Abstract

We propose and demonstrate a novel design of contradirectional couplers in which a subwavelength grating (SWG) waveguide replaces one of the asymmetric waveguides of the conventional designs. The fabricated devices in silicon-on-insulator (SOI) platform show over 35 dB suppression of undesired codirectional coupling and larger than 120 nm operating range free from the interference of intrawaveguide reflections thanks to the large optical phase-mismatch between the segmented SWG waveguide and its nearby continuous waveguide. We study the effects of tailoring the period of the SWG waveguide, the gap distance between the two waveguides, and the coupling length on the spectral characteristics of the device where changing the gap distance from 100 nm up to 500 nm allows for bandwidths from 18.2 nm down to 0.9 nm.

© 2016 Optical Society of America

1. Introduction

Silicon photonics is a promising technology for nanoscale miniaturization and integration of optics with electronics in a common silicon-on-insulator (SOI) platform in order to achieve monolithically fabricated optoelectronic systems-on-a-chip. In SOI platform, basic optical filtering needs have been addressed by development of various structures of Bragg gratings, ring resonators, arrayed waveguide gratings, etc. Recently, designs based on ring resonators have received much attention for their compact size; however, they support closely-packed resonant modes and therefore have limited flat-top response and free spectral range (FSR) for broadband filtering applications [1]. In comparison, filters based on Bragg gratings take advantage of FSR free operation as well as great design flexibility in achieving desired filtering characteristics. That flexibility has been exploited in various structures such as ones with chirped grating [2], designs with phase-modulated apodization profiles for precise tailoring of amplitude and phase responses [3], devices having subwavelength gratings with an additional design flexibility of tailoring the duty cycle [4], and cavities formed by phase-shifted gratings [5].

The general drawback of two-port Bragg grating devices is that they mostly operate in reflection mode, which usually translates to the need for optical circulators or isolators and increases the complexity of integration by requiring off-chip components or bonded nonreciprocal magneto-optic materials [6]. Researchers have demonstrated that some of the wideband Bragg-grating-defined filtering requirements can be fulfilled by compact SOI grating-assisted contradirectional couplers which are intrinsically four-port devices (add/drop devices) circumventing the need for the circulation or isolation of the optical power, and therefore, have great integration potential [7–9].

The reported grating-assisted contradirectional couplers basically consist of two closely placed waveguides and some form of periodic refractive index perturbations along the waveguides. Since improving the performance of the device requires increased optical phase mismatch between the involved eigenmodes of the coupler, designers have chosen the widths of the two waveguides to be as geometrically asymmetric as technically possible, limited by reasonably low insertion loss and single-mode operation [10]. On the other hand, dielectric perturbations provide the grating mechanism and the required phase factor in order to enable the intended contradirectional coupling between the two waveguides. Perturbations have been in the form of corrugations in the sidewall of either one or both of the coupled rib or strip waveguides [11, 12] or are created by the lack or excess of silicon in the space between the two waveguides [9, 13].

Recently, a novel antireflection design could overcome the bandwidth limitations arising from inter-waveguide Bragg reflections in silicon contradirectional couplers [8]; however, limited asymmetry has still kept the reported share of codirectionally coupled power up to 15 dB lower than that of contradirectionally coupled power [14]. Another issue is that fabrication errors, especially in implementation of small dielectric perturbations, distort the spectral responses. Those fabrication imperfections might also be the reason that drop-port extinction ratios better than 20 dB have been difficult to achieve [8, 11, 15].

In this paper, we propose and demonstrate a different approach where the need for two asymmetric waveguides as well as the need for a grating mechanism (dielectric perturbations) is simultaneously addressed by placing a subwavelength grating (SWG) waveguide in proximity to a continuous waveguide. In SOI platform, because of the potentially large optical phase-mismatch between SWG and strip waveguides, undesired codirectional coupling can be effectively suppressed. At the same time, the SWG waveguide, by itself, provides the required grating mechanism and enables contradirectional coupling between the two waveguides. This strong on/off grating leads to having relatively strong coupling coefficient meaning that the required power transfer can be achieved over a short coupling length, which is appropriate for wide-bandwidth filtering applications (> 1 nm). The short coupling length and the fabrication-friendly size of silicon segments of the SWG waveguide make the device tolerant to fabrication imperfections and this fact allows for large values of extinction ratio without much distortion and ripple in spectral responses. Furthermore, changing the duty-cycle of the SWG waveguide enables an additional degree of freedom in design of contradirectional couplers.

2. Device design

The schematic diagram of the proposed design and the layout of a fabricated device in SOI platform is illustrated in Figs. 1(a) and 1(b). A strip waveguide is being brought gradually in a gap distance, g, of an SWG waveguide with the period of Λ to form the coupler waist over the coupling length of LC. The duty cycle of the SWG waveguide is defined as the ratio of the length of one silicon segment to the period and will be denoted by η.

 figure: Fig. 1

Fig. 1 (a) Schematic top view of the proposed contradirectional coupler in SOI; (b) layout of a device with LC = 50 μm; (c) cross-section of the strip waveguide; (d) schematic top view of the SWG tapers.

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The height and width of all the strip waveguides are 220 nm and 500 nm, respectively, as shown in the cross-section view in Fig. 1(c). The waveguides sit on a 3 µm thick buried oxide (BOX) layer and a 2.5 μm thick cladding of SiO2 is deposited on top. SWG tapers [Fig. 1(d)] at the input and output of the SWG waveguide are responsible for adiabatic transition from the mode propagating in the SWG waveguide to the mode in the strip waveguides; they are basically chirped gratings with a uniform period of 250 nm and a width variation from 500 nm down to 150 nm over a length of 15 µm [16]. Focusing vertical grating couplers (VGCs) optimized for TE operation are used to couple light into and out of the chip [17].

The devices were fabricated using electron beam lithography, in a single, full, plasma etching process, at the University of Washington Nanofabrication Facility (WNF) [18]. The occupied chip area for the device with the layout in Fig. 1(b) is 90 µm × 400 µm.

3. Principle of operation

In this work, we restrict our attention to basic add/drop filters, in which the SWG waveguide is uniform over the coupler waist. Since the structure is periodic lengthwise over the coupling region, the Bloch-Floquet formalism described in [19] should be able to identify the eigenmodes of the periodic structure [20]. While no closed-form analytic solution exists, our results validate that a simplified model can be used to explain the basic characteristics of the device. The model decomposes the longitudinal variations of the permittivity of the segmented SWG waveguide into two components: an average permittivity that is constant lengthwise (responsible for waveguiding effect) and the remaining part having alternatively positive and negative permittivity along the waveguide (responsible for grating mechanism) [21].

With εSi (nSi) and εSiO2 (nSiO2) denoting the permittivity (refractive indices) of the silicon core, and silicon-dioxide cladding, the average yet dispersive permittivity and the resultant equivalent refractive index are given by the following:

εavg=ηεSi+(1η)εSiO2neq2=ηnSi2+(1η)nSiO22
Another common practice is averaging with respect to the refractive indices as described by:
navg=ηnSi+(1η)nSiO2
However, there are arguments that it is in fact the permittivity that is being averaged [21, 22] and this shows better agreement with our results described in Section 4 and summarized in Table 1.

Tables Icon

Table 1. Effect of period, Λ, on central coupling wavelength, λC, extracted from measurements, 3D FDTD simulations, and predictions of the model using either average permittivity or average refractive index while LC = 400 μm, g = 400 nm and η = 50%.

If we replace the SWG waveguide of the device in Fig. 1 by a strip waveguide having the average permittivity given by Eq. (1), we are able to identify the two lowest-order transverse TE-like supermodes in the coupler waist (E1 and E2) and assign propagation constants of β1 and β2 to them. It should be noted that, for a value of duty cycle sufficiently smaller than one (η<1), each of these waist eigenmodes are mainly localized to one waveguide due to the high index-asymmetry of the two waveguides (neq < nSi). Thus, given the adiabatic mode transformation between the coupler arms and the waist, light injected into one of the ports excites predominantly only the appropriate one of the two supermodes [7].

Based on coupled-mode theory, the phase factor of the uniform grating enables three efficient couplings between counter propagating E1 and E2 supermodes that have significant effect on the add/drop filtering response of the device. At a specific wavelength, λC, the phase-matched coupling requirement between E1 and E2 is satisfied by the condition given in Eq. (3a). Unwanted intra-waveguide back reflections of E1 and E2 also efficiently happen by the governing Bragg conditions in Eqs. (3b) and (3c) at the wavelengths denoted by 𝜆R1 and 𝜆R2, respectively [12].

β1(λC)+β2(λC)=2π/Λ
2β1(λR1)=2π/Λ
2β2(λR2)=2π/Λ

Figure 2 shows spectral responses obtained from 3D FDTD simulation of a device with Λ = 378 nm, g = 200 nm, and LC = 100 μm when light is injected into the input port. Three nulls in the through-port spectrum are centered at three wavelengths that are in good agreement with the predictions provided by Eqs. (3a)-(3c) (comparison in Fig. 2). The dispersion relations of E1 and E2 are required in calculations of Eqs. (3a)-(3c) and have been extracted from a finite difference eigenmode solver. The error in prediction of 𝜆R2 might be explained by the fact that the null at 𝜆R2 is attributed to the back reflection of E2 which is predominantly present in the SWG waveguide and the non-ideal tapers of this waveguide might be able to shift the central wavelength of the main back reflection. Due to the high degree of asymmetry in the effective refractive indices of the SWG and strip waveguides, there is a large spacing between 𝜆C and 𝜆R1 as well as 𝜆C and 𝜆R2. The simulations in Fig. 2 show the availability of a 200 nm operating span, although the limitations of our experimental setup allowed us to verify operation over only 120 nm of this 200 nm simulated range. Within the drop channel, Fig. 2 illustrates −16 dB power reflection to the input port because of mode mismatch and imperfect transitions between the modes of individual feed waveguides and supermodes of the coupler waist [7].

 figure: Fig. 2

Fig. 2 Full 3D FDTD simulation of the spectral responses of a device with Λ = 378 nm, LC = 100 μm and g = 200 nm. For comparison, values of λR2, λC, λR1 have also been calculated from the simplified model using the average permittivity.

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Based on the discussed simplified model and from the perspective of coupled mode theory, the surface integral of unperturbed mode profiles (normalized E1 and E2) and the first component of the Fourier series expansion of dielectric perturbation (ε1) determines the effective coefficient of power transfer (z-independent coupling coefficient, κ) between the coupled modes, E1 and E2, through Eq. (4a) where ω is the angular frequency. Over a coupling length of LC, Eq. (4b) determines the factor of the power transferring from the input port to the drop port (power coupling factor, |κc|2) and Eq. (4c) governs the factor that the power passes to the through port (power transmission factor, |tc|2) [23, 24], where:

s2=|κ|2(Δβ2)2and Δβ=β1+β22πΛ.κ=ω4E^1*(x,y).ε1(x,y)E^2(x,y)dxdy
|κc|2=P2(0)P1(0)=|κ|2sinh2(sLC)s2cosh2(sLC)+(Δβ2)2sinh2(sLC)
|tc|2=P1(LC)P1(0)=s2s2cosh2(sLC)+(Δβ2)2sinh2(sLC)

The direct calculation of κ using Eq. (4a) requires the determination of dielectric perturbation, and therefore, is sensitive to fabrication errors. A method to calculate effective |κ| of a fabricated device, at the central coupling wavelength, λC, has been developed in [24]; the method essentially requires the full width at half maximum (FWHM) bandwidth, Δλ3dB, to be extracted from the measured drop-port spectrum and the dispersion relations of E1 and E2 to be extracted from a mode solver.

The next steps are as follows: from Eq. (4b), maximum power coupling factor is given in Eq. (5) and corresponds to a specific wavelength, λC*, that satisfies the phase-match condition, Δβ(λC*) = 0 (λC* predicted from dispersion relation might be slightly different with the measurement of the central coupling wavelength in the drop-port spectrum, λC). For a still unknown but appropriate value of |κ| and at both wavelengths of (λ3dB,L* = λC* - 0.5 Δλ3dB) and (λ3dB,H* = λC* + 0.5 Δλ3dB), the power coupling factor is expected to be approximately half of its maximum value given in Eq. (5). By forcing this expectation at neither λ3dB,L* nor λ3dB,H* but at an average scenario that leads to Δβavg = 0.5 × (|Δβ(λ3dB,H*)| + |Δβ(λ3dB,H*)|), we end up with Eq. (6) that determines the appropriate coupling coefficient, |κ|, for a value of Δλ3dB extracted from measurements. Strong sidelobes might cause additional |κ| solutions for Eq. (6); however, the largest value of |κ| is the correct one [24].

|κc|max2=|κ|2sinh2(|κ|LC)|κ|2cosh2(|κ|LC)=tanh2(|κ|LC)
|κ|2sinh2(sLC)s2cosh2(sLC)+(Δβavg2)2sinh2(sLC)=12|κc|max2=12tanh2(|κ|LC)

4. Results

Design parameters of the coupling region play important roles on the performance of the coupler. We study the effects of tailoring the period of the SWG waveguide, the gap distance between the two waveguides, and the coupling length on the spectral characteristics of the device. Spectral responses are measurements on the optical power meter while a tunable laser scans the spectrum in steps of 50 pm. All the reported spectra are normalized to the back-to-back insertion loss of a test pair of vertical grating couplers (VGCs) with a central loss of 15 dB and a FWHM bandwidth of 30 nm.

For a gap of 400 nm and a coupling length of 400 μm, Table 1 summarizes the extracted values of central coupling wavelength, λC, for the couplers having SWG waveguides with different periods, but a fixed duty cycle of 50%. According to Table 1, the predictions of the simplified model in Eq. (3a) while using the average permittivity are within 1 nm of the measured results; however, using the average refractive index leads to calculated values that are ~20 nm off. Thus, the discussed simple model based on average permittivity enables quick and reliable prediction of the central coupling wavelength in comparison to computationally intensive 3D FDTD simulations of the entire device.

Next, we fix Λ at 378 nm and η at 50% for C-band operation and characterize devices with different values of g and LC. Figure 3 shows the spectral responses of four devices in this set while light is injected into the input port. Increasing the coupling length or decreasing the gap distance causes stronger extinction ratio as well as stronger presence of sidelobes as expected from Eqs. (4b) and (4c). The responses in Fig. 3 are asymmetric over wavelength and the side lobes at longer wavelengths are stronger than their counterparts at shorter wavelengths. This observation agrees with the fact, implicit in Eq. (4a), that the coupling coefficient is wavelength dependent in the sense that, at longer wavelengths, optical supermodes of the coupled waveguide system are less confined to the waveguide cores leading to higher coupling coefficient, and consequently, stronger side lobes at longer wavelengths.

 figure: Fig. 3

Fig. 3 Measured drop-port and through-port spectra for devices with labeled values of LC and g parameters while Λ = 378 nm, η = 50%, and light is injected into the input port. Arrows indicate the FWHM bandwidth.

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If strong sidelobes of uniform gratings is an issue for an application, apodization of the gap distance over the coupling length has been proven to be an effective way to suppress them [25]. We measured an insertion loss of up to 1.2 dB at λC for the devices; this is above and beyond the aforementioned VGC fiber-to-fiber losses of at least 15 dB.

Arrows in Fig. 3 indicate the FWHM bandwidth which is determined by considering the two wavelengths that the drop-port spectrum drops by 3 dB below its peak value while light is injected into the input port. The main FWHM bandwidth associated with the main lobe of the drop-port response is of special interest in order to study the dependence of the bandwidth to the design parameters of LC and g. However, in the measured drop-port spectrum of the devices with very strong coupling, such as the device with LC = 400 μm and g = 150 nm in Fig. 3, the main lobe becomes hardly mixed with the nearby side lobes and we believe that the measured FWHM bandwidth cannot be attributed to the main lobe. Thus, such devices are not part of the following parametric study. Tracking the evolution of the main lobe, among the devices that their gap distance decreases in appropriately small steps, help distinguish those cases.

The add/drop operation of the filter is further investigated in Fig. 4 where the spectral responses of two of the devices in Fig. 3 are illustrated this time by injecting light into the add port and measuring the optical power at the through port and the drop port. Results imply that, at the coupling wavelength, when the input power is mainly redirected toward the drop port, the injected add-port power is able to effectively transfer to the through port. The through-port responses in Fig. 4 are similar to their corresponding drop-port responses in Fig. 3 within the repeatability of the measurements and the left-right symmetry of the fabricated devices, which is consistent with the Lorentz reciprocity theorem that the source and the detector of a reciprocal device can be interchanged [23].

 figure: Fig. 4

Fig. 4 Measured through-port and drop-port spectra for devices with labeled values of LC and g parameters while Λ = 378 nm, η = 50%, and light is injected into the add port.

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Figure 5(a) illustrates the dependence of Δλ3dB on the gap distance and coupling length. Increasing the coupling length or increasing the gap distance leads to a narrower drop-channel bandwidth. Based on the measured Δλ3dB values and Eq. (6), we determine the coupling coefficients as shown in Fig. 5(b), which are almost independent of coupling length (as expected) but show an inverse exponential dependence to the gap distance, consistent with previous demonstrations of contradirectional couplers [12, 24].

 figure: Fig. 5

Fig. 5 (a) Measured FWHM bandwidth versus gap distance for devices with four different values of coupling length while Λ = 378 nm and η = 50%; (b) calculated coupling coefficients using values of FWHM bandwidth.

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For the same devices, Fig. 6(a) illustrates the effect of changing the gap distance and coupling length on the extinction ratio of the add/drop filter. Longer coupling length or narrower gap distance push the device from weak-coupling regime into a strongly coupled state with extinction ratios greater than 30 dB. As another important figure of merit, Fig. 6(b) shows a typical add-port spectral response implying that over 35 dB suppression of codirectional coupling is achievable due to the high optical phase-mismatch between the SWG and strip waveguides (while η<<100%).

 figure: Fig. 6

Fig. 6 (a) Extinction ratio versus gap distance for four different values of coupling length while Λ = 378 nm and η = 50%; (b) Measured add-port and drop-port spectra for a device with LC = 200 μm, g = 250 nm, Λ = 378 nm, and η = 50%; dash line highlights −35 dB normalized power.

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5. Discussion and summary

We propose and experimentally demonstrate a new design of silicon contradirectional couplers that make use of SWG waveguides. We also suggest a model that predicts the central coupling wavelength of the device and explains the effects of the gap distance and coupling length on its spectral characteristics. Although this work is a proof-of-concept demonstration of basic add/drop filters with uniform SWG waveguides that have a fixed duty cycle of 50%, future investigations might exploit all of the potential of the design in configurations having spatial non-uniformities, such as devices with chirped, apodized, or phase-shifted segments. The design is able to provide extinction ratios over 30 dB while suppressing the undesired codirectional coupling better than 35 dB; however, the suppression of the input power reflection is limited to ~15 dB. A device with short coupling length and large gap distance operates within the weak coupling regime and is suitable for implementation of wideband Bragg-defined filters (> 1 nm) in optical signal processing and microwave photonics applications. On the other hand, for a desired bandwidth, the operation of the devices with a coupling length greater than a corresponding threshold is within the strong coupling regime useful for applications such as add/drop multiplexers and demultiplexers of coarse wavelength division multiplexed (CWDM) interconnects. The strong sidelobes of the uniform gratings cause considerable cross talk between adjacent CWDM channels, but previous demonstrations of silicon contradirectional couplers with the idea of apodized gap distance could effectively address the issue.

Funding

Fonds de Recherche du Québec - Nature et Technologies (FRQNT); Natural Sciences and Engineering Research Council of Canada (NSERC); CREATE SiEPIC Program.

Acknowledgments

The devices were fabricated by Richard Bojko at the University of Washington, Washington Nanofabrication Facility, part of the National Science Foundation’s National Nanotechnology Infrastructure Network (NNIN).

References and links

1. M. Soltani, Q. Li, S. Yegnanarayanan, and A. Adibi, “Toward ultimate miniaturization of high Q silicon traveling-wave microresonators,” Opt. Express 18(19), 19541–19557 (2010). [CrossRef]   [PubMed]  

2. J. Wang, R. Ashrafi, M. Rochette, and L. R. Chen, “Chirped microwave pulse generation using an integrated SiP Bragg grating in a Sagnac loop,” IEEE Photonics Technol. Lett. 27(17), 1876–1879 (2015). [CrossRef]  

3. A. D. Simard, M. J. Strain, L. Meriggi, M. Sorel, and S. LaRochelle, “Bandpass integrated Bragg gratings in silicon-on-insulator with well-controlled amplitude and phase responses,” Opt. Lett. 40(5), 736–739 (2015). [CrossRef]   [PubMed]  

4. J. Wang, I. Glesk, and L. R. Chen, “Subwavelength grating filtering devices,” Opt. Express 22(13), 15335–15345 (2014). [CrossRef]   [PubMed]  

5. X. Wang, J. Flueckiger, S. Schmidt, S. Grist, S. T. Fard, J. Kirk, M. Doerfler, K. C. Cheung, D. M. Ratner, and L. Chrostowski, “A silicon photonic biosensor using phase-shifted Bragg gratings in slot waveguide,” J. Biophotonics 6(10), 821–828 (2013). [PubMed]  

6. M.-C. Tien, T. Mizumoto, P. Pintus, H. Kromer, and J. E. Bowers, “Silicon ring isolators with bonded nonreciprocal magneto-optic garnets,” Opt. Express 19(12), 11740–11745 (2011). [CrossRef]   [PubMed]  

7. W. Shi, X. Wang, C. Lin, H. Yun, Y. Liu, T. Baehr-Jones, M. Hochberg, N. A. Jaeger, and L. Chrostowski, “Silicon photonic grating-assisted, contra-directional couplers,” Opt. Express 21(3), 3633–3650 (2013). [CrossRef]   [PubMed]  

8. W. Shi, H. Yun, C. Lin, M. Greenberg, X. Wang, Y. Wang, S. T. Fard, J. Flueckiger, N. A. Jaeger, and L. Chrostowski, “Ultra-compact, flat-top demultiplexer using anti-reflection contra-directional couplers for CWDM networks on silicon,” Opt. Express 21(6), 6733–6738 (2013). [CrossRef]   [PubMed]  

9. W. Shi, V. Veerasubramanian, D. Patel, and D. V. Plant, “Tunable nanophotonic delay lines using linearly chirped contradirectional couplers with uniform Bragg gratings,” Opt. Lett. 39(3), 701–703 (2014). [CrossRef]   [PubMed]  

10. W. Shi, M. Greenberg, X. Wang, Y. Wang, C. Lin, N. A. F. Jaeger, and L. Chrostowski, “Single-band add-drop filters using anti-reflection, contradirectional couplers,” IEEE Group IV Photonics Conference (San Diego, CA, USA 2012), paper WA7 (2012). [CrossRef]  

11. H. Qiu, G. Jiang, T. Hu, H. Shao, P. Yu, J. Yang, and X. Jiang, “FSR-free add-drop filter based on silicon grating-assisted contradirectional couplers,” Opt. Lett. 38(1), 1–3 (2013). [CrossRef]   [PubMed]  

12. K. Ikeda, M. Nezhad, and Y. Fainman, “Wavelength selective coupler with vertical gratings on silicon chip,” Appl. Phys. Lett. 92(20), 201111 (2008). [CrossRef]  

13. W. Shi, X. Wang, W. Zhang, L. Chrostowski, and N. A. Jaeger, “Contradirectional couplers in silicon-on-insulator rib waveguides,” Opt. Lett. 36(20), 3999–4001 (2011). [CrossRef]   [PubMed]  

14. M. Caverley, R. Boeck, L. Chrostowski, and N. A. Jaeger, “High-speed data transmission through Silicon contra-directional grating coupler optical add-drop multiplexers,” in Conference on Lasers and Electro-Optics (Optical Society of America, San Jose, California, 2015), paper JTh2A.41. [CrossRef]  

15. D. T. H. Tan, K. Ikeda, S. Zamek, A. Mizrahi, M. P. Nezhad, A. V. Krishnamoorthy, K. Raj, J. E. Cunningham, X. Zheng, I. Shubin, Y. Luo, and Y. Fainman, “Wide bandwidth, low loss 1 by 4 wavelength division multiplexer on silicon for optical interconnects,” Opt. Express 19(3), 2401–2409 (2011). [CrossRef]   [PubMed]  

16. P. J. Bock, P. Cheben, J. H. Schmid, J. Lapointe, A. Delâge, S. Janz, G. C. Aers, D.-X. Xu, A. Densmore, and T. J. Hall, “Subwavelength grating periodic structures in silicon-on-insulator: a new type of microphotonic waveguide,” Opt. Express 18(19), 20251–20262 (2010). [CrossRef]   [PubMed]  

17. Y. Wang, X. Wang, J. Flueckiger, H. Yun, W. Shi, R. Bojko, N. A. F. Jaeger, and L. Chrostowski, “Focusing sub-wavelength grating couplers with low back reflections for rapid prototyping of silicon photonic circuits,” Opt. Express 22(17), 20652–20662 (2014). [CrossRef]   [PubMed]  

18. R. J. Bojko, J. Li, L. He, T. Baehr-Jones, M. Hochberg, and Y. Aida, “Electron beam lithography writing strategies for low loss, high confinement silicon optical waveguides,” J. Vacuum Sci. Technol. B Microelectron. Nanometer Struct. 29, 06F309 (2011).

19. R. Kashyap, Fiber Bragg Gratings (Academic, 1999).

20. R. Halir, P. J. Bock, P. Cheben, A. Ortega-Moñux, C. Alonso-Ramos, J. H. Schmid, J. Lapointe, D. X. Xu, J. G. Wangüemert-Pérez, Í. Molina-Fernández, and S. Janz, “Waveguide sub‐wavelength structures: a review of principles and applications,” Laser Photonics Rev. 9(1), 25–49 (2015). [CrossRef]  

21. Z. Weissman and A. Hardy, “Modes of periodically segmented waveguides,” J. Lightwave Technol. 11(11), 1831–1838 (1993). [CrossRef]  

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23. J.-M. Liu, Photonic Devices (Cambridge University, 2009).

24. R. Boeck, M. Caverley, L. Chrostowski, and N. A. F. Jaeger, “Process calibration method for designing silicon-on-insulator contra-directional grating couplers,” Opt. Express 23(8), 10573–10588 (2015). [CrossRef]   [PubMed]  

25. W. Shi, H. Yun, C. Lin, J. Flueckiger, N. A. Jaeger, and L. Chrostowski, “Coupler-apodized Bragg-grating add-drop filter,” Opt. Lett. 38(16), 3068–3070 (2013). [CrossRef]   [PubMed]  

References

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  1. M. Soltani, Q. Li, S. Yegnanarayanan, and A. Adibi, “Toward ultimate miniaturization of high Q silicon traveling-wave microresonators,” Opt. Express 18(19), 19541–19557 (2010).
    [Crossref] [PubMed]
  2. J. Wang, R. Ashrafi, M. Rochette, and L. R. Chen, “Chirped microwave pulse generation using an integrated SiP Bragg grating in a Sagnac loop,” IEEE Photonics Technol. Lett. 27(17), 1876–1879 (2015).
    [Crossref]
  3. A. D. Simard, M. J. Strain, L. Meriggi, M. Sorel, and S. LaRochelle, “Bandpass integrated Bragg gratings in silicon-on-insulator with well-controlled amplitude and phase responses,” Opt. Lett. 40(5), 736–739 (2015).
    [Crossref] [PubMed]
  4. J. Wang, I. Glesk, and L. R. Chen, “Subwavelength grating filtering devices,” Opt. Express 22(13), 15335–15345 (2014).
    [Crossref] [PubMed]
  5. X. Wang, J. Flueckiger, S. Schmidt, S. Grist, S. T. Fard, J. Kirk, M. Doerfler, K. C. Cheung, D. M. Ratner, and L. Chrostowski, “A silicon photonic biosensor using phase-shifted Bragg gratings in slot waveguide,” J. Biophotonics 6(10), 821–828 (2013).
    [PubMed]
  6. M.-C. Tien, T. Mizumoto, P. Pintus, H. Kromer, and J. E. Bowers, “Silicon ring isolators with bonded nonreciprocal magneto-optic garnets,” Opt. Express 19(12), 11740–11745 (2011).
    [Crossref] [PubMed]
  7. W. Shi, X. Wang, C. Lin, H. Yun, Y. Liu, T. Baehr-Jones, M. Hochberg, N. A. Jaeger, and L. Chrostowski, “Silicon photonic grating-assisted, contra-directional couplers,” Opt. Express 21(3), 3633–3650 (2013).
    [Crossref] [PubMed]
  8. W. Shi, H. Yun, C. Lin, M. Greenberg, X. Wang, Y. Wang, S. T. Fard, J. Flueckiger, N. A. Jaeger, and L. Chrostowski, “Ultra-compact, flat-top demultiplexer using anti-reflection contra-directional couplers for CWDM networks on silicon,” Opt. Express 21(6), 6733–6738 (2013).
    [Crossref] [PubMed]
  9. W. Shi, V. Veerasubramanian, D. Patel, and D. V. Plant, “Tunable nanophotonic delay lines using linearly chirped contradirectional couplers with uniform Bragg gratings,” Opt. Lett. 39(3), 701–703 (2014).
    [Crossref] [PubMed]
  10. W. Shi, M. Greenberg, X. Wang, Y. Wang, C. Lin, N. A. F. Jaeger, and L. Chrostowski, “Single-band add-drop filters using anti-reflection, contradirectional couplers,” IEEE Group IV Photonics Conference (San Diego, CA, USA 2012), paper WA7 (2012).
    [Crossref]
  11. H. Qiu, G. Jiang, T. Hu, H. Shao, P. Yu, J. Yang, and X. Jiang, “FSR-free add-drop filter based on silicon grating-assisted contradirectional couplers,” Opt. Lett. 38(1), 1–3 (2013).
    [Crossref] [PubMed]
  12. K. Ikeda, M. Nezhad, and Y. Fainman, “Wavelength selective coupler with vertical gratings on silicon chip,” Appl. Phys. Lett. 92(20), 201111 (2008).
    [Crossref]
  13. W. Shi, X. Wang, W. Zhang, L. Chrostowski, and N. A. Jaeger, “Contradirectional couplers in silicon-on-insulator rib waveguides,” Opt. Lett. 36(20), 3999–4001 (2011).
    [Crossref] [PubMed]
  14. M. Caverley, R. Boeck, L. Chrostowski, and N. A. Jaeger, “High-speed data transmission through Silicon contra-directional grating coupler optical add-drop multiplexers,” in Conference on Lasers and Electro-Optics (Optical Society of America, San Jose, California, 2015), paper JTh2A.41.
    [Crossref]
  15. D. T. H. Tan, K. Ikeda, S. Zamek, A. Mizrahi, M. P. Nezhad, A. V. Krishnamoorthy, K. Raj, J. E. Cunningham, X. Zheng, I. Shubin, Y. Luo, and Y. Fainman, “Wide bandwidth, low loss 1 by 4 wavelength division multiplexer on silicon for optical interconnects,” Opt. Express 19(3), 2401–2409 (2011).
    [Crossref] [PubMed]
  16. P. J. Bock, P. Cheben, J. H. Schmid, J. Lapointe, A. Delâge, S. Janz, G. C. Aers, D.-X. Xu, A. Densmore, and T. J. Hall, “Subwavelength grating periodic structures in silicon-on-insulator: a new type of microphotonic waveguide,” Opt. Express 18(19), 20251–20262 (2010).
    [Crossref] [PubMed]
  17. Y. Wang, X. Wang, J. Flueckiger, H. Yun, W. Shi, R. Bojko, N. A. F. Jaeger, and L. Chrostowski, “Focusing sub-wavelength grating couplers with low back reflections for rapid prototyping of silicon photonic circuits,” Opt. Express 22(17), 20652–20662 (2014).
    [Crossref] [PubMed]
  18. R. J. Bojko, J. Li, L. He, T. Baehr-Jones, M. Hochberg, and Y. Aida, “Electron beam lithography writing strategies for low loss, high confinement silicon optical waveguides,” J. Vacuum Sci. Technol. B Microelectron. Nanometer Struct. 29, 06F309 (2011).
  19. R. Kashyap, Fiber Bragg Gratings (Academic, 1999).
  20. R. Halir, P. J. Bock, P. Cheben, A. Ortega-Moñux, C. Alonso-Ramos, J. H. Schmid, J. Lapointe, D. X. Xu, J. G. Wangüemert-Pérez, Í. Molina-Fernández, and S. Janz, “Waveguide sub‐wavelength structures: a review of principles and applications,” Laser Photonics Rev. 9(1), 25–49 (2015).
    [Crossref]
  21. Z. Weissman and A. Hardy, “Modes of periodically segmented waveguides,” J. Lightwave Technol. 11(11), 1831–1838 (1993).
    [Crossref]
  22. S. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).
  23. J.-M. Liu, Photonic Devices (Cambridge University, 2009).
  24. R. Boeck, M. Caverley, L. Chrostowski, and N. A. F. Jaeger, “Process calibration method for designing silicon-on-insulator contra-directional grating couplers,” Opt. Express 23(8), 10573–10588 (2015).
    [Crossref] [PubMed]
  25. W. Shi, H. Yun, C. Lin, J. Flueckiger, N. A. Jaeger, and L. Chrostowski, “Coupler-apodized Bragg-grating add-drop filter,” Opt. Lett. 38(16), 3068–3070 (2013).
    [Crossref] [PubMed]

2015 (4)

J. Wang, R. Ashrafi, M. Rochette, and L. R. Chen, “Chirped microwave pulse generation using an integrated SiP Bragg grating in a Sagnac loop,” IEEE Photonics Technol. Lett. 27(17), 1876–1879 (2015).
[Crossref]

A. D. Simard, M. J. Strain, L. Meriggi, M. Sorel, and S. LaRochelle, “Bandpass integrated Bragg gratings in silicon-on-insulator with well-controlled amplitude and phase responses,” Opt. Lett. 40(5), 736–739 (2015).
[Crossref] [PubMed]

R. Halir, P. J. Bock, P. Cheben, A. Ortega-Moñux, C. Alonso-Ramos, J. H. Schmid, J. Lapointe, D. X. Xu, J. G. Wangüemert-Pérez, Í. Molina-Fernández, and S. Janz, “Waveguide sub‐wavelength structures: a review of principles and applications,” Laser Photonics Rev. 9(1), 25–49 (2015).
[Crossref]

R. Boeck, M. Caverley, L. Chrostowski, and N. A. F. Jaeger, “Process calibration method for designing silicon-on-insulator contra-directional grating couplers,” Opt. Express 23(8), 10573–10588 (2015).
[Crossref] [PubMed]

2014 (3)

2013 (5)

2011 (4)

2010 (2)

2008 (1)

K. Ikeda, M. Nezhad, and Y. Fainman, “Wavelength selective coupler with vertical gratings on silicon chip,” Appl. Phys. Lett. 92(20), 201111 (2008).
[Crossref]

1993 (1)

Z. Weissman and A. Hardy, “Modes of periodically segmented waveguides,” J. Lightwave Technol. 11(11), 1831–1838 (1993).
[Crossref]

1956 (1)

S. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).

Adibi, A.

Aers, G. C.

Aida, Y.

R. J. Bojko, J. Li, L. He, T. Baehr-Jones, M. Hochberg, and Y. Aida, “Electron beam lithography writing strategies for low loss, high confinement silicon optical waveguides,” J. Vacuum Sci. Technol. B Microelectron. Nanometer Struct. 29, 06F309 (2011).

Alonso-Ramos, C.

R. Halir, P. J. Bock, P. Cheben, A. Ortega-Moñux, C. Alonso-Ramos, J. H. Schmid, J. Lapointe, D. X. Xu, J. G. Wangüemert-Pérez, Í. Molina-Fernández, and S. Janz, “Waveguide sub‐wavelength structures: a review of principles and applications,” Laser Photonics Rev. 9(1), 25–49 (2015).
[Crossref]

Ashrafi, R.

J. Wang, R. Ashrafi, M. Rochette, and L. R. Chen, “Chirped microwave pulse generation using an integrated SiP Bragg grating in a Sagnac loop,” IEEE Photonics Technol. Lett. 27(17), 1876–1879 (2015).
[Crossref]

Baehr-Jones, T.

W. Shi, X. Wang, C. Lin, H. Yun, Y. Liu, T. Baehr-Jones, M. Hochberg, N. A. Jaeger, and L. Chrostowski, “Silicon photonic grating-assisted, contra-directional couplers,” Opt. Express 21(3), 3633–3650 (2013).
[Crossref] [PubMed]

R. J. Bojko, J. Li, L. He, T. Baehr-Jones, M. Hochberg, and Y. Aida, “Electron beam lithography writing strategies for low loss, high confinement silicon optical waveguides,” J. Vacuum Sci. Technol. B Microelectron. Nanometer Struct. 29, 06F309 (2011).

Bock, P. J.

R. Halir, P. J. Bock, P. Cheben, A. Ortega-Moñux, C. Alonso-Ramos, J. H. Schmid, J. Lapointe, D. X. Xu, J. G. Wangüemert-Pérez, Í. Molina-Fernández, and S. Janz, “Waveguide sub‐wavelength structures: a review of principles and applications,” Laser Photonics Rev. 9(1), 25–49 (2015).
[Crossref]

P. J. Bock, P. Cheben, J. H. Schmid, J. Lapointe, A. Delâge, S. Janz, G. C. Aers, D.-X. Xu, A. Densmore, and T. J. Hall, “Subwavelength grating periodic structures in silicon-on-insulator: a new type of microphotonic waveguide,” Opt. Express 18(19), 20251–20262 (2010).
[Crossref] [PubMed]

Boeck, R.

Bojko, R.

Bojko, R. J.

R. J. Bojko, J. Li, L. He, T. Baehr-Jones, M. Hochberg, and Y. Aida, “Electron beam lithography writing strategies for low loss, high confinement silicon optical waveguides,” J. Vacuum Sci. Technol. B Microelectron. Nanometer Struct. 29, 06F309 (2011).

Bowers, J. E.

Caverley, M.

Cheben, P.

R. Halir, P. J. Bock, P. Cheben, A. Ortega-Moñux, C. Alonso-Ramos, J. H. Schmid, J. Lapointe, D. X. Xu, J. G. Wangüemert-Pérez, Í. Molina-Fernández, and S. Janz, “Waveguide sub‐wavelength structures: a review of principles and applications,” Laser Photonics Rev. 9(1), 25–49 (2015).
[Crossref]

P. J. Bock, P. Cheben, J. H. Schmid, J. Lapointe, A. Delâge, S. Janz, G. C. Aers, D.-X. Xu, A. Densmore, and T. J. Hall, “Subwavelength grating periodic structures in silicon-on-insulator: a new type of microphotonic waveguide,” Opt. Express 18(19), 20251–20262 (2010).
[Crossref] [PubMed]

Chen, L. R.

J. Wang, R. Ashrafi, M. Rochette, and L. R. Chen, “Chirped microwave pulse generation using an integrated SiP Bragg grating in a Sagnac loop,” IEEE Photonics Technol. Lett. 27(17), 1876–1879 (2015).
[Crossref]

J. Wang, I. Glesk, and L. R. Chen, “Subwavelength grating filtering devices,” Opt. Express 22(13), 15335–15345 (2014).
[Crossref] [PubMed]

Cheung, K. C.

X. Wang, J. Flueckiger, S. Schmidt, S. Grist, S. T. Fard, J. Kirk, M. Doerfler, K. C. Cheung, D. M. Ratner, and L. Chrostowski, “A silicon photonic biosensor using phase-shifted Bragg gratings in slot waveguide,” J. Biophotonics 6(10), 821–828 (2013).
[PubMed]

Chrostowski, L.

R. Boeck, M. Caverley, L. Chrostowski, and N. A. F. Jaeger, “Process calibration method for designing silicon-on-insulator contra-directional grating couplers,” Opt. Express 23(8), 10573–10588 (2015).
[Crossref] [PubMed]

Y. Wang, X. Wang, J. Flueckiger, H. Yun, W. Shi, R. Bojko, N. A. F. Jaeger, and L. Chrostowski, “Focusing sub-wavelength grating couplers with low back reflections for rapid prototyping of silicon photonic circuits,” Opt. Express 22(17), 20652–20662 (2014).
[Crossref] [PubMed]

W. Shi, H. Yun, C. Lin, J. Flueckiger, N. A. Jaeger, and L. Chrostowski, “Coupler-apodized Bragg-grating add-drop filter,” Opt. Lett. 38(16), 3068–3070 (2013).
[Crossref] [PubMed]

X. Wang, J. Flueckiger, S. Schmidt, S. Grist, S. T. Fard, J. Kirk, M. Doerfler, K. C. Cheung, D. M. Ratner, and L. Chrostowski, “A silicon photonic biosensor using phase-shifted Bragg gratings in slot waveguide,” J. Biophotonics 6(10), 821–828 (2013).
[PubMed]

W. Shi, X. Wang, C. Lin, H. Yun, Y. Liu, T. Baehr-Jones, M. Hochberg, N. A. Jaeger, and L. Chrostowski, “Silicon photonic grating-assisted, contra-directional couplers,” Opt. Express 21(3), 3633–3650 (2013).
[Crossref] [PubMed]

W. Shi, H. Yun, C. Lin, M. Greenberg, X. Wang, Y. Wang, S. T. Fard, J. Flueckiger, N. A. Jaeger, and L. Chrostowski, “Ultra-compact, flat-top demultiplexer using anti-reflection contra-directional couplers for CWDM networks on silicon,” Opt. Express 21(6), 6733–6738 (2013).
[Crossref] [PubMed]

W. Shi, X. Wang, W. Zhang, L. Chrostowski, and N. A. Jaeger, “Contradirectional couplers in silicon-on-insulator rib waveguides,” Opt. Lett. 36(20), 3999–4001 (2011).
[Crossref] [PubMed]

Cunningham, J. E.

Delâge, A.

Densmore, A.

Doerfler, M.

X. Wang, J. Flueckiger, S. Schmidt, S. Grist, S. T. Fard, J. Kirk, M. Doerfler, K. C. Cheung, D. M. Ratner, and L. Chrostowski, “A silicon photonic biosensor using phase-shifted Bragg gratings in slot waveguide,” J. Biophotonics 6(10), 821–828 (2013).
[PubMed]

Fainman, Y.

Fard, S. T.

W. Shi, H. Yun, C. Lin, M. Greenberg, X. Wang, Y. Wang, S. T. Fard, J. Flueckiger, N. A. Jaeger, and L. Chrostowski, “Ultra-compact, flat-top demultiplexer using anti-reflection contra-directional couplers for CWDM networks on silicon,” Opt. Express 21(6), 6733–6738 (2013).
[Crossref] [PubMed]

X. Wang, J. Flueckiger, S. Schmidt, S. Grist, S. T. Fard, J. Kirk, M. Doerfler, K. C. Cheung, D. M. Ratner, and L. Chrostowski, “A silicon photonic biosensor using phase-shifted Bragg gratings in slot waveguide,” J. Biophotonics 6(10), 821–828 (2013).
[PubMed]

Flueckiger, J.

Glesk, I.

Greenberg, M.

Grist, S.

X. Wang, J. Flueckiger, S. Schmidt, S. Grist, S. T. Fard, J. Kirk, M. Doerfler, K. C. Cheung, D. M. Ratner, and L. Chrostowski, “A silicon photonic biosensor using phase-shifted Bragg gratings in slot waveguide,” J. Biophotonics 6(10), 821–828 (2013).
[PubMed]

Halir, R.

R. Halir, P. J. Bock, P. Cheben, A. Ortega-Moñux, C. Alonso-Ramos, J. H. Schmid, J. Lapointe, D. X. Xu, J. G. Wangüemert-Pérez, Í. Molina-Fernández, and S. Janz, “Waveguide sub‐wavelength structures: a review of principles and applications,” Laser Photonics Rev. 9(1), 25–49 (2015).
[Crossref]

Hall, T. J.

Hardy, A.

Z. Weissman and A. Hardy, “Modes of periodically segmented waveguides,” J. Lightwave Technol. 11(11), 1831–1838 (1993).
[Crossref]

He, L.

R. J. Bojko, J. Li, L. He, T. Baehr-Jones, M. Hochberg, and Y. Aida, “Electron beam lithography writing strategies for low loss, high confinement silicon optical waveguides,” J. Vacuum Sci. Technol. B Microelectron. Nanometer Struct. 29, 06F309 (2011).

Hochberg, M.

W. Shi, X. Wang, C. Lin, H. Yun, Y. Liu, T. Baehr-Jones, M. Hochberg, N. A. Jaeger, and L. Chrostowski, “Silicon photonic grating-assisted, contra-directional couplers,” Opt. Express 21(3), 3633–3650 (2013).
[Crossref] [PubMed]

R. J. Bojko, J. Li, L. He, T. Baehr-Jones, M. Hochberg, and Y. Aida, “Electron beam lithography writing strategies for low loss, high confinement silicon optical waveguides,” J. Vacuum Sci. Technol. B Microelectron. Nanometer Struct. 29, 06F309 (2011).

Hu, T.

Ikeda, K.

Jaeger, N. A.

Jaeger, N. A. F.

Janz, S.

R. Halir, P. J. Bock, P. Cheben, A. Ortega-Moñux, C. Alonso-Ramos, J. H. Schmid, J. Lapointe, D. X. Xu, J. G. Wangüemert-Pérez, Í. Molina-Fernández, and S. Janz, “Waveguide sub‐wavelength structures: a review of principles and applications,” Laser Photonics Rev. 9(1), 25–49 (2015).
[Crossref]

P. J. Bock, P. Cheben, J. H. Schmid, J. Lapointe, A. Delâge, S. Janz, G. C. Aers, D.-X. Xu, A. Densmore, and T. J. Hall, “Subwavelength grating periodic structures in silicon-on-insulator: a new type of microphotonic waveguide,” Opt. Express 18(19), 20251–20262 (2010).
[Crossref] [PubMed]

Jiang, G.

Jiang, X.

Kirk, J.

X. Wang, J. Flueckiger, S. Schmidt, S. Grist, S. T. Fard, J. Kirk, M. Doerfler, K. C. Cheung, D. M. Ratner, and L. Chrostowski, “A silicon photonic biosensor using phase-shifted Bragg gratings in slot waveguide,” J. Biophotonics 6(10), 821–828 (2013).
[PubMed]

Krishnamoorthy, A. V.

Kromer, H.

Lapointe, J.

R. Halir, P. J. Bock, P. Cheben, A. Ortega-Moñux, C. Alonso-Ramos, J. H. Schmid, J. Lapointe, D. X. Xu, J. G. Wangüemert-Pérez, Í. Molina-Fernández, and S. Janz, “Waveguide sub‐wavelength structures: a review of principles and applications,” Laser Photonics Rev. 9(1), 25–49 (2015).
[Crossref]

P. J. Bock, P. Cheben, J. H. Schmid, J. Lapointe, A. Delâge, S. Janz, G. C. Aers, D.-X. Xu, A. Densmore, and T. J. Hall, “Subwavelength grating periodic structures in silicon-on-insulator: a new type of microphotonic waveguide,” Opt. Express 18(19), 20251–20262 (2010).
[Crossref] [PubMed]

LaRochelle, S.

Li, J.

R. J. Bojko, J. Li, L. He, T. Baehr-Jones, M. Hochberg, and Y. Aida, “Electron beam lithography writing strategies for low loss, high confinement silicon optical waveguides,” J. Vacuum Sci. Technol. B Microelectron. Nanometer Struct. 29, 06F309 (2011).

Li, Q.

Lin, C.

Liu, Y.

Luo, Y.

Meriggi, L.

Mizrahi, A.

Mizumoto, T.

Molina-Fernández, Í.

R. Halir, P. J. Bock, P. Cheben, A. Ortega-Moñux, C. Alonso-Ramos, J. H. Schmid, J. Lapointe, D. X. Xu, J. G. Wangüemert-Pérez, Í. Molina-Fernández, and S. Janz, “Waveguide sub‐wavelength structures: a review of principles and applications,” Laser Photonics Rev. 9(1), 25–49 (2015).
[Crossref]

Nezhad, M.

K. Ikeda, M. Nezhad, and Y. Fainman, “Wavelength selective coupler with vertical gratings on silicon chip,” Appl. Phys. Lett. 92(20), 201111 (2008).
[Crossref]

Nezhad, M. P.

Ortega-Moñux, A.

R. Halir, P. J. Bock, P. Cheben, A. Ortega-Moñux, C. Alonso-Ramos, J. H. Schmid, J. Lapointe, D. X. Xu, J. G. Wangüemert-Pérez, Í. Molina-Fernández, and S. Janz, “Waveguide sub‐wavelength structures: a review of principles and applications,” Laser Photonics Rev. 9(1), 25–49 (2015).
[Crossref]

Patel, D.

Pintus, P.

Plant, D. V.

Qiu, H.

Raj, K.

Ratner, D. M.

X. Wang, J. Flueckiger, S. Schmidt, S. Grist, S. T. Fard, J. Kirk, M. Doerfler, K. C. Cheung, D. M. Ratner, and L. Chrostowski, “A silicon photonic biosensor using phase-shifted Bragg gratings in slot waveguide,” J. Biophotonics 6(10), 821–828 (2013).
[PubMed]

Rochette, M.

J. Wang, R. Ashrafi, M. Rochette, and L. R. Chen, “Chirped microwave pulse generation using an integrated SiP Bragg grating in a Sagnac loop,” IEEE Photonics Technol. Lett. 27(17), 1876–1879 (2015).
[Crossref]

Rytov, S.

S. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).

Schmid, J. H.

R. Halir, P. J. Bock, P. Cheben, A. Ortega-Moñux, C. Alonso-Ramos, J. H. Schmid, J. Lapointe, D. X. Xu, J. G. Wangüemert-Pérez, Í. Molina-Fernández, and S. Janz, “Waveguide sub‐wavelength structures: a review of principles and applications,” Laser Photonics Rev. 9(1), 25–49 (2015).
[Crossref]

P. J. Bock, P. Cheben, J. H. Schmid, J. Lapointe, A. Delâge, S. Janz, G. C. Aers, D.-X. Xu, A. Densmore, and T. J. Hall, “Subwavelength grating periodic structures in silicon-on-insulator: a new type of microphotonic waveguide,” Opt. Express 18(19), 20251–20262 (2010).
[Crossref] [PubMed]

Schmidt, S.

X. Wang, J. Flueckiger, S. Schmidt, S. Grist, S. T. Fard, J. Kirk, M. Doerfler, K. C. Cheung, D. M. Ratner, and L. Chrostowski, “A silicon photonic biosensor using phase-shifted Bragg gratings in slot waveguide,” J. Biophotonics 6(10), 821–828 (2013).
[PubMed]

Shao, H.

Shi, W.

Shubin, I.

Simard, A. D.

Soltani, M.

Sorel, M.

Strain, M. J.

Tan, D. T. H.

Tien, M.-C.

Veerasubramanian, V.

Wang, J.

J. Wang, R. Ashrafi, M. Rochette, and L. R. Chen, “Chirped microwave pulse generation using an integrated SiP Bragg grating in a Sagnac loop,” IEEE Photonics Technol. Lett. 27(17), 1876–1879 (2015).
[Crossref]

J. Wang, I. Glesk, and L. R. Chen, “Subwavelength grating filtering devices,” Opt. Express 22(13), 15335–15345 (2014).
[Crossref] [PubMed]

Wang, X.

Wang, Y.

Wangüemert-Pérez, J. G.

R. Halir, P. J. Bock, P. Cheben, A. Ortega-Moñux, C. Alonso-Ramos, J. H. Schmid, J. Lapointe, D. X. Xu, J. G. Wangüemert-Pérez, Í. Molina-Fernández, and S. Janz, “Waveguide sub‐wavelength structures: a review of principles and applications,” Laser Photonics Rev. 9(1), 25–49 (2015).
[Crossref]

Weissman, Z.

Z. Weissman and A. Hardy, “Modes of periodically segmented waveguides,” J. Lightwave Technol. 11(11), 1831–1838 (1993).
[Crossref]

Xu, D. X.

R. Halir, P. J. Bock, P. Cheben, A. Ortega-Moñux, C. Alonso-Ramos, J. H. Schmid, J. Lapointe, D. X. Xu, J. G. Wangüemert-Pérez, Í. Molina-Fernández, and S. Janz, “Waveguide sub‐wavelength structures: a review of principles and applications,” Laser Photonics Rev. 9(1), 25–49 (2015).
[Crossref]

Xu, D.-X.

Yang, J.

Yegnanarayanan, S.

Yu, P.

Yun, H.

Zamek, S.

Zhang, W.

Zheng, X.

” J. Vacuum Sci. Technol. B Microelectron. Nanometer Struct. (1)

R. J. Bojko, J. Li, L. He, T. Baehr-Jones, M. Hochberg, and Y. Aida, “Electron beam lithography writing strategies for low loss, high confinement silicon optical waveguides,” J. Vacuum Sci. Technol. B Microelectron. Nanometer Struct. 29, 06F309 (2011).

Appl. Phys. Lett. (1)

K. Ikeda, M. Nezhad, and Y. Fainman, “Wavelength selective coupler with vertical gratings on silicon chip,” Appl. Phys. Lett. 92(20), 201111 (2008).
[Crossref]

IEEE Photonics Technol. Lett. (1)

J. Wang, R. Ashrafi, M. Rochette, and L. R. Chen, “Chirped microwave pulse generation using an integrated SiP Bragg grating in a Sagnac loop,” IEEE Photonics Technol. Lett. 27(17), 1876–1879 (2015).
[Crossref]

J. Biophotonics (1)

X. Wang, J. Flueckiger, S. Schmidt, S. Grist, S. T. Fard, J. Kirk, M. Doerfler, K. C. Cheung, D. M. Ratner, and L. Chrostowski, “A silicon photonic biosensor using phase-shifted Bragg gratings in slot waveguide,” J. Biophotonics 6(10), 821–828 (2013).
[PubMed]

J. Lightwave Technol. (1)

Z. Weissman and A. Hardy, “Modes of periodically segmented waveguides,” J. Lightwave Technol. 11(11), 1831–1838 (1993).
[Crossref]

Laser Photonics Rev. (1)

R. Halir, P. J. Bock, P. Cheben, A. Ortega-Moñux, C. Alonso-Ramos, J. H. Schmid, J. Lapointe, D. X. Xu, J. G. Wangüemert-Pérez, Í. Molina-Fernández, and S. Janz, “Waveguide sub‐wavelength structures: a review of principles and applications,” Laser Photonics Rev. 9(1), 25–49 (2015).
[Crossref]

Opt. Express (9)

D. T. H. Tan, K. Ikeda, S. Zamek, A. Mizrahi, M. P. Nezhad, A. V. Krishnamoorthy, K. Raj, J. E. Cunningham, X. Zheng, I. Shubin, Y. Luo, and Y. Fainman, “Wide bandwidth, low loss 1 by 4 wavelength division multiplexer on silicon for optical interconnects,” Opt. Express 19(3), 2401–2409 (2011).
[Crossref] [PubMed]

P. J. Bock, P. Cheben, J. H. Schmid, J. Lapointe, A. Delâge, S. Janz, G. C. Aers, D.-X. Xu, A. Densmore, and T. J. Hall, “Subwavelength grating periodic structures in silicon-on-insulator: a new type of microphotonic waveguide,” Opt. Express 18(19), 20251–20262 (2010).
[Crossref] [PubMed]

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Figures (6)

Fig. 1
Fig. 1 (a) Schematic top view of the proposed contradirectional coupler in SOI; (b) layout of a device with LC = 50 μm; (c) cross-section of the strip waveguide; (d) schematic top view of the SWG tapers.
Fig. 2
Fig. 2 Full 3D FDTD simulation of the spectral responses of a device with Λ = 378 nm, LC = 100 μm and g = 200 nm. For comparison, values of λR2, λC, λR1 have also been calculated from the simplified model using the average permittivity.
Fig. 3
Fig. 3 Measured drop-port and through-port spectra for devices with labeled values of LC and g parameters while Λ = 378 nm, η = 50%, and light is injected into the input port. Arrows indicate the FWHM bandwidth.
Fig. 4
Fig. 4 Measured through-port and drop-port spectra for devices with labeled values of LC and g parameters while Λ = 378 nm, η = 50%, and light is injected into the add port.
Fig. 5
Fig. 5 (a) Measured FWHM bandwidth versus gap distance for devices with four different values of coupling length while Λ = 378 nm and η = 50%; (b) calculated coupling coefficients using values of FWHM bandwidth.
Fig. 6
Fig. 6 (a) Extinction ratio versus gap distance for four different values of coupling length while Λ = 378 nm and η = 50%; (b) Measured add-port and drop-port spectra for a device with LC = 200 μm, g = 250 nm, Λ = 378 nm, and η = 50%; dash line highlights −35 dB normalized power.

Tables (1)

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Table 1 Effect of period, Λ, on central coupling wavelength, λC, extracted from measurements, 3D FDTD simulations, and predictions of the model using either average permittivity or average refractive index while LC = 400 μm, g = 400 nm and η = 50%.

Equations (10)

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ε avg =η ε Si +( 1η ) ε Si O 2 n eq 2 =η n Si 2 +( 1η ) n Si O 2 2
n avg =η n Si +( 1η ) n Si O 2
β 1 ( λ C )+ β 2 ( λ C )= 2π /Λ
2 β 1 ( λ R1 )= 2π /Λ
2 β 2 ( λ R2 )= 2π /Λ
s 2 = | κ | 2 ( Δβ 2 ) 2 and Δβ= β 1 + β 2 2π Λ . κ= ω 4 E ^ 1 * (x,y). ε 1 (x,y) E ^ 2 (x,y)dxdy
| κ c | 2 = P 2 (0) P 1 (0) = | κ | 2 sin h 2 (s L C ) s 2 cos h 2 (s L C )+ ( Δβ 2 ) 2 sinh 2 (s L C )
| t c | 2 = P 1 ( L C ) P 1 (0) = s 2 s 2 cos h 2 (s L C )+ ( Δβ 2 ) 2 sin h 2 (s L C )
| κ c | max 2 = | κ | 2 sin h 2 (| κ | L C ) | κ | 2 cos h 2 (| κ | L C ) =tan h 2 (| κ | L C )
| κ | 2 sinh 2 (s L C ) s 2 cos h 2 (s L C )+ ( Δ β avg 2 ) 2 sinh 2 (s L C ) = 1 2 | κ c | max 2 = 1 2 tan h 2 (| κ | L C )

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