Add-drop filters in three-dimensional layer-by-layer photonic crystals using waveguides and resonant cavities Preeti Kohli, Caleb Christensen, Jason Muehlmeier, Rana Biswas, Gary Tuttle et al. Citation: Appl. Phys. Lett. 89, 231103 (2006); doi: 10.1063/1.2400398 View online: http://dx.doi.org/10.1063/1.2400398 View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v89/i23 Published by the AIP Publishing LLC.
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APPLIED PHYSICS LETTERS 89, 231103 共2006兲
Add-drop filters in three-dimensional layer-by-layer photonic crystals using waveguides and resonant cavities Preeti Kohli, Caleb Christensen, and Jason Muehlmeier Department of Electrical and Computer Engineering, Iowa State University, Ames, Iowa 50011 and Microelectronics Research Center, Iowa State University, Ames, Iowa 50011
Rana Biswasa兲 Departments of Electrical and Computer Engineering, Iowa State University, Ames, Iowa 50011; Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011; Microelectronics Research Center, Iowa State University, Ames, Iowa 50011; and Ames Laboratory, Iowa State University, Ames, Iowa 50011
Gary Tuttle Department of Electrical and Computer Engineering, Iowa State University, Ames, Iowa 50011; Ames Laboratory, Iowa State University, Ames, Iowa 50011; and Microelectronics Research Center, Iowa State University, Ames, Iowa 50011
Kai-Ming Ho Ames Laboratory, Iowa State University, Ames, Iowa 50011 and Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011
共Received 7 September 2006; accepted 22 October 2006; published online 4 December 2006兲 A three-dimensional layer-by-layer photonic crystal with a complete photonic band gap is used to experimentally and theoretically demonstrate a sharp tunable bandpass filter. The structure consists of input and output waveguide sections coupled through a nearby cavity. The authors show experimentally and verify with finite difference time domain simulations that this configuration is a bandpass filter where a particular resonant frequency of the cavity is selected from the input guide and transmitted to the output guide leaving out other input frequencies. An excellent coupling efficiency near 100% between the waveguide and the cavity is found for the drop frequencies. © 2006 American Institute of Physics. 关DOI: 10.1063/1.2400398兴 Photonic crystals 共PCs兲 have various emerging applications to optical communications, switching, and manipulation of photons. PC-based optical devices have been demonstrated and there is continuing interest in PC structures for channel add-drop filters.1,2 Trapping and emission of photons by cavities have been demonstrated in two-dimensional 共2D兲 structures experimentally3 and various add-drop filters in two-dimensional PCs have been realized. Out-of-plane losses in 2D PC’s are a major limitation. To eliminate out-of-plane losses it is advantageous to form filters in three-dimensional 共3D兲 PCs with complete band gaps. Coupling properties in 3D structures were simulated4 and the dropping of photons in a 3D layer-by-layer PC was demonstrated.5 This motivated the realization of a channel drop filter and bandpass filter in a 3D PC with a complete band gap. We find enhanced coupling between two PC waveguides through a nearby cavity defect. The coupled mode frequencies between waveguides are determined by the resonant frequencies of the cavity. Our experiments were performed on a microwave-scale PC structure. The structure is the layer-bylayer crystal, studied extensively at Iowa State University. The crystal has square cross-section 共3.2⫻ 3.2 mm2兲 dielectric rods of alumina 共measured n = 3 ± 0.1兲. The stacking sequence repeats every four layers, corresponding to a single unit cell in the stacking direction.6 The center-to-center spacing between rods is a = 1.07 cm, giving a filling ratio of 29%. The crystal has a full three-dimensional photonic band gap from 11 to 12.9 GHz. a兲
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X waveguides are formed 共Fig. 1兲 by removing single rods.7 Cavities are formed by removing a portion of a rod less than a few lattice constants. Utilizing an HP 8510 network analyzer, microwaves were coupled into and out of the waveguides using a conventional horn antenna matched to an in-line PC horn antenna formed within the PC. This coupling scheme was very effective.8 Our structure consists of an entrance and exit waveguide 共Fig. 1兲 separated by a dielectric rod of length d. A cavity of length L is one unit cell above the waveguide layer. The centerline of the cavity is aligned with the centerline of the waveguides. d controls the cavity waveguide interaction and the cross-talk between the waveguides without the cavity. Figure 2 shows transmission measurements for a straight waveguide, two waveguides separated by d = 8a between the ends and no cavity, and two waveguides with separation d = 8a and a cavity with L = 0.75a one unit cell above it. The straight waveguide has a strong transmission band from 11.8 to12.8 GHz, as seen in our earlier measurements, simi-
FIG. 1. Configuration of the PC with separation d between the waveguides and cavity of length L, shown in an x-z cross section. 89, 231103-1
© 2006 American Institute of Physics
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FIG. 2. Measured transmission for the straight waveguide 共solid兲 compared to transmission for waveguides separated by 8a, with 共dotted兲 and without a cavity of L = 0.75a 共dashed兲. The cavity-induced resonance is ⬃1 dB below the straight guide.
lar to the band at 1.5 in a submicron GaAs PC.9 By introducing the gap into the waveguide, the transmission is reduced by ⬎20 dB over all frequencies. With the cavity, a narrow transmission peak appears at 12.22 GHz 共Fig. 2兲. The peak transmission of this band is nearly that of the straight waveguide, suggesting excellent coupling. The 12.22 GHz mode is the resonant frequency of a cavity of length L = 0.75a. We studied the coupling for different cavity sizes 共L兲 and waveguide separations 共d兲. Each cavity has a different resonant frequency. For larger cavities, the cavity can support more than one mode. L = 0.75a corresponds to a singlemode cavity. The transmission 共Fig. 3兲 with a larger multimode cavity of size L = 5.5a 共and separation d = 9a兲 that supports three cavity modes shows three strong transmission peaks. For any localized mode with a resonant frequency, critical coupling can occur between the localized mode and an incident mode, where all energy from the input channel transfers to the localized mode or vice versa. For such critical coupling there should be ⬃100% transmission from one waveguide to the other. The localized mode can couple to other radiative or guided modes. For each of these available modes we define a quality factor Q such that 1 / Q is proportional to the coupling strength. Critical coupling occurs if Qinput / Qrest = 1, where Qinput is associated with the input channel, and Qrest with all other modes.10 The localized mode is the cavity mode. The input channel is the incident waveguide, and all other modes include the output waveguide and leakage from the cavity to outside
FIG. 3. Measured transmission for a cavity of length L = 5.5a 共dashed兲 displaying three resonant frequencies, compared to the straight waveguide and no cavity 共d = 9a兲.
FIG. 4. Effect of reducing cavity confinement on the resonant mode at 12.82 GHz. The PC size is decreased from 12 layers to 4 layers above the cavity, corresponding to the geometry of Fig. 3.
the crystal. The output and input waveguides can be interchanged by symmetry. If 1 / Qw is proportional to the coupling between the cavity and the waveguide, and 1 / Ql to the leakage out of the crystal, then 1/Qinput = 1/Qw,
and 1/Qrest = 1/Qw + 1/Ql .
共1兲
The critical coupling condition becomes Qinput/Qrest = 共Qw + Ql兲/Ql = 1 + Qw/Ql = 1.
共2兲
Since leakage always exists in a finite crystal, we cannot get perfect coupling. If the ratio 共2兲 is ⬃1, there should be very high transmission. Ql / Qw should be small for good transmission from the input guide to the cavity, and by symmetry, from the cavity to the output waveguide. Not surprisingly, this requires strong coupling between waveguides and cavities. Even small amounts of external leakage can drastically reduce the transmission across the cavity. Leakage from the cavity to radiative modes outside the crystal reduces the cavity Q. The number of cladding layers above the cavity was decreased from 12 to 4 layers 共Fig. 4兲 for the multimode cavity structure with d = 9a and L = 5.5a. Interestingly, the two transmission peaks, near the midgap, show opposite trends. The peak transmission of the dominant mode at 12.82 GHz drops by ⬃8 dB as the cavity becomes leakier, but the peak transmission of the secondary mode at 12.88 GHz increases slightly. To obtain a physical picture of the add-drop filter, the defect cavity, and external leakage, we performed finite difference time domain 共FDTD兲 simulations for a real-space understanding of the mode. The simulated PC had 20 layers and had 19 rods/layer, to minimize leakage, similar to the measurement 共Fig. 3兲. Two straight waveguide sections were separated by d = 6a. A cavity 共length L = 3a兲 was created symmetrically four layers above the waveguides, retaining the yz mirror plane. Computational constraints necessitated using smaller L and d than experiment. The frequency response was obtained by exciting the input guide with a pulsed dipole source and observing the E fields. The E field was sampled over a spatial grid in the exit guide, as a function of time and Fourier transformed to obtain the transmission. Simulated transmission peaks 共Fig. 5兲 at ⬃11.5, 12.0, and 12.5 GHz indicated resonant frequencies that couple the input and output waveguides through the cavity. The multipeak response has many features in reasonable agreement with the experiment, especially the peaks at 12 and
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FIG. 5. FDTD simulations of the transmission through the waveguides separated by d = 6a and coupled by a cavity 共length L = 3a兲, four layers above the waveguides. The input guide was excited by a pulsed source to obtain the transmission in the output waveguide.
12.5 GHz. Calculated peak frequencies may be shifted from measurement due to a slightly different simulation geometry. After identifying resonant modes, we obtained a visual understanding of the cavity mode by exciting the input guide with a constant frequency source, tuned to the resonance 共12.5 GHz兲 and simulated fields at progressive times. Initially 共1000⌬t兲, 共where the time step ⌬t = 2.06 ps兲, large fields exist only in the input waveguide 共Fig. 6兲. As time progresses 共3000⌬t, Fig. 6兲, the fields grow within the cavity, indicating input waveguide-cavity coupling. At a later time 共5000⌬t兲, the intensity gradually builds in the exit guide. At longer time 共9000⌬t兲, a large excitation of the output guide is accompanied by the excitation of the cavity 共Fig. 6兲. The slow coupling requires ⬎5000 time steps to excite the cavity mode and then couple to the exit guide. Simulations validate the waveguide-defect coupling found experimentally and confirm the low external leakage. Such add-drop filters are an alternative to recent filters with 2D PC’s.2,11,12 In conclusion, we have demonstrated experimentally and theoretically that waveguides in 3D PC’s can couple through defect cavities. The resonant frequency of the cavity can be selected from the input guide and dropped to the output guide. Controlling the geometry and location of defect cavities can lead to realistic add-drop filter for selected frequencies. The authors thank J. R. Cao for useful discussions. This research was supported by the Office of Basic Energy Sciences of the Department of Energy. Ames Laboratory is operated for the Department of Energy by Iowa State University under Contract No. W-7405-Eng-82. S. Kim, I. Park, and H. Lim, Opt. Express 12, 5518 共2004兲. S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou,
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