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We study theoretically modal properties and parametric dependence of guided-mode resonance bandpass filters operating in the mid- and near-infrared spectral domains. We investigate three different device architectures consisting of single, double, and triple layers based on all-transparent dielectric and semiconductor thin films. The three device classes show high-performance bandpass filter profiles with broad, flat low-transmission sidebands accommodating sharp transmission peaks with their efficiencies approaching 100% with appropriate blending of multiple guided modes. We present three modal coupling configurations forming complex mixtures of two or three distinct leaky modes coupling at different evanescent diffraction orders. These modal compositions produce various widths of sidebands ranging from ~30 nm to ~2100 nm and transmission peak-linewidths ranging from ~1 pm to ~10 nm. Our modal analysis demonstrates key attributes of subwavelength periodic thin-film structures in multiple-modal blending to achieve desired transmission spectra. The design principle is applicable to various optical elements such as high-power optical filters, low-noise label-free biochemical sensor templates, and high-density display pixels.
© 2014 Optical Society of America
1. Introduction
Nanostructured films with subwavelength periods exhibit guided-mode resonance (GMR) effects enabling applications including wavelength-selective mirrors [1,2] and biochemical sensors [3]. Attendant resonant thin-film lattices constructed with dielectric, metals, or semiconductors support spectral band profiles and local photonic field distributions in great variety. By tailoring multiple leaky modes in desired spectral domains, a host of useful photonic devices and components can be realized [4,5]. In early works, GMR devices operated principally in reflection [2,6–8]. In 1995, the first transmission, or bandpass, GMR filters were presented, which were designed with multilayer structures [9]. In subsequent research, Tibuleac et al. provided numerical transmission filter designs in the optical region and experimentally verified their performance in the microwave region [10–12]. Kanamori et al. reported transmission color filters with broad bandwidths [13]. Sang et al. considered the resonant transmission properties of Ge-based membranes [14]. Furthermore these authors presented mechanically tunable GMR devices by applying a dual membrane structure [15]. Most recently, experimental results have been provided in the telecommunications region near the 1550-nm wavelength [16], in the mid-IR 7- to 14-μm region [17], and in the 3- to 9-μm region [18].
To advance thin-film filter technology, GMR-based bandpass filters are of great interest as they can be seen as being complementary with metallic or multilayer-based systems. Key attributes of high-performance GMR transmission filters include a narrow-line transmission peak, high efficiency, and low sidebands. We note that designing GMR bandpass filters with broad low-transmission sidebands typically involves parametric optimization because of the optically transparent nature of the thin-film materials.
Initially, GMR transmission filters applied a waveguide grating to create a narrow resonance line with attached high-reflection multilayers to implement broad low-transmission sidebands [9]. More advanced GMR transmission filter designs with only a few thin-film layers and even a single-layer waveguide grating were found using a genetic algorithm [11]. Thus, in 2001, a single-layer transmission filter was presented that resonates at the CO2 laser wavelength of 10.6 μm and exhibits extensive <1% transmission sidebands spanning 2.21 μm while operating in TE polarization [11]. Corresponding to a part of this low transmission band, there is a 99% flat-reflection band covering 1.91 μm with the relative sideband width generated by this high-reflection band being ΔλR/λC = 18% where λC is the center wavelength. As designed, a narrow transmission peak emerges and splits the band in the middle [11]. This wideband high-reflectance/low-transmittance response is not explainable by homogeneous effective thin-film interference. Indeed, a complex modal interaction is involved as Ding and Magnusson identified in their theory of doubly resonant GMR bandpass filters [19]. Considering the versatile governing physics and substantial parametric design space associated with this device class, a host of unexplored aspects remains.
Accordingly, in this paper, we investigate the modal coupling mechanisms of key resonant bandpass-filter architectures in a quest to uncover new high-performance devices in this class. These include a single-layer binary Ge-Se waveguide grating, a two-layer partially etched Si grating structure, and a three-layer embedded Si waveguide structure. We show that devices with experimentally realistic geometric parameters can be designed to yield flat low-transmission sidebands with bandwidths ranging from ~40 nm to ~2100 nm and sharp transmission peaks with full-width at half-maximum (FWHM) linewidths ranging from ~1 pm to ~10 nm in the mid- and near-infrared spectral domains. We introduce a systematic method to achieve a practical transmission profile with an arbitrarily narrow pass bandwidth.
2. Ge-Se single-layer device
The first device is a single-layer waveguide grating structure similar to the one presented in [11]. The device schematic and diffraction spectra are illustrated in Figs. 1(a) and 1(b), respectively. We use rigorous coupled-wave analysis [20] for the numerical calculations. We excite the device with a TE-polarized plane wave under normal incidence; transverse-electric polarization refers to the electric field vector of the incident wave being normal to the plane of incidence and parallel to the grating lines.
Fig. 1 (a) Device schematic and (b) spectral performance of a single-layer GMR bandpass filter with period Λ = 6.91 µm, fill factor f = 0.42, and grating thickness d = 3.8 μm. Refractive indices are nC = 1 (air), nS = 1.4 (SiO2), nH = 4 (Ge), and nL = 2.64 (Se). T0 and R0 denote the zero-order transmittance and zero-order reflectance, respectively. The dashed line in (b) represents the optical response for the grating layer replaced with the effective homogeneous layer. The inset in (b) shows the distribution of the total electric field over 2Λ at the T0-peak wavelength. The TE polarization state prevails.
In Fig. 1(b), the high-reflection band shown with R0>99% covers 2.165 μm or 20.4%. In this case, the low sidebands are entirely generated by the high-reflectance band. This was not the case in [11] as discussed above, since propagating ± 1 diffraction orders below the Rayleigh wavelength aided the device performance; there the R0>99% band provided ~86% of the low band. Other performance parameters for this device are summarized in Table 1. Note that the side bandwidth is defined as T0≤1%.
Table 1. List of Performance Parameters for the Three Devices Treated
The total electric field distribution at the T0 peak wavelength of 10.6 μm shown in the inset of Fig. 1(b) indicates that the transmission peak arises from a second order coupling to the TE0 mode. As clearly revealed by the totally different response (red dotted curve) of an effective-medium homogeneous film, the low-transmission band is attributed to the broadband multiple-resonance effect rather than homogeneous thin-film interference. In the subwavelength region, a rectangular grating layer such as that in Fig. 1 may be approximately modeled as a homogeneous negative uniaxial layer with an ordinary index of refraction, corresponding to TE polarization, given by
In this computation, the periodic film is replaced by a film with refractive index thus found.We identify the resonant modes responsible for the flat sidebands by computing the amplitudes of the coupling orders for various wavelengths as summarized in Fig. 2. To place this in context, we recall that the y-component of the electric field in the grating in Fig. 1 can be expressed as [21]
where σq = ρ – qK with ρ being the wave vector of the refracted input wave, K is the grating vector with magnitude K = 2π/Λ, and r = (x,z) is the position vector. This is the coupled-wave expression for the internal field where the Sq is the amplitude of the q-th space harmonic in the inhomogeneous plane-wave expansion; Fig. 2 displays these amplitudes. At λ = 9.8 μm as shown in Fig. 2(a), we observe simultaneous first- and second-order coupling to a TE2-type mode. The zero-order propagating wave, with amplitude S0, is efficiently reflected with a resulting peak amplitude of 2 as there is a standing wave in the input half-space. The dominant contribution to the internal modal field is due to the evanescent diffraction orders with amplitudes S±1 as seen in Fig. 2(a). In Fig. 2(b), the S±2 contribution morphs into a TE0-like distribution; further variations are observed in Fig. 2(c) at the 50% transmittance wavelength on the short-wavelength side. Generally, the amplitudes of the first and second diffraction orders grow as the wavelength approaches the T0 peak. In Fig. 2(d), the second-order coupling amplitude far exceeds that of the first order and the distribution is in the shape of a TE0 mode; note that the transmitted zero-order amplitude is near unity, consistent with Fig. 1. For the longer wavelength range, the amplitudes of the first and second orders decrease as shown in Fig. 2(e). At λ = 11.2 μm, the first order has a typical TE2 mode character whereas the second order dominantly couples to a TE1-like mode as shown in Fig. 2(f).Fig. 2 Amplitudes of different coupling orders at (a) 9.80 μm, (b) 10.40 μm, (c) 10.59 μm, (d) 10.60 μm, (e) 10.65 μm and (f) 11.20 μm. S0 (green line), S±1 (blue line), and S±2 (red line) represent the zero-, first-, and second-order amplitudes, respectively. The inset on the right-top side of each plot indicates the corresponding wavelength with respect to the transmission spectrum.
In summary, TE2 and TE1 types of modes under first- and second-order coupling are involved in the formation of Fig. 1’s flat low-transmission sidebands, and a TE0 mode under dominant second-order coupling induces the narrow T0 peak in this design. This coupling configuration expresses the detailed modal properties of the doubly resonant bandpass filters proposed by Ding and Magnusson [19].
Figure 3 shows the variation of transmittance relative to key parameters. The grating layer thickness controls both the location of the transmission peak and the sidebands as shown in Fig. 3(a). In Fig. 3(b), the sideband properties are insensitive to the grating fill factor while the peak wavelength is highly dependent upon it. Therefore, by controlling the fill factor, the peak wavelength can be finely tuned without a significant effect on the sideband properties.
Fig. 3 T0 spectrum as a function of (a) grating thickness and (b) fill factor of the device in Fig. 1(a). The white circles in Fig. 3(a) and 3(b) correspond to the filter in Fig. 1.
3. Partially etched Si thin-film device
The second device architecture studied is a partially etched single-layer structure. In this device, the grating ridges match identically to the underlying homogeneous layer made of the same material such that no reflections or phase changes arise upon transition from the grating ridge into the layer. We call this structure a “zero-contrast” grating [22] in distinction to a high-contrast grating [23]. This structure is highly favorable for standard mask-based nanofabrication processes. The film medium is Si, and the device structure and T0 spectrum are presented in Fig. 4; Table 1 lists the performance parameters.
Fig. 4 (a) Structure and (b) performance spectrum of a Si-based GMR bandpass filter with the following parameters: grating period Λ = 1.018 µm, fill factor f = 0.21, grating thickness d = 0.355 μm, and homogeneous sublayer thickness d1 = 0.297 μm. Refractive indices are nC = 1 (air), nS = 1.45 (SiO2), and nH = 3.48 (Si). The dashed line in (b) represents the optical response with the grating layer replaced with an effective homogeneous layer. The inset in (b) shows the total electric field at the T0-peak wavelength.
The total electric field at the T0-peak wavelength in Fig. 4(b) arises dominantly from the second-order coupling to a TE0 mode in the homogeneous sublayer. The vertical oscillation of the field maxima in the sublayer is a result of interference between a TE0 mode under the second-order coupling and a TE1 mode at first-order coupling as shown in Fig. 5(c). The low-transmission sideband is again, as in Figs. 1 and 2, attributed to the broadband multiple-resonance effect rather than homogeneous thin-film interference as the effective homogeneous film shows a totally different response (red dotted curve) than the patterned device.
To identify the operative modes responsible for the flat sidebands and resonant peak, we compute amplitudes of coupling orders at different wavelengths ranging from 1.5 μm to 1.6 μm. Figures 5(a)–5(f) show the amplitudes at different wavelengths in increasing order. In the short wavelength range, for example at λ = 1.5 μm in Fig. 5(a), the dominant modes associated with excitation by the first and second evanescent diffraction orders, respectively, are TE2 and TE1 modes with classic mode shapes. For the first order, the excited mode has a significant amplitude in both the Si grating and homogeneous sublayer. In Fig. 5(b) for the longer wavelength λ = 1.545 μm, the amplitudes of both the first and second orders increase. The increase in the second-order amplitude is attributed to the coupling to a TE0 type mode. At the T0-peak, at wavelength of 1.55 μm, the second-order coupling gains dominance involving a TE0 mode and the first-order coupling is reduced to a TE1 mode. For wavelengths exceeding the T0-peak wavelength, the first and second order amplitudes decrease as shown in Figs. 5(e) and 5(f). In Fig. 5(f) for λ = 1.6 μm, a TE2 mode becomes dominant for the first-order coupling and the second order couples to a TE1 mode; this is similar to the coupled mode configuration for λ = 1.5 μm shown in Fig. 5(a). Like the single-layer device in Sec. 2, this device exhibits the double resonance mode configuration in [17]. In contrast to the single-layer device, however, the second-order excitation in this partially etched two-layer device involves TE1 excitation for the formation of the broad low transmission sidebands.
Fig. 5 Amplitudes of coupling orders at (a) 1.50 μm, (b) 1.545 μm, (c) 1.549 μm, (d) 1.55 μm, (e) 1.552 μm and (f) 1.6 μm. S0 (green line), S±1 (blue line), and S±2 (red line) represent the zero-, first-, and second-order amplitudes, respectively. The inset in each plot indicates the corresponding wavelength with respect to the transmission spectrum.
In contrast to the single-layer device in Sec. 2, the partially etched thin-film, single-layer Si architecture provides better controllability in parametric tuning of the spectral profile. Figure 6 shows the T0 spectrum as a function of the grating and homogeneous Si layer thickness. In Fig. 6(a), the sidebands and resonance wavelength are tolerant against significant changes in the grating layer thickness because the mode responsible for the narrow transmission peak dominantly resides in the homogeneous sublayer. Therefore, the homogeneous Si layer thickness has a significant impact on both the sidebands and the resonance position as shown in Fig. 6(b). Also in Fig. 6(b), we observe alternating low and high transmission bands with increasing homogeneous Si layer thickness. These low transmission bands can accommodate a transmission peak with appropriate sets of geometrical parameters.
Fig. 6 T0 spectrum as a function of (a) grating layer thickness and (b) homogeneous layer thickness. The white circles in 6(a) and 6(b) denote the location of the device parameters.
4. Three-layer embedded Si-waveguide device
The final device architecture under study is a three-layer structure consisting of a low-index cladding layer between the high-index grating and waveguide layers as reported by Tibuleac et al. [11]. The device structure and its T0 spectrum are presented in Fig. 7, and the performance parameters for this device are listed in Table 1.
Fig. 7 (a) Structure and (b) T0 spectrum of the three-layer GMR transmission filter with the optimized parameters of grating period Λ = 1.029 µm, fill factor f = 0.152, grating thickness d = 0.262 μm, upper cladding thickness d1 = 1.460 μm, and embedded Si waveguide thickness d2 = 2.221 µm. Refractive indices are nC = 1 (air), nS = n1 = 1.45 (SiO2), and nH = n2 = 3.2 (Si). The dashed line in (b) represents the optical response for the grating layer replaced with the effective homogeneous layer. The inset in (b) shows the total electric field at the T0-peak wavelength.
The field pattern in Fig. 7(b)’s inset shows a TE8 mode structure excited by first-order diffraction at the T0 peak wavelength. To identify the resonance modes responsible for the flat sidebands, we again compute the amplitudes of coupling orders at example wavelengths; the results are presented in Fig. 8. The flat sideband in the shorter wavelength range is attributed to the excitation of a TE9 mode coupling [24] with the evanescent first orders as shown in Figs. 8(a)–8(c). The first node of the TE9 mode is located in the homogeneous SiO2 layer while the remaining eight nodes are located in the homogeneous Si layer. As the wavelength increases from 1.5470 nm to 1.5496 nm, the location of the first node in the homogeneous SiO2 layer shifts toward the grating layer and the mode’s amplitude in the homogeneous Si layer increases as shown in Figs. 8(a)–8(c). At the peak wavelength λ = 1.550 nm, the first node in the homogeneous SiO2 layer is completely pushed out of the device as shown in Fig. 8(d). Therefore, the mode morphs from TE9 to TE8, forming the flat sidebands and the narrow T0 peak; at longer wavelengths, the relative size of the structure is smaller and cannot support the higher modes. For longer wavelengths, the amplitude of the first-order coupled TE8 mode is operative continuously and decreases with wavelength as shown in Figs. 8(e) and 8(f).
Fig. 8 Amplitudes of the main coupling orders at (a) 1.5470 μm, (b) 1.5496 μm, (c) 1.5499 μm, (d) 1.55 μm, (e) 1.5502 μm, and (f) 1.5600 μm. S0 (green line) and S±1 (blue line) represent the zero- and first-order amplitudes, respectively. The inset in each plot indicates the corresponding wavelength with respect to the transmission spectrum.
The operation of this device proceeds with a mechanism different from that in [19]. Figure 8 clearly shows how spectral metamorphosis of the multiple modes forms the broad sidebands and the narrow transmission peak. In Figs. 8(a)–8(f), the relative field amplitude of the dominant mode (blue curves) in the top grating layer first decreases with wavelength, then becomes maximal at the peak wavelength, and finally decreases again for the longer wavelength range. The coupled mode in the low-transmission sidebands has strong diffractive leakage to the outside radiation as it has a large overlap in the diffraction grating layer as demonstrated in Figs. 8(a) and 8(f). The broadband resonance feature providing the low-transmission sidebands is a natural consequence of the excitation of a short lived, highly leaky mode. In contrast, the coupled mode at the T0 peak has a small overlap with the grating layer and, thereby, has a long lifetime causing the narrow spectral resonance shown.
Interestingly, over the former two device structures, this design has remarkable advantages in controlling major spectral properties and internal field patterns. At a fixed wavelength of 1.55 μm, when the thickness of the waveguide layer increases from 0 to 3000 nm, we observe 11 discrete transmission states as shown in Fig. 9(a). Each resonant state corresponds to a different coupled guided mode from TE0 to TE10. Figures 9(b)–9(e) show electric field patterns associated with several selected transmission peaks in Fig. 9(a). For example, the first occurrence of such a transmission state is at a thickness of 25 nm. The total field at the T0 peak wavelength for a 25-nm thick waveguide layer shows a first-order coupling to a TE0 mode as shown in Fig. 9(b). The T0 spectrum for the 25-nm thick waveguide layer operating in a TE0 mode state is shown in Fig. 9(a). A series of transmission peaks appears with 274-nm spacing in the homogeneous Si waveguide layer thickness without significantly changing the low-transmission sideband properties. The total field plots at the T0-peak wavelengths for different transmission states exhibit similar characteristics. Figures 9(b)–9(e) show that the amplitudes of total fields across the devices with different waveguide layer thicknesses are similar. The fields in the grating layer and in the homogeneous SiO2 cladding are comparable and, to an extent, independent of the field distribution in the waveguide layer. In Figs. 8(a)–8(f), the amplitudes of all 8 nodes in the waveguide layer are constant for a given wavelength. Thus, with every 274-nm increase (or decrease) in waveguide layer thickness, the field configuration meets the boundary conditions for excitation by a higher- (or lower-) order mode at the fixed wavelength of 1550 nm in this example. We note that the free spectral range in thickness computed numerically in Fig. 9(a) is consistent with an approximate expression given previously [2]; for the present case this formula is
Fig. 9 (a) Zero-order transmittance for different values of homogeneous Si layer thickness (d2) at λ = 1.550 μm and total field at T0 peak wavelength when the homogeneous Si layer thickness is: (b) d2 = 0.025 μm operating at TE0 mode, (c) d2 = 0.300 μm operating at TE1 mode, (d) d2 = 1.398 μm operating at TE4 mode, and (e) d2 = 2.495 μm operating at TE9 mode.
The peak linewidth and side bandwidth for the TE0–10 modes are plotted in Fig. 10(a). The T0 peak linewidth is generally narrower for a thicker waveguide as shown in the same figure; this is because the added optical path in the thicker waveguide leads to a longer resonance lifetime. Moreover, in Fig. 10(a), the change in waveguide thickness does not have a significant impact on the level or bandwidth of the low-transmission sidebands as the sideband property is dominantly determined by the top grating patterns as observed in Fig. 7(b). The transmittance spectra for TE0, TE4, and TE8 modes plotted in Fig. 10(b) show the decreasing T0 peak linewidth for the higher-order mode while their side bandwidths are comparable.
Fig. 10 (a) Characteristic bandwidths and (b) selected T0 spectra of transmission peaks in Fig. 9(a). Inset in (b) provides passband comparison for the selected transmission states.
We proceed to further study the interesting physical properties provided by the homogeneous SiO2 upper cladding layer. The T0 spectrum is shown as a function of the SiO2 layer thickness in Fig. 11(a); the transmission peak location asymptotically increases, settling at 1550 nm for the SiO2 layer thickness larger than 1 μm. Figures 11(b) and 11(c) show the dependence of the T0 peak efficiency and linewidth, respectively, as functions of the SiO2 layer thickness. The bandwidth of the transmission peak exhibits exponential decay as the thickness increases. The increase in thickness, however, has no effect on the sideband properties as observable in Fig. 11(a). The T0 peak efficiency shows a recurring pattern with thickness. The recurrence has a 534-nm period in the homogeneous SiO2 layer thickness, and this value corresponds to the half wavelength (λ/2n1) in SiO2. Therefore, this particular property is related to a weak Fabry-Perot (FP) resonance in the homogeneous SiO2 layer. In more detail, fields in the grating layer can be enhanced with the FP resonance in the SiO2 layer, and the enhancement increases the radiation probability to the cover as evidenced by local peaks in the linewidth in Fig. 11(c). In addition, the recurring double peaks in the FP undulation in Fig. 11(b) may be explained by the association of the T0-peak efficiency with radiation probabilities to the cover and substrate [25,26]. This unique feature allows for transmission peaks with infinitesimal bandwidth while maintaining broad, flat sidebands.
Fig. 11 (a) T0 spectrum, (b) peak efficiency, and (c) peak linewidth (FWHM) as functions of homogeneous SiO2 layer thickness.
5. Conclusions
Table 1 summarizes the performance parameters for the three device architectures analyzed in comparison with the device studied in [19]. These performance parameters are supported by the different mode-coupling configurations as listed in Table 2. The device analyzed in [19] is a single-layer device with binary gratings similar in structure to the device in Sec. 2 in this analysis. The bandpass property in both device designs arises from the double-resonance GMR effect. The partially etched two-layer device in Sec. 3 shows different modal configurations with an additional leaky mode contributing to the formation of the low transmission sidebands. In the three-layer embedded Si-waveguide device in Sec. 4, the operative mode morphs from one guided mode (TEn) to the neighboring guided mode (TEn-1) stimulated by the same evanescent diffraction order. The narrow transmission peak is obtained upon the spectral mode transition.
Table 2. Mode Configurations Discussed in the Paper*
In summary, we study the modal properties of three GMR bandpass filter designs. We show that the broad low-transmission sidebands and narrow transmission peak with high efficiency approaching 100% can be obtained by appropriately blending multiple guided modes for a variety of thin-film grating structures based on all-transparent materials. More specifically, the single-layer binary grating structure consisting of Ge and Se yields bandpass filter performance in the mid-infrared spectral domain under the simultaneous excitations of the TE1 and TE2 modes for the broad low-transmission sidebands and the TE0 mode for the narrow transmission peak. For the partially etched Si grating structure in the optical communication band, a mixture of TE1 and TE2 modes contributes to form the broad low-transmission sidebands while the narrow transmission peak is induced by a combination of TE0 and TE1 modes with different coupling orders. The final three-layer device consists of a top Si grating, SiO2 upper cladding, and an un-patterned Si slab waveguide. In this structure, any combination of TEn and TEn+1 modes generates a spectral profile for a bandpass filter provided by an appropriate layer thickness. These modal properties provide novel design principles for GMR bandpass filters with narrow peak bandwidths, high efficiency, and low-transmission flat sidebands in a desired spectral domain.
Acknowledgments
This research was supported, in part, by the UT System Texas Nanoelectronics Research Superiority Award funded by the State of Texas Emerging Technology Fund as well as by the Texas Instruments Distinguished University Chair in Nanoelectronics endowment. Additional support was provided by the National Science Foundation (NSF) under Award No. ECCS-0925774. M. Niraula acknowledges NSF support under REU Supplement Award No. ECCS-0925774.
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