172 Theory of Fiber Bragg Gratings 4.8.2 Transfer Matrix Method An analytical solution for a grating of length L g , with an arbitrary coupling constant k(z) and chirp L(z), is desirable but no simple form exists. The vari- ables cannot be separated since they collectively affect the transfer function. In the T-matrix method, the coupled mode equations [for example, Eq. (4.3.9)] are used to calculate the output fields of a short section dl 1 of grating for which the three parameters are assumed to be constant. Each may possess a unique and independent functional dependence on the spatial param- eter z. For such a grating with an integral number of periods, the analytical solution results in the amplitude reflectivity, transmission, and phase. These quantities are then used as the input parameters for the adjacent section of grating of length dl 2 (not necessarily ¼ dl 1 ). The input and output fields for a single grating section are shown in Fig. 4.30. The grating may be considered to be a four-port device with four fields: input fields R(­dl 1 /2) and R(dl 1 /2) and output fields S(­dl 1 /2) and S(dl 1 /2). A transfer matrix T 1 represents the grating amplitude and phase response. For a short uniform grating, the two fields on the RHS of the following equation are transformed by the matrix into the fields on the LHS as ! ! Rðdl 1 =2Þ RðÀdl 1 =2Þ ¼ ½T 1 : ð4:8:1Þ