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[1] The spatial confinement of light is an important prerequisite for the excitation of light-matter interactions and nonlinear optics. Photonic crystal nanocavities can have very high quality factors that lead to the desired confinement. We study the feasibility of a hybrid time and frequency domain formulation for the computation of the resonance frequencies and the external quality factors of photonic crystal nanocavities. The time domain approach yields estimates for quality factor and resonance frequency by a special postprocessing technique as well as an approximate eigenvector. In the eigenvalue computation the approximate results from time domain analysis are used for the accelerated solution of the eigenvalue problem.

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[2] The spatial confinement of light is an important prerequisite for the excitation of light-matter interactions and nonlinear optics [Vahala, 2003]. From an engineering point of view a cavity is able to ensure spatial confinement of light at discrete energies. The corresponding Q-factors of the resonances are a measure of the achieved confinement from which the Purcell factor can be computed [Purcell, 1946]. An example of a photonic crystal nanocavity is shown in Figure 1. It consists of a dielectric substrate which has a regular lattice of holes with a hexagonal unit cell inside. The cavity itself is formed by a so-called defect: Three air holes in the center of the crystal have been removed. Depending on the frequency there are eigenmodes within the nanocavity that can radiate to surrounding free-space. The resonances and Q-factors of the photonic crystal nanocavity have been studied experimentally [Akahane et al., 2003] by slight fabrication variations. In this paper we focus on computational methods which are feasible for the design and optimization of the resonance frequencies. Both, time domain and frequency domain approaches are generally suitable for this task. As there are radiating modes, any numerical method for the calculation of the resonance frequencies inside the nanocavity has to provide a kind of absorbing boundary condition (ABC) in order to truncate the computational domain appropriately.

[3] Here we concentrate on the radiation losses of the eigenmodes, neglecting all dispersive and lossy material issues, and we aim at predicting not only the resonance frequencies but also the (external) quality factors in the range 10^{3} < Q < 10^{4}. As a consequence, the ABC has to be modeled very carefully, and the total simulation has to meet highest accuracy requirements. Moreover, in any driven simulations we must make sure that the desired modes are excited at all.

2. Computational Approach

[4] In this computational approach we employ the finite integration technique (FIT) [Weiland, 1977, 1996]. FIT is based on a Yee-like grid pair, the primary grid G and the dual grid . The degrees of freedom of the method are the so-called integral state variables, defined as integrals of the electric and magnetic field vectors over primary and dual edges L_{i}, _{j} and their corresponding dual and primary facets _{i}, A_{j}, respectively (see Figure 2):

Collecting all components in the computational domain in algebraic vectors, the Maxwell grid equations can then be given using the discrete curl operator C [Weiland, 1985]

The corresponding constitutive relations are given by the material equations and read

This formulation is as general as Maxwell's equations themselves. This general nature makes the equations very versatile and allows for all formula derivations from statics to wave propagation. Important physical properties such as energy and charge conservation as well as the orthogonality of eigenmodes are also maintained [Schuhmann and Weiland, 2001].

2.1. Formulations

[5] We use time and frequency domain formulations of the FIT in order to calculate the resonant modes.

2.1.1. Time Domain (TD) Formulation

[6] The energy conservation property of the time-continuous FIT formulation is preserved if the leapfrog scheme is used for time integration. This leads to an explicit algorithm, which is computationally equivalent to the finite-difference time-domain (FDTD) method [Weiland, 1996].

[7] The excitation loops and recording positions must be designed such that all sought modes are properly excited and recorded; this does however require at least some a-priori knowledge about the resulting field patterns. Different problems are caused by inadequate excitation and recording positions. For instance, if a sought mode is not excited or the excitation couples only weakly with its field pattern, the accuracy will be decreased. Also the recording positions have to be chosen very carefully, in order to avoid that an excited mode being accidentally missed during the recording process. Additionally, some limitations in our simulation code, especially in combination with symmetry conditions have to be complied with. A remedy can be the placement of many recording positions in the computational domain. Working with a too broad frequency band may cause issues when too many modes are excited at once as they cannot be distinguished from each other.

[8] The high-Q resonances of the photonic crystal nanocavity lead to very long transients, corresponding to the long settling times of the resonant fields. For the extraction of resonance frequencies and Q-factors, advanced signal processing tools can be used to accomplish this [Schuhmann et al., 2001], although they typically are not successful in a black-box manner. In this case we use the harmonic inversion technique introduced by Mandelshtam and Taylor [1997] and implemented in the freely available code provided by Johnson [2010]. In this technique the recorded time domain signal f(t) is expanded as a finite sum of decaying sinusoids:

The coefficients {A_{k}, ω_{k}, Φ_{k}, α_{k}} can be obtained solving a small generalized eigenvalue problem. The resonance frequencies are immediately given by the ω_{k} and the Q-factors can be calculated as ω_{k}/2α_{k}. In realistic applications the harmonic inversion works quite well as long as the explored signal consists of a modest number of decaying and oscillating modes. The overall length of the time-domain simulation has to last only a few pulse widths (i.e. some multiple lengths of the stimulation signal which is typically defined by a modulated Gaussian pulse with well-defined bandwidth). Degenerated modes which commonly appear in resonant structures with regular shapes have not been observed in our experiments, and the ability of the harmonic inversion technique to identify such modes has not been explicitly tested.

[9] The reduced length of the transients can also be expressed in terms of the energy contained within the computational domain; this is in the range of only −5 dB below its maximum value (compared to a typical value of −60 dB which is used as the standard stop criterion for accurate time domain results). It should be noted that the time domain approach typically requires two simulation runs: In the first run, using broadband excitation pulses, only the resonance frequencies are obtained by means of harmonic inversion. In a second run a small number of so-called ‘DFT monitors’ are applied in order to record the field distributions at the resonance frequencies using an ‘on-the-fly’ discrete Fourier transform (DFT) during the time stepping process.

[10] The time domain simulation is performed using CST Studio Suite [Computer Simulation Technology AG (CST), 2009] and a PML-type boundary condition which is implemented as the standard ABC according to the documentation.

2.1.2. Frequency Domain (FD) Formulation

[11] In frequency domain we solve the curl-curl eigenmode formulation which can be derived from (3) and (4). In the lossless case it reads:

As an absorbing boundary condition we use a formulation which is based on perfectly matched layers (PML). The PML technique was introduced by Berenger as an ABC for FDTD, i.e. in the time domain [Berenger, 1994]. A slightly different formulation is based on complex metric stretching [Zhao and Cangellaris, 1996; Teixeira and Chew, 2000] and can be introduced within the FIT in a straight-forward manner. However, the PML is only theoretically perfectly matched, since the discretization of the damping conductivities introduces a remaining reflection error which can be controlled by the number of layers. Formulated in frequency domain, the PML coefficients depend on frequency, which makes the eigenvalue equation (6) non-linear. The radiation losses introduced by the PML (or any other type of ABC) lead to complex and frequency-dependent material parameters () and (). The resulting system matrix becomes non-symmetrizable with complex eigenvalues ^{2}

Since eigenvalues of the interior spectrum of () are sought, a reasonable estimation frequency is necessary in any case. So the nonlinear eigenvalue problem (7) can be linearized by the evaluation of the complex system matrix at the estimation frequency ω_{e}. However, the solution remains computationally expensive, since several linear systems of equations have to be solved during the eigensolver run. For this purpose we use a Jacobi-Davidson eigenvalue solver [Sleijpen and Van der Vorst, 1996; Hochstenbach, 2010]. For fast convergence, good preconditioners, well estimated frequencies and starting vectors are necessary. The final result yields the modal fields as well as their resonance frequencies and quality factors, independent on any excitation issues.

2.1.3. Hybrid Approach

[12] For a hybrid approach the starting vectors for the eigenvalue solver are computed by means of a short time-domain simulation. The main aim is to solve the eigenvalue problem as fast as possible. The application flow is given in Figure 3. We start with a short time-domain simulation of a modest number of pulse widths. The excitation is done by small dipoles or current loops, whose shape and spatial position are oriented according to the field pattern of the sought eigenmode. During the first steps shown in Figure 3 the positions for excitation and recording have to be adjusted in order to match with the sought field pattern (as mentioned above). The decaying time signal is analyzed by means of the harmonic inversion which yields approximations of the frequencies and Q-factors. If the sought resonance is not found, the field distribution will be investigated and excitation and recording loops will be adjusted. This procedure has to be repeated until the sought resonance occurs. Next, a second simulation in the time-domain of some pulse widths is necessary, in which a discrete Fourier transform (DFT) at the frequency ω_{e} yields the frequency dependent field distribution within the whole computational domain. Since this so-called field monitor is based on a quite short transient simulation (with the remaining energy in the computational domain still close to its maximum), it is only a rough approximation of the sought eigenmode. However, it can be used as a starting vector in the following eigenvalue computation by means of the Jacobi-Davidson eigenvalue solver. Moreover, the estimation frequency ω_{e} obtained by harmonic inversion is used as the initial guess for the eigenvalue solver. Within the Jacobi-Davidson eigenvalue solver a series of linear systems has to be solved. Therefore, an iterative linear solver like the bicgstab [Sleijpen and Fokkema, 1993] can be used with or without a preconditioner. The preconditioner has to meet some requirements: The computation of the preconditioner matrix must not take more time than the solving process saves time due to the use of a preconditioner. Moreover, the use of a preconditioner based on triangular factorizations in an iterative linear solver like the bicgstab typically requires two linear systems to be solved per iteration. This leads to a slightly higher effort when a preconditioner is used compared to the case in which no preconditioner is used.

[13] The computational complexity of the propagated hybrid approach is essentially the computational complexity of the eigensolver with some additional matrix-vector products which yield the starting vector by means of time-domain simulation. Therefore, the asymptotic complexity stays the same. A good starting vector provides a factor less than one for the effort of solving the eigenvalue problem.

2.2. Postprocessing

[14] The quality factor Q is defined as the ratio of the energy stored within the resonator to the dissipated energy during one period of time. In frequency domain it reads

with the complex angular eigenfrequency . Another measure that leads to Q is the half-power bandwidth Δf of a signal around a peak at frequency f_{0}

or equivalently the center wavelength λ_{0} and line width Δλ [Slater, 1946].

3. Computational Setup

[15] For our computational setup we use a photonic crystal nanocavity similar to that used by Akahane et al. [2003]. The structure is shown in Figure 4 including all geometric data. The dielectric slab has a relative permittivity ε_{r} = 11.56 (n = 3.4) and is free of losses. In order to generate two different Q factors, the shifting parameter s is set to s = 0 and s = 0.15a, respectively.

[16] Since we are interested in the computational methods rather than performing a detailed comparison to experimental data, we only model a cut-out of the photonic crystal cavity shown in Figure 1 (see Figure 4), and this truncation will have some influence on the resulting eigenfrequencies. It is known from the literature [Akahane et al., 2003] that the sought resonances occur between 180 and 200 THz and have a maximum of E_{y} in the middle of the structure (origin). Moreover, we apply three symmetry conditions (xy, yz: magnetic, xz: electric) to reduce the number of degrees of freedoms to one eighth of the original setup. At the remaining boundaries PML's are applied. Including the PML, the first numerical model has 190,400 degrees of freedom, corresponding to a rather coarse discretization with 10 lines per wavelength. Note that time and frequency domain problems are solved on the same mesh.

[17] All following computation times refer to a Intel Xeon 7350 system with 16 computing cores and 128 GByte of random access memory. In this work, we rely on the parallel features of Matlab 2010b and CST Studio Suite only without doing any effort in order to maximize the speed-up due to parallelization.

3.1. Time Domain

[18] Small dipoles are used as excitation sources for the transient simulation. Their location is of major importance, and ideally there should be at least one dipole source near a point of maximum field strength for each searched mode. Thus, we form a square current loop with edge length 0.1 μm according to Figure 5, which lays in the xy-plane and is centered at (a/2; 0; 0) to excite an H_{z} field component at the center of the loop.

[19] The recorded field signal is the E_{y}(t) component at the origin, where it is known to show a local maximum [Akahane et al., 2003]. This signal is transformed to frequency domain, and finally the Q factor is calculated using (9).

3.2. Frequency Domain

[20] In the frequency domain simulation we use exactly the same computational model as before (geometrical data and mesh created again by CST Studio Suite) and add seven mesh cells of PML using laboratory Matlab [The MathWorks, 2010] code. Since the PML is known to operate in a broadband frequency range, we evaluate its coefficients at an estimation frequency of 190 THz in order to linearize the eigenvalue problem. For the solution of this complex, non-Hermitian eigenproblem we apply the Jacobi-Davidson eigensolver (JD) [Sleijpen and Van der Vorst, 1996] implementation available from Hochstenbach [2010]. The JD eigensolver requires a good estimation of the eigenvalue and a good preconditioner for the possibly ill-conditioned correction equation. The system matrix is reordered by Matlab's implementation of symmetric reverse Cuthill-McKee method [Cuthill and McKee, 1969] to reduce the fill-in during factorization, and the spectrum of the resulting matrix is normalized before we start the JD eigensolver. An incomplete LU decomposition is applied as a preconditioner and we use a drop tolerance of 10^{−6}. In conjunction with this preconditioner the bicgstab iterative linear solver [Sleijpen and Fokkema, 1993] is able to solve the correction equation within the JD eigensolver well enough to ensure good convergence of the JD eigensolver. The JD eigensolver iterates until the relative residual is less than 10^{−8} or until more than 200 iterations are needed. Finally, the Q factor is calculated using (8).

4. Numerical Results and Discussion

[21] In this section we show some specific properties of the time and frequency domain formulations as well as a mesh study and a possibility for tuning the resonances of the photonic crystal nanocavity.

4.1. Time Domain Approaches

[22] By using the correct setup according to section 3.1, the time domain calculation allows a first spectral view of the resulting modes. Figure 6 shows the DFT spectra of recorded E_{y}(t) time signals with different numbers of pulse widths. Obviously, the DFT needs a well-decayed signal to work properly, at least in terms of the evaluation of the Q-factor. For high-Q resonances it is especially to mention, that this procedure has long transients and thus long simulation times. In order to calculate these Q factors (9), we calculate the half-power bandwidth Δ f and the resonant frequency f_{0} in Matlab from the DFT result. For accurate results the energy-related stopping criterion for the time domain simulation has been set to −60 dB, and the required simulation time is roughly 1 hour for the mesh leading to 190,400 degrees of freedom.

[23] In order to reduce the simulation times, harmonic inversion is taken into account. This allows for calculating the Q-factor as well as the resonance frequency quite accurately from a signal that has just started to decay and with the energy within the calculation domain is remaining relatively high. Using the results in Table 1 we can observe that the resonant frequency can be estimated after a simulating time range of about three pulse-widths in our case, corresponding to a decay of the stored energy within the calculation domain of only −5 dB.

Table 1. Results of the Time Domain Simulation Using Harmonic Inversion and DFT for a Shift s = 0^{a}

Harmonic Inversion

DFT

3

8

20

40

263

40

263

a

The relative energy inside the computational domain E/E_{max}, resonant frequency f_{0}, Q factor and simulation time t are shown for different numbers of pulse widths. For corresponding DFT spectra, see Figure 6.

f_{0}/THz

191.92

191.91

192.15

192.04

191.91

192.17

192.18

Q

6092.62

3539.10

3515.93

3516.41

3516.65

1184

3487

E/E_{max}/dB

−4.1

−5.5

−8.3

−12.4

−60

−12.4

−60

t/s

30

83

198

399

3675

399

3675

[24] Thus, the usage of the harmonic inversion speeds up the simulation time enormously since there is no need to wait until an energy criterion is met, as long as the three dimensional field distribution is not sought with highest accuracy requirements. For an accurate calculation of the three dimensional electric field distribution of the resonant mode, the energy criterion should still be chosen as strict as −60 dB. Only in that case will other excited modes have sufficiently decayed. Yet, the short simulations (about eight pulse-widths, see Table 1) in combination with harmonic inversion are very helpful to find the sought resonances within the spectrum.

4.2. Frequency Domain Approach and Impact of Preconditioners

[25] For the computation of the eigenvalue problem (6) the setup from section 3.2 is able to reliably retrieve the desired eigenvalues. All eigenvalue computations have been done in Matlab 2010b.

[26] Three time consuming tasks are required during the eigenvalue solver run. Firstly, the matrix is scaled by the spectral radius in order to normalize the spectrum. Secondly, the incomplete LU (ILU) preconditioner is computed, and finally the JD iteration itself is time-consuming since a sequence of linear systems has to be solved.

[27] For the exemplary setup with shift s = 0, the matrix scaling takes 52 seconds and the JD iteration without preconditioning with random starting vector v_{0} needs 2101 seconds. The convergence history of the JD eigensolver is quite smooth, no large peaks are found, and the process reaches the relative residual of 10^{−8} after 61 iterations.

[28] A strong influence on the convergence behavior is observed in terms of the initial guess or starting vector v_{0} for the JD eigensolver, which usually is a random vector. However, when we choose a starting vector v_{0} that is generated from a frequency monitor near the estimation frequency ω_{e}, a very short (and rather inaccurate) time domain simulation yields enough information about the dominant mode to accelerate the JD eigensolver convergence to 27 instead of 61 iterations. This combination of a rough starting vector v_{0} obtained in 25 seconds from time domain simulation reduces the JD eigensolver runtime from 2101 down to 934 seconds.

[29]Figure 7 shows the field distribution for shift s = 0 in the z = 0 cutting plane, which is a direct result of the eigenvalue simulation. Thus, all kinds of applications that require the accurate field distribution of an eigenmode (e.g. the computation of the modal volume) can benefit from this approach.

[30] In Figure 8 a comparative study of different ILU preconditioners for the correction equation within the JD eigensolver is shown. The numbers of JD iterations needed for convergence are shown in dependency on different ILU drop tolerances t_{ILU}, which are related to the fill-in of the factors L and U. Moreover, the two different data series show the impact of the starting vector v_{0}. It turns out that the number of JD iterations needed for convergence are reduced drastically by a good starting vector resulting from short time-domain simulations. The average time for a single JD iteration step increases from 35 s without ILU preconditioner to more than 197 s when the best preconditioner is used, since better textitL and U factors are usually denser. Details on the total computation times are given in the first section of Table 2. The effort for the computation of an ILU preconditioner pays off until drop tolerances as low as t_{ILU} = 10^{−2}. For lower drop tolerances the overall computational time increases again, since the JD iteration steps work with a shifted linear system and do not fully benefit from the increased accuracy of the ILU of the pure system matrix. When more than one single eigenmode is sought or the computation of starting vectors is too time-consuming, the more accurate ILU preconditioners may pay off.

Table 2. Performance Study of the Jacobi–Davidson Eigensolver for Different ILU Preconditioners and Starting Vector Types

4.3. Feasibility for Larger Computational Problems

[31] In order to demonstrate the feasibility of the hybrid approach for larger computational models we present an extensive parameter study of different starting vectors v_{0} and ILU drop tolerances t_{ILU} in Tables 2 and 3. The computational problems were obtained by mesh refinement while the structure stays the same (Figure 4). The drop tolerances of the ILU govern the quality of the preconditioner as well as the fill-in nonzeros(L + U)/nonzeros(()) and the time that is spent for its computation. The eigenvalue and eigenvector retrieved by the JD eigensolver do not change for the different parameter sets but the number of iterations, the number of calculated matrix vector products (#MVP) and the time needed. The number of calculated matrix vector products increases for larger computational models as expected since the iterative linear solver needs more (inner) iterations to converge. Fastest overall total computing times are achieved for ILU drop tolerances of 10^{−2} and 10^{−3}. The use of a good starting vector resulting from time domain simulation reduces the overall computation time by a factor between 0.5 and 0.7. That is the reason why we report for the largest models in Table 3 (having 4458381 and 11606705 degrees of freedom) the times with a good starting vector v_{0} from TD only. The deviations of resonance frequencies calculated with the harmonic inversion are in the order of 10^{−5}. The achieved accuracy of the Q-factors is better than necessary for practical applications. Moreover, the accuracy of the harmonic inversion's results are sufficient for the generation of input data for the Jacobi-Davidson method.

Table 3. Performance Study of the Jacobi–Davidson Eigensolver for Different ILU Preconditioners and Starting Vector Types^{a}

Vector v_{0}

Incomplete LU

Jacobi-Davidson Eigensolver

Total Time (h)

t_{ILU}

Fill-in

Time (s)

#it.s

#MVP

Time (h)

a

The numerical model results from finer discretizations of the photonic crystal nanocavity from Figure 1.

[32] In the work of Akahane et al. [2003], different resonance frequencies and Q-factors have been measured from manufactured structures. Therefore, the variation of the shift parameter s (see Figure 4) has been proposed. We investigate the tuning behavior of photonic crystal nanocavity for s = 0 and s = 0.15. Figure 9 shows the spectra obtained from discrete Fourier transform of the time response and in Table 4 a comparison of the results of the different approaches is given. The resulting deviations in the approaches are small, which demonstrates the feasibility of the finite integration formulations for the design of photonic crystal nanocavities. The small differences between the time domain and frequency domain results can be well explained by the different implementations of the PML boundary condition in CST Studio Suite and our own Matlab code.

Table 4. Results From Accurate Time and Frequency Domain Approaches for Two Different Values of the Shift s

[33] A hybrid approach of time domain and frequency domain for the solution of the Maxwellian eigenvalue problem has been introduced and analyzed for a simple photonic crystal nanocavity.

[34] The estimate results from short time domain simulation are useful for a speed-up of the iteration of the Jacobi-Davidson eigensolver. The time domain simulation needs a well-chosen excitation in order to find the desired eigenmode. Especially for high Q factors is the entire simulation time-consuming. When the computation of an eigenvector is not required we find that the harmonic inversion is a very fast tool, since it only requires a small part of the decaying time domain signal.

[35] However, when an accurate eigenvector is sought, the computational effort for the eigenvalue formulation is less than for a long time domain iteration, at least with high Q-factors. The combination of a start vector for the eigenmode computation obtained from very short, unconverged time domain simulation can improve the convergence behavior of the Jacobi-Davidson eigenvalue solver significantly.

[36] In future work the capability of the proposed method for the computation of degenerated modes will be investigated. There is also potential for an additional reduction of the computation time when the code is parallelized for a specific hardware by hand.

Acknowledgments

[37] B.B. and C.C. wish to acknowledge the support of the Deutsche Forschungsgemeinschaft: GRK 1464.