained by twisting a homogeneous crystal about an axis parallel to the light path.