1. INTRODUCTIONThe phenomenological coupled-mode theory (CMT) has beenwidely applied in optoelectronics, photonics, and quantum op-tics to describe the linear and nonlinear properties of resona-tors [1–3]. It played an importation role in the development ofdevices that can be embedded into photonic crystals: fromwaveguide splitters and add–drop filters [4,5] to optical diodesand transistors [6,7]. Moreover, it was used to achieve anefficient generation of harmonics and difference frequenciesin microcavities [8,9], to describe bistable and multistableswitching [10,11], and to explain self-pulsations and chaoticbehavior in coupled microcavities [12,13]. The primary advan-tage of CMT comes from the fact that it allows one to under-stand properly the physical interactions between differentmodes of a system that are often hidden in the strictly numer-ical simulations.
The CMT equations can be obtained from general physicalconcepts like conservation of energy and time-reversal sym-metry [14,15]. As a consequence, these equations contain sev-eral fitting parameters: decay rates of resonances, couplingcoefficients to different scattering channels, and characteris-tic powers that describe the strength of the nonlinear effects.These parameters can be extracted from the experimentaldata, but most often they are determined after performing ad-ditional simulations in the time domain [16,17]. The main goalof this paper is to show that the CMT equations for the three-dimensional microcavities can be derived directly from theMaxwell equations without resorting to the phenomenologicalconcepts. As a result, we were able to obtain the explicit for-mulas for all fitting parameters in terms of the mode profiles.We restrict our attention to the on-channel (or resonantlycoupled) microcavities as opposed to the off-channel (side-coupled) microcavities [18], since the former case can bereadily applied to multilayered structures, and the results canbe simplified considerably.
The paper is organized as follows. In Section 2, we formu-late the eigenvalue problem for a microcavity with two cou-pling ports and construct a complete set of orthogonal modes
to expand an arbitrary field in it. To treat the electric and mag-netic fields on equal footing, the Maxwell equations are writ-ten in a form that is similar to the Schrödinger equation. It isemphasized that, due to the time-reversal symmetry, the Max-well equations are doubly degenerate, and two fundamentalmodes exist for any resonant frequency. In Section 3, we usethese modes as a basis to describe the behavior of the micro-cavity in the vicinity of resonance. The CMT equations arederived for both cases when the fundamental modes are re-presented by traveling and standing waves. It is shown thatthe standing waves basis is particularly suitable for micro-cavities of high-quality factors. Section 4 explains how to takeinto account perturbations caused by the Kerr nonlinearityand how to define the transfer matrix for the nonlinear micro-cavities. It provides the full set of equations for the time do-main and frequency domain simulations, including the explicitformulas for all fitting parameters. Section 5 presents severalnumerical examples and compares the accuracy of the CMTequations with other methods. It is also demonstrated how toapply the CMT equations to describe the nonlinear propertiesof microcavities with several localization centers.
2. GENERALIZED EIGENVALUE PROBLEMFOR MAXWELL’S EQUATIONSA. Analogy with the Schrödinger EquationTo work with the electric and magnetic fields on equal footing,the Maxwell equations
c∇ × E ¼ −μ∂tH; ð1Þ
c∇ ×H ¼ ε∂tE; ð2Þ
where ε is permittivity, μ is permeability, and c is the speed oflight in a vacuum, can be rewritten in a form that is similar tothe Schrödinger equation [19,20]
i ρ ∂tjΨi ¼ LjΨi: ð3Þ
V. Grigoriev and F. Biancalana Vol. 28, No. 9 / September 2011 / J. Opt. Soc. Am. B 2165
The wave function jΨi combines the electric and magneticfields in a single vector
jΨi ¼�EH
�; ð4Þ
with the inner product defined as
hΨajΨbi ¼ ∭ ðE�a · Eb þH�
a ·HbÞdV; ð5Þ
where the integration is performed over the volume of themicrocavity as shown in Fig. 1(a).
The operator ρ is used as a weighting function, and it takesthe following form for isotropic materials:
ρ ¼�εI 00 μI
�; ð6Þ
where I is the identity tensor. The generalization to the case ofmore complicated constitutive relations such as those presentin anisotropic or bianisotropic media is straightforward[21,22]. It is important that the weighting operator is Hermi-tian (ρ† ¼ ρ) for lossless media.
The operator L can be interpreted as a Hamiltonian and isdefined as
L ¼ ic
�0 ∇×
−∇× 0
�: ð7Þ
To prove that this operator is Hermitian, it is sufficient toshow that hΨajLΨbi ¼ hLΨajΨbi for two arbitrary solutionswith the same boundary conditions. This relation can be trans-formed to a surface integral around the microcavity [Fig. 1(a)]by using the identity divðA × BÞ ¼ BrotA − ArotB:
∯ ðE�a ×Hb þ Eb ×H�
aÞds ¼ 0: ð8Þ
The integral is nonzero only at the input and output ports sothat it can be reduced to
ðE�a ×Hb þ Eb ×H�
aÞjx¼Lx¼0 ¼ 0: ð9Þ
This equation is always valid for solutions that satisfy the per-iodic (Bloch) boundary conditions at the coupling ports:
jΨðx ¼ LÞi ¼ eiφjΨðx ¼ 0Þi; ð10Þ
where φ is a real constant, which corresponds to the propa-gating modes.
B. Orthogonality Relations between ModesAssuming the time dependence of the form jΨðr; tÞi ¼jψðrÞie−iωt, the generalized eigenvalue problem for Eq. (3)can be formulated as
Ljψmi ¼ ωmρjψmi: ð11Þ
Since both operators L and ρ are Hermitian, the eigen-values, or resonant frequencies ωm, should be real. Moreover,the eigenvectors, or modes corresponding to differentfrequencies ωn ≠ ωm, should satisfy the orthogonality relation
hψnjρψmi ¼ ∭ ðεE�n · Em þ μH�
n ·HmÞdV ¼ 0: ð12Þ
These modes can be normalized to a volume of the microcav-ity, but we prefer to preserve their meaning as the total energystored in the mode and will normalize the boundary condi-tions instead.
It is useful to introduce another Hermitian operator
σz ¼�I 00 −I
�; ð13Þ
which satisfies the following commutation relations σz L ¼−Lσz and σz ρ ¼ ρσz. By using these properties, it can beshown that the solutions of Eq. (11) come in pairs, and forany solution with a positive frequency, one can immediatelyconstruct another solution with a negative frequency:
Ljσzψmi ¼ −ωmρjσzψmi: ð14Þ
It is possible to return back to the positive frequencies by ap-plying the operation of complex conjugation and to obtain twoindependent solutions for the same frequency ωm:
jψþmi ¼
�Em
Hm
�and jψ−
mi ¼�
E�m
−H�m
�: ð15Þ
In homogeneous media, these solutions can be considered asplane waves moving in the opposite directions ðEy;�HzÞTexp½ið�kxx − ωtÞ�.
Similar to the derivation of the orthogonality relations, itcan be proved that, for ωn ≠ −ωm,
hψnjσzρψmi ¼ ∭ ðεE�n · Em − μH�
n ·HmÞdV ¼ 0: ð16Þ
For modes of the same frequency ωn ¼ ωm, it shows that theaveraged electric energy stored in the mode is equal to themagnetic energy ∭ εjEmj2dV ¼ ∭ μjHmj2dV . If Eqs. (12)and (16) are combined, the orthogonality relations can be se-parately formulated for the electric and magnetic fields(ωn ≠ �ωm):
∭ εE�n · EmdV ¼ ∭ μH�
n ·HmdV ¼ 0: ð17Þ
3. SLOWLY VARYING ENVELOPESA. Traveling Waves BasisIn the vicinity of the resonant frequency ωm, the field can besearched in the following form:
jΨi ¼ aþðr; tÞjψþmie−iωmt þ a−ðr; tÞjψ−
mie−iωmt; ð18Þ
where a�ðr; tÞ are slowly varying envelopes of forward- andbackward-moving waves. Substituting the approximate solu-tion (18) into the Schrödinger Eq. (3) gives
i ρð∂taþjψþmi þ ∂ta−jψ−
miÞ ¼ LðaþÞjψþmi þ Lða−Þjψ−
mi; ð19Þ
where the notation Lða�Þ means that the derivatives in theoperator L are applied only to the scalar envelopes a�:
2166 J. Opt. Soc. Am. B / Vol. 28, No. 9 / September 2011 V. Grigoriev and F. Biancalana
LðaÞ ¼ ic
�0 ð∇aÞ×
−ð∇aÞ× 0
�: ð20Þ
Projecting Eq. (19) on the vectors hψ�mj gives a set of
two coupled equations that describe the propagation of theenvelopes
∂taþ þ ðvg ·∇Þaþ ¼ −g�∂ta−; ð21Þ
∂ta− − ðvg ·∇Þa− ¼ −g∂taþ: ð22Þ
The coupling terms appear due to the fact that the two funda-mental modes jψ�
mi are not orthogonal:
hψ−mjρψþ
mi ¼ 2∭ εE2mdV: ð23Þ
It is convenient to measure their overlap by using theircommon norm,
hψ�mjρψ�
mi ¼ 2∭ εjEmj2dV; ð24Þ
and to introduce a special parameter,
g ¼ ∭ εE2mdV
∭ εjEmj2dV: ð25Þ
Since the inequality j R f ðxÞdxj2 ≤ R jf ðxÞj2dx holds for anycomplex function f ðxÞ, the parameter g is limited to the rangejgj ≤ 1. In homogeneous media, the propagating modes have arapidly oscillating phase, and the amplitude does not changesignificantly over many periods. This makes the averagedvalue of the integrand in the numerator of Eq. (25) close tozero, and the parameter g also tends to zero. As a result, thecoupling terms in Eqs. (21) and (22) disappear, and the twoenvelopes propagate independently. On the contrary, the fieldin nonhomogeneous media is localized around the defectregions, the amplitude varies very quickly in comparison tothe phase, and this leads to large values of the parameterg. As a consequence, g can describe the localization strengthof a resonance.
In the derivation of Eqs. (21) and (22),
hψ�mjLðaÞψ�
mi ¼ ∓2ic∇a ·∭ ReðE�m ×HmÞdV; ð26Þ
hψ�mjLðaÞψ∓
mi ¼ 0; ð27Þ
and the parameter vg was introduced
vg ¼ c∭ ReðE�
m ×HmÞdV∭ εjEmj2dV
: ð28Þ
The integrand in the numerator of Eq. (28) contains the aver-aged energy flow SW ¼ ðc=8πÞRe½E ×H��, which is asolenoidal vector field divSW ¼ 0 for any resonant fre-quency, because the averaged energy density W ¼ ð1=16πÞðεjEj2 þ μjHj2Þ does not change in a stationary state. In multi-layered structures, the energy flow depends only on the x co-ordinate, which means that divSW ¼ ∂xðSW Þx ¼ 0, and as a
result ðSW Þx ¼ const can be easily integrated. Therefore,the parameter vg can be interpreted as an effective group ve-locity. The same considerations should hold even for morecomplicated structures if the center of the microcavity ison the x axis. Because of the symmetry, the dominant energyflow is also directed along the x axis, and the directionalderivative in Eqs. (23) and (24) can be approximated asðvg ·∇Þ ¼ ðvg∂xÞ.
It is convenient to choose the position of reference planesin such a way that the boundary conditions (10) have φ ¼ 0.Using these reference planes as channels for the ingoing andoutgoing waves, it is possible to develop a scattering formal-ism [Fig. 1(b)] and to find the transmission spectrum of themicrocavity in the vicinity of the resonance ωm. Assuming thatthe reference planes are located at x ¼ 0 and x ¼ L, theboundary conditions for the forward- and backward-movingenvelopes can be written as a�ð0Þ ¼ u�, a�ðLÞ ¼ v∓. Approx-imating the spatial and temporal derivatives in Eqs. (21) and(22) with finite differences ∂xa� ¼ ðv∓ − u�Þ=L and ∂ta� ¼−iδωðu� þ v∓Þ=2, one can show that the scattering matrixSuv, defined as
u−v−
� �¼ Suv
uþvþ
� �¼
�ru t
t rv
�uþvþ
� �; ð29Þ
up to the first order of the detuning from the resonanceδω ¼ ω − ωm is
Suv ¼1
1 − iðδω=γÞ�igðδω=γÞ 1
1 ig�ðδω=γÞ�; ð30Þ
where γ ¼ vg=L or
γ ¼ cσ∭ εjEmj2dV
; ð31Þ
Fig. 1. (Color online) (a) Photonic crystal waveguide with an em-bedded defect that behaves as a microcavity. The region where theelectromagnetic filed is negligible has a dark gray background. Theintegration volume for the eigenvalue problem is shown by the dashedline. The dominant flow of energy goes through the reference planesat x ¼ 0 and x ¼ L. (b) Sketch of the model used in the scatteringformalism. The amplitudes of the ingoing waves (uþ and vþ) are re-lated to the outgoing waves (u− and v−) through the amplitude of themicrocavity mode (A).
V. Grigoriev and F. Biancalana Vol. 28, No. 9 / September 2011 / J. Opt. Soc. Am. B 2167
which is obtained from Eq. (28) in the assumption that themode profiles are considered as dimensionless quantitiesand normalized in such a way that
σ ¼ZZ
Re½ðE�m ×HmÞx�dydz ¼ 1 cm2: ð32Þ
The transmission tðωÞ and reflection ru;vðωÞ spectra forwaves incident on the ports U or V can be found by directcomparison of the matrix elements in Eqs. (29) and (30).For example, the transmission coefficient is given by
tðωÞ ¼ ½1 − iðδω=γÞ�−1; ð33Þ
and thus the resonance contour has the Lorentzian shape withthe half-width at half-maximum equal to γ.
B. Standing Waves BasisAs was mentioned before, the basis formed by two modesjψ�
mi is not orthogonal. However, a linear combination ofthese modes can be used to construct a new basis that willhave such a property. This is particularly easy to do for mirrorsymmetric structures:
jψAi ¼�EA
HA
�¼ jψþ
mi þ jψ−mi
2¼
�Re½Em�iIm½Hm�
�; ð34Þ
jψBi ¼�EB
HB
�¼ jψþ
mi − jψ−mi
2i¼
�Im½Em�
−iRe½Hm��: ð35Þ
The new modes jψAi and jψBi correspond to standing wavesbecause the energy flow SW for them is zero by definition. Itcan be shown that, when one of them is exponentially growingaround a defect region, the other mode is exponentially decay-ing [Fig. 2]. As a result, the norms of these modes can differsignificantly for resonances with strong localization. By usingthe property jψ�
mi ¼ jψAi � ijψBi, the parameter g in Eq. (25)can be rewritten as
g ¼ ∭ εðE2A − E2
BÞdV∭ εðE2
A þ E2BÞdV
: ð36Þ
Therefore, g is real for mirror symmetric structures and tendsto �1 depending on which mode dominates.
The total field can be expanded in the new basis as
where Aðr; tÞ and Bðr; tÞ are slowly varying amplitudes of themodes. The equations that describe their propagation can befound by substituting Eq. (37) into Eq. (3):
i ρð∂tAjψAi þ i∂tBjψBiÞ ¼ LðAÞjψAi þ iLðBÞjψBi: ð38Þ
Making projection of this equation on hψAj and then on hψBjleads to
ð1þ gÞ∂tAþ ðvg ·∇ÞB ¼ 0; ð39Þ
ð1 − gÞ∂tBþ ðvg ·∇ÞA ¼ 0; ð40Þ
where the following properties were used:
hψAjLðAÞψAi ¼ hψBjLðBÞψBi ¼ 0; ð41Þ
hψBjLðAÞψAi ¼ c∇A ·∭ ReðE�m ×HmÞdV; ð42Þ
hψAjLðBÞψAi ¼ −c∇B ·∭ ReðE�m ×HmÞdV: ð43Þ
Equations (39) and (40) can be applied to develop the scat-tering formalism in the standing waves basis. The boundaryconditions can be obtained by using the following relationswith the traveling waves basis: A ¼ aþ þ a− and B ¼ aþ − a−.It is convenient to consider A and B as a function of time onlyby taking their average value on the boundaries,
A ¼ Að0Þ þ AðLÞ2
¼ ðuþ þ u−Þ þ ðv− þ vþÞ2
; ð44Þ
B ¼ Bð0Þ þ BðLÞ2
¼ ðuþ − u−Þ þ ðv− − vþÞ2
; ð45Þ
which can be written in the matrix form as
u−v−
� �¼ −
�0 11 0
�uþvþ
� �þ A
�11
�þ B
−11
� �: ð46Þ
The spatial derivatives can be approximated then by
Fig. 2. (Color online) Example of two fundamental solutions that ex-ist at the same resonance frequency and satisfy the boundary condi-tions of standing waves with zero energy flow. The electric andmagnetic fields are shown by solid and dashed lines, respectively.(a) For one of the solutions, they are exponentially growing towardthe defect region in the middle, (b) while for the other independentsolution they are exponentially decaying. The background shows al-ternation of layers with a higher (H) and lower (L) index of refractioninside the structure.
2168 J. Opt. Soc. Am. B / Vol. 28, No. 9 / September 2011 V. Grigoriev and F. Biancalana
∂A∂x
¼ AðLÞ − Að0ÞL
¼ 2B − ðuþ − vþÞ
L; ð47Þ
∂B∂x
¼ BðLÞ − Bð0ÞL
¼ 2A − ðuþ þ vþÞ
L: ð48Þ
Therefore, Eqs. (39) and (40) take the following form:
1þ g
2dA
dt¼ −γAþ γðuþ þ vþÞ; ð49Þ
1 − g
2dB
dt¼ −γBþ γðuþ − vþÞ: ð50Þ
where the decay rate γ is given by
γ ¼ cσ∭ εðE2
A þ E2BÞdV
: ð51Þ
A considerable simplification can be made for resonanceswith large quality factors. In this case, the parameter g tendsto �1 depending on which mode dominates. If the even modehas a larger norm ∭ εE2
AdV ≫ ∭ εE2BdV , then, according to
Eq. (36), g → 1 and Eq. (50) reduces to B ¼ uþ − vþ. The fullset of the CMT equations is thus
dAdt
¼ −γAþ γðuþ þ vþÞ; ð52Þ
u−v−
� �¼ − uþ
vþ
� �þ A
�11
�: ð53Þ
On the contrary, if the odd mode has a larger norm, then g →
−1 and Eq. (49) reduces to A ¼ uþ þ vþ. The full set of theCMT equations is thus
dB
dt¼ −γBþ γðuþ − vþÞ; ð54Þ
u−v−
� �¼ uþ
vþ
� �þ B
−11
� �: ð55Þ
4. PERTURBATIONS CAUSED BY THE KERRNONLINEARITY
A. Time DomainThe nonlinear effects can be treated as perturbation terms inEq. (3), and the standing waves basis is particularly suitablefor this purpose because the dynamics of the system can bedescribed only by one variable. The left-hand side of Eq. (38)should be modified to include the perturbation term
ωmΔρðAjψAi þ iBjψBiÞ: ð56Þ
As a source of Δ ρ, the Kerr nonlinearity will be considered,which corresponds to the following constitutive relation be-tween the electric field and the displacement vector:
D ¼ εEþ εKE3: ð57Þ
If the effect of the third harmonic generation, which is repre-sented by the first term in the expansion
ðRe½Eωe−iωt�Þ3 ¼ ð1=4ÞRe½E3
ωe−3iωt� þ ð3=4ÞjEωj2Re½Eωe
−iωt�;ð58Þ
can be neglected, the constitutive relation is reduced to
D ¼ εEþ ð3=4ÞεK jEj2E; ð59Þ
which means that
Δ ρ ¼ 34
�εK jEj2 I 0
0 0
�: ð60Þ
Since the electric field can be written as E ¼ AEA þ iBEB, theprojection of Eq. (56) on hψAj and then on hψBj leads to theoverlap integrals of the following form:
ΓApBq ¼ 3ωm
4
∭ εK ðEAÞpðEBÞqdV∭ εðE2
A þ E2BÞdV
; ð61Þ
where p and q are nonnegative integers with an additional re-striction pþ q ¼ 4. There is only a small number of nonzerocoefficients Γ that should be taken into account due to thefact that EA and EB are functions of the opposite parity. Equa-tions (39) and (40) in the presence of the Kerr nonlinearity areconsequently
ð1þ gÞ∂tAþ ðvg ·∇ÞB ¼ iAjAj2ΓAAAA
þ ið2AjBj2 − A�B2ÞΓAABB; ð62Þ
ð1 − gÞ∂tBþ ðvg ·∇ÞA ¼ iBjBj2ΓBBBB
þ ið2BjAj2 − B�A2ÞΓAABB: ð63Þ
The nonlinear effects described by these equations include theself-phase modulation (terms proportional to AjAj2 and BjBj2)as well as the cross-phase modulation (AjBj2 and BjAj2). If thecoefficients Γwere equal, the relative strength of these effectswould be given by a factor of 2, which coincides with the re-sults obtained in the case of shallow gratings [23]. It is worthnoting that there are a few additional terms (A�B2 and B�A2),which are responsible for the phase conjugation and do notexist in the shallow gratings [24].
It turns out however that only the self-phase modulation isimportant for resonances with large quality factors. If the evenmode jψAi dominates, the overlap integrals differ significantlyΓAAAA ≫ ΓAABB ≫ ΓBBBB. Therefore, to take into account theKerr nonlinearity, the CMT Eqs. (52) and (54) should be mod-ified in the following way:
dA
dt¼ −
�i
�ωm − γ jAj
2
IA
�þ γ
�Aþ γðuþ þ vþÞ; ð64Þ
dB
dt¼ −
�i
�ωm − γ jBj
2
IB
�þ γ
�Bþ γðuþ − vþÞ: ð65Þ
The main influence of the Kerr nonlinearity results in theshift of the resonant frequency. To emphasize this, the time
V. Grigoriev and F. Biancalana Vol. 28, No. 9 / September 2011 / J. Opt. Soc. Am. B 2169
dependence expð−iωmtÞ used as a factor in Eq. (37) was ex-plicitly included in the amplitudes A and B. Two new para-meters were introduced IA and IB, which have the meaningof characteristic intensities. Before giving the explicit formu-las for them, it is important to choose proper units for theelectric field.
The intensities will be measured in MW=cm2, and the non-linear refractive index n2 caused by the Kerr nonlinearity willbe specified in cm2=MW for consistency. The transition fromthe intensity-dependent refractive index nðIÞ ¼ nþ n2I to thepermittivity can be performed as εðEÞ ¼ εþ 2εn2jEj2, wherethe units for the electromagnetic field were defined so asto produce a unit energy flow in vacuum, namely Iunit ¼ðc=8πÞjEunitj2 ¼ 1MW=cm2. Comparing with Eq. (59) givesεK ¼ ð8=3Þεn2. Therefore, the characteristic intensities are
IA;B ¼ cσωm∭ εn2ðEA;BÞ4dV
: ð66Þ
B. Frequency DomainIt is worth noting that the CMT Eqs. (64) and (53) for reso-nances with even modes [or Eqs. (65) and (55) for odd modes]represent an extension of the scattering matrix method to thetime domain. It is convenient to combine these equations
dA
dt¼ −
�i
�ω0 − γ jAj
2
I0
�þ γ
�Aþ γðuþ � vþÞ; ð67Þ
u−v−
� �¼ ∓ uþ
vþ
� �þ A
�11
� �; ð68Þ
where AðtÞ denotes the amplitude of the dominating mode andthe upper (lower) sign should be used if this mode is even(odd). The scattering matrix in the frequency domain is
Suv ¼1
1 − iðδωeff=γÞ�iðδωeff=γÞ 1
1 �iðδωeff=γÞ� �
; ð69Þ
where the effective frequency detuning was introduced,δωeff ¼ ω − ω0 þ γjAj2=I0, which takes into account the shiftof the resonant frequency due to the influence of the Kerr non-linearity. The amplitude A depends on the amplitudes of theingoing waves and can be found as a solution of the followingequation:
�1 − i
�ω − ω0
γ þ jAj2I0
��A ¼ uþ � vþ: ð70Þ
It is a cubic equation that has three different roots in the gen-eral case. This agrees with the fact that the system can showseveral stable states for the same input signals and explains itsbistable behavior from the mathematical point of view. It isalso possible to find A by using Eq. (68), which does not in-volve any nonlinear equations; however, A becomes a func-tion of both ingoing and outgoing signals. This can beparticularly suitable when signals are incident only from oneside. For example, if the incidence from the left is considered,vþ ¼ 0 and A ¼ −v−. Since the transmitted and input powersare given by Pout ¼ σjv−j2 and Pin ¼ σjuþj2, the nonlinear
transmission spectrum in the vicinity of the resonance canbe found as
TðωÞ ¼ Pout
Pin¼
�1þ
�ω − ω0
γ þ Pout
P0
�2�−1: ð71Þ
For a fixed value of the frequency detuning, Eq. (71) can beused to compute the hysteresis curve. It can be checked thatthis curve is equivalent to the polynomial of the third degreewith real coefficients, and thus it can describe only bistableresonances. More complex structures that demonstrate multi-stable behavior can be constructed by combining severalstrictly bistable microcavities [11]. To treat the nonlinearproperties of such structures, it is useful to define the non-linear transfer matrix Tuv,
uþu−
� �¼ Tuv
v−vþ
� �; ð72Þ
Tuv ¼ I − i
�ω − ω0
γ þ jvþ � v−j2I0
��1 �1∓1 −1
�; ð73Þ
where I is the identity matrix. The transfer matrices of singlemicrocavities can be multiplied producing the total transfermatrix of the structure. This leads to a hysteresis curve ofmore complicated shape that can be different for the oppositedirections of incidence because the transfer matrices do notcommute and the order of multiplication plays an importantrole [6].
5. NUMERICAL EXAMPLESA. Bragg Gratings with Symmetrically Placed DefectMultilayered structures can be considered as a one-dimensional (1D) realization of on-channel microcavitiesand are particularly suitable to check the accuracy of the CMTequations. As a test case, we use a Bragg structure with a sym-metrically placed defect, which can be described by the sym-bolic formula ðHLÞpðLHÞp. In what follows, the letters “L” and“H” correspond to quarter-wave layers of polydiacetylene 9-BCMU with linear (nonlinear) refractive index nL ¼ 1:55(n2L ¼ 2:5 × 10−5 cm2=MW) and rutile with nH ¼ 2:3 (n2H ¼10−8 cm2=MW), respectively [25,26]. The quarter-wave condi-tion is set to λq ¼ 0:7 μm,
nLdL ¼ nHdH ¼ λq=4; ð74Þ
which gives the thicknesses of the layers dL ¼ 112nm anddH ¼ 76nm. The main advantage of this structure is that ithas a well-defined resonance at ωq ¼ 2πc=λq, which is sur-rounded by bandgap regions. Because of the mirror symmetryof the structure, this resonance always shows perfect trans-mission, and its half-width can be adjusted by the numberof periods p in the Bragg mirrors.
The formulas for the decay rate [Eq. (31)] and the charac-teristic intensity [Eq. (66)] of the microcavity at the resonanceλ0 ¼ 2πc=ω0 can be simplified in the 1D case to
γ ¼�ð1=cÞ
ZL
0εE2
0dx
�−1; ð75Þ
2170 J. Opt. Soc. Am. B / Vol. 28, No. 9 / September 2011 V. Grigoriev and F. Biancalana
I0 ¼�ð2π=λ0Þ
ZL
0εn2E
40dx
�−1: ð76Þ
Instead of the decay rate γ, it is often convenient to use a di-mensionless quality factor defined as Q ¼ ω0=ð2γÞ:
Q ¼ ðπ=λ0ÞZ
L
0εE2
0dx: ð77Þ
In some simple cases, the integration can be performedanalytically, and an explicit formula for the quality factorcan be obtained [see Appendix A].
For the structure ðHLÞ8ðLHÞ8, the quality factor computedby Eq. (77) is Q ¼ 2060, and the characteristic intensity ac-cording to Eq. (76) is I0 ¼ 0:05695MW=cm2. The two para-meters together with Eq. (71) fully determine the hysteresisand the nonlinear transmission spectrum of the structure inthe vicinity of the resonant frequency ω0 ¼ ωq. The quality fac-tor can be also computed with the linear transfer matrix byfinding the full-width at half-maximum of the resonanceω0=ΔωFWHM ¼ 2058. Therefore, the accuracy of the CMTequations in this case can be estimated as 0.1%.
It is worth noting that the CMT parameters of a similarstructure ðLHÞ8ðHLÞ8 are different. It has a smaller quality fac-tor Q ¼ 581:3, and a significantly larger characteristic inten-sity I0 ¼ 1:593MW=cm2. This can be explained by adifferent localization of the electric field in the structure[Figs. 3(a) and 3(b)]. Nevertheless, the usage of normalized
units ensures that both structures have the same shape ofthe hysteresis and the nonlinear transmission spectrum[Figs. 3(c)–3(f)].
The Maxwell equations for nonlinear multilayeredstructures,
∂xEy ¼ iðω=cÞHz; ð78Þ
∂xHz ¼ iðω=cÞεð1þ 2n2jEyj2ÞEy ð79Þ
can be also solved by using strictly numerical methods [27,28]or semianalytical techniques [29,30]. A detailed comparisonwith the results obtained by the Runge–Kutta method ispresented in Figs. 3(c)–3(f).
B. Thue–Morse Multilayered StructuresAs a more complex example, we consider a Thue–Morse qua-sicrystal. It has a nonperiodic arrangement of layers that isgoverned by a deterministic set of inflation rules and featuresa number of pseudo-bandgap regions with resonances of com-plete transmission [11]. We choose one such resonance, whichis located at ω0 ¼ 0:705465ωq. It has two localization centersin the field profile so that the full structure can be dividedinto two parts that can be treated as coupled microcavities[Fig. 4(a)]. These microcavities, which will be denoted as αand β, have the same resonant frequencies ωα ¼ ωβ ¼ ω0,
Fig. 3. (Color online) Structures (a) ðHLÞ8ðLHÞ8 and (b) ðLHÞ8ðHLÞ8show a strong resonance at the quarter-wave frequency ω0 ¼ ωq buthave a different distribution of the electric field. (c), (e) The hysteresisof transmission for a fixed value of the frequency detuning can becomputed by Eq. (71), which follows from the CMT (solid lines),and by solving the Maxwell Eqs. (78) and (79) directly with theRunge–Kutta method (circles). Not all parts of the hysteresis curvesare stable, and arrows show how the switching between differentbranches occurs. The usage of normalized units ensures that bothstructures have the same shape of the hysteresis curve. The resultsof the two methods are in good agreement [the comparison forðHLÞ8ðLHÞ8 is shown; ðLHÞ8ðHLÞ8 is similar]. (d), (f) The same meth-ods can be applied to compute the nonlinear transmission spectrumfor a fixed value of the input intensity.
Fig. 4. (Color online) (a) Example of the resonance (the Thue–Morse structure of the seventh generation number at the frequencyω0 ¼ 0:705465ωq) that has two localization centers in the electric fieldprofile denoted as α and β. The (b) hysteresis and (c) nonlinear trans-mission spectrum of these parts can be computed by Eq. (73), whichfollows from the CMT (solid lines), and by solving the MaxwellEqs. (78) and (79) directly with the Runge–Kutta method (circles).The discrepancy between the two methods is noticeable only for thepart αbecause it has a relatively small localization strength. The Thue–Morse structure, which combines the parts α and β together, showsthe nonreciprocal behavior. For the same frequencyω ¼ 0:7046ωq andinput intensity I ¼ 1:5MW=cm2 as used in (b) and (c), respectively,the (c) hysteresis and (e) nonlinear transmission spectrum dependon the propagation direction (αβ or βα).
V. Grigoriev and F. Biancalana Vol. 28, No. 9 / September 2011 / J. Opt. Soc. Am. B 2171
but their parity and CMT parameters are different. For the leftpart, the quality factor is Qα ¼ 318:2, and the characteristicintensity is Iα ¼ 5:147MW=cm2, while for the right part,Qβ ¼ 1110, and Iβ ¼ 0:4078MW=cm2.
The nonlinear response of these microcavities is qualita-tively similar and can be described by the same analytic for-mula [Eq. (71)] [Figs. 4(b) and 4(c)]. The discrepancy with thenumerical results is noticeable only for the microcavity α,which has a relatively small quality factor. Since the accuracyis worse on the higher frequency side of the resonance wherethe bandgap is less pronounced, this suggests that the influ-ence of other resonances causes additional perturbations.
The coupling between microcavities leads to more com-plex hysteresis curves and nonlinear transmission spectra[Figs. 4(d) and 4(e)]. They can be obtained by multiplyingthe nonlinear transfer matrices of the microcavities [Eq. (73)]for a fixed output intensity and then restoring the input inten-sity. The order of multiplication plays an important role in thenonlinear case because the transfer matrices do not commuteand the result strongly depends on the direction of incidence.Apart from the nonreciprocal behavior, there is also the pos-sibility of multistable behavior since the hysteresis curve inthe case of coupled microcavities is described by a polynomialof a higher degree. It is very important that a simple modelbased on the CMT equations is able not only to explain thenonlinear properties of complex resonances like this onebut also shows a good quantitative agreement with computa-tionally intensive numerical methods.
6. CONCLUSIONSThe phenomenological CMT is a very efficient tool for study-ing the nonlinear behavior of microcavities both in the fre-quency and time domain. It considers the interactionbetween microcavity and waveguide modes in a way that issimilar to the scattering formalism. Therefore, the complexwave dynamics can be separated from a relatively simplepicture of coupling, and this gives a significant advantagein comparison to strictly numerical methods. The dynamicalproperties of the microcavities can be fully determined by asmall set of parameters, which includes the decay rate, cou-pling coefficients, and characteristic intensities.
By using on-channel microcavities with two coupling portsas an example, we provided for the first time a systematic de-rivation of the CMT equations starting directly from the Max-well equations and obtained the explicit formulas for allphenomenological parameters. Our derivation is particularlysuitable for microcavities embedded in photonic crystal wave-guides of various dimensionality and multilayered structures.The accuracy of the results depends on the quality factor of aspecific resonance and is mostly limited by the influence ofother resonances.
APPENDIX A: QUALITY FACTORSOF BRAGG GRATINGS WITHSYMMETRICALLY PLACED DEFECTSBy using the fact that the energy density is a constant in eachlayer of the structure, the formula for the quality factor[Eq. (77)] can be rewritten as
Q ¼ π2λ0
Xk
ðn2kjEkj2 þ jHkj2Þdk; ðA1Þ
where dk is the thickness of the layer k with the refractive in-dex nk and the sum is taken over all layers in the structure.Fields on opposite sides of the layer k can be related by thecharacteristic matrix Mk,
�Ek−1Hk−1
�¼ Mk
�Ek
Hk
�; ðA2Þ
Mk ¼�
cos ξk in−1k sin ξk
ink sin ξk cos ξk
�; ðA3Þ
where ξk ¼ nkdkω=c or ξk ¼ ðπωÞ=ð2ωqÞ if all layers satisfy thequarter-wave condition [Eq. (74)] at the frequency ωq. The M
matrix for the single period of the Bragg gratings in the struc-ture ðHLÞpðLHÞp can be obtained as a multiplication of M ma-trices corresponding to layers L and H. It takes a particularlysimple form at the resonance ω0 ¼ ωq,
MLMH ¼ −�nH=nL 0
0 nL=nH
�; ðA4Þ
which shows that the fields are exponentially growing or de-caying toward the center of the structure as ðnH=nLÞp, where pis the number of periods in the Bragg mirrors. The contribu-tion of each period to the quality factor is
π8
��nH þ n2
H
nL
�jEkj2 þ
�1nH
þ nL
n2H
�jHkj2
�; ðA5Þ
which makes in total
Q ¼ π4ðnH þ nLÞ
Xp−1k¼0
��nH
nL
�2kþ1
þ 1n2H
�nL
nH
�2k�: ðA6Þ
The sum of the geometric progressions can be found asPp−1k¼0 r
k ¼ ð1 − rpÞ=ð1 − rÞ, and keeping only the largest termleads to the following formula for the quality factor of thestructure ðHLÞpðLHÞp:
Q ¼ πnHnL
4ðnH − nLÞ�nH
nL
�2p: ðA7Þ
It is worth noting that the quality factor of a similar structure,ðLHÞpðHLÞp,
Q ¼ π4ðnH − nLÞ
�nH
nL
�2p
ðA8Þ
is smaller in nHnL times.
ACKNOWLEDGMENTSThis work was supported by the German Max Planck Societyfor the Advancement of Science (MPG).
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