Controlling branching angle of waveguide splitters based on GRIN lenses
CControlling branching angle of waveguide splitters based on GRIN lenses
S. Hadi Badri , M. M. Gilarlue , and S. G. Gavgani Department of Electrical Engineering, Sarab Branch, Islamic Azad University, Sarab, Iran and Department of Electrical Engineering, Azarbaijan Shahid Madani University, Tabriz, Iran (Dated: August 16, 2019)Designing beam splitting structures with wide branching angles is of great significance. Thebranching angle of conventional Y-junctions is limited. In this paper, we investigate the possibilityof utilizing gradient index (GRIN) lenses with two focal points such as the generalized Maxwellsfisheye (GMFE) and Eaton lenses in controlling the branching angle of power splitters. The GMFElens can provide a wide range of branching angles, however, we present only splitting angles of25 ◦ , 45 ◦ , and 65 ◦ . Furthermore, we propose a 90 ◦ splitter structure by employing the Eaton lens.We evaluate the performance of the proposed power splitters by ray-tracing and full-wave finiteelement method. While GRIN lenses provide a broad range of splitting angles, they require isotropicmetamaterials to implement high refractive indices at the center of these lenses. I. INTRODUCTION
Splitting and combining the optical signals in pho-tonic integrated circuits (PICs) rely on power splitters.Power splitters or Y-branch structures are the key ele-ments in Mach-Zehnder interferometers, optical switches,optical phase arrays, mode multiplexers, semiconductorlasers, samplers, logic gates, and hybrid-integrated op-tical transceivers [1–4]. The branching angle of conven-tional power splitters is usually lower than 12 ◦ [5]. Theconventional power splitters suffer from severe radiationloss as the branching angle increases. Reducing radia-tion loss can be achieved by decreasing the branchingangle and increasing the length of the splitting struc-ture, resulting in a larger footprint [6]. Various meth-ods have been studied to control the branching angleof splitters. T-junctions have been implemented basedon the left-handed properties of the metamaterial [7, 8].Transformation optics (TO) offers unprecedented controlover the flow of electromagnetic fields [9]. Beam split-ters with various branching angles have been designedby TO [10–14]. These designs are usually implementedby anisotropic metamaterials. Recent advances in meta-materials and nanofabrication techniques have turned theattention of researchers to the classical GRIN lenses suchas Maxwells fisheye [15, 16], Luneburg [17, 18], and Eaton[19, 20] lenses. In this paper, we present novel opticalpower splitters by employing GRIN lenses with dual fo-cal points. These lenses provide a flexible structure whichcould be considered as a proper choice for a wide rangeof splitting angle. Splitters with branching angles of 25 ◦ ,45 ◦ , and 65 ◦ are presented by the GMFE lens while a90 ◦ branching angle is designed by the Eaton lens. Thecost of this fle xibility is a high refractive index at thecenter of the lens which requires isotropic metamateri-als in the fabrication process. On the other hand, thesplitters designed by TO require anisotropic metamate-rials. The proposed splitters are evaluated by ray-tracingand full-wave simulations. Recently, the performance ofthe GMFE as a beam splitter has been studied in theGHz range [21] where a point source is used to evaluatethe performance of the lens. A point source can only be a valid estimate of the performance of the GMFElens as waveguide splitter if only the radius of the lensis considerably larger than the width of the waveguides.Therefore, we take a different approach by using an arrayof point sources in ray-tracing simulations to achieve amore reliable result. Moreover, conditions for maximumtransmission and minimum reflection are discussed. II. GMFE LENS AS SPLITTER
Maxwells fisheye (MFE) lens is a circular lens of radius R lens with refractive index profile of n lens ( r ) = 2 × n edge r/R lens ) , (0 ≤ r ≤ R lens ) (1)where r is the radial distance from the center of thelens, and n edge is the refractive index of the lens at itsedge. When rays leave a point source on the edge of theMFE lens, they are focused to the diagonally oppositepoint of the lens. The MFE lens can be generalized tohave two focal points. The refractive index profile forthis lens is [22] n lens ( r ) = 2 × n edge ( r/R lens ) − m + ( r/R lens ) m , (0 ≤ r ≤ R lens )(2)where m is a variable parameter. Eq. (2) reduces toEq. (1) for m = 1. For 0 . ≤ m ≤
1, the GMFE lenssplits the rays emitting from a point source on its edgeinto two points on its edge. For m = 0 .
5, the lens fo-cuses the rays back to its source after a single revolu-tion around the center [23]. For three values of m, theray diagrams are shown in Fig 1. By decreasing m, theangular separation between the focal points as well asthe refractive index of the lens increases. The refrac-tive index of the GMFE approaches to infinity at thecenter. Therefore, we limit the refractive index profilesin the figures presented in this paper in order to make a r X i v : . [ phy s i c s . op ti c s ] A ug FIG. 1. Ray trajectory based on a point source for GMFElenses with a) m = 0 .
95, b) m = 0 .
87, and c) m = 0 . them more distinguishable for the readers. In subsec-tion II A, we present ray-tracing calculation results forthree branching angles. And in subsection ‘II B, the two-dimensional (2D) full-wave simulations are presented forthe same structures. A. Geometrical Optics
In this section, ray-tracing calculations are performedwith Comsol Multiphysics to validate the proposed split-ters. Three splitters with branching angles of 25 ◦ , 45 ◦ ,and 65 ◦ are presented by the GMFE lens in Fig 2. Inthis study, we suppose a 250 nm -thick SiN guiding layerwhich is surrounded by
SiO substrate and upper aircladding. The effective refractive index of this waveg-uide is about 1.57. In two-dimensional (2D) simulations,we consider a waveguide with n core = 1 .
57 which is sur-rounded by air. For reducing reflection from the interfaceof the waveguide and lens, the refractive indices of thewaveguides core ( n core ) and the edge of the lens shouldbe equal. Therefore, minimizing the reflection is achievedby considering n edge = 1 .
57 in Eq. (2). The radius of thelens is R lens = 4 µm while the width of the source waveg-uide is 2 µm . And the width of branching waveguides is1 µm . A point source may be a reasonable approxima-tion for an input waveguide with a very narrow width.Therefore, in Fig 2, an array of point sources located inthe core of the waveguide is used to accurately evaluatethe performance of the GMFE lens as a splitter. It shouldbe noted that, while the refractive index profiles of thelenses are the same in Fig 1 and 2, the splitting anglesdiffer due to the difference between the light sources. Forinstance, the branching angle in Fig 1(c) is 90 ◦ while it is65 ◦ in Fig 2(c). Therefore, a single point source cannotbe used to determine the branching angle of the lenses,and an array of point sources in the core of the waveguideshould be employed to design waveguide splitters.As seen in Fig 2, the rays are limited to a small areaof the lens, therefore, the lens can be truncated withoutany degradation of its performance. This truncation re-duces the footprint of the splitters considerably. Fig 2(a)corresponds to a truncated lens with m = 0 .
95 wherethe splitting angle is 25 ◦ . The splitting angle can be in-creased by reducing m. The splitting angle is 45 ◦ and65 ◦ for m = 0 .
87 and 0 .
80, respectively.
FIG. 2. Three splitters based on GMFE lenses with branchingangles of a)25 ◦ b)45 ◦ , and c)65 ◦ determined by an array ofpoint sources.FIG. 3. The circular lenses of Fig 2 are truncated to reducethe footprint of the beam splitters. The branching angles area) 625 ◦ b) 45 ◦ , and c) 65 ◦ ..FIG. 4. Full-wave simulation results at the wavelength of1550 nm for the splitting angle of 25 ◦ with the a) completeGMFE lens, b) truncated GMFE lens, and c) without lens. B. Wave Optics
In this section, the 2D finite element method (FEM)simulations of the designed structures for the TE modeare presented at the wavelength of 1550 nm . We evalu-ate the performance of the proposed splitters based ontheir splitting efficiency ( P split /P in ). P split is the powerthat transmits through each branching waveguide while P in is the power entering the splitting structure. For thesplitter with m = 0 .
95, where the branching angle is 25 ◦ ,the full-wave simulation results of the complete and trun-cated lenses are displayed in Fig 4. The splitter based onthe complete lens with splitting efficiency of 32% is shownin Fig 4(a). The splitting efficiency of the truncated lensdisplayed in Fig 4(b) is 33%. We also investigate theperformance of the splitting structure without the lensin Fig 4(c). In this case, the splitting efficiency is 35%which is slightly better compared to the splitting struc-tures of complete and truncated lenses. In low branchingangles, the splitting structures based on lenses are noteffective compared to the simple Y-junctions.For the splitting structure with m = 0 .
87, the resultsof the full-wave simulations are shown in Fig 5. In thisdesign, the simulation results of complete and truncated
FIG. 5. Full-wave simulation results at the wavelength of1550 nm for the splitting angle of 45 ◦ with the a) completeGMFE lens, b) truncated GMFE lens, and c) without lensFIG. 6. Full-wave simulation results at the wavelength of1550 nm for the splitting angle of 65 ◦ with the a) completeGMFE lens, b) truncated GMFE lens, and c) without lens lenses with the branching angle of 45 ◦ are displayed inFig 5(a) and Fig 5(b), respectively. The splitting effi-ciency of 29% is achieved for the complete lens of Fig 5(a).The truncated lens of Fig 5(b) has the splitting efficiencyof 22%. Wave propagation and the splitting performanceof the structure without lens is shown in Fig 5(c). In thiscase, due to the fact that the optical signal is converted tohigher mode, the splitting efficiency is 8%. As expected,the performance of the Y-junction without lens degradesconsiderably as the branching angle increases.For the splitter of Fig 3(c), the results of simulationsare shown in Fig 6. This structure comes with m = 0 . ◦ . Simulations for both thecomplete and truncated lenses are carried out. The split-ter based on the complete lens with splitting efficiency of35% is shown in Fig 6(a). For the truncated lens, shownin Fig 6(b), the efficiency is 29%. We also investigatethe performance of the splitting structure without thelens in Fig 6(c) where the splitting efficiency is 0 . III. EATON LENS AS SPLITTER
The Eaton lens can bend the light waves trajectory by90 ◦ , 180 ◦ , or 360 ◦ . For a 90 ◦ bend, the refractive indexof the Eaton lens is given by [19–24] n lens = R lens n lens r + (cid:115)(cid:18) R lens n lens r (cid:19) − FIG. 7. Eaton lens as T-junction. a) ray trajectories for theEaton lens. Full-wave simulation result at the wavelength of1550 nm for the b) complete lens and c) truncated lens.
The refractive index of the lens ranges from unity at itsedge to infinity at the center of the lens. Minimizing theinsertion loss is achieved by matching the refractive in-dices at the interface of the waveguides and the lens, thusthe calculated is multiplied by n core . When an off-centerbeam is incident on the Eaton lens, power beam propa-gates inside the lens along a 90 ◦ bending path which canbe used to design waveguide bends [19, 20]. On the otherhand, when an on-center beam is incident on the lens, itbehaves like a T-junction [25]. As shown in Fig 7(a), anarray of point sources covers a width of the waveguidecore in ray-tracing calculations. Full-wave simulation re-veals that for the splitting structure with the completelens, the maximum splitting efficiency of 30% is achievedwhen the branching angle is 80 ◦ . However, for a branch-ing angle of 90 ◦ with the truncated lens, the splittingefficiency of 41% is achieved.GRIN lenses have been implemented by graded pho-tonic crystals [26, 27], multilayer structures [28], vary-ing the guiding layer thickness [29]. However, the pro-posed splitters cannot be implemented by these meth-ods due to the extreme values at the center of the lensesand isotropic metamaterials should be used to implementthese splitters. IV. CONCLUSION
In summary, we investigated the possibility of design-ing power splitters for moderate index-contrast waveg-uides based on GRIN lenses. We designed splitters withbranching angles of 25 ◦ , 45 ◦ , and 65 ◦ based on com-plete and truncated GMFE lenses. We also truncated theEaton lens to design a 90 ◦ splitter. The GRIN lenses helpto achieve a wide range of branching angles, however, thisflexibility comes at the price of large values of refractiveindices at the center of the lenses. The designed split-ting structures cannot be implemented by conventionalmethods and they should be implemented by isotropicmetamaterials. The advantage of the proposed designscompared to the TO-based splitters is the use of isotropicmetamaterials instead of anisotropic metamaterials. [1] Q. Wang, J. Lu, and S. He, Applied optics , 7644(2002).[2] H. Hatami-Hanza, P. Chu, and M. Lederer, IEEE pho-tonics technology letters.[3] X. Li, H. Xu, X. Xiao, Z. Li, J. Yu, and Y. Yu, Opticsletters , 4220 (2013).[4] S. Safavi-Naeini, Y. Chow, S. Chaudhuri, and A. Goss,Journal of lightwave technology , 567 (1993).[5] Z. Huang, H. P. Chan, and M. A. Uddin, Applied optics , 1900 (2010).[6] P. Wang, G. Brambilla, Y. Semenova, Q. Wu, andG. Farrell, (2011).[7] H. Chen, L. Ran, J. Huangfu, X. Zhang, K. Chen, T. M.Grzegorczyk, and J. A. Kong, Journal of applied physics , 3712 (2003).[8] C. Caloz, C.-C. Chang, and T. Itoh, Journal of AppliedPhysics , 5483 (2001).[9] U. Leonhardt and T. G. Philbin, in Progress in Optics ,Vol. 53 (Elsevier, 2009) pp. 69–152.[10] M. Rahm, D. Roberts, J. Pendry, and D. Smith, OpticsExpress , 11555 (2008).[11] M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry,and D. R. Smith, Physical Review Letters , 063903(2008).[12] Y.-L. Wu, Z. Zhuang, L. Deng, and Y.-A. Liu, Scientificreports , 24495 (2016).[13] S. Viaene, V. Ginis, J. Danckaert, and P. Tassin, Physi-cal Review B , 155412 (2017).[14] S. Viaene, V. Ginis, J. Danckaert, and P. Tassin, Physi-cal Review B , 085429 (2016).[15] M. Gilarlue and S. H. Badri, Optics Communications , 308 (2019). [16] S. H. Badri, H. R. Saghai, and H. Soofi, Applied Optics , 4647 (2019).[17] A. Sayanskiy, S. Glybovski, V. P. Akimov, D. Filonov,P. Belov, and I. Meshkovskiy, IEEE Antennas and Wire-less Propagation Letters , 1520 (2017).[18] O. Quevedo-Teruel, J. Miao, M. Mattsson, A. Algaba-Brazalez, M. Johansson, and L. Manholm, IEEE Anten-nas and Wireless Propagation Letters , 1588 (2018).[19] S. H. Badri and M. Gilarlue, JOSA B , 1288 (2019).[20] S. H. Badri, H. R. Saghai, and H. Soofi, Applied Optics , 5219 (2019).[21] Q. Lei, R. Foster, P. S. Grant, and C. Grovenor, IEEETransactions on Microwave Theory and Techniques ,4823 (2017).[22] T. Tyc, L. Herz´anov´a, M. ˇSarbort, and K. Bering, NewJournal of Physics , 115004 (2011).[23] H. Eskandari, M. S. Majedi, A. R. Attari, andO. Quevedo-Teruel, New Journal of Physics , 063010(2019).[24] G. Du, M. Liang, R. A. Sabory-Garcia, C. Liu, andH. Xin, IEEE Antennas and Wireless Propagation Let-ters , 1487 (2016).[25] M. Yin, X. Yong Tian, L. Ling Wu, and D. Chen Li,Applied Physics Letters , 094101 (2014).[26] M. Gilarlue, S. H. Badri, H. R. Saghai, J. Nourinia,and C. Ghobadi, Photonics and Nanostructures-Fundamentals and Applications , 154 (2018).[27] S. H. Badri and M. Gilarlue, Optik , 566 (2019).[28] M. Gilarlue, J. Nourinia, C. Ghobadi, S. H. Badri, andH. R. Saghai, Optics Communications , 385 (2019).[29] S. H. Badri, H. R. Saghai, and H. Soofi, Journal of Optics21