The broadband Green’s function with low wavenumber extraction (BBGFL) is applied to the calculations of band diagrams of two-dimensional (2D) periodic structures with dielectric scatterers. Periodic Green’s functions of both the background and the scatterers are used to formulate the dual surface integral equations by approaching the surface of the scatterer from outside and inside the scatterer. The BBGFL are applied to both periodic Green’s functions. By subtracting a low wavenumber component of the periodic Green’s functions, the broadband part of the Green’s functions converge with a small number of Bloch waves. The method of Moment (MoM) is applied to convert the surface integral equations to a matrix eigenvalue problem. Using the BBGFL, a linear eigenvalue problem is obtained with all the eigenmodes computed simultaneously giving the multiband results at a point in the Brillouin zone Numerical results are illustrated for the honeycomb structure. The results of the band diagrams are in good agreement with the planewave method and the Korringa Kohn Rostoker (KKR) method. By using the lowest band around the Γ point, the low frequency dispersion relations are calculated which also give the effective propagation constants and the effective permittivity in the low frequency limit.
© 2016 Optical Society of America
The band solutions of waves in periodic structures are important in physics and engineering and in the design of photonic, electronic, acoustic, microwave and millimeter wave devices such as that in photonic crystals and metamaterials. The common approaches to calculate the bands include the planewave method [1–7], and the Korringa Kohn Rostoker (KKR) method [8, 9]. The planewave method casts the problem into a linear eigenvalue problem and provides multiple band solutions simultaneously for a point in the Brillouin zone. However, in the planewave method, the permittivity (or potential) profile is expanded in Fourier series. For the case of abrupt profiles of large contrasts between background and scatterer, many terms are required in the Fourier series expansion. An artificially smeared dielectric function was also employed . The convergence issue of the planewave method has been reported [5, 7]. The KKR method is a multiple scattering approach [10, 11] by forming the surface integral equations with periodic Green’s function and the equation is solved by cylindrical or spherical waves expansions. The method is only convenient for scatterers with shapes of separable geometries such as circular cylinders or spheres. In the KKR/multiple scattering method, the eigenvalue problem is nonlinear requiring the root seeking procedure. One needs to repeat the non-linear root seeking one by one to find multiple band solutions. Numerical approaches such as the finite difference time domain method (FDTD) [12–13] and finite element method (FEM) [14–16] have also been used.
In this paper, the periodic Green’s function is used to formulate the dual surface integral equations. The periodic Green’s function has slow convergence. If the periodic Green’s function is used for the surface integral equation to calculate the eigenvalues, the equation is nonlinear and an iterative search needs to be performed with one band at a time. Recently, the Broadband Green’s Function with Low wavenumber extraction (BBGFL) [17–20] has been applied to wave propagation in waveguide/cavity of arbitrary shape. By using a single low wavenumber extraction, the convergence of the modal expansion of the Green’s function is accelerated. Also the singularity of the Green’s function has been extracted. The method has been shown to be efficient for broadband simulations of wave propagation in waveguides/cavity. It was noted that the expression of modal expansion of Green’s function in a waveguide is similar to the Floquet expansion of Green’s function in a periodic structure . In a recent paper  we adapted the BBGFL to calculate band diagrams in periodic structures where the scatterers obey Dirichlet boundary conditions. We use a low wavenumber extraction to accelerate the convergence of the periodic Green’s function. The BBGFL is used to formulate the surface integral equations. Next applying the Method of Moments (MoM) gives a linear eigenvalue equation that gives all the multi-band solutions simultaneously for a point in the first Brillouin zone. We label this as “broadband simulations” as the multi-band solutions are calculated simultaneously rather than searching the band solution one at a time. The BBGFL method was shown to be accurate and computationally efficient. The choice of the low wavenumber was shown to be robust as the choice of the low wavenumber can be quite arbitrary. The application of MoM makes the method applicable to arbitrary shapes. It is noted that the Boundary Integral Resonant Mode Expansion (BIRME) method  was used to find the modes of an arbitrary shaped waveguide with PEC or perfect magnetic conductor (PMC) boundary conditions, using the Greens functions of a rectangular waveguide or circular waveguide. In BIRME, a DC extraction is used for the Greens function. In this paper, we calculate band diagrams of periodic structures with general shapes of the unit cells and with dielectric scatterers embedded. Also, a low wavenumber extraction is used and the choice of the low wavenumbers is shown to be robust.
This paper is an extension of the recent paper of BBGFL on perfect electric conductor  to periodic dielectric scatterers. The case of dielectric scatterers have much wider applications in devices of photonic crystals and metamaterials. For the case of dielectric scatterers, the dual surface integral equations are derived. A distinct feature in the formulation is that we use the periodic Green’s function of the scatterers, in addition to using the periodic Green’s function of the background. The application of the broadband periodic Green’s function separates out the wavenumber dependence of Green’s function, and allows the conversion of the non-linear root searching problem into a linear eigenvalue analysis problem. The low wavenumber extractions are applied to both periodic Green’s functions for fast Floquet mode convergence and extraction of Green’s function singularities. The use of MoM makes the method applicable to scatterers of arbitrary shape. In section 2, we formulate the dual surface integral equation using the extinction theorems and the periodic Green’s functions. In Section 3, we apply BBGFL to the surface integral equations and derive the linear eigenvalue problem. In section 4, numerical results are illustrated for the band solution of circular air voids forming a honeycomb structure. The results are in good agreement with the plane wave method and the KKR method. We also describe the quick rejection of spurious modes by using extinction theorems. In Section 5, we use the lowest band around the Γ point to calculate the low frequency dispersion curves, the effective permittivity, and the effective propagation constants, all of which have wide applications in devices of metamaterials.
2. Extinction theorem and surface integral equations
Consider a periodic array in the xy plane of 2D dielectric scatterers embedded in a 2D lattice as illustrated in Figure 1. Let the primitive lattice vectors be and with lattice constant a and the primitive cell area . The scatterer has dielectric constant εrp and the background host εrb. Let Spq denote the boundary of the scatterer in the (p,q)-th cell and Apq is the domain of the scatterer in the (p,q) cell bounded by Spq.
We denote the field outside the scatterer as ψ and inside as ψ1, where ψ represents Ez for TMz polarization and Hz for TEz polarization. TMz means that the magnetic field is perpendicular to the z direction while TEz means that the electric field is perpendicular to the z direction. The extinction theorems state that
In (1a) and (1b) and are the 2D free space Green’s functions in the the background and in the scatterer, respectively. The wavenumbers are , and in the background and in the scatterer, respectively, where ω is the angular frequency and c is the speed of light.
Applying the Bloch wave condition to Eq. (1a), we have the surface integral to be only over S00 of the (0,0) cell. if is inside any scatterer, then10], the integral equations in Eqs. (2a) and (1b) are used to calculate the band solutions meaning that the background Green’s function is periodic while the scatterer Green’s function is the free space Green’s function. Instead of using (1b), we use periodic Green’s function also for the scatterer and using extinction theorem, it can be shown that (2b) is the periodic Green’s function of the scatterer. Thus, in the BBGFL formulations, we have periodic Green’s functions for both integral equations. An interpretation is that the periodic Green’s function is for an empty lattice, so that one can uses both gP and g1P.
Let be a wavevector in the first irreducible Brillouin zone , 0 ≤ β1, Β2 ≤ 1/2. and are the reciprocal lattice vectors. , . Then in the spectral domain the periodic Green’s functions for, respectively, background and domain scatterer are
The surface unknowns are and , and . There are boundary conditions for the TM case and the TE case giving a net of two surface unknowns and [10,23]. Let approach the boundary of S00, the coupled surface integral equations in and are given as follows,Eqs. (4a) and (4b) s = 1 for TMz polarization and for TEz polarization.
Applying MoM with pulse basis functions and point matching, we discretize S00 into N patches. In matrix form, we obtain
3. BBGFL in surface integral equations
We want to find the wavenumbers that satisfy both the surface integral equations in Eq. (4). These wavenumbers are the band solutions, and the corresponding surface field ψ and are the modal currents. This is an eigenvalue problem. The technique of BBGFL is described below.
3.1. Low wavenumber extraction in the Broadband Periodic Green’s function
In the BBGFL method, we choose a single low wavenumber kL, which is quite arbitrary, to decompose into and the difference . They shall be labelled as the low wavenumber part and the broadband part, respectively.
Using the spectral domain of from (3a), we then have
Note that is independent of wavenumber k, and the series in converges as instead of .
For the case of dielectric scatterer, we also decompose the periodic Green’s function of the scatterer into a low wavenumber part with k1L and a broadband part
We next express gB in the following form
It follows that
Both gB and are smooth for arbitrary and they converge everywhere including the self-point of . Because of fast convergence, we will truncate the series with M Floquet modes. Note the two dimensional index in α. (In our numerical implementation, each index is from −10 to 10, so that M = (21)2 = 441.)
With the decomposition , the matrices and (Eq. 6) are also in two parts which are the low wavenumber part and the broadband part
and are N × M matrices. The superscript † means Hermitian adjoint. The matrices and do not have wavenumber dependence. The only wavenumber dependence lies in the factor of the M × M diagonal matrix
With the low wavenumber separation and , the matrix form of the surface integral equation Eq. (5a) becomes
Using the surface integral equations with periodic Green’s functions of the scatterer which is also decomposed into a low wavenumber part and a broadband part, we have from Eq. (5b)
We also choose
In Appendix A, we describe efficient methods for calculating the matrix elements of , , and , at low wavenumbers of kL and k1L.
3.2. Eigenvalue problem
We next convert the matrix equation Eq. (14) into an eigenvalue problem. Let
Then from Eq. (13a)
Next, using the definitions of , , , , the matrix equation Eq. (14) becomes,
Next we express , in , , , ,Eq. (18) and (20), we obtain the eigenvalue problem,
The matrix dimension of the eigenvalue problem in (23a) is 4M × 4M. In the above equation, only λ depends on wavenumber while all the other matrices are independent of wavenumber. Thus the eigenvalue problem is linear with all the eigenvalues and eigenvectors solved simultaneously giving the multi-band solutions.
Knowing the eigenvalues λ, the mode wavenumbers k2 are obtained from the relation of , (Eq. (9c)). The modal surface currents distributions and are calculated from the eigenvectors through Eq. (20). Knowing and , we can also compute the modal field distribution ψ and ψ1 everywhere in the lattice by Eq. (2). The authenticity of the eigenmodes can also be checked by the extinction theorem (Eq. (2)) away from the boundaries.
4. Numerical results
We consider the triangular lattices with lattice constant a as shown in Figure 1. The dielectric background has permittivity εb, and the scatterers are circular air voids of radius b drilled in the background with εp = ε0, where ε0 is the free space permittivity. We consider two cases with case 1 b = 0.2a, εb = 8.9ε0 and filling ratio of 14.5% and case 2 of b = 0.48a, εb = 12.25ε0 and filling ratio of 83.6%. We calculate the solutions using BBGFL method. We also calculate the solutions using the plane wave method  and the KKR method  and compare the results of BBGFL with these two methods.
The Γ, M, and K points are
We plot the band solutions with following Γ → M→ K → Γ.
The normalized frequency fN (k) is defined as
In the following computations, we choose the low wavenumber kL such that fN (kL) = 0.2. The choice of kL is quite arbitrary as kL values in a range will work. This makes BBGFL method robust. In the examples, the maximum number of two-dimensional (2D) Bloch waves in BBGFL is 441 corresponding to −10 ≤ m, n ≤ 10, so that the highest m, n index is 10. This leads to an eigenvalue problem of maximum dimension 4M = 1764.
Case 1: b = 0.2a, εb = 8.9ε0, πb2/Ω0 = 14.5%
The band diagram is plotted in Figure 2, with MoM discretization N = 80, and Dn (as defined in Eq. (27)) at order 4 to compute gP and ∇′gP at the single low wavenumber kL. We compute solutions using the plane wave method with 1681 Bloch waves. Note that we only use 441 Bloch waves in the BBGFL method. The agreements between the two methods are excellent. In Figure 3, the surface modal currents, unnormalized, are plotted for the first few modes near Γ point with .
In Table 1, the convergence of the lowest mode with respect to the number of Bloch waves used in BBGFL using different low wavenumber kL are tabulated for TMz polarization with . It is noted that the choice of kL is robust, and fewer Floquet modes are needed as one choses a lower kL.
Case 2: b = 0.48a, εb = 12.25ε0, πb2/Ω0 = 83.6%
The parameters for this case are the same as in . The area ratio of scatterers to background is high at 83.6%. The dielectric constant ratio of εb/εp is as high as 12.25. In this case, the field varies more rapidly along the perimeter of the scatterer than the previous case and we choose the MoM discretization to be N = 180, and Dn (as defined in Eq. (27)) is selected at order 8 to compute gP and at the single low wavenumber kL. The band diagram is plotted in Figure 4. For the planewave method we used 6561 Bloch waves which is significantly larger than M = 441 Floquet modes for BBGFL. The KKR solution of the first few modes at the points of M, and K are also given (Ref.  has the complete KKR results). The band gap between the third and fourth bands is noted. The agreements between the three methods are in general good. Note that the planewave method predicts slightly larger higher order modes than the BBGFL. The differences decrease as more Bloch waves are included in the planewave expansion. Note that the planewave method results shown are not exactly the same as the results reported in , possibly due to the different treatment of the Fourier series expansion of the dielectric function [4, 5]. In our implementation, we do not apply any smearing function to the permittivity profile ε(r), and directly compute the Fourier transform of ε−1 ( ). For the KKR method, more terms in the Dn series are needed as frequency increases. In Figure 5, the unnormalized surface modal currents are plotted for the first few modes near Γ point with . The modal currents tend to have more variations as the modal wavenumbers increase.
The eigenvalues of the BBGFL approach include spurious modes. All the spurious modes are quickly rejected based on the following simple calculations. The spurious modes do not satisfy the extinction theorem as given in Eq. (2). If we evaluate the field inside the scatterers using gP, or evaluate the field outside the scatterers using g1P, we get non-zero values for the spurious modes. These modes can be identified by several points calculations inside or outside the scattering using the surface integrals. Some of the spurious modes do not satisfy one of the two surface integral equations Eq. (4), and for these modes, their modal currents ψ and on the boundary are essentially zeros and trivial. For the rest of the spurious modes, they satisfy the surface integral equations Eq. (4), but do not obey the extinction theorem Eq. (2), and these modes are shown to be nearly invariant with respect to , with values close to the roots of Jn (kb), where Jn is the n-th order Bessel function.
The CPU time is recorded when running the BBGFL code in Mat lab R2014b to compute the band diagram on an HP ProDesk 600 G1 desktop with Intel Core i7-4790 CPU @ 3.60 GHz and 32 GB RAM. Since the computation for different are independent, the CPU time for one will be described. For case 2 with the larger N and higher order of Dn, it takes a total of ~96 sec to complete the mode analysis at one , of which ~11s is spent on the eigenvalue analysis. More than 80% CPU time is spent on computing the matrices , and at the single low wavenumbers kL and k1L respectively for background periodic Green’s function and scatterer periodic Green’s function. The calculation of D (and by changing k to k1) as defined in Appendix A to facilitate the computation of gP and g1P takes ~24s, and the calculation of the low wavenumber matrices using Dn takes ~27s on the boundary. Another ~28s are needed for solutions away from the boundary as needed in checking the extinction theorem. The time recorded is for one polarization, and for the second polarization the added time is very small since only the eigenvalue problem is to be resolved. The approach is much more efficient than the KKR approach which is based on the evaluation of Dn at every frequency that needed to be used to search the band solution k, with one band at a time. The BBGFL can also be more efficient than the planewave method when the planewave method requires significantly larger number of Bloch waves as in the second case with large dielectric contrast and large filling ratio. Note that for the case of infinite contrast of perfect electric conductor (PEC), our previous paper  shows that BBGFL also works.
5. Low frequency dispersion relations, effective permittivity and propagation constants
In this section, we illustrate the low frequency dispersion relations which are useful for the design of devices in photonics and metamaterials.
In metamaterials, the scatterers and the lattice spacings are subwavelengths. Effective permittivities and effective propagation constants have been calculated for random media of dielectric mixtures [23–28]. In random medium, the positions of scatterers are random creating random phase in propagating waves. Every realization has different positions of scatterers although each realization has the same statistics and the field solutions of different realizations are different. The waves are decomposed into coherent waves and incoherent waves. The coherent wave has definite amplitude and phase while the incoherent waves have random amplitudes and phases giving speckle. In low frequency, the coherent waves dominate while at higher frequencies, the incoherent waves dominate. In random medium, the effective permittivity and the effective propagation constant are that of the coherent waves. On the other hand, the geometry in a periodic structure is deterministic with a single solution. The quasistatic method has been used to calculate the effective permittivity of a periodic medium . It is interesting to note that at very low frequency, the effective permittivities as derived for random medium and for periodic medium are the same. In the following, we associate effective permittivity and effective propagation constant with the low frequency dispersion relation of the lowest band in the vicinity of the Γ point.
In Figures 6 and 7, we plot the ω − k lowest band near the Γ point for case a and case b, respectively. These will be labeled as low frequency dispersion curves. The dispersion relation curves are plotted for moving along the line of ΓM, and along the line of ΓK in the first Brillouin zone, respectively. The points M and K represents the largest anisotropy in the first irreducible Brillouin zone. In Appendix B we derived the effective permittivity and effective propagation constants . The corresponding dispersion curves are straight lines and are also plotted in Figures 6 and 7.
The low frequency dispersion relations of the lowest band are close to the effective permittivity curves at low frequency and deviate as increases. At higher frequencies, towards the M and K points, they show significant departures when are close to M and K point. Thus as frequency increases, the effective propagation constants and effective permittivity are dispersive and anisotropic. The dispersion relations are plotted for both TM polarization and TE polarization. For the same ω, TM wave in general has a slightly larger k. The difference between the two polarization increases as the permittivity contrast and the scatterer filling ratio increases.
The BBGFL method also works for infinite permittivity contrast as in the case of PEC . In Figure 8, we plot the ω − k dispersion relation for the PEC scatterer case, with the PEC cylinder radius b = 0.2a, and the background permittivity εb = 8.9ε0. The behavior of the TE wave is similar to that of the dielectric case. But for the TM wave, as indicated by the quasistatic mixing formula that the mixture does not have a finite quasistatic effective permittivity. The ω − k dispersion relation from the lowest band has similar behavior to that of the plasma . The periodic structure behaves like a plasma for the TM wave that the wave could only propagate when its frequency is larger than the corresponding plasma frequency.
In this paper we have extended the BBGFL approach, previously used for Dirichlet boundary conditions , to calculate band solutions and the low frequency dispersion relations of dielectric periodic structures in 2D problem with 2D periodicity. In the BBGFL approach, we solve the dual surface integral equations with MoM using the broadband periodic Green’s function of both the background and the scatterer. The Broadband periodic Green’s functions are fast convergent because of low wavenumber extraction. Because MoM is applied, the method is applicable to arbitrary scatterer shapes and arbitrary filling ratio and permittivity contrasts. The BBGFL approach has the form of a linear eigenvalue problem. The method is shown to be efficient and accurate. The choice of the low frequency wavenumber has been shown to be robust because MoM is applied. The method, in principle, is applicable to arbitrary scaterer shapes and arbitrary filling ratio and permittivity contrasts. We are studying these cases. We are also extending the method to 3D problems with 3D periodicity.
Appendix A Evaluation of periodic Green’s function and the matrix elements of and at the single low wavenumber
We only need to compute Smn, Lmn and , as defined in Eq. (6), at the respective single low wavenumbers of kL and k1L. The spatial and spectral series summations as given in Eq. (3) converge slowly especially when . Thus we seek to subtract the primary contribution [10, 21], separating into the primary part and the response part .
It follows that
Expanding the Bloch wave exp into Bessel functions in the spectral domain expression of gP (Eq. (3)),Eq. (27) and (29), it follows that
Note that ρ is arbitrarily chosen in this expression. ρ should avoid the zeros of Jn(kρ) and not be too close to 0. The number of Floquet modes used in the above expression is much larger than in evaluation of gB using Eq. (8, 10). However, the series of gR in Eq. (26) converges in a few terms, usually |n|≤8 for moderately low wavenumber kL.
With addition theorem,
Eq. (34) is readily evaluated,
The summation over m also has fast convergence. In practice, it suffices to use the same upper limit as n of Eq. (26).
The expressions for the elements of and are of the same form by changing k to k1, except that
Note that the matrices and are Hermitian.
Appendix B 2D effective permittivity from quasistatic mixing formula
Consider scatterers with permittivity εp embedded in background media with permittivity ε. We derive the effective permittivity from the Lorentz-Lorenz law which states the macroscopic field is the sum of the exciting field and the dipole field [23, 26].
The effective permittivity εeff relates the macroscopic flux and the macroscopic field ,
Equation (46) is known as the Clausius-Mossotti relation. Consider 2D cylindrical scatterers with circular cross section of radius b and cross section area A0 = πb2, we derive the expressions of α and χ by solving the 2D Laplace equation.
For TE polarization,
Thus,Equation (48) is known as the Maxwell-Garnett mixing formula.
Note that when scatter is of PEC material εp → ∞, y → 1, thus εeff = ε(1 + fv)/(1 – fv) is finite.
For TM polarization,
Note that when the scatterers are PEC, εp → ∞, εeff → ∞. Thus εeff does not exist for TM wave with PEC scatterers. This is also clear from the dispersion relation of the lowest band case for the Dirichlet boundary condition that was treated previously .
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