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We investigate the dispersion properties of leaky-mode resonance elements with emphasis on slow-light applications. Using particle swarm optimization, we design three exemplary bandpass leaky-mode devices. A single-layer silicon-on-insulator leaky-mode element shows a time-delay peak of ~8 ps at the resonance wavelength. A double membrane element exhibits an average delay of ~6 ps over ~0.75 nm spectral bandwidth with a relatively flat dispersion response. By cascading five double-membrane elements, we achieve an accumulative delay of ~30 ps with a very flat dispersion response over ~0.5 nm bandwidth. Thus, we show that delay elements based on leaky-mode resonance, by proper design, exhibit large amount of delay yet very flat dispersion over appreciable spectral bandwidths, making them potential candidates for optical buffers, delay lines, and switches.
©2010 Optical Society of America
1. Introduction and background
Materials that are artificially structured on a nanoscale exhibit electronic and photonic properties that differ dramatically from those of the corresponding bulk entity. In particular, subwavelength photonic lattices are of immense interest owing to their applicability in numerous optical systems and devices including communications, medicine, and laser technology [1–3]. When the lattice is confined to a layer, thereby forming a periodic waveguide, an incident optical wave may undergo a guided-mode resonance (GMR) on coupling to a leaky eigenmode of the layer system. The external spectral signatures can have complex shapes with high efficiency in both reflection and transmission [4–10]. It has been shown that subwavelength periodic leaky-mode waveguide films with one-dimensional periodicity provide diverse spectral characteristics such that even single-layer elements can function as narrow-line bandpass filters, polarized wideband reflectors, wideband polarizers, polarization-independent elements, and wideband antireflection films [11-12]. The spectra can be further engineered with additional layers [13]. The relevant physical properties of these elements can be explained in terms of the structure of the second (leaky) photonic stopband and its relation to the symmetry of the periodic profile. The interaction dynamics of the leaky modes at resonance contribute to sculpting the spectral bands. The leaky-mode spectral placement, their spectral density, and their levels of interaction strongly affect device operation and functionality [11]. There has been a considerable amount of research performed on the spectral attributes of these elements. In contrast, little has been done to understand their dispersive properties and related slow-light applications.
Optical delay lines have important roles in communication systems and in radio-frequency (RF) photonics. They are common in optical time-division multiplexed communication systems for synchronization and buffering and in RF phased arrays for beam steering [14]. These delay lines are implemented, for example, as free-space links, fiber-based links, fiber-Bragg gratings, and ring resonators. The delay properties are based on the phase response of the medium or filter in which the delay line is implemented as discussed in detail by Lenz et al. in [14]. Many examples of practical delay systems are given in [15–18] using all-pass optical filters.
Advances in nanofabrication and nanolithography are placing the long-awaited photonic integrated circuit in view as an attainable goal. The all-optical networks of the future will bypass optical-to-electrical converters, thereby eliminating associated noise and considerably reducing the attendant bit error rates. As discussed by Parra and Lowell, all-optical processing requires slow-light-enabled synchronizers, buffers, switches, and multiplexers [19]. The technology needed to generate these functions for mass deployment is not available today, but there is a considerable amount of research being devoted to develop it. Many means are currently used to realize these valuable slow-light applications in the laboratory. This includes stimulated (Brillouin, Raman) scattering in fibers, semiconductor optical amplifiers, 2D photonic crystals, atomic vapors, and high-finesse ring resonators that many groups are pursuing [19].
In this paper, we introduce a new concept for slow-light applications. We envision compact chips that contain leaky-mode resonance elements capable of buffering light across considerable bandwidths. Our concept can be viewed as a counterpart to other small-footprint devices such as 2D photonic-crystal (PhC)-based micro-resonator chips that are receiving much attention [20, 21]. These coupled-resonator optical waveguides (CROW) may contain hundreds of high-Q cavities within a PhC lattice. These cavities, formed by slightly offset lattice holes, are perhaps ~2000 nm in diameter. Experimentally, the structure can attain group velocity below 0.01c and long group delays as shown recently by Notomi et al. [20].
Whereas the amplitude-based spectral response of leaky-mode elements has been studied extensively, their phase response has received less attention. We briefly review the relevant papers published so far; most of these apply the usual reflection-type resonance. Schreier et al. treated a sinusoidally modulated waveguide grating at oblique incidence, computing the phase variation of the reflectance near resonance relative to modulation strength [22]. They quantified the degree to which the structural parameters control the amount of delay achievable with computed values of delay ranging from sub-ps to ~40 ps depending on conditions [22]. Using a finite-difference time-domain computational approach, Mirotznik et al. [23] evaluated the temporal response of a subwavelength dielectric grating that we designed previously as a reflection-type GMR element [24]. The model input pulse was Gaussian with center wavelength of 510 nm, spectral width of 5000 nm, and temporal pulse width of ~5 fs. They noted that the reflected energy persisted for ~1 ps after the incident field decayed [23]. Later, Suh et al. designed a 2D photonic-crystal-slab-type GMR transmission filter computing the resonance amplitude, transmission spectrum, and group delay. For a 1.2 μm thick slab, a peak delay of about 10 ps was obtained at 1550 nm; the spectral width of the response was ~0.8 nm [25]. Nakagawa et al. presented a method to model ultra-short optical pulse propagation in periodic structures, based on the combination of Fourier spectrum decomposition and rigorous coupled-wave analysis (RCWA) [26]. They simulated an incident pulse (167 fs) on a resonant grating supporting two modes and found that two pulses were transmitted with shape similar to the excitation pulse shape. Vallius et al. modeled spatial and temporal pulse deformations generated by GMR filters. They illuminated the structure with a Gaussian temporal pulse of 2 ps duration and 633 nm wavelength. Lateral spread and temporal decompression were observed in the reflected and transmitted pulses [27]. As the spectrum of the pulse was not well accommodated by the GMR element, the reflection efficiency of the pulse was relatively low.
In this paper, we aim for results that reach considerably beyond these initial works. We show that GMR elements can be designed to provide a phase response suitable for slow-light applications. We provide examples of resonance structures operating in transmission as cascaded transmissive elements yield more compact geometry than reflective elements. We emphasize attainment of considerable time delay and flat dispersion. For ease of fabrication, we focus on elements that are locally one-dimensional (1D) although they may be contained within 2D photonic slabs as shown by examples.
2. Computational basics
In this paper, the examples presented consist of structures with one-dimensional (1D) binary modulation. For simplicity, it is assumed that the periodic layers are transversely infinite and the materials are lossless. The spectra and phase are calculated with computer codes based on rigorous coupled-wave analysis (RCWA) of wave propagation in periodic media [28, 29]. We also use RCWA to compute the time response; we summarize the method as follows. A transform-limited TE-polarized Gaussian pulse is represented as
where E0 is the amplitude of the pulse; T is the temporal pulse width (T =Δτ(2ln2)-1/ 2; Δτ is the full width at half maximum (FWHM) of |Еy(t)|2); t0 is the pulse-peak offset; ω0 = 2πc/λ0 is the central angular frequency and c and λ0 are the speed of light and the wavelength in vacuum, respectively. To use RCWA for analysis, the incident Gaussian pulse is decomposed into its monochromatic Fourier components (plane waves), which is performed by the Fourier transformation and discretization. These discrete monochromatic components are then treated independently by our established RCWA analysis technique, which at a given incident angle provides the complex reflection coefficients R(ωn) (or R(λn)) and complex transmission coefficients T(ωn) (or T(λn)) of each diffraction order. In addition, the independent analysis of each monochromatic component can facilitate the inclusion of material dispersion effects. The reflected pulse ЕR(ωn) and transmitted pulse ЕT(ωn) in the frequency domain for a specific diffraction order are thus given byTo obtain the time domain representation of the reflected and transmitted pulses, an inverse Fourier transform is performed. Since the frequency domain representation of the fields is discrete and finite, a Riemann sum can take the place of the integral in the inverse Fourier transformation. In other words, the reflected and transmitted fields can be obtained by superimposing the resulting spectral components from Eqs. (2) and (3), assuming that the Fourier kernel is included in the expression for the fields ЕR(t; ωn) and ЕT(t; ωn).
Figure 1 clarifies the computational method. Utilizing this technique, we find the output pulse shape and its delay with respect to the input pulse over a wide range of pulse widths (~several fs to hundreds of ps). The time delay (τ) and delay dispersion (D) are calculated by
where φ is the wavelength (λ)-dependent phase in reflection or transmission [2,14].
Fig. 1 Flow chart of the computational procedure utilized to obtain the output pulse shapes in wavelength and time domains. FFT: Fast Fourier Transform, RCWA: Rigorous Coupled-Wave Analysis, and IFFT: Inverse Fast Fourier Transform.
3. Leaky-mode resonance dispersive device examples
As a first example, we provide a silicon-on-insulator (SOI) GMR transmission filter with 0.5 nm spectral width and minimal sidelobes. This filter is designed using the particle swarm optimization (PSO) technique [30, 31]. This device is illustrated in Fig. 2 with parameters Λ = 979 nm, d = 465 nm, and a period that is divided into four parts with fill factors [F1, F2, F3, F4] = [0.071, 0.275, 0.399, 0.255]. Also, nH = 3.48, nS = 1.48, and nL = ninc = 1.0 (air). Figure 3(a) shows the transmittance, phase response, delay, and dispersion of this filter under normal incidence with TE polarization. This filter provides delays as high as ~10 ps at the transmission resonance; however, the dispersion width is narrow and zero dispersion is obtainable only near 1524.5 nm. Figures 3(b) and (c) display the response of this filter to excitation with a pulse in the spectral (wavelength) and time domains, respectively. The input pulse has a width of 30 ps (FWHM) in time. The output pulse experiences a delay of ~8.25 ps with respect to the input pulse. It has reduced amplitude on account of the incomplete transmission and limited passband noted in Fig. 3(b).
Fig. 2 A schematic view of a subwavelength guided-mode resonance element under normal incidence. A single layer with thickness d, fill factors Fi, and a multi-part period Λ is shown. I, R, and T denote the incident wave, reflectance, and transmittance, respectively. TE polarized light has its electric field vector normal to the plane of incidence.
Fig. 3 (a) Transmittance, phase, delay, and dispersion of a 0.25 nm-wide (FWHM) SOI GMR transmission filter. Λ=979 nm, d = 465 nm, and [F1, F2, F3, F4] = [0.071, 0.275, 0.399, 0.255]. (b), (c) Response of this filter to excitation with a pulse in wavelength and time domains, respectively.
As a second example, we use two free-standing membranes with an air gap (cavity) between them to realize a ~0.75 nm (FWHM) flat-top transmission band as shown in Fig. 4(a) . Again, this structure is designed by PSO and its structural parameters are: Λ = 1103.9 nm, d = 432.2 nm, [F1, F2, F3, F4] = [0.0626, 0.3013, 0.4576, 0.1785], and dCavity = 2000 nm. Figure 4(b) illustrates the transmittance, phase, delay, and dispersion of this device. This element shows a flat-top transmission bandwidth, which actually is a result of merging two adjacent narrow transmission resonances. In addition, the delay response exhibits an average of ~7 ps in the transmission band. In comparison to the previous case, the dispersion is flatter. Figures 5(a) and (b) show the pulse response of this filter in wavelength and time domains, respectively. The input pulse has a full-width half-maximum (FWHM) of 20 ps in time and spectrally fits well inside the transmission bandwidth of the filter. The input pulse is delayed by ~6.1 ps by being transmitted through this filter in good agreement with the delay in Fig. 4(b).
Fig. 4 (a) Structure of a single-cavity GMR transmission filter. (b) Transmittance, phase, delay, and dispersion of the filter. Λ=1103.9 nm, d = 432.2 nm, [F1, F2, F3, F4] = [0.0626, 0.3013, 0.4576, 0.1785], and dcavity= 2000 nm.
Fig. 5 Pulse response of the filter in Fig. 4. (a) Spectrum of the input pulse in relation to the filter spectrum. (b) Time domain response. FWHM of the input pulse is 20 ps.
By cascading the structure in Fig. 4, we can build a structure resembling a multi-cavity photonic crystal waveguide [20]. To illustrate, we cascade five GMR subunits (NCavity = 5) with spacing dB= 5.0 μm. Figure 6(a) shows the computed results. Although the high-transmission bandwidth is smaller than it is for the single-cavity structure, cascading the cells results in a flat delay response of ~30 ps over a ~0.5 nm wavelength band. Moreover, the flat low-dispersion response illustrates that such structures are promising for imposing constant (and almost dispersion-free) delays on optical pulses. Theoretically, this ~30 ps group delay for the ~34 μm long structure designed here corresponds to a group velocity of ~0.0038c. Figures 6(b) and (c) display the response of this filter to pulse excitation. The input pulse has a FWHM of 30 ps, and the output pulse preserves its shape with a delay of ~30 ps with respect to the input pulse. For comparison, Notomi et al. reported 75 ps delay with 60 cavities each being 2100 nm in diameter; the total structure length was 175 μm [20].
Fig. 6 (a) Transmittance, phase, delay, and dispersion of a five-cavity GMR transmission filter. Λ = 1103.9 nm, d = 432.2 nm, [F1, F2, F3, F4] = [0.0626, 0.3013, 0.4576, 0.1785], dCavity = 2000 nm, dB = 5000 nm, and NCavity = 5. Pulse response of this filter (b) in wavelength, and (c) in time domain. The output pulse experiences a ~30 ps delay with respect to the input pulse.
Figure 7 shows a conceptual implementation of GMR slow-light devices. If the device height dD is large, the device functions as a bulk element. Each sub-ridge layer in Fig. 7 is a GMR element like in Fig. 2. If we design the sublayer to operate as a bandpass filter, the input light will resonate transversely and be reradiated forward to the next GMR layer. This idea can also be implemented with a series of GMR filters on numerous substrates cascaded as a stack. The concept in Fig. 7 is convenient in that a large number of cascaded resonant units can be fabricated in a few steps by e-beam lithography (EBL) and deep reactive ion etching (DRIE), resulting in a compact system of resonant delay units. Certainly, the dimensions of the device and the input beam size should be specified with practical limitations in mind. On the other hand, if thickness dD is small, such as on the order of 100-300 nm, this can be a waveguide device. In that case, the functionality of the device employs waveguiding is a dual sense. First, there is the waveguide that guides light from one resonant layer to the next. For that to work, the structure requires a higher average refractive index than that of the surrounding media as usual. A membrane in air will satisfy this requirement with additional considerations if the device sits on a substrate. Second, each GMR cell forms a resonant waveguide, again similar to the one shown in Fig. 2. In principle, we can cascade a large number of these GMR cells to achieve a specified delay. Indeed, multiple-cell cascading is the basis for the new coupled-resonator optical waveguide (CROW) technology being developed [20] as noted above.
Fig. 7 Schematic view of a conceptual implementation of an example GMR slow-light device. Multiple resonant units can be cascaded to realize a specified delay. Only three subunits, each based on two transversely-resonant GMR elements, are shown. Here, dD is the device thickness, dC is the cavity length, and dB is the buffer length. Other parameters are defined in Fig. 2.
4. Conclusions
A new class of dispersive optical elements is proposed. It is founded on the phase response of resonant leaky-mode subwavelength periodic structures. Phase-related properties of such devices including time delay and dispersion are important in high-speed optical communications that utilize very short optical pulses for information transmission. With appropriate design, these devices exhibit a large amount of delay yet very low dispersion across considerable spectral widths, making them potential candidates for optical buffers, delay lines, and switches.
Acknowledgements
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. ECCS-0925774.
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