In this study, we adopted the finite-difference time-domain (FDTD) method, to fully solve the Maxwell's equations to obtain the accurate description of laser propagation. For the calculation of the beam absorption in materials: 1) Due to the dispersive characteristic of dispersive materials, the standard FDTD is no longer suitable and cannot describe the material behavior. In this study, a method is discussed to simulate dispersive materials by changing the refractive index and extinction number of the material.
In this regard, it is possible to use only one simple standard FDTD code to simulate a general material, regardless of whether it is dielectric or dispersive. A method to increase the wavelength while maintaining the same reflectance is developed in this study, so a larger domain can be simulated.
Development and applications of FDTD algorithm
Review the history of FDTD
Maxwell’s equation
Standard FDTD alogrithm
When coding the FDTD algorithm, a "normalization" technique can be applied to the H components to obtain the same magnitude as the E components when dealing with a plane wave source.
Drude model for dispersive materials (metals)
Because of that, the electron orbits the metal ion and is affected by a damping force proportional to its speed. When external E fields are applied, electrons are governed by Newton's law of motion, E being the electric fields. where v is the velocity of the electron; t is the momentum relaxation time, describes the interaction between the electrons and metal ions, the second term in the above equation is the damping term.
When studying the physical characteristics of a certain object, only a frequency band is needed, so the permittivity at infinite frequency e¥ is no longer 1. Where wpis plasma frequency of the metal and gpis the electron collision frequency or damping constant[18].
FDTD for Drude model
The second component of the ADE algorithm involves solving En+1, and then again using a semi-implicit scheme centered on n+1/2 [21]. Thus, the ADE-FDTD algorithm for simulating a distributed medium with only one Drude pool is only a fully explicit three-step procedure[21]. The E and H fields have the same order of magnitude which provides an advantage in formulating ABC and has higher computational efficiency.
As for the magnetic field components, since there is no change in permeability, the update equations should remain the same as the standard FDTD algorithm except that H = H Z / 0 according to the normalization process, while the update equations for J are as follows: .
Numerical stability and temporal interval
This is the correlation between time and the special interval for stable solutions for the FDTD method, called the Courant stability condition.
Numerical dispersion and spatial interval
As we can see from here that the k and w are not in a linear relationship, it will definitely cause a dispersion to this system, which is called numerical dispersion. From the previous discussion that the scattering can occur even for non-scattering materials due to the difference approximation we have done to the wave equation. In fact, when the difference approach is applied, anisotropy, which means that the phase velocity is related to the propagation direction, is also introduced into the system.
Absorption Boundary conditions (CPML)
The CPML formulation is more accurate, efficient and suitable for the use of generalized material domains [21] and will be adopted for our problems. We now need to derive the explicit update equations for the E/M fields in the context of the FDTD algorithm. The discrete convolution of y at time n t D can be computed with knowledge of the nth-order floating-point operations.
A recursive relation that efficiently calculates the time progress of y can be derived, which in turn can ease the computational burden instead of doing it directly. One notable thing is that the discrete coefficients bzk,czka are non-zero only in the CPML region z with normal bounds.byjand cyj are the same except that they are non-zero in the CPML region with y normal bounds.
Introducing the laser source (Gaussian beam)
The length of the quiver represents the size of the E-fields, and the direction of the quiver represents the direction of the E-fields. In FDTD encoding, there are three common methods to introduce the source into the code: hard source, soft source, Total-Field/Scattered-Field (TF/SF)[21]. Hard source will not be part of the iteration, so no additional memory and computation time is required;
In other words, the source is part of the repeater, so no waves will be reflected when they hit the source plane. It is impossible to separate the total field and the scattered field into hard source or soft source, but it is possible in TF/SF. But this method will need more memory for the distributed area and more computing time.
Reflectance calculation by using Poynting Vector
To calculate the reflectance of a given geometric structure, we must first calculate the reflected and incoming beam energy. Calculate the power at the boundaries marked in red (3 is shown in the figure, but there will be 2 more, one at the front and one at the back since it is a three D simulation). Since for most metals, the reflectance is quite high and there is no light passing through the material (Transmittance=0.0), the absorption can be obtained from:.
Memory requirement for FDTD simulations
Since the entire FDTD code may contain key fields, properties, and most importantly ABC, there will be a fairly large number m, usually up to tens, and there are some other variables such as temporary variables that are not included.
A method of reducing the Memory requirement for single incident angle
In most cases, n is greater than one and so is k, making lmin less than l0. For FDTD simulations, we need the smallest possible distance between the grids to get accurate results. Given that only two parameters n and k control the reflectance, we can try to choose another set of n and k values to meet the R= reflectance requirement.
As illustrated in the figure above, the horizontal axis is k and the vertical axis is n; the contour lines are the reflections. For a given reflectivity, there are infinite sets of n and k that all satisfy the reflectance requirement. Meanwhile, the skin depth of metal is very small. To capture an accurate reflection, there must be at least one full grid spacing in one skin depth.
For the same memory requirement, we can run 20 gratings in one wavelength with the new parameters, while only 10 with the original ones. As we can see from fig. 3.4, the incident angle-dependent reflectance patterns for the two numerical results are quite comparable, although the smaller n and k seem to have slightly larger errors, this is because the lattice density is smaller than the original values in the vacuum side.
A method of changing wavelength to simulate large scale problems
In this case n>k, e1 is a positive number (and greater than 1), so the standard FDTD can be applied. We can choose another set of n,k that satisfies the requirement of n > kwhile keeping the same reflectance pattern. After choosing a proper set of n,k values, we can use the standard FDTD code to simulate dispersive materials as well.
For both P and S polarization the n-k value relations are the same, so we only have the same set of n-k for both S and P polarization. Red lines are original values (n=3.81,k =4.44), blue and green lines are results by changing n,k while keeping the same reflectance at 0°. For P and S polarization the n-k value relations are different, there are 2 different sets of n-k for S and P polarization.
For P and S polarization the n-k value relations are different, there are 2 different sets of n-k for S and P polarization. And it is clear from fig. 3.5, while large errors are generated at larger angles (> 60°). 2). Reference angle at small angles gives us the best results and they are almost the same.
But for an angle of 0°, only one set of n,k is required since the P and S polarizations share the same n,k. In the above figure, square mark lines are P-polarization and triangle mark lines are S-polarization. Blue lines are standard FDTD numerical results for modified values, while brown lines are dispersive FDTD numerical results for original values.
The propagation of the S and P polarization beam of Fe at 1.06 um with the original parameters in the steady state for different incident angles (from 0°- 60°) is given in Figure 3.21 (S-polarization) and Figure 3.22 (P. polarization) as a demonstration , the scheme is the same as Figure 3.1, only the location of the source is different and the vacuum-material interface tilts. Dashed lines are S-polarization, solid lines are P-polarization, and crossed lines are circular polarizations.
Keyhole absorption simulation
So the method is valid and we can use this method to calculate dispersive materials by changing n, k with standard FDTD algorithm.
Main research contents and conclusion
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Taylor, Lam DH, Shumpert TH, "EM pulse scattering in time-varying inhomogeneous media", Ieee Transactions on Antennas and Propagation, vol. Taflove A., “Numerical solution of steady-state EM scattering problems using the time-dependent Maxwell equations,” ieee Trans. Mur, "Boundary-Conditions for the Finite-Difference Approximation of the Time-Domain Electromagnetic-Field Equations", Ieee Transactions on Electromagnetic Compatibility, vol.
Beker, "Calculation and experimental validation of induced currents on coupled wires in an arbitrary shape cavity," IEEE Transactions on Antennas and Propagation, vol. S., “Detailed FDTD Analysis of EM Fields Penetrating Narrow Gaps and Covered Junctions in Thick Conductive Screens,” ieee Trans. Choi and Hoefer W J R, "Finite difference time domain method and its application to the eigenvalue problem," Ieee Trans.
S, "A numerical example of a two-dimensional scattering problem using a subgrid," Applied Computational Electromagnetics Society Journal vol. Kupka, "Light scattering in metallic nanowire arrays: Finite-difference time-domain studies of silver cylinders," Physical Review B, vol. Schatz, "Finite-difference time-domain studies of light transmission through nanohole structures," Applied Physics B- Lasers and Optics, vol.
