The performance of the perfectly matched layer absorption boundary condition is fully exploited when it is applied to the planewave based transfer-scattering matrix method in photonic crystal device simulation. The mode profile of one dimensional dielectric waveguide and the optical properties of sub-wavelength aluminum grating with semi-infinite substrate are studied to illustrate the accuracy and power of this approach.
© 2008 Optical Society of America
The introduction of the photonic crystal (PC) concept leads to a new research field of applications of electromagnetic (EM) propagation in periodic dielectric and metallic structures [1,2]. Existence of photonic band gaps and controllable non-linear dispersion relation are two examples of novel EM properties of such structures . Numerous studies over a wide frequency range from microwave to visible light have been performed. Structures with various embedded defects such as waveguide and resonant cavity are excellent candidates for future components of optical integrated circuits [4–6]. Many numerical simulation methods have been developed to understand the principle of different PC structures and to design devices with specific functions. Typical numerical simulation methods include finite-difference time-domain method (FDTD) , finite element method (FEM) , planewave expansion method (PWE) , and real space transfer matrix method (RTMM) . However, there is still a high demand for faster and more versatile simulation tools capable of larger calculation domains both in scientific research and industrial applications.
The recently developed planewave based transfer-scattering matrix method (PTMM) which combines the advantage of PWE and RTMM has the potential to be a good simulation tool [11,12]. As a frequency domain method, PTMM can accurately calculate each individual frequency without introducing finite time span convolution or transient effects. This advantage can be used to efficiently and accurately retrieve resonant frequencies and Q values in very high Q resonant cavities . PTMM can also treat dispersive, magnetic and anisotropic material naturally in the same framework. The newly developed aspects of PTMM such as spectrum interpolation , higher-order incidence  and curvilinear coordinate transformation  have extended PTMM’s versatility in application.
The intrinsic boundary condition used in standard PTMM is the Bloch periodic boundary condition. For perfect PC simulation, PTMM returns the properties (such as spectrum, band structure) for the infinite PC structure. For more interesting defective PC structures, the supercell concept and periodic boundary condition (i.e. an infinite array of defects) can be adopted to approximate a single defect embedded in the infinitely large PC. The convergence of increasing the size of the supercell for a resonant cavity embedded in the layer-by-layer woodpile PC has been studied recently . Periodic boundary conditions are also used in FDTD and other popular numerical methods to simulate infinitely large PC structures.
In experiments and real applications PC structures are all finite in size, and it is usually important to study the defect structure within a finite PC. The perfectly matched layer (PML) absorption boundary condition first introduced in FDTD can be adapted to PTMM to simulate finite PC structures . Although the performance of PML has been extensively studied in FDTD, there have been very few reports on PML applications to the PTMM method [11,12]. In this paper, we systematically studied the performance of PML in PTMM with detailed benchmark results and related applications. Section 2 of this paper reviews the concept of PML and studies the performance of the Z-axis PML in PTMM. Section 3 of this paper studies the performance of the XY-axis side PML with the calculations of one-dimensional dielectric waveguide structure. Section 4 of this paper applies PML into sub-wavelength grating (one-dimensional PC structure) to simulate infinitely long substrate. Section 5 concludes the paper.
2. A review of the concept of PML and benchmark for the Z-axis PML
In FDTD simulations, absorption boundary conditions must be applied to terminate the calculation domain to simulate an unbounded region . The concept of PML was first introduced in 1994 by J.P. Berenger to improve the performance of absorption boundary conditions [16,17]. Further studies derived directly from the Maxwell equations showed that perfect matching with no reflection can be realized with an artificial material called uniaxial PML with appropriate tensor form of electric permittivity (dielectric constant) and magnetic permeability [18,19]. In PTMM, magnetic and anisotropic material can be treated naturally which makes the idea of uniaxial PML with permittivity and permeability both tensors very easy to implement [11,12].
The illustration of Z-axis PMLs with s wave and p wave incidence are given in Fig. 1(a) and Fig. 1(b). The propagation direction is from left to right with incident angle θ, and the interface of the Z-axis PMLs (XY plane at Z=0) is perpendicular to Z axis. The isotropic region in the left side of the interface has dielectric constant ε 1 and magnetic permeability μ 1.
The anisotropic PML region right to the interface will be perfectly matched to the isotropic region if the dielectric constant tensor ε 2 and magnetic permeability tensor μ 2 are given by Eq. (1) with sz = a + bi, any complex number . For ideal Z-axis PML, there will be no reflection for planewaves of any incidence angle at all frequencies, and the transmitted wave in the PML will be exponential attenuated by a factor exp(-α) with α ∼ bz cos θ/λ (b imaginary part of sz , z thickness of PML, λ wavelength, and θ incident angle).
To test the performance of the PML at PTMM, we put a slab of Z-axis PML with thickness of 0.5a0 (a0 the lattice constant) after air. The parameter sz is chosen to be 4+4i. A wide range of normalized frequency a0/λ is calculated from 0.01 to 10 for normal incidence. The reflectance and transmittance amplitudes of s wave and p wave are similar and only s wave spectra are shown at Fig. 2(a). The reflectance amplitudes for both s and p waves are lower than 10-10; and the perfectly matched condition (no reflection at all frequency) is achieved at normal incidence. The transmittance amplitudes are determined by the attenuation factor exp(-α) with α ∼ bz cosθ/λ; and the exponential decrease of transmittance is expected with respect to normalized frequency a0/λ.
In photonic crystal research, one particular interesting frequency range is close to the first band gap which is usually around the normalized frequency 0.3–0.5. We will focus on the performance of PML at normalized frequency ϖa0 /(2πc) = 0.4 with thickness 0.5a0 and sz =4 + 4i;. The transmittance and reflectance amplitudes as functions of the incident angle are shown in Fig. 2(b) the reflectance amplitudes are always below 10-10; and the perfectly matched condition is achieved for all incidence angles. The transmittance amplitudes increases exponentially for large incidence angle and approaches 100% due to the fact that the attenuation factor α ∼ bz cos θ/λ approach zero as θ approaches 90 degree; but for a moderate incidence angle (for example 30 degrees), the transmittance amplitudes are still below 1%.
Two strategies are studied to improve the performance of PML in term of attenuating the transmission: one is to increase the imaginary part of sz and the other is to increase the thickness of the PML. The transmittance and reflectance amplitudes of both methods are illustrated in Fig. 3: double the thickness of PML and double the imaginary part of sz. As shown in Fig. 3(a), the performance of PML is improved for all incident angles, but when the incident angle approaches 90 degree, the transmittance amplitudes always approaches 100%; while the perfectly matched condition (no reflection) still remains valid for all angles [Fig. 3 (b)]. Further studies such as grading of PML are required to further improvement of the transmittance attenuation performance. Even without grading, the reflectance amplitudes are below 10-10 which is already better than many of the complicated grading PML approaches in FDTD .
3. XY-side PML and its application on one-dimensional dielectric waveguide structure
To get perfect matching in the X or Y boundary within the XY plane is desirable in some situations (for example, to eliminate the crosstalk between neighboring supercells in the XY plane). We can adopt a similar approach: to match X boundary, Eq. (2) should be used; to match Y boundary, Eq. (3) should be used; and to match the corner of XY boundary, the matrix product of right side of Eqs. (2) and (3) should be used . Fig. 1(c) and 1(d) illustrates three types of XY plane PMLs applied to a 3x3 supercell structure with a defect embedded at center. It is tricky to get direct performance of the X- or Y-axis side PML because in PTMM we can only collect spectra information in the propagation direction (Z-axis), while the structure is periodic along the X and Y axis direction. To get an idea of the performance of X or Y side PML, the indirect approach of electric field distribution comparison with analytical results will be utilized.
The one-dimensional infinite long dielectric single waveguide is a standard example used in many text books to illustrate guided wave properties . Analytical results of the EM field distribution can be derived with the knowledge of geometry and material parameters: width of the waveguide, the dielectric constants of the waveguide and background, and the frequency of guided EM wave. Meanwhile, PTMM can be used to calculate the EM field distribution of periodic one-dimensional waveguides without side PMLs and with side PMLs. Then those EM field distributions can be compared to check the performance of side PMLs. In the comparison the parameters of the one-dimensional dielectric waveguide are as following: EM wavelength 2.5μm, dielectric constant of waveguide 9.0, dielectric constant of background 1.0 (air), and the width of the waveguide 2.5 μm.
First, PTMM is used to calculate a finite length waveguide which is periodic along Y axis and uniform along X axis [Fig. 4(a)]. Due to the reflection at the interface between the right end of the waveguide and the air background, strong interference should be expected. Fig. 4 (a) illustrates the interference pattern of electric field with the dashed box representing the position of the waveguide. Next we put a layer of Z-axis PML of thickness 0.5μm at the end of the waveguide to simulate an infinite long waveguide; at this stage the waveguide is still periodic along Y axis. Fig. 4 (b) shows the electric field profile with dashed boxes representing waveguide and PML locations; the strong interference disappears. Then side PMLs and corner PMLs are added on both Y axis boundaries and the YZ corners, and the electric field profile is shown on Fig. 4 (c). To get more precise numerical comparison between the electric field profiles, we take the Y-axis cross section at the Z-axis position 0.1μm in front of the Z-axis PML.
The Y-axis cross section of electric field magnitude for Fig. 4 (b) and Fig. 4 (c) are plotted in Fig. 4 (d). The blue curve in Fig. 4 (d) is the electric field magnitude for waveguide with only Z-axis PML (i.e. the periodic along Y-axis infinite long in Z-axis waveguide array case). The red curve in Fig. 4 (d) is the electric field magnitude when Z-axis, Y-axis side and YZ-corner PMLs are used, i.e. the single infinite long waveguide case. To get the performance of the Y-axis side and YZ-corner PMLs, the analytical solution of the electric field magnitude of the single infinite long dielectric waveguide is also plotted [black curve in Fig. 4 (d)]. The difference between the analytical solution and the PTMM calculation with Y-side and YZ-corner PMLs is much smaller than the difference between the analytical solution and the PTMM calculation without Y-side and YZ-corner PMLs, which indicates the Y-side and YZ-corner PMLs do efficiently cut the cross talk between neighborhood supercells.
4. Application of PML to sub-wavelength metal grating at visible wavelengths
One of the simplest PC structure is the one-dimensional grating which has been studied in the optical community for decades. In this section, we simulate the performance of an aluminum sub-wavelength grating structure as polarized beam splitter for visible light. The height of the grating is 136nm, and the period is 170nm with the filling ratio of aluminum 40% (i.e. the grating metal width is 68nm and the grating air width is 102nm). Both the finite thickness (50μm) substrate case and the infinite thick substrate case (which is realized by putting a Z-axis PML at the end of the substrate) are calculated.
The incidence angle θ of the incident planewave is set as 7 degree (inset of Fig. 5). Due to the strong reflection at the end of the substrate, the reflection spectrum of p wave shows strong interference at the finite substrate case [Fig. 5(a)]. This strong interference disappears for the infinite substrate and a smooth reflection of p wave is observed [Fig. 5(b)] with a minimum reflection rate (approximately zero) at wavelength 0.45μm. The corresponding results of s-polarized waves for both finite and infinite substrates are approximately constant (90%) over the whole visible frequency range. At a wavelength of 0.45μm the s and p polarization beams can be separated. To get a better understanding of the EM propagation at this wavelength, the electric field mode profiles for the infinite thick substrate grating are plotted for both s and p waves at Fig. 6. The s wave is Ey dominated, and strong reflection occurs at the front of the grating which corresponds to high reflectance. On the other hand, the p wave is Ex dominated and the electromagnetic energy can propagate through the grating and substrate.
In this paper, we systematically review the performance of perfectly matched layer in planewave based transfer-scattering matrix method, and apply it to sub-wavelength metal grating at visible wavelength. One of our recent research shows that a coordinate transformation can be applied to PTMM to simulate curved waveguide structure. The result of the coordinate transformation is very similar to the PMLs studied in this paper except that the parameter of sz is a real function of coordinate . Although PML is an artificial material first introduced for numerical simulation purpose, researchers are enthusiastically seeking PML-like materials or structures of controllable electric permittivity and magnetic permeability [21, 22]. PTMM can also be used in the design of such permittivity and permeability controllable materials just like the implementation of PML absorption boundary condition.
References and links
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