## The finite element method applied to the study of two-dimensional photonic crystals and resonant cavities |

Optics Express, Vol. 21, Issue 4, pp. 4072-4092 (2013)

http://dx.doi.org/10.1364/OE.21.004072

Acrobat PDF (3555 KB)

### Abstract

A critical assessment of the finite element (FE) method for studying two-dimensional dielectric photonic crystals is made. Photonic band structures, transmission coefficients, and quality factors of various two-dimensional, periodic and aperiodic, dielectric photonic crystals are calculated by using the FE (real-space) method and the plane wave expansion or the finite difference time domain (FDTD) methods and a comparison is established between those results. It is found that, contrarily to popular belief, the FE method (FEM) not only reproduces extremely well the results obtained with the standard plane wave method with regards to the eigenvalue analysis (photonic band structure and density of states calculations) but it also allows to study very easily the time-harmonic propagation of electromagnetic fields in finite clusters of arbitrary complexity and, thus, to calculate their transmission coefficients in a simple way. Moreover, the advantages of using this real space method in the context of point defect cluster quality factor calculations are also stressed by comparing the results obtained with this method with those obtained with the FDTD one. As a result of this study, FEM comes out as an stable, robust, rigorous, and reliable tool to study light propagation and confinement in both periodic and aperiodic dielectric photonic crystals and clusters.

© 2013 OSA

## 1. Introduction

4. E. Istrate and E. H. Sargent, “Photonic crystal heterostructures,” Rev. Mod. Phys. **78**, 455–481, (2006). [CrossRef]

5. A. J. Garcia-Adeva, “Band gap atlas for photonic crystals having the symmetry of the kagome and pyrochlore lattices,” New J. Phys. **8**, 86/1–14 (2006). [CrossRef]

6. A. J. Garcia-Adeva, “Band structure of photonic crystals with the symmetry of a pyrochlore lattice,” Phys. Rev. B. **73**, 0731071 (2006). [CrossRef]

7. E. Yablonovitch, T. J. Gmitter, and K. M. Leung, “Photonic band structure: the face-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett. **67**, 2295–2299, (1991). [CrossRef] [PubMed]

8. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express **8**, 173–190 (2001). [CrossRef] [PubMed]

9. MPB on-line manual, http://ab-initio.mit.edu/wiki/index.php/MPB\_manual.

*Q*factors: using real-space harmonic propagation transmittance computations —together with Brent’s method [24] for determining the width of the resonance— allows for a more efficient and reliable calculation than analyzing the

*Q*-factor in these structures using a time domain transient propagation environment.

## 2. The mathematics behind the propagation of EM waves in inhomogeneous media

*E*(

_{z}*r*⃗) is the

*z*-th component of the electric field at position

*r*⃗, ∊

*(*

_{r}*r*⃗) is the inhomogeneous relative dielectric constant of the photonic crystal, and

*k*

_{0}=

*ω*/

*c*with

*ω*the angular frequency of the incident electric field and

*c*the speed of light in free space. In writing down equation Eq. 1, it has been assumed that the photonic crystal is non-magnetic (μ

*= 1) and non-conducting (σ = 0). Once equation Eq. 1 is solved, the time-harmonic electric and magnetic fields are easily calculated as On the other hand, in the case of TM polarization, the magnetic field is perpendicular to the photonic crystal plane —whereas the electric field is constrained into this plane— and the sourceless Maxwell’s equations reduce to a Hemholz equation for the magnetic field given by where*

_{r}*H*(

_{z}*r*⃗) is the

*z*-th component of the magnetic field at position

*r*⃗. The time-harmonic electric and magnetic fields are easily calculated once equation 4 is solved and they are given by An important aspect of any electromagnetic simulation is the use of appropriate boundary conditions at the interfaces. On the one hand, for the photonic band structure calculations reported below, it is necessary to implement boundary conditions that mimic an infinite simulation domain together with the periodicity of the photonic crystal lattice. An infinite medium is simulated by using perfect magnetic (PMC) or perfect electric conductor (PEC) boundary conditions that mirror the simulation domain. The former condition is used for TE polarization of the EM field and ensures that the component of the magnetic field tangent to the boundary is identically zero at the outer interface, that is, where

*n̂*is a unit vector perpendicular to the outer simulation domain surface at each point. The later is used for TM polarization of the EM field and ensures that the component of the electric field tangent to the boundary is identically zero at the outer interface, that is, The periodicity of the photonic crystal lattice is ensured by an adequate use of Bloch’s theorem at the boundaries of the photonic crystal unit cell. This theorem states that when the electric (magnetic) field propagates from one point in the PC to another one separated from the previous one by a lattice vector,

*R*⃗, the only effect on the EM field is a change of its phase, and for TE and TM polarization, respectively. In these expressions,

*R*⃗ is a vector of the photonic crystal lattice and

*k*⃗ is the wavevector of the electromagnetic wave. On the other hand, for the transmittance calculations reported below, finite clusters in the direction parallel to the incident wave vector were used, whereas the clusters are of infinite extension in the perpendicular direction. In order to mimic such a material, PMC and PEC boundary conditions were used for the outer interfaces that limit the simulation domain in the direction perpendicular to the incident wave vector for TE and TM polarization, respectively. For the interfaces at which the EM wave enters and exits the cluster, the situation is a little more involved because it is necessary to avoid unphysical reflections due to the finite size of the cluster and, thus, perfectly matched layers were used at these interfaces. The equations that describe such boundaries are given by where

*E*

_{0z}and

*H*

_{0z}are the initial values of the electric and magnetic fields at the boundaries, respectively, and

*β*=

*k*

_{0}is the propagation constant. The upper condition applies to TE polarization, whereas the lower one corresponds to TM polarization. If the electric field is an eigenmode of the boundary, the boundary is exactly non-reflecting.

## 3. Computational methods

12. M. -C. Lin and R. -F. Jao, “Finite element analysis of photon density of states for two-dimensional photonic crystals with in-plane light propagation,” Opt. Express **15**, 207–218 (2007). [CrossRef] [PubMed]

22. P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency”, Phys. Rev. B **54**7837–7842, (1996). [CrossRef]

25. Elmer – Finite Element Software for Multiphysical Problems, http://www.csc.fi/elmer/index.phtml.

27. The EMAP Finite Element Modeling Codes, http://www.emclab.umr.edu/emap.html.

^{®}multiphysics package [28

28. Comsol multiphysics and Electromagnetics module, http://www.comsol.com.

^{®}).

*mesh elements*. In two dimensions, this program uses an unstructured mesh (triangular mesh elements) generator based on the Delaunay algorithm. Once a mesh is created, the dependent variables are approximated by a known function (shape functions) that can be described with a finite number of parameters called

*degrees of freedom*. Inserting this approximation into the original equations generates a set of equations for the degrees of freedom that is then solved with an appropriate

*solver*. In particular, for the present problem, the shape functions are second order Lagrange elements and the solver for the resulting linear problem is the so called UMFPACK solver, which is a very efficient direct solver for unsymmetric systems [29

29. T. A. Davis, UMFPACK 4.6: Unsymmetric MultiFrontal sparse LU factorization package, http://www.cise.ufl.edu/research/sparse/umfpack/.

8. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express **8**, 173–190 (2001). [CrossRef] [PubMed]

23. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comp. Phys. Comm. **181**, 687–702, (2010). [CrossRef]

## 4. Simulation results for photonic crystals based on square and triangular lattices

*r*is the radius of the cylinders and

*a*the lattice parameter, was taken as 0.38. This well known structure was first investigated by McCall and coworkers [31

31. S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, “Microwave propagation in two dimensional dielectric lattices,” Phys. Rev. Lett. **67**, 2017–2020, (1991). [CrossRef] [PubMed]

*ωa*/2

*πc*=

*a/λ*has been used to characterize the frequency of the incident EM wave, where

*ω*is the frequency of the incident EM wave and

*λ*the associated wavelength. The corresponding eigenmodes of the

*z*-component of the electric field,

*E*, were also calculated at the Γ,

_{z}*M*, and

*K*points of the 1BZ and they are shown also in Fig. 1. It is clear from that figure that the band structure calculated with the FE method faithfully reproduces the one calculated with MPB to its minimum details. There are three photonic gaps in the band structure of this lattice whose sizes coincide with the calculated ones with MPB. Also, the modes calculated with the FE method closely resemble those calculated with MPB up to a trivial symmetry operation or linear combination of degenerate modes. In addition, a quantity that can be readily calculated with the FEM method is the transmittance of a finite photonic crystal. The transmittance of the cluster is calculated using the usual approach of power integration along a domain boundary. The boundary conditions for the simulation domain were set as follows: on the input and output boundaries (left and right boundaries) a perfectly matched layer was used to avoid spurious reflections from non-physical boundaries. The

*z*-component of the electric field was set to 1 and 0 at the initial time of the simulation on the input and output boundaries, respectively. On the other hand, perfectly magnetic conductor boundary conditions that set the tangential magnetic component of the magnetic field to zero were used for the upper and lower boundaries (simulations using different boundary conditions for the upper and lower boundaries were performed and no significant differences in the calculated transmission spectra were found) in order to mimic an infinite stripe in that direction.

*X*point for the ninth band for different discretizations of the square unit cell. In the FE case, meshes with 254, 414, 928, and 1502 elements were used, whereas for the calculations done with MPB, resolutions of 16 ×16 (=256) grid elements, 32 ×32 (= 1024) grid elements, 48 ×48 (= 2304) grid elements, and 64 ×64 (= 4096) grid elements were used. We took the eigenvalue calculated with MPB by using a resolution of 256 ×256 and a mesh size of 25 as the exact one. The result of this comparison can be seen in Fig. 3(a). It is noticeable that the FE method gets a better accuracy with coarser discretizations of the lattice than the PW does. This is due to the use of second order Lagrange elements. However, the differences between both methods should be negligible for most applications. Of course, this better accuracy comes at a price and the simulation times for the FE calculations are longer by a noticeable factor than those for the MPB ones, as can be seen in Fig. 3(b), where we have depicted the evolution of the simulation runtime with the number of mesh elements and grid elements for the FE and MPB calculations, respectively.

## 5. Defect states in 2D photonic crystal superlattices

32. E. Waks and J. Vuckovic, “Coupled mode theory for photonic crystal cavity-waveguide interaction,” Opt. Express **13**, 5064–5073 (2005). [CrossRef] [PubMed]

33. C. Sibilia, T. M. Benson, M. Marciniak, and T. Szoplik, *Photonic Crystals: Physics and Technology* (Springer, Milano, 2008). [CrossRef]

34. B. Temelkuran, M. Bayindir, E. Ozbay, R. Biswas, M. M. Sigalas, G. Tuttle, and K. M. Ho, “Photonic crystal-based resonant antenna with a very high directivity,” J. Appl. Phys. **87**, 603–605, (2000). [CrossRef]

35. Takasumi Tanabe, Masaya Notomi, Eiichi Kuramochi, Akihiko Shinya, and Hideaki Taniyama, “Trapping and delaying photons for one nanosecond in an ultrasmall high-Q photonic-crystal nanocavity,” Nat. Photonics **1**, 49–52, (2007). [CrossRef]

36. T. Baba, T. Kawasaki, H. Sasaki, J. Adachi, and D. Mori, “Large delay-bandwidth product and tuning of slow light pulse in photonic crystal coupled waveguide,” Opt. Express **16**, 9245–9253, (2008). [CrossRef] [PubMed]

37. D. -S. Song, S. -H. Kim, H. -G. Park, C. -K. Kim, and Y. -H. Lee, “Single-fundamental-mode photonic-crystal vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. **80**, 3901–3903, (2002). [CrossRef]

38. R. V. Nair and R. Vijaya, “Photonic crystal sensors: an overview,” Prog. Quant. Electron. **34**, 89–134, (2010). [CrossRef]

### 5.1. A square lattice of dielectric rods with a point defect: the McCall’s experiment

31. S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, “Microwave propagation in two dimensional dielectric lattices,” Phys. Rev. Lett. **67**, 2017–2020, (1991). [CrossRef] [PubMed]

8. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express **8**, 173–190 (2001). [CrossRef] [PubMed]

*a*. The rod dielectric constant has been set to 9, as in the original McCall’s work. A 5 ×5 supercell has been used for the FEM calculations reported in this section. It has been discretized into 95903 mesh elements. A 3096 element grid has been used for the corresponding MPB calculations. The real part of the z-component of the electric field for the defect mode located at

*a/λ*=0.4686 was calculated at the Γ point of the 1BZ.

*N*times (

*N*being the linear dimension of the supercell). This fact leads to larger computational times since the amount of eigenvalues to be solved grows ∼

*N*. We have therefore solved eigenvalues for 211 points in

*K*-space for 100 bands by both methods. Experimental results obtained by McCall and coworkers, MPB calculations, and FEM results fully agree, as shown in Fig. 4(a).

*k*-points constrained to the irreducible portion of the 1BZ for each lattice and they are also reported in Fig. 4(a). In the long wavelength limit, this quantity clearly exhibits the linear behavior expected for propagation in an homogeneous two-dimensional dielectric medium. The band gaps are also clearly visible as the frequency regions of zero DOS.

### 5.2. A triangular lattice of dielectric rods with a point defect

39. D. R. Smith, R. Dalichaouch, N. Kroll, S. Schultz, S. L. McCall, and P. M. Platzman, “Photonic band structure and defects in one and two dimensions,” J. Opt. Soc. Am. B **10**, 314–321, (1993). [CrossRef]

*z*-component is well localized around the defect neighborhood and rapidly decays to small amplitudes as one moves further from the defect. Also, the eigenfunction around the lattice irregularity clearly shows an inherent symmetry. Indeed, both lattices present a monopole pattern with a single nodal plane through each dielectric rod [1]. The symmetry of such point defect modes is analyzed in detail in [40

40. S. -H. Kim and Y. -H. Lee, “Symmetry relations of two-dimensional photonic crystal cavity modes,” IEEE J. Quantum Electron. **39**, 1081–1085, (2003). [CrossRef]

## 6. Defect states in finite 2d photonic crystal clusters

*N*increases.

## 7. Time domain approach: FDTD

*Q*factors calculated using FEM. For the present manuscript, the FDTD simulations were performed using MEEP [23

23. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comp. Phys. Comm. **181**, 687–702, (2010). [CrossRef]

*Q*factors in PC topologies. However, computing

*Q*-factors using Eq. 15 and FDTD can be cumbersome. In FDTD one needs to compute the transmittance spectrum at the end of the computational cell, somewhere before the PML, running the simulation until the fields evolve and decay significantly (|

*E*|

_{z}^{2}or |

*H*|

_{z}^{2}need to decay by at least a factor of

23. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comp. Phys. Comm. **181**, 687–702, (2010). [CrossRef]

*Q*-factors by means of the transmittance spectrum obtained from FDTD is not a fast, nor reliable, approach.

*Q*-factors (rather than the transmittance data directly), there are still other complications associated to using FDTD. Indeed, in FDTD space and time are divided into a finite rectangular grid and then fields are evolved in time using discrete steps. However, FDTD has serious limitations when the computational domain is finite whilst FEM convergence time is rather insensitive to this fact. In FDTD the computational domain must be terminated with some boundary conditions as it is the case in the FEM and, in order to simulate open boundaries in finite clusters, a wave absorbing mechanism, such as the split-field PML proposed by Berenger [45

45. J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comp. Phys. **114**, 185–200, (1994). [CrossRef]

22. P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency”, Phys. Rev. B **54**7837–7842, (1996). [CrossRef]

**181**, 687–702, (2010). [CrossRef]

*Q*-factors results obtained by both methods, a broadband source has been used for FDTD simulations, whereas a monochromatic excitation was used for the FEM ones, as explained above. In particular, for the present FDTD computations, a point Gaussian source has been placed inside the cavity. This excitation must be short enough (broad bandwidth) to excite the defect mode for each cluster. When this Gaussian source is switched on, the field grows and after some time this source is extinguished. Subsequently, a resonance effect occurs and the electromagnetic fields bounce back and forth for a limited amount of optical periods. Meanwhile, the energy trapped around the defect exhibits an exponential time decay (see Fig. 9). At this point, if one takes into account the slowly varying component only, i.e., the envelope of the electric field norm, the quality factor is determined by the ratio of the power stored at a given time

*t*divided by the power lost after one period

*T*

46. A. F. Oskooi, L. Zhang, Y. Avniel, and S. G. Johnson, “The failure of perfectly matched layers and towards their redemption by adiabatic absorbers,” Opt. Express **16**, 113761 (2008). [CrossRef]

47. Z. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Ant. Prop. **43**, 1460–1463, (1995). [CrossRef]

*ν*[23

**181**, 687–702, (2010). [CrossRef]

*a*surrounds each structure. Besides, the distance between the UPML layer and the edge of the computational grid must be adequately tuned in order to avoid an unphysical field decay. According to this, a distance of 0.8 ≤

*d*≤ 1.1 has been used. Sampling the electric field response with a period relative to the resonance frequency, results in a quasi-periodic step function that induces some uncertainties in the

*Q*factor determination if Eq. 16 is directly applied. These fluctuations could be due to the abruptly broken translational symmetry of the clusters. However, the marked decaying behavior of the electric field data can be easily filtered and interpolated. Therefore, by iteratively applying Eq. 16 to the interpolated sample field, an average

*Q*factor and the corresponding theoretical error is obtained. Finally, in order to compare with previous

*Q*factor calculations obtained via FEM transmittance results and FDTD decaying field value relations, an harmonic inversion of time signals (

`Harminv`) [48

48. V. A. Mandelshtam and H. S. Taylor, “Harmonic inversion of time signals,” J. Chem. Phys. **107**, 6756–6769, (1997). Erratum, ibid. **109**, 4128 (1998). [CrossRef]

`Harminv`performs the signal processing of the fields in the cavity. This way, it identifies the frequencies and decay rates of the excited resonant modes. This method is a fast and accurate way to determine the

*Q*factors but still requires waiting until the fields have evolved and decayed to a certain small value. Computation time scales also with the cluster size, but this fact can often be overcome if symmetries are used to reduce the size of the computational cell. Besides, choosing a broad Gaussian source can yield to the appearance of spurious solutions but, at the same time, the source must be broad enough to excite the resonant frequency.

*Q*factor. Table 1 summarizes the results obtained using the three methods described in this report. Two dimensional FEM calculation of

*Q*factors offers a very satisfactory agreement with FDTD and Harminv results whilst providing substantial gain in solution robustness and efficiency. Indeed, FDTD is a powerful tool to calculate resonant frequencies and quality factors for complex cavity structures. However, it is very inefficient because one must discard many simulation cycles before reaching the stationary regime. Also, there are many heuristic factors that enter into the FDTD simulation process, such as the thickness of the UPML, the excitation source location and both its central frequency and width, that make the results from this method not as reliable as one would want. Moreover, when computing resonant modes in time domain especial care must be taken to avoid choosing an excitation source nearly orthogonal to the resonant mode because FDTD is likely to miss it, otherwise. In this regard, FEM can be seen as an efficient, reliable and more rigorous alternative to FDTD for the analysis of quality factors and resonant modes in complex dielectric material.

## 8. Conclusions

`Harminv`procedures for different cluster dimensions.

*Q*factor calculation in a speedy and computationally effective way. In addition, the leakage mechanism of PC cavities transforms the transmittance spectra into an almost Lorentzian peak and, therefore, with few transmittance calculations one can obtain the entire point defect transmittance response. It is noteworthy, with regards to the the point defect PCs addressed above, the fact that the three methods accurately reproduce the

*Q*factor for each topology. However, FDTD based methods have some serious drawbacks. On the one hand, the width and the proximity of the absorbing layers significantly determine the decaying behavior of the trapped fields inside the cavity, and so, the parameters of the PMLs must be carefully chosen for each structure. In the aforementioned point defect structures, the source must be placed inside the cavity, which in some cases can be an unrealistic situation. Furthermore, the bandwidth of the source must be set intuitively so as to excite only the defect mode. On the other hand, we believe that among these methods FEM is desirable for the calculation of

*Q*factor due to its high numerical efficiency and stability because in FDTD, after the source is extinguished, one must wait an uncertain amount of time until the fields evolve and decay. In fact, FDTD gives quite accurate values for both the resonant frequency and the

*Q*factor but, for higher

*Q*values, the slope of the electromagnetic field inside the defect is nearly zero and hence, convergence time and and numerical errors increase drastically.

## Acknowledgments

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48. | V. A. Mandelshtam and H. S. Taylor, “Harmonic inversion of time signals,” J. Chem. Phys. |

**OCIS Codes**

(050.1755) Diffraction and gratings : Computational electromagnetic methods

(140.3945) Lasers and laser optics : Microcavities

(160.5293) Materials : Photonic bandgap materials

(160.5298) Materials : Photonic crystals

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: November 28, 2012

Revised Manuscript: January 4, 2013

Manuscript Accepted: January 15, 2013

Published: February 11, 2013

**Citation**

Imanol Andonegui and Angel J. Garcia-Adeva, "The finite element method applied to the study of two-dimensional photonic crystals and resonant cavities," Opt. Express **21**, 4072-4092 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-4-4072

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