Abstract

Because of the high index contrast, current silicon photonics based optical phased arrays cannot achieve small beam divergence and large field-of-view simultaneously without increasing fabrication complexity. To resolve the dilemma, we propose an ultra-long waveguide grating antenna formed by placing subwavelength segments within the evanescent field of a conventional strip waveguide. Bound state in the continuum effect is leveraged to suppress the sidewall emission. As a proof of concept, we theoretically demonstrated a millimeter-long through-etched waveguide grating antenna with a divergence angle of 0.081° and a feature size compatible with current silicon photonics foundries.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

The ability to steer a light beam toward any desired direction is of essential importance to a wide range of free-space optical systems, such as point-to-point free-space communication and light detection and ranging (LiDAR) [13]. With the rapid advancement of the large-scale integrated silicon photonics technology, silicon photonics based optical phased array (OPA) has been considered as a promising approach to realize ultra-compact, high stability (without moving parts), and low-cost beam steering systems [1,411]. However, due to the strong mode confinement induced by the high index contrast between silicon and cladding materials, a trade-off has to be made between beam divergence and field-of-view for silicon photonics based OPAs [12]. As the beam divergence angle is determined by the length of the waveguide grating antennas(WGAs), many attempts have been made to increase the length of WGAs on silicon-on-insulator(SOI), such as shallow-etched gratings(SEGs) [1315] and sidewall corrugated gratings (SCGs) [1618]. Despite the substantial efforts, SEGs and SCGs still result in strong gratings with limited length. Besides, the sideward emission in SCGs assists the crosstalk between the adjacent WGAs, which deteriorates the far-field and the energy efficiency [16]. In lower index contrast platforms such as Si3N4/SiO2, WGAs with millimeter in length can be easily achieved as the optical mode is less confined within the waveguide core [11,1921]. However, the minimal spacing between adjacent waveguides required to avoid crosstalk increases as a result of the reduced mode confinement [12]. The field-of-view(FoV) reduces as it is reversely proportional to the minimum spacing. Multilayer gratings (and overlay gratings) are investigated to solve the dilemma by placing waveguides and gratings on different layers [12,2123]. Although a small beam divergence angle and large FoV can potentially be achieved simultaneously, the fabrication complexity increases significantly. Therefore, a long and fabrication-friendly WGA without sacrificing the array density is still highly desirable for silicon photonic OPAs.

In this paper, we propose an ultra-long and easy-to-fabricate WGA incorporating subwavelength structures. Subwavelength structures have emerged as a compelling approach to engineer macroscope optical properties of natural existing materials [2428], which brings remarkable flexibility to the design of integrated photonic components [29]. Many innovative components with unprecedented performance have been demonstrated, such as grating couplers [2935], microring resonators [3638], and spectral filters [3943]. While the subwavelength structure has been proven to be an effective method to reduce grating strength, which leads to the realization of narrow band filters on SOI [39,40,44], its potential in forming long WGAs has rarely been discussed [45,46].

The ultra-long WGA proposed in this paper is achieved by placing subwavelength silicon segments within the evanescent field of conventional strip waveguide. A similar structure with subwavelength grating waveguide has been reported in [46]. However, similar as Si3N4 gratings, subwavelength grating waveguide has poorer mode confinement compared to Silicon strip waveguide. Thus, the antenna array pitch has to be increased to avoid crosstalk, resulting in a small FoV.

In our design, wide range of grating strength can be obtained by tuning the dimensions and positions of the segments while keeping the feature size compatible with current silicon photonics foundries. WGAs with a few centimeters in length can be realized. At the same time, the lateral emission is suppressed by leveraging the bound state in the continuum (BIC) effect. The proposed WGA design offers a promising alternative for the realization of small divergence angle and large FoV integrated OPAs.

2. Subwavelength structure enabled waveguide grating antenna

The schematic of the proposed structure is shown in Fig. 1(a)-(c). The optical antennas are designed for SOI with a 220 nm single crystal silicon layer and a 2 μm buried oxide (BOX) layer. In this paper, the discussion concentrates on structures with a critical dimension larger than 100 nm to assure the design is compatible with typical fabrication processes in silicon photonics foundries. As shown in Fig. 1(a), subwavelength segments are placed in close proximity to a conventional strip waveguide to generate tunable refractive index perturbation. The grating strength could be conveniently adjusted by tuning the dimensions of the subwavelength segments and their distance to the strip waveguide. Strip waveguide is chosen because of its high mode confinement. It can be replaced with a subwavelength waveguide to obtain more design flexibility [46]. The transverse-electric (TE) field intensity distribution of the fundamental modes in WGAs with and without subwavelength segments are plotted in Fig. 1(d) and Fig. 1(e). The mode profiles are simulated at 1.55 μm. It can be seen that subwavelength segments only interact with the evanescent field of the strip waveguide, resulting in significantly weaker grating strength compared to SCGs. The effective index difference within a grating period, which indicates the grating strength, reduces to ∼ 1/20 of the value in SCGs with the same feature sizes [16].

 figure: Fig. 1.

Fig. 1. (a) The schematic of the proposed grating. (b)Top and (c) cross-section of the proposed WGA. (d)Electric field (|E|) distribution of the fundamental mode supported by the sections with (Top) and without (Bottom) subwavelength segments. (e) Electric field (|E|) distribution of the fundamental mode in WGA sections with and without subwavelength segments along the y-axis.

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3. Waveguide grating antenna design

3.1 Subwavelength segments

To reduce the strength of a grating, the mode mismatch between the two consecutive sections within a grating period must be small, meaning the presence of subwavelength segments must not cause an abrupt change of the mode profile of the strip waveguide. Thus, the dimensions of silicon segments should satisfy two requirements. On one hand, the segments should be small to avoid supporting resonant modes by themselves [44]. On the other hand, waveguide sections with subwavelength segments should only support fundamental modes to avoid the excitation of higher-order modes. As WGAs are expected to operate over a wavelength range as large as 100 nm, it is also of essential importance to assure the sections with subwavelength segments operate as a single-mode waveguide over the entire wavelength range.

The number of modes supported by the WGA sections with subwavelength segments varies with operating wavelength and the geometric parameters of the subwavelength segments. Fig. 2(a) shows the effective refractive indices of the TE modes supported by WGA sections with subwavelength segments in the wavelength range of 1.5 ∼ 1.6 μm. The sections with subwavelength segments tend to support higher-order modes at short wavelengths. The number of modes also depends upon the gap d between subwavelength segments and the strip waveguide. When d=200 nm, the single-mode cut-off wavelength is 1.522 μm, while the cut-off wavelength becomes 1.51 μm when d reduces to 100 nm.

 figure: Fig. 2.

Fig. 2. (a) Effective refractive indices of the modes supported by WGA sections with subwavelength segments when the gap d equals 100 nm (blue), 150 nm (purple), and 200 nm (red), respectively (w1=400 nm, w2=300 nm, dc=0.25). (b) Effective refractive indices in relation to w2 when gap d is 100 nm (blue), 150 nm (purple), and 200 nm (red) at the wavelength of 1.5 μm. Inset: |E| distribution of the fundamental mode with w2=300 nm (top), w2=400 nm (middle), and the 2nd-order mode with w2=300 nm (bottom). The back dash line indicates the effective index of a strip waveguide with a width of 400 nm. The results are simulated by the finite-difference eigenmode (FDE) method.

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To assure the WGA is single mode, we simulate the modes supported by the sections with subwavelength segments at the shortest wavelength (1.5 μm) of the operating wavelength range with different values of d. The results are summarized in Fig. 2(b). The sections with subwavelength segments begin to support higher order modes when the width of the silicon segment w2 is larger than 300 nm. As shown by the bottom figure in the inset of Fig. 2(b), the 2nd-order mode has a mode profile entirely different from the fundamental mode. It can also be observed that the increase of the effective index of the fundamental mode accelerates when w2 is over 350 nm (Green line), indicating that the structure can no longer be considered as a subwavelength structure beyond this dimension. As shown by the middle figure in the inset of Fig. 2(b), the difference between the field distributions of the fundamental mode of the sections with subwavelength segments and the conventional strip waveguide become eminent. Thus, in order to introduce a weak perturbation to the mode inside the strip waveguide, w2 should be less than 300 nm, which prevents the existence of higher-order modes and the eminent change of the profile of the fundamental mode.

3.2 Sideward emission suppression by bound state in the continuum

As it has been studied theoretically and experimentally [16], the through-etched gratings such as SCG based WGAs emit in both vertical and lateral directions, as illustrated in Fig. 3(a). While the downward emission caused energy loss could be salvaged by bottom reflectors or asymmetric designs [20,22,47], it is more challenging to suppress sideward emission due to the lack of asymmetricity along the vertical direction. The sideward emission not only decreases the energy efficiency of OPAs, but also deteriorates the far-field quality [16]. Especially, when sideward emission is stronger than upward emission, two spots appear along ψ direction in the far-field as a result of the interference between the sideward and upward emission [48].

 figure: Fig. 3.

Fig. 3. (a) The possible emitting directions in WGA with subwavelength segments. (b) Normalized upward emission (orange dot), sideward emission (orange circle) and its ratio (black dot) in relation with w1 (d=100 nm, w2=160 nm, dc=0.2). Field profiles of WGAs with subwavelength segments when (c) w1 = 370 nm and (d) w1=500 nm. All the simulation is done at 1.55 μm.

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In order to avoid the occurrence of two spots in the far-field, we propose to suppress the sideward emission by the BIC effect in the lateral direction, which does not require additional fabrication steps to create asymmetric structures in the vertical direction [46]. BIC was first predicted by von Neumann and Wigner by mathematically constructing a 3D potential extending to infinity and oscillating in a way that was tailored to support an electronic BIC [49]. While BIC has never been experimentally demonstrated in the proposed quantum system [50], it has been observed in other physical systems [50], including photonics [51]. BIC provides a new light confinement mechanism distinct from conventional approaches relying on forbidding outgoing waves [52]. The BIC based waveguide could be interpreted as that the wave leakage is extinguished by the destructive interference among multiple leakage channels [52,53]. Thus, BIC could enable unprecedented devices with properties that could not be achieved by conventional waveguiding mechanisms, such as low-refractive index waveguide [53], diffraction engineering [48], and ultra-high quality factor Q [54]. The unique nature of BIC has led to many applications, such as lasers [55] and sensors [56].

In this paper, the BIC effect in WGAs is constructed by building a destructive interference between the scattered waves excited by nanostructures [48]. The interference among multiple leakage channels along the lateral direction could be controlled by tuning the width of the core waveguide w1. As the width of w1 has negligible impact on the interference in the vertical direction, the upward emission would not be affected significantly. As shown in Fig. 3(b), the emission in all directions decreases with the increase of w1 because the mode is more confined in the strip waveguide when its width increases. As a result of the BIC effect in the lateral direction, sideward emission curve has three local minima while upward emission decreases without significant variation. When w1 equals 370 nm, the ratio between the sideward and upward emission is minimized, indicating that the sideward emission is effectively suppressed. As a comparison, we simulate the field distribution when w1 = 370 and 500 nm and plot in Figs. 3(c) and 3(d), respectively. It can be seen that the top/bottom emission dominants when w1=370 nm, while light emits both sides and top/bottom direction when w1=500 nm.

To prove that the two-spot far-field phenomenon shown in Fig. 4(a) can be prevented by the suppression of lateral emission, the relative difference ΔI between the two peaks and the valley is used to quantify the two-spot far-field pattern, as shown in Fig. 4(b). Since the device structure is mirror-symmetric with respect to the plane y=0, the two spots should be identical. Therefore, ΔI >0 means that two spots exist in the far-field and the value of ΔI indicates the degree of splitting. The impact of dc and w1 on ΔI is simulated and plotted in Fig. 4(c). The white dotted contour line represents ΔI=0, indicating the threshold between single spot/two spots in the far field. In the region to the left of the line, the far-field has one spot, while to the right of the line, the far-field splits into two spots. Fig. 4(d) shows the corresponding sideward to upward emission ratio. The black dotted line in Fig. 4(d) represents the contour line when sideward to upward emission ratio equals 0.9. As a comparison, the dotted line in Fig. 4(c) (ΔI = 0) is also plotted in the same figure. The two lines exhibit a similar pattern, meaning the sideward emission is the dominant factor leading to the two-spot phenomenon. Thus, it is necessary to make sideward to upward emission ratio less than 0.9. To assure the two-spot phenomenon being suppressed over the entire operating range of 1.5 μm ∼ 1.6 μm, the sideward to upward emission ratio at 1.5, 1.55, and 1.6 μm is simulated and plotted in Fig. 4(e). The emission ratio varies as much as 0.6 when the wavelength changes 100 nm as a result of the wavelength dependency of the BIC effect. Therefore, the emission ratio should be kept below 0.25 at 1.55 μm to ensure the far-field has only one spot over the entire operation wavelength range, as shown by the black line in Fig. 4(d).

 figure: Fig. 4.

Fig. 4. (a) Typical two-spot far field pattern and (b) normalized field distribution along ψ direction at 1.55μm ( w1=500 nm, dc=0.4, d=100 nm ). (c) ΔI in relation to w1 and dc (d=100 nm,w2=160 nm). (d) The ratio of sideward to upward emission in relation to w1 and dc (d=100 nm, w2=160 nm). (e) The ratio of sideward to upward emission in relation to w2 (red) (w1=400 nm, dc=0.25, d=100 nm) and d (blue) (w1=400 nm, dc=0.25, w2=160 nm) with wavelength of 1.5, 1.55 and 1.6 μm.(f) Coupling length of strip waveguide(w1=400 nm) and proposed waveguide grating(w1=400 nm,w2=190 nm,d=100 nm,dc=0.25,Λ=800 nm) whcn the center-to-center distance equals 1.3 μm and 1.5 μm.

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Sideward emission suppression is also crucial to reducing the crosstalk between adjacent WGAs. Fig. 4(f) shows the coupling length of typical strip waveguide(220 nm x 400 nm cross-section) and the proposed WGA. It can be noticed that the coupling length of the proposed WGA is close to that of the strip waveguide when the center-to-center distance is 1.3 μm. When the center-to-center distance becomes 1.5 μm, the coupling length of the proposed WGA deviates from strip waveguide at short wavelengths, which could be caused by the sideward emission. To assure the crosstalk is below 10% [21], the minimum distance between two 1-mm-long WGAs is around 1.5 μm for an operating wavelength range of 1.5-1.6 μm. According to Fig. 4(f), the minimum distance for two 1-mm-long strip waveguide to satisfy the same requirements is similar. Therefore, the subwavelength segments will not jeopardize the array density.

3.3 Grating strength engineering

According to Fig. 3(b), the sideward emission minimizes at w1 = 360 nm. However, such a narrow width inevitably increases the propagation loss. Thus, w1 = 400 nm is selected for the discussion below. As aforementioned, the targeted grating length and corresponding grating strength can be achieved by tuning the sizes and positions of the subwavelength segments. The grating strength is expressed with α in the formula pz = p0 exp(-2αz) with the unit of mm-1, which is also referred to as the decay constant of the electric field [12].

The correlation between grating strength and geometrical parameters of the subwavelength segments is simulated with the three-dimensional (3D) Finite-Division Time-Domain (FDTD) method. The period of the grating Λ is selected as 800 nm in the simulation to ensure that the diffraction angle is about 10° at 1550 nm. The results are summarized in Fig. 5. The grating strength for different w2 and d at 1550 nm is plotted in Fig. 5(a). The dc is kept at 0.2 to ensure the subwavelength segments satisfy the subwavelength condition in the light propagation direction [24]. The white line indicates the single-mode condition discussed in the previous section. Figure 5(a) shows that the grating strength can be reduced by increasing d, as the evanescent field decreases rapidly outside the waveguide. In the meantime, controlling w2 can also effectively tuning the grating strength when d is small. However, as d increases, the grating strength becomes less sensitive to w2. The minimum value of α in Fig. 5(a) is around 0.04 mm-1, meaning the light can transmit 2.5 cm long before its power attenuates to 1/e2 of its initial power. Fig. 5(b) shows the grating strength in relation to w2 and dc at 1550 nm. It can be found that the grating strength can be weakened by reducing dc. From Fig. 5(a) and (b), it can be concluded that through the combination of these design parameters, grating strength could be customized to any value between 0.04 mm-1 and 25 mm-1, making the approach an intriguing tool for grating strength control with large flexibility. α can be further decreased to infinitely close to zero with larger d, smaller dc, or smaller w2. However, for OPAs, the on-chip coherent length needs to be taken into consideration [57]. Thus, we keep the grating strength larger than 0.04 mm-1 in this paper.

 figure: Fig. 5.

Fig. 5. Grating strength variation in relation to (a) w2 and d (w1=400 nm, dc=0.2), (b) w2 and dc (w1=400 nm, d=100 nm), and (c) wavelength and w2 (w1 = 400 nm, dc=0.2, d=100 nm). The white line indicates the single mode condition.

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As the operating wavelength range of the WGA is broad, we simulated the wavelength dependence of the grating strength. Fig. 5(c) shows the grating strength in relation to wavelength and w2, when w1=400 nm, dc=0.2, and d=100 nm. It can be found that the grating strength remains stable when operating at wavelengths from 1.5 μm to 1.6 μm.

3.4 Ultra-long waveguide grating antenna

In this section, a WGA leveraging subwavelength segments is designed as a proof of concept demonstration. Due to the limitation of computation capacity, a length of 1 mm is chosen. The results could be readily extended to the design of longer WGAs. To assure the far-field of the WGA has only one spot over the entire operation wavelength range, the width of the waveguide core w1 is set as 400 nm to suppress sideward emission with BIC effect, as proposed and proved in previous sections. Other structural parameters (w2=190 nm, d=100 nm, dc=0.25, Λ=800 nm) are selected to obtain the desired grating strength of 1 mm-1 and assure the bandgap of the periodic structure locates outside of the operation wavelength range. The simulation results are shown in Fig. 6.

 figure: Fig. 6.

Fig. 6. (a) Power attenuation within the proposed subwavelength segments (blue) and SCG (red) based WGAs. Inset: the structure of the SCG based WGA used for comparison. The black dash line corresponds to 10% of the initial power. (b) The far-field profile of the proposed WGA at 1.55 μm. (c) The far-field profile along θ direction. The blue dots are simulated with FDTD. The divergence angle is estimated by Gaussian fitting. (d) Typical far-field profiles along θ direction when wavelength sweeps from 1.5 μm to 1.6 μm.

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The blue and red lines in Fig. 6(a) show the power attenuation within the WGA incorporating subwavelength segments and SCG, respectively. The typical structure of SCGs reported in related literature is shown in the inset of Fig. 6(a) [16]. For comparison, the period Λ, dc, and the width of core waveguide w1 are the same as the proposed WGA. The corrugation width w2 is selected as 100 nm. It can be seen that in the proposed WGA, guided light decays exponentially and propagates one millimeter before attenuating to ∼10% of its initial value, while in SCG based WGAs, the remaining power falls below 10% within less than 100 μm. The far-field distribution at 1.55 μm is plotted in Fig. 6(b) and its inset, which proves that single spot operation could be achieved by the method proposed in this paper. Figure 6(c) shows that the full-width-half-maximum (FWHM) of the divergence angle, which is estimated to be ∼0.081° through Gaussian fitting. A narrower beam could be achieved by further increasing the length of the WGA. However, the maximum WGA length is subject to the limitation of the on-chip coherent length [57]. As shown in Fig. 6(d), the emission angle of the WGA could be tuned between 2.6° to 19.7° by sweeping the wavelength from 1.5 μm to 1.6 μm, corresponding to an angular dispersion of 0.17°/nm. The emission efficiency varies with wavelength which is the nature of grating emitters and can potentially be improved with more sophisticated structures.

4. Conclusion

In conclusion, in this paper, the potential of using subwavelength segments to tailor the grating strength is investigated. We demonstrate that through optimizing the dimensions and positions of the subwavelength segments, a centimeter-long grating antenna could be realized. In combination with the waveguide dimensions, the diffraction properties of the antenna could be engineered to suppress the sideward emission. As a proof of concept demonstration, we designed a millimeter-long optical antenna with a feature size larger than 100 nm. The waveguide width is adjusted to reduce the sideward emission. The designed antenna shows an angular dispersion of 0.17°/nm with a beam width of 0.081° at 1550 nm. We also compared the proposed structure with typical waveguide grating antennas, as shown by Table 1 in the Appendix. Only a few have a feature size larger than 100 nm. Among these approaches, through-etched Silicon gratings usually have a limited length [16]. For Si3N4 gratings, the antenna pitch is typically 4 μm [11,19] and the angular dispersion reduces to 0.074°/nm [11], meaning the FoV reduces significantly. The approach proposed in this paper could achieve long grating with feature size larger than 100 nm. In the meantime, the antenna pitch could be 1.5 μm for 1 mm long waveguide gratings if 10% crosstalk is tolerable. Therefore, the proposed approach achieves small divergence beams without sacrificing the FoV, making it a promising alternative to conventional WGAs for OPAs.

Appendix

Tables Icon

Table 1. Typical waveguide grating antennas for OPAs

Funding

Science, Technology and Innovation Commission of Shenzhen Municipality; Harbin Institute of Technology (JA11409001); National Natural Science Foundation of China (61705099); China Academy of Engineering Physics.

Disclosures

The authors declare no conflicts of interest.

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28. P. Cheben, R. Halir, J. H. Schmid, H. A. Atwater, and D. R. Smith, “Subwavelength integrated photonics,” Nature 560(7720), 565–572 (2018). [CrossRef]  

29. P. Cheben, P. J. Bock, J. H. Schmid, J. Lapointe, S. Janz, D. X. Xu, A. Densmore, A. Delage, B. Lamontagne, and T. J. Hall, “Refractive index engineering with subwavelength gratings for efficient microphotonic couplers and planar waveguide multiplexers,” Opt. Lett. 35(15), 2526–2528 (2010). [CrossRef]  

30. A. Sanchez-Postigo, J. Gonzalo Wanguemert-Perez, J. M. Luque-Gonzalez, I. Molina-Fernandez, P. Cheben, C. A. Alonso-Ramos, R. Halir, J. H. Schmid, and A. Ortega-Monux, “Broadband fiber-chip zero-order surface grating coupler with 0.4 dB efficiency,” Opt. Lett. 41(13), 3013–3016 (2016). [CrossRef]  

31. D. Benedikovic, P. Cheben, J. H. Schmid, D. X. Xu, B. Lamontagne, S. Wang, J. Lapointe, R. Halir, A. Ortega-Monux, S. Janz, and M. Dado, “Subwavelength index engineered surface grating coupler with sub-decibel efficiency for 220-nm silicon-on-insulator waveguides,” Opt. Express 23(17), 22628–22635 (2015). [CrossRef]  

32. J. D. Sarmiento-Merenguel, A. Ortega-Monux, J. M. Fedeli, J. G. Wanguemert-Perez, C. Alonso-Ramos, E. Duran-Valdeiglesias, P. Cheben, I. Molina-Fernandez, and R. Halir, “Controlling leakage losses in subwavelength grating silicon metamaterial waveguides,” Opt. Lett. 41(15), 3443–3446 (2016). [CrossRef]  

33. J. Wang, R. Ashrafi, R. Adams, I. Glesk, I. Gasulla, J. Capmany, and L. R. Chen, “Subwavelength grating enabled on-chip ultra-compact optical true time delay line,” Sci. Rep. 6(1), 30235 (2016). [CrossRef]  

34. T. Watanabe, M. Ayata, U. Koch, Y. Fedoryshyn, and J. Leuthold, “Perpendicular Grating Coupler Based on a Blazed Antiback-Reflection Structure,” J. Lightwave Technol. 35(21), 4663–4669 (2017). [CrossRef]  

35. P. Cheben, D. X. Xu, S. Janz, and A. Densmore, “Subwavelength waveguide grating for mode conversion and light coupling in integrated optics,” Opt. Express 14(11), 4695–4702 (2006). [CrossRef]  

36. V. Donzella, A. Sherwali, J. Flueckiger, S. M. Grist, S. T. Fard, and L. Chrostowski, “Design and fabrication of SOI micro-ring resonators based on sub-wavelength grating waveguides,” Opt. Express 23(4), 4791–4803 (2015). [CrossRef]  

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

38. Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435(7040), 325–327 (2005). [CrossRef]  

39. S. Zamek, D. T. Tan, M. Khajavikhan, M. Ayache, M. P. Nezhad, and Y. Fainman, “Compact chip-scale filter based on curved waveguide Bragg gratings,” Opt. Lett. 35(20), 3477–3479 (2010). [CrossRef]  

40. J. Ctyroky, J. Gonzalo Wanguemert-Perez, P. Kwiecien, I. Richter, J. Litvik, J. H. Schmid, I. Molina-Fernandez, A. Ortega-Monux, M. Dado, and P. Cheben, “Design of narrowband Bragg spectral filters in subwavelength grating metamaterial waveguides,” Opt. Express 26(1), 179–194 (2018). [CrossRef]  

41. B. Naghdi and L. R. Chen, “Silicon photonic contradirectional couplers using subwavelength grating waveguides,” Opt. Express 24(20), 23429–23438 (2016). [CrossRef]  

42. D. Perez-Galacho, C. Alonso-Ramos, F. Mazeas, X. Le Roux, D. Oser, W. Zhang, D. Marris-Morini, L. Labonte, S. Tanzilli, E. Cassan, and L. Vivien, “Optical pump-rejection filter based on silicon sub-wavelength engineered photonic structures,” Opt. Lett. 42(8), 1468–1471 (2017). [CrossRef]  

43. H. Podmore, A. Scott, P. Cheben, A. V. Velasco, J. H. Schmid, M. Vachon, and R. Lee, “Demonstration of a compressive-sensing Fourier-transform on-chip spectrometer,” Opt. Lett. 42(7), 1440–1443 (2017). [CrossRef]  

44. D. T. Tan, K. Ikeda, and Y. Fainman, “Cladding-modulated Bragg gratings in silicon waveguides,” Opt. Lett. 34(9), 1357–1359 (2009). [CrossRef]  

45. J. Chen, W. Li, J. Tian, Y. Sun, Y. Yao, and X. Xu, “Millimeter-Long Optical Antenna for Optical Phased Array,” in 14th Pacific Rim Conference on Lasers and Electro-Optics (CLEO PR 2020), (2020), pp. 1–3.

46. P. Ginel-Moreno, D. Pereira-Martin, A. Hadij-ElHouati, W. N. Ye, D. Melati, D. X. Xu, S. Janz, A. Ortega-Monux, J. G. Wanguemert-Perez, R. Halir, I. Molina-Fernandez, J. H. Schmid, and P. Cheben, “Highly efficient optical antenna with small beam divergence in silicon waveguides,” Opt. Lett. 45(20), 5668–5671 (2020). [CrossRef]  

47. D. Benedikovic, C. Alonso-Ramos, D. Perez-Galacho, S. Guerber, V. Vakarin, G. Marcaud, X. Le Roux, E. Cassan, D. Marris-Morini, P. Cheben, F. Boeuf, C. Baudot, and L. Vivien, “L-shaped fiber-chip grating couplers with high directionality and low reflectivity fabricated with deep-UV lithography,” Opt. Lett. 42(17), 3439–3442 (2017). [CrossRef]  

48. H. N. Xu and Y. C. Shi, “Diffraction engineering for silicon waveguide grating antenna by harnessing bound state in the continuum,” Nanophotonics 9(6), 1439–1446 (2020). [CrossRef]  

49. J. von Neumann and E. P. Wigner, “Über merkwürdige diskrete Eigenwerte,” in The Collected Works of Eugene Paul Wigner: Part A: The Scientific Papers, A. S. Wightman, ed. (Springer Berlin Heidelberg, Berlin, Heidelberg, 1993), pp. 291–293. [CrossRef]  

50. C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1(9), 16048 (2016). [CrossRef]  

51. B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljacic, “Topological nature of optical bound states in the continuum,” Phys. Rev. Lett. 113(25), 257401 (2014). [CrossRef]  

52. C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, “Observation of trapped light within the radiation continuum,” Nature 499(7457), 188–191 (2013). [CrossRef]  

53. C.-L. Zou, J.-M. Cui, F.-W. Sun, X. Xiong, X.-B. Zou, Z.-F. Han, and G.-C. Guo, “Guiding light through optical bound states in the continuum for ultrahigh-Qmicroresonators,” Laser Photonics Rev. 9(1), 114–119 (2015). [CrossRef]  

54. D. C. Marinica, A. G. Borisov, and S. V. Shabanov, “Bound States in the continuum in photonics,” Phys. Rev. Lett. 100(18), 183902 (2008). [CrossRef]  

55. S. T. Ha, Y. H. Fu, N. K. Emani, Z. Pan, R. M. Bakker, R. Paniagua-Domínguez, and A. I. Kuznetsov, “Directional lasing in resonant semiconductor nanoantenna arrays,” Nat. Nanotechnol. 13(11), 1042–1047 (2018). [CrossRef]  

56. J. Lee, B. Zhen, S.-L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and Differentiation of Unique High-QOptical Resonances Near Zero Wave Vector in Macroscopic Photonic Crystal Slabs,” Phys. Rev. Lett. 109(6), 067401 (2012). [CrossRef]  

57. Y. Yang, Y. Ma, H. Guan, Y. Liu, S. Danziger, S. Ocheltree, K. Bergman, T. Baehr-Jones, and M. Hochberg, “Phase coherence length in silicon photonic platform,” Opt. Express 23(13), 16890–16902 (2015). [CrossRef]  

58. K. Van Acoleyen, W. Bogaerts, J. Jagerska, N. Le Thomas, R. Houdre, and R. Baets, “Off-chip beam steering with a one-dimensional optical phased array on silicon-on-insulator,” Opt. Lett. 34(9), 1477–1479 (2009). [CrossRef]  

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    [Crossref]
  50. C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1(9), 16048 (2016).
    [Crossref]
  51. B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljacic, “Topological nature of optical bound states in the continuum,” Phys. Rev. Lett. 113(25), 257401 (2014).
    [Crossref]
  52. C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, “Observation of trapped light within the radiation continuum,” Nature 499(7457), 188–191 (2013).
    [Crossref]
  53. C.-L. Zou, J.-M. Cui, F.-W. Sun, X. Xiong, X.-B. Zou, Z.-F. Han, and G.-C. Guo, “Guiding light through optical bound states in the continuum for ultrahigh-Qmicroresonators,” Laser Photonics Rev. 9(1), 114–119 (2015).
    [Crossref]
  54. D. C. Marinica, A. G. Borisov, and S. V. Shabanov, “Bound States in the continuum in photonics,” Phys. Rev. Lett. 100(18), 183902 (2008).
    [Crossref]
  55. S. T. Ha, Y. H. Fu, N. K. Emani, Z. Pan, R. M. Bakker, R. Paniagua-Domínguez, and A. I. Kuznetsov, “Directional lasing in resonant semiconductor nanoantenna arrays,” Nat. Nanotechnol. 13(11), 1042–1047 (2018).
    [Crossref]
  56. J. Lee, B. Zhen, S.-L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and Differentiation of Unique High-QOptical Resonances Near Zero Wave Vector in Macroscopic Photonic Crystal Slabs,” Phys. Rev. Lett. 109(6), 067401 (2012).
    [Crossref]
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    [Crossref]
  58. K. Van Acoleyen, W. Bogaerts, J. Jagerska, N. Le Thomas, R. Houdre, and R. Baets, “Off-chip beam steering with a one-dimensional optical phased array on silicon-on-insulator,” Opt. Lett. 34(9), 1477–1479 (2009).
    [Crossref]

2020 (6)

M. Chul Shin, A. Mohanty, K. Watson, G. R. Bhatt, C. T. Phare, S. A. Miller, M. Zadka, B. S. Lee, X. Ji, I. Datta, and M. Lipson, “Chip-scale blue light phased array,” Opt. Lett. 45(7), 1934–1937 (2020).
[Crossref]

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S. A. Miller, Y. C. Chang, C. T. Phare, M. C. Shin, M. Zadka, S. P. Roberts, B. Stern, X. C. Ji, A. Mohanty, O. A. J. Gordillo, U. D. Dave, and M. Lipson, “Large-scale optical phased array using a low-power multi-pass silicon photonic platform,” Optica 7(1), 3–6 (2020).
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P. Ginel-Moreno, D. Pereira-Martin, A. Hadij-ElHouati, W. N. Ye, D. Melati, D. X. Xu, S. Janz, A. Ortega-Monux, J. G. Wanguemert-Perez, R. Halir, I. Molina-Fernandez, J. H. Schmid, and P. Cheben, “Highly efficient optical antenna with small beam divergence in silicon waveguides,” Opt. Lett. 45(20), 5668–5671 (2020).
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H. N. Xu and Y. C. Shi, “Diffraction engineering for silicon waveguide grating antenna by harnessing bound state in the continuum,” Nanophotonics 9(6), 1439–1446 (2020).
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2019 (4)

R. Fatemi, A. Khachaturian, and A. Hajimiri, “A Nonuniform Sparse 2-D Large-FOV Optical Phased Array With a Low-Power PWM Drive,” IEEE J. Solid-State Circuits 54(5), 1200–1215 (2019).
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Y. Zhang, Y. C. Ling, K. Zhang, C. Gentry, D. Sadighi, G. Whaley, J. Colosimo, P. Suni, and S. J. Ben Yoo, “Sub-wavelength-pitch silicon-photonic optical phased array for large field-of-regard coherent optical beam steering,” Opt. Express 27(3), 1929–1940 (2019).
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Q. Wang, S. Wang, Y. Zeng, W. Wang, Y. Cai, Z. Tu, W. Yue, X. Wang, Q. Fang, and M. Yu, “Dual-layer waveguide grating antenna with high directionality for optical phased arrays,” Appl. Opt. 58(21), 5807–5811 (2019).
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J. G. Wangüemert-Pérez, A. Hadij-ElHouati, A. Sánchez-Postigo, J. Leuermann, D.-X. Xu, P. Cheben, A. Ortega-Moñux, R. Halir, and Í. Molina-Fernández, “[INVITED] Subwavelength structures for silicon photonics biosensing,” Opt. Laser Technol. 109, 437–448 (2019).
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2018 (7)

P. Cheben, R. Halir, J. H. Schmid, H. A. Atwater, and D. R. Smith, “Subwavelength integrated photonics,” Nature 560(7720), 565–572 (2018).
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K. Han, V. Yurlov, and N. E. Yu, “Highly directional waveguide grating antenna for optical phased array,” Curr. Appl. Phys. 18(7), 824–828 (2018).
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M. Zadka, Y. C. Chang, A. Mohanty, C. T. Phare, S. P. Roberts, and M. Lipson, “On-chip platform for a phased array with minimal beam divergence and wide field-of-view,” Opt. Express 26(3), 2528–2534 (2018).
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R. Halir, A. Ortega-Monux, D. Benedikovic, G. Z. Mashanovich, J. G. Wanguemert-Perez, J. H. Schmid, I. Molina-Fernandez, and P. Cheben, “Subwavelength-Grating Metamaterial Structures for Silicon Photonic Devices,” Proc. IEEE 106(12), 2144–2157 (2018).
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J. Ctyroky, J. Gonzalo Wanguemert-Perez, P. Kwiecien, I. Richter, J. Litvik, J. H. Schmid, I. Molina-Fernandez, A. Ortega-Monux, M. Dado, and P. Cheben, “Design of narrowband Bragg spectral filters in subwavelength grating metamaterial waveguides,” Opt. Express 26(1), 179–194 (2018).
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2017 (9)

D. Benedikovic, C. Alonso-Ramos, D. Perez-Galacho, S. Guerber, V. Vakarin, G. Marcaud, X. Le Roux, E. Cassan, D. Marris-Morini, P. Cheben, F. Boeuf, C. Baudot, and L. Vivien, “L-shaped fiber-chip grating couplers with high directionality and low reflectivity fabricated with deep-UV lithography,” Opt. Lett. 42(17), 3439–3442 (2017).
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D. Perez-Galacho, C. Alonso-Ramos, F. Mazeas, X. Le Roux, D. Oser, W. Zhang, D. Marris-Morini, L. Labonte, S. Tanzilli, E. Cassan, and L. Vivien, “Optical pump-rejection filter based on silicon sub-wavelength engineered photonic structures,” Opt. Lett. 42(8), 1468–1471 (2017).
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H. Podmore, A. Scott, P. Cheben, A. V. Velasco, J. H. Schmid, M. Vachon, and R. Lee, “Demonstration of a compressive-sensing Fourier-transform on-chip spectrometer,” Opt. Lett. 42(7), 1440–1443 (2017).
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C. V. Poulton, M. J. Byrd, M. Raval, Z. Su, N. Li, E. Timurdogan, D. Coolbaugh, D. Vermeulen, and M. R. Watts, “Large-scale silicon nitride nanophotonic phased arrays at infrared and visible wavelengths,” Opt. Lett. 42(1), 21–24 (2017).
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M. Raval, C. V. Poulton, and M. R. Watts, “Unidirectional waveguide grating antennas with uniform emission for optical phased arrays,” Opt. Lett. 42(13), 2563–2566 (2017).
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M. J. R. Heck, “Highly integrated optical phased arrays: photonic integrated circuits for optical beam shaping and beam steering,” Nanophotonics 6(1), 93–107 (2017).
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C. V. Poulton, A. Yaacobi, D. B. Cole, M. J. Byrd, M. Raval, D. Vermeulen, and M. R. Watts, “Coherent solid-state LIDAR with silicon photonic optical phased arrays,” Opt. Lett. 42(20), 4091–4094 (2017).
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D. Benedikovic, M. Berciano, C. Alonso-Ramos, X. L. Roux, and L. J. O. E. Vivien, “Dispersion control of silicon nanophotonic waveguides using sub-wavelength grating metamaterials in near- and mid-IR wavelengths,” Opt. Express 25(16), 19468–19478 (2017).
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2016 (6)

2015 (7)

C.-L. Zou, J.-M. Cui, F.-W. Sun, X. Xiong, X.-B. Zou, Z.-F. Han, and G.-C. Guo, “Guiding light through optical bound states in the continuum for ultrahigh-Qmicroresonators,” Laser Photonics Rev. 9(1), 114–119 (2015).
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Y. Yang, Y. Ma, H. Guan, Y. Liu, S. Danziger, S. Ocheltree, K. Bergman, T. Baehr-Jones, and M. Hochberg, “Phase coherence length in silicon photonic platform,” Opt. Express 23(13), 16890–16902 (2015).
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H. Abediasl and H. Hashemi, “Monolithic optical phased-array transceiver in a standard SOI CMOS process,” Opt. Express 23(5), 6509–6519 (2015).
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J. C. Hulme, J. K. Doylend, M. J. Heck, J. D. Peters, M. L. Davenport, J. T. Bovington, L. A. Coldren, and J. E. Bowers, “Fully integrated hybrid silicon two dimensional beam scanner,” Opt. Express 23(5), 5861–5874 (2015).
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D. Benedikovic, P. Cheben, J. H. Schmid, D. X. Xu, B. Lamontagne, S. Wang, J. Lapointe, R. Halir, A. Ortega-Monux, S. Janz, and M. Dado, “Subwavelength index engineered surface grating coupler with sub-decibel efficiency for 220-nm silicon-on-insulator waveguides,” Opt. Express 23(17), 22628–22635 (2015).
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V. Donzella, A. Sherwali, J. Flueckiger, S. M. Grist, S. T. Fard, and L. Chrostowski, “Design and fabrication of SOI micro-ring resonators based on sub-wavelength grating waveguides,” Opt. Express 23(4), 4791–4803 (2015).
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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).
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2014 (5)

J. Wang, I. Glesk, and L. R. Chen, “Subwavelength grating filtering devices,” Opt. Express 22(13), 15335–15345 (2014).
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D. Kwong, A. Hosseini, J. Covey, Y. Zhang, X. Xu, H. Subbaraman, and R. T. Chen, “On-chip silicon optical phased array for two-dimensional beam steering,” Opt. Lett. 39(4), 941–944 (2014).
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D. Kwong, A. Hosseini, J. Covey, X. Xiaochuan, Z. Yang, S. Chakravarty, and R. T. Chen, “Corrugated Waveguide-Based Optical Phased Array With Crosstalk Suppression,” IEEE Photonics Technol. Lett. 26(10), 991–994 (2014).
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S. Jie, E. Timurdogan, A. Yaacobi, S. Zhan, E. S. Hosseini, D. B. Cole, and M. R. Watts, “Large-Scale Silicon Photonic Circuits for Optical Phased Arrays,” IEEE J. Sel. Top. Quantum Electron. 20(4), 264–278 (2014).
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B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljacic, “Topological nature of optical bound states in the continuum,” Phys. Rev. Lett. 113(25), 257401 (2014).
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2013 (3)

C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, “Observation of trapped light within the radiation continuum,” Nature 499(7457), 188–191 (2013).
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J. Sun, E. Timurdogan, A. Yaacobi, E. S. Hosseini, and M. R. Watts, “Large-scale nanophotonic phased array,” Nature 493(7431), 195–199 (2013).
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G. Weihua, P. R. A. Binetti, C. Althouse, M. L. Masanovic, H. P. M. M. Ambrosius, L. A. Johansson, and L. A. Coldren, “Two-Dimensional Optical Beam Steering With InP-Based Photonic Integrated Circuits,” IEEE J. Sel. Top. Quantum Electron. 19(4), 6100212 (2013).
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2012 (1)

J. Lee, B. Zhen, S.-L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and Differentiation of Unique High-QOptical Resonances Near Zero Wave Vector in Macroscopic Photonic Crystal Slabs,” Phys. Rev. Lett. 109(6), 067401 (2012).
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2010 (2)

2009 (2)

2008 (1)

D. C. Marinica, A. G. Borisov, and S. V. Shabanov, “Bound States in the continuum in photonics,” Phys. Rev. Lett. 100(18), 183902 (2008).
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2006 (1)

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Alonso-Ramos, C.

D. Perez-Galacho, C. Alonso-Ramos, F. Mazeas, X. Le Roux, D. Oser, W. Zhang, D. Marris-Morini, L. Labonte, S. Tanzilli, E. Cassan, and L. Vivien, “Optical pump-rejection filter based on silicon sub-wavelength engineered photonic structures,” Opt. Lett. 42(8), 1468–1471 (2017).
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D. Benedikovic, M. Berciano, C. Alonso-Ramos, X. L. Roux, and L. J. O. E. Vivien, “Dispersion control of silicon nanophotonic waveguides using sub-wavelength grating metamaterials in near- and mid-IR wavelengths,” Opt. Express 25(16), 19468–19478 (2017).
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D. Benedikovic, C. Alonso-Ramos, D. Perez-Galacho, S. Guerber, V. Vakarin, G. Marcaud, X. Le Roux, E. Cassan, D. Marris-Morini, P. Cheben, F. Boeuf, C. Baudot, and L. Vivien, “L-shaped fiber-chip grating couplers with high directionality and low reflectivity fabricated with deep-UV lithography,” Opt. Lett. 42(17), 3439–3442 (2017).
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J. Wang, R. Ashrafi, R. Adams, I. Glesk, I. Gasulla, J. Capmany, and L. R. Chen, “Subwavelength grating enabled on-chip ultra-compact optical true time delay line,” Sci. Rep. 6(1), 30235 (2016).
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P. Cheben, R. Halir, J. H. Schmid, H. A. Atwater, and D. R. Smith, “Subwavelength integrated photonics,” Nature 560(7720), 565–572 (2018).
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G. Weihua, P. R. A. Binetti, C. Althouse, M. L. Masanovic, H. P. M. M. Ambrosius, L. A. Johansson, and L. A. Coldren, “Two-Dimensional Optical Beam Steering With InP-Based Photonic Integrated Circuits,” IEEE J. Sel. Top. Quantum Electron. 19(4), 6100212 (2013).
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P. Cheben, P. J. Bock, J. H. Schmid, J. Lapointe, S. Janz, D. X. Xu, A. Densmore, A. Delage, B. Lamontagne, and T. J. Hall, “Refractive index engineering with subwavelength gratings for efficient microphotonic couplers and planar waveguide multiplexers,” Opt. Lett. 35(15), 2526–2528 (2010).
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D. C. Marinica, A. G. Borisov, and S. V. Shabanov, “Bound States in the continuum in photonics,” Phys. Rev. Lett. 100(18), 183902 (2008).
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J. Wang, R. Ashrafi, R. Adams, I. Glesk, I. Gasulla, J. Capmany, and L. R. Chen, “Subwavelength grating enabled on-chip ultra-compact optical true time delay line,” Sci. Rep. 6(1), 30235 (2016).
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Cassan, E.

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D. Kwong, A. Hosseini, J. Covey, X. Xiaochuan, Z. Yang, S. Chakravarty, and R. T. Chen, “Corrugated Waveguide-Based Optical Phased Array With Crosstalk Suppression,” IEEE Photonics Technol. Lett. 26(10), 991–994 (2014).
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J. G. Wangüemert-Pérez, A. Hadij-ElHouati, A. Sánchez-Postigo, J. Leuermann, D.-X. Xu, P. Cheben, A. Ortega-Moñux, R. Halir, and Í. Molina-Fernández, “[INVITED] Subwavelength structures for silicon photonics biosensing,” Opt. Laser Technol. 109, 437–448 (2019).
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R. Halir, A. Ortega-Monux, D. Benedikovic, G. Z. Mashanovich, J. G. Wanguemert-Perez, J. H. Schmid, I. Molina-Fernandez, and P. Cheben, “Subwavelength-Grating Metamaterial Structures for Silicon Photonic Devices,” Proc. IEEE 106(12), 2144–2157 (2018).
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P. Cheben, R. Halir, J. H. Schmid, H. A. Atwater, and D. R. Smith, “Subwavelength integrated photonics,” Nature 560(7720), 565–572 (2018).
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J. Ctyroky, J. Gonzalo Wanguemert-Perez, P. Kwiecien, I. Richter, J. Litvik, J. H. Schmid, I. Molina-Fernandez, A. Ortega-Monux, M. Dado, and P. Cheben, “Design of narrowband Bragg spectral filters in subwavelength grating metamaterial waveguides,” Opt. Express 26(1), 179–194 (2018).
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D. Benedikovic, C. Alonso-Ramos, D. Perez-Galacho, S. Guerber, V. Vakarin, G. Marcaud, X. Le Roux, E. Cassan, D. Marris-Morini, P. Cheben, F. Boeuf, C. Baudot, and L. Vivien, “L-shaped fiber-chip grating couplers with high directionality and low reflectivity fabricated with deep-UV lithography,” Opt. Lett. 42(17), 3439–3442 (2017).
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H. Podmore, A. Scott, P. Cheben, A. V. Velasco, J. H. Schmid, M. Vachon, and R. Lee, “Demonstration of a compressive-sensing Fourier-transform on-chip spectrometer,” Opt. Lett. 42(7), 1440–1443 (2017).
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D. Benedikovic, P. Cheben, J. H. Schmid, D. X. Xu, B. Lamontagne, S. Wang, J. Lapointe, R. Halir, A. Ortega-Monux, S. Janz, and M. Dado, “Subwavelength index engineered surface grating coupler with sub-decibel efficiency for 220-nm silicon-on-insulator waveguides,” Opt. Express 23(17), 22628–22635 (2015).
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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).
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P. Cheben, P. J. Bock, J. H. Schmid, J. Lapointe, S. Janz, D. X. Xu, A. Densmore, A. Delage, B. Lamontagne, and T. J. Hall, “Refractive index engineering with subwavelength gratings for efficient microphotonic couplers and planar waveguide multiplexers,” Opt. Lett. 35(15), 2526–2528 (2010).
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P. Cheben, D. X. Xu, S. Janz, and A. Densmore, “Subwavelength waveguide grating for mode conversion and light coupling in integrated optics,” Opt. Express 14(11), 4695–4702 (2006).
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Chen, L. R.

Chen, R. T.

D. Kwong, A. Hosseini, J. Covey, X. Xiaochuan, Z. Yang, S. Chakravarty, and R. T. Chen, “Corrugated Waveguide-Based Optical Phased Array With Crosstalk Suppression,” IEEE Photonics Technol. Lett. 26(10), 991–994 (2014).
[Crossref]

D. Kwong, A. Hosseini, J. Covey, Y. Zhang, X. Xu, H. Subbaraman, and R. T. Chen, “On-chip silicon optical phased array for two-dimensional beam steering,” Opt. Lett. 39(4), 941–944 (2014).
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Chrostowski, L.

Chua, S. L.

C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, “Observation of trapped light within the radiation continuum,” Nature 499(7457), 188–191 (2013).
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Chua, S.-L.

J. Lee, B. Zhen, S.-L. Chua, W. Qiu, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Observation and Differentiation of Unique High-QOptical Resonances Near Zero Wave Vector in Macroscopic Photonic Crystal Slabs,” Phys. Rev. Lett. 109(6), 067401 (2012).
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Chul Shin, M.

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J. C. Hulme, J. K. Doylend, M. J. Heck, J. D. Peters, M. L. Davenport, J. T. Bovington, L. A. Coldren, and J. E. Bowers, “Fully integrated hybrid silicon two dimensional beam scanner,” Opt. Express 23(5), 5861–5874 (2015).
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G. Weihua, P. R. A. Binetti, C. Althouse, M. L. Masanovic, H. P. M. M. Ambrosius, L. A. Johansson, and L. A. Coldren, “Two-Dimensional Optical Beam Steering With InP-Based Photonic Integrated Circuits,” IEEE J. Sel. Top. Quantum Electron. 19(4), 6100212 (2013).
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Figures (6)

Fig. 1.
Fig. 1. (a) The schematic of the proposed grating. (b)Top and (c) cross-section of the proposed WGA. (d)Electric field (|E|) distribution of the fundamental mode supported by the sections with (Top) and without (Bottom) subwavelength segments. (e) Electric field (|E|) distribution of the fundamental mode in WGA sections with and without subwavelength segments along the y-axis.
Fig. 2.
Fig. 2. (a) Effective refractive indices of the modes supported by WGA sections with subwavelength segments when the gap d equals 100 nm (blue), 150 nm (purple), and 200 nm (red), respectively (w1=400 nm, w2=300 nm, dc=0.25). (b) Effective refractive indices in relation to w2 when gap d is 100 nm (blue), 150 nm (purple), and 200 nm (red) at the wavelength of 1.5 μm. Inset: |E| distribution of the fundamental mode with w2=300 nm (top), w2=400 nm (middle), and the 2nd-order mode with w2=300 nm (bottom). The back dash line indicates the effective index of a strip waveguide with a width of 400 nm. The results are simulated by the finite-difference eigenmode (FDE) method.
Fig. 3.
Fig. 3. (a) The possible emitting directions in WGA with subwavelength segments. (b) Normalized upward emission (orange dot), sideward emission (orange circle) and its ratio (black dot) in relation with w1 (d=100 nm, w2=160 nm, dc=0.2). Field profiles of WGAs with subwavelength segments when (c) w1 = 370 nm and (d) w1=500 nm. All the simulation is done at 1.55 μm.
Fig. 4.
Fig. 4. (a) Typical two-spot far field pattern and (b) normalized field distribution along ψ direction at 1.55μm ( w1=500 nm, dc=0.4, d=100 nm ). (c) ΔI in relation to w1 and dc (d=100 nm,w2=160 nm). (d) The ratio of sideward to upward emission in relation to w1 and dc (d=100 nm, w2=160 nm). (e) The ratio of sideward to upward emission in relation to w2 (red) (w1=400 nm, dc=0.25, d=100 nm) and d (blue) (w1=400 nm, dc=0.25, w2=160 nm) with wavelength of 1.5, 1.55 and 1.6 μm.(f) Coupling length of strip waveguide(w1=400 nm) and proposed waveguide grating(w1=400 nm,w2=190 nm,d=100 nm,dc=0.25,Λ=800 nm) whcn the center-to-center distance equals 1.3 μm and 1.5 μm.
Fig. 5.
Fig. 5. Grating strength variation in relation to (a) w2 and d (w1=400 nm, dc=0.2), (b) w2 and dc (w1=400 nm, d=100 nm), and (c) wavelength and w2 (w1 = 400 nm, dc=0.2, d=100 nm). The white line indicates the single mode condition.
Fig. 6.
Fig. 6. (a) Power attenuation within the proposed subwavelength segments (blue) and SCG (red) based WGAs. Inset: the structure of the SCG based WGA used for comparison. The black dash line corresponds to 10% of the initial power. (b) The far-field profile of the proposed WGA at 1.55 μm. (c) The far-field profile along θ direction. The blue dots are simulated with FDTD. The divergence angle is estimated by Gaussian fitting. (d) Typical far-field profiles along θ direction when wavelength sweeps from 1.5 μm to 1.6 μm.

Tables (1)

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Table 1. Typical waveguide grating antennas for OPAs

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