## Abstract

A significant improvement on the basic design of a channel add-drop multiplexer of the in-plane type, based on the two-dimensional photonic-crystal membrane structure of triangular-lattice holes, has been made to increase the channel-selectivity *Q* factor as high as 7300, which demonstrates the viability of the original basic design. The three-dimensional finite-difference time-domain simulation shows that theoretically it is possible to design a channel add-drop multiplexer with better than -0.7 dB of the forward-drop insertion loss, -29 dB of the pass-through cross-talk at the center frequency. A revised coupled-mode analysis with an augmented directional coupling gives a good agreement between its parametric analysis and the finite-difference time-domain analysis in regard to the detailed asymmetric forward-drop frequency response of the multiplexer.

© 2005 Optical Society of America

## 1. Introduction

In-plane-type channel-drop filters, viz., channel add-drop multiplexers (ADMs), have been designed or demonstrated on the two-dimensional (2D) photonic-crystal (PhC) platform with periodic arrangements of infinite-height rods [1] and holes [2], or of triangular-lattice holes on a thin dielectric plate [3, 4]. If they are ever realized, the design would provide a platform for a channel ADM of the smallest physical dimension. With the original concept being created on an ideal 2D PhC structure of an array of infinite-height dielectric rods [1], the device can achieve nearly complete transfer of optical power to the drop port at the center frequency of the selected channel. Evidently, with no confinement along the direction of rods or holes, the original structure is not practical. However, any practical in-plane implementation of the 2D PhC configuration tends to introduce a significant drop of the resonator *Q* factor due to power leak from out-of-plane radiation of the electromagnetic fields, which was not mentioned in those early papers [1, 5] but was quickly recognized [6].

Having a resonator of high *Q* is one of the important conditions for a channel ADM with correspondingly high channel selectivity. Additionally, such a high *Q* design should be compatible with the following basic design rules for such a device [5, 7]:

1. In order to achieve the original channel-drop configuration, we need two symmetric resonators side-by-side, so as to support symmetric and antisymmetric resonance modes which are “accidentally [1]” degenerate, i.e., by an elaborate attunement. We will call it the *condition of attuned degeneracy*.

2. To achieve near-unity forward-drop at the resonance frequency, the out-of-plane *Q*_{out}
should be orders of magnitude greater than the in-plane *Q*_{in}
, which we will call the *condition of low-insertion loss*.

3. The Q factors of the even- and odd-resonance modes, *Q*_{even}
and *Q*_{odd}
should match, which will be shown later to be the *phase-matching condition*.

In the implementation on the PhC structure, a few additional rules should be considered:

4. The effective distance between the two symmetric resonators should match with the basic lattice-constant of the PhC structure.

5. Lastly, the line-defect structure of the bus should work as a single-mode waveguide without out-of-plane power leak from the membrane structure.

The in-plane-type resonator filter in the 2D PhC membrane structure, *without the add-drop feature* was successfully demonstrated in the work of Ref. [3] with the *Q* factor measured around 13,000. However, among the listed design rules of a PhC channel-drop filter, the structure forewent all the rules except items 2 and 5 in the above list. In other words, since a single resonator supporting a single localized mode was used as their resonance system, the resonating field was evenly coupled into all four ports, thereby limiting collection efficiency below 25 % in either drop port as well as unavoidable feedback and crosstalk into input and pass-through ports each of nearly the same 25 %.

The PhC channel ADM using the resonator configuration of two symmetric hexapole resonators in order to enhance the forward drop efficiency and to eliminate reflection, transmission, and backward drop was reported by the present authors [4]. Unfortunately, as it turned out, the structure did not meet the fifth rule in the list. The reported channel-selectivity *Q* factor was around to be 3,500.

In this paper, by making an elaborate tuning with an additional resonator through three-dimensional (3D) finite-difference time-domain (FDTD) simulation, we have extended the configuration with a single 2D PhC stick-shaped resonator of Ref. [3] into the channel add-drop configuration and have made a design prototype which provides a decent channel selectivity as a channel optical filter. The final design, as illustrated in Fig. 1(b), consists of 1) the bus design of two parallel waveguides of unholed lines with the modified holes in the siding layers and 2) a fine-tuned resonance system with two symmetric stick-shape resonators supporting degenerate even- and odd-symmetric localized resonance modes, whose temporal phase patterns are illustrated in Fig. 2.

With the coupled-mode theory in our background understanding, we will discuss on the design procedures in detail for the new configurations meeting the afore-mentioned list of the design rules. Reporting on the FDTD-simulation results showing asymmetric spectral responses, we will provide a parametric coupled-mode analysis in Sec. 2, in which directional coupling between the two line-defect waveguide buses is newly included.

## 2. Coupled-mode theory

Much insight into the basic design and some essential guidance toward the final design are gained by the coupled-mode theory. We consider a four-port wavelength-selective ADM of the configuration containing two symmetric loosely-coupled resonators in the middle between two parallel waveguides as shown schematically in Fig. 3. Here, it is required that each resonator supports only a single-resonance mode near the operating frequency.

Let *𝓐*_{L}
(*t*) and *𝓐*_{R}
(*t*) represent the complex amplitudes for the resonators in the left and right sides, respectively, oscillating with the ‘positive’ frequency only as *e*
^{-i2πνt} in response to the incident time-harmonic wave at frequency *ν*. Whereas *𝒮*
_{+j}(*t*) [*𝒮̂*_{+j}(*t*)] and *𝒮*
_{-j}(*t*) [*𝒮̂*_{-j}(*t*)] represent the similarly oscillating amplitudes of the incoming and the outgoing waves, respectively, at the entrance-points [mid-points] of port *j* of two cascaded four-port devices in the back-to-back configuration of Fig. 3. Then, those amplitudes at the two resonators will evolve in time according to a set of coupled-wave equations [7];

where *ν*
_{0} is the resonance frequency of a single resonator in isolation, τ_{out} is the decay time by loss mostly due to out-of-plane radiation, τ_{bus} is the decay time constant for coupling into one of the two waveguide buses, *µ* is the coefficient for coupling between the localized modes of the two resonators, and σ_{1} and σ_{2} are the coupling coefficients between the resonator and the waves propagating in the opposite directions along the bus. We take σ_{1}=σ_{2}≡σ for our configuration, while [8]

We have observed that all the FDTD simulation of the device of the similar configurations produces a non-negligible degree of asymmetry in all spectral response curves with respect to the center filtering frequency for the incident wave of a given polarization. We predict that this anomaly is due to the unaccounted directional coupling between the two parallel waveguide buses among other possibilities. To improve the modeling capability of the method using the solutions from the coupled-wave analysis, we thus augment the coupled-wave solutions with a pair of additional terms which gives a right amount of directional coupling. In the steady state, using the impulse response functions such as

from implied impulse inputs of {*𝒮*
_{+j}(*t*)}, we may write

where *β* is the propagation constant of the guided mode; *δβ* is the newly introduced coefficient for lossless directional coupling between the two symmetric parallel waveguides; and *L* is the half-length of the entire structure.

If the incoming wave is given only at port 1, then *S*
_{+2}, *S*
_{+3}, and *S*
_{+4} will vanish. Each of reflection, transmission, backward-drop, and forward-drop coefficients is obtained by calculating the ratio between the outgoing wave at the respective port and the incoming wave at port 1;

with its absolute square as the corresponding power spectrum.

Consideration of the involved symmetry at the resonance system with the two resonators can be inferred from the schematic diagram of Fig. 2. The explicit *condition for attuned degeneracy* was given a new interpretation from the viewpoint of coupled-mode theory in Ref. [7]. There, it was expressed as

where *d* is the effective distance between the two symmetric resonators. Additionally, the so-called *phase-matching condition* of [1]

with *m* being a positive integer must be satisfied to achieve a maximized power transfer to the forward-drop port.

The *Q* factor of a *single* resonance mode is defined as the ratio of the stored energy *U* and the energy leak dissipation per unit radian of oscillation;

This definition gives a well-known formula of

where Δ*ν* is the width of the full-width half-maximum (FWHM) of the power spectrum in the case of a well-isolated single resonance mode. Below, we use Eq. (17) for our estimation of the channel selectivity in the channel ADM by picking up Δ*ν* as the FWHM bandwidth from the channel-drop spectrum after performing the digital Fourier transform (DFT) of the FDTD data, although, strictly speaking, two resonance modes are involved in the latter spectrum.

## 3. Simulation and results

The 3D FDTD method with perfectly-matched layers (PML) of artificially introduced anisotropic media [9] has been used for the design and simulation of the PhC channel ADM. The computational domain is discretized to eight grid points per lattice constant *a* for each of the electromagnetic field components in the 3D FDTD simulation with 60 discretized time steps per one cycle [10]. Five grid points are assigned to each anisotropic PML in the faces, the edges, and the corners of the 3D computational domain [11]. In-plane confinement of the electromagnetic fields in the dielectric membrane of thickness 0.75*a* is so strong that the air regions of the width of only 2*a* below and above the dielectric membrane before the PML of five grid points, as shown schematically in Fig. 4, appear to be sufficient for the analysis of resonators of *Q* up to 100, 000.

The electric permittivity profile has been discretized and been interpolated (with proper volume weighting) at every data points for each of the six field components of the electromagnetic fields. This amounts interpolation at eight locations per cubic 3D FDTD cell; 8/8 corner, 12/4 edge, 6/2 face, and 1 center points in every cubic cell in which each field component is updated [11].

A 2D PhC semiconductor membrane structure with a periodically arranged triangular-lattice holes of radius *r*=0.300*a* creates a relatively large bandgap in the frequency domain for the incident waves of the chosen TE-like polarization with the electric field vibrating on the plane of the membrane of thickness 0.75*a* with the electric-permittivity contrast of *ε*=11.56*ε*
_{0} versus *ε*
_{0}. The frequency responses in Eq. (13) of the PhC structure can be obtained by performing DFT on the output FDTD data from the Gaussian-enveloped input wave packet. Analysis on the data from a similar procedure with even narrower an envelope has given reliable numbers for the bandgap range of 0.244<*aν/c*≡*a*/λ<0.312, in the scale of normalized frequencies, with *c* being the light speed in vacuum.

#### 3.1. Line-defect waveguide buses

The dispersion curve in the band-structure diagram for a line-defect waveguide can be obtained by analyzing the supercell on the dielectric membrane structure using the MPB program [12], as illustrated in Fig. 5(b). The bus waveguide is designed to support a single guided mode for the given input polarization. As it turns out, there exist two single-mode frequency ranges inside the bandgap. Between the two, the lower one supports lossless propagation of the guided wave, while the upper one does not due to the curve submerging into the inner side of the light cone. Note that, unlike the hexapole resonator in Ref. [4] in which the resonance frequency shows up very close to the upper edge of the bandgap, the stick-shape resonator system allows the resonance frequency which matches comfortably with the lower single-mode range.

In order to obtain the lower single-mode range of the curve totally outside the light cone, the structure has indeed been further modified by reducing the radius of each air hole in first layers ‘siding’ the line of missed holes, i.e., *r*_{s}
=0.265*a* opposed to the default radius of *r*=0.300*a*, for which one can refer to Fig. 1(a) for the illustration. With such modifications, the two single-mode frequency ranges are found from 0.247 to 0.261 and from 0.273 to 0.301, which are shown in Fig. 5(b) with yellow shadings.

As shown by the intersection of the vertical and horizontal dotted lines in Fig. 5(b), the lower one is also advantageous because a sizable group velocity is observed near *aν/c*=0.25338 with the corresponding propagation constant of ±17/52×2*π/a* in the first Brillouin zone. Being in the lower range, the wave at the above frequency has been confirmed to be free from out-of-plane radiation by the 3D FDTD simulation.

#### 3.2. The resonator system

It is important to apprehend the implication of the condition of Eq. (14), which suggests that the required accidental degeneracy is obtained by merely adjusting the mutual coupling between the two symmetric resonators. For this attunement, the radius of all the twelve holes of the stick-shape resonator, *r*_{m}
, is set commonly to 0.267*a*. The two edge holes of the single resonator, denoted by *l* in Fig. 1(a), have been displaced from their original places by 0.15*a* from the default positions in order to enhance the *Q* factor of the resonator. All the *Q* factors of the resonator modes are obtained according to Eq. (16) by plotting the decaying oscillation of the field at the center of the resonator which was let loose from the initial excitation.

In the case of a single stick-shape resonator, the computational domain is set to include 33 holes to the x direction and 25 hole-layers to the y direction. While the *Q* factor of a single stick-shape resonator in the absence of other defect structures is measured to be 141,000, that of the same resonator in the presence of the two waveguide buses is measured to be merely 6,600. Since *Q*
^{-1}=${Q}_{\text{in}}^{-1}$+${Q}_{\text{out}}^{-1}$, by measuring the ratio ${Q}_{\text{in}}^{-1}$ : ${Q}_{\text{out}}^{-1}$ betwee n the in-plane power leak and the out-of-plane power leak from the Poynting-vector integration, we estimate *Q*
_{in}=7,100 and *Q*
_{out}=93,700.

We now consider the final design with the two symmetric stick-shape resonators: The size of the computational domain is increased to 42 holes in the x direction to reflect the size increase of the structure. It has been found that the phase-matching condition of Eq. (15) is well satisfied with *d*=13*a* for the distance between the centers of the two stick-shape resonators. The individual *Q* factors of the even- and the odd-symmetric modes of the two symmetric stick-shape resonators in the presence of the waveguide buses are separately calculated by applying appropriate boundary conditions on the half of the aforementioned FDTD computational domain. A sizable, thus undesirable difference in the two *Q* factors have been measured; *Q*
_{even}=8,120 and Q_{odd}=5,560 for the even- and the odd-symmetric modes, respectively. It is a yet-to-be-corrected compromise in the prototype design failing rule 3 in the list of design rules in Sec. 1. As was implied in [7], their inverse numbers yield the average of

which coincides the value directly measured over the entire structure.

#### 3.3. Channel add-drop multiplexer

Figure 6(a) shows two sets of spectral curves; the solid curves from the 3D FDTD simulation and the dotted curves from our version of a parametric coupled-mode analysis, which correctly shows the effect of directional coupling between the two parallel waveguide buses. The parameters used for the dotted curves are

for *aν*
_{0}/*c*=0.25338, where the mutual coupling *µ* between the two symmetric resonators are slightly off-tuned by the factor of 1.005 from the recipe value in Eq. (14) for a better FDTD result. The channel-selectivity *Q* factor by Eq. (17), estimated from the DFT analysis with the data from the 3D FDTD simulation is found to be around 7,320.

Note that the pronounced asymmetry in the power spectrum of the forward drop from the 3D FDTD simulation is adequately traced by the result from the coupled-mode theory. Among many possibilities [13], the main cause turned out to be the often-ignored directional coupling between the guided modes of the two line-defect waveguides. Indeed, if we ignore such directional coupling, the asymmetry disappears from the result of the parametric coupled-mode theory as the dot-dash curve in Fig. 6(b), while its inclusion gives a substantial improvement in the agreement of the two results in Fig. 6(a); one from the FDTD simulation and the other from the parametric coupled-mode analysis.

Other observations include the two almost-equal side peaks measured around -21 dB in the backward-drop power spectrum. Their appearance is believed to be the result of the mistuned phase matching condition, i.e., 8.565*π* in Eq. (20) instead of the exact eight and a half times *π* from Eq. (15). It was once reported that the separation of the two center frequencies of the even- and the odd-symmetric resonance modes [14] should be responsible. However, we have confirmed that, in our work, those two side peaks have shown up in the backward-drop spectrum because the bandwidths of the two resonance modes do not coincide. We make this statement because their center frequencies have actually been synchronized quite well. Specifically, the two resonance frequencies are found to be 0.25337991 and 0.25338009, whose difference is much smaller than the respective FWHM bandwidths; 0.25337991/*Q*
_{even} and 0.25338009/*Q*
_{odd}. In such a circumstance, the phase matching condition of Eq. (15) has been sacrificed. We thus believe that the aforementioned phase mismatch is the primary reason for the sidelobes in the spectral response of the backward-drop port.

The transmitted power and the less-important backward-drop power are reduced below -29 dB and -31 dB, respectively, at the very center frequency, as illustrated in Fig. 6. More than 85 % (i.e., -0.7 dB) of the incident power at the input port is transferred to the forward-drop port at the center frequency. Such high forward-drop efficiency has been made possible basically by the high out-of-plane *Q* factor of the resonance system.

## 4. Conclusion

By an extensive 3D FDTD simulation, we have made a design prototype for a channel ADM of the in-plane type based on the PhC semiconductor membrane structure, consisting of two parallel bus waveguides and two symmetric stick-shape resonators in the middle. By tuning the positions and the radii of the air holes siding the bus waveguide and the two resonators located apart at a specific distance, our best-effort optimization pursuing all the design rules for an ideal configuration of a channel ADM has been carried out. The 3D FDTD simulation on the prototype design has shown the channel-selectivity *Q* of 7,300 with -0.7 dB of the forward-drop power and the -29 dB of the pass-through crosstalk at the center of the filtering frequency.

Observed peculiar asymmetry in the forward-drop spectrum from the 3D FDTD simulation has been compared with the results obtained by the parametric coupled-mode analysis. From this comparative study, it was revealed that often-ignored directional coupling between the two parallel bus waveguides is a dominant mechanism for the asymmetric spectral response in this four-port waveguide configuration.

Although there still exist a few practical considerations for a practical channel ADM, the prototype design is believed to provide the add-drop functionality with the record channel-selectivity *Q* number for a PhC channel-drop filter. With the on-going development in modern nano-fabrication technology [3], the present work should be considered a significant improvement toward the viability of the basic design of the PhC channel ADM which was once believed extremely difficult to implement on any practical 2D PhC configurations.

## Acknowledgments

We thank the late Professor, H. A. Haus of MIT for an inspiring discussion. This work has been supported in part by the KOSEF through the UFON-ERC Program, by IITA through the CHOAN-ITRC Program and by MOST through the Program.

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