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We show by theory and experiment that silicon-dioxide-cladding gratings for Fabry-Pérot cavities on silicon-on-insulator channel (“wire”) waveguides provide a low-refractive-index perturbation, which is required for several important integrated photonics components. The underlying refractive index perturbation of these gratings is significantly weaker than that of analogous silicon gratings, leading to finer control of the coupling coefficient κ. Our Fabry-Pérot cavities are designed using the transfer-matrix method (TMM) in conjunction with the finite element method (FEM) for calculating the effective index of each waveguide section. Device parameters such as coupling coefficient, κ, Bragg mirror stop band, Bragg mirror reflectivity, and quality factor Q are examined via TMM modeling. Devices are fabricated with representative values of distributed Bragg reflector lengths, cavity lengths, and propagation losses. The measured transmission spectra show excellent agreement with the FEM/TMM calculations.
© 2011 Optical Society of America
1. Introduction
Integrated Fabry-Pérot (FP) resonator cavities have been utilized for a number of applications in silicon (Si) photonics including bonded III–V laser cavities [1, 2], optical filters [3], multiplexing and demultiplexing [4], optical switching [5, 6], and modulators [7, 8]. It is well known that such a cavity can be formed by introducing a defect (cavity), with a length equal to an integer number of quarter-waves, at the center of a distributed Bragg reflector (DBR). Silicon waveguide (SiWG) FP cavities for the silicon-on-insulator (SOI) platform have been researched extensively for high index contrast air/Si gratings on Si wires and rib/ridge structures [1,2, 4–14]. While these high-index-contrast gratings have the advantage of creating a small-footprint device, they make fine tuning of the grating coupling challenging due to their extreme sensitivity to fabrication tolerance. Conversely, gratings with weaker index perturbations allow for finer control of the grating parameters. Such precise grating control is of particular importance for narrow bandwidth DBR reflection filters, where the bandwidth of the filter depends strongly on effective index difference between the two grating sections. Similarly, precise control of the coupling coefficient is important for accurately tuning the Q of a SiWG FP cavity, which is commonly used as a transmission filter or as the laser cavity for a bonded III–V heterostructure gain medium. An additional and equally important advantage of weak-index-perturbation gratings is that they are known to allow longer cavity lengths, which, in turn, reduce the local device thermal load and, hence, enable a higher operating laser power [1]. Low index perturbation DBR’s on large-mode rib-ridge SiWG’s have been explored [15–18], including for the case of silicon-dioxide-(SiO2)-cladding gratings on a large-mode rib/ridge SiWG [3]. Similarly, one-step weak-to-moderate-index-perturbation gratings [19, 20] and FP cavities [21] have been fabricated with vertical air/Si corrugations.
In this paper, we show that the use of a low-refractive-index cladding on a highly confined channel (“wire”) SiWG [22, 23] enables the design of a high-Q integrated FP cavity, and that this materials choice enables precise tuning of the cavity parameters with standard e-beam fabrication methods. The paper is organized as follows: we begin with a short description of the basic device structure and its requirement, followed by a discussion of the theoretical approaches needed for its design. The theoretical performance of such SiWG FP cavities, simulated with the finite element method (FEM) in conjunction with the transfer-matrix method (TMM), is also presented. This modeling approach allows for the fast and accurate design of SiWG FP cavities, as well as an in-depth understanding of the influence of the main parameters characterizing the device on its optical response. The fabrication and testing of the device is described, and the experimentally measured transmission spectra are fit with the TMM model, showing excellent agreement between experiment and theory.
2. Device structure and motivation
Figure 1(a) shows a sketch of the integrated SiWG FP investigated here, with an SEM image of a fabricated device shown in Fig. 1(b). Notice that the device is formed by etching gratings into the SiO2 cladding on a highly confined channel (“wire”) SiWG, for the purpose of providing a weak index perturbation. Similar results are achievable with dielectric cladding gratings other than air/SiO2, such as SiN/SiO2, or SiOxNy/SiO2, which could potentially be employed for compatibility with a dielectric cladding layer over the entire chip. Previous demonstrations of similar devices have focused on patterning the SiWG itself to form a cavity, with the index contrast of the grating being between air and Si, thus resulting in a very large index perturbation. A comparison of the index perturbation caused by a SiO2 grating [Fig. 1(inset)] as opposed to a Si etched grating [Fig. 1(inset)] is shown in Fig. 1(c), where it can be seen that the coupling coefficient κ, defined by Eq. (1) below, is more than an order of magnitude larger for the Si grating than the SiO2 grating. It can clearly be seen that the SiO2 overlayer device geometry allows for a finer control of κ, which can be used to precisely tune the reflectivity of the DBR’s as well as the quality factor Q of the resonator. An important issue in such a low-index-contrast device is the achievable Q for the device and how well it is matched to the potential applications. Our results demonstrate that fabricated devices, shown in Fig. 1(b), have measured Q’s of ≈ 2,000 and calculations, see below, show that this number can be increased to ≈ 160,000 with a longer device geometry. These Q values are sufficient for applications in filter design for WDM systems [24], switching [6], modulation [7, 8], and hybrid laser cavities [1].
Fig. 1 (a) Schematic of the SiWG FP cavity (b) Tilted side view SEM image of a SiWG FP with Lc = 2a. (c) Comparison of κ for a SiO2-cladding grating and a Si etched grating with increasing h, plotted on a logarithmic scale. For a Si etched grating with h > 20 nm the fundamental quasi-TM (QTM) mode is no longer supported, thus κ is not calculated for this polarization. (inset) Cross-sectional schematic of a SiO2-cladding grating and a Si waveguide etched grating.
3. Modeling method
Cross-sectional schematics of both SiO2 and Si grating types are shown in Fig. 1(inset), with the coupling coefficient κ, calculated using Eq. (1), shown in Fig. 1(c). While Eq. (1), underestimates the grating coupling strength for strong perturbations, it can still be seen that the quasi-TE (QTE) mode experiences greater than one order of magnitude larger κ with the Si etched grating as compared to the SiO2 grating. Fig. 1(c) also shows that the coupling coefficient of the quasi-TM (QTM) mode is larger than in the case of the QTE mode. Our calculations suggest that this result is due to an increased overlap between the optical mode and grating (see also Fig. 2). Here κ has been calculated using first-order coupled-mode theory, with κ defined as follows [25]:
where states |1〉, |2〉 represent the modes of the waveguide in the unperturbed and perturbed region, respectively. The field E1,2 represents the vector time-harmonic electric-field components of the optical mode in each waveguide section, which have been calculated using the FEM (as described in the following section) and have been normalized to 1 W of power propagating in the z direction. The overlap integral is performed over the area of the grating, which has dimensions of w = 520 nm in the x direction and h = 10–100 nm in the y direction. The index of the perturbation nperturb is equal to noxide(λ) for SiO2 gratings, and nSi(λ) for Si gratings, where the material dispersions of Si and SiO2 have been modeled via the Sellmeier equation with coefficients taken from [26]. Although this approach provides a strong motivation for utilizing SiO2 gratings in applications where weaker perturbations are desired, it is well known that coupled-mode theory underestimates the grating coupling for situations where the grating strength is not weak, as is the case with our fabricated devices.Fig. 2 Modal profiles at λ = 1.55 μm for (a) QTE mode with cladding, (b) QTE mode without cladding, (c) QTM mode with cladding and (d) QTM mode without cladding, where Δneff,QTE = 0.015, Δneff,QTM = 0.035. (e) Δneff (λ) with h = 90 nm for both QTE and QTM modes.
Thus to provide an accurate design of the SiWG FP cavity, the TMM is employed to calculate the transmission characteristics of the structure. The waveguide is divided into two sections, labeled section a for a waveguide segment with the oxide cladding and section b for a waveguide segment with air cladding. The effective index neff (λ) of the waveguide mode is calculated for each section using the FEM. More specifically, neff (λ) is determined from the relation neff (λ) = β/k, where β is the mode propagation constant, found numerically by using the FEM, and k is the wave vector in free space. The FEM is also used to calculate the optical modes, whose spatial profiles are shown in Fig. 2(a)–2(d). The length of each waveguide section is chosen such that a = λ0/(4neff,a), b = λ0/(4neff,b) and Λ = a+b, which satisfies the Bragg condition for opening a spectral gap at the center wavelength, λ0. In the TMM calculations the two waveguide sections are treated as homogeneous media with the calculated neff values. Similar approaches have been discussed in the literature [27–30].
3.1. Finite element method
The neff (λ) of each waveguide mode is determined using the commercially available RSoft FemSIM software package [31], with material dispersion taken into account as described in the previous section. The waveguide dimensions in the simulation cell are 520 nm × 220 nm, with h = 90 nm for waveguide section a. Grid spacing of 20 nm in the x direction and 10 nm in the y direction are used, and are found to converge with an neff accuracy of 10−4. The FEM calculated modal profiles at λ = 1.55 μm for each waveguide section are shown for both polarizations in Fig. 2(a)–2(d), with the outline of the waveguide cross-section shown in white. The effective index difference between the modes of both grating sections, calculated as a function of wavelength, Δneff (λ), is shown for each polarization in Fig. 2(e), where Δneff has been calculated in 5 nm steps of λ. The FEM calculated neff (λ) is fit with a 7th order polynomial function for the TMM calculations presented in the following section. Our calculations show that at lower wavelengths the effective index difference is larger in the case of the QTM mode as compared to the QTE mode, whereas the opposite holds at longer wavelengths.
3.2. Transfer-Matrix method
To obtain transmission and reflection characteristics of the FP structure, a transfer matrix is defined to relate the forward and backwards propagating field amplitudes in each waveguide section with length equal to the period Λ:
where M is the transfer matrix of the unit cell of the grating and and are the forward and backward traveling incoming and outgoing complex wave amplitudes, respectively. Depending on the polarization, the field amplitudes in Eq. (2) represent either the electric field (QTE case) or the magnetic field (QTM case). The total transfer matrix, Mt, which describes the relationship between the incoming and outgoing waves in the system, is found by cascading the individual transfer matrices for each section in the FP structure:Here, A,B,C,D, represent the elements of the transfer matrix for one period of the DBR structure, which must be raised to the total number of grating periods N1, N2, in each DBR. The cavity adds an optical phase of kamLc (m = 0, 1, where 0 indicates the QTE mode and 1 indicates the QTM mode) for both forward and backward traveling waves as indicated by the middle matrix. For a lossless system and due to time reversal symmetry, allowing for the transfer matrices of both polarizations to be defined completely by the following elements [25]:
The use of the above matrix elements can be extended to include modest losses by substituting klm = βlm – iα/2 (for l = a,b) which are the complex propagation constants for the modes in each waveguide section (note that time reversal symmetry no longer holds once this substitution is made). Using these definitions the power transmission and reflection spectra for a given geometry can be calculated by solving for the elements of Mt(λ) using equations (3)–(7). The power transmission and reflection coefficients are T = (1/M11)2 and R = (M21/M11)2, respectively, where Mjk indicates the element of the matrix Mt.
Using the TMM, the DBR reflectivity R with increasing N for h = 90 nm, λ0 = 1.55 μm, and α = 3 dB/cm (α being the modal loss coefficient, as defined previously) is calculated as shown in Fig. 3(a). The QTM mode has a higher reflectivity as compared to QTE mode with the same number of gratings due to the larger field overlap with the grating for the QTM mode, as seen in Fig. 2(a)–2(d). The spectral width of the DBR stop band, Δλgap, as a function of h is also calculated, as shown in Fig. 3(b). The results of this calculation are plotted alongside those obtained by using an analytical expression [25]:
Fig. 3 (a) DBR reflectivity for the QTE and QTM modes as a function of number of grating periods N1 = N2 = N, with h = 90 nm, λ0 = 1.55 μm, and α = 3 dB/cm, (b) Spectral width of the DBR stop band Δλgap as a function of h, calculated with TMM and with the analytical expression (8), (inset) SEM micrograph of a fabricated DBR.
From Fig. 3(b) it is clear that reflection filters can be designed with bandwidths ranging from < 5 nm to > 20 nm by utilizing this type of DBR grating without the defect cavity. With the inclusion of a cavity extremely narrow linewidth transmission filters can be designed. For a short device with a total length of ≈ 80 μm and the waveguide geometry presented previously (α = 3 dB/cm, Lc = 2a, h = 90 nm, λ0 = 1.55 μm), Q = λ0/Δλ is calculated to be 2,580 for the QTM mode with N1 = N2 = 90. For a ≈ 155 μm long device, Q is increased to 161,500 for the QTM mode with N1 = N2 = 178, however, the peak transmission will fall to ≈ 50%. An innate trade-off exists between Q and the peak transmission of the cavity; higher Q requires larger N1, N2, which reduces the peak transmission in the presence of realistic propagation losses. However, higher Q could potentially be achieved without a transmission penalty by extending Lc, or by employing grating tapering for Bloch mode matching [32, 33].
4. Fabrication and measurement
The basic SiWGs are fabricated using deep-UV-stepper lithography on the CMOS line at MIT Lincoln Laboratory as described in [34, 35]; these waveguides have been characterized to have a propagation loss of 2–3 dB/cm. To validate our design method for the case of lossy cavities, gratings have also been fabricated on waveguides with 100 dB/cm propagation loss. The increased propagation loss is induced by implanting the waveguides with Si+ ions, and is mainly due to increased material absorption via sub-bandgap defect states [34, 35]. The ability to accurately model cavities with large loss due to absorption is essential for the accurate design of compact modulators [7, 8] and detectors.
Several post-processing steps are performed at the Center for Functional Nanomaterials at Brookhaven National Laboratory to make the SiO2 gratings. The 90 nm SiO2 hardmask, which is left atop the SiWG as a consequence of the etch process used to define the waveguide, is etched to create the oxide gratings. The gratings are formed by patterning the SiO2 hardmask using a 30 keV scanning electron microscope with ZEP-520A resist and a subsequent reactive-ion etch using CH4/CHF3/Ar chemistry. A side view of the fabricated device is shown in Fig. 1(b), and a tilted view of the DBR is shown in Fig. 3(inset).
Transmission measurements are performed using a tunable C-band CW laser source, which is chopped at a frequency of 1 kHz. Light is coupled on- and off-chip using single-mode lensed tapered fibers with an in-line polarization rotator to control the input polarization. Transmitted light is measured with a cooled InGaAs photodiode using lock-in detection with the chopped input signal. The measured transmission spectra are normalized by the transmission spectrum of an unpatterned waveguide with the same length and similar loss characteristics to the waveguide under test.
Four different grating structures have been fabricated to test the validity of our design method; the parameters of which are shown in Table 1. All fabricated devices are designed for the QTM mode at λ0 = 1.55 μm with N1 = N2, a = 216 nm, b = 220 nm, and Λ = 436 nm. Each device was fabricated on a separate chip, with the electron beam lithography and SiO2 etch being performed independently, hence the fabrication errors for each chip differ slightly. The measured transmission spectra from these devices are fit with TMM by adjusting the neff of each stack such that the difference between the two curves is minimized in the least squares sense. To accomplish this fitting, a wavelength independent effective index change is added to the FEM calculated neffa,b(λ) of both waveguide sections, denoted Δneff,a,b. These adjustments are necessary to take into account various sources of non-ideality in the waveguide fabrication process, including: 1) deviation in the waveguide design width due to inherent variations in the fabrication process; 2) deviation in the waveguide design height due to variations in the SOI device layer; and 3) deviations in the quarter-wave stack lengths due to over/under dosing from the electron beam. The fitting adjustments Δneff,a,b are also given in Table 1. For very small changes in waveguide width and height the corresponding change Δneff can be approximated as linear, and can be calculated using FEM. It is found that Δneff changes by 2 × 10−3 nm−1 with waveguide width and by 3.1 × 10−3 nm−1 with waveguide height. The quoted variation in the SOI device layer thickness from the manufacturer[36] is ± 12.5 nm, which corresponds to a worst case Δneff = ±0.025. Considering the variation in device layer thickness, along with other fabrication errors, adjustments of less than 0.056 to the effective indices are reasonable.
Table 1. Device parameters for fabricated devices. Effective indices of each quarter-wave stack section are adjusted to fit TMM transmission calculations to the experimentally measured data, given as Δneff,a and Δneff,b.
The measured and TMM fit spectra for Cases 1–4 are shown in Fig. 4(a)–4(d), while the measured and TMM calculated values for λ0, Q, and extinction are compared in Table 2. Device characteristics are extracted from the measured transmission data by using cubic spline interpolation to fit a smooth curve. For Case 1, where the measured wavelength of ≈ 1.574 μm and design center wavelength of 1.55 μm show the largest discrepancy, the effective indices are adjusted by Δneff,a = .0526 and Δneff,b = .0558. Case 2 exhibits an increased extinction with a narrower Δλgap, as compared to the other cases due to the increased N1 = N2 = 90. This result clearly illustrates the trade-off between these two quantities, which can be tuned via R(N). The Q of Case 2 is also substantially higher than the other three cases, owing to longer mirror lengths and lower propagation loss than Case 3 and Case 4. Case 3 shows a device where the performance was non-ideal due to an imperfect oxide etch, yet the TMM fit is still able to predict λ0 within 0.3 nm, the extinction within ≈ 36%, and all other parameters with a similar margin of error to the other cases. In Case 4 the device was designed with a cavity, which was sufficiently long to reduce the free spectral range such that multiple transmission peaks fit within Δλgap. An odd number of quarter-wave stacks were used in the cavity construction, hence causing λ0 to be placed in between the two transmission resonances rather than directly on a resonance. It should be also noted that, as expected, in the cases with larger optical absorption (Case 3 and Case 4) the corresponding resonant transmission is smaller. The TMM fit has been shown to accurately model the fabricated device characteristics for SiWG FP structures with different N1, N2, α, and Lc. With refinements in the fabrication process, it is clear that the FEM/TMM can act as a fast and accurate tool for designing SiO2-cladding gratings for SiWG cavities and DBR filters.
Fig. 4 Experimentally measured transmission spectra with TMM calculated curve fits for (a) Case 1: Lc = 2a, α = 3 dB/cm, N1 = N2 = 50, (b) Case 2: Lc = 2a, α = 3 dB/cm, N1 = N2 = 90, (c) Case 3: Lc = 2a, α = 100 dB/cm, N1 = N2 = 52, (d) Case 4: Lc = 123a, α = 100 dB/cm, N1 = N2 = 52.
Table 2. Comparison of measured and theoretical device characteristics. Experimental data is fit with a cubic spline interpolation, TMM calculations are performed with neff,a and neff,b adjusted such that the calculated transmission spectrum fits the experimental data in the least squares sense.
5. Conclusion
The design and fabrication of a weak-perturbation SiO2-cladding grating SiWG FP has been presented. A fast and accurate modeling approach for the design of this structure has been demonstrated, and the device parameter space for cavity Q, DBR R, and DBR Δλgap have been calculated. We have fabricated four different device geometries using CMOS-compatible processes, and Q’s of ≈ 2,000 have been shown experimentally. Our model shows excellent agreement with the measured transmission spectrum; the only fitting parameters being adjustments to the calculated effective indices Δneff,a,b. Our calculations show that κ can be tuned over a wide range of values by changing h, providing finer control of the DBR R as compared to analogous Si gratings. The versatility, ease of design, and tunability of these SiO2-cladding gratings provides an important building block for optical filters, modulators, and laser cavities in the SOI platform.
Acknowledgments
The authors thank Michael W. Geis and Steven J. Spector at MIT Lincoln Laboratory for waveguides and fabrication advice. This research was funded by National Science Foundation (NSF) Grant DMR-08-06682 and Engineering and Physical Sciences Research Council (EPSRC), and was carried out in part at the Center for Functional Nanomaterials, Brookhaven National Laboratory, which is supported by the U.S. Department of Energy, Office of Basic Energy Science, under Contract No. DE-AC02-98CH10886.
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