We demonstrate that guided resonant modes can be readily observed in asymmetrical photonic crystal slabs on high-index substrates. In spite of the high radiative loss associated with all optical modes in these cases, the guided resonant modes are found to give rise to strong high-Q features in the transmission spectra. Since these photonic crystal structures are far more robust and easier to fabricate than the free-standing photonic crystal membranes used in previous studies of guided resonant modes, detailed studies of relevant optical phenomena and the implementation of proposed applications are greatly simplified.
©2005 Optical Society of America
Recent theoretical work has identified and characterized a class of modes in photonic crystal slabs dubbed guided resonances . These resonant modes are guided in the sense of having their electric field distribution confined within the photonic crystal slab. However, unlike other resonances that exist in photonic crystals, guided resonances are also strongly coupled to radiation modes. The latter property of guided resonances holds promise for several potential applications, such as the efficient extraction of light originating from emitting layers that could be incorporated into photonic crystal slabs and the filtering of externally incident light .
In a photonic crystal slab, the index mismatch between the slab and the surrounding medium (usually air) gives rise to the necessary conditions for confining the electric field in all three dimensions. The in-plane confinement is the result of the guided modes that make up the in-plane band structure, while the out-of-plane confinement results from the discontinuity in index at the slab faces. The previous work on guided resonances, both theoretical and experimental, has focused exclusively on symmetrical (free-standing) photonic crystal slabs, which consist of a suspended membrane composed of a high-index optical material in which a photonic crystal has been patterned [1–5]. By contrast, in this paper we consider an asymmetrical situation in which the photonic crystal slab is in contact with a substrate (see Fig. 1). The substrate has a refractive index intermediate between the index of the photonic crystal slab and that of the surrounding medium. In such asymmetrical structures, the substrate was initially suspected of increasing the radiative coupling of the guided resonances to such an extent that they would no longer be observable (because the field might then decay too quickly). However, we have found that guided resonances persist and are readily observable in transmission spectra in spite of the increased radiative coupling. The substrate does have a noticeable influence on the lineshapes and positions of these modes, but high-Q features are readily apparent in the transmission spectra nevertheless.
The photonic crystal slabs investigated in this study consist of patterned films of GaN (a wide band gap semiconductor) on sapphire substrates. As shown in Fig. 2, these asymmetrical structures allow almost no guided modes below the light line (which defines the lower-frequency limit of the radiation continuum). This is quite different from the case of the symmetrical (free-standing) membranes studied in the past [1–5], which can show band gaps in the guided modes even for membranes with moderate values of refractive index. Note also that since so few guided modes exist in the asymmetrical case studied here, no band gaps appear for the guided modes, as seen in Fig. 2. The absence of band gaps results both from the mixing of odd- and even-parity modes, caused by the asymmetry, and also from the relatively low refractive index contrast between the photonic crystal slab and the substrate (e.g., approximately 2.37 for GaN and 1.8 for sapphire).
The presence of a substrate has several experimental advantages in this situation, both for the experimental studies of guided resonances and for future applications envisioned to rely on their properties. As discussed above, the photonic crystal slabs which support these guided resonances typically consist of arrays of holes in a thin optical-quality film. In order for high-Q guided resonances to be observable in transmission, the hole diameter is required to be significantly smaller than the wavelengths of interest, as discussed previously . In addition, the film thickness must also be comparable to the wavelengths of interest in order to minimize the density of modes available in the photonic crystal slabs. This is because nearby modes can couple to each other, thus leading to lower Q’s as manifested in the corresponding resonant spectral features in transmission spectra. Therefore, fabrication of such structures in suspended membranes (as was done in previous studies [3–5]) becomes a challenge at wavelengths in the optical region, and particularly in the visible spectrum. Such free-standing membranes, generally obtained by post-patterning release of the thin films from their substrates , must then be very thin (typically, approximately 0.5 μm thick for the visible spectrum), while simultaneously spanning approximately 10–100 μm (or more) in order to enable optical measurements. Such suspended membranes are extremely fragile and, although amenable to fabrication in certain material systems, are strongly affected by intrinsic material defects as well as any residual stress in the thin films. Suspended membranes are also well known to have poor thermal conductivity and dissipation. In practice, these disadvantages tend to limit the number of applications in which suspended membranes can play a role. However, all of these practical limitations are mitigated if the guided resonances discussed above can be accessed experimentally while the photonic crystal slab is in contact with a substrate, as demonstrated here.
For our studies, we fabricated photonic crystal structures consisting of high-density, sub-micron diameter hole patterns in optical-quality GaN films on sapphire substrates. The 260 nm thick GaN films were grown by MOCVD on 2” diameter a-plane sapphire wafers . A thin (approximately 25 nm) AlN nucleation layer was also present (see Fig. 1). The etch mask used to create the photonic crystal patterns (e.g., square arrays of holes in this case), was a Ni film patterned by e-beam lithography and ion milling as described previously . The photonic crystal patterns were obtained by plasma etching in a Cl2/Ar atmosphere (with a nearly 1:1 ratio of etch to buffer gas). Reproducible feature sizes well below 100 nm could be obtained in the GaN films by these techniques, with high precision and low edge and surface roughness. Images of typical fabricated structures obtained by SEM and AFM are shown in Fig. 1. Typical patterns covered areas of 100×100 μm in size in order to facilitate the optical transmission measurements. The high degree of circularity and sidewall verticality of the GaN photonic crystal patterns are evident in these images, and are responsible for the observation of relatively narrow (high-Q) spectral features, as discussed below. The precision of our fabrication process enables a high degree of control over the spectral positions of the guided resonances. A small raised area (“ring”) is observed around the circumference of each hole in the AFM image of Fig. 1(c) and is likely the result of redeposition of GaN during the etching process. However, the height of these rings is only a small fraction of the total film thickness. We note that GaN structures similar to ours have been investigated previously , although our fabrication techniques appear to give superior results. We also note that the previous results  were analyzed in terms of a 2D photonic band structure, which corresponds only approximately to the experimental case, and did not include a detailed analysis of guided resonant modes.
To identify the guided resonant modes present in our photonic crystal slabs, we examined the E-field distribution patterns of the Γ-point modes identified in Fig. 2(a). As described previously , the guided resonances are distinguished by having their field distribution strongly confined to the spatial region occupied by the photonic crystal slab. By contrast, non-resonant modes have a field intensity distribution that quickly radiates away from the slab and into the surrounding medium. Examples of resonant field patterns are shown in Figs. 2(b) and (c). Note that the field patterns of the resonant modes displayed in Figs. 2(b) and (c) do not share the full (square) symmetry of the photonic crystal lattice.
Normal-incidence transmission measurements were carried out by illuminating the photonic crystal slabs with a weakly focused halogen-bulb source. The transmitted light was collected by an infinity-corrected near-IR microscope objective and analyzed in two fiber-coupled spectrometers equipped with array detectors, covering the wavelength range between 400 nm and 1.7 μm. Since the transmission was measured relative to an unpatterned area of the same GaN film, transmission results in excess of 1 indicate spectral regions where the GaN photonic crystal pattern serves to reduce the impedance mismatch between the unpatterned GaN film on sapphire and the ambient. Several different samples with various hole sizes and array spacings were measured. In each case, the optical transmission results were compared to transmission calculations for the specific sets of parameters determined by imaging the fabricated structures with SEM and AFM instruments. The transmission calculations were performed by FDTD techniques, using perfectly matched layer boundary conditions on the surfaces of the computational domain parallel to the slab and repeating boundary conditions in the four other directions. For the purposes of this study, we considered only the normal incidence direction in both the measurements and simulations. We found the experimental transmission results to be very sensitive to both propagation direction and focusing conditions, and we took great care to ensure that the measurements were done along the direction normal to the slab faces and that the optical beam was as weakly focused as possible while still allowing for the efficient collection of transmitted light. We estimate both the divergence of the beam at the sample position and any deviations from normal incidence to be less than 1°.
Typical transmission results for a GaN on sapphire photonic crystal slab are shown in Fig. 3. Several spectral features associated with the guided resonant modes discussed above are easily identified in the measured spectrum and are matched well by corresponding features in the calculated transmission spectrum. In particular, the inset of Fig. 3 shows, in greater detail, the two resonant modes identified as ω1 and ω2 in Fig. 2. The agreement between the measured and calculated spectra is generally very good. (Note that both the measured and calculated spectra in Fig. 3 are for unpolarized light.) The asymmetrical (Fano) lineshapes observed in some cases result from the coupling between the resonant modes and the continuum of radiation states, as described previously . At shorter wavelengths, the reduced broadband transmission in the measured spectra may be due to enhanced scattering resulting from residual fabrication errors in the photonic crystal array, which are not included in the simulations. Nevertheless, the detailed agreement between the measured and calculated spectra, including spectral positions and lineshapes (or linewidths), is excellent in general, as exemplified by the inset of Fig. 3.
Careful examination of the E-field patterns for the several Γ-point modes appearing in Fig. 2(a), combined with a comparison of Figs. 2(a) and 3, indicates that only a subset of the Γ-point modes which are resonant also give rise to corresponding spectral features (resonances) in the normal-incidence transmission spectrum. The resonant modes which are “missing” from this spectrum (in both experiments and FDTD simulations) are those which share the full symmetry of the photonic crystal lattice. These modes cannot be excited by externally incident plane waves [9,10]. This observation is analogous to the null-coupling effect observed previously for certain photonic bands in 2D photonic crystals .
We have also investigated the effect of hole size within a specific pattern (i.e., a square array with fixed nearest-neighbor spacing but varying hole sizes) on the positions of the guided resonances. As shown in Fig. 4, we find excellent agreement between the measured and calculated positions of the spectral features corresponding to these modes. This agreement clearly validates both our FDTD simulation results and the precision of the fabrication process developed for the GaN photonic crystal slabs studied here.
In order to clarify the role of the substrate in the observation of these guided resonant modes, it is interesting to consider more closely the effect of the substrate index in the asymmetric photonic crystal slabs investigated here. The effect of substrate index is demonstrated by the calculated transmission spectra in Fig. 5, for two resonant modes corresponding to ω1 and ω2 in Fig. 2. (This correspondence of modes holds true in spite of the fact that the band structure in Fig. 2 is calculated for r=0.27a, while Fig. 5 is associated with r=0.15a.) A series of spectra is shown in Fig. 5 for (hypothetical) substrate indices ranging from 2.135 (relatively close to the GaN index of 2.37) to 1.8 (the approximate value for the sapphire substrate). At low index contrast (high substrate index), only a very weak resonance appears, corresponding to ω2 in Fig. 2. (In the limiting case where the substrate index matches that of the photonic crystal slab, no resonances are expected (or observed).) As shown in Fig. 5, this resonance shifts to higher frequency with increasing index contrast (decreasing substrate index), and at a certain value of index contrast (near 2.0) a second resonance appears, corresponding to ω1 in Fig. 2. Thus, there appears to be a minimum index contrast value required to observe a particular resonance in the transmission spectrum, and this minimum index contrast is specific to each resonance. The second resonance does not appear to be due to a “splitting” in the traditional sense, since the separation between the two resonances in Fig. 5 does not extrapolate to a vanishing value at any size of index contrast. We do not currently have an explanation for what appears to be a non-monotonic change in the strengths of the resonances as the substrate index is varied. Although not included in Fig. 5, a closer look at spectra corresponding to much lower substrate index values (i.e., higher index contrast), approaching the symmetric case of a free-standing membrane, shows that the frequency separation has a mild dependence on index contrast, implying that each resonance shifts with index contrast at a slightly different rate. Also, we note in Fig. 5 that the asymmetry of the lineshapes, discussed in detail in Ref. , is stronger at larger index contrast and shows an apparent sign reversal at low index contrast.
Although we have not yet done a detailed study of the lineshapes (which are sometimes strongly asymmetrical, as expected ) and Q values associated with these resonances, we note that the narrow modes observed in Fig. 3 for our asymmetrical photonic crystal slabs are very similar to those found previously [1–5] for guided resonances in free-standing membranes (symmetrical structures), in spite of the increased coupling to radiation modes expected in the present case. For example, one of the narrowest resonances observed in this study (identified as ω1 in Figs. 2 and 3) has a measured Q of approximately 200 at r=0.17a, comparable to similar resonances observed in previous studies. The dependence of this mode’s Q on hole size is included in the inset of Fig. 4. The relatively high Q’s of the observed modes suggest a correspondingly low radiative coupling for these modes. Furthermore, a comparison between the lineshapes obtained in the FDTD transmission simulations and the experimental results, as shown in the inset of Fig. 3, suggests that fabrication errors do not dominate the measured linewidths, except possibly for the photonic crystal slabs with the smallest hole sizes fabricated for this study. These residual fabrication errors at the smallest hole (feature) sizes can be attributed to the fact that techniques for the sub-micron scale patterning of GaN are far less optimized at present than analogous techniques for more widely investigated semiconductors, such as Si and GaAs, for example. Therefore, optimizing the fabrication of these patterns in GaN is expected to improve the agreement between the calculated and measured lineshapes of even the narrowest guided resonances.
In conclusion, we have demonstrated that guided resonances can be readily observed in asymmetrical photonic crystal slabs, where the presence of a substrate greatly increases the radiative coupling of the modes in the photonic crystal slab. These structures typically do not exhibit band gaps in the guided mode spectrum, particularly when the index of refraction contrast between the slab and the substrate is only moderately high, as is the case (examined here) for a GaN photonic crystal slab on a sapphire substrate. In addition to the effects on optical properties discussed above, asymmetrical structures consisting of photonic crystal slabs supported by a substrate also have a number of practical advantages over their symmetrical counterparts (free-standing membranes) for potential applications, including ease of fabrication, mechanical robustness and improved thermal properties. Our results demonstrate that it is not necessary to fabricate very thin large-area free-standing membranes in order to access many of the properties of guided resonant modes discussed previously. Therefore, optical components or devices based on these narrow (high-Q) resonances are far easier to fabricate and hence much more practical than previously believed.
We acknowledge technical assistance from S. Bounnak (Sachs-Freeman Associates, Inc.) and J.A. Murakowski (Univ. of DE).
References and Links
1. Shanhui Fan and J.D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B65, 235112 (2002); Shanhui Fanet al, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A20, 569 (2003). [CrossRef]
2. See, for example, Wojoo Suh, M.F. Yanik, Olav Solgaard, and Shanhui Fan, “Displacement-sensitive photonic crystal structures based on guided resonance in photonic crystal slabs,” Appl. Phys. Lett. 82, 1999 (2003). [CrossRef]
3. K.B. Crozieret al, “Two-dimensional photonic crystals at visible wavelengths,” in CLEO/QELS and PhAST 2004 (OS A Washington, DC, 2004), CWG2.
5. F. Raineriet al, “Nonlinear optical manipulation of Fano resonances in 2D photonic crystal slabs,” in CLEO/QELS and PhAST 2003 (OS A Washington, DC, 2004), QThPDA1.
6. C.R. Eddy Jr., R.T. Holm, R.L. Henry, J.C Culbertson, and M.E. Twigg, “Investigation of a three-step epilayer growth approach of GaN thin films to minimize compensation,” J. Electron. Mater. (to be published).
7. David S.Y. Hsuet al, “Using Ni masks in inductively coupled plasma etching of high density hole patterns in GaN,” J. Vac. Sci. Technol. B (to be published).
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10. Shanhui Fan, Dept. of Electrical Engineering, Stanford University, Stanford, CA 94305 (personal communication, 2005).