Abstract

We apply inverse numerical methods to design compact wideband reflectors in which a periodic silicon layer supports resonant leaky modes. Using particle swarm optimization to determine appropriate device thickness, period, and fill factors, we arrive at example reflector designs for both TE and TM polarized input light. As a properly configured grating profile provides added design freedom, we design reflectors with two and four subparts in the period. In TM polarization, a particular single-layer two-part reflector has 520 nm bandwidth whereas the four-part device reaches 600 nm bandwidth. In TE polarization, the corresponding numbers are 125 nm and 495 nm, respectively. We provide a qualitative explanation for the smaller TE-reflector bandwidth. We quantify the effects of deviation from the design parameters and compute the angular response of the elements. As the angle of incidence deviates from normal incidence, narrow transmission channels emerge in the response yielding a bandpass filter with low sidebands. The effects of adding a silica sublayer between a silicon substrate and the periodic silicon layer is investigated. It is found that a properly designed sublayer can extend the reflection bandwidth significantly.

© 2008 Optical Society of America

1. Introduction

Efficient reflection of light across wide spectral bands is essential in a plethora of common photonic systems. Classic mirrors are made with evaporated metal films and dielectric multilayer stacks. These ordinary devices have been widely studied for a long time and are well understood. A new method to achieve effective wideband reflection response has recently emerged. This approach is based on guided-mode resonance (GMR) effects [119] that are native to one-dimensional (1D) and two-dimensional (2D) waveguide gratings, also called photonic crystal slabs. Indeed, it is increasingly being recognized by researchers around the world that even a single resonant layer can supply an extraordinary variety of spectra that are controlled by the layer’s patterning and refractive-index contrast as shown in [8].

In this paper, we employ resonant leaky modes to spectrally engineer reflective elements with emphasis on realizing wide flat bands. Briefly reviewing the relevant literature, the pursuit of resonant wideband response can be traced to Gale et al. [20] and to Brundrett et al. [21] who achieved experimental full-width half-maximum (FWHM) linewidths near 100 nm albeit not for flat spectra. Applying cascaded resonance structures, Jacob et al. designed narrow-band flattop filters which exhibited also lowered sidebands and steepened stopbands [22]. Alternatively, by coupling several diffraction orders into corresponding leaky modes in a two-waveguide system, Liu et al. found a widened spectral response and steep filter sidewalls generated by merged resonance peaks [23]. Suh et al. designed a flattop reflection filter using a 2D-patterned photonic crystal slab [10]. Emphasizing new modalities introduced by asymmetric profiles, Ding et al. presented single-layer elements exhibiting both narrow and wide flat-band spectra [24]. Using a subwavelength grating with a low-index sublayer on a silicon substrate, Mateus et al. designed flattop reflectors with linewidths of several hundred nanometers operating in TM polarization [25]. Subsequently, they fabricated a reflector with reflectance exceeding 98.5% over a 500 nm range and compared the response with numerical simulations [26]. To emphasize the numerous new device possibilities afforded by properly designed resonant leaky-mode elements, we extended ref. [24] and showed single-layer elements with ~600 nm flattop reflectance in both TE and TM polarization [8]. Most recently, we further addressed the detailed physical basis for such reflectors by treating the simplest possible case which is a single-layer waveguide grating patterned in 1D with a two-part period [27]. We quantified the bandwidth provided by a single resonant layer by illustrative examples for both TE and TM polarized incident light and showed that reported experimental [26] wideband reflectors operate under leaky-mode resonance. This work defines the minimal structure capable of extensive reflectance bands [27].

Additionally, as a related topic, we point out that flat reflection bands enable transmission (bandpass) guided-mode resonance filters, which have been reported previously [28]. Here, the reflection bands form the low-transmission sidebands of the bandpass device. Recently, bandpass filters based on excitation of resonant leaky-mode pairs have been presented in which the transmission peak resides within a ~100% reflection band exceeding 100 nm in width [29, 30].

This paper has two main aims. First, to provide context, we summarize the main results of [27] and add numerous new results relevant to that case. Second, we characterize the spectral response of resonant wideband reflectors that have more complex architectures than the fundamental device treated in [27]. Thus, we treat GMR-based broadband reflectors with interesting profiles having four-part periods that may also sit on a silica sublayer above the substrate. We quantify the pertinent spectral response and modal behavior for these elements under both TE- and TM-polarized incident light illumination. Effects of deviations from optimal parameters are studied. The reflectors are designed for the 1.45-2.0 µm spectral band using the particle swarm optimization (PSO) technique [31, 32].

2. Device structure and design

Figure 1 shows the most general structure treated in this article. It is basically a silicon-oninsulator (SOI) element. For most of the designs, a silica substrate is used such that dL→∞. However, in some of our examples a silica film deposited on the silicon wafer defines a sublayer with low refractive index. The grating layer, which also acts as the waveguide, is defined by its period (Λ), thickness (d), and fill factors (Fi; ∑i Fi=1) that are the fractions of the period filled with each constituent material, as shown in Fig. 1. We emphasize the case of normal incidence for both TE and TM polarizations while also examining the angular sensitivity of the reflection spectra. The grating period has two or four sections consisting of two materials with different refractive indices nH and nL where nH > nL.

An inverse numerical method, the particle swarm optimization (PSO) technique is utilized to design the broadband reflectors. PSO is a robust, readily implemented optimization method with roots in swarm intelligence [31]. In this technique, particles, which comprise the swarm and represent the optimization parameters, move around the search space looking for the proper solutions. The optimal solution found is based on the experience of each particle and the best experience found by all particles through an iterative procedure. Introduction to the basics of PSO and its application to the design of diffraction gratings and GMR devices are reported elsewhere [32]. To attain the objectives of the present study, we set the design target to be 100% reflectance (R0) across the 1.45-2.0 µm band (550 nm bandwidth). The parameters to be optimized to achieve the target reflectance are the period (Λ), thickness (d), and fill factors (Fi). Also, when studying the effect of a sublayer, its thickness (dL) can be taken as a design parameter.

 

Fig. 1. Basic structure and parameters of the GMR reflector. Λ, d, and F1 to F4 denote the grating period, thickness, and fill factors, respectively. dL is the silica sublayer thickness. The incidence medium is air, substrate is silicon, nH=nSi=3.48, nL=nair=1.0, and nsilica=1.48.

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3. Resonant broadband reflectors with two-part period and no sublayer

3.1. TM polarization

As reported in [27], we designed a wideband reflector with optimized parameters for TM polarization being Λ=0.766 µm, d=0.490 µm, F1=0.7264, F2=1-F1, F3=F4=0, and dL →∞. Figures 2(a) and (b) show the pertinent spectra on linear and logarithmic scales. The width of the reflection band with R0 > 0.99 is ~520 nm over the 1.45-2.0 µm range. As seen in Fig. 2(b), there exist three transmittance dips inside the reflection band, each of which corresponding to a leaky-mode resonance. This shows that the broad reflection band is a result of co-existence and interaction of three TM leaky modes in this optimally-designed structure. Figures 3(a-c) show the amplitude of the magnetic (modal) field (Hy(z)) inside the grating structure and in the surrounding media for the resonances at 1.495, 1.620, and 1.839 µm for the zeroth, first (evanescent), and second (evanescent) diffraction orders. As displayed, the modal profiles corresponding to the first diffraction order (S1) at these resonant wavelengths show mixed modal states of TM0 to ~TM2.

 

Fig. 2. Reflectance and transmittance spectra (a) linear and (b) logarithmic of a PSO-designed broadband reflector for TM polarization. The resonance wavelengths are (i) 1.495 µm, (ii) 1.620 µm, and (iii) 1.839 µm.

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Fig. 3. Amplitude of the magnetic (modal) field (Hy(z)) inside the grating structure and in the surrounding media for the three resonances in Fig. 2 at (a) 1.495 µm, (b) 1.620 µm, and (c) 1.839 µm.

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Figure 4(a) displays a color-coded reflectance map R0(λ,d) drawn versus wavelength and grating thickness. This map quantifies the resonance characteristics relative to thickness which is especially useful for high-refractive-index contrast elements. There appears a semi-repetitive behavior for reflectance against thickness with reflection bands evolving from a single narrow resonance to a broad one. Figure 4(b) illustrates the associated transmittance versus wavelength and thickness in dB thus emphasizing the precise high-reflectance locus. The calculations for these spectra are done using rigorous coupled-wave analysis (RCWA) [33, 34] and modal analysis techniques [35]. RCWA and modal techniques are two well-known rigorous methods for solving diffraction problems involving optical periodic structures. Using these techniques, reflected and transmitted fields and their corresponding diffraction efficiencies are calculated. These two methods approach the same diffraction problem in two different ways, i.e. in RCWA the field is expanded into space harmonics while in the modal technique it is represented by modal functions. Therefore, the analysis results with these two techniques should be identical [36]. In this paper, the results found with these two formalisms exhibit complete agreement.

 

Fig. 4. (a) Reflectance map R0(λ,d) drawn versus wavelength and grating thickness. (b) Corresponding transmittance map T0(λ,d) in dB.

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Figure 5(a) provides a magnified view of the transmission map (in dB) for the device with thickness between 0.2 µm and 0.9 µm around the optimum design thickness (dotted line). For thicknesses between 0.485 µm and 0.499 µm, the reflection band is maximum and three resonances take part to realize the broadest reflectance. Figure 5(b) compares the reflection spectra for two thicknesses (0.455 and 0.520 µm) close to the optimal thickness (0.490 µm). Figures 5(c,d) show the modal field profiles for two points far from the broadband reflection region (d=0.217 µm and d=0.843 µm). For thicknesses well below the design thickness, the main excited leaky mode has TM0 characteristics, while for larger thickness values hybrid mode shapes appear. These field profiles imply a modal evolution (transition) in the grating layer with maximum reflectance bandwidth occurring where there is a blend of leaky modes. These modes do not provide the maximum reflectance linewidth alone as shown in Fig. 5(b).

 

Fig. 5. (a) Magnified transmission map (in dB, Fig. 4(b)) of the reflector for thicknesses between 0.2 µm and 0.9 µm around the optimum design thickness (dotted line). (b) Reflection spectra for thicknesses at two points near the optimal thickness (0.455 and 0.520 µm) in comparison to that for the optimal thickness (0.490 µm); note vertical axis scale change. Amplitude of the leaky-mode magnetic field for two points on the resonance locus in Fig. 5(a) but far from the optimal condition: (c) d=0.217 µm, λ=1.363 µm, {point (i)}, and (d) d=0.843 µm, λ=1.883 µm {point (ii)}.

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The color-coded map of R0(λ,d) in Fig. 4(a) corresponds qualitatively to modal curves of an “equivalent” homogeneous slab waveguide, since the resonance arises when the incident light beam is coupled to a leaky mode. The modal curves can be obtained by solving the eigenvalue equation of the equivalent homogenous slab waveguide system [7]. To clearly reveal this connection, we reduce the refractive index contrast of the structure while keeping the average refractive index of the waveguide layer fixed at 1.7314 (zero-order effective index). This reduces the resonance linewidth, enhancing the visibility of the modal curves. Taking nH=2.0 and nave=1.7314, nL is calculated to be 1.3417. Figure 6(a) shows the R0(λ,d) map for this reduced-contrast structure while Fig. 6(b) depicts calculated modal curves corresponding to the first four leaky modes excited by the first diffraction order in an equivalent homogenous film with refractive index of nf=1.92. There is excellent qualitative agreement between the maps in Fig. 6(a,b). Figures 6(c,d) show the magnetic field amplitudes for curves (I) and (IV) in Fig. 6(a) as examples. These profiles clearly approximate those in classic slab waveguides.

Figure 7(a) estimates the angular sensitivity of the reflection spectra associated with this device. The response is highly sensitive to the angle of incidence and ±1° deviation from normal incidence splits the band into two shorter bands. Figure 7(b) provides sampled spectra under normal (θ=0°) and oblique (θ=+5°) incidence. A narrow transmission channel emerges within the reflection spectra under off-normal incidence, yielding a resonance bandpass filter response [28].

 

Fig. 6. (a) R0(λ,d) map for a low-refractive-index contrast structure (nH=2.0 and nL=1.3417). (b) Calculated modal curves for the first four leaky modes excited by the first diffraction order in an equivalent homogenous film with refractive index of 1.92. (c,d) Magnetic modal field amplitudes corresponding to curves I (at λ=1.1849 µm and d=0.3293 µm (TM0)) and IV (at λ =1.1466 µm and d=1.913 µm (TM3)) in Fig. 6(a), respectively.

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Fig. 7. (a) Angular sensitivity of the reflection spectra of the broadband reflector. ~ ±1° deviation from normal incidence induces a transmission channel in the reflection band. (b) Samples of reflection spectra under normal and off-normal incidence.

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It is known that the spectral response of GMR-based devices is highly dependent on the structural parameters. Thus, we evaluate the response of our reflector under variation in the period, thickness, and fill factor. Figures 8(a-c) show the sensitivity of the spectra when each parameter deviates from its optimal value by up to ±10%. Such results are used to estimate fabrication tolerances. A statistical sensitivity analysis can also be performed by randomly deviating all of the parameters together in a predetermined parameter space.

 

Fig. 8. Sensitivity of the reflectance R0 to the structural parameters. In each part only one parameter is variable and the other two parameters are kept at their optimal values (a) Reflectivity map with period deviation, (b) Reflectivity map with thickness deviation, and (c) Reflectivity map with fill factor deviation. In each case, the optimum value is shown by a dotted line.

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3.2. TE polarization

Broadband high reflectors can be also designed to operate under TE-polarized incident light. Applying the PSO algorithm, a two-part-period reflector is designed for the 1.45-2.0 µm band using a SOI structure. The resulting parameters are found to be Λ=0.986 µm, d=0.228 µm, F1=0.3294, F2=1-F1, F3=F4=0, and dL →∞. Figure 9(a) illustrates the spectra of the device on linear and logarithmic scales. The only transmission dip, which shows the resonance wavelength, falls at 1.558 µm. The reflection bandwidth of the filter for R0 > 0.99 is ~125 nm. In addition, Fig. 9(b) displays the amplitude distribution of the total electric field (Ey(z)) in the grating and surrounding media. This figure clearly shows that the leaky-mode has ~TE0 characteristic and how the total field concentrates in parts of the grating period.

Figure 10(a) displays a map of R0(λ,d) showing the response of the reflector to thickness change. The reflection spectrum is widest at the design thickness. The shape of the highreflection regions differs from those in Fig. 4(a) for TM polarization. Further, in comparison to the TM case, the Rayleigh wavelength (λ R=nsilicaΛ) is considerably larger. For λ > λ R, the ±1 diffraction orders in the substrate are cut off (zero-order regime dominates). The Rayleigh wavelengths for the TM and TE cases are ~1.13 µm and ~1.45 µm, respectively. The zeroorder effective medium refractive index (average refractive index) of the structure is 2.159 and the second-order index is 3.365 (at 1.7 µm). Keeping the average refractive index fixed while reducing the refractive-index contrast, the reflection map evolves towards the leakymode curves of the “equivalent” homogenous layer, as in the TM case. Figures 10(b,c) show the reflection maps for the reduced-modulation structures ({nH=3.0, nL=1.5898} and {nH=2.3, nL=2.086}), respectively. Figure 10(d) illustrates the modal characteristic curves for the equivalent homogeneous slab corresponding well to the reflection map in Fig. 10(c).

 

Fig. 9. (a) Reflectance and transmittance spectra of a broadband reflector for TE polarization on linear and logarithmic scales. (b) Map of the amplitude of the total electric field (Ey(z)) in the grating and surrounding media at the resonance wavelength (λ=1.558 µm).

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Fig. 10. (a) Color-coded R0(λ,d) map for the TE reflector. (b) Reflectance map for reduced refractive contrast with nH=3.0 and nL=1.5898. (c) Reflectance map for reduced contrast with nH=2.3 and nL=2.0856, and (d) Modal characteristic curves for the equivalent homogeneous slab waveguide corresponding to (c) with the layer’s refractive index set to 2.167. The optimal thickness is shown by the dotted line in part (a).

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Figures 6(b) and 10(d) show modal curves corresponding to relatively weakly modulated structures to emphasize the modal origin of the leaky resonances. The spectrum of the TE reflector in Fig. 9(a) shows that a single resonance generates the reflection band. This contrasts with the TM reflector in Fig. 2 where three leaky modes participate. Therefore, the TE bandwidth is not as wide. A qualitative explanation for this bandwidth difference is based on the modal curves of the equivalent homogenous slab waveguides corresponding to the heavily modulated silicon-air GMR reflectors in Figs. 2 (TM) and 9 (TE). We establish the second-order effective refractive index in each case. Figures 11(a,b) display the resulting TE/TM modal curves in which the equivalent homogenous layers have effective refractive indices of 3.365 and 3.1577, respectively. We note that the curves are quite flat for TE polarization such that for a given thickness a ~single mode is excited in a reasonable wavelength band. The excitation of additional modes may be possible, however, the modes will be spaced far apart and effective resonance cooperation may not be achieved to form a continuous band. In contrast, the modal curves in the TM case provide sufficient curvature and spectral proximity such that ~two or more modes can become resonant and yield flat spectra.

Figure 12(a) shows the angular sensitivity of this reflector. The reflection response is highly dependent on the angle of incidence and ~±0.7° deviation from normal incidence generates a transmission channel and thus a bandpass filter response. Figure 12(b) displays sample spectra for normal and off-normal incidence. Figures 13(a–c) illustrate reflectance maps under variations in period, thickness, and fill factor. These maps show the sensitivity of the reflection spectra when each parameter deviates from its optimal value (dotted lines) up to ~±10%.

 

Fig.11. a) Modal curves for equivalent homogeneous slab waveguides corresponding to the high-contrast reflectors in Figs. 2 (TM) and 9 (TE) using second-order effective indices. (a) TE reflector (nf=3.365), and (b) TM reflector (nf=3.1577).

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Fig.12. a) Angular sensitivity of the spectra of the TE reflector. (b) Samples of the reflection spectra for normal (θ=0°) and off-normal (θ=+3°) incidence.

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Fig. 13. Maps showing the sensitivity of the reflection spectra to (a) period, (b) thickness, and (c) fill factor for TE polarization. The optimal parameters are denoted by the dotted line in each case.

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4. Resonant broadband reflectors with multi-part periods and sublayers

4.1. TM and TE reflectors with two-part periods and sublayers

In this section, we study the effects of a silica sublayer (with thickness dL) inserted between the resonant grating and a silicon substrate. The optimal sublayer thickness can be obtained either by utilizing a direct search (with known grating parameters) or an optimization technique (simultaneously design the grating with the sublayer). As a first example, we utilize PSO to design a broadband reflector for the 1.4-2.2 µm band with a silica sublayer over a silicon wafer for TM polarization. The optimal parameters are found as Λ=0.786 µm, d=0.500 µm, F=0.720, and dL=0.919 µm. Figure 14(a) illustrates the reflectance and transmittance of this device. The bandwidth for R0 > 0.99 is ~605 nm. To quantify the effect of the sublayer, we let dL→∞ (silica substrate) and calculate the reflection spectra again. The result is shown in Fig. 14(b); by removing the sublayer, the bandwidth decreases by ~49 nm from ~605 nm to ~556 nm.

Similarly, a reflector with a silica sublayer is designed for TE polarization using PSO. An optimal parameter set is found as Λ=0.965 µm, d=0.221 µm, F=0.326, and dL=3.354 µm. Figure 15(a) shows the spectra of this device. Figure 15(b) depicts the spectral response of the reflector with and without a silica sublayer. The sublayer with optimal thickness enhances the reflection bandwidth from 118 nm to 128 nm in this example.

 

Fig. 14. (a) Reflectance and transmittance of a resonant reflector with a silica sublayer for TM polarization (logarithmic scale). (b) Reflection spectra with and without the sublayer.

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Fig. 15. (a) Reflectance and transmittance spectra of the TE reflector with silica sublayer on a logarithmic scale. (b) Reflection spectra with and without the sublayer.

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4.2. TM and TE reflectors with four-part periods and without sublayers

Dividing the period into more than two parts increases the number of design parameters and provides design freedom and control over the Fourier series component distribution of the structure resulting in enhanced spectra for broadband reflectors. Two-part and four-part period gratings can have the same zero-order effective index albeit with completely different spectral response. Also, multi-part period gratings provide both symmetric and asymmetric leaky-mode structures, which can result in excitation of degenerate and non-degenerate leaky modes [24]. In this section, GMR elements whose period is divided into four parts (with fill factors F1, F2, F3, and F4; F1+F2+F3+F4=1.0) are designed to provide maximum reflection bandwidth coverage. The first design is for TM polarization with parameters Λ=1.0 µm, d=0.81 µm, and [F1, F2, F3, F4]=[0.5, 0.125, 0.25, 0.125] established by our PSO algorithm; it is identical to the corresponding device reported in [8]. The device layer is again comprised of silicon and air parts and the substrate is silica. This reflector has a symmetric structure. Figure 16(a) displays the spectra of this filter on linear and logarithmic scales. A bandwidth of ~600 nm for R0 > 0.99 over the target wavelength range is found. In comparison to the TM reflector with two-part period with spectrum in Fig. 2, this design exhibits ~80 nm enhancement in the bandwidth.

 

Fig. 16. (a) Reflectance (solid line) and transmittance (dashed line) spectra of the four-part broadband reflector for TM polarization. (b) R0(λ,d) map for this device. (c) Transmittance map T0(λ,d) in dB. (d-f) Magnetic field amplitude distribution in the device and surrounding media for three leaky-mode resonances at 1.627 µm, 1.744 µm, and 2.015 µm, respectively.

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Figure 16(b) shows the R0(λ,d) map, which illustrates the effect of thickness change on the reflectance behavior. The associated transmittance map T0(λ,d) on a dB scale is shown in Fig. 16(c) displaying quantitatively the leaky-mode resonance locations where the transmittance approaches zero. Figures 16(d-f) represent the magnetic field amplitude distribution in the device and surrounding media for the three main resonances in Fig. 16(a). The leaky modes contributing to the broad spectra show TM1 and ~TM2 features. In addition, as seen in Fig. 16(a), there is another resonance outside the target bandwidth at 2.44 µm with TM0 modal characteristics. The angular sensitivity of this broadband reflector is shown in Figs. 17(a,b). As in our prior examples, a broadband reflection response is obtainable within ~±1.0° deviation from normal incidence.

 

Fig. 17. (a) Angular sensitivity of the four-part broadband reflector for TM polarization. (b) Samples of reflection spectra for normal and off-normal incidence.

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Fig. 18. (a) Reflectance and transmittance spectra of the four-part broadband reflector for TE polarization. (b) Color-coded R0(λ,d) map. (c) T0(λ,d) map in dB. (d–f) Amplitudes of electric field modal profiles for the three resonance leaky-modes at 1.489 µm, 1.642 µm, and 1.872 µm, respectively.

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Similarly, a four-part TE reflector has design parameters Λ=0.979 µm, d=0.465 µm, and [F1, F2, F3, F4]=[0.071, 0.265, 0.399, 0.265], which is a symmetric profile. Figure 18(a) shows the associated spectra. This reflector has a bandwidth of ~495 nm for R0 > 0.99. Considerable enhancement (~370 nm) is achieved relative to the two-part structure in Fig. 9. The effect of thickness on the spectra is displayed in Figs. 18(b) and (c). The electric-field profiles of the modes contributing to the bandwidth of the reflector are displayed in Figs. 18(d–f); these show ~TE0 and ~TE1 mode features. Figure 19(a) displays the angular sensitivity of this structure, which is ~±0.6° whereas Fig. 19(b) shows samples of reflection spectra for normal and off-normal incidence. It is interesting to note that the spectrum splits at its extremes rather than in the center.

 

Fig. 19. (a) Angular sensitivity of the four-part broadband reflector. (b) Samples of reflection spectra for normal (θ=0°) and oblique (θ=+3°) incidence.

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4.3. TM and TE reflectors with four-part periods and sublayers

Additional layers directly affect the resonance response. Thus, to study the influence of sublayers on the performance of GMR reflectors, we directly simulate insertion of a silica layer between the periodic device layer and a silicon substrate for the four-part reflectors designed in Sec 4.2. Thus, in this case, the sublayer thickness is not included in the PSO optimization. The results are shown as R0(λ,dL) maps in Figs. 20(a,b). Proper sublayer thickness enhances the bandwidth to some extent. The periodic undulation seen as function of sublayer thickness is principally a thin-film effect.

 

Fig. 20. (a) Color-coded R0(λ,dL) map showing the effect of adding a silica sublayer to the fourpart reflector designed in section 4.2 for TM polarization. (b) Same for TE polarization.

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5. Conclusions

In this paper, we have designed numerous guided-mode resonance (GMR) reflectors possessing one or two layers and studied many of their properties in detail by numerical simulations. The particle swarm optimization technique (PSO) is found to be an effective design method to establish the device parameters. We design reflectors with two and four subparts in the period operating in TE and TM polarization and evaluate their spectral response and leaky-mode structure emphasizing the 1.45-2.0 µm spectral band. We present computed leaky-mode profiles as well as 3D color-based reflectance and transmittance maps elucidating the spectral response in detail for the designed reflectors. On reduction of the refractive-index contrast in the periodic device layer, the resonance linewidths fall and the spectral behavior takes on features approaching those defined by the mode structure of classical slab waveguides. Similarly, for the fully modulated silicon-air TE/TM GMR reflectors under study, the physical insights provided by connecting with the modal curves of the equivalent slab waveguides permit qualitative explanation as to why the TE reflector exhibits a smaller bandwidth. Moreover, we address the effect of a low refractive-index sublayer inserted between a silicon substrate and the device layer. This sublayer with optimal thickness can provide significant reflection band enhancements. For example, relative to a two-part, single-layer TM reflector, the sublayer increases the bandwidth from 520 nm to 605 nm. Finally, we treat the effects of deviation from the design parameters on the reflection spectra. These reflectors show considerable sensitivity to angular and parametric variations which should be taken into account in fabrication of the devices.

Acknowledgments

The authors thank Y. Ding for his contributions in developing parts of the analysis codes used. This material is based, in part, upon work supported by the National Science Foundation under Grant No. ECS-0524383.

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16. Y. Kanamori, T. Kitani, and K. Hane, “Guided-mode resonant grating filter fabricated on silicon-on-insulator substrate,” Jpn. J. Appl. Phys. 45, 1883–1885 (2006). [CrossRef]  

17. O. Parriaux, V. A. Sychugov, and A. Tishchenko, “Coupling gratings as waveguide functional element,” Pure Appl. Opt. 5, 453–469 (1996). [CrossRef]  

18. C.-L. Hsu, Y.-C. Liu, C.-M. Wang, M.-L. Wu, Y.-L. Tsai, Y.-H. Chou, C.-C. Lee, and J.-Y. Chang, “Bulk-micromachined optical filter based on guided-mode resonance in silicon-nitride membrane,” J. Lightwave Technol. 24, 1922–1928 (2006). [CrossRef]  

19. V. A. Astratov, I. S. Culshaw, R. M. Stevenson, D. M. Whittaker, M. C. Skolnick, T. F. Krauss, and R. M. De La Rue, “Resonant coupling of near-infrared radiation to photonic band structure waveguides,” J. Lightwave Technol. 17, 2050–2057 (1999). [CrossRef]  

20. M. T. Gale, K. Knop, and R. Morf, “Zero-order diffractive microstructures for security applications,” Proc. SPIE 1210, 83–89 (1990). [CrossRef]  

21. D. L. Brundrett, E. N. Glytsis, and T. K. Gaylord, “Normal-incidence guided-mode resonant grating filters: design and experimental demonstrations,” Opt. Lett. 23, 700–702 (1998). [CrossRef]  

22. D. K. Jacob, S. C. Dunn, and M. G. Moharam, “Normally incident resonant grating reflection filters for efficient narrow-band spectral filtering of finite beams,” J. Opt. Soc. Am. A 18, 2109–2120 (2001). [CrossRef]  

23. Z. S. Liu and R. Magnusson, “Concept of multiorder multimode resonant optical filters,” IEEE Photon. Technol. Lett. 14, 1091–1093 (2002). [CrossRef]  

24. Y. Ding and R. Magnusson, “Use of nondegenerate resonant leaky modes to fashion diverse optical spectra,” Opt. Express 12, 1885–1891 (2004). [CrossRef]   [PubMed]  

25. C. F. R. Mateus, M. C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang-Hasnain, “Ultrabroadband mirror using low-index cladding subwavelength grating,” IEEE Photon. Technol. Lett. 16, 518–520 (2004). [CrossRef]  

26. C. F. R. Mateus, M. C. Y. Huang, L. Chen, C. J. Chang-Hasnain, and Y. Suzuki, “Broad-band mirror (1.12-1.62 µm) using a subwavelength grating,” IEEE Photon. Technol. Lett. 16, 1676–1678 (2004). [CrossRef]  

27. R. Magnusson and M. Shokooh-Saremi, “Physical basis for wideband resonant reflectors,” Opt. Express 16, 3456–3462 (2008). [CrossRef]   [PubMed]  

28. S. Tibuleac and R. Magnusson, “Narrow-linewidth bandpass filters with diffractive thin-film layers,” Opt. Lett. 26, 584–586 (2001). [CrossRef]  

29. R. Magnusson, Y. Ding, K. J. Lee, D. Shin, P. S. Priambodo, P. P. Young, and T. A. Maldonado, “Photonic devices enabled by waveguide-mode resonance effect in periodically modulated films,” Proc. SPIE 5225, 20–34 (2003). [CrossRef]  

30. Y. Ding and R. Magnusson, “Doubly resonant single-layer bandpass optical filter,” Opt. Lett. 29, 1135–1137 (2004). [CrossRef]   [PubMed]  

31. R. Eberhart and J. Kennedy, “Particle swarm optimization,” in Proceedings of IEEE Conference on Neural Networks (IEEE, 1995), pp. 1942–1948.

32. M. Shokooh-Saremi and R. Magnusson, “Particle swarm optimization and its application to the design of diffraction grating filters,” Opt. Lett. 32, 894–896 (2007). [CrossRef]   [PubMed]  

33. T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985). [CrossRef]  

34. M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: Enhanced transmittance matrix approach,” J. Opt. Soc. Am. A 12, 1077–1086 (1995). [CrossRef]  

35. S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 23, 123–133 (1975). [CrossRef]  

36. R. Magnusson and T. K. Gaylord, “Equivalence of multiwave coupled-wave theory and modal theory for periodic-media diffraction,” J. Opt. Soc. Am. 68, 1777–1779 (1978). [CrossRef]  

References

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  • |

  1. P. Vincent and M. Neviere, "Corrugated dielectric waveguides: A numerical study of the second-order stop bands," Appl. Phys. 20, 345-351 (1979).
    [CrossRef]
  2. L. Mashev and E. Popov, "Zero order anomaly of dielectric coated gratings," Opt. Commun. 55, 377-380 (1985).
    [CrossRef]
  3. E. Popov, L. Mashev, and D. Maystre, "Theoretical study of anomalies of coated dielectric gratings," Opt. Acta 33, 607-619 (1986).
    [CrossRef]
  4. G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, and A. V. Tishchenko, "Total reflection of light from a corrugated surface of a dielectric waveguide," Sov. J. Quantum Electron. 15, 886-887 (1985).
    [CrossRef]
  5. I. A. Avrutsky and V. A. Sychugov, "Reflection of a beam of finite size from a corrugated waveguide," J. Mod. Opt. 36, 1527-1539 (1989).
    [CrossRef]
  6. R. Magnusson and S. S. Wang, "New principle for optical filters," Appl. Phys. Lett. 61, 1022-1024 (1992).
    [CrossRef]
  7. S. S. Wang and R. Magnusson, "Theory and applications of guided-mode resonance filters," Appl. Opt. 32, 2606-2613 (1993).
    [CrossRef] [PubMed]
  8. Y. Ding and R. Magnusson, "Resonant leaky-mode spectral-band engineering and device applications," Opt. Express 12, 5661-5674 (2004).
    [CrossRef] [PubMed]
  9. S. Peng and M. Morris, "Resonant scattering from two-dimensional gratings," J. Opt. Soc. Am. A 13, 993-1005 (1996).
  10. W. Suh and S. Fan, "All-pass transmission or flattop reflection filters using a single photonic crystal slab," Appl. Phys. Lett. 84, 4905-4907 (2004).
    [CrossRef]
  11. S. Boonruang, A. Greenwell, and M. G. Moharam, "Multiline two-dimensional guided-mode resonant filters," Appl. Opt. 45, 5740-5747 (2006).
    [CrossRef] [PubMed]
  12. A.-L. Fehrembach, A. Talneau, O. Boyko, F. Lemarchand, and A. Sentenac, "Experimental demonstration of a narrowband, angular tolerant, polarization independent, doubly periodic resonant grating filter," Opt. Lett. 32, 2269-2271 (2007).
    [CrossRef] [PubMed]
  13. A. R. Cowan, P. Paddon, V. Pacradouni, and J. F. Young, "Resonant scattering and mode coupling in two-dimensional textured planar waveguides," J. Opt. Soc. Am. A. 18, 1160-1170 (2001).
    [CrossRef]
  14. D. Gerace and L. C. Andreani, "Gap maps and intrinsic diffraction losses in one-dimensional photonic crystal slabs," Phys. Rev. E 69, 056603 (2004).
    [CrossRef]
  15. Y. Ding and R. Magnusson, "Band gaps and leaky-wave effects in resonant photonic-crystal waveguides," Opt. Express 15, 680-694 (2007).
    [CrossRef] [PubMed]
  16. Y. Kanamori, T. Kitani, and K. Hane, "Guided-mode resonant grating filter fabricated on silicon-on-insulator substrate," Jpn. J. Appl. Phys. 45, 1883-1885 (2006).
    [CrossRef]
  17. O. Parriaux, V. A. Sychugov, and A. Tishchenko, "Coupling gratings as waveguide functional element," Pure Appl. Opt. 5, 453-469 (1996).
    [CrossRef]
  18. C.-L. Hsu, Y.-C. Liu, C.-M. Wang, M.-L. Wu, Y.-L. Tsai, Y.-H. Chou, C.-C. Lee, and J.-Y. Chang, "Bulk-micromachined optical filter based on guided-mode resonance in silicon-nitride membrane," J. Lightwave Technol. 24, 1922-1928 (2006).
    [CrossRef]
  19. V. A. Astratov, I. S. Culshaw, R. M. Stevenson, D. M. Whittaker, M. C. Skolnick, T. F. Krauss, and R. M. De La Rue, "Resonant coupling of near-infrared radiation to photonic band structure waveguides," J. Lightwave Technol. 17, 2050-2057 (1999).
    [CrossRef]
  20. M. T. Gale, K. Knop, and R. Morf, "Zero-order diffractive microstructures for security applications," Proc. SPIE 1210, 83-89 (1990).
    [CrossRef]
  21. D. L. Brundrett, E. N. Glytsis, and T. K. Gaylord, "Normal-incidence guided-mode resonant grating filters: design and experimental demonstrations," Opt. Lett. 23, 700-702 (1998).
    [CrossRef]
  22. D. K. Jacob, S. C. Dunn, and M. G. Moharam, "Normally incident resonant grating reflection filters for efficient narrow-band spectral filtering of finite beams," J. Opt. Soc. Am. A 18, 2109-2120 (2001).
    [CrossRef]
  23. Z. S. Liu and R. Magnusson, "Concept of multiorder multimode resonant optical filters," IEEE Photon. Technol. Lett. 14, 1091-1093 (2002).
    [CrossRef]
  24. Y. Ding and R. Magnusson, "Use of nondegenerate resonant leaky modes to fashion diverse optical spectra," Opt. Express 12, 1885-1891 (2004).
    [CrossRef] [PubMed]
  25. C. F. R. Mateus, M. C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang-Hasnain, "Ultrabroadband mirror using low-index cladding subwavelength grating," IEEE Photon. Technol. Lett. 16, 518-520 (2004).
    [CrossRef]
  26. C. F. R. Mateus, M. C. Y. Huang, L. Chen, C. J. Chang-Hasnain, and Y. Suzuki, "Broad-band mirror (1.12-1.62 μm) using a subwavelength grating," IEEE Photon. Technol. Lett. 16, 1676-1678 (2004).
    [CrossRef]
  27. R. Magnusson and M. Shokooh-Saremi, "Physical basis for wideband resonant reflectors," Opt. Express 16, 3456-3462 (2008).
    [CrossRef] [PubMed]
  28. S. Tibuleac and R. Magnusson, "Narrow-linewidth bandpass filters with diffractive thin-film layers," Opt. Lett. 26, 584-586 (2001).
    [CrossRef]
  29. R. Magnusson, Y. Ding, K. J. Lee, D. Shin, P. S. Priambodo, P. P. Young, and T. A. Maldonado, "Photonic devices enabled by waveguide-mode resonance effect in periodically modulated films," Proc. SPIE 5225, 20-34 (2003).
    [CrossRef]
  30. Y. Ding and R. Magnusson, "Doubly resonant single-layer bandpass optical filter," Opt. Lett. 29, 1135-1137 (2004).
    [CrossRef] [PubMed]
  31. R. Eberhart and J. Kennedy, "Particle swarm optimization," in Proceedings of IEEE Conference on Neural Networks (IEEE, 1995), pp. 1942-1948.
  32. M. Shokooh-Saremi and R. Magnusson, "Particle swarm optimization and its application to the design of diffraction grating filters," Opt. Lett. 32, 894-896 (2007).
    [CrossRef] [PubMed]
  33. T. K. Gaylord and M. G. Moharam, "Analysis and applications of optical diffraction by gratings," Proc. IEEE 73, 894-937 (1985).
    [CrossRef]
  34. M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, "Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: Enhanced transmittance matrix approach," J. Opt. Soc. Am. A 12, 1077-1086 (1995).
    [CrossRef]
  35. S. T. Peng, T. Tamir, and H. L. Bertoni, "Theory of periodic dielectric waveguides," IEEE Trans. Microwave Theory Tech. 23, 123-133 (1975).
    [CrossRef]
  36. R. Magnusson and T. K. Gaylord, "Equivalence of multiwave coupled-wave theory and modal theory for periodic-media diffraction," J. Opt. Soc. Am. 68, 1777-1779 (1978).
    [CrossRef]

2008

2007

2006

2004

D. Gerace and L. C. Andreani, "Gap maps and intrinsic diffraction losses in one-dimensional photonic crystal slabs," Phys. Rev. E 69, 056603 (2004).
[CrossRef]

C. F. R. Mateus, M. C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang-Hasnain, "Ultrabroadband mirror using low-index cladding subwavelength grating," IEEE Photon. Technol. Lett. 16, 518-520 (2004).
[CrossRef]

C. F. R. Mateus, M. C. Y. Huang, L. Chen, C. J. Chang-Hasnain, and Y. Suzuki, "Broad-band mirror (1.12-1.62 μm) using a subwavelength grating," IEEE Photon. Technol. Lett. 16, 1676-1678 (2004).
[CrossRef]

W. Suh and S. Fan, "All-pass transmission or flattop reflection filters using a single photonic crystal slab," Appl. Phys. Lett. 84, 4905-4907 (2004).
[CrossRef]

Y. Ding and R. Magnusson, "Use of nondegenerate resonant leaky modes to fashion diverse optical spectra," Opt. Express 12, 1885-1891 (2004).
[CrossRef] [PubMed]

Y. Ding and R. Magnusson, "Doubly resonant single-layer bandpass optical filter," Opt. Lett. 29, 1135-1137 (2004).
[CrossRef] [PubMed]

Y. Ding and R. Magnusson, "Resonant leaky-mode spectral-band engineering and device applications," Opt. Express 12, 5661-5674 (2004).
[CrossRef] [PubMed]

2003

R. Magnusson, Y. Ding, K. J. Lee, D. Shin, P. S. Priambodo, P. P. Young, and T. A. Maldonado, "Photonic devices enabled by waveguide-mode resonance effect in periodically modulated films," Proc. SPIE 5225, 20-34 (2003).
[CrossRef]

2002

Z. S. Liu and R. Magnusson, "Concept of multiorder multimode resonant optical filters," IEEE Photon. Technol. Lett. 14, 1091-1093 (2002).
[CrossRef]

2001

1999

1998

1996

S. Peng and M. Morris, "Resonant scattering from two-dimensional gratings," J. Opt. Soc. Am. A 13, 993-1005 (1996).

O. Parriaux, V. A. Sychugov, and A. Tishchenko, "Coupling gratings as waveguide functional element," Pure Appl. Opt. 5, 453-469 (1996).
[CrossRef]

1995

1993

1992

R. Magnusson and S. S. Wang, "New principle for optical filters," Appl. Phys. Lett. 61, 1022-1024 (1992).
[CrossRef]

1990

M. T. Gale, K. Knop, and R. Morf, "Zero-order diffractive microstructures for security applications," Proc. SPIE 1210, 83-89 (1990).
[CrossRef]

1989

I. A. Avrutsky and V. A. Sychugov, "Reflection of a beam of finite size from a corrugated waveguide," J. Mod. Opt. 36, 1527-1539 (1989).
[CrossRef]

1986

E. Popov, L. Mashev, and D. Maystre, "Theoretical study of anomalies of coated dielectric gratings," Opt. Acta 33, 607-619 (1986).
[CrossRef]

1985

G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, and A. V. Tishchenko, "Total reflection of light from a corrugated surface of a dielectric waveguide," Sov. J. Quantum Electron. 15, 886-887 (1985).
[CrossRef]

L. Mashev and E. Popov, "Zero order anomaly of dielectric coated gratings," Opt. Commun. 55, 377-380 (1985).
[CrossRef]

T. K. Gaylord and M. G. Moharam, "Analysis and applications of optical diffraction by gratings," Proc. IEEE 73, 894-937 (1985).
[CrossRef]

1979

P. Vincent and M. Neviere, "Corrugated dielectric waveguides: A numerical study of the second-order stop bands," Appl. Phys. 20, 345-351 (1979).
[CrossRef]

1978

1975

S. T. Peng, T. Tamir, and H. L. Bertoni, "Theory of periodic dielectric waveguides," IEEE Trans. Microwave Theory Tech. 23, 123-133 (1975).
[CrossRef]

Andreani, L. C.

D. Gerace and L. C. Andreani, "Gap maps and intrinsic diffraction losses in one-dimensional photonic crystal slabs," Phys. Rev. E 69, 056603 (2004).
[CrossRef]

Astratov, V. A.

Avrutsky, I. A.

I. A. Avrutsky and V. A. Sychugov, "Reflection of a beam of finite size from a corrugated waveguide," J. Mod. Opt. 36, 1527-1539 (1989).
[CrossRef]

Bertoni, H. L.

S. T. Peng, T. Tamir, and H. L. Bertoni, "Theory of periodic dielectric waveguides," IEEE Trans. Microwave Theory Tech. 23, 123-133 (1975).
[CrossRef]

Boonruang, S.

Boyko, O.

Brundrett, D. L.

Chang, J.-Y.

Chang-Hasnain, C. J.

C. F. R. Mateus, M. C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang-Hasnain, "Ultrabroadband mirror using low-index cladding subwavelength grating," IEEE Photon. Technol. Lett. 16, 518-520 (2004).
[CrossRef]

C. F. R. Mateus, M. C. Y. Huang, L. Chen, C. J. Chang-Hasnain, and Y. Suzuki, "Broad-band mirror (1.12-1.62 μm) using a subwavelength grating," IEEE Photon. Technol. Lett. 16, 1676-1678 (2004).
[CrossRef]

Chen, L.

C. F. R. Mateus, M. C. Y. Huang, L. Chen, C. J. Chang-Hasnain, and Y. Suzuki, "Broad-band mirror (1.12-1.62 μm) using a subwavelength grating," IEEE Photon. Technol. Lett. 16, 1676-1678 (2004).
[CrossRef]

Chou, Y.-H.

Cowan, A. R.

A. R. Cowan, P. Paddon, V. Pacradouni, and J. F. Young, "Resonant scattering and mode coupling in two-dimensional textured planar waveguides," J. Opt. Soc. Am. A. 18, 1160-1170 (2001).
[CrossRef]

Culshaw, I. S.

De La Rue, R. M.

Deng, Y.

C. F. R. Mateus, M. C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang-Hasnain, "Ultrabroadband mirror using low-index cladding subwavelength grating," IEEE Photon. Technol. Lett. 16, 518-520 (2004).
[CrossRef]

Ding, Y.

Dunn, S. C.

Fan, S.

W. Suh and S. Fan, "All-pass transmission or flattop reflection filters using a single photonic crystal slab," Appl. Phys. Lett. 84, 4905-4907 (2004).
[CrossRef]

Fehrembach, A.-L.

Gale, M. T.

M. T. Gale, K. Knop, and R. Morf, "Zero-order diffractive microstructures for security applications," Proc. SPIE 1210, 83-89 (1990).
[CrossRef]

Gaylord, T. K.

Gerace, D.

D. Gerace and L. C. Andreani, "Gap maps and intrinsic diffraction losses in one-dimensional photonic crystal slabs," Phys. Rev. E 69, 056603 (2004).
[CrossRef]

Glytsis, E. N.

Golubenko, G. A.

G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, and A. V. Tishchenko, "Total reflection of light from a corrugated surface of a dielectric waveguide," Sov. J. Quantum Electron. 15, 886-887 (1985).
[CrossRef]

Grann, E. B.

Greenwell, A.

Hane, K.

Y. Kanamori, T. Kitani, and K. Hane, "Guided-mode resonant grating filter fabricated on silicon-on-insulator substrate," Jpn. J. Appl. Phys. 45, 1883-1885 (2006).
[CrossRef]

Hsu, C.-L.

Huang, M. C. Y.

C. F. R. Mateus, M. C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang-Hasnain, "Ultrabroadband mirror using low-index cladding subwavelength grating," IEEE Photon. Technol. Lett. 16, 518-520 (2004).
[CrossRef]

C. F. R. Mateus, M. C. Y. Huang, L. Chen, C. J. Chang-Hasnain, and Y. Suzuki, "Broad-band mirror (1.12-1.62 μm) using a subwavelength grating," IEEE Photon. Technol. Lett. 16, 1676-1678 (2004).
[CrossRef]

Jacob, D. K.

Kanamori, Y.

Y. Kanamori, T. Kitani, and K. Hane, "Guided-mode resonant grating filter fabricated on silicon-on-insulator substrate," Jpn. J. Appl. Phys. 45, 1883-1885 (2006).
[CrossRef]

Kitani, T.

Y. Kanamori, T. Kitani, and K. Hane, "Guided-mode resonant grating filter fabricated on silicon-on-insulator substrate," Jpn. J. Appl. Phys. 45, 1883-1885 (2006).
[CrossRef]

Knop, K.

M. T. Gale, K. Knop, and R. Morf, "Zero-order diffractive microstructures for security applications," Proc. SPIE 1210, 83-89 (1990).
[CrossRef]

Krauss, T. F.

Lee, C.-C.

Lee, K. J.

R. Magnusson, Y. Ding, K. J. Lee, D. Shin, P. S. Priambodo, P. P. Young, and T. A. Maldonado, "Photonic devices enabled by waveguide-mode resonance effect in periodically modulated films," Proc. SPIE 5225, 20-34 (2003).
[CrossRef]

Lemarchand, F.

Liu, Y.-C.

Liu, Z. S.

Z. S. Liu and R. Magnusson, "Concept of multiorder multimode resonant optical filters," IEEE Photon. Technol. Lett. 14, 1091-1093 (2002).
[CrossRef]

Magnusson, R.

R. Magnusson and M. Shokooh-Saremi, "Physical basis for wideband resonant reflectors," Opt. Express 16, 3456-3462 (2008).
[CrossRef] [PubMed]

M. Shokooh-Saremi and R. Magnusson, "Particle swarm optimization and its application to the design of diffraction grating filters," Opt. Lett. 32, 894-896 (2007).
[CrossRef] [PubMed]

Y. Ding and R. Magnusson, "Band gaps and leaky-wave effects in resonant photonic-crystal waveguides," Opt. Express 15, 680-694 (2007).
[CrossRef] [PubMed]

Y. Ding and R. Magnusson, "Resonant leaky-mode spectral-band engineering and device applications," Opt. Express 12, 5661-5674 (2004).
[CrossRef] [PubMed]

Y. Ding and R. Magnusson, "Doubly resonant single-layer bandpass optical filter," Opt. Lett. 29, 1135-1137 (2004).
[CrossRef] [PubMed]

Y. Ding and R. Magnusson, "Use of nondegenerate resonant leaky modes to fashion diverse optical spectra," Opt. Express 12, 1885-1891 (2004).
[CrossRef] [PubMed]

R. Magnusson, Y. Ding, K. J. Lee, D. Shin, P. S. Priambodo, P. P. Young, and T. A. Maldonado, "Photonic devices enabled by waveguide-mode resonance effect in periodically modulated films," Proc. SPIE 5225, 20-34 (2003).
[CrossRef]

Z. S. Liu and R. Magnusson, "Concept of multiorder multimode resonant optical filters," IEEE Photon. Technol. Lett. 14, 1091-1093 (2002).
[CrossRef]

S. Tibuleac and R. Magnusson, "Narrow-linewidth bandpass filters with diffractive thin-film layers," Opt. Lett. 26, 584-586 (2001).
[CrossRef]

S. S. Wang and R. Magnusson, "Theory and applications of guided-mode resonance filters," Appl. Opt. 32, 2606-2613 (1993).
[CrossRef] [PubMed]

R. Magnusson and S. S. Wang, "New principle for optical filters," Appl. Phys. Lett. 61, 1022-1024 (1992).
[CrossRef]

R. Magnusson and T. K. Gaylord, "Equivalence of multiwave coupled-wave theory and modal theory for periodic-media diffraction," J. Opt. Soc. Am. 68, 1777-1779 (1978).
[CrossRef]

Maldonado, T. A.

R. Magnusson, Y. Ding, K. J. Lee, D. Shin, P. S. Priambodo, P. P. Young, and T. A. Maldonado, "Photonic devices enabled by waveguide-mode resonance effect in periodically modulated films," Proc. SPIE 5225, 20-34 (2003).
[CrossRef]

Mashev, L.

E. Popov, L. Mashev, and D. Maystre, "Theoretical study of anomalies of coated dielectric gratings," Opt. Acta 33, 607-619 (1986).
[CrossRef]

L. Mashev and E. Popov, "Zero order anomaly of dielectric coated gratings," Opt. Commun. 55, 377-380 (1985).
[CrossRef]

Mateus, C. F. R.

C. F. R. Mateus, M. C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang-Hasnain, "Ultrabroadband mirror using low-index cladding subwavelength grating," IEEE Photon. Technol. Lett. 16, 518-520 (2004).
[CrossRef]

C. F. R. Mateus, M. C. Y. Huang, L. Chen, C. J. Chang-Hasnain, and Y. Suzuki, "Broad-band mirror (1.12-1.62 μm) using a subwavelength grating," IEEE Photon. Technol. Lett. 16, 1676-1678 (2004).
[CrossRef]

Maystre, D.

E. Popov, L. Mashev, and D. Maystre, "Theoretical study of anomalies of coated dielectric gratings," Opt. Acta 33, 607-619 (1986).
[CrossRef]

Moharam, M. G.

Morf, R.

M. T. Gale, K. Knop, and R. Morf, "Zero-order diffractive microstructures for security applications," Proc. SPIE 1210, 83-89 (1990).
[CrossRef]

Morris, M.

Neureuther, A. R.

C. F. R. Mateus, M. C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang-Hasnain, "Ultrabroadband mirror using low-index cladding subwavelength grating," IEEE Photon. Technol. Lett. 16, 518-520 (2004).
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A. R. Cowan, P. Paddon, V. Pacradouni, and J. F. Young, "Resonant scattering and mode coupling in two-dimensional textured planar waveguides," J. Opt. Soc. Am. A. 18, 1160-1170 (2001).
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G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, and A. V. Tishchenko, "Total reflection of light from a corrugated surface of a dielectric waveguide," Sov. J. Quantum Electron. 15, 886-887 (1985).
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O. Parriaux, V. A. Sychugov, and A. Tishchenko, "Coupling gratings as waveguide functional element," Pure Appl. Opt. 5, 453-469 (1996).
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S. T. Peng, T. Tamir, and H. L. Bertoni, "Theory of periodic dielectric waveguides," IEEE Trans. Microwave Theory Tech. 23, 123-133 (1975).
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Tishchenko, A.

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G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, and A. V. Tishchenko, "Total reflection of light from a corrugated surface of a dielectric waveguide," Sov. J. Quantum Electron. 15, 886-887 (1985).
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R. Magnusson, Y. Ding, K. J. Lee, D. Shin, P. S. Priambodo, P. P. Young, and T. A. Maldonado, "Photonic devices enabled by waveguide-mode resonance effect in periodically modulated films," Proc. SPIE 5225, 20-34 (2003).
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Figures (20)

Fig. 1.
Fig. 1.

Basic structure and parameters of the GMR reflector. Λ, d, and F1 to F4 denote the grating period, thickness, and fill factors, respectively. dL is the silica sublayer thickness. The incidence medium is air, substrate is silicon, nH=nSi=3.48, nL=nair=1.0, and nsilica=1.48.

Fig. 2.
Fig. 2.

Reflectance and transmittance spectra (a) linear and (b) logarithmic of a PSO-designed broadband reflector for TM polarization. The resonance wavelengths are (i) 1.495 µm, (ii) 1.620 µm, and (iii) 1.839 µm.

Fig. 3.
Fig. 3.

Amplitude of the magnetic (modal) field (Hy(z)) inside the grating structure and in the surrounding media for the three resonances in Fig. 2 at (a) 1.495 µm, (b) 1.620 µm, and (c) 1.839 µm.

Fig. 4.
Fig. 4.

(a) Reflectance map R0(λ,d) drawn versus wavelength and grating thickness. (b) Corresponding transmittance map T0(λ,d) in dB.

Fig. 5.
Fig. 5.

(a) Magnified transmission map (in dB, Fig. 4(b)) of the reflector for thicknesses between 0.2 µm and 0.9 µm around the optimum design thickness (dotted line). (b) Reflection spectra for thicknesses at two points near the optimal thickness (0.455 and 0.520 µm) in comparison to that for the optimal thickness (0.490 µm); note vertical axis scale change. Amplitude of the leaky-mode magnetic field for two points on the resonance locus in Fig. 5(a) but far from the optimal condition: (c) d=0.217 µm, λ=1.363 µm, {point (i)}, and (d) d=0.843 µm, λ=1.883 µm {point (ii)}.

Fig. 6.
Fig. 6.

(a) R0(λ,d) map for a low-refractive-index contrast structure (nH=2.0 and nL=1.3417). (b) Calculated modal curves for the first four leaky modes excited by the first diffraction order in an equivalent homogenous film with refractive index of 1.92. (c,d) Magnetic modal field amplitudes corresponding to curves I (at λ=1.1849 µm and d=0.3293 µm (TM0)) and IV (at λ =1.1466 µm and d=1.913 µm (TM3)) in Fig. 6(a), respectively.

Fig. 7.
Fig. 7.

(a) Angular sensitivity of the reflection spectra of the broadband reflector. ~ ±1° deviation from normal incidence induces a transmission channel in the reflection band. (b) Samples of reflection spectra under normal and off-normal incidence.

Fig. 8.
Fig. 8.

Sensitivity of the reflectance R0 to the structural parameters. In each part only one parameter is variable and the other two parameters are kept at their optimal values (a) Reflectivity map with period deviation, (b) Reflectivity map with thickness deviation, and (c) Reflectivity map with fill factor deviation. In each case, the optimum value is shown by a dotted line.

Fig. 9.
Fig. 9.

(a) Reflectance and transmittance spectra of a broadband reflector for TE polarization on linear and logarithmic scales. (b) Map of the amplitude of the total electric field (Ey(z)) in the grating and surrounding media at the resonance wavelength (λ=1.558 µm).

Fig. 10.
Fig. 10.

(a) Color-coded R0(λ,d) map for the TE reflector. (b) Reflectance map for reduced refractive contrast with nH=3.0 and nL=1.5898. (c) Reflectance map for reduced contrast with nH=2.3 and nL=2.0856, and (d) Modal characteristic curves for the equivalent homogeneous slab waveguide corresponding to (c) with the layer’s refractive index set to 2.167. The optimal thickness is shown by the dotted line in part (a).

Fig.11.
Fig.11.

a) Modal curves for equivalent homogeneous slab waveguides corresponding to the high-contrast reflectors in Figs. 2 (TM) and 9 (TE) using second-order effective indices. (a) TE reflector (nf=3.365), and (b) TM reflector (nf=3.1577).

Fig.12.
Fig.12.

a) Angular sensitivity of the spectra of the TE reflector. (b) Samples of the reflection spectra for normal (θ=0°) and off-normal (θ=+3°) incidence.

Fig. 13.
Fig. 13.

Maps showing the sensitivity of the reflection spectra to (a) period, (b) thickness, and (c) fill factor for TE polarization. The optimal parameters are denoted by the dotted line in each case.

Fig. 14.
Fig. 14.

(a) Reflectance and transmittance of a resonant reflector with a silica sublayer for TM polarization (logarithmic scale). (b) Reflection spectra with and without the sublayer.

Fig. 15.
Fig. 15.

(a) Reflectance and transmittance spectra of the TE reflector with silica sublayer on a logarithmic scale. (b) Reflection spectra with and without the sublayer.

Fig. 16.
Fig. 16.

(a) Reflectance (solid line) and transmittance (dashed line) spectra of the four-part broadband reflector for TM polarization. (b) R0(λ,d) map for this device. (c) Transmittance map T0(λ,d) in dB. (d-f) Magnetic field amplitude distribution in the device and surrounding media for three leaky-mode resonances at 1.627 µm, 1.744 µm, and 2.015 µm, respectively.

Fig. 17.
Fig. 17.

(a) Angular sensitivity of the four-part broadband reflector for TM polarization. (b) Samples of reflection spectra for normal and off-normal incidence.

Fig. 18.
Fig. 18.

(a) Reflectance and transmittance spectra of the four-part broadband reflector for TE polarization. (b) Color-coded R0(λ,d) map. (c) T0(λ,d) map in dB. (d–f) Amplitudes of electric field modal profiles for the three resonance leaky-modes at 1.489 µm, 1.642 µm, and 1.872 µm, respectively.

Fig. 19.
Fig. 19.

(a) Angular sensitivity of the four-part broadband reflector. (b) Samples of reflection spectra for normal (θ=0°) and oblique (θ=+3°) incidence.

Fig. 20.
Fig. 20.

(a) Color-coded R0(λ,dL) map showing the effect of adding a silica sublayer to the fourpart reflector designed in section 4.2 for TM polarization. (b) Same for TE polarization.

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