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A compact in-plane channel drop filter design in 2D hexagonal lattice photonic crystal slabs is presented in this paper. The system consists of two photonic crystal waveguides and a single cavity with two degenerate modes. Both modes are able to confine light strongly in the vertical dimension and prove to couple equally into the waveguides. Three dimensional finite difference time domain simulations show that the quality factor is around 3,000. At resonance, power transferred to the drop waveguide is 78% and only 1.6% remains in the bus waveguide. We also show that by carefully tuning the drop waveguide boundary, light remaining in the bus can be further reduced to below 0.4% and thus the channel isolation is larger than 22dB.
©2005 Optical Society of America
1. Introduction
Two-dimensional photonic crystal slabs (2D PCS’s), thin dielectric slabs with two-dimensional photonic crystal pattern, have attracted much attention as they provide an inplane photonic band gap, are able to confine light in the vertical dimension with index guiding and promise easier fabrication than their 3D counterparts [1–6]. Various miniature photonic devices can be realized in 2D PCS’s, such as high-Q resonators [7–10], waveguides [11–12] and channel drop filters [13–20], which are the essential components of photonic integrated circuits (PICs) and dense wavelength division multiplexing (DWDM) optical communication systems.
Over the past few years, there have been different designs of channel drop filters in 2D PCS’s. One design is the surface-emitting type [15–18], which consists of a waveguide and a cavity system. Input signal from the waveguide tunnels into the cavity and is emitted in the vertical direction. Though in principle all light can be dropped in the vertical direction by using a cavity system with two degenerate modes [18], this design brings up a real challenge when collecting light vertically. Usually light emitted downward cannot be collected due to structural restrictions. The vertical radiation pattern has to be optimized in order to efficiently collect the other 50% of light emitted upward. The in-plane design involves two waveguides (bus and drop) and a cavity system. Channel to be selected comes from the bus waveguide, tunnels into the cavity and is eventually transferred to the drop waveguide. Theoretical analysis of the in-plane channel drop filter has matured over the years [21–23] and in principle 100% transfer between two waveguides can be realized if the cavity system provides at least a pair of degenerate modes of opposite symmetry with infinitely high vertical Q factor. Yet very few in-plane designs in 2D PCS’s have been reported [19–20]. The bottleneck lies in the cavity system, which largely limits the drop efficiency (loss) and line-width of the filter.
As mentioned earlier, in 2D PCS’s light is confined vertically by index-guiding. Though theoretically 2D PCS waveguides can be lossless within a narrow frequency region, cavity modes are always leaky in the vertical direction. For a symmetric in-plane channel drop filter system where the degenerate modes have equal frequency and equal decay rates into both waveguides, the power transmission |T|2 and power dropped |D|2 at resonance can be expressed by [22],
where 1/τo is the decay rate due to loss and 1/τe is the decay rate into either waveguide. The decay rate is related to Q factor by Qo=ω0τo/2 and Qe=ω0τe/2. If the resonant frequency ω0 of the cavity modes lies within the lossless region of the waveguide, the total loss of the system is mainly from the vertical loss of the cavity, i.e., Qo ≈Q⊥ , where Q⊥ is the vertical Q factor of the cavity. The in-plane Q factor (Q◃ ) of the cavity is related to Qe by Q◃=Qe /2 if we assume the cavity modes decay into both waveguides equally and ignore the in-plane loss due to the limited number of lattice surrounding the system. From equation (1) and (2), we can see the direct relation between the drop efficiency and the ratio Q⊥/Q◃ . For example, in order to have |D|2>80%, Q⊥ should at least be 10 times larger than Q⊥ . The choice of high Q cavity is critical and has not yet been well addressed in the previous designs. In Ref. [19] Q⊥ is only a few times larger than Q◃ , the power transferred to the drop waveguide is only 61% and 5% still remains in the bus. In Ref. [20], the resonant frequency of the cavity mode lies beyond the lossless region of the waveguide, which further increases vertical loss. In both designs the degenerate modes are provided by two identical single-mode cavities separated by a distance, which not only makes the device larger and more loss-prone, but also raises difficulties for fabrication, since it is not yet likely to produce two identical cavities or to determine their positions precisely using present etching techniques.
In this paper, we come up with a compact design of in-plane channel drop filter in 2D PCS’s with a single cavity that supports two high Q modes. We will show the process of tuning the two modes, even and odd, into exact frequency degeneracy with similar Q⊥ and Q◃ values. When the transmission spectra are achieved, we will discuss how to improve the channel isolation efficiently. For computations of cavity modes and simulations of wave propagation, we use our home-made 3D finite difference time domain (3D FDTD) code with perfect matched layer boundary conditions [25–26]. When we analyze the individual cavity modes, we apply mirror boundary conditions in the x, y, and z directions, which reduces the computation size to one-eighths and also allows us to study the even and odd modes separately [9]. A combination of FDTD techniques and Padé approximation with Baker’s algorithm [27–28] is used for the cavity mode Q factor calculations. The total computation domain is 18a×19.06a×4a along the x, y and z directions, where a is the lattice constant. The grid size is 0.05a in the x, z direction and 0.0433a in the y direction.
2. Cavity design and modes tuning
Though there have been designs of cavities in 2D PCS’s with Q⊥ larger than 106 [8–9], we find them deteriorate tremendously when used in the in-plane channel drop filter system. The reason lies in the presence of the waveguides. Most high Q cavities reported involve some missing air holes and their high Q⊥ is achieved by carefully modifying the surrounding boundaries so that in the momentum (k) space, most of the components are pushed away from the light cone [7–9, 24]. However, when the waveguides are placed close to the cavity, the boundaries of the cavity are changed and so is the k space distribution of the cavity mode. As a result, more k components are introduced into the light cone and thus Q⊥ is reduced greatly.
Though increasing the distance between the waveguide and the cavity can reduce the waveguide’s impact on Q⊥ of the cavity, this will reduce the coupling between the waveguide and the cavity, Q◃ is also increased and the ratio Q⊥/Q◃ may not improve at all.
To avoid the dilemma mentioned above, we seek to use a cavity whose vertical light confinement does not rely much on the surrounding boundaries [10]. The structure of the cavity is shown in Fig. 1(a). This cavity is constructed by a central large air hole with radius R0 , surrounded by 4 periods of air holes with decreasing radii, along the outward direction denoted by R1, R2, R3 , and R4 respectively. The radius of the air holes on the same hexagon stays the same and R1 to R4 follows a parabolic pattern as shown in equation (3). The radius of air holes on the outmost hexagon R4 is the same as the regular air hole radius R=0.30a, where a is the lattice constant and set to 420nm in our case. The slab thickness t is 0.6a and the refractive index n of the slab is 3.4, corresponding to silicon at 1.55µm. If R0 and R1 are given, the cavity structure is determined.
Fig. 1. (a) Top view of the system. Selected channel is transferred along the forward direction of the drop waveguide. The cavity involves a central large air hole surrounded by 4 hexagons of lattice. The radii of air holes on the hexagons are tuned into a graded pattern following equation (3). (b) Hz field distribution of the even mode at central slab plane. (c) Hz field distribution of the odd mode at central slab plane. The mirror planes are shown as the dashed lines.
This cavity consists of two-levels of vertical light confinement [10]. The first level of confinement has a centrally enlarged air hole (R0 ) followed by a relatively large decrease in hole radius (R1 ) for the nearest neighbor holes. The hole radii are then parabolically decreased in moving radially outwards down to R4=R at the edge of the cavity, forming the second level of confinement. Though increasing the number of hexagons surrounding the central large hole can enhance the vertical light confinement, it also reduces waveguide-cavity coupling. We choose the number of surrounding hexagons to be 4, which is sufficient to keep good waveguide-cavity coupling while maintaining high vertical Q. The hexagonal symmetry of the cavity itself makes it possible to support two modes of opposite symmetries (though only the mode with Hz field even in both in-plane directions was reported in [10]). As shown in Fig. 1(b) and 1(c), the first mode has Hz field even in both x and y directions and the second mode has Hz field even along y direction but odd along x direction. Their symmetry properties decide a forward drop when the two modes are degenerate [22].
The cavity structure has to be fine-tuned in order to force the even and odd mode into degeneracy, i.e., the two modes should have the same resonant frequency and similar, if not exactly the same, Q factors. We can see from Fig. 1(b) and 1(c) that the even mode has strong field distribution on the central air hole region, while the odd mode tends to concentrate on the first-round neighboring holes, so we expect that modification of R0 parameter will have more impact on the even mode and so it is with R1 on the odd mode. In order to maintain the vertical light confinement of the cavity, we keep R0 > R1 and follow Eq. (3) strictly during the tuning process. We start with R0 =0.365a and increase R0 by step of 0.005a to 0.395a. For each value of R0 , we choose three values for R1 , i.e., R1 =R0 -0.005a, R1=R0 -0.01a and R1=R0 -0.015a. The results are shown in Fig. 2.
Fig. 2. Tuning of the cavity shown in Fig. 1(a). The lattice constant a is 420nm. (a) Variations of central wavelengths of the two modes. (b) Variations of total Q factors. (c) Central wavelength difference shifts. (d) Q factor difference shifts. The region inside the dashed circle is the work region, where the two modes have closest central wavelengths and Q factors.
Figure 2(a) shows the central wavelengths of the two modes with respect to R0 and R1 . The central wavelength of the even mode λ1 is labeled with diamond maker and λ2 of the odd mode is labeled with circle maker. The Q factor variations of the two modes are shown in Fig. 2(b). Q1 is the total Q factor of the even mode and Q2 is the total Q factor of the odd mode. Fig. 2(c) and 2(d) show the central wavelength and Q factor difference respectively. Examine Fig. 2(c) and 2(d) in detail and we find the two modes have the closest central wavelengths and Q factors when R0=0.38a and R0 - 0.005a>R1>R0 - 0.1a. We set R0=0.38a and tune R1 in smaller steps. The result shows when R1 is 0.372a, the two modes have closest central wavelengths as well as Q factors, with λ1=1554.51nm, λ2=1554.52nm, Q1=3,060 and Q2=3,020. Further computation separates Q⊥ and Q◃ from Q total by Q-1 . For the even mode, Q⊥1=40,500 and Q◃1=3,100. For the odd mode, Q⊥2=35,000 and Q◃2 =3,050. The central wavelength difference is smaller than 0.01nm and is negligible compared to the line width.
We have already seen from Fig. 2 that the even and odd modes have a strong tendency to degenerate. For a few tests, we vary the refractive index n of the silicon slab by ±1% and also the slab thickness t by ±0.05a, the two modes stay degenerate with Δλ<0.01nm and ΔQ<100, despite the absolute shift in central wavelength and Q factors. This makes our design insensitive to some of the fabrication uncertainties, such as slab thickness and material index.
Furthermore, the waveguide mode is single, non-leaky and not highly dispersive in the frequency range 0.267~0.28 (c/a), corresponding to 1500~1573 (nm) in wavelength assuming lattice constant a=420 nm. The waveguide loss is avoided in our system as the central wavelengths of the cavity modes lie within this range.
The 3D FDTD simulation result for the transmission computation is shown in Fig. 3. At resonance, 78% of light is transferred along the forward direction of the drop waveguide, 1.75% is transferred along the backward direction of the drop waveguide, and 1.6% of light is still left in the bus waveguide. The Q factor measured as full width half maximum (FWHM) of the transmission spectrum is around 3,000, relevant to the previous Q◃ calculations.
Fig. 3. Intensity spectra for the structure shown in Fig. 1(a). The wavelengths correspond to a lattice constant of 420nm.
3. Waveguide tuning to improve the drop effect
One important factor characterizing a channel drop filter is its channel isolation (CI), defined as CI=10log(P1/P2), where P1 and P2 are the power of the selected channel transferred to the drop waveguide and the power remaining in the bus respectively. The channel isolation should be as large as possible to avoid the cross talk. The CI value of the system we designed above is only 16.9 dB. We try to improve it by further reducing the 1.6% of light that remains in the bus while keeping the power transferred high. We re-write Eq. (1) in more detail. In equation (4) below, 1/τe1 and 1/τ’e1 are the decay rate of the even mode into bus and drop waveguide respectively, 1/τo1 is the decay rate of the even mode due to loss and similarly defined are 1/τe2, 1/τ’e2 and 1/τo2 for the odd mode.
When the system is symmetric and each mode decay equally into bus and drop waveguide, we have 1/τe1=1/τ’e1 and 1/τe2=1/τ’e2 . In this case, |T|2 will always be larger than zero according to equation (4) and the selected channel cannot be 100% dropped. However, if we modify the waveguide boundary and let τ’e1, 2 slightly larger than τe1, 2 so that the nominator of Eq. (4) becomes zero, the input signal power at resonant wavelength is completely removed from the bus.
As shown in Fig. 4, we reduce the radius Rw of the air holes at the upper boundary of the drop waveguide. This modification turns out to affect both modes equally. The in-plane Q factors of both modes are improved and thus τ’e1, 2 increased to the same extent. The two modes stay degenerate though their central wavelengths shift slightly by the same amount. There are other ways to change the decay rates, such as enlarging the air hole radius of the bus waveguide boundaries and increasing the drop waveguide width. These changes, however, are too abrupt and difficult to control, so we have skipped them in this paper.
Fig. 4. The radius of air holes on the upper boundary of the drop waveguide is modified into Rw in order to reduce the signal power remained in the bus waveguide.
The simulation result is shown in Table 1 below. At Rw=0.27a, only 0.39% of signal power remains in the bus waveguide and the channel isolation is enhanced to 22.7dB. It is worth noting that due to the Lorentian line shape of the transmission spectra [22], the filter undergoes fairly large interchannel cross talk. The Q factor of the filter is around 3,000 and if we use 100GHz channel spacing centered at the resonant frequency, the interchannel cross talk level is only 13dB below the desired channel. A higher Q factor is desired in the future channel drop filter designs using 2D PCS’s.
Table 1. Signal power remained and transferred at resonance with different Rw
Figure 5 shows the oscillation of the steady-state field distribution (Hz ) at the resonant frequency for the case Rw=0.27a. A silicon wire is connected to each end of the photonic crystal waveguides. Most of the field is transferred along the forward direction of the drop waveguide. The field along the forward direction of the bus waveguide and along the backward direction of the drop waveguide is negligible. Also note that the field is much stronger in the cavity region due to the high Q factor (~3,000). The filter is compact and only occupies an area of a few wavelengths square.
Fig. 5. Steady state Hz field oscillation at resonant frequency for the structure shown in Fig. 1(a) and 4, with Rw=0.27. (Movie 1,293 KB)
4. Conclusions
To conclude, we have successfully designed an in-plane channel drop filter in 2D hexagonal lattice photonic crystal slab. The system is compact and only involves a single cavity, which is constructed by arranging the radii of air holes on a few hexagons into a graded pattern. The cavity supports two modes of opposite symmetry. Both modes keep high vertical Q and the system undergoes low vertical loss. By carefully tuning the cavity, the two modes can achieve degeneracy. For a lattice constant of 420nm, the selecting wavelength is 1554.52nm and the line width of the filter is around 0.52nm corresponding to a Q factor of 3,000. At resonance, the signal power transferred along the forward direction of the drop waveguide is 78% and 1.6% remains in the bus. By modifying the upper boundary of the drop waveguide, light remaining in the bus can be further reduced to below 0.4%, with channel isolation enhanced to 22.7 dB. We believe this novel and compact design is adequate for dense wavelength division multiplexing applications in modern optical networks.
Acknowledgments
This work was supported by the Swedish Foundation for Strategic Research (SSF) on INGVAR program, the SSF Strategic Research Center in Photonics, and the Swedish Research Council (VR) under project 2003-5501.
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