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We measure the normal-incidence transmission coefficient of photonic crystal slabs with hexagonal arrays of air holes in silicon. The transmission spectra exhibit sharp resonant features with Fano line shapes. They are produced due to the coupling of the leaky photonic crystal modes, called guided resonances, to the continuum of free-space modes. We investigate the effects of several types of structural disorder on the spectra of these resonances. Our results indicate that guided resonances are very tolerant to disorder in the hole diameter and to interface roughness, but very sensitive to disorder in the lattice periodicity.
©2007 Optical Society of America
1. Introduction
Photonic crystals are materials with a periodically varying refractive index [1–3]. The refractive index can vary in one or more dimensions, so that photonic crystals can manipulate the flow of electromagnetic waves in a variety of novel ways [3]. In particular, a photonic crystal slab has a two-dimensional lattice structure in the plane of the slab, and uses index guiding to confine light in the third dimension [3, 4]. Because of their two-dimensional periodicity, photonic crystal slabs are capable of supporting in-plane modes, called guided modes. These guided modes lie below the light line in the photonic band structure, and are strictly confined in the plane of the slab. They have been studied extensively and used to implement various waveguide structures and photonic devices [3, 5–8].
Besides the in-plane guided modes, photonic crystal slabs also exhibit modes that can leak out of the plane of the slab and interact with the external radiation modes. These discrete photonic crystal modes lie above the light line, and are called leaky modes or guided resonances [9, 10]. They couple to the continuum of free-space modes to produce sharp resonant features in the optical transmission spectrum of the slab. This is an example of an interference phenomenon between discrete and continuum states, with the well-known Fano resonance line shape [11] often observed in atomic physics and condensed matter systems [12]. The Fano resonances in photonic crystal slabs [13] have been the subject of several recent studies [14–21]. Analogous to changes in Fano lines shapes due to impurities in the condensed matter lattice models [22], the Fano line shapes in photonic crystals can also be controlled by doping [23]. It has also been shown that resonances otherwise not present in the perfect photonic crystal [24] can appear due to lattice disorder [25].
The subject of using controlled lattice defects to engineer photonic crystal properties has always been very important [26–28]. However, a perfectly periodic dielectric structure could also be disturbed due to the presence of unintentional lattice disorder, affecting both the inplane guided modes and the out-of-plane guided resonances. Disorder of this kind can occur due to imperfect fabrication processes, and can appear as irregularities in the size, shape, and position of the structural elements of the photonic crystal [29]. There have been numerous theoretical and experimental studies on the effect of disorder in one-dimensional [30], two-dimensional [31–42], and three-dimensional photonic crystals [43–46]. All of these disorder studies, including those on photonic crystal slabs, have analyzed how the in-plane guided modes and band gaps are affected. The effect of structural disorder on out-of-plane propagation in a one-dimensional photonic crystal was presented only very recently [47]. There is also only one existing report considering the effects of weak structural disorder on guided resonances [25]. A detailed study on how lattice disorder affects the out-of-plane propagation is lacking. Such results can reveal, for example, the extent to which the observed resonances, measured in the far field, can be described either as a cooperative phenomena involving many lattice sites or as a superposition of many individual independent resonators.
In this paper, we systematically evaluate the effect of several different types of lattice disorder on guided resonances in terahertz photonic crystal slabs. The photonic crystal slabs have a hexagonal lattice of air holes in silicon, and are fabricated by standard microfabrication techniques. The normal-incidence transmission spectrum, perpendicular to the plane of the slab, is measured using terahertz time-domain spectroscopy [48, 49]. We study the effects of inhomogeneity in the size of the holes as well as sub-wavelength-scale surface roughness. We also study the effect of disruption of the periodicity of the photonic lattice, resulting from displacement of the hole centers. These data reveal that the guided resonances are tolerant to disorder in the hole diameter and to interface roughness, but are very sensitive to disorder in the lattice periodicity.
2. Experiment
A schematic of the experimental setup is shown in Fig. 1. Single-cycle terahertz pulses are generated and detected using photoconductive antennas. The terahertz radiation propagates in a direction perpendicular to the plane of the photonic crystal slab, and is focused into the crystal by a polyethylene lens. A similar setup is used to collect the transmitted radiation. Using this technique, one measures the transmitted electric field as a function of time. The complex transmission coefficient of a sample is given by the ratio of the Fourier transform of a pulse transmitted through the sample to that of a reference, which is usually the freely propagating pulse, measured without the sample in the beam path. In our measurements, a frequency resolution of 1.8 gigahertz is obtained by scanning the terahertz pulse over a 555 picosecond window. In all the measurements presented in this paper, the electric field is polarized along the Γ-K direction of the hexagonal lattice, although identical results are obtained for the Γ-M orientation.
Fig. 1. A schematic of the experimental setup. The terahertz radiation propagates in the direction perpendicular to the plane of the photonic crystal slab. The terahertz beam spot size is roughly 8 mm. The samples are all very thin compared to the Rayleigh range of the focused THz beam.
The photonic crystal slabs consist of a hexagonal array of circular holes etched all the way through a high-resistivity (ρ>10 kΩ cm) silicon slab, using deep reactive ion etching [49, 50]. High-resistivity silicon is chosen because of its low absorption and frequency-independent refractive index (n=3.42) in the terahertz regime [49, 51]. Each photonic crystal slab is characterized by three parameters: hole radius r, lattice parameter a, and slab thickness t. Figure 2 shows one of the samples, with r=150 µm, a=400 µm, and t=250 µm. For a hexagonal lattice, the holes occupy a fractional area of the slab ϕh which is given by ϕ=(2π/√3)(r/a)2. So, for the sample shown in Fig. 2, ϕh=0.51. As the feature sizes are in the order of several hundred microns, it is possible to fabricate crystals with essentially perfect structures using standard lithographic and etching techniques.
Figure 3 shows the normal-incidence transmission spectrum for the sample shown in Fig. 2. The overall spectrum is the superposition of two different responses: the Fabry-Perot oscillations that are prominent at lower frequencies and arise because of the finite slab thickness, and the photonic crystal modes which correspond to the portions of the band structure lying above the light line, that are excited due to normal incidence illumination of the crystal. The dips in the spectrum corresponding to transmission minima are the signatures of the guided resonances [10, 17]. The widths of these resonant features represent the lifetimes of the modes. We see that the experimental results match very well with the finite elements simulations [17, 52], confirming that the fabricated crystals are of essentially perfect quality.
Fig. 2. (a). A top view of the photonic crystal slab. The scale bar is 5 mm. (b) A magnified portion from (a), with a scale bar of 300 µm. The crystal parameters of this sample are r=150 µm, a=400 µm, and t=250 µm. For a hexagonal lattice, the Γ-K direction points along the axis connecting the centers of adjacent (nearest neighbor) holes, while the Γ-M direction points along the axis connecting the centers of next-nearest neighbors.
Fig. 3. Normal-incidence transmission spectrum of a photonic crystal slab with r=150 µm, a=400 µm, and t=250 µm. The open black circles are experimental results, while the solid red curve is obtained from simulations based on the finite elements method (FEM).
3. Effect of disorder in the hole diameter
In photonic crystal slabs formed by etching of cylindrical holes into the silicon substrate, the diameter of the holes can be nonuniform if the etching conditions are not appropriately optimized. Also, the walls of the etched silicon may not be perfectly vertical and can display surface roughness. We have studied several such samples, either with diameter nonuniformity or interface roughness. In both cases, the lattice periodicity was not perturbed; the centers of the holes remained periodic on a hexagonal lattice with a=400 µm.
Fig. 4. Distribution of the hole diameter for perfect (top) and imperfect (middle and bottom) samples. The perfect sample has a mean hole diameter of 297 µm, which is close to the design parameter of 300 µm. The three samples have hole diameters with standard deviations of 1 µm, 32.0 µm, and 39.6 µm, from top to bottom. The samples in the lower two images have different slab thicknesses. The bin size of each bar in the histogram is 5 µm. Images show sample regions within a 8 mm circular area, roughly equal to the THz beam spot size. All scale bars are 1.5 mm.
3.1 Disorder in hole size
Figure 4 shows histograms depicting the distributions of the values of the hole diameters for one nominally perfect and two imperfect samples. These data are obtained by acquiring 25 high-resolution optical images of each sample, and then using image processing software to extract the individual hole diameters. The diameters of about 400 holes are measured for each of the three samples; this is about the number of holes illuminated by the ~8 mm diameter terahertz beam. For the perfect sample, all the holes have nearly identical diameters of 297±1 microns. This 1 micron uncertainy is actually an upper limit, reflecting the limit of the resolution of the image processing software and of the images. In the imperfect samples, the hole diameters are not uniform. Although most of the hole diameters are close to the target size (300 µm), there is a distribution of holes of different sizes. It can also be seen from the images that holes of almost half the ideal size occur every 2–3 mm.
Fig. 5. Normal-incidence transmission spectra of photonic crystal slabs with disorder in the hole size. (a) The solid black curve is the transmission spectrum of the ideal crystal with r=150 µm, a=400 µm, and t=250 µm (top picture, Fig. 4), while the dashed red curve is the spectrum of the corresponding imperfect sample (middle picture, Fig. 4). (b) Same as (a) but for a sample with slightly different slab thickness of 275 µm. The corresponding imperfect sample is shown in Fig. 4 (bottom picture).
Figure 5(a) shows the transmission spectrum of the perfect sample with r=150 µm, a=400 µm, t=250 µm, and also of the imperfect sample shown in second row of Fig. 4. The resonances in the imperfect crystal are only very slightly broadened and shifted in frequency, but clearly well-defined resonances remain. Similar behavior is observed (Fig. 5(b)) in a sample with slightly different slab thickness (t=275 µm) shown in the last row of Fig. 4. The small shifts in the spectra can be attributed to the fact that the variable hole size produces a small shift in the average refractive index of the photonic crystal slab. This results in a small shift of the photonic bands, which in turn shifts the resonances by a small amount [25].
It has been shown that broader resonances with low Q-factors are not substantially affected by lattice disorder [25]. But from Fig. 5, it is clear that even the sharpest resonances are not adversely affected. This degree of tolerance to disorder in the hole size suggests that the size of the air holes is not the critical factor in determining the lifetime of the Fano resonance modes. Similar results were observed in photonic crystal slabs at visible and infrared frequencies that exhibited very small fabrication disorders [16].
3.2 Disorder in hole shape
In addition to nonuniformity in hole diameter, a poorly optimized deep etching process can also lead to substantial surface roughness on the internal air-silicon interfaces within the etched holes. Figure 6 shows top-view images of a nominally perfect photonic crystal, as well as a photonic crystal with roughness disorder. From the image in the third column of Fig. 6(b), it is clear that the length scale of the boundary roughness is about 20 µm. This value is small compared to the wavelength of the incident terahertz radiation, ranging from about λ/15 to λ/150 within the measured spectral band. Even so, the impact of this relatively small degree of interface roughness on the optical spectrum is not immediately obvious. For example, Maskaly et al. studied the effects of interface roughness on a one-dimensional photonic crystal, showing that the change in reflectivity can be a sensitive function of the degree of roughness [53]. In photonic crystal fibers, interfacial roughness can be the dominant loss mechanism [54].
Fig. 6. Top-view images of photonic crystal slabs showing disorder in the hole shape. Images from left to right increase in magnification. (a) Sample with nominally perfectly circular holes, of diameter 250 µm. (b) Sample with holes of imperfectly circular shape, exhibiting interface roughness.
Figure 7 shows the transmission spectrum of the perfect and imperfect crystals shown in Fig. 6. We notice that for the sample that has holes with rough boundaries, the lowest two resonances at 0.436 THz and 0.472 THz are shifted by 8.3 GHz and 13 GHz, respectively. These shifts, although small, are very reproducible. As the holes with rough boundaries are still almost circular and do not differ in sizes by large amounts, one can measure their diameters and assign an average diameter to the entire sample. Using the same analysis method described above (Fig. 4), we find that the average hole diameter of the sample with rough boundaries is about 4.7 µm more than that of the nominally perfect sample. This difference in diameter is very small, less than 2%. However, from a detailed study on how the resonant peaks shift in frequency with change in the hole diameter [52], a 4.7 µm difference in the hole diameter is predicted to shift the aforementioned resonances by about 11.8 GHz and 14.7 GHz, respectively. Within experimental limits, the observations are very close to this prediction. Very small but observable shifts in the resonant features for less than 2% change in the hole diameter can also be extrapolated from the studies on chalcogenide photonic crystals at near-infrared frequencies [18].
Fig. 7. Normal-incidence transmission spectra of photonic crystal slabs with r=125 µm, a=400 µm, and t=300 µm. The solid black curve is for the sample with perfectly circular holes [shown in Fig. 6(a)], while the dashed red curve is for a sample with slight disorder in the hole shape [shown in Fig. 6(b)].
Therefore, the frequency shifts observed in the transmission spectrum of the imperfect sample can be attributed to the slight difference in the average hole diameter. Provided that the surface roughness is small compared to the wavelength of the radiation, the only outcome of the rough hole boundaries is to produce a slight difference in the average hole diameter. This in turn causes the resonances to shift in frequencies by a very small amount. The resonances at higher frequencies shift by larger amounts, as observed in the detailed study of guided resonances in terahertz photonic crystal slabs [52]. In any event, it is clear that the overall features of the narrow and pronounced guided resonances are largely insensitive to the presence of moderate amounts of surface roughness.
4. Effect of disorder in the lattice parameter
As noted above, in the samples with lattice disorder due to imperfect shape and size of the holes, the long-range crystal periodicity is maintained. This is because the hole centers are not displaced from their ideal lattice sites. To study the effect of crystal disorder caused by disruption of the lattice periodicity, randomness in the hole positions are required. Figure 8 shows samples that were designed to have disorder in the lattice periodicity. The lattice parameter of the nominally perfect crystal (i.e. the center-to-center distance between adjacent holes) is a=400 µm. For the imperfect samples, the hole centers are randomly displaced by values either in the range 0–50 µm [Fig. 8(b)], or in the range 0–80 µm [Fig. 8(c)]. There is no disorder present due to hole shape and size: the hole diameter in all the samples are 250±1 microns, and the sidewalls of the etched silicon are perfectly smooth and vertical within our ability to measure them.
Fig. 8. Top-view images of photonic crystal slabs with disorder in the lattice parameter. The holes have a diameter of 250 µm, and the lattice parameter of the nominally perfect sample is 400 µm. (a) Sample with no disorder in the lattice parameter. The first column shows a sample region within a 7 mm circular area. The second column shows a magnified portion of the image in the first column. The last column shows the pair correlation function (red) computed from the image in the first column. The (black) vertical lines depict the pair correlation function of a perfect lattice of infinite size. (b) Same as (a), but for a sample with slight disorder in the lattice parameter. (c) Same as (b), but with a greater degree of disorder in the lattice parameter. All scale bars in the first column are 1.5 mm, while those in the second column are 300 µm.
The extent of disorder in the lattice parameter can be quantified by calculating the pair correlation function (PCF), g(r), from the sample images. The PCF is a dimensionless quantity used for characterizing the lattice ordering through statistical mapping of the two-dimensional radial distribution [45]. The rightmost column in Fig. 8 shows the PCF for the corresponding perfect and imperfect crystals. The common x-axis of the plots is the distance normalized by the hole diameter D. The black vertical lines depict the PCF of an ideal hexagonal lattice of infinite size. Because of the finite size of the actual samples, the peaks in the calculated PCF are not ideal delta functions but have a small spread. The position of the first peak signifies the mean center-to-center distance between the adjacent holes averaged over the entire imaged area. For all the three samples, the first peak occurs at 1.6 times the hole diameter, equal to 400 µm, the ideal lattice parameter. However, the PCFs of samples with lattice disorder display peaks of reduced heights and substantial broadening, indicative of the disruption of long-range lattice periodicity.
Figure 9 shows the transmission spectra of the samples shown in Fig. 8. The transmission coefficient is plotted on a logarithmic scale to emphasize the differences between the spectra of the three samples. The transmission spectrum of the nominally perfect sample is shown in Fig. 9(a), and contains several sharp resonant features between 0.4 and 0.85 THz. Due to disorder in the lattice parameter, most of the resonant features in the spectra of the imperfect crystals are either completely washed out or substantially suppressed [Figs. 9(b) and 9(c)]. For the imperfect crystal with lesser degree of disorder, very broadened yet still noticeable resonant dips are observed at around 0.6, 0.7, and 0.8 THz [Fig. 9(b)]. This is because the perfect crystal has several resonances cluttered around these frequencies. As these collections of resonances broaden and overlap each other, the result is the observed small signatures. However, as the translational symmetry is further destroyed, the resonances are broadened even further [Fig. 9(c)]. These results indicate that the guided resonances are very sensitive to disorder in the lattice parameter, and can be completely washed out with a disruption of the lattice periodicity.
Fig. 9. Normal-incidence transmission spectra of photonic crystal slabs with r=125 µm, a=400 µm, and t=300 µm. (a) A perfect sample with no disorder in the lattice parameter [shown in Fig. 8(a)]. (b) A sample with slight disorder in the lattice parameter [shown in Fig. 8(b)]. (c) A sample with greater degree of disorder in the lattice parameter [shown in Fig. 8(c)].
The data presented in Fig. 9 provide insight into the nature of the modes which give rise to the guided resonances. In most cases, these modes have been characterized by observation of the transmission spectrum of a nominally ideal photonic crystal slab, in the far field. In this case, it is not immediately obvious that the periodicity of the lattice is required – one could imagine a collection of uncoupled resonators giving rise to a sharp dip in the measured far-field transmission spectrum, independent of their relative locations in the plane of the slab. However, these data demonstrate that the periodicity of the array is required, which emphasizes the strongly cooperative nature of these modes. Such cooperative behavior is reminiscent of the sensitivity of waveguide modes to lattice disorder near the edges of a photonic crystal waveguide [55, 56].
5. Conclusion
We have studied the effect of lattice disorder on guided resonances in terahertz photonic crystal slabs with a hexagonal lattice of air holes in silicon. The normal-incidence transmission spectrum is obtained using terahertz time-domain spectroscopy. Moderate amounts of disorder in the hole shape and size does not appear to affect the lifetime of guided resonance modes significantly. On the other hand, the introduction of disorder in the lattice parameter severely broadens the resonant features in the spectrum. These results illustrate the extreme sensitivity of the guided resonance modes to lattice periodicity. Evidently, these modes are very cooperative in nature, with a high degree of spatial coherence extending over many lattice sites of the photonic crystal slab.
Our data also clarify an important point about the nature of this cooperativity. It is tempting to describe the observed far-field transmission resonances using the language of a phased array, in which each resonator contributes to the measured far-field spectrum according to its relative location in the two-dimensional array. However, the guided resonances are much more sensitive to the periodicity of the array than to the frequencies of the individual resonators (which depend on the hole shape and radius). This fact casts doubt on the validity of the phased array picture. In a phased array, disorder in the location of the array elements should be just as important as disorder in the resonant frequency of each array element. Evidently, the guided resonances should not be described as merely a phased array effect. Near-field measurements of the transmitted terahertz field could help to further clarify the nature of the cooperative resonant modes.
Acknowledgments
This work is supported in part by the National Science Foundation and the R. A. Welch Foundation.
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