Optical coherence calculations with the split-step fast Fourier transform method.

Björn Hermansson is with Swedish Telecommunications Administration, Technology Department, S-123 86 Farsta, Sweden; D. Yevick is with Pennsylvania State University, Department of Electrical Engineering, University Park, Pennsylvania 16802, and A. T. Friberg is with Helsinki University of Technology, Department of Technical Physics, SF-02150 Espoo 15, Finland. Received 4 February 1986. 0003-6935/86/162645-02$02.00/0. © 1986 Optical Society of America. Although a large number of problems involving the paraxial propagation of spatially coherent electric fields have recently been analyzed with the aid of the split-step fast Fourier transform (SSFFT) method, the technique has not yet been applied to incoherent or partially coherent light beams. An obvious reason for this omission is that to describe incoherent light propagation through an inhomogeneous optical medium with the SSFFT, many individual realizations of the noncoherent electric field must be propagated, resulting in an unreasonable expenditure of computer time. A resolution of this difficulty, recently proposed by two of us (Yevick and Hermansson), is to reformulate the SSFFT in terms of Green's function matrices as follows. First, an equidistant set of N transverse grid points χ1 = XL, X2 = xL + Δ χ , . . . ,ΧΝ = xL + (N 1)Δχ is specified along a line or plane at z = 0 perpendicular to the optical axis. For a given transverse grid point Xj, we consider the electric field distribution Ei(xj) = δij, where δij, is the Kronecker delta function which is one on the given grid point and zero on the remaining points. After propagating this electric field a distance Z through an inho­ mogeneous optical medium with the aid of the SSFFT, we obtain an output electric field vector Ep(xq). The SSFFT method involves repeated application of a three-step procedure according to which the electric field is first fast Fourier transformed and propagated a distance Δz/ 2 in a homogeneous medium with a refractive index equal to some representative index n0 of the optical medium. Next, the field is inverse Fourier transformed and multiplied by a phase term obtained by exponentiating i n 0 zπ/λ times the average of [n(x,y,z)/n0 1] over the interval zi < z < zi + Δz. Finally the field is propagated again in the homogeneous medium a distance z/2. Here λ denotes the vacuum wave­ length of the monochromatic incoming light beam. Special­ izing for simplicity to a 2-D system, we may summarize this procedure in the following formula, valid to order (Δz):