Chapter 3
Simulation and optimization of T - shaped dielectric gratings
While standard dielectric coatings could be simulated well by the Transfer Matrix Method (TMM), a Surface Relief Grating (SRG) needs to be numerically described by two- or three-dimensional simulational methods. Such grating structures are presented in Sec. 1.1.2. In Chapter 2 four different numerical methods are described in details, all of them can be applied for two- or three-dimensional simulations, but depending on the requested physical quantity some of them fit better to the problem.
The purpose of this chapter is to explore thoroughly the reflection and transmission properties of a T-shaped grating structure introduced in [19,20] under the variation of its geometry. The operation range for the wavelength is selected in the Near-InfraRed (NIR) range:1−2µm. First we demonstrate how the spectral range of maximal (nearly full) reflection of a broadband mirror can be increased for normal incidence. Then spe-cial mirrors are being worked out, which are obtained from the basic T-shaped SRG by optimizing the dimensions of the T-shaped ridges: a bandpass filter at normal incidence, a short- and a longpass dichroic filter for 45◦ angle of incidence. The organization of the chapter is as follows: The properties of the initial surface relief grating are summarized in Sec. 3.1. The optimization procedures and the obtained mirror structures are presented in Sec. 3.2.
3.1 Geometry and optical properties of the initial
periodic in the x direction with period lengthΛ = 700 nm , translation invariant in the z direction, and the normal incidence is parallel to the y direction in accordance with the notations for gratings in Sec. 1.1.1. The T-shaped structure can be divided into layers, the bottom layer refers to the stem of the T, while the top layer refers to the bar of the T. The width of the top layer is wtop = 392 nm, the width of the bottom layer is wbottom = 182 nm, the height of the top layer is dtop = 350 nm, and the height of the bottom layer is dbottom = 430 nm.
For this SRG an equivalent layer structure (two homogeneous layers on a substrate) can be defined by the effective medium approximation. Here effective refractive indices are used, which depend on the duty cycles of the layers. The value of the applied effective refractive index is the integral average of the refractive index inside the layer over the length of period. The duty cycle is the ratio of the width of the structure inside the layer to the length of period (w/Λ). We get a high-low-high refractive index sequence as the top layer acquires high duty cycle and the bottom layer has low duty cycle. The index contrast can be changed via optimization of the structure. As the structure characteristic size gets close to the wavelength, the effective medium approximation provides just rough, inaccurate result, thus in order to obtain fine and accurate details of the spectra, rigorous numerical methods are applied, which are detailed in Chapter 2.
Fig. 3.1. Initial geometry of the SRG under consideration.
The optical properties of this SRG were detailed in [19,20]. First, to validate their findings the light wave propagation has been simulated through the structure for different angles of incidence in the NIR (1100 - 1900 nm) spectral range for Transverse Magnetic (TM) polarization. The resulting reflection map is shown in Fig. 3.2. The source code
Fig. 3.2. Reflectivity map of theSRG: the horizontal and vertical axes are the wavelength of the illuminating field and the angle of incidence, respectively.
of the program resulting this figure is listed in Appendix A as PRG Fig. 3.2. There is a wide reflective plateau around 1550 nm in the spectrum for normal incidence. The figure also indicates that the structure has quite good tolerance to the variation of the angle of incidence: between 0◦ and 15◦ the maximal reflection plateau barely changes.
Energy conservation was always considered and checked in the course of simulations.
Convergency of the simulational results when higher resolution is used by the simulation was also taken into account, and so the critical parameters for different methods were optimized in terms of the accuracy. However, only energy conservation and convergence do not warrant the validity of numerical results. Therefore, independent numerical methods were used and the results were directly compared validating each other. Altogether four methods were used for simulations: the Finite Difference Time Domain (FTDT) method, the Finite Difference Frequency Domain (FDFD) method , the Rigorously Coupled Wave Analysis (RCWA), and the Method of Lines (MoL) (detailed in Chapter 2).
In case of theFTDTmethod the simulation is done by stepping in time and refreshing field values according to Eqs. (2.6a) - (2.6c). Let us denote the elementary step size in space as ∆x, then the resolution of the distance corresponding to one wavelength
Fig. 3.3. Simulation layout for theFDTD method.
is Nλ = ∆xλ0. In order to achieve reasonable numerical accuracy, usually Nλ ≥ 20 is chosen. The elementary time step is strongly connected to the space step via the numerical stability condition (see Sec. 2.1.2), in two-dimensional simulations it is usually selected as c∆t = ∆x√
2. Implementation of a source in the system is not trivial, the Total Field - Scattered Field (TF-SF) method is used [81]. Periodic Boundary Condition (PBC) is applied in the periodic spatial direction of the structure (xaxis) and Uniaxially Perfectly Matched Layer (UPML) as boundary condition to close the simulation area in the air and in the substrate (see Fig. 3.3) (y axis). For the simulation of the SRG structure
∆x = ∆y = 5 nm resolution and a grid size of of 140×240 is selected. The source is implemented in the source plane as
Ex(t) = E0sin (2πf(t−T3))e−
(t−T3)2 2T2 ,
where T3 = 3T, T = 1/f, f = c/λ0, λ0 = 1550 nm and E0 = 1V/m. Detector planes are applied to determine the flux of the reflected and transmitted light. The reflectivity is computed by integrating the Poynting vector after Fourier transformation in the time domain. The simulation layout can be seen in Fig. 3.3.
The same simulation setup can be applied for the FDFD method (Fig. 3.3), and the reflectivity can be computed again by integrating the Poynting vector as in the FDTD method. The field distribution for each frequency component is directly obtained by in-version of theFDFDmatrix in Eq. (2.41). As an example the field distribution is shown in Fig. 3.4 around a T-shaped ridge at 1100 nm (R ∼= 0%, left-hand panel, the appropriate
Fig. 3.4. The distribution of the electric and magnetic fields in the structure at 1100 nm (R ∼= 0%, left-hand panel) and at 1550 nm (R ∼= 100%, right-hand panel). The magnetic field strength is displayed as a density plot and the electric field as a vector field, the magnitude of the fields are normalized to the incident magnetic and electric fields, respectively.
source code in Appendix A is PRG Fig. 3.4) and at 1550 nm (R ∼= 100%, right-hand panel, the appropriate source code in Appendix A is PRG Fig. 3.4). The magnetic field strength is displayed as a density plot and the electric field is represented as a vector field, the magnitudes of the fields are normalized to the incident magnetic and electric fields, respectively. For the simulation of the SRG structure ∆x= ∆y = 5nm resolution is applied, the grid size is 140×280. The resolution of the wavelength range is ∆λ = 10 nm, the source is a plane wave implemented by theTF-SFmethod described in Sec.2.2.3.
For the simulation of theSRGstructure with the MoLmethod,∆x= 1 nm resolution is chosen, so the vector size is 700. The resolution of the wavelength range is ∆λ = 10 nm, the source is a plane wave, implemented via the cinc parameter detailed in Sec. 2.3.
For the RCWA the resolution is selected in the Fourier space Nf = 30, that means one structure in real space is built up in Fourier space from 61 harmonics: −Nf. . . Nf. The resolution of the wavelength range is ∆λ = 10 nm, the source is implemented the same way like for MoL. Table 3.1 includes a summary chart of simulational parameters for the methods described above, where BC stands for Boundary Condition.
Fig.3.5 exhibits the correctness of the numerical results. In case of normal incidence, the reflection spectra computed with four different methods were captured and plotted simultaneously with colored curves. The source code for all the four methods are provided
parameter FDTD FDFD MoL RCWA
∆x/Nf 5 nm 5 nm 1 nm 30
c∆t/∆λ ∆x/√
2 10 nm 10 nm 10 nm
BC inx PBC PBC PBC PBC (inherently)
BC inz UPML UPML Refl. and trans. layer Refl. and trans. layer
Source TF-SF TF-SF Plane wave Plane wave
Table 3.1. Summary chart of simulational parameters for the methods.
Fig. 3.5. Direct comparison of reflectivity plots resulting from independent numerical methods forTMpolarized field (colored curves). Throughout the computations the initial SRG geometry was used and the grating was illuminated at normal incidence. In this structure the reflection of TEpolarized field (dashed curve) is much smaller.
in Appendix A: PRG Fig. 3.5 FDTD for the FDTD method, PRG Fig. 3.5 FDFD for the FDFD method, PRG Fig. 3.5 MoL for the MoL method, and PRG Fig. 3.5 RCWA for the RCWA method. The curves nearly coincide, the total reflection plateau, the low and high ends of the spectrum are in good agreement. For binary waveguide SRGs, the efficiency of the ±1 diffraction order for TE polarized light is significantly lower than for TM polarization [17]. This result indicates that the reflectivity forTE polarized light (dashed line in Fig. 3.5) is much smaller than for TMpolarized field incident on the same T-shaped grating.