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The resonant frequencies of a single cavity embedded in the three-dimensional layer-by-layer photonic crystal are studied with microwave experiments and transfer-scattering matrix method simulations. The effects of the number of cladding layers and the size of the embedded cavity on resonant frequencies and Q values are carefully examined. The fine increments of cavity size indicate a new pattern of relation between resonant frequencies and cavity sizes.
©2008 Optical Society of America
1. Introduction
In recent years photonic crystals (PCs) have attracted extensive interest due to their potential to engineer the properties of electromagnetic (EM) wave propagation. The existence of photonic band gap where radiation of a particular frequency range is completely forbidden makes PCs useful in a number of applications, like controlling spontaneous emission, low-threshold lasing and single mode LEDs.1-4 These applications involve introduction of defects or light emitters to the crystal. Good progress has been made in two-dimensional PCs, and there is increasing interest in three-dimensional (3D) PCs.5,6,7 The presence of a complete and robust band gap in the 3D layer-by-layer PC structure made of alumina rods was demonstrated early, and the defect cavity modes within the band gap were also studied. 8,9,10
Fine tuning defect parameters is usually needed to achieve certain function with desired performance, for example tuning cavity for extremely high quality factor.11,12 For 3D PC devices design at visible or infrared frequency range, it is usually difficult to change the defect size or geometry during experiments for real time responses. But due to the scaling property of photonic crystal structures, we can alternatively extract useful information by microwave frequency experiments with defect sizes and geometry well controlled and fine tuned. In this paper, physical properties of fine tuning cavities embedded in 3D layer-by-layer PC are studied by both microwave experiments and transfer-scattering matrix method (TMM) simulations.13-16
2. Experimental and simulation setup
The structure that we examine in this paper is a layer-by-layer PC built of square cross section alumina rods of width 3.2mm. These rods are commercially available with measured refractive index 3.0. The whole photonic crystal has a rectangular cross section of 15.24cm by 30.48cm with height determined by the number of layers of dielectric rods stacked on top of each other. The stacking sequence repeats every four layers, corresponding to a single unit cell along the stacking direction. The center-to-center spacing between neighborhood rods is 10.7mm, giving a filling ratio of 29%.
The defect cavity is introduced by removing portion of the rod material at the center of the crystal’s XY plane and in the middle of the stacking direction, for example the 7-1-6 configuration representing 14 layers in total along the stacking direction and the defect layer is located at the 8th layer from the top. The geometry of the photonic crystal, cavity and the experimental setup are conceptually illustrated at Fig. 1.
The transmission properties are measured by Hewlett Packard 8510A network analyzer with standard gain horn antennas to transmit and receive EM radiation. The electric field polarization is parallel to the rods of the defect layer for all these measurements. The losses in the cables and the horns are normalized by calibrating all measurements up to the ends of the horn antennas. The numerical calculation is performed on a 5-by-5 super cell structure using TMM with interpolation and higher-order incidence techniques. The 5-by-5 super cell structure is same as the experiment structure except that there are repeating cavities in both X and Y directions in the defect layer, which is introduced by the periodic boundary condition required by TMM. The measured directional band gap for this configuration lies between 10GHz and 16GHz, which agrees well with TMM calculations’ results (10.7GHz to 15.4GHz). Typical transmission characteristics for propagation along stacking direction are shown at Fig. 2 for both experiments (red line) and calculations (blue line) along with the calculated directional band gap (gray line). The transmission rate is usually less than 10% and the zoom in view of the spectra at Fig. 2 shows good agreement between microwave experiments and numerical simulations.
Fig. 1. (a) shows the schematic picture of 3D layer-by-layer PC (not the actual experiment unit); (b) shows the cavity generated by removing portion of the rod material and the stacking sequence along with the polarization direction of electric field; and (c) shows the microwave experiment set up with two horn antennas used for coupling.
Fig. 2. A typical transmission response of a defect cavity for both microwave experiment (red) and numerical simulation (blue), along with the directional band gap of the perfect crystal (gray). The inset is the zoom in view of the transmission around the resonant frequency.
3. Increasing the cladding layers
First, we investigate the effects of increasing the number of cladding layers above and below the cavity layer on the resonant frequency and the cavity quality factor (Q value). A cavity of size d/a=1.0 is studied where ‘d’ is the length of rod removed and ‘a’ is the center-to-center spacing between adjacent rods at XY plane. The resonant frequencies and Q values from experiments (solid black line with square) and calculations (gray dashed line with circles) of configuration 5-1-4, 6-1-5, 7-1-6, 9-1-8 and 11-1-10 are illustrated at Fig. 3.
The trends of resonant frequency and Q value when the cladding layers are increased agree well between experiments and simulations. The resonant frequency oscillates and approaches to a constant while the Q value increases experientially. The systematic lower simulated resonant frequencies may due to the uncertainty of refractive index of the rod material and the rod dimension of the experiment structures; and the systematic lower experimental Q values may due to the edge loss from the finite size of experiment structures and misalignment of the rods.
Fig. 3. Comparison of numerical simulation (gray circles with dashed lines) and microwave experiments (black squares with solid lines) for resonant frequency and Q value when the number of cladding layers are increased with fixed cavity size (d/a=1).
4. Varying the cavity size
Then we study the effects of the defect cavity size (d) to the resonant frequency by increasing the cavity size from 0.5a to 8.0a in steps of 0.25a. There were no visible response peaks for cavities of size smaller than 0.5a for microwave experiments. For calculation, the size of the defect is increased up to 2.75a. When cavity sizes are larger than 2.75a, interactions among neighborhood cavities are not negligible for the TMM 5-by-5 super cell configurations.
The resonant frequencies of increasing defect sizes do not follow a regular pattern. Modes split at intervals as shown at Fig. 4: each square symbol corresponds to a resonant peak, and dark squares are peaks with highest transmission (dominant modes) for a particular defect size. The resonant frequencies increase as the defect size increases for both dominant and non-dominant modes. Considering defects with size from 2a to 5.5a, the resonant frequency of dominant mode increases up to size 3.5a, then it shifts to a lower range and starts to increase again to cavity size up to 5.5a. Before the shifting of the dominant modes at 3.5a, new modes arise and become dominant gradually from the fact of gradual change of coupling coefficient across different resonant modes (i.e. around 3.5a the coupling coefficients are almost the same for the two competing resonant modes). As the cavity size increases, the rate of resonant frequency increment reduces and resonant frequencies become almost flat for defect size close to 7a. The change of dominant mode resonant frequency of defect sizes from 0.5a to 8a is only 3.5%, and the change of lower transmission mode resonant frequency is around 8%. The measured Q values of all modes in this configuration range from 500 to 1000. At one of our previous work10 the deceiving decreasing trends of resonant frequencies as the cavity size increased was due to the relative larger tuning steps, which is illustrated by the blue arrows shown at Fig. 4. The fine tuning reveals more detailed properties of 3D photonic crystal cavities which are essential for future devices design.
Fig. 4. Experimental response of peak resonant frequency for cavities with fine tuned size from 0.5a to 8a. The black square is the highest transmission peak among all resonant peaks for fixed cavity size, the gray square is the relatively low transmission peaks, and the blue line with arrow is the trends of dominant mode if relative larger tuning steps are used.
Fig. 5. Comparison between the microwave experiments (dots) and TMM calculations (lines) with different color representing two different modes (A and B).
Figure 5 shows the calculation results for increasing cavity size up to 2.75a. Both the measurement and calculation resonant peak frequencies are directly obtained from the spectra as shown at Fig. 2, and there will be more than one peak in the spectra for cavities larger than 1.5a. The trend of frequency increment is the same as experiments. The small discrepancy in frequency between the experiments and calculations may due to the uncertainty of refractive index of the rod material and the cavity dimension of the experiment structures. The calculation also shows that the electric fields for both modes at Fig. 5 are X-component dominant and the mode profiles of electric field X-component are illustrated at Fig. 6.
Fig. 6. Electric field mode profile from TMM calculation for resonant mode A and B (defined at Fig. 5) of cavity size 2a with the solid rectangle showing the location of the cavity. The x and y scales are in unit of lattice constant a.
The trend of resonant frequency increasing with cavity size increase can be attributed to the increase in k vector as the size of the cavity increases.17 For two-dimensional photonic crystals at reference 17 where the slope of dispersion is negative, the resonant frequency decreases on increasing d/a, but our three-dimensional layer-by-layer photonic crystal has a positive defect dispersion slope18 and we see an opposite trend in the frequencies. This may be attributed to the abnormal modal reflectivity at the ends of the cavity with the present of 2D photonic crystal background.
5. Conclusions
In summary, we have systematically studied the behaviors of cavity embedded three-dimensional layer-by-layer photonic crystal structure through both microwave experiments and numerical simulations. Consistent results of resonant frequency and Q value for structures with different cavity size or different number of cladding layer are obtained between experiments and simulations. The correct trends of increasing cavity size have been found through fine tuning steps. Those quantitative results of fine tuning cavity embedded in 3D photonic crystal structure can be very useful for future photonic crystal devices design.
Acknowledgments
This work is supported by the Director for Energy Research, Office of Basic Energy Sciences. The Ames Laboratory is operated for the U.S. Department of Energy by Iowa State University under contract No. DE-AC02-07CH11358.
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