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Anticrossing of Whispering Gallery Modes in microdisk resonators embedded in an anisotropic environment

the material show to numerically investigate the behavior of Whispering Gallery Modes (WGMs) in circularly shaped resonators like microdisks,with diameters in the range of optical vacuum wavelengths
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  Anticrossing of Whispering Gallery Modes in microdisk resonatorsembedded in an anisotropic environment S. Declair*, C. Meier, T. Meier, J. Fo¨rstner  Department of Physics and CeOPP, University of Paderborn, Warburger Str. 100, D-33098 Paderborn, Germany Received 26 January 2010; received in revised form 24 February 2010; accepted 8 March 2010Available online 15 March 2010 Abstract Wenumerically investigate thebehaviorofWhispering Gallery Modes(WGMs)incircularly shapedresonators like microdisks,with diameters in the range of optical vacuum wavelengths. The microdisk is embedded in an uniaxial anisotropic dielectricenvironment. By changing the optical anisotropy, one obtains spectral tunability of the optical modes. The degree of tunabilitystrongly depends on the radial (azimuthal) mode order M (N). As the modes approach each other spectrally, anticrossing isobserved, leading to a rearrangement of the optical states. # 2010 Elsevier B.V. All rights reserved. Keywords:  Microdisk; Whispering Gallery Mode; Anticrossing; FDTD 1. Introduction Optical resonators in general are of great interestbecause of their capability to store light in well definedspectralintervals.Thisgivesrisetousetheseresonatorsin optical circuits for quantum information processing,laser applications [1] and investigation of the strongcoupling regime of confined electromagnetic modeswith themselves or with semiconductor heterostruc-tures like quantum dots. The microdisk systeminvestigated in this study has the big advantage of comparably easy experimental realization with highaccuracy. Microdisk resonators provide high qualityfactors ( Q -factors) while generally lacking in smallmode volume (e.g. in comparison to photonic crystalcavities). To overcome the lack of small mode volume,this work concentrates on the use of a submicronmicrodisk resonator with a radius of   R ¼ 361 nm,height of   h ¼ 265 nm as used in the experiment bySong et al. [2], see Fig. 1. For the microdisk we use a model material without any resonances in theconsidered spectral range which therefore can bemodeled via a constant value for the dielectric constantof   ffiffiffiffi e d  p  ¼ 3 : 4.Tunable resonators are of special interest for modeswitching devices or broadband applications. Since theresonator geometry itself is usually fixed, one has tomodify the environment of the resonator which can beinfluenced more easily. In our case, the microdisk resonator is embedded in an uniaxial anisotropicenvironment to model effects like in a liquid crystal(LC) [3]. LCs have the property to make a phasetransition from isotropic to nematic (anisotropic) whenapplying a bias at temperatures below a criticaltemperature [4], also known as the clearing tempera-ture, which then changes the dielectric features of theentire system. As a model, we use an artificial uniaxialanisotropic dielectric environment which can be tunedfromvacuumpermittivitybeyondthe permittivity oftheperfectly dielectric resonator material.  Available online at Photonics and Nanostructures – Fundamentals and Applications 8 (2010) 273–277* Corresponding author. Tel.: +49 5251602325. E-mail address: (S. Declair).1569-4410/$ – see front matter # 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.photonics.2010.03.002  In this work, we deal with the spectral shift of Whispering Gallery Modes (WGMs) [5] in microdisk resonators in the presence of an uniaxial anisotropicenvironment, where confinement of the modes isobserved even when the permittivity of the anisotropicenvironmentexceedstheresonator—aregime,whereanisotropic environment does not allow confinement.Additionally, anticrossing of the WGMs is observedwhen the modes approach each other spectrally, leadingto a rearrangement of the optical states. 2. Theory One can solve the electromagnetic problem of themicrodisk system analytically in a perfect electricconducting environment (PEC) in three dimensions aswell as with isotropic dielectric environment. Assumingcylindrical coordinates ( r ; f ;  z ) in Fig. 1, one canseparatethe  z -componentinthethree-dimensionalwaveequation with even (odd) solutions (localized modes)inside and zero field outside the resonator with PECboundarycondition.Inthecaseofanisotropicdielectricenvironment one gets even (odd) solutions inside andexponentially decaying solution outside of the micro-disk. The remaining two-dimensional problem yieldsthe well-known  Bessel  differential equation. The in-plane solution for the transversal electric (TE) mode( ~  H   ¼ð 0 ; 0 ;  H   z Þ T  ; ~ E  ¼ð E   x ; E   y ; 0 Þ T  ) is a  Bessel  functionof first kind (  J   N  ) inside the microdisk and a super-position of the  Bessel  function of first and second kind( Y   N  ) outside [6]:  H   z  ˜  r ¼  ffiffiffiffi e d  p  2 plr   R ; f   ¼  J   N  ð ˜  r Þ e iM  f (1)  H   z  ˜  r ¼  ffiffiffiffi e e p  2 plr >  R ; f   ¼ð  J   N  ð ˜  r Þþ iY   N  ð ˜  r ÞÞ e iM  f ¼  H   N  ð ˜  r Þ e iM  f (2)with  R ;  N  ;  M  ; l ;  e d   and  e e  being the radius of the micro-disk resonator, the radial mode order, azimuthal modeorder, vacuum wavelengths of the resonance whichfulfill the boundary condition, electric permittivity of the resonator and electrical permittivity of the environ-ment, respectively.  H   N   denotes the  Hankel  function. Asimilar derivation holds for transversal magnetic (TM)modes, but they are not considered here due to ingeneral lower  Q -factor and lower amplitude. Addition-ally, TE modes would primarily excite embedded quan-tum dots due to high carrier confinement perpendicularto the microdisks’ plane, and vice versa.The uniaxial anisotropy of the environment,however, breaks the symmetry and the analyticalsolution is no longer applicable (except for certainsymmetry conditions [4]). Instead, increasing compo-nents of dielectric tensor of the environment,  e e , causea lower contrast of the refractive indices inside andoutside of the resonator. Considering this refractiveindex contrast as a potential for the confinement of thefield, this decreased potential yields in an increase of the leakage of the modes into the uniaxial anisotropicenvironment. Thus, the WGMs couple stronger to thepropagating modes outside of resonator. The effect is alarger effective radius for the maxima of the fielddistribution and a decreasing  Q -factor. Consideringmodes of lower radial order, the effective radius islarger than for higher order modes. As an illustrationone canimagine,thata largerpart ofthe circumferenceof the microdisk is covered by the mode maxima,which yields in an increased bending loss of thesemodes due to curvature (see Fig. 2). Hence, the  Q -factor is also decreased.Due tothefactthatthe leakageofWGMs ofdifferentmode order into the uniaxial anisotropic environment isdifferent, the spectral shift is also different. Thus thebirefringent property of an anisotropy can be used totune WGMs of different mode order spectrally intoresonance. Further tuning can be achieved throughchanges of in-plane resonator parameters perpendicularto its main axis like the radius, an edge profile [7] or anelliptic shape. 3. Methods For the numerical investigations, an in-house Finite-Difference Time-Domain ( FDTD ) code [8] has beenused. The  FDTD  code supports uniaxial anisotropicdielectric material and dynamic equations for nonlocal,nonlinear semiconductor heterostructures like quantumdots. Additionally,the code is tightly linked to the filter-diagonalisation method  Harmonic Inversion  [9] forefficient extraction of resonances out of time signalsconsisting of a (finite) number of resonances. S. Declair et al./Photonics and Nanostructures  –  Fundamentals and Applications 8 (2010) 273 – 277  274Fig. 1. Geometrical setup for numerical investigation of microdisk resonator in uniaxial anisotropic environment. Dimensions of themicrodisk: radius  R ¼ 361 nm, height  h ¼ 265 nm, microdisk per-mittivity  ffiffiffiffi e d  p  ¼ 3 : 4. Uniaxial anisotropic environment permittivity:  ffiffiffiffiffiffi e  xx p  ¼  ffiffiffiffiffiffi e  yy p  ¼ 1,  ffiffiffiffiffi e  zz p   1.  The effectiveness of the  Harmonic Inversion  incomparison to the discrete Fourier Transform (DFT) isthat the extraction of resonances can be done alreadyafter a short simulation time compared with the decaytime of the resonances. 4. Numerical results Fig. 3 shows the TE-like spectral response of themicrodisk system, where the peaks are labeled with  N  ;  M  , according to their mode order. The radial andazimuthal mode orders have been extracted by countingthe maxima in the mode profiles. The label  N  ðÞ ;  M  belongs to the mode TE ðÞ  N  ;  M  . Modes labeled with anasterisk have their field maxima only inside of themicrodisk and have therefore higher  Q -factors thanmodes labeled without asterisk, where the field maximaare localized at the interface. These surface-localizedmodes are similar to  D’yakonov surface waves  [10],which are lossless, but the conditions for excitation of surface waves propagating along the interface of isotropic and anisotropic medium are not fulfilled inthis system. Nevertheless, these modes can be attractivefor future applications. Energetically almost equidistantresonances with the same radial mode order areobserved. For the TE   N  ;  M   the energies are higher thanfor the TE  N  ,  M   due to confinement only within the disk.The coupling of the TE  N  ,  M   modes to the environmentwill therefore be larger because the global maxima of the field distribution is located at the interface of themicrodisk. Hence, they have a stronger interaction withthe environment.In Fig. 4, numerical results in a changing uniaxialanisotropic environment are shown. The mode mapshows the absolute square of the  H   z -component of theTE-like resonancesinthe system onalogarithmicscale.The changing  ffiffiffiffiffi e  zz p   component of the uniaxial aniso-tropic environment is plotted on the vertical axes. Sucha range is beyond current experimental realization, butinteresting as a model study (including the rangeachievable today, 1 : 4 9  ffiffiffiffiffi e  zz p   9 1 : 8, see for example [4]or [11]) for potential future materials. The energy rangewas chosen because of the high  Q  of the modes. Forvacuum permittivity the resulting spectrum is equal toFig. 3. With increasing  ffiffiffiffiffi e  zz p   , all resonances are redshifting. This effect is due to the decreased index S. Declair et al./Photonics and Nanostructures  –  Fundamentals and Applications 8 (2010) 273 – 277   275Fig.2. LeakageofWGMsinvacuumenvironmentduetobendinglossin the plane perpendicular to the main axis of the microdisk resonator(here:  x   y -plane). The color coding is: blue: negative, white: zero,red: positive amplitude of TE-like mode (real part of   H   z -component).The mode with  M   ¼ 5 (left) has more leakage of the field componentinto the (uniaxial anisotropic) environment than the modewith  M   ¼ 8(left) due to less bending loss. (For interpretation of the references tocolor in this figure legend, the reader is referred to the web version of the article.) [ Fig. 3. Top: Broadband spectrum of the TE response in vacuumenvironment and specification of radial (azimuthal) mode orders  N  ð  M  Þ  of the WGMs, log j  H   z j 2 . Examples of the in-plane fielddistribution are shown in the bottom plots for  M   ¼ 7. Left: TE 1,7 .Middle:TE 2,7 .Right:TE  2 ; 7 .Thecolorcodingis:blue:negative,white:zero, red: positive amplitude of TE-like mode (real part of   H   z -component). (For interpretation of the references to color in thisfigure legend, the reader is referred to the web version of the article.). [ Fig. 4. Wavelength map for resonant WGMs in the microdisk systemFig. 1, log j  H   z j 2 . The spectral line at vacuum permittivity refers to theenergetic range from 2 : 1    E     2 : 6eV from Fig. 3. With increasing  z -component of the electric permittivity tensor of the uniaxial aniso-tropic environment, resonant modes are red shifting and show clearanticrossing behavior when modes of different mode order approacheach other.  contrast, hence lower confinement, provided by theincreased  ffiffiffiffiffi e  zz p   of the environment. Thus, WGMs canleak deeper into the environment with preservedconfinement (but lower  Q ), so the effective radius of the WGM increases, yielding a lower energy. If theenvironmental potential is too low for the mode TE  N  ,  M  ,the mode is no longer supported by the microdisk. Forthe sake of validity of the continuity condition at theinterface of resonator and environment, continuousconnection of the tangential component of the solutioninside and outside is only possible if the mode orderchanges to lower radial, azimuthal mode order or bothto still maintain confinement in the system. This effectappearsintheanticrossingbehaviorofthemodesshownin Fig. 4 which is a typical sign for strongly coupledoscillators (normal mode splitting). Fig. 5 shows theanticrossing behavior from 1 : 54   ffiffi e p   zz   1 : 72(experimentally accessible range) and 2 : 32    E    2 : 36eV in more detail.In our system, only the  z -component is varied whilethe other components are equal to 1. Since theresonance is a non-perfect TE mode, the  H   z -componentis driven by all components of the electric field, hencethe  z -component of the electric field brings the  z -component of the electric permittivity into play.However, since the other components of the electricfield couple to the  H   z -component too, the confinementis maintained due to the TE-components of the resonantmode, but the anticrossing behavior is due to theinteraction of the non-TE-like component with  ffiffiffiffiffi e  zz p   .The effect of the uniaxial anisotropy can be seen inFig. 6, where the spectrum is shown on a broad energyrange for two cases of the environmental electricpermittivity:First,fortheuniaxialanisotropiccase(top)and second for the isotropic case (bottom). In bothcases, the values for  ffiffiffiffiffi e  zz p   and  ffiffi e p   were chosen to be1 ; 1 : 8 ; 3 : 4and3 : 6. The spectra for the first case are redshifted and confinement is preserved for almost allWGMs within the investigated energy range. Modeswith low azimuthal mode order  M   shift stronger thanmodes with higher azimuthal mode order due to thestronger interaction with the environmental uniaxialanisotropy. Confinement of the WGMs is maintained,when the electric permittivity tensor componentapproaches the electric permittivity of the microdisk or even exceeds it. This changes dramatically when theenvironmental electric permittivity is increased iso-tropically. For  ffiffi e p  ¼ 1 : 8 low energy modes are nolonger confined since the bending of the higherwavelengths causes the field components to leak moreinto the environment. Thus, they can couple stronger tothe propagating modes outside like in the anisotropiccase. All field components interact now stronger withthe environment. Accordingly, only higher energymodes are still confined in the system as it is expecteddue to less bending loss and less leakage into theenvironment. For even higher values of the permittivityclose to the permittivity of the microdisk, the modesvanish, because the system becomes more and morehomogeneous. Hence, there is no interface availablewhere the conditions for total internal reflection have tobe fulfilled. S. Declair et al./Photonics and Nanostructures  –  Fundamentals and Applications 8 (2010) 273 – 277  276 [ Fig. 5. Spectrum of WGMs (log j  H   z j 2 ) in an uniaxial anisotropicrange 1 : 54    ffiffiffiffiffi e  zz p    1 : 72 (experimentally accessible). The antic-rossing behavior is clearly visible. Dotted lines are guides for the eye. [ Fig. 6. Broadband spectrum of the TE response (log j  H   z j 2 ) of themicrodisk system Fig. 1 for different electric permittivity tensors of the environment for uniaxial anisotropic (top) and isotropic (bottom)with  ffiffiffiffiffi e  zz p   and  ffiffi e p   equal to 1 ; 1 : 8 ; 3 : 4and3 : 6. The difference betweenthe isotropic and uniaxial anisotropic environment variation isclearly visible since the WGMs are red shifting for the uniaxialanisotropic case (lower energy modes shift more) with preservedconfinement (but lower  Q -factors). For the isotropic case only highenergymodesarestillconfinedfor  ffiffi e p  ¼ 1 : 8whileforhigherelectricpermittivity all modes vanish since the index contrast is getting less.The dotted lines are guides for the eye to follow the fundamentalmode with  N   ¼ 1.
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