This work analyzes a new simulation approach to the evaluation of the time-domain electromagnetic (EM) field by reducing the number of equations to solve. Scalar Helmholtz-equations are utilized in order to determine the electric and magnetic Hertzian-potentials that yield the EM field. The method is applied to the example of optical waveguide arrays by considering the field-perturbation effect due to high dielectric contrast and dielectric discontinuities. The rigorous Hertzian potentials formulation is extended to bi-dimensional structures.
©2006 Optical Society of America
Numerical simulation has become a powerful and important technique for solving electromagnetic problems. This is due to increased computer performance as well as to growing complexity of problems that must be solved. Time domain methods such as finite-difference time-domain (FDTD) , transmission line matrix (TLM) , and wavelet- Galerkin  methods are gaining importance by virtue of their versatility and the natural way in which they simulate what happens in reality. Nevertheless, at present these methods are limited by available computer memory and computational time required in the implementation of Maxwell’s equations. Several beam propagation method (BPM) algorithms have been proposed – for handling dielectric structures. However, all those algorithms are based on formulations in the frequency domain and are therefore not directly applicable for the simulation of pulse propagation and reflection. FDTD can be used naturally for simulation of full-wave propagation and reflection in the time domain, but it is time- and memory-intensive and not suitable for structure of large optical dimensions. In this direction we propose an efficient numerical algorithm to solve the EM field by using the Hertz vectors – starting from the Helmholtz scalar equation. –. The number of equations to solve is reduced to two scalar equations instead of six in the one-dimensional case. Subsequently all the EM components are obtained by the Hertzian potentials –. In our work, we analyze the example of one and bi-dimensional waveguide dielectric arrays at high frequency. The first structure is a polarizing beam splitter (see Fig. 1) – ,in which two type of thin dielectric films, with high (n1) and low (n2) refractive index alternate periodically (i.e., grating period is much smaller than the wavelength of the incident field). The second structure analyzed is an antireflection (AR) coatings for photonic devices – (see Fig. 2). AR coatings are required for devices as the semiconductor optical amplifiers (SOA), the tunable external –cavity semiconductor laser, and the master oscillator power amplifiers (MOPAs) –. In the frequency domain, the transmission and reflection properties of dielectric discontinuities may be derived by means of an equivalent circuit , ,  that automatically ensures continuity of the fields and their first derivatives along the axis z. If potentials are used, instead, second derivatives are involved and generators are necessary at each dielectric interface (see Fig. 3). In this work we (i) describe the time-domain field solution using Hertzian Potentials approach: one of the reasons for introducing the Hertzian potentials is the possibility of reducing the number of equations to be solved, in fact, only two scalars are needed to represent the field; (ii) we simulate one-dimensional dielectric multilayer optical waveguides by considering the accurate transmission line model, inclusive of generators, of Fig. 3 in the time domain, and finally (iii) we extend the model to an ASR bi-dimensional structure.
2. Hertzian field produced by dielectric discontinuities
where a is unit vector. From (1) it is possible to evaluate all the components of the EM field as
By writing (2) explicitly in the Cartesian coordinates all the electromagnetic field components are given by
In the next section, we report the discretised solution of the scalar potentials Ψe, h(x, y, z, and t) and the implementation formulae. It is known that the scalar wave equation may lead to inconsistencies because, in an inhomogeneous medium, it is, in general, not equivalent to Maxwell’s equations. Electromagnetic scattering problems, including free space, involve the calculation of the fields produced in the presence of geometrical discontinuities by arbitrary currents. Such discontinuities may be replaced by equivalent generators  (see Fig. 3), giving an accurate solution of the EM field for structures with high dielectric contrast. In fact, the scalar wave Eq. (5) for a non-dissipative medium can be rewritten as 
represent the dielectric polarization and for the one-dimensional case:
Therefore we solve Eq. (6) in the dielectric array and in proximity of the dielectric interfaces, and Eq. (5) in the homogenous region. In appendix A, we report the iterative solution of Eq. (6) and the parametric differences with the solution of Eq. (5) for the mono-dimensional case.
3. Algorithm description: finite-difference (FD) method implementation and ABC Mur condition.
In this section, we describe the algorithm of Fig. 4 used for the simulations. In order to reduce the computational time before the starting of the FD kernel we set the permittivity mask of the structure and establish the discretized region in which we solve the homogeneous Helmhotz scalar equation for a non-dissipative medium , 
Or the inhomogeneous scalar equation
and for the second order by
where x=0, h and y=0, h represents the boundary-coordinates of a bi-dimensional domain (x, y). We observe that for a bi-dimensional structure matrix Eq. (13) is three-dimensional. After the temporal loop, we use the values of Ψe, h to evaluate the EM field by Eqs. (2), (3), and (4). By performing the Discrete Fourier transform (DFT), we obtain the reflectivity by considering the Fourier transform of the reflected electric field. In fact, we evaluate the reflection coefficient by using the ratio Ey reflected/ Ey homogeneous (transmission without the layers), where the reflected field Ey reflected is given by the difference between Ey and Ey homogeneous.
4. Iterative solution of inhomogeneous scalar equation
In order to highlight the parametric differences between the homogeneous and inhomogeneous solution of the wave Eq. (9) and Eq. (10) we use Eq. (11) and Eq. (12). In this manner, the solution of Eq. (5) in the iterative form can be expressed as:
Instead, the inhomogeneous solution of Eq. (10) is expressed by:
The first simulated structure is the dielectric multilayer structure of Fig. 1. In particular Fig. 5 shows the Ey component evaluated one cell before the multilayer structure, for 15-layer structure of Si and SiO2 materials, in particular n1=3.48, d1=0.11μm, n2=1.44, d2=0.26μm. The structure is discretized by 70 domain cells with Δz=4∙10-8 m, the first five cells and the last five ones are filled by air, Δt =1.33∙10-16 sec. The input signal is a sinusoidal signal with wavelength 1.523 μm modulated by a Gaussian pulse. We compare in Fig. 6 (20-layer structure with n1=3.48, d1=0.1μm, n2=1.465, d2=20 nm) and Fig. 7 (20-layer structure with n1=3.48, d1=0.1μm, n2=1.465, d2=30 nm) the theoretical values of the reflection coefficient with the numerical ones (“FDP” finite difference perturbed solution). We observe that the reflection coefficient of a dielectric multilayer is obtained from
where l is the number of the layers, pj = njcos (θj) (j=0, 1, 2… l+1), and m 11, m 12, m 21, m 22 are the elements of the characteristic matrix M=M1∙M2…Mj…Ml . We then extend the solution of (10) to a bi-dimensional structure: the ASR simulated structure of Fig. 2 is characterized by ns=3.17, nc=3.524, dc=0.11μm, d1=0.1816 μm, and λ0=1.54 μm. We use 80×80 cells of domain, 11x35 cells for the central slab, 18 cells of the thickness of the first layer, 5 cells of the thickness of the second layer, Δx=Δy=0.01μm, and Δt=3.335∙10-17sec. In Fig. 8 is shown the time evolution and the contour plot of the Ez-field component. In Fig. 9, we report the comparisons between the simulated reflectivity and the experimental one by varying the thicknesses of the second layer d2. In this case, we have a good agreement with the measured data .
In this work, we suggest an alternative approach to the numerical simulation of behavior of open waveguides. The reduction of the number of equations to solve is very useful in obtaining low computational costs. The model of the dielectric discontinuities provides accurate EM field solutions close to the real behavior of the optical dielectric array. The proposed algorithm can be also extended to 3D cases and non-linear dielectric layers.
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