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Abstract
We propose and theoretically demonstrate an ultrashort multimode waveguide taper based on the all-dielectric metamaterial. Attributed to the gradient index distribution of the metamaterial, the spot sizes of the four lowest-order transverse magnetic (TM) modes can be expanded in a short distance of 6 μm with negligible mode conversions. Numerical results prove that the insertion losses of the taper are lower than 1 dB, 1.12 dB, 1.26 dB and 1.66 dB for the TM0 - TM3 modes, respectively, and the intermodal crosstalk values are below -15 dB for the four modes, both in the wavelength range of 1.5 μm - 1.6 μm. To the best of our knowledge, this is the first multimode waveguide taper that has low intermodal crosstalk of < -15 dB over a 100-nm bandwidth.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Mode-division multiplexing (MDM), where signals are carried by multiple guided modes in parallel, provides a promising and cost-effective approach for achieving more channels and increasing the link capacity with a single wavelength carrier [1,2]. In recent years, great efforts have been devoted to silicon photonic multimode devices for on-chip MDM systems. For example, a number of key components have been developed, including mode (de)multiplexers [3–6], multimode switches [7–10], multimode waveguide bends [11–14], multimode waveguide crossings [15–17], multimode chip-fiber couplers [18,19], etc. The multimode waveguide taper is essential for connecting the waveguides of different widths in the MDM systems [5,18]. However, the conventional tapers are usually very long because they employ the adiabatic mode evolution scheme to suppress the intermodal crosstalk [20–22]. Although short waveguide tapers based on subwavelength structures [23–27] and flattened lenses [28–32] have been proposed recently, most of them focus on the fundamental modes and pay little attention to the high-order modes [20–30]. Therefore, a compact taper working for multiple modes is very much desired in the multimode transmission systems.
Gradient-index (GRIN) optics offers a large degree of freedom to manipulate the propagation of light in the integrated photonic chips. Combined with transformation optics (TO) [33–35], it enables a plethora of innovative designs such as invisible cloaks [36,37], spot size converters [31,38], Maxwell’s fisheye [15,17] and Luneburg lenses [39–41], dual-function “Janus” devices [42], etc. Thanks to nowadays high-resolution lithography techniques, the GRIN materials can be obtained by gray-scale electron-beam lithography (EBL) or subwavelength metamaterials [15,16,43]. In this paper, we propose a metamaterial-based ultrashort waveguide taper with low intermodal crosstalk, which can be implemented on the silicon-on-insulator (SOI) platform with CMOS-compatible processing. We use the quasi-conformal (QC) mapping technique to derive the refractive index distribution and approximate the index profile with a metamaterial consisting of silicon nanopillars. Attributed to the adiabatic mode evolution in the metamaterial region, the multiple input modes are preserved throughout the taper simultaneously, leading to low intermodal crosstalk and broad bandwidths in the telecom band. We investigate the behaviors of the four lowest-order transverse magnetic (TM) modes of the strip waveguide when they propagate through the metamaterial taper. The principal component of the electric field of the TM modes is Ez, which is parallel to the axes of the nanorods that compose the metamaterial. Numerical results prove that for these four TM modes the intermodal crosstalk can be controlled below -15 dB and the insertion losses are smaller than 1.66 dB in the wavelength range of 1.5 μm - 1.6 μm within a metamaterial taper of only 6 μm connecting a 3-μm-wide waveguide to a 4.5-μm-wide waveguide. The method we use in designing the waveguide taper shows potential in manipulating the mode evolution in the multimode devices and realizing the on-chip MDM systems with high efficiency and integration density.
2. Design and analysis
Figure 1 shows a traditional waveguide taper built on the SOI platform and how the TM modes evolve in the ultrashort taper. The 3D schematic view and the top view of the device are shown in Figs. 1(a) and 1(b), respectively. It is based on the SOI wafer with a 340-nm-thick top silicon layer and a 2-μm-thick buried oxide layer. The top cladding material is silica. The taper connects a 3-μm-wide waveguide to a 4.5-μm-wide waveguide with a length of 6 μm, which corresponds to a taper angle of 7.1°. For the waveguide modes to evolve adiabatically without converting into other modes, the taper angle is required to be no larger than 1.8° [3]. We use the 3D finite element method (FEM) to simulate the propagation and evolution of the TM0 - TM3 modes in the ultrashort taper. Clear mode conversion and mode profile distortion can be observed due to the large taper angle, as shown in Fig. 1(c). The mode purities drop to 92.57%, 83.15%, 71.33% and 64.29% for these four modes, respectively, after passing the taper region. That explains why the refractive index engineering is necessary in an ultrashort taper to mold the flow of light and suppress the conversions between the guided modes.
Fig. 1. (a) 3D schematic view and (b) top view of a traditional ultrashort waveguide taper based on the SOI platform. The taper connects a 3-μm-wide waveguide to a 4.5-μm-wide waveguide with a length of 6 μm. (c) Mode evolutions of the TM0 - TM3 modes in the ultrashort taper. The Ez fields are depicted because they are the principal electric field components of the TM modes.
To design a GRIN ultrashort taper with low intermodal crosstalk, we consider the coordinate transformation that can map the virtual space to the physical space shown in Fig. 2(a). The virtual space contains a homogenous medium with a uniform refractive index. Mode conversions will not happen when waves propagate in such space. Since the physical laws of electromagnetic waves are preserved during the transformation, the absence of mode conversions could also be expected in the physical space which is a GRIN ultrashort taper structure. The dimensions of the transformation domain are the same as those of the taper in Fig. 1(b). The QC transformation optics (QCTO) approach is employed to minimize the anisotropy of the constitutive materials, allowing the all-dielectric metamaterial implementation [33,35]. The mesh grids in the physical space are numerically generated by solving Poisson’s equation with Neumann-Dirichlet boundary conditions [44–46]. As a result, the longitudinal and latitudinal geodesic lines created by the QC mapping possess strong orthogonality over the entire transformation domain, except at the four turning points A, B, C and D indicated in Fig. 2(a). The refractive index distribution of the GRIN medium can be found by [35,47–49]
where nref is the refractive index of the background material and g is the covariant metric. Here we focus on the TM modes of which the Ez components are the principal ones, so the propagation of light is mostly affected by the εzz element of the permittivity tensor. Figure 2(b) plots the neff distribution within the taper and the waveguides. Here nref is chosen to be 2.2 so that neff changes from 1.76 to 2.4 in most of the area of the taper. The effective index range can be fully covered by the metamaterial we will show later. The extreme value of index appears only in the small areas around the turning points and therefore has little influence on the guided modes [33,35,46]. The effective refractive index of the waveguides is 2.77 while the effective refractive indices at the input and output ends of the taper are ∼1.9 and ∼2.3, respectively. The refractive index mismatch causes reflections at the interfaces, which increases the insertion loss of the designed taper. To verify the mode-preserving property of the index-engineered taper, we use the 3D FEM to simulate the propagation and evolution of the TM0 - TM3 modes in the medium with exactly the same index distribution as shown in Fig. 2(b). The results are displayed in Fig. 2(c) where the profiles (Ez) of the TM0 - TM3 modes are restored compared to those in Fig. 1(c). This indicates that the GRIN taper designed by the QCTO approach can expand the spot sizes of multiple modes in a short length while preventing the mode conversions caused by the abrupt variation of the waveguide width. The physical mechanism behind this phenomenon can be explained as follows. After the QCTO transformation, normally incident waves will propagate along the latitudinal geodesic lines in the physical space with the equal-phase surfaces following the longitudinal ones. The field distribution on the equal-phase surface is then enlarged gradually without introducing any other mode distributions. Thus, the mode profile at the output end of the taper is just the expanded version of that at the input end, leading to a high mode purity and suppressed intermodal crosstalk after tapering.Fig. 2. (a) QCTO method transforms the virtual space to the physical space. The quasi-orthogonal grids are obtained after transformation. (b) Refractive index distribution of neff derived from QCTO. (c) Propagation and evolution of the TM0 - TM3 modes in the medium with the gradient index shown in Fig. 2(b). The Ez fields are depicted here.
To realize the index distribution on the SOI platform, we propose a metamaterial composed of an array of nanopillars whose effective index can be engineered by varying the nanopillar filling factor, as shown in Fig. 3. Figure 3(a) presents the 3D schematic view of the proposed metamaterial device. Each pixel of the array consists of a nanopillar with the position-dependent radius R sitting on a 250 nm × 250 nm square tablet. Due to the limitation of the minimum pixel size, it is only a close approximation of the GRIN material and structure shown in Fig. 2(b). The top view and the side view of a pixel are shown in Fig. 3(b). The heights of the nanopillar and the tablet are both 170 nm. The top cladding material is silica. The structure can be fabricated on a standard SOI wafer with a 340-nm-thick top silicon layer by using the EBL to define the structure, the inductively coupled plasma (ICP) dry etching to transfer the pattern onto the silicon wafer, and finally the plasma enhanced chemical vapor deposition (PECVD) to deposit the silica [43]. The equivalent model of the slab is shown in the inset of Fig. 3(c), where nSi = 3.46 and nSiO2 = 1.45 are the refractive indices of silicon and silica, respectively. According to the effective medium theory (EMT), the effective index of the metamaterial nmeta is given by [16,50–52]
where η is the filling factor of the nanopillars with space-variant radii. Figure 3(c) depicts the effective index of the TM slab mode calculated using the FEM eigen mode solver as a function of the nanopillar radius. It changes from 1.74 to 2.6 as the nanopillar radius R increases from 25 nm to 125 nm. To make it feasible for fabrication, we only choose the nanopillar radius in the range of 30 nm - 105 nm so that the minimum gap in the structure is larger than 40 nm and the minimum diameter of the nanorod is no smaller than 60 nm [16,52]. It corresponds to an effective index of 1.76 - 2.4, which can basically meet the index requirement of the proposed device except in the small neighborhoods close to the turning points. In the following section, we will show the transmission of several TM modes through the metamaterial taper and discuss its performance in maintaining the mode profile and suppressing the intermodal crosstalk.Fig. 3. (a) 3D schematic view of the metamaterial-based ultrashort taper. (b) Top view (top subplot) and side view (bottom subplot) of the pixels that comprise the metamaterial section. (c) Effective index of the TM slab mode as a function of nanopillar radius R. The inset shows the cross section of the equivalent waveguiding slab.
However, the application of this method to the design of a transverse electric (TE) mode taper is limited due to the following reason. After the QC transformation is applied, the derived permittivity and permeability components out of the plane of propagation are varied while the in-plane components are approximately constant. For TM polarized waves, the principal component of the electric field (Ez) is out of the plane while that of the magnetic field (Hy) is in the plane. Therefore, only the variation of the out-of-plane permittivity and the approximately unity in-plane permeability are required for TM modes, which can be implemented easily with all-dielectric metamaterials. However, for TE polarized waves, the principal component of the electric field (Ey) is restricted in the plane of propagation with that of the magnetic field (Hz) being out of the plane. A spatially variant out-of-plane permeability is needed to control the flow of TE modes, which cannot be realized with dielectric-only materials. Thus, by using the QCTO approach and patterning the dielectric waveguides, one cannot manipulate the TE modes as we did with the TM modes.
3. Results and discussions
In order to characterize the performance of the metamaterial-based ultrashort waveguide taper, full-wave numerical simulations are performed using the 3D finite-difference time-domain (FDTD) methods. The light propagation profiles (Ez) in Figs. 4(a-d) correspond to the TM0 - TM3 modes in the proposed device at the central wavelength of 1.55 μm. The input TM modes are expanded adiabatically in the metamaterial taper region and emerge from the output end with nearly planar equal-phase surfaces. Therefore, the guided mode in the narrow waveguide can couple to the same mode in the wide waveguide with a high efficiency, resulting in depressed intermodal crosstalk. It is further verified by the transmission responses of the multimode waveguide taper based on such a metamaterial, as presented in Fig. 4. We use the mode overlapping between the output field and the waveguide eigen modes to calculate the mode losses and crosstalk (CT) levels [53]. The insertion losses (ILs) are lower than 1 dB, 1.12 dB, 1.26 dB and 1.66 dB for the TM0 - TM3 modes over a 100-nm wavelength span from 1.5 μm to 1.6 μm. The losses originate from the abrupt change of the effective index at the interfaces between the conventional straight waveguides and the metamaterial taper. The IL increases with mode order, which can be explained as follows. The metamaterial device is designed based on the εzz distribution found by the QCTO approach, so it works well when the Ez component of the mode is principal. For higher-order modes, electric components other than Ez become non-negligible, which means the GRIN metamaterial has less control over the flow of light as the mode order increases. It gives rise to the degradation of performance such as ILs. As expected, the CT values are smaller than -15 dB for the four lowest-order TM modes at wavelengths ranging from 1.5 μm to 1.6 μm. The broad bandwidth benefits from the wavelength-insensitive property of the adiabatic mode evolution process taking place in the GRIN metamaterial region. It is also worth mentioning that a larger taper angle will cause severer refractive index mismatch between the taper and the waveguides, which in turn deteriorates the performance of the taper further. Also, a larger taper angle will result in a wider range of refractive index according to the QCTO method, which could be hard to implement with all-dielectric metamaterials.
Fig. 4. Ez field profiles and transmission spectra for the metamaterial-based multimode waveguide taper when (a) TM0, (b) TM1, (c) TM2 and (d) TM3 modes are launched from the left narrow waveguide.
To make a comparison between the conventional ultrashort waveguide taper and the metamaterial counterpart, we calculate the mode purities of the TM0 - TM3 modes at the output end of the taper at the wavelength of 1.55 μm for both cases, as exhibited in Fig. 5. The dimensions of the metamaterial taper are consistent with those of the conventional taper shown in Figs. 1(a) and 1(b), sharing the same taper angle of 7.1°. We find that the mode purity drops from 92.57% to 64.29% for the traditional taper as the input mode changes from TM0 to TM3. Large intermodal CT exists as a result of the strong diffraction and scattering in the fast expanding region. So the performance of the multimode taper deteriorates seriously for higher-order modes. In contrast, the mode purities of the metamaterial taper are 94.89%, 91.59%, 89.79% and 87.22% for the TM0 - TM3 modes, respectively. It shows clear evidence that the mode conversions are largely suppressed in the proposed device compared to those in the conventional waveguide taper. Consequently, the intermodal crosstalk is controlled below -15 dB for the four lowest-order TM modes supported by the multimode waveguide, as shown in Fig. 4. We also observe a minor decrease of the mode purity as the mode order gets higher, which can be explained by the same reason that causes larger ILs for higher-order modes. Nevertheless, it still outperforms the traditional ultrashort waveguide taper in expanding the spot sizes of multiple modes simultaneously with high mode purities.
Fig. 5. Mode purities of the TM0 - TM3 modes after passing the metamaterial-based waveguide taper (black solid line with square symbols) and the traditional waveguide taper (red solid line with round symbols).
Table 1 summaries several reported results of the compact waveguide tapers and compares them with our device. Our work can handle four modes at the same time while most of the designs can only handle a single mode. Besides, the proposed device has very low intermodal crosstalk and relatively low ILs in a broad bandwidth of 100 nm. Most importantly, it can be fabricated on the SOI platform with CMOS-compatible processing. The proposed taper could find its application in the mode demultiplexers based on asymmetric directional couplers. Very long adiabatic tapers are usually used in this kind of demultiplexers to connect the waveguides of different widths, which makes the total length of the device extremely long (on the order of 100 μm to 1 mm). By using the metamaterial taper, the size of the demultiplexer can be greatly reduced with the intermodal crosstalk still kept at a low level.
Table 1. Comparison of various compact waveguide tapers
4. Conclusion
In conclusion, we have proposed and theoretically demonstrated a metamaterial-based ultrashort waveguide taper. The QC mapping technique is adopted to calculate the refractive index profile required to suppress the intermodal crosstalk. The metamaterial composed of a nanopillar array is used to realize the gradient index distribution on the SOI platform. Numerical results show that the metamaterial taper has insertion losses lower than 1 dB, 1.12 dB, 1.26 dB and 1.66 dB for the TM0 - TM3 modes and intermodal crosstalk values below -15 dB for the four modes over a 100-nm bandwidth spanning from 1.5 μm to 1.6 μm. The method we used in this work can be applied to the designs of other compact and low-crosstalk multimode devices which may play important roles in realizing on-chip MDM systems with high efficiency and integration density.
Funding
National Key Research and Development Program of China (2019YFB1803903); National Natural Science Foundation of China (62035016).
Disclosures
The authors declare no conflicts of interest.
References
1. D. Dai, J. Wang, and S. He, “Silicon multimode photonic integrated devices for on-chip mode-division-multiplexed optical interconnects,” Prog. Electromagn. Res. 143, 773–819 (2013). [CrossRef]
2. C. Li, D. Liu, and D. Dai, “Multimode silicon photonics,” Nanophotonics 8(2), 227–247 (2018). [CrossRef]
3. D. Dai, J. Wang, and Y. Shi, “Silicon mode (de)multiplexer enabling high capacity photonic networks-on-chip with a single-wavelength-carrier light,” Opt. Lett. 38(9), 1422–1424 (2013). [CrossRef]
4. D. Guo and T. Chu, “Silicon mode (de) multiplexers with parameters optimized using shortcuts to adiabaticity,” Opt. Express 25(8), 9160–9170 (2017). [CrossRef]
5. Y. He, Y. Zhang, Q. Zhu, S. An, R. Cao, X. Guo, C. Qiu, and Y. Su, “Silicon high-order mode (de)multiplexer on single polarization,” J. Lightwave Technol. 36(24), 5746–5753 (2018). [CrossRef]
6. D. Dai, C. Li, S. Wang, H. Wu, Y. Shi, Z. Wu, S. Gao, T. Dai, H. Yu, and H.-K. Tsang, “10-channel mode (de)multiplexer with dual polarizations,” Laser Photonics Rev. 12(1), 1700109 (2018). [CrossRef]
7. B. Stern, X. Zhu, C. P. Chen, L. D. Tzuang, J. Cardenas, K. Bergman, and M. Lipson, “On-chip mode-division multiplexing switch,” Optica 2(6), 530–535 (2015). [CrossRef]
8. Y. Xiong, R. B. Priti, and O. Liboiron-Ladouceur, “High-speed two-mode switch for mode-division multiplexing optical networks,” Optica 4(9), 1098–1102 (2017). [CrossRef]
9. Y. Zhang, Y. He, Q. Zhu, C. Qiu, and Y. Su, “On-chip silicon photonic 2×2 mode-and polarization-selective switch with low inter-modal crosstalk,” Photonics Res. 5(5), 521–526 (2017). [CrossRef]
10. L. Yang, T. Zhou, H. Jia, S. Yang, J. Ding, X. Fu, and L. Zhang, “General architectures for on-chip optical space and mode switching,” Optica 5(2), 180–187 (2018). [CrossRef]
11. L. H. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun. 3(1), 1–6 (2012). [CrossRef]
12. X. Jiang, H. Wu, and D. Dai, “Low-loss and low-crosstalk multimode waveguide bend on silicon,” Opt. Express 26(13), 17680–17689 (2018). [CrossRef]
13. H. Xu and Y. Shi, “Ultra-sharp multi-mode waveguide bending assisted with metamaterial-based mode converters,” Laser Photonics Rev. 12(3), 1700240 (2018). [CrossRef]
14. H. Wu, C. Li, L. Song, H.-K. Tsang, J. E. Bowers, and D. Dai, “Ultra-sharp multimode waveguide bends with subwavelength gratings,” Laser Photonics Rev. 13(2), 1800119 (2019). [CrossRef]
15. S. Li, Y. Zhou, J. Dong, X. Zhang, E. Cassan, J. Hou, C. Yang, S. Chen, D. Gao, and H. Chen, “Universal multimode waveguide crossing based on transformation optics,” Optica 5(12), 1549–1556 (2018). [CrossRef]
16. H. Xu and Y. Shi, “Metamaterial-based Maxwell's fisheye lens for multimode waveguide crossing,” Laser Photonics Rev. 12(10), 1800094 (2018). [CrossRef]
17. S. H. Badri, H. R. Saghai, and H. Soofi, “Polygonal Maxwell’s fisheye lens via transformation optics as multimode waveguide crossing,” J. Opt. 21(6), 065102 (2019). [CrossRef]
18. D. Dai and M. Mao, “Mode converter based on an inverse taper for multimode silicon nanophotonic integrated circuits,” Opt. Express 23(22), 28376–28388 (2015). [CrossRef]
19. I. Demirtzioglou, C. Lacava, A. Shakoor, A. Khokhar, Y. Jung, D. J. Thomson, and P. Petropoulos, “Apodized silicon photonic grating couplers for mode-order conversion,” Photonics Res. 7(9), 1036–1041 (2019). [CrossRef]
20. Y. Fu, T. Ye, W. Tang, and T. Chu, “Efficient adiabatic silicon-on-insulator waveguide taper,” Photonics Res. 2(3), A41–A44 (2014). [CrossRef]
21. R. Takei, E. Omoda, M. Suzuki, S. Manako, T. Kamei, M. Mori, and Y. Sakakibara, “Ultranarrow silicon inverse taper waveguide fabricated with double-patterning photolithography for low-loss spot-size converter,” Appl. Phys. Express 5(5), 052202 (2012). [CrossRef]
22. R. Takei, M. Suzuki, E. Omoda, S. Manako, T. Kamei, M. Mori, and Y. Sakakibara, “Silicon knife-edge taper waveguide for ultralow-loss spot-size converter fabricated by photolithography,” Appl. Phys. Lett. 102(10), 101108 (2013). [CrossRef]
23. J. Zhang, J. Yang, H. Xin, J. Huang, D. Chen, and Z. Zhang, “Ultrashort and efficient adiabatic waveguide taper based on thin flat focusing lenses,” Opt. Express 25(17), 19894–19903 (2017). [CrossRef]
24. P. Sethi, R. Kallega, A. Haldar, and S. K. Selvaraja, “Compact broadband low-loss taper for coupling to a silicon nitride photonic wire,” Opt. Lett. 43(14), 3433–3436 (2018). [CrossRef]
25. J. Zou, Y. Yu, M. Ye, L. Liu, S. Deng, X. Xu, and X. Zhang, “Short and efficient mode-size converter designed by segmented-stepwise method,” Opt. Lett. 39(21), 6273–6276 (2014). [CrossRef]
26. D. Vermeulen, K. Van Acoleyen, S. Ghosh, S. Selvaraja, W. A. D. de Cort, N. A. Yebo, E. Hallynck, K. de Vos, P. P. P. Debackere, P. Dumon, G. Roelkens, D. Van Thourhout, and R. Baets, “Efficient tapering to the fundamental quasi-TM mode in asymmetrical waveguides,” in Proceedings of European Conference on Integrated Optics, Cambridge, 2010, paper WeP16.
27. M. Asaduzzaman, M. Bakaul, E. Skafidas, and M. R. H. Khandokar, “A compact silicon grating coupler based on hollow tapered spot-size converter,” Sci. Rep. 8(1), 1–9 (2018). [CrossRef]
28. S. Abbaslou, R. Gatdula, M. Lu, A. Stein, and W. Jiang, “Ultra-short beam expander with segmented curvature control: the emergence of a semi-lens,” Opt. Lett. 42(21), 4383–4386 (2017). [CrossRef]
29. J. M. Luque-González, R. Halir, J. G. Wangüemert-Pérez, J. de-Oliva-Rubio, J. H. Schmid, P. Cheben, I. Molina-Fernández, and A. Ortega-Moñux, “An ultracompact GRIN-lens-based spot size converter using subwavelength grating metamaterials,” Laser Photonics Rev. 13(11), 1900172 (2019). [CrossRef]
30. K. Van Acoleyen and R. Baets, “Compact lens-assisted focusing tapers fabricated on silicon-on-insulator,” in Proceedings of IEEE Conference on Group IV Photonics (IEEE, 2011), pp. 157–159. [CrossRef]
31. S. H. Badri and M. M. Gilarlue, “Silicon nitride waveguide devices based on gradient-index lenses implemented by subwavelength silicon grating metamaterials,” Appl. Opt. 59(17), 5269–5275 (2020). [CrossRef]
32. W. Qi, Y. Yu, and X. Zhang, “On-chip arbitrary-mode spot size conversion,” Nanophotonics 9(14), 4365–4372 (2020). [CrossRef]
33. Q. Wu, J. P. Turpin, and D. H. Werner, “Integrated photonic systems based on transformation optics enabled gradient index devices,” Light: Sci. Appl. 1(11), e38 (2012). [CrossRef]
34. L. Xu and H. Chen, “Conformal transformation optics,” Nat. Photonics 9(1), 15–23 (2015). [CrossRef]
35. J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008). [CrossRef]
36. L. H. Gabrielli, J. Cardenas, C. B. Poitras, and M. Lipson, “Silicon nanostructure cloak operating at optical frequencies,” Nat. Photonics 3(8), 461–463 (2009). [CrossRef]
37. J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009). [CrossRef]
38. S. H. Badri and M. M. Gilarlue, “Ultrashort waveguide tapers based on Luneburg lens,” J. Opt. 21(12), 125802 (2019). [CrossRef]
39. A. Di Falco, S. C. Kehr, and U. Leonhardt, “Luneburg lens in silicon photonics,” Opt. Express 19(6), 5156–5162 (2011). [CrossRef]
40. L. H. Gabrielli and M. Lipson, “Integrated Luneburg lens via ultra-strong index gradient on silicon,” Opt. Express 19(21), 20122–20127 (2011). [CrossRef]
41. J. Hunt, T. Tyler, S. Dhar, Y.-J. Tsai, P. Bowen, S. Larouche, N. M. Jokerst, and D. R. Smith, “Planar, flattened Luneburg lens at infrared wavelengths,” Opt. Express 20(2), 1706–1713 (2012). [CrossRef]
42. T. Zentgraf, J. Valentine, N. Tapia, J. Li, and X. Zhang, “An optical “Janus” device for integrated photonics,” Adv. Mater. 22(23), 2561–2564 (2010). [CrossRef]
43. L. Sun, Y. Zhang, Y. He, H. Wang, and Y. Su, “Subwavelength structured silicon waveguides and photonic devices,” Nanophotonics 9(6), 1321–1340 (2020). [CrossRef]
44. N. B. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Mater. 9(2), 129–132 (2010). [CrossRef]
45. N. B. Kundtz, D. R. Smith, and J. B. Pendry, “Electromagnetic design with transformation optics,” Proc. IEEE 99(10), 1622–1633 (2011). [CrossRef]
46. D. R. Smith, Y. Urzhumov, N. B. Kundtz, and N. I. Landy, “Enhancing imaging systems using transformation optics,” Opt. Express 18(20), 21238–21251 (2010). [CrossRef]
47. D. H. Kwon and D. H. Werner, “Transformation electromagnetics: an overview of the theory and applications,” IEEE Antennas Propag. Mag. 52(1), 24–46 (2010). [CrossRef]
48. Z. Chang, X. Zhou, J. Hu, and G. Hu, “Design method for quasi-isotropic transformation materials based on inverse Laplace’s equation with sliding boundaries,” Opt. Express 18(6), 6089–6096 (2010). [CrossRef]
49. N. I. Landy and W. J. Padilla, “Guiding light with conformal transformations,” Opt. Express 17(17), 14872–14879 (2009). [CrossRef]
50. B. Vasić, G. Isić, R. Gajić, and K. Hingerl, “Controlling electromagnetic fields with graded photonic crystals in metamaterial regime,” Opt. Express 18(19), 20321–20333 (2010). [CrossRef]
51. L. Sun and C. Jiang, “Quantum interference in a single anisotropic quantum dot near hyperbolic metamaterials,” Opt. Express 24(7), 7719–7727 (2016). [CrossRef]
52. J. H. Lee, J. Blair, V. A. Tamma, Q. Wu, S. J. Rhee, C. J. Summers, and W. Park, “Direct visualization of optical frequency invisibility cloak based on silicon nanorod array,” Opt. Express 17(15), 12922–12928 (2009). [CrossRef]
53. A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford University, 2006).
