1. Introduction

Resonant coupling of external radiation to leaky modes of a slab waveguide via a dielectric grating or a two-dimensional (2D) photonic crystal (PC), frequently dubbed guided-mode resonance (GMR), has been widely studied [1–4]. Due to a relatively simple fabrication process and a straightforward design concept, numerous device applications of the grating-based GMR have been proposed and demonstrated. This includes optical filters, wideband reflectors, polarizers, display pixels, and biosensors [2,5–9]. The GMRs in PC slabs have also found various applications such as extraction efficiency enhancement of spontaneous emission or radiation mode control of lasers exploiting large density of states (DOS) of their leaky modes [10–12].

When two GMR elements interact with each other, the resonance characteristics can be tuned. The interaction between two GMR elements occurs in two possible routes: one is a direct evanescent coupling of the slab waveguides and the other is an indirect coupling through free-space propagation. Depending on relative strengths of those coupling mechanisms, different resonance characteristics are observed. Suh et al. have numerically investigated a widely tunable resonant transmission and displacement sensitivity of a two PC slab structure [13]. They have also shown that a characteristic of a similar structure can be switched between an all-pass filter or a band rejection filter depending the gap between the PC slabs [14]. The tunability of grating-based structures has also been studied, and wideband tuning ranges have been demonstrated [8,15,16]. These works have mainly treated the effect of configuration changes of closely located periodic diffraction structures, and aforementioned two coupling schemes have not been fully investigated. Tunable filters based on those evanescent and propagation wave couplings of slab waveguide gratings may have advantages over PC slabs due to their economic wide area fabrication. Besides relatively simple band structures of the slab waveguide gratings will allow conceptually easier control of physical parameters and straightforward design.

In this paper, we recapitulate the general characteristics of two coupled GMR elements by using the temporal coupled-mode theory [17], and based on that, characteristics and tunability of GMR in coupled gratings are discussed. The tunability of the GMR in coupled gratings is numerically demonstrated, which reveals that both strong and weak (negligible) evanescent coupling regimes can be adopted to realize wide-range wavelength tunable filters in good agreement with the coupled-mode theory description. In this work, the numerical analysis of gratings is conducted using the modal method [18].

2. Theory

Figure 1(a) shows the coupled grating structure treated in this work. A leaky guided-mode in each waveguide grating is resonantly excited by incident waves, and due to the leaky nature the excited guided-mode radiates outgoing waves as it propagates through the waveguide grating. The GMR in a lossless single grating brings about 100% reflection at the resonance. At the frequencies near the resonance, this characteristic can be modeled as a resonator that is side-coupled to a wave propagating channel with physical parameters such as resonance frequency and decay rate of the resonator [17]. The decay rate of the grating in the GMR is related to an index modulation of the grating and can be phenomenologically determined from a quality factor of the GMR, that is, $Q={\omega }_o\tau /4$, where ${\omega }_o$ is a resonance frequency and $1/\tau$is a decay rate in one direction. When two identical waveguide gratings are placed close to each other as depicted in Fig. 1(a), the leaky guided-modes are coupled evanescently and thus, two super-modes of even and odd symmetries are formed. The different dispersion properties of the two super-modes will cause two split resonances in the coupled gratings. In addition to the evanescent coupling of the guided-modes, the leakage wave (outgoing wave) from each grating will excites the other with a phase retardation. These two interactions result in modification of the resonance characteristic of the single grating. The interaction between two gratings and the resulting resonance characteristics can be described in a more insightful way by the aforementioned resonator model which is frequently dubbed the temporal coupled-mode theory. Therefore, before analyzing the resonance characteristics of the coupled grating by using a robust numerical Maxwell’s equation solver, we investigate the general resonance characteristics of the coupled resonator by using the coupled-mode theory.

 

Fig. 1 (a) Coupling of two identical gratings. (b) General structure of the two resonators with both evanescent and propagation wave couplings which is equivalent to (a). The field profile of the guided-mode of each waveguide grating is depicted in (a).

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Figure 1(b) shows a general structure of two coupled identical resonators in which both direct and indirect coupling routes exist. This is equivalent to the two coupled gratings in Fig. 1(a). The general wave propagation channel in Fig. 1(b) corresponds to free space in the coupled grating in Fig. 1(a). The temporal change of the normalized mode amplitudes of the resonators, $a_1$and $a_2$ are described by [15]

If we define the amplitudes of the super-modes with even and odd symmetries as $a_{even}=(a_1+a_2)/\sqrt{2}$ and $a_{odd}=(a_1-a_2)/\sqrt{2}$, which correspond to the super-modes formed by the evanescent coupling between the leaky guided-modes of the gratings, (4a) and (4b) are decoupled and rewritten as

From (6a) and (6b), we can see that the two super-modes will have different resonant frequencies and they can be tuned by controlling μ, $1/\tau$, and θ. In order to illustrate this tunability, the reflection coefficient, which is defined as the ratio between the amplitudes of the incident and the reflected waves, spectrum is calculated. When $s_{+4}=0$, from (2) and (4) the reflection coefficient is given by

Figure 2(a) shows the calculated reflection coefficient as a function of a normalized frequency, $(\omega -{\omega }_o)\tau /2=\Delta \omega \tau /2$ for different values of μ with θ = π/2. As μ increases, separation between two symmetric resonance peaks increases. Figures 2(b) and 2(c) show the calculated reflection coefficient for different values of θ with $\mu =10/\tau$. We can see that if the direct evanescent coupling between the resonators is strong ($\mu \tau$ >> 1), the phase retardation does not change the resonance peaks much but mainly affects the linewidths of the peaks. As expected from (7a) and (7b), for 0 < θ < π, the higher frequency peak corresponding to the odd super-mode gets broader and the even super-mode peak gets sharper. This characteristic provides a means to control the linewidth (Q-factor) and the center frequency of the resonance separately, which may find many applications. Note that a linewidth can be arbitrarily small according to (7a) and (7b). In this strong evanescent coupling regime, the resonance tuning is dominantly governed by μ and for more efficient tuning, high Q-factor resonators are preferred.

 

Fig. 2 Reflection coefficient spectra in evanescently coupled resonators as a function of a normalized frequency, (ω−ω_ο)τ/2 = Δωτ/2 (a) for θ = π/2, (b) and (c) for μ = 10.

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If two resonators are separated far enough such that evanescent coupling is negligible (μ≈0), the phase retardation plays an important role in tuning. In Fig. 3 , the reflection coefficient spectra are plotted for different values of θ with μ = 0. In this case, a reflection dip (transmission peak) occurs, and from (8), the normalized frequency of zero reflection is given by

 

Fig. 3 Reflection coefficient spectra as a function of a normalized frequency, (ω−ω_ο)τ/2 = Δωτ/2 in the case of negligible evanescent coupling (μ = 0).

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We can see that the dip becomes sharper as θ gets closer to 0 and the linewidth can be arbitrarily small. For θ = 0, the dip disappears. This implies that we can obtain a tunable resonant transmission in the negligible evanescent coupling regime (negligible direct coupling, μ = 0). The tuning range is determined by the decay rate of the resonator ($1/\tau$). For a wide tuning range, low Q-factor (broadband) resonators are preferred in this case.

3. Design and numerical calculation

Based on the theory presented above, we designed GMR systems with coupled gratings in regimes of strong and negligible evanescent coupling. Their characteristics are calculated using the modal method that has been proven to be accurate for periodic diffraction problems [18]. Material dispersion was not included in our calculation. In each following calculation, a normally incident wave with a certain polarization (TE or TM) was assumed. However, the theory discussed in this paper is not limited to a certain polarization or incident angle of light.

3.1 Guided-mode resonance system in strong evanescent coupling regime

Figure 4(a) depicts the coupled grating structure, where two identical gratings face each other with a gap of d. Each grating is made of 50 nm thick SiN (n = 1.85) layer with a period of 800 nm and a fill factor of 50% on a slab waveguide that is made of a 330 nm thick chalcogenide glass (n = 2.38) layer. The glass substrate (n = 1.5) is assumed to be thick enough and have an anti-reflection coating layer on the bottom. Figure 4(b) shows the reflection spectrum of a single grating for a normally incident wave of TE polarization. Since a high Q-factor resonator is preferred in a strong evanescent coupling regime, we designed the grating to have a narrow linewidth of 1.5 nm (λ_o = 1599.2 nm). The control parameters for tuning of the resonance characteristics are the gap distance d and the horizontal shift s between the two gratings as depicted in Fig. 4(a).

 

Fig. 4 (a) Structure of two identical coupled grating. (b) Reflection spectrum of the single grating in (a). Reflection spectra for the coupled gratings under different alignment conditions: (c) complete alignment (s = 0), (d) quarter-period shifted (s = P/4), and (e) half-period shifted (s = P/2). The calculations are done for a normally incident TE polarized wave.

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The evanescent coupling strength can be easily controlled by changing the gap size. For a normally incident TE polarized wave, we calculated reflection spectra for various values of d, and we also considered different alignment conditions between the gratings; a perfect alignment in Fig. 4(c), a quarter-period shifted alignment in Fig. 4(d), and a half-period shifted alignment in Fig. 4(e). The horizontal shift is supposed to mainly change the phase retardation with the evanescent coupling strength remaining almost the same. A horizontal shift over a half-period may cause a θ change of π roughly. Note that θ is a function of both wavelength and effective distance between two guided modes that is different from d. Therefore, determination of exact values of θ is difficult.

In all the different alignment conditions shown in Figs. 4(c), 4(d), and 4(e), two resonant reflection peaks are clearly observed and their separation increased approximately from 8 nm to 180 nm as d is varied from 500 nm to 0, and the longer wavelength resonant peak is changed over about 95 nm. A tuning range of a tunable filter can be defined as the maximum variable range of a center wavelength. Our calculation reveals that the structure in Fig. 4(a) can provide a tunable filter with a tuning range of 95 nm. We can see that for the same d, the alignment condition causes almost no change in resonant peak wavelengths, and the linewidth of the longer wavelength peak (even super-mode) degreases as s varies from 0 to P/2. Whereas, the shorter wavelength peak (odd super-mode) shows an increase of the linewidth as s increases from 0 to P/2. For example, for d = 0, the even mode peak wavelengths for three different alignments (s = 0, P/4, P/2) are respectively 1697.7 nm, 1698.1 nm, and 1698.2 nm and their linewidths are 9.4 nm, 3.2 nm, and 0.16 nm, respectively. This characteristic is qualitatively in good agreement with the coupled-mode theory description in the previous section and provides a means to control the resonant wavelength and the linewidth independently in tunable filer applications by controlling both the gap size and the alignment. The gap size and the horizontal shift can be controlled by means of micro/nano-electro-mechanical syatem (M/NEMS) technologies [8,19].

Although the linewidth dependence on s is well explained, in order to understand the linewidth change completely, field distributions should be also considered since each single grating is not symmetric along z direction and the gap size variation changes not only μ but also the decay rates ($1/{\tau }_{even}$ or $1/{\tau }_{odd}$). Note that this effect is not explicitly included in the coupled-mode theory in the previous section. Figures 5(a) and 5(b) show field (E_y) distributions for d = 0 at two resonance peaks of λ = 1697.7 nm and λ = 1517.1 nm, respectively. We can see that the field distributions show even symmetry at the longer wavelength peak in Fig. 5(a) and odd symmetry at the shorter wavelength peak in Fig. 5(b) as expected. Note that only amplitudes are plotted here and the odd super-mode shows 0 field amplitude at the center between two gratings. In the structure shown in Fig. 4(a), the odd super-mode experiences very little of the gratings due to the very small field amplitude in the central region and thus, has a much smaller decay rate than the even super-mode. This decay rate difference can be seen clearly in the field distributions in Figs. 5(a) and 5(b). The odd super-mode shows about one order of magnitude larger field amplitude confined in the waveguides than the even super-mode as a result of a longer photon life time. (See the color scales carefully.) That is why the odd-modes show much narrower linewidths in the cases of perfect alignment (Fig. 4(c)) and quarter-period shift (Fig. 4(d)). In these cases, as d increases, a larger portion of the odd mode experiences the gratings and thus, its linewidth increases. Whereas, the even mode shows an opposite tendency since the field amplitude decreases as d increases.

 

Fig. 5 Field (E_y) distributions in the coupled grating shown in Fig. 4(a) for d = 0: (a) at λ = 1697.7 nm (even super-mode) and (b)at λ = 1517.1 nm(odd super-mode). Field amplitudes are normalized to the incident wave.

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In the case of half-period shift alignment (Fig. 4(e)), the phase retardation θ plays a dominant role. The half-period shift causes θ ≈π for very small d, which makes the decay rate of an even mode ($1/{\tau }_{even}$) close to 0 from (7). That is why the even modes also show very small linewidths in Fig. 4(e). This phenomenon can be explained in terms of the interaction of the field and the gratings. When two weak gratings are very close with half-period shift, the effect of the periodicity is minimum, which means the effect of the gratings on the field is very small. This is interpreted as the cancellation of the scattering of each grating (or resonator in general) due to a phase shift close to π in the coupled mode theory. The increase of the linewidth of the even mode as d increases in Fig. 4(e) is also explained by (7).

In Fig. 4(d) with s = P/4, a very sharp reflection dip (a transmission peak) is observed near the even mode peak for d < 100 nm, which is not expected from the coupled-mode theory discussed in the previous section. The dip results from a destructive interference between two nondegenerate GMRs associated with the even mode. In an asymmetric waveguide grating, both edges of the second stop band support GMRs, while only one edge supports a GMR in a symmetric grating [1,20]. In our coupled grating structure, the grating shape becomes asymmetric when horizontal misalignment exists, and consequently, each super-mode will have two GMRs. The relative strengths of two GMR will depend on the extent of asymmetry and the strength of the grating. If two nondegenerate GMRs have comparable strength and their center wavelengths of resonance are close enough, they will interfere forming a transmission peak. Note that s = P/4 is the case of the strongest asymmetry and the dip is observed only for d < 100 nm even at s = P/4. The modulation of index (or grating strength) in the structure shown in Fig. 4(a) is rather weak and thus, only the even mode can support two nondegenerate GMRs strong enough under these limited conditions while the odd mode cannot. The dip may be avoided with a proper design of the grating.

Another coupled grating architecture is depicted in Fig. 6(a) . In this structure, a grating is located between a waveguide layer and a substrate. Materials and dimensions are the same as in the previous example. Due to the changed location of the gratings, two super-modes are expected to experience similar amount of interaction with the gratings. Figure 6(b) shows the reflection spectrum of a single grating for a normally incident wave of TE polarization, whose center wavelength and lindwidth are 1574.0 nm and 2.4 nm, respectively. Figures 6(c), 6(d), and 6(e) show the reflection spectra of the coupled gratings with three different alignment conditions as before for various d. A normally incident TE polarized wave is assumed. The tuning range is increased since the two waveguides can get closer. For the even mode, the resonance peak wavelength is tuned from 1750 nm to 1576 nm as d varies from 0 to 500 nm. The dependence of the linewidth on s (or θ) and the independence of the resonant peak wavelength of s are observed as in the previous structure. We can see that the odd mode interacts with the gratings more strongly than the even mode from Fig. 6(d), in which balanced interactions would bring about similar linewidths for both modes for θ ≈π/2 according to (7). In this structure, however, as a result of the changed grating location, the decay rate imbalance between two super-modes seems to be reduced compared to the previous structure.

 

Fig. 6 (a) Structure of two identical coupled grating. (b) Reflection spectrum of the single grating in (a). Reflection spectra for the coupled gratings under different alignment conditions: (c) complete alignment (s = 0), (d) quarter-period shifted (s = P/4), and (e) half-period shifted (s = P/2). The calculations are done for a normally incident TE polarized wave.

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Another important thing to note is that the dip near the even mode peak is seldom observed in this case. According to our investigation which is not shown in Fig. 6(d), the dip is observed for d < ∼10 nm at s = P/4. This can be explained from the reduced interaction of the even mode with the gratings. There is still no dip for the odd mode peak despite of its enhanced interaction to the gratings. We surmise that it is associated with the characteristics of the second stop band of the odd mode. However, a clear explanation to this matter awaits additional study.

3.2 Guided-mode resonance system in negligible evanescent coupling regime

As previously mentioned, a resonator with a low Q-factor (broad bandwidth) is preferred for a wide tuning range in a negligible evanescent coupling regime. A GMR grating with a wideband reflection spectrum that has been considered in other works [4] is used to investigate the tuning of resonant transmission. GMR transmission filters have been presented previously using a single grating [21,22]. Figure 7(a) shows the coupled grating structure, where the substrate (n = 1.48) is included to deal with a practical structure. In numerical analysis, the substrates are assumed to be thick enough and have anti-reflection coatings on the bottoms. The reflection spectrum of the single grating for a normally incident wave of TM polarization is plotted in Fig. 7(b). The calculated transmission spectra for a normally incident TM polarized wave are plotted in Fig. 7(c), where the distance between the gratings d is varied from 1350 nm to 2150 nm. It has been confirmed from the mode profile of the single grating structure that the evanescent coupling between the guided modes of the gratings becomes negligible for d > 1000 nm. The calculated spectra are independent of the horizontal alignment of two grating since free space propagation is a dominant coupling mechanism. In Fig. 7(c), one can see that there are two wide tuning windows; one is from λ = 1656.7 nm to λ = 2194.3 nm for 1350 nm < d < 2150nm and the other, from λ = 1327.6 nm to λ = 1648.5 nm for 1500 nm < d < 2150 nm. The linewidth of the transmission is determined by the reflection of the single grating at the transmission peak wavelength. The linewidth becomes as small as 0.06 nm at λ = 1656.7 nm for d = 1350 nm.

 

Fig. 7 (a) Structure of two identical coupled grating. (b) Reflection spectrum of the single grating in (a). Transmission spectra for the coupled gratings for various d. The calculations are done for a normally incident TM polarized wave.

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The resonant transmission can be simply understood as a result of Fabry-Perot (F-P) like interference between two reflectors. However, the real spectral characteristic is rather complicated as seen in Fig. 7(c) because of the resonant nature of the reflection gratings. It is obvious that the transmission peaks for a certain d cannot be related by a simple equation of 2d = mλ, where m is an integer. If the spectral characteristic of the reflection grating is close to Lorentzian, the transmission peaks can be found by solving (9) derived from the coupled-mode theory. In the grating shown in Fig. 7(a), however, multiple GMRs are involved, which is the origin of the broad reflection band [6]. Besides, it is not easy to determine the reference plane location as in Fig. 1 due to the vertical asymmetry of the single grating. Nonetheless, Eq. (9) is useful in approximate estimation of a range of d variation and a maximum tunable range of the transmission. If the wavelength of the lower limit (short) in a tuning range is given as λ_S for d = d_S, then the upper limit of the d variation is estimated as d_L = d_S + λ_S/2 from the periodic nature of (9), and the longest wavelength of the tuning range is given by the peak wavelength for d = d_L. For example, in Fig. 7(c), we have a transmission peak at λ = 1327.6 nm for d = 1500 nm and if this is chosen as λ_S, d_L is estimated as 2163.8 nm. This looks like a good estimation since we have a transmission peak at λ = 1327.6 nm for d = 2150 nm in Fig. 7(c). The actual transmission peak for d = 2163.8 nm occurs at λ = 1327.1 nm, which is not shown in Fig. 7(c).

4. Conclusion

In this work, a general theory on tunable spectral characteristics of coupled resonators has been recapitulated and two possible tuning schemes have been discussed using the temporal coupled-mode theory. When two resonators are evanescently coupled, resonant reflection peaks can be tuned by controlling the coupling strength, and their spectral linewidth can be also controlled by the phase retardation. When two resonators are coupled via propagating waves, a resonant transmission peak appears within a high reflection band and it can be tuned by the phase retardation of the propagation. These two schemes have been numerically studied in GMRs of coupled gratings. In the GMR system of two identical gratings with high Q-factors, the evanescent coupling of the gratings was controlled with a gap size and a tuning of the reflection peak wavelength from 1576 nm to 1750 nm was observed for the gap size change of 500 nm. The tuning corresponds to about 72 times of the linewidth of the single grating. Under very strong evanescent coupling conditions, an almost independent control of the lindwidth was possible via horizontal alignment change. In the GMR system in which two identical gratings with broad reflection bands were coupled via free-space propagation, a tunable resonant transmission was observed as the distance between the gratings was varied. A wide tuning range over 400 nm (1656.7 nm < λ < 2194.3 nm) was obtained for a distance variation of 800 nm (1350 nm < d < 2150 nm). These tuning ranges are comparable to those of the previously reported tunable filter schemes based on configuration changes of closely located periodic diffraction structures [15,16]. In ref [15], the tuning range was about 8% of the normalized wavelength, which corresponds to 124 nm if a center wavelength of 1550nm is chosen. In ref [16], the tuning of 300 nm (1400 ~1700 nm) was achieved.

In conclusion, we have shown a wide range tunability of the GMR in coupled gratings using a rigorous numerical method. The independent controllability of the resonance wavelength and the linewidth in a strong evanescent coupling regime is expected to find many useful applications with help of M/NEMS technologies.