Introduction to Operations Management - Modeling Method and Equations

CHAPTER 5

MODELING METHOD AND EQUATIONS

1. Modeling of PCF

A convenient synergy between theory and practical experimental works is decisive for every developing research field. Theoretical study is very relevant to know various aspects of the issues for proper understanding of the research field. It is equally relevant in developing new fibers based devices. Numerical modeling plays an indispensable role for theoretical studies of PCFs because of the complex nature of the electromagnetic waves.

1. Computational Methods for Electromagnetics

Wave equations in fiber-optics are based on Maxwell’s equations.  Various computational methods distinctively or in combination used to explain wave equations along with the plane wave expansion method (PWEM) [93, 94] convenient for periodic structures,biorthonormal basis method [95,96] suitable for accurate dispersion calculations,  multipole method (MPM) [97]-[99] convenient for forecasting leakage losses of PCFs, Fourier decomposition method [100], beam propagation method (BPM) [101]-[103], localized function method [104], boundary element method [105], finite element method (FEM) [106]-[111] and finite difference method (FDM) in time domain [112] or in frequency domain [113, 114]. The index contrast between the silica and air in PCFs is generally high and therefore, the scalar wave analysis methods are not accurate to predict their propagation properties, a full-vector approach is required [115]. Among the full vector methods, the FDM is simple, accurate, and is suitable for arbitrary structured waveguides as the PCF [115]. In this thesis the APSSTM software based on the FDM in time domain is used for PCFs’ modal computations.  

1. Finite Difference Method in Time Domain

The FDM in time domain (FDTD) arranges direct time domain solutions of Maxwell’s differential equations by discretizing both the time and space using convenient number of grids. This is a powerful method which can explain almost all kinds of physical electromagnetic (EM) environments [115]. Fig. 5.1 depicts the popularity of the method.

First, the time dependent Maxwell’s equations in partial differential form are discretized using central-difference approximations to the space and time partial derivatives. Second, the emanating equations are explained in software. The electric field vectors in a volume of space are solved at a given instant of time and then the magnetic field vectors in the same spatial volume are solved at the next instant of time. The process repeats again and again until the desired field response is fully evolved.

[pic 1]

Fig.5.1 Widespread usage of FDTD method.

1. Working of FDTD Method

The FDM approximates the EM fields on grids by interlacing the E and H fields.  It is perceived that both the electric and magnetic fields are dependent in Maxwell’s differential equations.  A modification in the E-field in time is dependent on a change in the H-field in space. It means at any point in space, the updated value of the time dependent E-filed is dependent also on the earlier value of the E-filed and of the local sharing of the H-field in space [112]. Equivalently, at any point in space, the updated value of the time varying H-field is dependent on the previous value of the H-filed plus the local distribution of the E-filed in space.

Fig. 5.2(a) displays one unit cell of the physical space-time domain discretized using the Yee’s algorithm [116]. The E and H field components are interlaced in space as well as in time so that every E component is enveloped by four circulating H components and every H components is surrounded by four circulating E components. The associated time domain formulations are illustrated below. Interlacing between E and H fields during computation is shown in Fig.5.2 (b)

1. Governing Equations for FDTD

Differential forms of Maxwell’s equations for non-conductive medium can be addressed as [112, 115]:

[pic 2]                                         (5.1)

[pic 3]                                             (5.2)

Where, the fundamental relations[pic 4]are used. Equations (5.1) and (5.2) receipt the following forms for a linear, isotropic, and non-dispersive medium:

[pic 5]                                                             (5.3)

[pic 6]                                                               (5.4)

As illustrated in Fig. 5.2(b) that all the E components are updated at NΔt and all the H components are updated at (N/2) Δt.