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We demonstrate spiral Bragg grating waveguides (BGWs) on the silicon-on-insulator (SOI) platform for the fundamental transverse magnetic (TM) mode. We also compare TM spiral waveguides to equivalent transverse electric (TE) spiral waveguides and show that the TM spiral waveguides have lower propagation losses. Our spiral waveguides are space-efficient, requiring only areas of 131×131 µm2 to accommodate 4 mm long BGWs, and, thus, are less susceptible to fabrication non-uniformities. Due to the lengths and reduced susceptibility to fabrication non-uniformities, we were able to obtain narrow bandwidth, large extinction ratio (ER) devices, as narrow as 0.09 nm and as large as 52 dB, respectively. Finally, we demonstrate a 4 mm long TM chirped spiral Bragg grating waveguide with a negative, average, group delay slope of −11 ps/nm.
© 2015 Optical Society of America
1. Introduction
There is an increasing need to develop components for processing transverse magnetic (TM) modes in waveguides fabricated on a silicon-on-insulator (SOI) platform. Numerous components can be made more compactly or to give better performance when designed for TM mode operation. For example, directional couplers can be made smaller due to the larger coupling coefficients achieved for TM modes [1]. Optical sensors utilizing the evanescent fields [2–5] of TM modes can have higher sensitivities, as compared to sensors using transverse electric (TE) modes. Given the increased packing efficiency and robustness possible using edge coupling to SOI chips, there is current interest in separating, and separately processing, the TE and TM modes that are coupled from standard fibers onto SOI chips using edge couplers [6]. To this end, existing TE and TM polarizers [7–9], polarization rotators (PRs) [10–14], polarization splitters (PSs) [15, 16], and polarization splitter-rotators (PSRs) [17–19] allow us to efficiently integrate TM devices with TE devices. For example, the low loss PR in Ref. [10] is able to convert the fundamental TE mode to the fundamental TM mode with an insertion loss below 0.5 dB, and converts the fundamental TM mode to the fundamental TE mode with an insertion loss below 1 dB, over a 200 nm wavelength range that includes the C and L bands. The relatively low insertion loss and large wavelength range allows TM devices to be efficiently integrated with TE devices.
In this work, we have found several advantages to using TM modes, as compared to TE modes. The TM modes show smoother spectral responses and lower propagation losses than TE modes in the same waveguides. Also, the coupling coefficients in Bragg gratings can be smaller for TM modes than for TE modes. As a result, uniform Bragg gratings can benefit by using TM modes to obtain high quality spectral responses, such as narrow bandwidths and large extinction ratios (ERs). Waveguides using TM modes are also suited for chirped Bragg gratings, since these gratings require small coupling coefficients and low propagation losses. Such TM chirped Bragg gratings can be designed to achieve linear group delays with negative slopes over broad passbands. A negative group delay slope can be used for optical dispersion compensation applications [20–23]. Additionally, integrated microwave photonics signal processing requires high quality, low propagation loss, and low coupling coefficient gratings [24] and, thus, can benefit by using TM modes.
Bragg grating waveguides (BGWs) on the SOI platform have been attracting interest for optical communication and sensing applications. Until now, most of the work has focused on BGWs designed for TE mode operation [25–27]. Typically, such devices have relatively large grating coupling coefficients, which means that they can be relatively short and have relatively large bandwidths. However, small coupling coefficients and long grating lengths are required to achieve narrower bandwidths (< 1 nm) and large ERs. There are several approaches for TE BGWs to achieve small coupling coefficients. For instance, one approach is to use small sidewall corrugations on the strip waveguide, i.e., below 10 nm [26]. Such small corrugations are challenging to manufacture as they are limited by the fabrication lithography. Another approach is to use rib waveguides with larger corrugations on the rib and/or the slab sidewalls. However, using rib waveguides requires two etch steps [28,29], which increases the fabrication cost. In contrast, TM mode Bragg gratings, implemented in typical 500 nm wide by 220 nm thick waveguides, allow for small coupling coefficients to be achieved for relatively large corrugations on strip waveguides due to the fact that the TM mode is less optically confined to the waveguide. Figures 1(a) and 1(b) illustrate the confinement of the fundamental TE and TM modes of a 500 nm×220 nm strip waveguide with SiO2 cladding, respectively. As mentioned above, such TM BGWs with small coupling coefficients require long grating lengths, on the order of millimeters, in order to achieve large ERs in their spectral responses. However, the main challenge in making long TM gratings on a straight silicon waveguide is the non-uniformity in the waveguide thickness [30, 31], which adversely affects the effective refractive indices over the length of the device. To reduce the effects of the waveguide thickness variations, we have made our long TM gratings into spirals.
Fig. 1. Electric field distributions of (a) the fundamental TE mode and (b) the fundamental TM mode in a 500 nm×220 nm strip waveguide with SiO2 cladding.
Spiral configurations have been shown to be very efficient for packing long waveguides in compact areas [32–36]. By wrapping a long grating into a spiral, the straight-line distance between any two points on the grating is minimized which, in turn, minimizes the waveguide thickness variation in the grating. Hence, the adverse effects caused by waveguide thickness variations can be significantly reduced using spirals. TE spiral BGWs (SBGWs) have been demonstrated on rib waveguides [34] and relatively wide (larger than 1000 nm) strip waveguides [35] that reduce the sensitivity to sidewall roughness as compared to a TE mode in a regular 500 nm×220 nm strip waveguide. It should be mentioned that the sidewall roughness not only increases scattering losses, but also results in Fabry-Perot effects which cause undesirable variations in the spectral responses [37, 38]. Instead of having to use rib waveguides or relatively wide strip waveguides to achieve these improvements, a TM mode propagating in a regular 500 nm×220 nm strip waveguide will have a similar reduction in the scattering losses and, therefore, reduced variations in the spectral responses as compared to a TE mode propagating in the same waveguide. These improvements are realized because the TM mode interacts mostly with the smooth top and bottom surfaces of the waveguide and not as much with the rough sidewalls [39]. Hence, by implementing TM Bragg gratings on spiral strip waveguides, the adverse effects of non-uniformity caused by both sidewall roughness and waveguide thickness variations can be reduced. Therefore, TM SBGWs are able to obtain high quality spectral responses and low propagation losses.
In this paper, we demonstrate uniform SBGWs (U-SBGWs) and chirped SBGWs (C-SBGWs) for TM mode operation using SOI strip waveguides. First, we present a method to wrap long BGWs into compact areas using spiral waveguide. Then, we analyze the refractive index variation as a function of the radius of curvature to justify modeling a spiral waveguide using a straight waveguide approximation. Such a straight waveguide approximation was applied to numerically simulate our TM U-SBGWs and C-SBGWs and to design the devices reported on herein. Then, the designed SBGWs were fabricated and their spectral responses were measured. We present the transmission spectra of TM U-SBGWs that had various corrugation widths and grating periods. In addition, we present the measurement results for a TM C-SBGW having a negative, average, group delay slope of −11 ps/nm.
2. Design
2.1. Spiral grating schematic
The goal is to wrap long Bragg gratings into compact areas using spiral waveguides. Figures 2(a) and 2(b) are, respectively, a scanning electron microscope (SEM) image and a schematic representation of an SBGW. There are two semi-circular waveguides forming an S-shaped waveguide in the center of the device. Another two interleaved Archimedean spirals are connected to the S-shaped waveguide. Both semi-circular waveguides have the same radius of curvature, R0. Each SBGW has two ports, Port A and Port B, see Fig. 2(b). Depending on the intended function of the grating, either reflected light or transmitted light can be emitted from either port.
Fig. 2. (a) SEM image and (b) schematic of a 4 mm long TM SBGW wrapped into an area of 131×131 µm2. (c) Zoom-in showing the grating period, Λ, the corrugation width, Wcorr = (Wmax-Wmin)/2, and the spacing between two spiral waveguides, g.
The complete spiral waveguide consists of the S-shaped waveguide and the two Archimedean spirals. We start our design by laying out the S-shaped waveguide. In Cartesian coordinates, the center of the waveguide for one of the semi-circular sections, see the red semi-circular section in Fig. 2(a), can be expressed as:
where R0 is designed to be 15 µm (see below). Since the two semi-circular sections are centrosymmetric with respect to the center of the spiral, the coordinates of the green semi-circular section in Fig. 2(b) can be obtained from those of the red semi-circular section by simply inverting the coordinates in Eq. (1).The two interleaved Archimedean spirals are also centrosymmetric with respect to the center of the spiral, as can be seen by the red and green Archimedean spirals in Fig. 2(b). The coordinates of the center of the waveguides for the red Archimedean spiral are given by:
where the parameter B controls the distance between the two Archimedean spirals through the expression B = (g +Wmax)/π, where g is the spacing between the spirals and Wmax is the maximum width of the grating, as shown in Fig. 2(c). The parameter Rc controls the starting point of the red Archimedean spiral, and is equal to 2R0. Due to the symmetry between the red and the green Archimedean spirals, the coordinates of the green spiral can be calculated by inverting the coordinates in Eq. (2).Once the layout of the centers of the waveguides for the complete SBGW was determined, the normal vectors were added to the inner and outer sides of the paths of the centers of the waveguides, which allowed us to calculate the exact positions of the sidewalls and/or corrugations for each waveguides. For SBGWs, the amplitude of each normal vector is determined by half of the grating width, i.e., Wmax/2 and Wmin/2, as shown in Fig. 2(c). Using this approach, we were able to place arbitrary Bragg gratings onto the spiral waveguides, by appropriately adjusting the grating periods and widths.
2.2. Spiral waveguide design
The main challenge encountered when designing a grating on a spiral waveguide is that the effective refractive index, neff, of the waveguide varies with the radius of curvature of the spiral. The radius of curvature of the spiral needs to be designed to reduce the neff variations. Using MODE Solutions by Lumerical Solutions, Inc., [40], we calculated the spiral waveguide index variation, δneff, for various radii of curvature, as compared to a straight waveguide. As can be seen from the blue curve in Fig. 3, δneff dramatically increases as the radius of curvature decreases. However, when the radius of curvature is greater than or equal to 15 µm, δneff is small and exhibits a low rate of change as the radius of curvature increases. Also, the reflection caused by the mode mismatch at the center of the S-shaped waveguide is calculated to be smaller than -28.4 dB using MODE Solutions. Hence, as long as R0 ≥15 µm, δneff and the mode mismatch reflection are small. neff can be approximated to be that of a straight waveguide. However, increasing R0 also reduces the packing efficiency of a spiral waveguide (see below) and, thus, increases the non-uniformity effects caused by the waveguide thickness variation. The packing efficiency, α, can be defined as , where L is the grating length and A is the area of the spiral waveguide [41]. As can be seen in Figs. 2(b) and 2(c), the spiral waveguide area is mainly affected by g and R0. In order to increase α, while preventing evanescent coupling of the optical modes between the two Archimedean spirals, g was chosen to be 2 µm. By considering a 4 mm long SBGW, the packing efficiencies for various R0 values are shown in Fig. 3. The goal is to keep δneff small while achieving a large packing efficiency. As can be seen from Fig. 3, this criterion is best satisfied for R0=15 µm.
Fig. 3. Simulated effective refractive index variations (blue curve) and calculated packing efficiencies (green curve) for spiral waveguides with R0s ranging from 5 µm to 60 µm.
For R0=15 µm, a small δneff of 0.78×10−3 was obtained. Theoretically, such a small δneff can be compensated for by an appropriate change of the grating period, Λ, based on [42]:
where the Λc is the local period with compensation, and the neffc and neff are the effective refractive indices of the curved waveguide and the straight waveguide, respectively. However, for a TM SBGW with a radius of curvature greater than 15 µm, the change of the designed grating period, Λ-Λc, is less than 0.27 nm. Such a small change in the grating period is below the fabrication resolution for a single period and, therefore, has been ignored in our design. Also, according to our calculations, using the transfer matrix method (TMM) [43], the simulated spectrum is not greatly affected by the small value of δneff, in fact, the δneff calculated here causes a bandwidth increase of only 0.02 nm. In other words, since R0=15 µm, we have no need to compensate for the small waveguide index variations caused by the increasing radius of curvature.2.3. Grating design and simulation
We have two types of TM SBGWs: U-SBGWs and C-SBGWs. To predict the device characteristics and performance, we have simulated the U-SBGWs using Finite-difference time-domain (FDTD) simulations [44, 45], and the C-SBGWs using the TMM. As our spiral waveguides have small waveguide index variations, as compared to straight waveguides, in our simulations, we used straight BGWs.
The TM U-SBGWs are realized by introducing periodic corrugations on both sidewalls of the spiral waveguide, see Fig. 2(c). Since a larger corrugation width, Wcorr, gives stronger coupling and a larger spectral bandwidth, Wcorr was varied from 40 nm to 160 nm for the U-SBGWs. The grating period was kept constant, which determined the center wavelength of the spectral response, λ0, according to the phase-match condition, λ0=2Λneff. The duty cycle of the grating was 50%. In order to achieve a center wavelength of 1550 nm, the period was designed to be 438 nm. To account for variations in the fabrication processes, we also varied the grating period by ±6 nm, i.e., included two extra gratings with periods of 432 nm and 444 nm. FDTD simulations were performed using Bloch boundary conditions, which allowed an infinitely long uniform Bragg grating to be simulated using only one unit cell [40]. In the simulations, Bloch boundaries were used to locate the band gap of the U-SBGWs, from which λ0 and the bandwidth, ∆λ, can be found [46]. Then, the coupling coefficient, κ, can be calculated using [46,47]:
where ng is the group index of a waveguide having a width, W, equal to the average width of our grating, i.e., having W = (Wmax +Wmin)/2.The C-SBGWs are chirped, such that in them Λ varies linearly along the length of the device. In principle, as Λ changes along the length, the wavelength of the reflected light also changes according to the phase-match condition. As stated previously, a C-SBGW can be designed to obtain negative dispersion. This can be done by introducing a negative chirp rate, dΛ/dL, in the C-SBGW. In other words, due to the increased propagation distance, the optical waves at shorter wavelengths experience larger group delay as compared to those at longer wavelengths. Here, we have designed a 4 mm long TM C-SBGW that has dΛ/dL = −6 nm/cm starting from the grating period of 444 nm and using Wcorr = 60 nm. Using the TMM, the reflection spectrum and group delay for the TM C-SBGW were calculated, as shown in Fig. 4. As expected, the average slope of the group delay is negative, with a slope of −11 ps/nm over the passband. It can be seen that the group delay exhibits ripples. These ripples are due to our use of constant corrugation widths. These ripples can be eliminated by using apodization [22, 48–50].
Fig. 4. Simulated transmission spectrum and group delay of the proposed TM C-SBGW. The propagation loss is assumed to be 2 dB/cm.
3. Fabrication and experimental results
3.1. Fabrication
The SBGWs were fabricated using electron beam (eBeam) lithography at the University of Washington [51]. The fabrication used a single etch process on an SOI wafer with 220 nm thick silicon on a 3 µm thick buried oxide layer. A 2 µm thick silicon dioxide cladding layer was then deposited on the etched sample. Figures 5(a)–5(c) show SEM images of a fabricated U-SBGW device. Three full-etched grating couplers (GCs) [52], fabricated on a 127 µm pitch, were used to couple light into and out of our devices from a fiber array with a 127 µm fiber-to-fiber pitch. A low loss Y-junction power splitter [53] was used to connect the input and output GCs to the SBGWs in order to transfer the injected light to the device and to collect the reflected light. Also, the designed rectangular corrugations on the SBGWs have been accurately fabricated, as shown by the zoom-in SEM images, Figs. 5(b) and 5(c).
Fig. 5. SEM images of an U-SBGW. (a) The complete U-SBGW with input and output GCs. (b) Zoom-in of the center of the S-shaped grating waveguide. (c) Zoom-in of the Archimedean spiral grating waveguides.
3.2. Spiral waveguide propagation loss
In order to verify that TM spiral waveguides are less sensitive to sidewall roughness and have lower propagation losses than the TE spiral waveguides, spiral waveguides without gratings were designed, for both the TE and TM modes, with various lengths, ranging from 0.5 cm to 3 cm. Figures 6(a) and 6(b) compare the measured transmission spectra between three TE spiral waveguides and three TM spiral waveguides over the C-band. They have equivalent lengths of 1 cm, 2 cm, and 3 cm. It can be seen that the TM spiral waveguides have much smoother transmission spectra, which means that they have reduced sensitivities to Fabry-Perot effects caused by the sidewall roughness. As will be seen below, this increased smoothness in the transmission spectra is observed in the results presented for all TM SBGWs.
Fig. 6. Measured transmission spectra, for the both (a) TE and (b) TM spiral waveguides, with lengths of 1 cm, 2 cm, and 3 cm.
Using the measured transmission spectra, the average transmission losses were extracted in the C-band. Figure 7 shows that the average transmission losses, for both the TE and TM spiral waveguides, increase as the length increases. The rate of change of the average transmission loss, as a function of length, gives the average propagation loss of a spiral waveguide. Using linear fits to the data in Fig. 7, the average propagation loss of the TM spiral waveguides and of the TE spiral waveguides were found to be 2.6 dB/cm and 6 dB/cm, respectively. This confirms that TM spiral waveguides have lower propagation losses than TE spiral waveguides.
Fig. 7. Transmission losses of both the TE and TM spiral waveguides for lengths, ranging from 0.5 cm to 3 cm.
Since our devices were fabricated using a low-cost relatively large shot pitch of 6 nm for rapid prototyping, the loss is relatively large. Spiral waveguides fabricated with electron beam lithography have slightly higher roughness, hence, the average propagation loss is larger, as compared to straight waveguides [51]. However, using a reduced shot pitch, such as 1 nm or 2 nm, the propagation losses can be reduced, see Ref. [51]. Also, using more advanced fabrication technologies, both TE and TM waveguide propagation losses can be reduced. For example, in Ref. [36], the TM spiral waveguides fabricated using 193 nm DUV lithography showed 1 dB/cm propagation loss, which is smaller than the propagation loss of 2.5 dB/cm obtained for the TE spiral waveguides.
3.3. Uniform spiral Bragg grating waveguides
According to the phase-match condition, the center wavelength of spectral response is determined by the chosen grating period. This was observed experimentally. The center wavelengths of the stop bands shift linearly to longer wavelengths as the grating period increases, as shown in Fig. 8(a). The shift of the center wavelength as a function of the grating period was calculated to be 1.7. When the period is increased to 444 nm, the center wavelength reaches 1549.3 nm, which closely matches the simulated value, as shown in Fig. 8(b). Also, since the gratings have long lengths, the transmission spectra exhibit large ERs, as large as 52 dB, as shown in Fig. 8(a).
Fig. 8. (a) Measured transmission spectra for the TM U-SBGWs, with grating periods of 432 nm, 438 nm, and 444 nm. The devices have the same corrugation widths, 60 nm and the same lengths, 1 mm. (b) The measured and simulated transmission spectra, where the average neff of the waveguide was adjusted by 0.84×10−3 in the simulation.
On the other hand, the bandwidths of the TM U-SBGWs are controlled by the corrugation widths of the gratings. Figure 9 shows the measured transmission spectra for TM U-SBGWs with various corrugation widths. As expected, the larger corrugation widths resulted in larger bandwidths due to the increased coupling coefficients.
Fig. 9. Measured transmission spectra of TM U-SBGWs with various corrugation widths. The grating period of each TM U-SBGW is 432 nm. The grating lengths of the TM U-SBGWs with Wcorr > 40 nm were 1 mm. The TM U-SBGW with Wcorr = 40 nm had a grating length of 3 mm.
Figure 9 also shows that the center wavelengths of the gratings shift to shorter wavelengths as the corrugation widths increase. Such wavelength shifts are also observed in the simulation results, as shown in Fig. 10(a). This indicates that larger corrugation widths result in smaller average neff values due to the non-linear relationship between neff and waveguide width [54]. This effect can be compensated for by adjusting the grating widths, Wmax and Wmin, to maintain a fixed, average neff, and, thus, a fixed center wavelength. The wavelength shift can also be compensated for by adjusting the grating period to keep the center wavelength constant.
Fig. 10. Simulation and experimental results for the Bragg gratings. (a) The Bragg grating’s center wavelength shifts to shorter wavelengths as the corrugation width increases. The vertical offset between simulation and experimental results is due to a fabrication difference as compared to the target waveguide design. (b) The grating strength (coupling coefficient) increases as a function of the corrugation width.
Using the measured center wavelengths and bandwidths, the coupling coefficients were extracted using Eq. (4), and then compared to FDTD simulation results. Figure 10(b) shows the simulated and measured coupling coefficients as functions of the corrugation width. Also shown in Fig. 10(b) is a second-order polynomial fit to the simulated coupling coefficients.
Since TM U-SBGWs can have small coupling coefficients for relatively large corrugations, we were able to design our U-SBGWs with narrow bandwidths. Figure 11 shows the reflection spectrum from a TM U-SBGW with a 3-dB bandwidth of 0.09 nm, which is smaller than the bandwidth of 0.14 nm obtained by a state-of-the-art TE spiral grating using a 1200 nm wide strip waveguide in Ref. [55].
Fig. 11. Measured reflection spectrum of a TM U-SBGW with a corrugation width of 20 nm, a length of 4 mm, and a period of 444 nm.
3.4. Linear chirped spiral Bragg grating waveguide
According to simulations, the proposed TM C-SBGW can have a negative, average, group delay slope over a broad passband. This was observed experimentally. Figure 12 shows the measured reflection spectrum and the group delay of the TM C-SBGW, which has a corrugation width of 60 nm and a grating length of 4 mm. It can be seen that the reflection spectrum has a 3-dB bandwidth of 11.7 nm and a center wavelength near 1548 nm. The group delay was measured using an Optical Vector Analyzer™ STe by Luna Innovations, Inc. As shown in Fig. 12, the group delay decreases as the wavelength increases over the passband. As per the design, the C-SBGW has a group delay with an average slope of −11 ps/nm, as compared to the 5 ps/nm obtained by a TE straight chirped grating using a 800 nm wide strip waveguide in Ref. [56]. As discussed in Ref. [23], devices with such negative group delay slopes can be used to compensate for positive dispersion. The undesired ripples were also noticed in the group delay in Fig. 12. As mentioned previously, the ripples can be reduced or eliminated by using apodization [22, 48–50].
Fig. 12. Reflection spectrum of the TM C-SBGW. The group delay decreases linearly as the wavelength increases within the passband.
4. Conclusion
In conclusion, we have demonstrated uniform Bragg gratings and chirped Bragg gratings for the fundamental TM mode in compact spiral SOI strip waveguides fabricated by eBeam lithography. We verify that TM spiral waveguides have smoother transmission spectra and lower propagation losses as compared to TE spiral waveguides. The U-SBGWs can be used as notch filters with narrow bandwidths and large ERs, as narrow as 0.09 nm and as large as 52 dB, respectively. In addition, we have demonstrated a linear chirped Bragg grating that is compatible with our TM spiral waveguides. We presented the measurements taken on the TM C-SBGW with a negative, average, group delay slope of −11 ps/nm. TE and TM polarizers, PRs, PSs, and PSRs provide design flexibility and the possibility of integrating both TE and TM devices in SOI-based systems.
Acknowledgments
We acknowledge the Natural Sciences and Engineering Research Council of Canada, particularly the SiEPIC CREATE program, and CMC Microsystems for their financial support. We would like to thank Richard Bojko for fabrication at the University of Washington, Washington Nanofabrication Facility (WNF), part of the National Science Foundation’s National Nanotechnology Infrastructure Network (NNIN). We also acknowledge Lumerical Solutions, Inc., and Mentor Graphics, Corp., for the design software.
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