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The infrared propagation character of metallic woodpile photonic crystals made of gold in the <110> direction was studied. The bandgap in metallic woodpile was found to be interrupted by a new propagating mode far beyond the waveguide cutoff. This new passband cannot be explained by the defect mode of the photonic crystal or the waveguide theorem. However, studies of surface plasmons within the structure showed that 3-D metallic photonic crystals can expand the propagating mode far beyond waveguide cutoff. This passband mode can be adjusted by design allowing for new optical filtering devices using 3-D photonic crystals. Finite-difference-time-domain (FDTD) simulations were applied to explain this phenomenon and matched well with experimental results.
©2011 Optical Society of America
1. Introduction
The study of photonic crystals over the past 10 years has led to new ways of manipulating electromagnetic waves [1,2]. With the help of modern microfabrication techniques, larger photonic crystals with periodic features on the micrometer or nano scale can now be built over multiple square inches. This article examines some unique properties of the woodpile structure, a face-centered tetragonal [3,4] photonic crystal that was theoretically found to have a full band gap if the dielectric constant contrast is large enough [5]. This structure is normally fabricated with metallic beams in an air, oxide or polymer dielectric [4–16] because metallic photonic crystals have more interesting advantages compared to dielectrics and semiconductors due to the special optical character of metals [6,12]. Fleming was among the first to report measurements and simulations of a tungsten woodpile photonic crystal in 2002 [6]. In their letter, Fleming built a metallic woodpile structure whose surface normal is in the <001> direction, so the measurements and the simulations are also based on this direction. In the <001> direction, the stop band can be explained by waveguide cutoff. There is no propagating mode beyond the metallic waveguide cutoff [7]. Sang theoretically predicted the existence of the passband mode beyond waveguide cutoff if dielectric spacers are inserted into the metallic layers [17,18] and Chang experimentally proved the existence of these propagating modes [12]. Up to now, most discussions and analyses about the metallic woodpile photonic crystal were based on <001> direction. However, it is complicated and time consuming to fabricate this layer-by-layer metallic photonic crystal. Multiple attempts have been made to simplify the fabrication and analysis of 3-D photonic crystals [13,14] by developing a top-down approach to fabricate the woodpile structure that is much easier than layer-by-layer method [15,16]. One variation of this approach is a kind of tilted-woodpile, which means the surface normal is along the <110> direction. Recently, Williams and his colleagues fabricated a group of large area of tilted-woodpile metallic (Au) photonic crystals using deep x-ray lithography and subsequent electroforming technique, the detailed process of which is described in [19]. This production scheme for metallic photonic crystals makes studying metallic tilted-woodpile structures both experimentally and theoretically available. This article shows that metallic tilted-woodpiles exhibit a passband far beyond the waveguide cutoff without inserting any dielectric spacer between metallic layers. The experimental measurements and simulation results reported here demonstrate the existence of these beyond-cutoff propagating modes in an all-metallic woodpile photonic crystal. To study these propagating modes, the FDTD simulations were performed and surface plasmon effects were found to contribute to the modes.
2. Structure of the tilted-woodpile photonic crystal
Figure 1(a) is the schematic of the tilted-woodpiles created by the top-down method. Three parameters are used to engineer the structure: the rod spacing (d), rod width (w) and rod thickness (h). The first layer of the 3-D schematic is shown in Fig. 1(a) where the logs are arranged parallel in an inclined angle 45°. The second layer (next to the first layer) has the same parameters but an opposite inclined direction (inclined angle is 135°). This is different from the traditional woodpile which is layer-by-layer created in <001> direction (shown in Fig. 1(b)). This tilted-woodpile is fabricated by the top-down method in <110> direction so the testing light is launched out in the <110> direction with or without an oblique angle of θ or ϕ (shown in Fig. 1(a)).
Fig. 1 Schematics of (a) the tilted-woodpile photonic crystal created by top-down method, whose surface normal is in the <110> direction, and (b) the conventional photonic crystal created by layer-by-layer method, whose surface normal is in the <001> direction.
3. Results and analysis
A hemispherical directional reflectometer was used to measure the specular and total reflectance of the tilted-woodpile with the light being non-polarized. Figure 2 shows the total reflectance of a gold tilted-woodpile photonic crystal with rod thickness h = 2.4 μm, rod width w = 1 μm and rod spacing d = 4 μm. The incident light was inclined at an angle θ = 7° from the <110> direction with an air dielectric. The tested sample has a thickness greater than 2.5 unit cells with a reflective gold film underneath. Thus, the experiment closely represents an ideal photonic crystal lattice [6,20]. Different from the proceeding studies in the <001> direction [6,7], the stop band is interrupted by a passband (about 7 μm to 9 μm) where the reflectance falls down rapidly.
Fig. 2 Measured total reflectance of a tilted-woodpile depicted in Fig. 1(a) made of gold with h = 2.4 μm, w = 1 μm and d = 4 μm.
In attempt to understand the propagation mode data measured from the tilted-woodpile structure and study the physical origin of the observed passband modes, a 3-D FDTD method was applied to simulate the tilted-woodpile and traditional woodpile structures, with different log size and oblique incident angles. A rectangular simulation domain was chosen and two kinds of boundary conditions were used in the simulation. Because the height is finite in a real sample, the top and bottom of the sample were set to be perfectly matched layer (PML) boundaries. The other four sides were set to be periodic boundaries since the real samples are large enough and also periodic. So for the tilted-woodpile, the two boundaries of <110> directions were set to be PMLs. For the traditional woodpile, the boundaries in <001> direction were set to be PMLs. To get good convergence and accurate results, the space grid size was set to be about 1/10 of the minimum feature size of the sample and the time step needed to be smaller than the stable limit (1.5 times less is enough). The convergence time largely depended on the simulation wavelength. Normally, the mid infrared (IR) (3 μm - 8 μm) takes much a longer time to get convergence than the short (1 μm - 3 μm) and long (8 μm - 12 μm) IR wavelengths. This is because the pass band is normally located in the mid IR spectrum and it takes more computational effort to simulate a photonic crystal near the band edge where large diffusive optical properties and changes in reflectivity are present. Great effort was paid to make the simulation as realistic as possible. Plane waves (with different wavelengths), which are large enough to cover the sample, were chosen as the light source by which all other shape of waves could be constituted. Complex permittivity of gold was used to model the metallic dielectric constant as a function of wavelength [21]. To avoid the influence of discontinuity at the top and bottom surfaces, all logs were cut to begin in the same top surface and end in the same bottom surface. The simulated sample has a total depth of 3 unit cells. Both TM and TE waves were simulated and the results were similar, so we used TM mode to demonstrate the prorogation character in the following paragraph.
Figure 3 shows the simulation’s total reflectance in two directions for the same woodpile structure. One is for the traditional layer-by-layer woodpile (<001>direction) and another is for the tilted-woodpile (<110> direction). In <110> direction (tilted-woodpile) the band gap is interrupted by a wide passband, similar with the experimental result in Fig. 2. Thus, the measured pass band was not due to surface roughness in the fabrication of the tilted-woodpile or the defect mode, since the simulation was performed to the ideal tilted-woodpile photonic crystal. One may be confused by the reflectance in the <001> direction: it seems there are also a similar passband (4 μm - 5.8 μm) and a peak (3.8 μm), which at first glance comes from the same origin as the <110> direction and just shift left off. Further analysis, however, showed that the similarity is just a coincidence. The same woodpiles, but made of silver and tungsten, were also simulated both in <110> and <001> directions. The similar dip and peak in the <001> direction cannot be found in these two woodpiles (the reflectance of tungsten woodpile can also be found in reference [6] and [7]). Except this, all spectra characters discussed in this article can also be applied to the silver and tungsten-woodpiles, indicating these results illustrate a general phenomenon for all metallic woodpile crystals.
Fig. 3 Normal incidence reflection from the gold woodpile photonic crystal with h = 1 μm, w = 1 μm and d = 4 μm using traditional fabrication approaches, <001> direction, and the tilt-woodpile, <110> direction.
If a slice is cut normal to <001> direction, the cross section is just a layer of the structure (as shown in Fig. 1(a)) and can be referred to as a 001-cut cross section. Viewing the structure along the logs’ direction, the structure looks like many metallic waveguides with a rectangular cross section except that for each waveguide the top and bottom sides are not totally closed. Therefore, it seems to be reasonable to explain the propagation in the frame of the waveguide. To reveal the origin of these propagating modes, an electromagnetic (EM) wave of wave length λ = 8.2 μm was launched into the tilted-woodpile with the parameters h = 1 μm, w = 1 μm and d = 4 μm in the <110> direction. The electric field contour map of FDTD simulation in Fig. 4(a) shows the wave propagation in a 001-cut cross section, where the electric field is represented by colors and the dark lines are the outlines of the structure. In this simulation, the electromagnetic wave was found to propagate just like in a traditional waveguide.
Fig. 4 Electric field contours for different cross sections of the gold woodpile photonic crystal with h = 1 μm, w = 1 μm and d = 4 μm. (a) Cross section normal to <001> direction; (b) cross section normal to <110> direction; (c) cross section normal to <1,-1,0> direction.
Deeper analysis of wave propagation within this structure, however, shows a dramatic difference between the tilted-woodpiles and conventional waveguides. According to waveguide theorem [22], the longest cutoff wavelength is λc = 2a, where a is the larger parameter among the air space between rod (d-w) and rod thickness h. In this case, (d-w) = 3 μm > h, so the cutoff wavelength λc = 2(d-w) = 6 μm. Any EM wave with the wavelength greater than λc will not be propagated in a waveguide. These assumptions cannot account for the observed passband from 6 μm to 9 μm. Thus the passband is due to other effects beyond the waveguide mode. One can also note that this passband is very similar to that of Chang [12], where dielectric material layers were inserted to get the beyond-cutoff propagating modes. It can also be found from Fig. 3 that in <001> direction, the stop band is continuous and the band edge is just located at the cutoff wavelength (6 μm), so the stop band in <001> direction can be explained by waveguide theory. One should note that the starting edge of the stop band in <001> direction is located at the starting edge of the pass band in <110> direction. Furthermore, calculations performed at multiple different angles of incidence showed that the band passband decreased in size and shifted left as the observed orientation of the crystal was changed from <110> to 60° off axis in both θ and φ orthogonal directions. Thus we propose that the full band 3-D bandgap proposed for the woodpile photonic crystal actually opens a previously un-observed passband at the <110> orientation that is not easily simulated in standard band gap calculations due to the highly dispersive optical properties of a metallic photonic lattice such as gold.
Since the propagating wave is not a waveguide mode, there must be other modes contributing to it. To find these modes, the propagating wave was watched from different points of view. The electric field contours of the other two directions’ cross sections are shown in Fig. 4(b) and Fig. 4(c). Figure 4(b) is the cross section normal to <110> direction, from where it can be noticed that the field is mainly distributed on the boundary of the structure, especially on the edges of the logs. This is a typical surface plasmon effect. These surface plasmon modes along the interface between gold logs and air can also been seen from Fig. 4(c), which is the cross section normal to <1,-1,0> direction. So it is clear that the passband beyond waveguide cutoff in the <110> direction is due to the surface plasmon propagation modes.
To study this beyond-cutoff propagating mode, several samples with different parameters were tested. Figure 5 shows measured total reflectance of three tilted-woodpiles with different rod sizes (h and w) but the same rod spacing (d). It can be found that the width and the depth of these dips change a lot when the log size changes. In Fig. 5, the smaller dip represents the larger log size (meaning smaller air space between logs) and the larger dip comes from the smaller log size. The result indicates that the passband can be tuned by changing the rod size.
Fig. 5 Measured total reflectance of different log size tilt-woodpiles, the smaller dip represents the larger rod size and the smaller space between rods.
Experimental data (Fig. 5) showed that the dip size and amplitude depend on the size of the rod. The bigger the rod size (or the smaller air space between rods), the smaller the dip. To study how to engineer the passband, different cases were simulated with only rod width (w) changing and then with only rod thickness (h) changing (d remained constant). The results with same log width (w = 1 μm) but different log thickness (h = 1, 1.5, 2.4 μm) are shown in Fig. 6(a) . They show a change of the dip position and width without significant change in depth of the transmission window. Figure 6(b) shows the reflectance of three tilted-woodpiles with the same rod thickness (h = 1 μm) but different rod width (w = 1, 1.5, 2.4 μm). It indicates the change of the dip size (width and depth). However, the initial wavelength of the dip changes very little. Therefore the measured reflectance is the combined result of different log thickness (a small dip position shift) and different log width. Changing one without changing the other would change the resonant conditions. This allows the ratio h/w to be optimized and to tune the passband. The trend of this passband can be predicted this way: larger log thickness will make the passband narrower and larger log width will shift the passband left. For example, if we want a 1μm-width passband centered at 6 μm, make the log thickness h = 2.5 μm and log width w = 1.5 μm.
Fig. 6 Simulation reflectance of tilted-woodpile with (a) different log thickness and (b) different log width.
Meanwhile, incident light from the surface normal of the tilted-woodpile (<110> direction) deviated both in θ and φ angle up to 60° (as shown in Fig. 1(a)) were simulated and experimentally tested. They showed a similar propagation character as the normal incidence case. This means that for the metallic woodpile photonic crystal, the band gap, which is continuous in the <001> direction, is divided into two parts by a passband in the <110> direction. This character is applicable in a large angle range.
4. Passband applications
The demonstration of a tunable IR transmissive passband in tilted-woodpile photonic crystals allows for the development of new optical filtering devices. Slight changes in the 2-D lithography mask design allow one to pattern discrete volumes of photonic crystal across tens of microns or millimeters in size that have either different IR passbands or no passband at all. This allows for the construction of discrete optical transmissions of IR light from varying wavelengths through a periodically structured material at the behest of the designer. Simple devices include a bullseye for targeting, and different diffractive imaging in both reflection and transmission. The simple, low cost, and large area fabrication scheme previously demonstrated [19,20] for these devices can therefore be utilized to open new application spaces for photonic crystal design.
5. Conclusion
It has been shown that woodpile metallic photonic crystals in the <110> direction possess a band gap broken into two parts by a passband, which is far beyond the cutoff wavelength. This passband, due to surface plasmon modes, is naturally present in metallic tilted-woodpile photonic crystal structures and can be tuned by changing the log size and the space between logs. Also, in <110> direction, the propagation characteristic stays roughly the same within a large solid angle. With these characters, 3-D photonic diffractive devices can be designed by tuning the passband in the <110> direction of metallic tilted-woodpile photonic crystals.
Acknowledgments
The authors would like to thank the Center for Advanced Microstructures and Devices (CAMD) at Louisiana State University and especially Mr. Zhong-Geng Ling for his work exposing these structures. The authors would further like to recognize Dr. William C. Sweatt of Sandia National Laboratories (SNL) for co-inventing the fabrication process for the tilted woodpile photonic crystal. We also wish to acknowledge Robert Ellis, of SNL for BDR testing. Device fabrication and testing was completed under Dr. Williams at Sandia National laboratories, a multi-program laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy under contract DEAC04-94AL 85000. Modeling and simulation were completed at University of Alabama in Huntsville and funded by the Alabama Graduate Scholars Research Program (GSRP).
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