Open Access
Add to BibSonomyAbstract
The guided mode resonances (GMRs) of diffraction gratings surrounded by low index materials can be designed to produce broadband regions of near perfect reflection and near perfect transmission. These have many applications, including in optical isolators, in hybrid lasers cavities and in photovoltaics. The excitation of rapid GMRs occurs in a background of slowly varying Fabry-Perot oscillation, which produces Fano resonances. We demonstrate the critical role of the polarity of adjacent Fano resonances in the formation of the broadband features. We design gratings for photovoltaic applications that operate at wavelengths where material absorption must be considered and where light is incident at non-normal angles.
© 2015 Optical Society of America
1. Introduction
Diffraction gratings are fundamental structures in optics, however their study has historically been focused on surface gratings [1], where the grooves of the grating do not extend all the way through the layer or the grating is placed on top of another high index material. Here we study high-contrast gratings (HCGs) [2], where the grating is surrounded by lower refractive index materials, such as air, and the inclusions penetrate the whole layer (see Fig. 1).
Fig. 1 Schematic of a grating of period d and thickness h, consisting of high and low index rulings: nH, nL. Also shown are the incident wave vector k0 at an angle of θ, its x-component and the propagation constant β of a slab waveguide mode. The field of the slab waveguide mode is illustrated in red.
The resonances of HCGs lead to rapid transitions in transmission/reflection from near-zero to near-unity, which can be used to make narrow and ultra-wide bandwidth filters [3–6]. HCGs have been applied as anti-reflection coatings [7], as optical isolators [8], and as cavities in hybrid lasers [2, 9]. It has also been shown that dispersion engineering the angular response of HCGs opens further possibility, including enhancing the Purcell effect, creating polariton-based lasers and quantum circuits, and exotic quantum phases in polaritons [10].
The resonances of gratings have been explained using three different conceptual frameworks: the excitation of leaky guided mode resonances (GMRs) [11], which are the waveguide modes of the homogenised slab that couple to the incident propagating waves [12]; the Fabry-Perot (F-P) resonances of the grating’s eigenmodes, which are quasi-periodic in the ±x-direction and propagate in the ±z-direction, and may be considered as either Bloch modes (BMs) [6] or approximated as the waveguide modes of the high index rulings [2]; and by studying the zeros and poles of the transmission and reflection functions [1]. The GMR analysis is consistent with the results of Magnusson showing that broadband near-unity reflections can be achieved using gratings placed upon anti-reflection layers that totally remove the bottom high contrast interfaces that are central to the vertical F-P resonance formulation [13]. Our analysis includes aspects of all three approaches, which clarifies the relationship between these currently disparate literatures [2, 13, 14].
Previous theoretical studies have focussed on predicting the locations of the resonances. An important issue that has remained unresolved is the relationship between isolated resonances and the occurrence of broadband regions of near perfect transmission or reflection. In this paper we answer this question by understanding the specular anomalies as Fano resonances [15], where the sharp resonant excitation of a leaky GRM (or equivalently a set of BMs) occurs in the presence of a slowly varying F-P resonance of the homogenized layer. We make the important step of showing that consecutive Fano resonances have opposite polarity, which allows them to form broadband regions of uniformly high reflection when the resonances broaden.
In the Sect. 5 we consider HCGs as wavelength selective filters operating at visible wavelengths, such as those being considered for photovoltaic applications [16], as well as those mentioned above [2, 7–9]. In the solar application their role is to direct different wavelength bands of the solar spectrum to solar cells that have different bandgap energies, and therefore convert their allocated wavelengths more efficiently than if the whole spectrum were incident upon any one cell [17]. This type of design has attracted much interest recently [18, 19] as it could lead to relatively cheap tandem cells with efficiencies greater than 30% [20], as well as ultra-efficient modules with efficiencies exceeding 50% [16]. The reduced cost of the tandem structures is achieved by using cheap silicon cells as the low bandgap subcell and a low-cost emergent material, such as a Perovskite [21, 22], in the other subcell.
These solar energy applications differ from those studied to date in two ways, they require: the grating to have minimal absorption; and to operate at angles of incidence far off-normal, often at θ = 45° where the reflected light exits at θ = 90° to the incident and transmitted light. In the wavelength range of primary interest for photovoltaic applications, 310 nm < λ < 1 µm, high refractive index materials are all, to some degree, lossy. In previous studies the materials were assumed to be lossless, which is valid at the wavelengths targeted, 1 µm < λ < 3 µm.
Our study was carried out using the freely available EMUstack package [23–25]. This is ideally suited to such a study because it provides access to the Bloch modes of the grating, and allows variations in thickness to be calculated with negligible computational cost.
The paper is organized as follows: Sect. 2 reviews the physics that governs where the resonances occur; in Sect. 3 we examine their spectral characteristics, focussing on the factors influencing their Fano line-shape; and in Sects. 4 and 5 we design HCGs as wavelength selective reflectors for applications where the effects of material absorption and angle of incidence are critical.
2. The modes of dielectric gratings
As our initial example we study a weak diffraction grating, where only 5% of the layer is the low index material (fL = 0.05, fH = 0.95). The grating period is chosen to be d = 400 nm so that, at the wavelengths considered, only the specular order propagates in the surrounding air. The materials are chosen to be dispersionless, with complex refractive indices nH = 3.5 + 0.005i, which is similar to silicon, and nL = 1.0. We include loss not only because it is important for practical applications, but also because the resonances of light are clearly visible as peaks in the absorption spectra. Throughout this paper we consider light that has Transverse Electric (TE) polarization (E-field along x-axis), and we begin by considering the case of normal incidence.
Figure 2 shows the reflection, transmission and absorption spectra of the example grating as a function of the grating thickness h. The reflection and transmission spectra are dominated by Fabry-Perot fringes that run diagonally through Figs. 2(a) and 2(b) respectively. The F-P resonances are also visible in the absorption spectra of Figs. 2(c) and 2(d), where absorption peaks occur on-resonance. There is also a second class of curves, where anomalies cause the spectra to vary rapidly. A striking feature of these curves, which are most obvious as peaks in the absorption spectra, is that their amplitudes vary strongly along the curves. These are the resonances that form the broadband reflection regions in stronger gratings, with their properties and underlying physics being examined in detail.
Fig. 2 (a) Reflection, (b) transmission, and (c) absorption spectra as a function of grating thickness. In (d) we magnify a section of (c) and overlay the results of the model described in the text: the dashed and solid white curves are the F-P resonances of the 0 and ±1 orders respectively; the green curves are like the solid white curves but uses heff; the cyan dots approximate the waveguide modes; the blue curve marks the trough in resonance amplitude; and the yellow dot-dashed line is the cut-off of the ±1 orders in nH.
We initially choose to study a weak grating because it simplifies the analysis, allowing us to approximate the BMs of the grating by plane wave diffraction orders (PWs). This is demonstrated in Fig. 3, where we show the Ey field distributions of the two dominant lowest order BMs modes, which we label BM-A, BM-B, and their Fourier decompositions into PWs. Figure 3(c) shows that BM-A is closely related to the specular diffraction order, while Fig. 3(d) shows that BM-B is associated with the ±1 orders. Approximating the BMs by PWs allows us to proceed analytically, approximating the propagation constants of the BMs by the propagation constants of their respective PW diffraction orders, within a homogenized grating layer.
Fig. 3 Ey(x) field distributions of (a) BM-A and (b) BM-B of the example grating for λd = 9/8. The edges of the inclusions are indicated in pink and the x-axis is normalised to the period. (c) and (d) show the Fourier decompositions of the fields of (a) and (b) respectively.
Figure 2(d) shows a magnified section of Fig. 2(c), where we overlay a number of analytic and semi-analytically calculated curves that capture all of the observed features. We now explain how these were derived.
The dashed white lines show where the specular PW order, m = 0, satisfies the F-P resonance condition, 2kz,mh = 2π p, with 1 < p < 6. The vertical propagation constant within the homogenized grating is
where θg is the angle of propagation within the grating, and the effective refractive index of the grating is calculated using the linear mixing formula of the permittivity for TE polarization, . Though we do not present results for TM polarization, the applicable effective index for this case is .The cyan dots correspond to the waveguide (WG) modes of the homogenized slab. They are calculated by solving the dispersion relations for the WG modes [12] for the transverse k-vector values of the grating,
where G is the reciprocal lattice vector. In the wavelength range shown, all dots correspond to coupling via the m = ±1 reciprocal lattice vector, which are degenerate at normal incidence. Each curve corresponds to a different order waveguide mode. The curves are a good fit to the observed resonances, with discrepancies only arising due to the approximation of an effective index, which is corroborated by observation that the fits improve as fL → 0.The solid white and green curves approximate the waveguide modes as F-P resonances of the ±1 PW orders, whose vertical propagation constants are
The quantitative accuracy of the green curves is improved by the inclusion of the phase shift acquired by the higher diffraction orders when they reflect off the top and bottom interfaces of the grating at oblique angles. This effect is similar to the Goos-Hänchen shift [12], and analogously we incorporate it into our model by introducing an effective thickness
where where m = ±1 for the ±1 PWs, is purely imaginary.The yellow vertical line marks the cut-off wavelength of the ±1 PW orders, above which these orders (and the corresponding BMs) are evanescent.
The blue curve indicates where the amplitude of the anomalies is at a minimum. This variation in the resonance amplitudes is due to the beating between the grating modes as they propagate with different kz, acquiring different phases. At the bottom of the grating, the electric field outside of the grating must match the superposition of the grating. If, at the bottom interface, the superposition of the grating modes has a waveform that matches that of a propagating PW order, then light is efficiently transmitted [26]. If, on the other hand, the superposition has little overlap with the waveform of any propagating order, then there is little transmittance and the strength of the resonance is enhanced. Since the superposition of the grating modes is equal to the incident PW at the top interface, we know that maximum transmission occurs when the phase difference of the grating modes is a multiple of 2π [26], i.e.,
where l is an integer. These points are marked by the blue curve of Fig. 2(d), where we approximate the kz of each BM by their respective PW order.The analytic calculations presented in Fig. 2 demonstrate how the complicated spectral response of HCGs can be understood in simple terms, and illustrate the relationships between the slab waveguide modes, the F-P resonances of the BMs of the grating, and the F-P resonances of non-specular diffraction orders within the homogenized layer. The PW approximation to the BMs breaks down as fL is increased, while the formalism that considers the grating as an array of waveguides becomes more accurate as the high index inclusions become increasingly separated [14].
We have so far focused on predicting the location of the resonances of grating. In the next section we examine the line shape of the resonances, which governs the creation of broadband regions of near total reflection or transmission. The analysis builds on our observations that the BMs can be accurately approximated by PW orders and that the waveguide modes can be approximated by F-P resonances of higher diffraction order PWs.
3. Analysis of Fano resonances
In Fig. 4 we show the transmission spectra of our initial grating when loss is (red), and is not (black), included. The thickness of the gratings is h = 520 nm, as indicated by the horizontal white dashed line in Fig. 2(b). We see that the superposition of phases from the F-P resonances and the waveguide modes produces resonances with a characteristic Fano line shape. The inclusion of loss dampens the resonances, with the absorption being highly localized to the resonant wavelengths.
Fig. 4 Transmission spectrum for the weak grating with h = 520 nm. The red curve includes loss (and is marked by the horizontal white dashed in line in Fig. 2(b)), while the black curve is calculated for the same grating but ignoring loss. Individual resonances are labelled.
We note that the polarity of the Fano resonance’s line shape changes sign for consecutive resonances; i.e., for the resonance labelled (iv) λ(t = 0) < λ(t = 1), whereas the resonance labelled (v) has the opposite symmetry with λ(t = 0) > λ(t = 1). Furthermore, we see that consecutive Fano resonances that occur on opposite sides of a F-P resonance (iii), such as those labelled (ii) and (iv), have the same polarity.
To clarify this effect, it is instructive to study spectra where the contribution of the F-P resonance to the complex transmission coefficient t is held constant. We do this by adjusting the thickness of the grating as a function of the wavelength,
Figure 5(a) shows the transmission along the diagonal white dotted line of Fig. 2(b), where p = 4.1. We chose this value of p because the resultant line lies between the resonance and anti-resonance conditions of the F-P oscillation intersects many GMRs. Keeping the F-P component of the transmission fixed highlights the opposite symmetry of the consecutive GMRs.
Fig. 5 (a) Transmission spectrum where the thickness of the grating is adjusted as a function of λ such that Eq. 7 is satisfied, indicated by the diagonal dotted line in Fig. 2(b). (b) Transmission along the same diagonal line, where the t corresponding to the F-P resonance has been subtracted, leaving the transmission due to the GMRs.
The opposite symmetry of consecutive Fano resonance is critical for the formation of the broadband reflection and transmission regions, as it enables the resonance spectra of two Fano resonances to merge without passing through a zero. Furthermore, since there are no resonances in these regions, the absorption is low.
In Sect. 2 we showed how the GMRs are related to F-P resonances of higher order plane waves in an homogenized medium. This allows us to make an analogy between the GMRs and a classic Fabry-Perot cavity. Figure 5(b) is for the same parameters as Fig. 5(a), where we now show the transmission that is due solely to the GMR. This is found by calculating t for the equivalent homogenized film and subtracting this from the t of the grating. We see that, precisely as in the case of a high finesse F-P cavity, the transmission is zero, except for on resonance where it goes briefly to unity. The finesse of the resonance is set by the coupling coefficient of the ±1 PW orders of the grating to the specular PW outside, directly analogously to the reflectivity of the mirrors of a F-P cavity.
In the following subsections we analytically analyse the F-P resonances, showing how the alternating polarity of consecutive orders is related to the poles of the transmission function, and then extend this analysis to the GMRs through numerical calculations. It is the superposition of these two effects that explains the features observed in Figs. 4, and 5.
3.1. Functional analysis of Fabry-Perot resonances
The complex transmission (t) and reflection (r) coefficients associated with light incident upon a homogeneous film are well known analytical functions [27] that can be derived using the transfer matrix method. For instance,
at normal incidence, where is the ratio of the refractive indices of the layer (nTF) and the index of the surrounding media (nout). The zeros of r occur at real frequencies, when kz,mh = π p, which reproduces the F-P result. r and t have common poles, which in contrast lie in the lower half of the complex frequency plane, as required by causality.For a homogeneous film we find that the poles occur at
where ρ = (noutnout −nTF)/(nout +nTF) is the Fresnel reflection coefficient for light within the layer incident upon the interface with the surrounds. The position of a pole directly relates to the Q-factor of the resonance,For real frequencies close to ωr,p the response is dominated by pole p, and is of the form
where is the residue of pole p [28]. The general result is that t(ωr) traces out a circle in the complex plane as ωr passes through ωr,p. As a function of increasing ωr the circle is followed in an anti-clockwise manner. The circle passes through the origin, and is centred at where . For a homogeneous film we find thatThis shows that t(ωr) traces out circles that are centred on the real axis, with consecutive F-P resonances (p) lying on opposite sides of the imaginary axis. The blue curves in Fig. 6 illustrate schematically how t(ωr) evolves in a frequency range that covers two consecutive F-P resonances, where the Q-factor of the resonance is sufficiently small that the effect of the two poles merge, forming a peanut shape. The arrows indicate how the curves are traced out for increasing wavelengths.
Fig. 6 Schematic of the trajectories of t(ωr) through the complex-plane, where blue curves represent the change in t due to the F-P resonance and red curves due to GMRs. The insets illustrate the line-shapes of the transmission associated with each trajectory (in (d) following just the red curve). (a), (b) Illustrate how GMRs whose residues lie on opposite sides of the real axis produce lines-shapes of opposite symmetry, despite the loops having the same starting point. (c) Shows the trajectory of two consecutive F-P resonances, whose low Q-factors makes them merge without going through the origin. (d) Features the same GMR as in (b), but occurring on the other side of a F-P resonance, which is seen to produce the opposite symmetry Fano resonance.
3.2. Functional analysis of guided mode resonances
In order to study the poles of the GMRs we construct a model structure that contains the same essential physics of the HCGs: where light is coupled to the leaky waveguide modes by a grating, but in which the GMRs are described exactly by PWs. The model structure consists of infinitesimally thin gratings placed at the top and bottom of a uniform thin film, which we refer to as the surface grating structure. The refractive indices of the grating must be very large to ensure that the grating scatters significant energy into higher diffraction orders. For simplicity we take all materials to be lossless in this model. We use a grating of thickness hg = 1 nm and composed of equal parts nH = 20, nL = 1 and with d = 400 nm. The uniform film is chosen to be hTF = 600 nm thick with nTF = 2, such that the ±1 PW orders propagate within the uniform film for λ < 800 nm.
The major advantage of the surface grating model is that the analytic expressions for the PWs within the uniform layer can be generalized to complex frequencies, allowing us to locate the poles of t, r and the associated residues. Here t and r are the elements of the scattering matrices corresponding to the single propagating order (i.e. the specular order). To do this we calculate the scattering matrices that describe the surface grating for a fixed real frequency (3.77 PHz) and then use these values for all other frequencies. The transmission spectrum of this surface grating structure, presented in Fig. 7(a) as a function of real frequencies, exhibits precisely the same types of features as the gratings in Fig. 4.
Fig. 7 Transmittance of the surface grating structure. In (a) the transmission is a function of real angular frequencies, while in (b) the transmission is calculated across a range of complex frequencies.
Figure 7(b) shows a contour plot of the transmission for complex frequencies, where two classes of poles can be seen. One class has small ωi,p and large Q-factors (from Eq. 10), and correspond to the GMRs. The other class correspond to F-P resonances and lie further away from the real axis with smaller Q-factors. The respectively high-Q and low-Q nature of these resonances is visible in the spectra of Fig. 7(a).
The residue of each pole is calculated using numerical contour integration. The results for the F-P resonances agree well with the analytic expressions; the residues are purely imaginary, and have opposite sign for consecutive resonances (i.e. their phase of the residues differs by π). This produce t(ωr) trajectories centred around points on the real axis with consecutive resonances lying on opposite sides of the imaginary axis. For the GMRs we find that the phase of the residues also differs for consecutive resonances, such that t(ωr) follows circles in the complex plane that lie on opposite sides of the imaginary axis. Physically, this difference is related to odd or even order resonances having an odd or even number of half wavelengths within the grating layer, which dictates the phase of the transmitted light.
We illustrate the trajectories of t(ωr) in the vicinity of a GMR by the red curves of Fig. 6. The trajectories of the GMRs shown in Figs. 6(a) and 6(b) begin from the same point, but the residues of the resonances lie on opposite sides of the real axis. Following the trajectories we see that in Fig. 6(a) the transmission goes through zero before going through its maximum value, while the opposite is true for Fig. 6(b). This is shown schematically in the inset. The effect of combining the GMRs with the F-P resonance is shown in Fig. 6(d), which features the same GMR as in Fig. 6(b), however the GMR now occurs on the other side of a F-P resonance, which produces a Fano line-shape of opposite symmetry.
In our weak gratings, the GMRs occur over frequency bandwidth of approximately 0.03 PHz and each F-P resonances spans approximately 0.8 PHz, which is consistent with the difference in ωi,p for the two classes of poles. The large difference in the frequency ranges of the resonances means that the combined effect of the blue and red curves is as follows: t(ωr) follows the blue curve until ωr approaches ωr,q for a WG resonance q; now t(ωr) completes a loop around a red curve, which is completed in such a small frequency interval that the phase of the F-P resonance has changed very little (causing little movement along the blue curve) and the red curve almost closes on itself; once the frequency range of the GMR is passed, t(ωr) resumes following the blue curve. The supplementary materials contain animations that shows the numerically calculated progression of t(ωr) corresponding to Fig. 4 ( Visualization 1), and Fig. 5 ( Visualization 2). Similar trajectories of have been measured experimentally at infrared frequencies by Botten et. al [29].
3.3. Stronger diffraction gratings
Our analysis has been focused on weak gratings, where the modes of a grating are well approximated by plane wave orders. Our general findings however hold for all gratings, because the existence and parity of the poles is topologically preserved. As the diffraction strength of the grating is increased, by increasing the refractive index contrast and/or increasing fL, light is coupled more efficiently into, and out of, the waveguide modes. This reduces the Q-factor of the GMR and broadens the spectral response of the Fano resonances.
The broadening is shown in Fig. 8(a) where fL = 0.05 (corresponding to the marked line in Fig. 2(a)), and in Fig. 8(b) where fL = 0.35, which is marked in Fig. 8(d). We note that adjacent Fano resonances continue to have opposite polarity, so that when they broaden to the extent that they overlap, they form broad regions of uniformly high reflectivity.
Fig. 8 (a) Reflection spectrum corresponding to the dot-dashed line in Fig. 2(a) (fL = 0.05, h = 210 nm). (b) Reflection spectrum of a grating with fL = 0.35 and h = 150 nm (indicated by dot-dashed line in (d)). (c) Reflection spectra with fL = 0.2 as a function of grating thickness. (d) Reflection spectra with fL = 0.35 as a function of grating thickness. Additional resonances occur at short wavelengths in (c), where the waveguide modes are excited by the ±2 PW orders.
Having understood both the location and the polarity of the Fano resonances, we can now fully explain the striking “checkerboard” pattern observed in Figs. 8(c) and 8(d) where fL = 0.2 and fL = 0.35 respectively: high reflectivity occurs between consecutive Fano resonances whose line-shapes have opposite symmetry; the transmission is high on the other side of these Fano resonances; and the areas are bounded by the specular F-P resonances, because these invert the polarity of the Fano resonances. Regions of high transmissivity are formed by the same process, between Fano resonances with the opposite symmetry. This fully explains the physics that causes the broadband regions of near total reflectivity or transmission that are the stand-out features of HCGs.
4. Dependence on incident angle
So far we have considered only normally incident light. For spectrum splitting applications, however, light is typically incident at θ = 45°, such that the transmitted and reflected light are at θ = 90° to each other. At normal incidence the ± m PW orders are degenerate ( in Eq. 2, when θ = 0). The major effect of moving to non-normal angles of incidence is that this degeneracy is lifted, with one PW having a larger, and the other a smaller, kz. This doubles the number of wavelengths at which the grating couples to waveguide modes, doubling the number of Fano resonances. The m = +1 set occurs at shorter wavelengths than at normal incidence, and the m = −1 set at longer wavelengths.
In Figs. 9(a) and 9(b) we show the absorption spectra of the grating from Fig. 2 illuminated at θ = 5° and θ = 20°, where we use the absorption peaks as proxies for the location of the Fano resonances. In Fig. 9(c) we show the analytic curves equivalent to the red curve of Fig. 2(d) at θ = 5°, which exhibits the splitting of the degeneracy, although their quantitative agreement to Fig. 9(a) is not as good as in Fig. 2. The −1 PW order are shown in dashed lines, the +1 PW order in dot-dashed lines, and the ±1 PWs at normal incidence are shown for comparison as solid lines. The different colours represent different order WG modes.
Fig. 9 Absorption for our canonical grating (fL = 0.05) at angles of incidence of (a) θ = 5°, (b) θ = 20°. (c) The F-P resonances of the m = +1 (dotted) and the m = −1 (dashed) PW order at θ = 5°. The solid lines are for normal incidence and different colours are different WG orders. (d) The transmission spectrum corresponding to the marked slice through (b).
Figure 9(d) shows the transmission spectrum of the grating at θ = 20° with h = 250 nm (marked in Fig. 9(b)). We see that the GMRs that correspond to coupling into the same order waveguide mode (labelled (ii), (iii)) have the same polarity, as they must, given that they are degenerate at normal incidence. Consecutive WG order resonances still have opposite polarity, as can be seen by comparing (i) and (iii). Because (ii) and (iii) have the same symmetry, the transmission dip that extended uniformly from (i) to (iii) at normal incidence now has an anomaly in it. As θ increases, (ii) shifts to shorter wavelengths, passing through (i), so that the broadband region is restored. In the next section θ = 45°, such that the resonances associated with the −1 PW orders occur at λ < 450 nm and are not observed in the range of interest.
5. Application as a wavelength selective reflector
Building on the results of the previous sections, we now design wavelength selective reflectors operating at the visible and near infrared wavelengths of importance in photovoltaic applications, with light incident at θ = 45°. We investigate gratings composed of annealed, amorphous TiO2, with air inclusions. For simplicity we follow previous studies [2, 6] and consider HCGs suspended in air, with similar results be obtainable in the presence of low index substrates. We focus on TiO2 as it is a standard material for nanofabrication that has relatively low absorptivity. The complex refractive index data for TiO2 is taken from [30], with values varying from at λ = 450 nm to at λ = 1200 nm (the long wavelength limit of the data). Figure 10 shows the reflection and absorption spectra of gratings composed of for 450 nm < λ < 1200 nm, which covers the peak of the solar spectrum.
Fig. 10 (a)–(c) Reflection spectra and (d)–(f) absorption spectra of gratings as a function of their thickness. In (a), (d) d = 450 nm, while in (b), (e) d = 540 nm. The spectra of the stacked structure; grating with d = 450 nm, 1 µm thick air spacer, grating with d = 540 nm, is shown in (c), (f), where the y-axis gives the thickness of each grating, which are equal. In all cases and the angle of incidence is 45°.
In Figs. 10(a) and 10(b) we see how reducing the refractive index and fH from our previous examples has significantly reduced the Q-factors of the Fano resonances, broadening the wavelength range of high reflectance, and slightly decreasing the amplitude of the reflection peak. The resonances associated with the fundamental WG mode and the first higher order WG mode overlap; and since, in the absence of an intermittent F-P resonance, their symmetries are opposite they form broadband reflection regions.
The bandwidths of the reflection regions in Figs. 10(a) and 10(b) are smaller than in previous studies [5], which is due to the lower refractive index. The only difference between the gratings in Figs. 10(a) and 10(b) is the period, which is d = 450 nm in Fig. 10(a) and d = 540 nm in Fig. 10(b). We see that this tunes the location of the Fano resonances, and of the cut-off wavelength, below which the Fano resonances associated with the ±1 orders disappear. This later cut-off is where the ±1 orders become propagating in air, and are therefore no longer highly reflected at the grating-air interface.
Figures 10(d) and 10(e) show the absorption spectra corresponding to Figs. 10(a) and 10(b). The substantial absorption at λ < 500 nm, particularly for thick gratings, is due to the imaginary part of increasing dramatically at shorter wavelengths. We limit the scale bar to α < 25% to bring out the absorption at longer wavelengths. Associated with each Fano resonance is an absorption peak, which occurs due to the light spending a prolonged time in a waveguide mode. These peaks are of relatively low amplitude and remain highly localized to the centre of the Fano resonance, even when the resonances broaden to form the broadband feature.
The reflection bandwidth may be increased to meet practical requirements by combining multiple gratings. This can be achieved while maintaining relatively low absorption in the region of interest. Figure 10(c) shows the results for a stack where the grating of Fig. 10(a) are placed above the grating of Fig. 10(b), separated by an air layer of thickness 1 µm. The spectra are quite insensitive to the separation distance as long as hspacer > 2d such that the gratings do not couple evanescently. The resultant reflection band spans approximately 300 nm (∆ λ/λ ≈ 0.3), similar to previous studies [5]. The absorption peak at approximately λ = 950 nm is strengthened in Fig. 10(e) because it corresponds to both the d = 450 nm and d = 540 nm HCGs being on resonance, coupling to the fundamental and first higher order WG modes respectively.
The Fano line-shape ensures that the reflection is close to zero on the other side of the reflection bands. At longer wavelengths, beyond the cut-off of the ±1 order (here around λ = 1200 nm), the grating acts as a homogenised film with slowly varying F-P resonances. These preliminary results show that dielectric gratings are well suited to being used as spectrum splitting elements with low loss and high wavelength selectivity. Because the HCG reflectors are only a single thin layer (in this case h < 750 nm), their parasitic absorption may be less than in a multi-layered Bragg stack composed of the same material.
6. Conclusion and discussion
We studied the Fano resonances that occur in high-contrast gratings when guided mode resonances are excited in the presence of slowly varying Fabry-Perot resonances. The broadband regions of near 100% reflection and near 100% transmission were shown to arise because consecutive GMRs have opposite polarity. The checker-board patterns, that are signatures of HCGs, were shown to be caused by the polarity of the Fano resonance flipping when a F-P resonance is passed. We also examined the effect of material absorption and non-normal angles of incidence, finding that good wavelength selectivity can be achieved with little parasitic absorption, even when using lossy materials.
Our results are for TE polarized light. Many applications, such as solar spectrum splitting, however use unpolarized light. One-dimensional structures are intrinsically highly polarization sensitive, and a recent study of HCGs showed that gratings optimized for TM polarized light were substantially thicker than gratings designed for TE polarized light [31]. This strengthens the case for investigating bi-periodic structures for applications involving unpolarized light. The insights developed in this paper are, however, quite general, and should apply also to bi-periodic structures, which support guided modes for both TE and TM polarized light.
Acknowledgments
We thank S. Darbe and H. Atwater from the Californian Institute of Technology, and T. White and K. Catchpole from the Australian National University for fruitful discussions. This work was supported by the Australian Renewable Energy Agency, and the Australian Research Council Discovery Grant and Centre of Excellence Schemes CE110001018. Computation resources were provided by the National Computational Infrastructure, Australia and the NeCTAR Research Cloud, Australia.
References and links
1. D. Maystre, “Diffraction gratings: An amazing phenomenon,” Comptes Rendus Phys. 14, 381–392 (2013). [CrossRef]
2. C. Chang-Hasnain and W. Yang, “High-contrast gratings for integrated optoelectronics,” Adv. Opt. Photonics 4, 379–440 (2012). [CrossRef]
3. S. S. Wang and R. Magnusson, “Theory and applications of guided-mode resonance filters,” Appl. Opt. 32, 2606–2613 (1993). [CrossRef] [PubMed]
4. S. Tibuleac and R. Magnusson, “Narrow-linewidth bandpass filters with diffractive thin-film layers,” Opt. Lett. 26, 584–586 (2001). [CrossRef]
5. C. F. R. Mateus, M. C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang-Hasnain, “Ultrabroadband mirror using low-index cladded subwavelength grating,” IEEE Photonics Technol. Lett. 16, 518–520 (2004). [CrossRef]
6. P. Lalanne, J. P. Hugonin, and P. Chavel, “Optical properties of deep lamellar gratings: A coupled bloch-mode insight,” J. Light. Technol. 24, 2442–2449 (2006). [CrossRef]
7. Y. Kanamori, K. Hane, H. Sai, and H. Yugami, “100 nm period silicon antireflection structures fabricated using a porous alumina membrane mask,” Appl. Phys. Lett. 78, 142–143 (2001). [CrossRef]
8. T. Glaser, S. Schröter, H. Bartelt, H.-J. Fuchs, and E.-B. Kley, “Diffractive optical isolator made of high-efficiency dielectric gratings only,” Appl. Opt. 41, 3558–3566 (2002). [CrossRef] [PubMed]
9. A. Taghizadeh, J. Mørk, and I.-S. Chung, “Ultracompact resonator with high quality-factor based on a hybrid grating structure,” Opt. Express 23, 17282–17287 (2015). [CrossRef]
10. Z. Wang, B. Zhang, and H. Deng, “Dispersion engineering for vertical microcavities using subwavelength gratings,” Phys. Rev. Lett. 114, 73601 (2015). [CrossRef]
11. S. S. Wang, R. Magnusson, J. S. Bagby, and M. G. Moharam, “Guided-mode resonances in planar dielectric-layer diffraction gratings,” J. Opt. Soc. Am. A 7, 1470 (1990). [CrossRef]
12. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).
13. R. Magnusson, “Wideband reflectors with zero-contrast gratings,” Opt. Lett. 39, 4337–4340 (2014). [CrossRef] [PubMed]
14. S. Collin, “Nanostructure arrays in free-space: optical properties and applications,” Reports Prog. Phys. 77, 126402 (2014). [CrossRef]
15. S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A. 20, 569–572 (2003). [CrossRef]
16. A. Polman and H. A. Atwater, “Photonic design principles for ultrahigh-efficiency photovoltaics,” Nat. Mater. 11, 174–177 (2012). [CrossRef] [PubMed]
17. L. C. Hirst and N. J. Ekins-Daukes, “Fundamental losses in solar cells,” Prog. Photovoltaics Res. Appl. 19(3), 286–293 (2010).
18. P. Löper, B. Niesen, S.-J. Moon, D. Nicolas, J. Holovsky, Z. Remes, M. Ledinsky, F.-j. Haug, J.-H. Yum, S. D. Wolf, and C. Ballif, “Organic-inorganic halide perovskites: Perspectives for silicon-based tandem solar cells,” IEEE J. Photovoltaics 4, 1545–1551 (2014). [CrossRef]
19. C. D. Bailie, M. G. Christoforo, J. P. Mailoa, A. R. Bowring, E. L. Unger, W. H. Nguyen, J. Burschka, N. Pellet, J. Z. Lee, M. Grätzel, R. Noufi, T. Buonassisi, A. Salleo, and M. D. McGehee, “Semi-transparent perovskite solar cells for tandems with silicon and CIGS,” Energy Environ. Sci. 8, 956–963 (2015). [CrossRef]
20. T. P. White, N. N. Lal, and K. R. Catchpole, “Tandem solar cells based on high-efficiency c-Si bottom cells: Top cell requirements for $¿ 30%$ efficiency,” IEEE J. Photovoltaics 4, 208–214 (2014). [CrossRef]
21. H. J. Snaith, “Perovskites: The emergence of a new era for low-cost, high-efficiency solar cells,” J. Phys. Chem. Lett 4, 3623–3630 (2013). [CrossRef]
22. M. Liu, M. B. Johnston, and H. J. Snaith, “Efficient planar heterojunction perovskite solar cells by vapour deposition,” Nature 501, 395–398 (2013). [CrossRef] [PubMed]
23. B. C. P. Sturmberg, K. B. Dossou, F. J. Lawrence, C. G. Poulton, R. C. McPhedran, C. M. de Sterke, and L. C. Botten, “EMUstack: an open source route to insightful electromagnetic computation via the Bloch mode scattering matrix method,” Comp. Phys. Comm., submitted.
24. “EMUstack: An open-source package for Bloch mode based calculations of scattering matrices,” physics.usyd.edu.au/emustack.
25. K. B. Dossou, L. C. Botten, A. A. Asatryan, B. C. P. Sturmberg, M. A. Byrne, C. G. Poulton, R. C. McPhedran, and C. M. de Sterke, “Modal formulation for diffraction by absorbing photonic crystal slabs,” J. Opt. Soc. Am. A 29, 817–831 (2012). [CrossRef]
26. K. R. Catchpole and M. A. Green, “A conceptual model of light coupling by pillar diffraction gratings,” J. Appl. Phys. 101, 063105 (2007). [CrossRef]
27. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 2005).
28. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge University Press, 1996). [CrossRef]
29. L. C. Botten, R. C. McPhedran, and J. M. Lamarre, “Inductive grids in the resonant region: Theory and experiment,” Int. J. Infrared Millimeter Waves 6, 949 (1985). [CrossRef]
30. B. S. Richards, “Single-material TiO2 double-layer antireflection coatings,” Sol. Energy Mater. Sol. Cells 79, 369–390 (2003). [CrossRef]
31. I.-S. Chung, “Study on differences between high contrast grating reflectors for TM and TE polarizations and their impact on VCSEL designs,” Opt. Express 23, 16730 (2015). [CrossRef] [PubMed]
