High-efficiency waveguide couplers via impedance-tunable transformation optics
aa r X i v : . [ phy s i c s . op ti c s ] A ug High-efficiency waveguide couplers via impedance-tunable transformationoptics
Jun Cao,
1, 2
Lifa Zhang, Weifeng Jiang, Senlin Yan, and Xiaohan Sun a) National Research Center for Optical Sensing/Communications Integrated Networking, Department of Electronics Engineering,Southeast University, Nanjing 210096, China Department of Physics, Nanjing Xiaozhuang University, Nanjing 211171, China Department of Physics, The University of Texas at Austin, Austin, Texas 78712, USA (Dated: 28 July 2014)
We design compact waveguide couplers via impedance-tunable transformation optics. By tuning impedance coef-ficients in the original space, two-dimensional metallic and dielectric waveguide couplers are designed with a highefficiency. Through tuning refractive index simultaneously, we find that the transformation medium inside a designedmetallic waveguide coupler can be a reduced-parameter material for coupling waves between waveguides with arbitrarydifferent cross sections and embedded media. In the design of dielectric waveguide couplers, we apply two differentschemes: one is that both core region and its cladding region are contained in a transformed space, the other is onlycore region contained in the transformed space. The former has a very high efficiency near ; the latter is lessefficient with a very small decline which can be a simplified candidate in the design of near-perfect dielectric waveg-uide couplers. The transformation medium for dielectric waveguide couplers can also be reduced-parameter materialby selecting appropriate refractive index coefficients. Two-dimensional numerical simulations confirm our design withgood performances.Couplers are very essential electromagnetic (EM) devices,which act as intermediary components to reduce mode mis-match in order to transfer light efficiently between waveguidesof different cross sections and embedded media. The conven-tional method to reduce the coupling loss is to introduce aconnected taper with a gradient change of size ; however, itis very space consuming. With the progress of integrated opti-cal circuits, the fiber-to-chip coupling remains a critical prob-lem. To solve it, various couplers via different mechanismshave been proposed, such as the grating coupler , parabolicreflector , and luneburg lens ; but the reported coupling effi-ciency are still very low. Thus to efficiently couple EM wavesin waveguides is still a challenging topic up to now.Transformation optics, as a unconventional theory reportedby Pendry and Leonhardt , provides a powerful tool todesign various optical devices . However, in the tech-nique applied in some cases to manipulate the intensityof the electromagnetic fields out of a transformation medium,there is a mismatch of impedance at boundaries and the re-sulted reflections limit its applications. Different taper struc-tures placed with transformation medium have been appliedto the design of waveguide couplers , but the impedance mis-match has not been solved thus the coupler performance arenot satisfied. In Ref. , impedance has been matched by in-serting another special impedance-matched media in one ofwaveguides, however it can not be easily generalized to ar-bitrary coupling waves. Very recently, we proposed a gener-alized theory of impedance-tunable transformation optics inthe geometric optics limit , which not only can manipulateimpedance match but also can provide more flexibilities to se-lect realizable transformation medium.In this letter, we design two-dimensional (2D) metallic anddielectric waveguide couplers using the impedance-tunable a) Electronic mail: [email protected] transformation optics. By tuning the impedance coefficientof the coupler, the mode waves can be coupled efficientlybetween two waveguides with arbitrarily different cross sec-tions and embedded media. Through tuning the refractive in-dex in the original space, the transformation medium can bereduced-parameter materials (magnetic-response-only materi-als for TE polarization or dielectric-response-only materialsfor TM polarization). Different schemes have been imple-mented and compared for the design of high-efficiency dielec-tric waveguide couplers.In the theory of impedance-tunable transformation optics ,the relative permittivity ε i ′ j ′ and permeability µ i ′ j ′ of thetransformation medium for a given coordinate transformation x ′ = x ′ ( x ) can be expressed as ε i ′ j ′ = | det ( A i ′ i ) | − A i ′ i A j ′ j εδ ij /k,µ i ′ j ′ = | det ( A i ′ i ) | − A i ′ i A j ′ j kµδ ij . (1)Here A i ′ i = ∂ ( x ′ ,y ′ ,z ′ ) ∂ ( x,y,z ) denotes a Jacobian tensor between thetransformed space ( x ′ , y ′ , z ′ ) and the original space ( x, y, z ) ,and the impedance coefficient k is a spatial continuous func-tion.Based on the strategy of impedance-tunable transformationoptics, a 2D compact waveguide coupler is designed as shownin Fig. 1, where waveguide WG1 and WG2 of different width h and h are connected by a linear taper structure ABCD oflength d as a waveguide coupler. The relative permittivity andpermeability of the media in WG1 and WG2 are ε , µ and ε , µ respectively. To couple waves efficiently from WG1to WG2 in the + z direction, the region ABCD are embeddedwith transformation medium, which compresses the rectangu-lar region ABC’D’ in the original space to trapezoidal regionABCD in the transformed space. We apply the transformationdefined in the 2D compressor/expander design : x ′ = x [1 − zd (1 − γ )] , y ′ = y, z ′ = z (2) FIG. 1. (Color online) Schematic diagram of the waveguide couplerusing the impedance-tunable transformation optics. The width, rela-tive permittivity, permeability of the waveguide WG1 and WG2 are h , ε , µ and h , ε , µ respectively; the length of the coupler is d ,and embedded in a transformation medium. where γ = h h is the compression coefficient.Different from Ref. where the background of transforma-tion media is supposed to be air, arbitrary background me-dia ε , µ and ε , µ are discussed here. Thus we reset thematerial parameters in the original space to calculate a newimpedance function k . Without changing the refractive in-dex, the permittivity and permeability of the original mediumare set to be ε /k and kµ . No unique function of k canbe selected, but there is no guarantee that the transformationmedium can be a reduced-parameter material. A normal in-cident plane wave illuminating upon the coupler is discussedin the geometrical optics limit. Similar to the derivation inRef. , the function k can be set as k = dd − z ′ (1 − γ q ε µ ε µ ) forTE polarization, and k = d − z ′ (1 − γ q ε µ ε µ ) d for TM polariza-tion. Note that only when the impedance of embedded mediain WG1 and WG2 are equal, ε µ = ε µ , the function k arethe same as that in Ref. , which leads to a reduced-parametertransformation medium.Reduced sets of material properties are generally favoredin the two-dimensional transformation-optical design dueto its easier realization. In this paper, in order to obtain areduced-parameter transformation medium for arbitrary con-ditions, we generalize the theory of impedance-tunable trans-formation optics to the design of waveguide coupler that therefractive index in the original space can also be tunable, andthe relative permittivity and permeability in the original spaceare reset to be ε ij = ε ( z ) δ ij /k and µ ij = kµ ( z ) δ ij . Therefractive index n = p ε ( z ) µ ( z ) in the original space is a z -dependent continuous function, thus it will not change thelight rays compared to the conventional transformation. ForTE polarization we set ε ( z ) = p , µ ( z ) = p µ + z ′ d ( µ − µ ) ε + z ′ d ( ε − ε ) andfor TM polarization µ ( z ) = p , ε ( z ) = p ε + z ′ d ( ε − ε ) µ + z ′ d ( µ − µ ) . Notethat coefficient p has not change the impedance but change itsrefractive index of the transformation medium, we define itas refractive index coefficient, large p will extend the applica-tion of the geometric optics approximation, and improve theperformance of the coupler at low frequencies. FIG. 2. (Color online) The normalized y direction electric field dis-tribution of a waveguide coupler. (a) A traditional case without trans-formation; (b) An impedance-nontunable transformation case; (c)and (d) are impedance-tunable transformation cases for TE modewith p = 1 and p = 3 , respectively; (e) and (f) are impedance-tunable transformation cases for TE mode with p = 1 and p = 3 ,respectively. Applying the continuity of the total tangential electric andmagnetic fields, one can obtain k = dd − z ′ (1 − γ ) for TE polar-ization and k = d − z ′ (1 − γ ) d for TM polarization, which satisfythe impedance match condition at the boundaries AB and CDof the coupler. The parameters of the transformation mediumassociated with TE polarization can be written as: ε yy = pµ xx = p (1 + (1 − γ ) d x ′ [ d − z ′ (1 − γ )] ) µ + z ′ d ( µ − µ ) ε + z ′ d ( ε − ε ) µ xz = µ zx = p − (1 − γ ) d x ′ [ d − z ′ (1 − γ )] µ + z ′ d ( µ − µ ) ε + z ′ d ( ε − ε ) µ zz = p d [ d − z ′ (1 − γ )] µ + z ′ d ( µ − µ ) ε + z ′ d ( ε − ε ) . (3)The material based on the above transformation is of magneticresponse only. We also can obtain a dielectric-response-onlytransformation medium associated with TM polarization as: µ yy = pε xx = p (1 + (1 − γ ) d x ′ [ d − z ′ (1 − γ )] ) ε + z ′ d ( ε − ε ) µ + z ′ d ( µ − µ ) ε xz = ε zx = p − (1 − γ ) d x ′ [ d − z ′ (1 − γ )] ε + z ′ d ( ε − ε ) µ + z ′ d ( µ − µ ) ε zz = p d [ d − z ′ (1 − γ )] ε + z ′ d ( ε − ε ) µ + z ′ d ( µ − µ ) . (4)To investigate the performance of the waveguide couplerembedded with impedance-tunable transformation medium,we do two-dimensional numerical simulations using COM-SOL Multiphysics for TE-mode waves incidence (similar cal-culation for TM-mode waves can also be done). For a metallicwaveguide, the calculation domain is bounded by a perfectlyelectric conductor. In our simulations, without loss of gener-ality, the width of the waveguide WG1 and WG2 are set to be C oup li ng e ff i c i en cy Frequency(GHz)
FIG. 3. (Color online) Coupling efficiency of a metallic waveg-uide coupler for TE mode as a function of frequency. The bluedotted and green dashed line correspond to couplers without transfor-mation and with impedance-nontunable transformation, respectively;the black dash-dotted and red solid line correspond to couplers withimpedance-tunable transformation for p = 1 and p = 3 , respectively. h = 0 . m and h = 0 . m respectively, and ε = 4 , µ = 1 ,and ε = 1 , µ = 9 respectively. The length of the coupler is d = 0 . m. TE mode waves are excited at a port with an in-cident frequency f = 3 GHz. Fig. 2 shows electric field sim-ulation results, in which the inset (a)-(d) are for TE mode,and the inset (e)-(f) are for TE mode. Figure 2(a) shows atraditional design case without transformation medium in thecoupler, where the coupling efficiency η = W /W is due to the obvious reflections ( W and W are the coupler in-put and output power, respectively). For a conventional trans-formation medium embedded in the coupler (an impedancenon-tunable case for k = 1 ), as shown in Fig. 2(b), althoughthe wave profile are preserved well, even more serious reflec-tions occur at the exit boundary of the coupler, resulting in anamplitude modulation of the incoming wave, and the simu-lated coupling efficiency is only . While the impedance-tunable transformation medium embedded in the coupler with p = 1 , Fig. 2(c) shows a good performance with the cou-pling efficiency near , and improved to in Fig. 2(d)with a larger refractive index coefficient p = 3 , which is al-most reflectionless. The coupler using the impedance-tunabletransformation optics can also be applied to couple high-ordermode waves efficiently; Fig. 2(e) and Fig. 2(f) are simulatedresults of TE mode with p = 1 and p = 3 respectively, witha similar coupling efficiency to that of TE mode.Figure 3 shows the coupling efficiency of TE modes foran impedance-tunable coupler, an impedance-nontunable oneand a traditional one without transformation simulated from . GHz to GHz. As can be clearly seen from Fig. 3, thetraditional transformation optics does not exhibit enough com-petitive advantages compared to the conventional case withouttransformation, and thus limits its applications in many cases.However, by introducing a tunable impedance, we can obtaina coupler with extremely high efficiencies except in the vicin-ity of the cutoff frequency. In the whole range of frequency,
FIG. 4. (Color online) The normalized y direction electric field dis-tribution of a dielectric waveguide coupler with total transformation(a), with only core transformation (refractive index nontunable) (b),with only core transformation (refractive index tunable with p = 1 )(c) and with only core transformation (refractive index tunable with p = 6 ) (d). the impedance-tunable coupler has a best performance com-pared to the impedance-nontunable one via traditional trans-formation optics and the traditional one without transforma-tion. The inevitable unperfect coupling performance at lowfrequencies due to the geometric optics limit can be improvedthrough setting large refractive index coefficient p , the cost isthat the relative permittivity of the transformation medium isnot 1 and extreme larger parameters will increase the difficultyof realization of designed transformation medium. Neverthe-less we still can properly increase p to increase the perfor-mance at low frequencies while it is not difficult to be realizedin designed medium.Beside a metallic waveguide, an efficient dielectric waveg-uide coupler is also very important in optical design. Usingthe impedance-tunable transformation optics method, a 2Dcompact dielectric waveguide coupler can be designed withhigh efficiency and less space occupation.For a dielectric waveguide, distribution of the EM field canbe divided into two regions: a concentrated dielectric core andan evanescent air cladding. Therefore to couple waves witha high efficiency, the transformed space can also be dividedinto an inner space (major contribution) and an outer space(minor contribution). For the total transformation the embed-ded medium can be obtained through setting different originalspaces and tuning different impedance coefficients for innerand outer parts. Note that in order to preserve the profile of themode waves in the dielectric waveguide coupler the refractiveindex in the inner original space can not be tunable, resulteda non-reduced sets of material properties in the inner space ofthe coupler. However at high frequencies or in high-index-contrast waveguide, a simple method is acceptable to omitevanescent energy and only focus on the core energy coupling.Thus only inner transformation medium (core transformation)can be considered with a slightly reduced coupling efficiencyas we will see in the simulations latter; and the transformationmedium also can be reduced-parameter material if we care-fully set the refractive index coefficient p . With total transformation
With core transformation
With core transformation (reduced-parameter material) C oup li ng e ff i c i en cy Frequency(GHz)
FIG. 5. (Color online) Coupling efficiency of a dielectric waveguidecoupler as a function of frequency. The blue dotted, green dashedand red solid lines correspond to the couplers with total transforma-tion, with only core transformation, and with only core transforma-tion while the transformation medium is reduced-parameter material(refractive index tunable with p = 6 ) , respectively. In total transformation, the relative permittivity and perme-ability in the inner and outer original space are set to be ε ij = ε δ ij /k , µ ij = k δ ij and ε ij = δ ij /k , µ ij = k δ ij respec-tively. One can obtain k = dd − z ′ (1 − γ q ε ε ) , k = dd − z ′ (1 − γ ) for TE polarization, and k = d − z ′ (1 − γ q ε ε ) d , k = d − z ′ (1 − γ ) d for TM polarization.In numerical simulations, the relative permittivity of coresin dielectric waveguide WG1 and WG2 are set as ε = 2 and ε = 20 , the width of the core in WG1 and WG2 are 0.4 m and0.1 m, respectively, the coupler length is 0.2 m. The incidentfrequency is set to be 0.6 GHz, thus only TE mode can existin WG1 and WG2. Fig. 4(a) shows that TE mode waveshave been coupled from WG1 to WG2 with a high couplingefficiency over . As a simplified method with only coretransformation, Fig. 4(b) shows a little difference comparedto Fig. 4(a), with a slightly reduced coupling efficiency butalso near to . Note that in simulations of Fig. 4(b), therefractive index in the original core space is fixed, resulting inno reduced-parameter transformation medium. If we let therefractive index tunable, reduced-parameter material can beobtained. But such reduced parameter may make refractiveindex of the core less than its of cladding thus cause largeradiation loss if we select a small refractive index coefficient p ; as shown in Fig. 4(c) the coupling efficiency is only with p = 1 . While if we set a large refractive index coefficient p = 6 in Fig. 4(d), the coupling efficiency near can beobtained.As a further step, the coupling efficiency with only coretransformation and total transformation have been simulatedat different incident frequencies from 0.3 GHz to 1.2 GHz inFig. 5. The coupling efficiency of the dielectric waveguidecoupler increases at high frequencies, where high-order mode waves may propagate in the waveguide, but the performancenever be affected with the coupling efficiency near to by total transformation design, and a very small decline of ef-ficiency for only core transformation, which can be a propercandidate in the design of near-perfect dielectric waveguidecoupler. 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