Description

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

All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.

Related Documents

Share

Transcript

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.
www.elsevier.com/locate/photonics
Available online at www.sciencedirect.com
Photonics and Nanostructures – Fundamentals and Applications 8 (2010) 273–277* Corresponding author. Tel.: +49 5251602325.
E-mail address:
sdeclair@mail.upb.de (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
ðÞ
;
M
belongs to the mode TE
ðÞ
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.

Search

Similar documents

Tags

Related Search

Separation Of Church And State In The United History of the German minority in RomaniaRepresentations of Race and Ethnicity In the The role of the European Union in the construDISTRIBUTION OF CALCIUM OXALATE CRYSTALS IN TDIVERSITY OF GENUS BULBOPHYLLUM THOUARS IN WEHistory of the Victorian Period in EnglandDiagnosis of Stargardt's disease in 2011The Use of the Old Testament in the NewThe Effect of Electronic Visual Feedback in L

We Need Your Support

Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us.

Thanks to everyone for your continued support.

No, Thanks

SAVE OUR EARTH

We need your sign to support Project to invent "SMART AND CONTROLLABLE REFLECTIVE BALLOONS" to cover the Sun and Save Our Earth.

More details...Sign Now!

We are very appreciated for your Prompt Action!

x