Table Of ContentResonant state expansion for Transverse Electric modes of two-dimensional open
optical systems
M.B. Doost
Independent Researcher, United Kingdom
(Dated: January 16, 2017)
The resonant state expansion (RSE), a rigorous perturbative method in electrodynamics, is for-
mulated for Transverse Electrodynamic modes of an effectively 2-dimensional system. The RSE is
a perturbation theory based on the Lippmann Schwinger Green’s function equation and requires
knowledge of the Green’s function of the unperturbed system constructed from Resonant-states. I
7 usetheanalyticGreen’sfunctionforthemagneticfieldtonormalizethemodesappearinginthecor-
1 respondingspectralGreen’sfunction. ThisuseoftheresidueandcutoftheanalyticGreen’sfunction
0 isasolutiontothefailureofthefluxvolumeintegralmethodofnormalizationforcontinuumstates,
2 a problem which is discussed in detail in this manuscript. In brief, the flux volume integral fails
n forcontinuumstatesbecausetheyarenottrueresonance. Ananalyticrelation betweennormalized
a magnetic and electric modes is developed as part of the solution to these difficulties. The complex
J eigenfrequenciesofmodesarecalculatedusingtheRSEforthecaseofahomogeneousperturbation.
3
1 PACSnumbers: 03.50.De,42.25.-p,03.65.Nk
]
s I. INTRODUCTION has to be replace with an appropriately calculated n
eff
c
which takes into account the thickness of the resonator,
i
t for the detailed mathematics, see Ref.[11, 12].
p Dielectric microcavities have attracted significant in-
o terest since 1989when they were found to support whis-
s. pering gallery modes (WGMs) [1], particularly with re- In this manuscript dispersion in the refractive index
c gards to the development of hybrid optoelectronic cir- is not treated directly, however it is straight-forward to
i
s cuits [2]. Perturbed microcavities have proved to be the add dispersion using the method from Ref.[20], which I
y mostpromisingdirectionofresearchsincetheiremissions developed in collaboration with E. A. Muljarov, based
h arenotisotropic[3–9]. Other usesof2-dimensionalopti- on my successful initial independent work incorporating
p
cal microcavities include miniature lasing devices [10]. dispersion into the RSE.
[
RecentlytherehasbeendevelopedtheResonant-state-
3 expansion (RSE) a theory for calculating the resonances
Interestingly the RSE is a near identical translation
v ofopenopticalsystems[15–20]. ThattheRSEissuitable
7 to electrodynamics of a much earlier theory from Quan-
for use with the RSE Born approximation, to calculate
0 tum Mechanics by More, Gerjuoy, Bang, Gareev, Gizza-
thefarfieldscattering,hasbeendemonstratedinRef.[21–
8 tkulov and Goncharov Ref.[13, 14]. The only difference
0 23]. The RSE Born approximation is a rigorous pertur- between the two approachesis the choice of RS normali-
0 bation method for electrodynamic scattering, which be- sationmethod. TheRSE[13,14]isaperturbationtheory
. comes the exact solution of Maxwell’s wave equations in
1 basedontheLippmannSchwingerGreen’sfunctionequa-
the far field, in the limit of an infinite number of reso-
0 tionandrequiresknowledgeoftheGreen’sfunction(GF)
7 nance being taken into account in the Born approxima- of the unperturbed system constructed from RSs. I was
1 tion expansion.
able to show in Ref.[21] that the general normalisation
: InthispaperIreportextendingtheRSEtotheTrans-
v of RSs which I derived in Ref.[17] is the most numeri-
i verse Electric (TE) modes of effectively two-dimensional callystableavailablenormalisationmethod. Thegeneral
X
(2D) systems (i.e. 3D systems translational invariant in
normalisation which I derived in Ref.[17] is based on a
r one dimension) which are not reducible to effective 1D
a prototype normalisationwhich appeared in Ref.[19].
systems. Ionlytreatsystemswithzerowavevectoralong
the translational invariant direction at present. I use
One difficulty in formulating an RSE for 2D systems,
a dielectric cylinder with uniform dielectric constant in
is the presence of a one-dimensional continuum in the
vacuum as unperturbed system and calculate perturbed
manifold of RSs. This continuum is specific to 2D sys-
Resonant-states(RSs) forhomogeneousperturbationsin
temsandisrequiredforthecompletenessofthebasisand
order to demonstrate the method.
thus for the accuracy of the RSE applied in 2D, as dis-
The treatment in this manuscript is restricted to 2-
cussed when similar observations were previously made
dimensional systems, however for the modes of a cavity
for Transverse Magnetic (TM) modes of a microcylinder
with mode wavelengthonly a smallfractionofthe thick-
in my Ref.[16].
ness of the cavity it is possible to make the mathemati-
calapproximationtoaneffectively 2-dimensionalsystem
[11, 12]. A consequence of this approximation for the In Ref.[16, 17] it wasshownthat the continuum states
mathematical treatment is that the refractive index n intheRSEcannotbenormalisedbythefluxvolumenor-
r
2
malisation, Inpractice,thecontinuumofnon-resonantstatesaredis-
cretisedandincludedascutpolesintheperturbationba-
∂ω2εˆ(r˙,ω)
1+δ = E (r˙) E (r˙)dr˙ (1) sis. This problem is treated in depth in Appendix A, B
kn,0 n · ∂ω2 n
ZA (cid:12)ω=ωn andC.Thecombinedindexn¯ isusedtodenotebothreal
+ lim En·∇E−(cid:12)(cid:12)(cid:12) E·∇EndL poleskn andcutpoles kα simultaneously. Theweighting
k→knILA k2−kn2 factors φn¯ are defined as follows
IwilldiscussthereasonforthisinmoredetailinSec.III. φ =1 for real poles,
Instead of using Eq.(1), it is necessary to make a com- φn¯ = φn for cut poles (5)
α
parison of the residue and contour integral between the (cid:26)
spectral GF After modifying the expansion of the perturbed wave
E (r) E (r′) φ E (r) E (r′) functions to include the cut poles to,
Gˆ (r,r′)= n ⊗ n n¯ n¯ ⊗ n¯
k
2k(k k ) ≈ 2k(k k )
Zn − n n¯ − n¯ E (r)= φ b E (r) (6)
X X ν n¯ n¯ν n¯
(2)
n¯
and the analytically derived GF. The electrodynamic X
wave function En¯ is the solution to which is the solution to the equation,
∇×∇×En¯(r)=kn2¯εˆ(r)En¯(r) (3) ∇×∇×Eν(r)=κν2[εˆ(r)+∆εˆ(r)]Eν(r) (7)
with outgoing boundary conditions (BCs) and Gˆ (r,r′)
k
withoutgoingBCs,IcanrepeatthederivationofRef.[16]
is the corresponding GF for Eq.(3).
and make use of the Lippmann Schwinger Green’s func-
Unfortunately it is not possible to derive the analytic
GF for the electric E-field since for 2-dimensional TE tion equation
modesofanopensystemMaxwell’swaveequationisnot
reducible to one dimensionwhilst in the electric field de- E (r)= κ2 Gˆ (r,r′)∆εˆ(r′)E (r′)dr′ (8)
scription. However the magnetic H-field GF is available ν − νZ k ν
and in Appendix A I use physical arguments to con-
in conjuncture with the spectral GF
struct an analytic formula linking normalised magnetic
lmytoidceGsrHeen¯n’stofunnocrtmioanlifsoerdtheelehcotrmicogmeondeoeussE2n¯-d.imTehnesiaonnaa-l Gˆk(r,r′)= En(r)⊗En(r′) φn¯En¯(r)⊗En¯(r′)
2k(k k ) ≈ 2k(k k )
microcylinder is given in Appendix B. Zn − n n¯ − n¯
X X
(9)
This manuscriptis organisedasfollowsSec.II outlines
to arrive at the modified the RSE Matrix equations,
the derivation of the RSE perturbation theory, Sec.III
explains the special provisions made for the normalisa-
tionoftheEn¯ modes,Sec.IVgivesthebasismodesofthe κν δn¯n¯′ + φn¯φ′n¯Vn¯n¯′/2 bn¯′ν =knbn¯ν (10)
2-dimensionalRSE,Sec.Vcontainsthenumericalvalida- Xn¯′ (cid:16) p (cid:17)
tionoftherigorousanalyticsprovidedinthismanuscript,
and
AppendixAandBderivethenormalisationoftheTEE
n¯
modes, Appendix C gives perturbation matrix elements
rpeeqnudiirxedDtogirveepsrtohdeuecveatlhueatniuonmoerfiacnalirmespuolrttsaonftSeexcp.rVe.ssAiopn- n¯′ δkn¯n¯n¯′′ + Vn¯2n¯′skφn¯n¯kφn¯′n¯′cn¯′ν = κ1νcn¯ν (11)
inAppendixA.AppendixEgivesanewandvaluablenu- X
mericalrecipeusedintheproductionofnumericalresults As always Vn¯n¯′ is given by
for this manuscript.
Vn¯n¯′ = En¯(r)∆εˆ(r)En¯′(r)dr (12)
II. THE RSE PERTURBATION METHOD Z
TheperturbationmethodwhichIhavebrieflyoutlined
InthissectionIgiveabriefaccountofthederivationof was recently translated from Quantum Mechanics [13,
theRSE[13,14]. Themethodofderivationisbuiltupon 14] and has now become known as the Resonant-state-
the Lippmann Schwinger Green’s function equation. For expansion.
a more detailed account of these derivations I direct the
interested reader to Ref.[13–23].
The spectral representation of the 2D system GF is III. MAXWELL’S WAVE EQUATION FOR
MAGNETIC FIELD
modified to include this cut contribution as integral [16]
E (r) E (r′) φ E (r) E (r′)
Gˆ (r,r′)= n ⊗ n n¯ n¯ ⊗ n¯ InthissectionIdiscussthenumericalandanalyticdif-
k
Zn 2k(k−kn) ≈ n¯ 2k(k−kn¯) ficulties associated with constructing an electrodynamic
X X
(4) RSE perturbation theory for magnetic resonances.
3
If we examine Maxwell’s wave equation for magnetic the outgoing wave boundary conditions and Maxwell’s
field H(r), wave equation with a delta function source term
−∇×∇×H(r)+εˆ(r)−1∇εˆ(r)×(∇×H(r))+k2εˆ(r)H(r) −∇×∇×GˆHk(r,r′)+ˆε(r)−1∇εˆ(r)×(∇×GˆHk (r,r′))
= J(r) εˆ(r)−1 εˆ(r) J(r) (13) +k2εˆ(r)GˆHk (r,r′)
∇× − ∇ ×
=1ˆδ(r r′) (16)
it would at first sight seem of no use to the RSE. In −
Eq.(13) we can see εˆ(r) which has prevented me from followingthederivationforthespectralrepresentationof
∇
formulating an equation analogous to Eq.(8) relating GˆE in Appendix B we find GˆH can be written as
k k
the perturbed eigenmodes of magnetic field to their un-
perturbed GF and perturbation in the dielectric profile. φ H (r) H (r′)
GˆH(r,r′)= n¯ n¯ ⊗ n¯ (17)
ThereforeasIamconsideringperturbationsinthedielec- k 2k(k k )
n¯
tric profile I am forced to formulate the RSE in terms of Xn −
electric modes En¯(r). where the field H (r) satisfies,
n¯
However even in light of the previous paragraph I do
need to consider Eq.(13) in this paper because the cut H (r)+εˆ(r)−1 εˆ(r) ( H (r))
poles cannot be normalised using [17, 22] −∇×∇× n¯ ∇ × ∇× n¯
+k2εˆ(r)H (r)=0 (18)
n¯
∂ω2εˆ(r˙,ω)
1+δkn,0 = En(r˙)· ∂ω2 En(r˙)dr˙(14) withoutgoingBCs. InAppendixAIwillmakeuseofHn¯
ZA (cid:12)ω=ωn definedinEq.(16),Eq.(17)andEq.(18)tonormaliseE .
E E (cid:12) E E n¯
+ lim n·∇ −(cid:12)(cid:12) ·∇ ndL I will briefly explain the idea behind the analytics in
k→knILA k2−kn2 Appendix A and direct the interested reader to that rel-
evant appendix where I give my detailed mathematics.
The method of normalisation will thus make use of The E and H fields are both generated by an electric
Eq.(13). ThereasonIcan’tuseEq.(14)forthecutpoles current J. By using an identical current to generate E
is that they are not true resonances of the system. This and H fields and letting the electric current amplitude
problem can be intuitively understood if we assume we tend to zero as its harmonic wave number of oscillation
cannormaliseasinglecutpolewithEq.(14)andobserve tends to k we can compare residues of GˆH and GˆE in
n¯ k k
the resulting logical contradiction. If the discretisation order to relate normalised E and H . The H modes,
n¯ n¯ n¯
of the cut is made finer by increasing the number of cut including the continuum modes, discussed in Appendix
poles by several orders of magnitude, logically the nor- B, can be normalised by comparing the residue of the
malisation of the chosen cut pole should be drastically spectral GF and the residue of the analytic GF, derived
reduced as it’s weight is further sharedbetween many of in Appendix B.
these extra poles. This final remark gives the contradic-
tion because the normalisation calculated from Eq.(14)
cannot change, it is fixed. IV. UNPERTURBED BASIS FOR 2D SYSTEMS
In light of the problems normalising cut poles I am
forced to normalise the basis modes in this chapter by
In this section I will use the approach outlined in
comparing spectral GFs of the form [16]
Sec.III to formulate the basis states for both TE, TM
polarizations. ForcompletenessofthebasisIincludelon-
E (r) E (r′) φ E (r) E (r′)
Gˆ (r,r′)= n ⊗ n n¯ n¯ ⊗ n¯ gitudinal electric modes which are curl free static modes
k
Zn 2k(k−kn) ≈ n¯ 2k(k−kn¯) satisfying Maxwell’s wave equation for kn =0. Longitu-
X X
(15) dinal electric modes have zero magnetic field.
which will still hold true for cut poles, with the analytic Splitting off the time dependence e−iωt of the elec-
GF of magnetic field calculated in Appendix B. Unlike tric fields E and D and magnetic fie∝ld H, the first pair
the TM case where the analytic GF for the electric field of Maxwell’s equations can be written in the form
GˆE(r,r′) is available, I can only derive the analytic GF
forkthe magnetic field GˆH(r,r′) in the case of transverse E=ikH, H= ikD J (19)
k ∇× ∇× − −
electric (TE) polarisation. The reason for this is that
a scalar GF equation is available for GˆH(r,r′) but not where k = ω/c and D(r) = εˆ(r)E(r). Combining them
k leads to
for GˆE(r,r′) in the TE case. Fortunately in Appendix
k
A I have been able to derive a method of expressing the E(r)+k2εˆ(r)E(r)=ikJ(r) (20)
electric field basis modes E in terms of the magnetic −∇×∇×
n¯
field basis modes Hn¯. and
The Green’s function for the magnetic component of
an electrodynamic system is a tensor GˆH which satisfies E (r)=k2εˆ(r)E (r) (21)
k ∇×∇× n n n
4
whichsatisfytheoutgoingwaveboundarycondition,and theperturbationleftthedielectricprofileasatensorthen
to all polarisations would mix.
The unperturbed RS wave functions factorise as
E (r) E (r′) φ E (r) E (r′)
Gˆ (r,r′)= n ⊗ n n¯ n¯ ⊗ n¯
k Zn 2k(k−kn) ≈ n¯ 2k(k−kn¯) TM: En(r)=ATMRm(ρ,kn)χm(ϕ)ez
X X
(22) TE: H (r)=ATE(k )R (ρ,k )χ (ϕ)e (30)
n m n m n m z
for the corresponding GF.
For kz = 0, these three groups of modes of a homoge- I normalized the wave functions from the analytic TM
neous dielectric cylinder can be written as (TE)Green’sfunctioninAppendixBwithnormalisation
constants A(k)
TM: E=fe ,
z
TE: H=fez, 1 2
ATM = (31)
LE: E=−∇f, Rsn2r −1
where f(r) is a scalar function satisfying the Helmholtz
equation
2k[J (n kR)]2
ATE = m r
∇2f +k2εf =0 (23) m vuu(n2r−1) mk2 [Jm(nrkR)]2+R2k[Jm′ (nrkR)]2
withthe dielectric susceptibility ofthe sphereinvacuum u (cid:20) (cid:21)
t (32)
The two boundary conditions atthe surface of the cylin-
n2 for ρ6R,
ε(ρ,ϕ)= r (24) der, the continuity of the electric field and its radial
1 for ρ>R
(cid:26) derivative, produce a secular equation for the RS wave
number eigenvalues k , which has the form
and n
f(r)=Rm(ρ,kn)χm(ϕ) (25) DmTM(knR)=0 (33)
The angular parts are defined by
DTE(k R)=0 (34)
m n
π−1/2sin(mϕ) if m<0,
χ (ϕ)= (2π)−1/2 if m=0, (26) where
m π−1/2cos(mϕ) if m>0 DTM(z)=n J′ (n z)H (z) J (n z)H′ (z) (35)
m r m r m − m r m
where there is nom = 0 mode for the TE case. The
χm(ϕ) are orthonormal according to DTE(z)=J′ (n z)H (z) n J (n z)H′ (z) (36)
m m r m − r m r m
2π
′ ′
χm(ϕ)χm′(ϕ)dϕ=δmm′ (27) and Jm(z) and Hm(z) are the derivatives of Jm(z) and
Z0 Hm(z), respectively. Here z represents a complex argu-
ment, as opposed to the spatial coordinate used earlier.
The radial components satisfy,
Incylindricalcoordinates,theelectricvectorfieldE(r)
∂2 1 ∂ m2 can be written as
+ +ε(ρ)k2 R (ρ,k)=0 (28)
∂ρ2 ρ∂ρ − ρ2 m
(cid:20) (cid:21) Eρ
and have the form E(ρ,ϕ,z)=Eρer+Eϕeϕ+Ezez =Eϕ (37)
E
z
J (n kρ)/J (n kR) for ρ6R,
R (ρ,k)= m r m r (29)
m H (kρ)/H (kR) for ρ>R Therefore the TM modes can be written as
m m
(cid:26)
0
in which J (z) and H (z) H(1)(z) are, respectively,
the cylindrmical Bessel amnd H≡ankeml functions of the first ETn¯M(r)=ATM 0 (38)
kind. Hm(z) is a multiple valued function, with multi- Rm(ρ,kn¯)χm(ϕ)
ple values corresponding to a range of possible bound-
ary conditions at infinity. In order to obtain the correct Since I am considering a homogeneous micro-cylinder
boundary conditions for the RSE I provide a numerical and TE modes, I can now calculate the normalised E
n¯
recipe in Appendix E. fromthe H using the followingsimple relations derived
n¯
I treat in this work all polarisations, however only in Appendix A
TM and LE polarisations mix due to the perturbations
treated being strictly scalar in dielectric permittivity. If H (r)=M k2E (r) (39)
∇× n¯ kn¯ n¯ n¯
5
f
which gives, inside the homogeneous cylinder only, a
-0.05 0.00 0.05
R (ρ,k )dχ (ϕ)
m n¯ m (a) (b)
ETn¯E(r)= MAkTn¯Ekn2¯ −∂Rmρ∂(ρρ,kn¯)χdmϕ(ϕ) (40) 0 m = 0 m = 0
0
TheunknownM iscalculatedanalyticallyinAppendix
kn¯ -5
A and Appendix D.
)
The k =0 LE modes are given by, inside the homoge- R
k
neous cylinder only, (
m
I m = 10
m = 10
ELn¯E(r)= |m|(n2r−1)klnim→0ETn¯E(r) (41) -10
p
which follows from Eq.(14), in the limit of k =0, noting
thefastattenuationoftheRSfarfieldinthisk =0case.
ForboththeTMandTEcase,treatedhere,thefullGF
m = 21
of the homogeneous dielectric cylinder, which is defined -15 m = 20
via Maxwell’s equation with a line current source term,
-10 0 10 -0.5 0.0 0.5
is given by Gˆ =G e e , in which
k k z ⊗ z Re(kR) Ims
m
′ ′ ′
G (ρ,ρ)= G (ρ,ρ;k)χ (ϕ)χ (ϕ) (42)
k m m m
FIG. 1: (a): Cut poles kα (stars) representing the cut of the
m
X GF of a homogeneous dielectric cylinder with nr =2, in the
and the radial components have the following spectral complex wave-number plane for m = 0, 11, and 20. Normal
representation poleskn (opensquares)arealsoshown. (b): Cutpoledensity
σm(k) (solid curves) and the cut pole strength φ(αm) (stars),
[A (k )]2R (ρ,k )R (ρ′,k ) for the same values of m.
′ m n m n m n
G (ρ,ρ) =
m
2k(k k )
n
n −
X
0 [A (k)]2R (ρ,k′)R (ρ′,k′)
+ m m m σ (k′)dk′ An example of cut poles assigned for m = 0, 10, and
Z−i∞ 2k(k−k′) m 20 is given in Fig.1(a). The cut poles contribute to the
[A (k )]2R (ρ,k )R (ρ′,k ) RSE in the same way as the normal poles.
m n m n m n
(43)
≡ 2k(k k ) In the numerical calculation in Sec.V, I use N(m)
XZn − n N(m), where N(m) is the number of normal polescin th∼e
derived in Appendix B. For TE modes I take Am(k) = basis for the given m and Nc(m) is the number of cut
ATmE(k) and for the TM modes I take Am(k) = ATmM, poles, in order to demonstrate the convergence towards
then the exact solution.
8J (n kR)2 1 The next section discusses results of the RSE for dif-
m r
σm(k)= π2kR2D+(kR)D−(kR)A (k)2 (44) ferent effective 2D systems. We consider a homogeneous
m m m
dielectriccylinderofradiusRandrefractiveindexn =2
r
Here D±(z) are the two limiting values of D (z′) for z′ (ε = 4) with a homogeneous perturbation of the whole
m m
approaching point z on the cut from its different sides cylinder in Sec.V. Explicit forms of the matrix elements
Rez′ ≷ 0. D±(z) are taken to be DTE or DTM in the for these perturbations and details of their calculation
m m m
case of TE or TM cut pole respectively. are given in Appendix C.
I use the same method to discretised this continuum
into cut poles k and cut pole strengths φ(m) as in my
α α
Ref.[18]
V. HOMOGENEOUS CYLINDER
qα(m+)1 PERTURBATION
k(m) = kσ (k)dk φ(m) (45)
α m α
Zqα(m) ,
In this section I present my numerical results which
where the cut pole strength φ(m) is defined as support the validity of the analytic results relating to
α
the 2-dimensionalTE RSE whichI haveprovidedin this
qα(m+)1 manuscript.
φ(m) = σ (k)dk (46)
α Zqα(m) m The perturbation we consider in this section is a ho-
6
mogeneous change of ε overthe whole cylinder, given by values κ(exact).
ν
We choose the basis of RSs for the RSE in such a
∆ε for ρ6R,
∆ε(ρ,ϕ)=∆εθ(R ρ)= (47) way that for the given azimuthal number m and the
− 0 for ρ>R given number of normal RSs N we find all normal poles
(cid:26)
k < k (N) with a suitably chosen maximum wave
with the strength ∆ε = 4 used in the numerical cal- n max
| |
vectork (N)andthenaddthecutpoles. Wefindthat
culation. For ϕ-independent perturbations, modes with max
different azimuthal number m are decoupled, and so are as we increase N, the relative error κν/κν(exact) 1 de-
even and odd (cosine and sine). I show only the sine creases as N−3. Following the procedure descr−ibed in
(cid:12) (cid:12)
modes here,and use for illustrationm=20. The matrix Ref.[18] we can extrapolate the pe(cid:12)rturbed wave n(cid:12)um-
elements of the perturbation are calculated analytically bers. Theresultingperturbedwavenumbersareshownin
and given by Eq.(12). The homogeneous perturbation Fig.2. The perturbation is strong, creating 3 additional
doesnotchangethe symmetryofthe system,sothatthe WGMs with m=20 having up to 4 orders of magnitude
perturbedmodesobeythesamesecularequationEq.(34) narrower linewidths. For N = 800, the RSE reproduces
with the refractive index n of the cylinder changed to about 100 modes to a relative error in the 10−7 range,
r
n2+∆ε,andthustheperturbedwavenumbersκ cal- which is decreasing by one or two orders of magnitude
r ν
culated using the RSE can be compared with the exact afterextrapolation. Thecontributionofthecutandzero
p
frequencymodeissignificant: Ignoringthesemodesleads
to a relative error of the poles in the 10−3 range. The
102
101 factthattherelativeerrorimprovesby4-5ordersofmag-
100 nitude after taking into account the cut in the form of
10-1 (a) thecutpolesshowsthevalidityofthereportedanalytical
10-2
treatment of cuts in the RSE, and the high accuracy of
10-3 TE polarisation l = 20
10-4 the discretisation method into cut poles.
10-5
R) 10-6 Unperturbed
- Im(k 111000---987 EPCxeuart tcautnrbde zde ro pole missing VI. SUMMARY
10-10 eD (r) = 4 De (r) I have applied the resonant state expansion (RSE) to
10-11
effectivetwo-dimensional(2D)openopticalsystems,such
0 e(r)
-4.0x10-9 as dielectric micro-cylinders and micro-disks with per-
-8.0x10-9 0 a turbations, by finding a formula linking the normalisa-
10-1 tion of resonant states appearing in the magnetic and
(b) electric Green’s functions which describe identical sys-
10-2
tems. Theseanalyticformulasallowustotakeadvantage
10-3 of an analytic Green’s function of the magnetic field of
error 1100--54 NECxoutttr aeapxnotdrla azpteeordloa tpeodle missing tahpepehaorminoggeinnetohuessmpieccrtor-aclylrienpdreersetnotnatoiromnaolifsethteheGmreoedne’ss
ve 10-6 function for the electric field, electric modes which when
elati 10-7 noUrmsianlgisetdhefoarnmaltyhteicablalysiskonfowthnebRaSsiEs.of resonant states
R 10-8
(RSs) of an ideal homogeneous dielectric cylinder – a
10-9 complete set of eigenmodes satisfying outgoing wave
10-10 boundary conditions – I have treated an effectively 1D
10-11 homogeneous perturbations. I investigated the conver-
0 10 20 30 40 50 genceforthisperturbationsandcomparedthe RSEwith
Re(kR) analytic solutions. I have found agreement between the
RSE and analytic solutions.
FIG. 2: (a): Perturbed RS wave numbers for the homo-
geneous perturbation Eq.(47) calculated via the RSE with
N = 800 (only sine modes are shown). The perturbed poles
Acknowledgments
with (+)andwithout (×)thecutcontributionarecompared
with the exact solution (open squares). Unperturbed wave
numbers are also show (open circles with dots). Inset: Di- The work in this paper is completely my own original
electric constant profile for the unperturbed and perturbed mathematicalderivationsandnumericaldata. Thiswork
systems. (b): Relativeerrorinthecalculatedperturbedwave was completed in 2013, and shortly after its completion
numbers with (heptagons) and without (triangles) contribu- I presented a draft of this manuscript to my then col-
tion of the cut. Relative error for a simulation including the laborator Dr E. A. Muljarov. Upon Dr E. A. Muljarov’s
cutandimprovedbyextrapolationisalsoshown(crossedcir- insistence (as my PhDsupervisorand manager),a refor-
cles).
mulated version of the mathematics in Appendix A was
7
generated in collaboration with Dr E. A. Muljarov and The formulas for the E(r)-field and H(r)-field in terms
presented in my PhD Thesis. I do not make use of any of their respective Green’s functions are given as the fol-
part of these reformulations of my original work in this lowing convolutions with the source,
manuscript, hence I am the sole author of this work.
E(r)= GˆE(r,r′)ikJ(r)dr′ (A6)
k
Z
Appendix A: Normalised electric modes expressed ε(r′)
in terms of normalised magnetic modes H(r)= GˆH(r,r′) J(r′) ∇ J(r′) dr′
k ∇× − ε(r′) ×
Z (cid:20) (cid:21)
(A7)
InthisappendixIprovidetheanalyticformularelating
SinceatthisstageJ(r)isarbitrarywecansetJ(r)=0
thenormalisedE andnormalisedH modes. Thebasic
n¯ n¯ when ρ > R− where R− is some radius R− < R. R is
principals ofmy derivationareas follows,let us consider
the radius of the micro-cylinder. Also since we choose
the case where the E and H fields are both generated
our basis system to be a homogeneous micro-cylinder,
by a oscillating electric current J. By using an identical
importantly for Eq.(A11) ε(ρ,ϕ) = 0 when ρ < R.
currenttogenerateEandHfieldsandlettingthecurrent ∇
At this stage k is also arbitrary so let it be k = k +
tend to zero as its harmonic wavenumber of oscillation n¯
∆k, ∆k 0. To simplify our discussion let W(r) be a
tends to the resonant wavenumber kn¯ we can compare function→related to J(r) by,
residues of GˆH and GˆE in order to relate normalised
k k
E and H . The H modes, including the continuum ikJ(r)=2k ∆kW(r) (A8)
n¯ n¯ n¯ n¯
modes, discussed in Appendix B, can be normalised by Eq.(A8)allowsJ(r)tomeettherequirementsthatIhave
comparingtheresidueofthespectralGFandtheresidue
just set out.
of the analytic GF, derivedin Appendix B.The detailed
In light of Eq.(A8) Eq.(A6) becomes
mathematics will now be thoroughly explained.
Our starting point for finding a relation between the φ E (r) E (r′)
correctnormalisationofthemodesappearinginthespec- lim m¯ m¯ ⊗ m¯ 2k ∆kW(r′)dr′
n¯
tral Green’s function for the magnetic H(r)-field and ∆k→0ZR−" m¯ 2km¯(kn¯ −km¯ +∆k)#
X
electric E(r)-field of a homogeneous dielectric micro-
cylinder is Maxwell’s equations of electrodynamics, =φn¯En¯(r) En¯(r′)·W(r′)dr′ =E(r) (A9)
Z
E=ikH, H= ikD J (A1) where we have replaced GˆEk(r,r′) with its spectral rep-
∇× ∇× − − resentation
where we have assumed harmonic oscillationof the driv-
ingcurrentwithrespecttotime, sothatthe timederiva- E (r) E (r′) φ E (r) E (r′)
Gˆ (r,r′)= n ⊗ n n¯ n¯ ⊗ n¯
tive in Eq.(A1) can be replaced with ik. k 2k(k k ) ≈ 2k(k k )
FortheE(r)-fieldwithacurrentsourceJ(r)oscillating XZn − n Xn¯ − n¯
(A10)
at frequency k we have according to Maxwell,
Similarly Eq.(A7) becomes
−∇×∇×E(r)+k2εE(r)=ikJ(r) (A2) H(r)= lim φm¯Hm¯(r)⊗Hm¯(r′)
and also the corresponding wave equation with a source ∆k→0ZR−" m¯ 2km¯(kn¯ −km¯ +∆k)#
for the H(r)-field, 2k X∆k W(r′)
n¯ ∇× dr′
· ik
−∇×∇×H(r)+ ∇εε(r(r)) ×(∇×H(r))+k2ε(r)H(r) =φn¯Hn¯(r(cid:20)) Hn¯(r′)·(∇×(cid:21)W(r′))dr′ (A11)
ik
ε(r) ZR− n¯
=∇×J(r)− ∇ε(r) ×J(r) (A3) where we have replaced GˆHk (r,r′) with its spectral rep-
resentation
The GF for Eq.(A2) is, φ H (r) H (r′)
GˆH(r,r′)= n¯ n¯ ⊗ n¯ (A12)
k 2k(k k )
GˆE(r,r′)+k2εˆ(r)GˆE(r,r′)=1ˆδ(r r′) (A4) Xn − n¯
−∇×∇× k k −
Eq.(A11) andEq.(A9) canbe substituted in Eq.(A1)
and for Eq.(A3) to give,
GˆH(r,r′)+ˆε(r)−1 εˆ(r) ( GˆH(r,r′)) [ En¯(r)] En¯(r′) W(r′)dr′
−∇×∇× k +k2εˆ(r)GˆH(∇r,r′) × ∇× k ∇× ZR− ·
k =1ˆδ(r r′) (A5) =Hn¯(r) Hn¯(r′)·(∇×W(r′))dr′ (A13)
− ZR−
8
We now make the substitution W(r) = Γ(ρ)En¯(r), having made use of
where,
H (r)=M k2E (r) (A21)
∇× n¯ kn¯ n¯ n¯
1 for ρ6R−, whichcomesfromtaking the curlofEq.(A15) andusing
Γ(ρ)= (A14)
0 for ρ>R− the definition ofRSs, specifically, inside the homogenous
(cid:26)
cylinder,
We can calculate the curl of W(r) analytically. First
note for a given mode we can evaluate Eq.(A1) to be ∇×∇×En¯(r)=kn2¯n2rEn¯(r) (A22)
equivalently (in this case)
Eq.(A20) is not instantly recognisable as a standard
integral, however if we make use of the two following
n2H (r)
E (r)= r n¯ (A15) general properties of Bessel function,
n¯
∇× M
kn¯ Jm−1(z)−Jm+1(z)=2Jm′ (z) (A23)
where M is an unknown to be found.
kn¯
UsingEq.(A15)wecanwritethefollowingidentityfor 2m
W(r): Jm−1(z)+Jm+1(z)= z Jm(z) (A24)
we see that
W(r)= Γ(ρ) E (r)+Γ(ρ) E (r)
n¯ n¯
∇× ∇ × ∇×
=−δ(ρ−R−)eρ×En¯(r)+Γ(ρ)n2rMHn¯(r) (A16) ZR−En¯(r′)2dr′
kn¯ A(k )2n2
= n¯ r J2 (n k ρ′)+J2 (n k ρ′) ρ′dρ′
Substituting Eq.(A16) into Eq.(A13) and cancelling 2M2 k2 m−1 r n¯ m+1 r n¯
common factors of Hn¯(r) gives kn¯ n¯ ZR−(cid:2) (cid:3) (A25)
which is a standard integral given in Appendix C.
n2
r E (r′)2dr′ Inthe lastintegralinEq.(A18)wecanmakesubstitu-
n¯
M
(cid:20) kn¯(cid:21)ZR− tions from Eq.(23) and Eq.(30) to obtain the following
= Hn¯(r′) δ(ρ′ R−)eρ En¯(r′)dr′+ standard integral given in Appendix C,
− · − ×
ZR− H (r′)2dr′ =A(k )2 J2(nk ρ′)ρ′dρ′ (A26)
n2[H (r′)]2 n¯ n¯ m n¯
+ Γ(ρ′) r n¯ dr′ (A17) ZR− ZR−
M
ZR− kn¯ Finally we use Eq.(A25) and Eq.(A26) to solve
Eq.(A18) for M
Substituting Eq.(40) into Eq.(A17) kn¯
2M2
kn¯ =
E (r′)2dr′ n4
n¯ r
ZR− J2 (n k ρ′)+J2 (n k ρ′) ρ′dρ′
=Z−ππR−dϕχm(ϕ)χm(ϕ)·A2n¯Jmk(n2¯nnk2rn¯R−)∂Jm(n∂kρn¯R−) = R−Jm(nRrk(cid:2)n¯mR−−1)∂Jrmn¯(n∂rrkn¯R−m)++1 krn2¯εn¯ Jm(cid:3)2(nkn¯ρ′)ρ′dρ′
+ Hn¯(r′)2dr′ (A18) =R n22k2 (A27)
ZR− r n¯
or
Eq.(40) was generated by Eq.(A21), where M is the
kn¯ n2
unknown we are trying to find. In Eq.(A18) we took M2 = r (A28)
kn¯ k2
n¯
H (r)=A J (nk ρ)χ (ϕ)e (A19)
n¯ n¯ m n¯ m z I evaluate analytically the integral in Eq.(A27) in Ap-
and re-calculated E according to Eq.(A21). pendix D.
n¯ Hence our analytics in this section have reached their
Calculating the first integral in Eq.(A18) by making
conclusion. We can now mathematically relate nor-
substitutions from Eq.(40)
malisedE modes to normalisedH modes withthe an-
n¯ n¯
alytic formula,
En¯(r′)2dr′ H (r)=M k2E (r) (A29)
ZR− ∇× n¯ kn¯ n¯ n¯
A(k )2 m2J2(nk ρ′) ∂J (nk ρ′) 2 I make use of this valuable derived Eq.(A29) in Ap-
= M2n¯k4 mρ′ n¯ +ρ′ m∂ρn¯ dρ′ pendix B, Sec.III, Sec.IV and Sec.V, to translate the
kn¯ n¯ ZR−" (cid:20) (cid:21) # correctlynormalisedHn¯ modesthereinderived,intocor-
(A20) rectly normalised E modes.
n¯
9
Appendix B: Green’s function of a homogeneous in which ρ = min ρ,ρ′ , ρ = max ρ,ρ′ , and the
< >
cylinder Wronskian W(f,g)={fg′} f′g. For TE{pola}rization, a
−
suitable pair of solutions is given by
Inthis appendix I nearlyexactly repeatthe derivation
I made with E. A. Muljarov in Ref.[16] and derive an f (ρ) = √ρ Jm(nrρk), ρ6R,
analytic GF for 2-dimensional cylindrical homogeneous m · amJm(ρk)+bmHm(ρk), ρ>R
(cid:26)
electromagnetic resonators. c J (n ρk)+d H (n ρk), ρ6R,
g (ρ) = √ρ m m r m m r
Theonlydifferencebetweentheanalyticshereandthe m · Hm(ρk), ρ>R
analytics of Appendix B Ref.[16] is that here I am re- (cid:26)
strictedtothe magnetic H-fieldGFofthe 2-dimensional where
TEmodesofthecylinder,whileinAppendixBofRef.[16]
we were restricted to the E-field TM modes of a mathe- a (k) = H (x)J′ (n x) n H′ (x)J (n x) πix/2n ,
m m m r − r m m r r
matically identical system. ′ ′
The reasonfor the difference inapproachbetweenthis bm(k) = (cid:2)nrJm(x)Jm(nrx)−Jm(x)Jm(nrx) π(cid:3)ix/2nr,
′ ′
appendix and Ref.[16] is that to formulate the E-field cm(k) = (cid:2)nrHm(x)Hm(nrx)−Hm(x)Hm(nr(cid:3)x) πinrx/2,
TreEntmlyoadnesuninstoolveeffdecptriovbellyem1.-dTimoegnesnieornaatleetqhueaatnioanlystiisccGurF- dm(k) = (cid:2)Hm(x)Jm′ (nrx)−nrHm′ (x)Jm(nrx) π(cid:3)inrx/2
oftheelectrodynamicproblem,usingthemethodsinthis with x=k(cid:2)R. The Wronskian is calculated to b(cid:3)e
appendix,requirestherecastingoftheequationintoaset
of decoupled 1-dimensional problems. W(f ,g )=2in2a (k)= nxD (x)/π, (B6)
m m r m − m
TheTEcomponentoftheGFofahomogeneouscylin-
der in vacuum satisfies the following equation for mag- with Dm(x) defined in Eq.(36). Inside the cylinder, the
netic field GF then takes the form
GˆH(r,r′)+ ∇ε(r) ( GˆH(r,r′)) G˜Hm(ρ,ρ′;k) = π ρρ′ Jm(nrkρ<)Hm(nrkρ>)
−∇×∇× k ε(r) × ∇× k 2i (cid:20)
p
+k2ε(r)GˆH(r,r′)=1ˆδ(r r′) (B1) cm(k)J (n kρ )J (n kρ ) (B7)
k − −a (k) m r < m r >
m (cid:21)
with
The GF has simple poles k in the complex k-plane
n
n2 for ρ6R, which are the wavevectors of RSs, givenby a (k )=0,
ε(ρ)= r (B2) m n
1 for ρ>R an equation equivalent to Eq.(34). The residues Res of
(cid:26) n
the GF at these poles are calculated using
Using the angular basis Eq.(26) the GF can be written
as
c (k)
m
r (k )=
GHk (ρ,ρ′)= √1ρρ′ G˜Hm(ρ,ρ′;k)χm(ϕ)χm(ϕ′) (B3) m n −ddkam(k)(cid:12)(cid:12)k=kn
Xm (cid:12)(cid:12) 2i
= (cid:12)
similartoEq.(42). Notethatweredefinedheretheradial π(n2 1) m2 [J (nk R)]2+R2k [J′ (nk R)]2
part as G˜H(ρ,ρ′;k)=√ρρ′GH(ρ,ρ′;k) which satisfies r − k m n n m n
m m (cid:20) n (cid:21)
(B8)
d2 m2 1/4 1 dε(ρ) 1
dρ2 − ρ−2 + ε(ρ) dρ 2ρ G˜Hm(ρ,ρ′;k)+ which can be seen by noting that inside the resonantor
(cid:20) (cid:21) Eq.(B5) and Eq.(A12) are equal as k k as they are
n
1 dε(ρ) d →
+ k2ε(ρ) G˜H (ρ,ρ′;k)=δ(ρ ρ′) equal for all k, at every point in space inside the res-
m
− ε(ρ) dρ dρ − onator.
(cid:20) (cid:21)
(B4) In addition to the poles, the GF has a cut in the com-
plexk-planealongthe negativeimaginaryhalf-axis. The
Using two linearly independent solutions f (ρ) and
m cut is due to the Hankel function H (z).
m
g (ρ)ofthecorrespondinghomogeneousequationwhich
m Owing to the cut of the Hankel function H (z) the
m
satisfy the asymptotic boundary conditions
GF also has a cut along the negative imaginary half-
axis in the complex k-plane, so that on both sides of the
f (ρ) ρm+1/2 for ρ 0,
gm(ρ)∝eikρ for ρ→ cut G˜m takes different values: G˜+m one the right-hand
m ∝ →∞ side and G˜− one the left-hand side of the cut. The step
m
the GF can be expressed as ∆G˜m = G˜+m −G˜−m over the cut can be calculated using
the corresponding difference in the Hankel function:
f (ρ )g (ρ )
G˜Hm(ρ,ρ′;k)= mW(<fm,mgm)> (B5) ∆Hm(z)=Hm+(z)−Hm−(z)=4Jm(z)
10
The result is where,
π
∆G˜H (ρ,ρ′;k)= ρρ′J (n kρ )J (n kρ )∆Q (k)
m 2ip m r < m r > (mB9) γ(kn¯)= |m|(1n2r −1) footrhekrwn¯i=se0, (C2)
(cid:26)p
where
and
c+ c− 4 2 1
∆Q (k)= 4 m + m =
m (cid:20) − a+m a−m(cid:21) −(cid:18)πkR(cid:19) Dm+(k)Dm−(k) Ij(kn¯,kn¯′)= Jj(nkn¯ρ′)Jj(nkn¯′ρ′)ρ′dρ′ (C3)
(B10) ZR
with D (k) given by Eq.(36).
m yielding for identical basis states (n¯ =n¯′)
G˜Hm(ρ,ρ′;k′)dk′ = −i∞ G˜H+mdk′ + 0 G˜H−mdk′ Im(kn¯,kn¯′)
I=2πi k−Rk′esn 2πiGZ˜0H (ρ,ρk′;−k)k′ Z−i∞ k(−Bk1′1) = R22 [Jm(nrkn¯R)]2−Jm−1(nrkn¯R)Jm+1(nrkn¯R)
k k − m (cid:2) (C(cid:3)4)
n
n −
X
′
Note that in second part of the above equation we have and for different basis states (n¯ =n¯ )
6
made use of the residue theorem, expressing the closed-
loopintegralin the left-hand side in terms of a sum over R kn¯′
residues at all poles inside the contour. Using Eq.(B11) k2 k2 · n Jm(kn¯nrR)Jm−1(kn¯′nrR)
n¯ − n¯′ r
the GF can be expressed as
R k
n¯
G˜H(ρ,ρ′;k)= Resn + 1 0 ∆G˜Hm(ρ,ρ′;k′)dk′ − kn2¯ −kn2¯′ · nrJm(kn¯′nrR)Jm−1(kn¯nrR)
m n k−kn 2πiZ−i∞ k−k′ =Im(kn¯,kn¯′) (C5)
X
(B12)
When k = 0 we can use the asymptotic form of the
which is a generalization of the Mittag-Leffler theorem. n¯
Bessel function,
In conclusion then, the residues Res of the GF con-
n
tributing to Eq.(B12) are calculated as (z/2)m
π Jm(z)= (C6)
Res = ρρ′J (n kρ )J (n kρ )r (k ) (B13) Γ(m+1)
n m r < m r > m n
2i
p
withr (k )foundinEq.(B8). Giventhatthespatialde-
m n Γ(m+1)=m! (C7)
pendence of the GF, as described by Eqs.(B12), (B13),
and(B9),isrepresentedbyproductsoftheRSwavefunc-
from which we see k 0 cancels out everywhere in
n¯
tionsR (ρ,k )andtheiranalyticcontinuationsR (ρ,k) →
m n m Vn¯n¯′.
with k-values taken on the cut, we arrive at the GF in
There is an important transformation to note,
the formof Eqs.(43) and(44)which arethenused inthe
RSE, once the modes have been converted into En¯ form ATE(kn¯) =AˆTE(k ) (C8)
using the analytics in appendix A. n¯
J (k n R)
m n¯ r
Hence with the analytic integrals in this appendix we
Appendix C: Homogeneous cylinder perturbation are able to completely reproduce the analytic results in
Sec.V.
In this appendix is given the perturbation matrix el-
ements for the homogeneous perturbation of the TE
modes of a homogeneous cylinder. I give the analytic Appendix D: Evaluating Equation A26
expressionfortheperturbationmatrixelementsrequired
to produce the numerical results in Sec.V, numerical re- In this appendix I analytically evaluate the important
sults which validate the analytics which I developed for integral equation Eq.(A27)
this manuscript.
The homogeneous perturbation Eq.(47) does not mix 2M2
different angular momentum m-values. The matrix ele- kn¯ =
n4
ments between RS with the same azimuthal number m r
J2 (n k ρ′)+J2 (n k ρ′) ρ′dρ′
are given by the radial overlap integrals = m−1 r n¯ m+1 r n¯
AˆTE(kn¯)AˆTE(kn¯′)γ(kn¯)γ(kn¯′)∆ǫ R−Jm(nRrk(cid:2)n¯R−)∂Jm(n∂rrkn¯R−) +kn2¯ε Jm(cid:3)2(nkn¯ρ′)ρ′dρ′
Vn¯n¯′ = 2 R 2
= (D1)
×[Im−1(kn¯,kn¯′)+Im+1(kn¯,kn¯′)] (C1) n2rkn2¯
