Alloptical switching, bistability, and slowlight transmission in photonic crystal waveguideresonator structures.
ABSTRACT We analyze the resonant linear and nonlinear transmission through a photonic crystal waveguide sidecoupled to a Kerrnonlinear photonic crystal resonator. First, we extend the standard coupledmode theory analysis to photonic crystal structures and obtain explicit analytical expressions for the bistability thresholds and transmission coefficients which provide the basis for a detailed understanding of the possibilities associated with these structures. Next, we discuss limitations of standard coupledmode theory and present an alternative analytical approach based on the effective discrete equations derived using a Green's function method. We find that the discrete nature of the photonic crystal waveguides allows a geometrydriven enhancement of nonlinear effects by shifting the resonator location relative to the waveguide, thus providing an additional control of resonant waveguide transmission and Fano resonances. We further demonstrate that this enhancement may result in the lowering of the bistability threshold and switching power of nonlinear devices by several orders of magnitude. Finally, we show that employing such enhancements is of paramount importance for the design of alloptical devices based on slowlight photonic crystal waveguides.

Article: Transmission resonance induced by a δlike defect in the FanoAnderson model with two Fano defects
physica status solidi (b) 09/2012; 249(9):17651770. · 1.49 Impact Factor  Optik  International Journal for Light and Electron Optics 10/2013; 124(19):39433945. · 0.77 Impact Factor
 SourceAvailable from: Almas Sadreev[Show abstract] [Hide abstract]
ABSTRACT: We consider a system of two or four nonlinear sites coupled with binary chain waveguides. When a monochromatic wave is injected into the first (symmetric) propagation channel, the presence of cubic nonlinearity can lead to symmetry breaking, giving rise to emission of antisymmetric wave into the second (antisymmetric) propagation channel of the waveguides. We found that in the case of nonlinear plaquette, there is a domain in the parameter space where neither symmetrypreserving nor symmetrybreaking stable stationary solutions exit. As a result, injection of a monochromatic symmetric wave gives rise to emission of nonsymmetric satellite waves with energies differing from the energy of the incident wave. Thus, the response exhibits nonmonochromatic behavior.Physical Review E 09/2013; 88(31):032901. · 2.31 Impact Factor
Page 1
arXiv:physics/0605156v1 [physics.optics] 18 May 2006
Alloptical switching, bistability, and slowlight transmission
in photonic crystal waveguideresonator structures
Sergei F. Mingaleev,1,2Andrey E. Miroshnichenko,3Yuri S. Kivshar,3and Kurt Busch1
1Institut f¨ ur Theoretische Festk¨ orperphysik, Universit¨ at Karlsruhe, Karlsruhe 76128, Germany
2Bogolyubov Institute for Theoretical Physics of the National Academy of Sciences of Ukraine, 03143 Kiev, Ukraine
3Nonlinear Physics Centre and Centre for Ultrahigh bandwidth Devices for Optical Systems (CUDOS),
Research School of Physical Sciences and Engineering,
Australian National University, Canberra ACT 0200, Australia
(Dated: May 17, 2006)
We analyze the resonant linear and nonlinear transmission through a photonic crystal waveg
uide sidecoupled to a Kerrnonlinear photonic crystal resonator. Firstly, we extend the standard
coupledmode theory analysis to photonic crystal structures and obtain explicit analytical expres
sions for the bistability thresholds and transmission coefficients which provide the basis for a detailed
understanding of the possibilities associated with these structures. Next, we discuss limitations of
standard coupledmode theory and present an alternative analytical approach based on the effective
discrete equations derived using a Green’s function method. We find that the discrete nature of the
photonic crystal waveguides allows a novel, geometrydriven enhancement of nonlinear effects by
shifting the resonator location relative to the waveguide, thus providing an additional control of res
onant waveguide transmission and Fano resonances. We further demonstrate that this enhancement
may result in the lowering of the bistability threshold and switching power of nonlinear devices by
several orders of magnitude. Finally, we show that employing such enhancements is of paramount
importance for the design of alloptical devices based on slowlight photonic crystal waveguides.
PACS numbers: 42.65.Pc; 42.70.Qs; 42.65.Hw; 42.79.Ta
I. INTRODUCTION
It is believed that future integrated photonic circuits
for ultrafast alloptical signal processing require different
types of nonlinear functional elements such as switches,
memory and logic devices. Therefore, both novel physics
and novel designs of such alloptical devices have at
tracted significant research efforts during the last two
decades, and most of these studies utilize the concepts of
optical switching and bistability [1].
One of the simplest bistable optical devices which can
find applications in photonic integrated circuits is a two
port device which is connected to other parts of a circuit
by one input and one output waveguide. Its transmis
sion properties depend on the intensity of light sent to
the input waveguide. Two basic realizations of such a
device can be provided by either direct or sidecoupling
between the input and output waveguides to an optical
resonator. In the first case, we obtain a system with res
onant transmission in a narrow frequency range, while in
the second case, we obtain a system with resonant reflec
tion. Both systems may exhibit optical bistability when
the resonator is made of a Kerr nonlinear material. The
resonant twoport systems of the first type, with direct
coupled resonator, can be realized in onedimensional
systems, and they have been studied in great details in
the context of different applications. In contrast, the res
onant systems of the second type, with sidecoupled res
onators, can only be realized in higherdimensional struc
tures, and their functionalities are not yet completely un
derstood.
Our goal in this paper is to study in detail the sec
ond class of resonant systems based on straight optical
waveguides sidecoupled to resonators as shown in Fig. 1.
Moreover, we assume that the waveguide and resonator
are created in two or threedimensional photonic crystal
(PhC) [2]. Due to a periodic modulation of the refractive
index of PhCs, such structures may possess complete pho
tonic band gaps, i.e. regions of optical frequencies where
PhCs act as ideal optical insulators. Embedding carefully
designed cavities into PhCs, one can create ultracompact
photonic crystal devices which are very promising for ap
plications in photonic integrated circuits. As an illustra
tion, sidecoupled waveguideresonator systems created
in PhCs through arrays of cavities are schematically de
picted in Fig. 1(b) and Fig. 1(c).
Practical applications of such PhC devices are becom
ing a reality due to the recent experimental success in
realizing both linear and nonlinear light transmission in
twodimensional PhC slab structures where a lattice of
cylindrical pores is etched into a planar waveguide. In
particular, Noda’s group have realized coupling of a PhC
waveguide to a leaky resonator mode consisting of a
defect pore of slightly increased radius [3–6]; Smith et
al. demonstrated coupling of a threeline PhC waveg
uide with a largearea hexagonal resonator [7]; Seassal
et al. have investigated the mutual coupling of a PhC
waveguide with a rectangular microresonator [8]; No
tomi et al. [9] and Barclay et al. [10] have observed
alloptical bistability in directcoupled PhC waveguide
resonator systems.
Photoniccrystal based devices offer two major advan
tages over corresponding ridgewaveguide systems: (i)
the PhC waveguides may have very low group veloci
Typeset by REVTEX
Page 2
2
Aα
in
1/2
I eikx
(ω)in
t I eikx
1/2
(ω)in
r I e−ikx
1/2
Ak
+
Ak
−
A−2
A2
A1
A0
Aα
A−1
A−2
A2
A1
A0
A−1
Aα
(c)
(b)
(a)
FIG. 1: (Color online) Three types of the geometries of a
straight photoniccrystal waveguide sidecoupled to a non
linear optical resonator, Aα.
ory is based on the geometry (a) which does not account
for discretenessinduced effects in the photoniccrystal waveg
uides. For instance, light transmission and bistability are
qualitatively different for (b) onsite and (c) intersite loca
tions of the resonator along waveguide and this cannot be
distinguished within the conceptual framework of structure
of type (a).
Standard coupledmode the
ties and, as a result, may significantly enhance the ef
fective coupling between short pulse and resonators, and
(ii) photonic crystals allow the creation of ultracompact
highQ resonators, which are essential for the further
miniaturization of alloptical nanophotonic devices. De
spite this, many researchers still believe that the basic
properties of devices based on ridge waveguides or PhC
waveguides are qualitatively identical, and that they can
be correctly described by the coupledmode theory for
continuous systems (see Refs. [11–21] and the discussion
in Sec. II).
However, an inspection of Figs. 1(ac) reveals, that a
major difference between the ridge waveguide in (a) and
PhC waveguides in (b,c) is that a PhC waveguide is al
ways created by an array of coupled smallvolume cav
ities and, therefore, exhibits an inherently discrete na
ture. This suggests that in these systems an additional
coupling parameter appears which relates the position
of the αresonator to the waveguide cavities along the
waveguide. As a matter of fact, we may (laterally) place
the αresonator at any point relative to two succesive
waveguide cavities., thus creating a generally asymmet
ric device which (in the nonlinear transmission regime)
should exhibit the properties of an optical diode, i.e.,
transmit highintensity light in one direction only. This
is an intriguing peculiarity of photoniccrystal based de
vices which we will analyze in a future publication. In
this paper, however, we restrict our analysis to symmetric
structures and study the cases of either onsite coupling of
the αresonator to the PhC waveguide, shown schemat
ically in Figs. 1(b), or intersite coupling, as shown in
Fig. 1(c).
To address these issues, we employ a recently devel
oped approach [22–24] and describe the photoniccrystal
devices via effective discrete equations that are derived by
means of a Green’s function formalism [25–29]. This ap
proach allows us to study the effect of the discrete nature
of the device on its transmission properties. In particular,
we show that the transmission depends on the location
of the resonance frequency ωα of the αresonator with
respect to the edges of the waveguide passing band. If
ωαlies deep inside the passing band, all devices shown in
Figs. 1(ac) are qualitatively similar, and can adequately
be described by the conventional coupledmode theory.
However, if the resonator’s frequency ωα moves closer
to the edge of the passing band, standard coupledmode
theory fails [30]. More importantly, we show that in
this latter case the properties of the devices shown in
Figs. 1(b) and Fig. 1(c) become qualitatively different:
light transmission vanishes at both edges of the passing
band, for the cases shown in Fig. 1(a) and Fig. 1(b), but
for the case shown in Fig. 1(c) it remains perfect at one
of the edges. Moreover, the resonance quality factor for
the structure (c) grows indefinitely as ωαapproaches this
latter band edge, accordingly reducing the threshold in
tensity required for a bistable light transmission. This
permits to achieve a very efficient alloptical switching in
the slowlight regime.
The paper is organized as follows. In Sec. II we sum
marize and extend the results of standard coupledmode
theory which accurately describes the system shown in
Fig. 1(a). Then, in Sec. III.A we derive a system of ef
fective discrete equations [25, 26] and utilize a recently
developed approach for its analysis [22, 23]. Specifically,
in Sec. III.B and Sec. III.C, respectively, we study the
two geometries of the waveguideresonator coupled sys
tems schematically depicted in Fig. 1(b) and Fig. 1(c).
In Sec. IV, we illustrate our main findings for several
examples of optical devices based on a twodimensional
photonic crystal created by a square lattice of Si rods. Fi
nally, in Sec. V we summarize and discuss our results. For
completeness as well as for justification of the effective
discrete equations employed, we include in Appendix A
an analysis of simpler cases of uncoupled cavities and
waveguides. The effects of nonlocal waveguide dispersion
and nonlocal waveguideresonator couplings are briefly
summarized in Appendix B.
II. COUPLEDMODE THEORY
In this Section, we first summarize the results of stan
dard coupledmode theory and other similar approaches
developed for the analysis of continuouswaveguide struc
tures similar to those displayed in Fig. 1(a). Then, we
extend these results in order to obtain analytical formu
las for the description of bistable nonlinear transmission
in such devices.
Page 3
3
A. Linear transmission
Transmission of light in waveguideresonator systems
is usually studied in the linear limit using a coupledmode
theory based on a Hamiltonian approach. This approach
has been pioneered by Haus and coworkers [11, 12] and
is similar to that used by Fano [13] and Anderson [14] for
describing the interaction between localized resonances
and continuum states in the context of an effect which
is generally referred to as “Fano resonance”.
analysis of the transmission of photoniccrystal devices,
this approach has been employed first by Fan et al. [15]
and has been elaborated on by Xu et al. [16].
Throughout this paper we consider the propagation of
a monochromatic wave with the frequency ω lying inside
the waveguide passing band; we assume that the waveg
uide is singlemoded as well as that the resonator α is
nondegenerate and losses can be neglected. In this case,
the complex transmission and reflection amplitudes, t(ω)
and r(ω), can be written in the form
For the
t(ω) =
σ(ω)
σ(ω) − i,
r(ω) =
eiϕr(ω)
σ(ω) − i,
(1)
with a certain realvalued and frequencydependent func
tion σ(ω) and the reflection phase ϕr(ω). Accordingly,
the absolute values of the transmission coefficient T =
t2and reflection coefficient R = r2are
σ2(ω)
σ2(ω) + 1
T(ω) =
andR(ω) =
1
σ2(ω) + 1, (2)
and it is easy to see that T + R = 1 for any σ(ω).
If the frequency ωα of the resonator α lies inside the
waveguide passing band, Fanolike resonant scattering
with zero transmission at the resonance frequency ωres,
lying in the vicinity of the resonator’s frequency, ωα,
should be observed [13, 22].
condition σ(ωres) = 0 and, based on the terminology de
veloped in Refs. [17, 18], σ(ω) may be interpreted as the
detuning of the incident frequency from resonance.
The results of standard coupledmode theory analysis
(for instance, see Ref. [16]) indicate that in the vicinity of
a highquality (or highQ) resonance, the detuning func
tion σ(ω) can be accurately described through the linear
function
σ(ω) ≃ωres− ω
γ
This corresponds to the
, whereγ =ωres
2Q,
(3)
which leads to a Lorentzian spectrum. Here, Q is the
quality factor of the resonance mode of the αresonator.
From the Hamiltonian approach [16], we find that the res
onance frequency ωresalmost coincides with the resonator
frequency ωα(see, however, Appendix A in Ref. [19] for
a more accurate estimate of ωres), the reflection phase is
ϕr = π/2, and the resonance width γ is determined by
the overlap of the mode profiles of waveguide and res
onator:
ω2
res
4WkWα
γ ≈
L
vgr
??
d? rδε(? r)?E∗
k(? r)?Eα(? r)
?2
. (4)
Here,?Eα(? r) is the normalized dimensionless electric field
of the resonator mode,?Ek(? r) is the corresponding field
of the waveguide mode at wavevector k = k(ωres), vgr=
(dω/dk) is the group velocity calculated at the resonance
frequency, and L is the length of the waveguide section
employed for the normalizing the modes to
?
wg section
d? rεwg(? r)?Ek(? r)2= Wk,
?
all space
d? rεα(? r)?Eα(? r)2= Wα. (5)
Furthermore, εα(? r) and εwg(? r) are the dielectric func
tions that describe the resonator α and waveguide, re
spectively. From Eqs. (4)–(5) it is easy to see that the
resonance width γ does not depend on the length L.
However, within the Hamiltonian approach, the func
tion δε(? r) in Eq. (4) remains undetermined. Generally,
it is assumed to be a difference between the total dielec
tric function and the dielectric function ε0(? r) “associated
with the unperturbed Hamiltonian” [16] which is an ill
defined quantity. A different approach based on a per
turbative solution of the wave equation for the electric
field [20] sheds some light on the resolution of this am
biguity and shows explicitly that ε0(? r) can be taken as
either εwg(? r) or εα(? r).
B. Nonlinear transmission
If the resonator α is made of a Kerrnonlinear ma
terial, increasing the intensity of the localized mode of
the resonator leads to a change of the refractive index
and, accordingly, to a shift of the resonator’s resonance
frequency. As a result, the nonlinear light transmission
in this case is described by the same Eqs. (1)–(2), with
the only difference that the frequency detuning parame
ter σ(ω) should be replaced by the generalized intensity
dependent frequency detuning parameter (σ(ω) − Jα).
Here, Jαis a new dimensionless parameter which is, as
we show below, proportional to the intensity of the res
onator’s localized mode. In particular, Eqs. (2) take the
form
T =
[σ(ω) − Jα]2
[σ(ω) − Jα]2+ 1, R =
In order to find an explicit expression for Jα, we as
sume that:(i) The dimensionless mode profiles?Eα(? r)
and?Ek(? r) introduced in Eqs. (4)–(5) are normalized to
their maximal values (as functions in real space), i.e.,
?Eα(? r)2
fields are described by amplitudes, Aαand Ak, multiply
ing the field profiles. Consequently, the maximum inten
sity of the electric field in the vicinity of the αresonator,
?E(? r) ≃ Aα?Eα(? r), is equal to Aα2; (iii) The αresonator
is made of a Kerrnonlinear material with the nonlinear
1
[σ(ω) − Jα]2+ 1. (6)
max= ?Ek(? r)2
max= 1; (ii) The physical electric
Page 4
4
susceptibility χ(3)
function θα(? r). This function is equal to unity for all ? r
inside the cavities which form the resonator structure and
vanishes outside. In this case, Jαtakes the form
Jα=12πQκ
W2
α
α and it covers the area described by the
χ(3)
αAα2, (7)
where κ is the dimensionless and scaleinvariant nonlin
ear feedback parameter (first introduced in similar form
in Refs. [17, 18]) which measures the geometric nonlinear
feedback of the system. It depends on the overlap of the
resonator’s mode profile with spatial distribution θα(? r)
of nonlinear material according to
κ =
3
W2
α
?
c
ωres
?d ?
all space
d? r θα(? r)εα(? r)?Eα(? r)4, (8)
where d is the system dimensionality.
The dependence of Jα on the power of the incoming
light has already been studied analytically in Refs. [18,
20, 21]. Here, we suggest a simpler form for this depen
dence
Jin= Jα
?[σ(ω) − Jα]2+ 1?
, (9)
where we have introduced the dimensionless intensity Jin
which is proportional to the experimentally measured
power of the incoming light
Pin=c2k(ω)
2πω
Iin= P0Jin. (10)
In this expression, we have abbreviated the incoming
light intensity as Iin = Ak2and introduced the char
acteristic power P0 of the waveguide defined as (see
Refs. [17, 18, 20, 21] for derivation):
P0=
?
c
ωres
?d−1
√εα
Q2καχ(3)
α
. (11)
Finally, the outgoing light power Pout= P0Joutcan be
determined through the dimensionless intensity of the
outgoing light Jout = T Jin with the transmission co
efficient T defined by Eq. (6).
It follows from Eqs. (6) and (9) that the nonlinear
transmission problem is completely determined by the
value of σ(ω) and the sign of the product σ(ω)·Jα. As is
illustrated in Fig. 2, for frequencies where (σ(ω)·Jα) < 0,
the transmission coefficient T and the outgoing light in
tensity Jout grow monotonically with Jin for all values
of σ(ω).
The situation becomes more interesting for frequencies
lying on the other side of the resonance where (σ(ω) ·
Jα) > 0. In this case T and, therefore, Joutbecome non
monotonic functions of Jin, as is illustrated in Fig. 3.
Moreover, for σ2(ω) > 3 these functions become multi
valued functions of Jin in the interval J(3,4)
in
≤ Jin ≤
0
0,5
1
1,5
2
Jin
0
0,2
0,4
0,6
0,8
1
0
0,5
1
1,5
2
Jin
0
0,25
0,5
0
0,5
1
1,5
2
Jin
0
0,5
1
1,5Jout(Jin)
Jα(Jin)
T(Jin)
σ2 = 1
FIG. 2: (Color online) Dependencies of the transmission co
efficient, T, the outgoing light intensity, Jout, and the res
onator’s mode intensity, Jα, on the incoming light intensity,
Jin, for σ2(ω) = 1 and negative product (σ(ω) · Jα).
J(1,2)
in
, where
J(1,2)
J(3,4)
in
=
2
27
2
27
?
?
σ3+ 9σ +?σ2− 3?3/2?
σ3+ 9σ −?σ2− 3?3/2?
,
in
=
, (12)
which are also shown in Fig. 4. In this interval the nonlin
ear light transmission becomes bistable: low and high
transmission regimes coexist at the same value of the
incoming light intensity Jin, as can be seen in Fig. 3
for σ2> 3 (intermediate parts of the curves correspond
to unstable transmission). Therefore, by increasing an
intially low intensity Jinwe obtain a hysteris where we
jump from the point (1) to (2), and then upon decreasing
Jin, we jump from the point (3) to (4). The transmission
coefficients at these characteristic points are
T(1,3)=
1
2σ2(1 ∓?1 − 3/σ2) − 2
(1 ∓ 2?1 − 3/σ2)2
,
T(2,4)=
5 − 3/σ2∓ 4?1 − 3/σ2,(13)
and they are depicted in Fig. 4. For completeness, we also
present the expressions for the resonator’s mode intensity
at these points
J(1,3)
α
=
2σ
3
2σ
3
∓1
±2
3
?
?
σ2− 3 ,
J(2,4)
α
=
3
σ2− 3 .(14)
From a practical point of view, these solutions have im
portant consequences. Firstly, the bistability condition
σ2> 3 corresponds to a linear transmission T > 3/4.
That is, the bistable transmission becomes possible only
for frequencies where (σ(ω) · Jα) is positive and linear
transmission exceeds 75%. As demonstrated in Fig. 4
and Eq. (12), when σ2grows, all threshold intensities
grow, too, starting with the minimum threshold inten
sity J(1,2,3,4)
For ideal nonlinear switching the coefficients T(1)and
T(4)should be close to unity while T(2)and T(3)should
in
= 8/31.5≈ 1.54 at σ2= 3.
Page 5
5
0123
Jin
0
0,2
0,4
0,6
0,8
1
0123
Jin
0
0,5
1
1,5
2
2,5
0123
Jin
0
0,2
0,4
0,6
0,8
Jout(Jin)
Jα(Jin)
T(Jin)
σ2 = 3
01234
Jin
0
0,2
0,4
0,6
0,8
1
01234
Jin
0
1
2
3
01234
Jin
0
0,4
0,8
1,2
Jout(Jin)
Jα(Jin)
(1)
(1)
(1)
T(Jin)
(2)
(3)
(4)
(2)
(2)
(3)
(3)
(4)
(4)
σ2 = 4
024
6
8 10
Jin
0
0,2
0,4
0,6
0,8
1
024
6
8 10
Jin
0
1
2
3
4
024
6
8 10
Jin
0
1
2
3
4
5
Jout(Jin)
Jα(Jin)
(1)
(1)
(1)
T(Jin)
(2)
(3)
(4)
(2)
(2)
(3)
(3)
(4)
(4)
σ2 = 10
0
0,5
1
1,5
Jin
0
0,2
0,4
0,6
0,8
1
0
0,5
1
1,5
Jin
0
0,5
1
1,5
0
0,5
1
1,5
Jin
0
0,05
0,1
0,15
Jout(Jin)
Jα(Jin)
T(Jin)
σ2 = 1
FIG. 3: (Color online) Dependencies of the transmission co
efficient, T, the outgoing light intensity, Jout, and the res
onator’s mode intensity, Jα, on the incoming light intensity,
Jin, for several different values of σ2(ω) and positive product
(σ(ω) · Jα).
vanish. However, as can be seen from Fig. 4 and the
asymptotic (for large σ2) expressions
T(1)≈ 1 −
9
σ2,
1
σ2,
T(2)≈ 1 −
9
4σ2,
T(3)≈ 1 −
T(4)≈
1
4σ2, (15)
of Eqs. (13), these conditions cannot be satisfied simul
taneously. In particular, the transmission coefficient T(2)
does not vanish but approaches unity for large σ2. More
24
6
8 10
σ2
0
0,2
0,4
0,6
0,8
1
24
6
8 10
σ2
0
0,2
0,4
0,6
0,8
1
24
6
8 10
σ2
1
2
3
4
5
6
T(4)
T(3)
T(1)
T(2)
Jin
(1,2)
Jin
(3,4)
FIG. 4: (Color online) Dependencies of the threshold incom
ing light intensity and the corresponding transmission coeffi
cients on σ2(ω) for the four critical points (1)(4) indicated
by circles in Fig. 3. Here, we assume that (σ(ω) · Jα) > 0.
over, there exists no condition under which T(2)and T(3)
vanish simultaneously. Therefore, it is impossible to cre
ate ideal nonlinear switches in these systems.
A reasonable compromise for realistic nonlinear switch
ing schemes of this type could be the usage of the fre
quency with σ2≃ 5, for which the linear light transmis
sion is close to 83%. For this case, the critical trans
mission coefficients T(2)≃ 3.7% and T(3)≃ 7% are suf
ficiently small, while T(1)≃ 60% and T(4)≃ 74% are
large enough for practical purposes. The threshold in
tensities J(1,2)
20% from each other, so that in this case one can achieve
a highcontrast and robust switching for sufficiently small
modulation of the incoming power.
The above analysis suggests that the optimal dimen
sionless threshold intensities are fixed around J(i)
so that the real threshold power of the incoming light,
P(i)
in, can only be minimized by minimizing the
characteristic power, P0, of the system. An inspection
of Eq. (11) shows that this can be facilitated by increas
ing the resonator nonlinear feedback parameter, κα, the
material nonlinearity, χ(3)
α , or the resonator quality fac
tor, Q. For smallvolume photonic crystal resonators, it
has been established that κ ∼ 0.2 (see [17, 18]), and this
value can hardly be further increased.
Therefore, only two practical strategies remain that
could lead to an enhancement of nonlinear effects in this
system. The first approach is based on specific mate
rial properties: We should create the resonator α from a
material with the largest possible value of χ(3)
index semiconductors, nearly instantaneous Kerr nonlin
earity reaches values of n2 ∼ 1.5 × 10−13cm2/W [31],
where n2∝ χ(3)/n0and n0is the linear refractive index
of the material. Even such relatively weak nonlinearity
is already sufficient for many experimental observations
of the bistability effect in the waveguideresonator sys
tems [9, 10]. However, using polymers with nearly in
stantaneous Kerr nonlinearity of the order of n2> 10−11
cm2/W and, at the same time, sufficiently weak two
photon absorption [32], one could potentially decrease
the value of P0by at least two orders of magnitude. Poly
mers, however, have a low refractive index which is insuf
in
≃ 2.53 and J(3,4)
in
≃ 2.11 differ about
in∼ 2.5
in= P0J(i)
α . In high
Page 6
6
ficient for creating a (linear) photonic bandgap required
to obtain good waveguiding and low losses. The solution
to this could be the embedding of such highly nonlin
ear but lowindex materials into a host photonic crystal
made of a highindex semiconductor. Optimized waveg
uding designs for the basic functional devices of this kind
are available [33–35] and recent experimental progress
[36, 37] may soon allow a realization of corresponding
linear and nonlinear devices.
The second approach is based on designing waveguide
resonator structures with the largest possible quality fac
tor, Q. Potentially, one can increase Q indefinitely by
mere increase of the distance between the waveguide and
the resonator. However, this leads to a corresponding
increase in the size of the nonlinear photonic devices. A
very attractive alternative possibility for increasing Q is
based on the adjustment of the resonator geometry [38].
In what follows, we suggest yet another possibility to
dramatically increase Q through an optimal choice of the
resonator location relative to the discrete locations of the
cavities that form the photoniccrystal waveguide.
C. Limitations of the coupledmode theory
Standard coupledmode theory exhibits a number of
limitations. Firstly, it gives analytical expression for the
detuning parameter σ(ω) only near the resonator fre
quency ωα. And this immediately highlights the second
limitation: standard coupledmode theory [16–21] can
not analytically describe resonant effects near waveguide
band edges. However, numerical studies [30] have re
cently demonstrated that the effects of the waveguide
dispersion become very important at the band edges and
may lead to nonLorentzian transmission spectra in cou
pled waveguideresonator systems.
As a matter of fact, the question “what happens if the
resonator frequency ωαlies near the edge of the waveg
uide passing band or even outside it?” may be of a great
practical importance due to two reasons. Firstly, in real
istic structures it is not always possible to appropriately
tune the frequency ωα, and therefore it is important to
understand properties of the system for any location of
the resonance frequency. Secondly, as we have already
mentioned in the Introduction, PhC waveguides can pro
vide us with a very slow group velocity of the propagating
pulses — but in most cases they do it exactly at the pass
ing band edges. Therefore, if we wish to utilize such a
slow light propagation for nonlinearity enhancement, we
should extend the above analysis to such cases, too.
In what follows, we describe an alternative analyti
cal approach to the coupled waveguideresonator struc
tures which allows us to correctly analyze both linear
and nonlinear transmission for arbitrary locations of the
resonator frequency ωαrelative to the waveguide passing
band, including the transmission near band edges in the
slow light regime.
III. DISCRETE MODEL APPROACH
Having
continuouswaveguide structure shown in Fig. 1(a), we
now take into account the discrete nature of the waveg
uding structure embedded in photonic crystals. In par
ticular, we analyze what will change in the system prop
erties when we move the resonator along the waveguide
from the onsite location shown in Fig. 1(b) to the inter
site location shown in Fig. 1(c). Our analysis is based on
effective discrete equations that have been derived for the
description of photonic crystal devices [25–29] in combi
nation with a recently developed discrete model approach
to nonlinear Fano resonances [22].
discussed theresults obtained forthe
A. Discrete Equation Approach
First, we derive an appropriate set of discrete equations
[see Eqs. (24) below], and show that they can be applied
to a variety of the photoniccrystal devices. We start
from the wave equation in the frequency domain for the
electric field
?
?∇ ×?∇ × −
?ω
c
?2
ˆ ε(? r)
?
?E(? r) = 0 , (16)
where the dielectric function ˆ ε(? r) = ˆ εpc(? r) + δˆ ε(? r) con
sists of the dielectric function ˆ εpc(? r) of a perfectly peri
odic structure and a perturbation δˆ ε(? r) that describes the
embedded cavities. It is convenient to introduce the ten
sorial Green function of the perfectly periodic photonic
crystal,
?
?∇ ×?∇ × −
?ω
c
?2
ˆ εpc(? r)
?
ˆG(? r,? r′ω) =ˆIδ(? r −? r′) (17)
and to rewrite Eq. (16) in the integral form,
?E(? r) =
?ω
c
?2?
? r′ ˆG(? r,? r′ω)δˆ ε(? r′)?E(? r′) ,(18)
where we assume that the frequency ω lies inside a com
plete photonic bandgap so that the electric field vanishes
everywhere except for areas inside and in the vicinity of
cavities. We enumerate the cavities by an integer index
n and introduce dimensionless functions θn(? r) which de
scribe the shape of the nth cavity. As a result, δˆ ε(? r)
may be represented as
δˆ ε(? r) =
?
n
?
δˆ εn+ χ(3)
n?E(? r)2?
θn(? r −?Rn) ,(19)
where?Rn, δˆ εn, and χ(3)
ear) dielectric function, and nonlinear thirdorder suscep
tibility of the nth cavity.
Similar to Sec. II, we describe the electric field of the
nth cavity mode via a dimensionless field profile?En(? r)
and a complex amplitude An. Taking into account that
n
are, respectively, position, (lin
Page 7
7
inside the cavities the electric field of the system is a
superposition
?E(? r) ≃
?
n
An?En(? r −?Rn) , (20)
Eq. (18) can be rewritten as a set of discrete nonlinear
equations
Dn(ω)An=
?
m?=n
Vn,m(ω)Am+ κn(ω)χ(3)
nAn2An, (21)
where Dn(ω) = 1−Vn,n(ω) is the dimensionless frequency
detuning from the resonance frequency, ωn, of the nth
cavity. Furthermore,
Vn,m(ω) =
δεm
Wn
?ω
c
?2?
d? r
?
d? r′?E∗
n(? r)ˆ εn(? r) (22)
× θm(? r′)ˆG(? r +?Rn−?Rm,? r′ω)?Em(? r′) ,
is the dimensionless linear coupling between the nth and
the mth cavity. Similarly,
κn(ω) =
1
Wn
?ω
c
?2?
d? r
?
d? r′?E∗
n(? r)ˆ εn(? r) (23)
× θn(? r′)ˆG(? r,? r′ω)?En(? r′)2?En(? r′) ,
is the dimensionless and scaleinvariant nonlinear feed
back parameter which should be compared with the
analogous parameter (8) introduced in the conventional
coupledmode theory analysis [17, 18]. Finally, Wn is
defined in exactly the same way as Wαin Eq. (5).
We remark that in deriving Eqs. (21) we have neglected
higherorder couplings proportional to the integrals of
?E∗
count the coupling coefficients which involve integrals of
ˆG(? r +?Rn−?Rm,? r′ω) with n ?= m. This approxima
tion is sufficiently accurate in most cases, as we demon
strate in Refs. [24, 26]. We would like to mention that
in Eqs. (21)–(23) we have used more accurate definitions
of the coupling coefficients than those that have been in
troduced earlier in Refs. [26–29]. They have also a more
generic form than those we used in Ref. [25].
Typical frequency dependencies of the parameters of
the discrete model, Eq. (21), are displayed in Figs. 9–
11 of Appendix A, where we also discuss the application
of Eqs. (21)–(23) to simple structures such as linear and
nonlinear photonic crystal resonators and straight waveg
uides. Here, we apply Eqs. (21)–(23) to study the more
complicated case of the nonlinear coupled waveguide
resonator systems shown in Figs. 1(b,c).
Eqs. (21) may be separated in this case according to
n(? r)?Em(? r +?Rn−?Rm) with n ?= m but take into ac
The set of
Dw(ω)An =
L
?
?
j=1
Vjw(ω)(An+j+ An−j) + Vn,α(ω)Aα,
Dα(ω)Aα =
j
Vα,j(ω)Aj+ κα(ω)χ(3)
αAα2Aα, (24)
where we assume that all cavities of the photoniccrystal
waveguide are identical and linear, so that we can de
note Dw(ω) ≡ Dn(ω) and Vjw(ω) ≡ Vn,n±j(ω) for any n
inside the waveguide. Furthermore, the index α defines
the parameters of the sidecoupled nonlinear resonator.
Below we show that the assumption of linear waveguide
cavities may be relaxed for frequencies near the resonator
resonance frequency ωαbecause then the amplitudes An
remain small in comparison with the amplitude Aα.
For the first equation in Eq. (24), we seek solutions of
standard form
An=
?
I1/2
I1/2
in
int(ω)eik(ω)sn
?
for n ≫ 1,
for n ≪ −1,(25)
eik(ω)sn+ r(ω)e−ik(ω)sn?
where s is the distance between the nearest waveguide
cavities and Iin is the intensity of the incoming light.
For both structures shown in Figs. 1(b,c), we obtain that
the transmission and reflection coefficients can formally
be described by the same expressions (1)–(2) as for the
structure depicted in Fig. 1(a). However, within the dis
crete equation approach the expression for the detuning
parameter σ(ω) can now be found for the entire frequency
range. Below, we discuss novel results for the structures
shown in Fig. 1(b) and Fig. 1(c) separately.
B.Onsite resonator
First, we obtain the solution of this problem for the
structure shown in Fig. 1(b).
sume that the only nonvanishing coupling coefficients in
Eq. (24) are V1w(ω), Vα,0(ω), and V0,α(ω) (see, however,
Appendix B for a more accurate analysis which takes into
account additional coupling coefficients).
we obtain the transmission and reflection coefficients de
scribed by Eqs. (1)–(2) with φr= π/2 and a correspond
ing expression for σ(ω):
For simplicity, we as
As a result,
σ(ω) = 2sin[k(ω)s]V1w(ω)
V0,α(ω)
A0
Aα
,(26)
which should be considered as a generalized intensity
dependent frequency detuning parameter σ(ω) + Jαin
troduced in Eq. (6) above. The amplitude A0in Eq. (26)
is given by
A0 = t(ω)I1/2
in
, (27)
while the waveguide dispersion relation k(ω) is deter
mined by Eq. (A6).
In the case of a linear αresonator (i.e.
the amplitude Aα= Vα,0(ω)A0/Dα(ω) is proportional to
the amplitude A0. Therefore, σ(ω) and, accordingly, the
transmission and reflection coefficients do not depend on
the light intensity. Upon introducing the abbreviation
χ(3)
α
≡ 0),
µ(ω) =
Dα(ω)V1w(ω)
V0,α(ω)Vα,0(ω),(28)
Page 8
8
0.37 0.38
Frequency, a/λ = ωa/2πc
0.390.4
0.41
0
0.2
0.4
0.6
0.8
1
T(ω)
FIG. 5: (Color online) Linear transmission through a photonic
crystal waveguide that is created by removing every second
rod in a row (? s = 2? a1) sidecoupled to a onesite resonator cre
ated by removing a single rod. The underlying 2D photonic
crystal is described in Appendix A1. We compare exact nu
merical results (solid line) with the analytical results based
on Eq. (29) (dotdashed line) and Eq. (B1) (dashed line).
the detuning parameter, Eq. (26), for a linear αresonator
reads as
σ(ω) = 2µ(ω)sin[k(ω)s] .(29)
This implies that σ(ω) vanishes when either Dα(ω) = 0
or k(ω) = πn/s with an arbitrary integer n. The first
condition reproduces Eq. (3) with ωres = ωα and the
resonance width γ given by
γ ≈
ωα∆α
sin[k(ωα)s]≈sωαωw
vgr
νανwV0,αVα,0, (30)
where ναand νware defined by Eq. (A1),
∆α=
V0,αVα,0
2ωαD′αV1w
=V0,αVα,0
2V1w
να, (31)
and the group velocity
vgr=dω
dk
????
ωα
≈ −2sωwνwV1wsin[k(ωα)s] ,(32)
can be found directly from Eq. (A6). Here and in what
follows, we assume that the values of all frequency
dependent parameters whithout explicitly stated fre
quency dependence are evaluated at the resonance fre
quency, ωres. Finally, we notice that the resonance width,
Eq. (30), is very similar to that described by the coupled
mode theory, Eq. (4).
It is important that the quality factor Q of the reso
nance
Q =ωα
2γ≈sin[k(ωα)s]
2∆α
≈
vgr
sωw
1
2νανwV0,αVα,0
, (33)
is multiplied by the factor sin[k(ωα)s] ∼ vgr, and, there
fore, becomes strongly suppressed near the edges of
waveguide passing band, k(ωα) = 0, ±π/s. Accordingly,
the detuning parameter (29) vanishes at these edges, too.
This means that, in agreement with the numerical calcu
lations shown in Fig. 5, the transmission coefficient T(ω)
vanishes not only at the resonance frequency, but also at
both edges of the waveguide passing band. Such an effect
was recently observed by Waks and Vukovic [30] in their
numerical calculations based on standard coupledmode
theory which takes into account the waveguide disper
sion. Therefore, the effect of vanishing transmission at
the spectral band edges may be attributed also to the
structure shown in Fig. 1(a).
Obviously, this enhancement of light scattering at the
waveguide band edges should be very important from the
point of view of fabrication tolerances since virtually any
imperfection contributes to scattering losses. Moreover,
as discussed in Sec. IV, this effect is detrimental to the
concept of alloptical switching devices based on slow
light photonic crystal waveguides.
We support this conclusion by another observation.
First, the light intensity at the 0th cavity, A02=
T(ω)Iin, vanishes at the resonancefrequency for arbitrary
large incoming light intensity, because T(ωα) ≡ 0. There
fore, the nonlinearity of this cavity may safely be ne
glected. In contrast, the light intensity at the αresonator
reaches its maximum value at ωα,
Aα(ωα)2≃ 4
?V1w
· Iin≃ (2QναVα,0)2· Iin,
V0,α
?2
sin2[k(ωα)s] · Iin
≃
?
vgr
sωwνwV0,α
?2
(34)
which may significantly exceed the incoming light inten
sity Iinwhen the coupling V0,αbetween the αresonator
and waveguide becomes small enough relative to the cou
pling V1w between the cavities in the waveguide. This
strong enhancement suggests a physical explanation for
the existence of the rather strong nonlinear effect of light
bistability at relatively low intensities of the incoming
light. However, when the resonance frequency ωα lies
close to any of the waveguide band edges, it is seen from
Eq. (34) that the light intensity at the αresonator be
comes (strongly) suppressed by a factor sin2[k(ωα)s] .
Details of an extension of the above discussion to the
case of more realistic nonlocal couplings, i.e., more than
nearest neighbors couplings, is presented in Appendix B
and here we only summarize the results. Both, a non
locality of the intercoupling between waveguide cavities
as well as a nonlocality of crosscoupling with the α
resonator lead to a small shift in the resonance frequency,
ωres, but do not change the main result about the sup
pression of the detuning σ(ω) and transmission T(ω) at
both edges of the waveguide passing band. However, we
would like to emphasize that for a fully quantitative anal
ysis, nonlocal couplings have to be taken into account,
for instance, within the framework of the recently devel
Page 9
9
oped Wannier function approach [39].
We now consider the case when the resonator α is non
linear, i.e.χ(3)
α
?= 0. As has been previously shown
in Ref. [22], this case, too, can be studied analytically
even for nonlocal couplings between the cavities and
resonator and novel effects originating solely from the
nonlocality may be expected when the nonlocal cou
pling strength exceeds one half of the local coupling. Un
fortunately, in realistic photonic crystals this limit may
hardly be realized so that here we restrict our analysis to
the localcoupling approximation. In this case, we obtain
from the second equation in Eqs. (24) that the amplitude
Aαuniquely determines the amplitude A0. Substituting
the latter expression into Eqs. (26)–(27), we find that
the nonlinear transmission is described by Eqs. (6) and
(9) with the detuning σ(ω) determined by Eqs. (28)–(29)
and the dimensionless intensities Jαand Jingiven by the
expressions
Jα ≃ 2Qκαναχ(3)
Jin ≃ 8sin[k(ωres)s]V1w
≃ −4vgr
sωw
αAα2, (35)
?δε0
Q2καναχ(3)
δεα
?
Q2καν2
αχ(3)
αIin
?δε0να
δεανw
?
αIin,
where Q is determined by Eq. (33). Therefore, all the
results for the nonlinear light transmission which are dis
played in Figs. 2–4 are directly applicable to the structure
of Fig. 1(b), too.
In an experiment, one measures not the light inten
sity in the waveguide, Iin, but the propagation power,
Eq. (10), where for the discrete structure of Fig. 1(b),
the characteristic power P0is
P0 ≃
c2k(ωα)
16πsin[k(ωα)s]ωαV1w
≃ −c2k(ωα)s
8πvgr
?δεα
?
δε0
?
1
Q2καν2
1
αχ(3)
α
?ωwδεανw
ωαδε0να
Q2καναχ(3)
α
. (36)
Again, this result is quite similar to Eq. (11) for the con
tinuous structure of Fig. 1(a). Nevertheless, our more
general analysis explicitly suggests that it should be bet
ter to use the αresonatorwith the resonance frequency at
the center of the waveguide passing band k(ωα) ≈ π/2s,
where the group velocity reaches its maximum.
tice, however, that this suggestion becomes wrong for
the structure of Fig. 1(c) studied in the next subsection.
No
C. Intersite resonator
In the system where the αresonator is placed sym
metrically between two cavities of the waveguide and,
therefore, couples equally to both of them, a qualitatively
different type of resonant transmission occurs. The cor
responding structure is schematically shown in Fig. 1(c).
Assuming that in this case the nonvanishing coupling co
efficients in Eq. (24) are V1w(ω), Vα,1(ω) ≡ Vα,0(ω), and
0.37 0.38
Frequency, a/λ = ωa/2πc
0.39 0.4
0.41
0
0.2
0.4
0.6
0.8
1
T(ω)
FIG. 6: (Color online) Liner transmission through a photonic
crystal waveguide created by removing every second rod in a
row (? s = 2? a1) sidecoupled to an intersite resonator created
by removing a single rod. The underlying 2D photonic crystal
described in Appendix A1. We compare exact numerical re
sults (solid line) with the analytical results based on Eq. (39)
(dashed line).
V1,α(ω) ≡ V0,α(ω), we seek solutions to the first equa
tion of the system (24) that are of the form of Eq. (25).
Again, we find that the transmission and reflection coeffi
cients are given by Eqs. (1)–(2) albeit with the frequency
dependent phase φr(ω) = π/2 + k(ω)s. Here, k(ω) is
determined by Eq. (A6), and the generalized intensity
dependent frequency detuning is
σ(ω) + Jα= i − i
?
eik(ω)s− 1
?V1w(ω)
V0,α(ω)
I1/2
in
Aα
. (37)
The corresponding amplitudes are
A0 = I1/2
in
−
1
[1 − e−ik(ω)s]
−
[1 − e−ik(ω)s]
V0,α(ω)
V1w(ω)Aα,
A1 = eik(ω)sI1/2
in
1
V0,α(ω)
V1w(ω)Aα. (38)
Despite the complex form of Eq. (37), we would like to
emphasize that the detuning σ(ω) determined by Eq. (37)
is a realvalued function (see also the discussion following
Eq. (1) above).
In the case of the linear αresonator (i.e., for χ(3)
we obtain
α ≡ 0),
σ(ω) = [1 + µ(ω)]tan
?k(ω)s
2
?
, (39)
where µ(ω) is given by Eq. (28). For a highquality α
resonator in the vicinity of the resonance frequency this
detuning parameter can be approximated by Eq. (3) with
ωres≃ ωα(1 − 2∆α) and
2ωres∆α
tan[k(ωres)s/2].
γ =
(40)
Here, ∆αis defined by Eq. (31). In contrast to Eq. (33),
the corresponding quality factor
Q =ωres
2γ
≈tan[k(ωres)s/2]
4∆α
≈V1wtan[k(ωres)s/2]
2ναV0,αVα,0
(41)
Page 10
10
is now multiplied by the factor tan[ks/2] which does not
vanish and even diverges as k(ωres) approaches the edge
of the transmission band k = ±π/s. At this band edge,
σ(ω) ∼ tan(k(ω)s/2) → ∞ and, therefore, light trans
mission is always perfect. This conclusion is supported
by the exact numerical calculations presented in Fig. 6.
At the other band edge, i.e., for k(ω) = 0, transmission
vanishes, similar to the structures shown in Figs. 1(a,b).
The light intensity at the αresonator reaches its max
imal value at the resonance frequency
Aα(ωres)2≃ 4
?V1w
?
V0,α
?2
sin2
?k(ωres)s
?k(ωres)s
2
?
??2
· Iin
≃4QναVα,0cos
2
· Iin(42)
Again, in contrast to the corresponding light intensity
(34) for the onsite coupled structure, Eq. (42) does not
vanish at the edge of the transmission band k = ±π/s.
Therefore, we can expect that for intersite coupled struc
ture nonlinear effects at the band edge k = ±π/s should
be sufficiently strong to allow bistable transmission and
switching.
To investigate this, we assume that the αresonator
is nonlinear (χ(3)
α
?= 0) and introduce the same dimen
sionless intensities Jαand Jinas in Eq. (35). However,
now the quality factor Q is defined by Eq. (41) and the
resonance frequency is ωres ≃ ωα(1 − 2∆α).
that this nonlinear problem, too, has a solution of the
form given by Eqs. (6) and (9). However, now the de
tuning σ(ω) is given by Eq. (39). Therefore, all results
presented above in Figs. 2–4 remain applicable to this
structure, too. The only but very important qualitative
difference of the structure shown in Fig. 1(c) is that the
transmission coefficient T(ω) and the corresponding light
intensity Aα2at the αresonator do not vanish at the
band edge k = ±π/s since the quality factor Q at this
band edge grows to infinity for the intersite structure of
Fig. 1(c). Therefore, this structure may be utilized for
realizing efficient alloptical switching devices based on
slowlight photonic crystal waveguides. This is in sharp
contrast to the structures shown in Figs. 1(a,b).
We find
IV. DISCUSSION OF RESULTS
In this section, we summarize our results and em
phasize their importance by applying them to spe
cific photoniccrystal structures.
dimensional photonic crystal created by a square lattice
of dielectric rods in air. The rods are made from Si or
GaAs (ε = 11.56) and have radius r = 0.18a.
First, we consider a waveguide created by removing
every second rod (s = 2a) in a straight line of rods cou
pled to a nonlinear resonator α created by replacing a
single rod of the twodimensional lattice with a highly
nonlinear polymer rod. The corresponding structure is
schematically shown in the insets in Fig. 7. The resonant
We consider a two
0.374
0.375
Frequency, a/λ = ωa/2πc
0.376
0.3770.378
0
0.2
0.4
0.6
0.8
1
T(ω)
εα=2.56
εα=2.50
εα=2.40
0.374
0.375 0.376
0.377 0.378
0
0.2
0.4
0.6
0.8
1
T(ω)
FIG. 7: (Color online) Linear transmission spectrum for a
photonic crystal waveguide created by removing every second
rod in a row (? s = 2? a1) sidecoupled to a single onsite (a)
or intersite (b) polymerrod resonator (marked by the open
circle in the insets). The underlying 2D photonic crystal is de
scribed in Appendix A1 and results for three different values
of the resonator dielectric constant εα are shown.
frequency of the polymerrod resonator lies very close to
the edge k = ±π/s of the waveguide passing band, and
can be tuned by changing the linear dielectric constant
εαof the rod.
In Fig. 7(a) and (b), respectively, we display the trans
mission spectra for both onsite and intersite positions
of the sidecoupled resonator for three different values of
resonator dielectric constant εα. We notice that in the
case of the onsite position of the resonator the trans
mission coefficient T(ω) remains below the critical value
of T = 75% required for bistable switching operation
for all frequencies ω below the resonance frequency ωres.
The condition ω < ωres corresponds to the condition
(σ(ω) · Jα) > 0 which should be satisfied to realize non
monotonic dependencies of the nonlinear transmission
shown in Fig. 4). Therefore, this onsite system cannot
exhibit bistability.
On the other hand, bistability may be realized for the
intersite position of the sidecoupled resonator for which,
in a full agreement with our analysis presented above, the
transmission remains perfect at the band edge k = ±π/s
and the quality factor Q increases as the resonant fre
quency approaches this band edge. In Fig. 8(b) (exam
ple A) we show that in this case bistable transmission
indeed occurs for the frequency marked by a filled circle
in Fig. 8(a). This corresponds to T(ω) = 80%, i.e., the
choice σ2(ω) = 4 for the detuning parameter.
We want to emphasize that the large value of the qual
ity factor (41) for the intersite structure at k(ω) close to
±π/s leads to very low bistability thresholds as compared
to the cases of onsite coupled and continous waveguide
Page 11
11
0.373
0.3735
Frequency, a/λ = ωa/2πc
0.374
0.37450.375 0.3755
0
0.2
0.4
0.6
0.8
1
T(ω)
A
B
C
0 0.001 0.0020.003
Pinχ(3)/c
0
0.2
0.4
0.6
0.8
T(ω)
A
B
C
A
B
C
(a)
(b)
(c)
FIG. 8:
and (b) nonlinear bistable transmission for three different
sidecoupled waveguideresonator photonic crystal structures
whose topology is shown in (c). The rods consist of Si or GaAs
(full black circles) or polymer (open red circles). Example A
represents a close to optimal structure with intersite loca
tion of the αresonator whose resonance frequency lies close
to the edge k = ±π/s of the passing band; example B rep
resents a suboptimal structure with an intersite location of
the αresonator whose resonance frequency lies near the cen
ter of the passing band; example C represents a suboptimal
but commonly used structure with an onsite location of the
αresonator whose resonance frequency lies near the center of
the passing band. Closed circles in (a) indicate frequencies
with T(ω) = 80% that are used for achieving highcontrast
bistability in (b). Red circles in (c) indicate positions of the
nonlinear polymer rods with εα = 2.56. Other parameters of
the 2D photonic crystal are described in Appendix A1.
(Color online) (a) Linear transmission spectrum
coupled structures.
and C of Fig. 8: Relative to the waveguide design in ex
ample A, the design in example B moves the resonance
frequency deeper into the passing band thus decreasing
the quality factor (41). Nevertheless, the intersite cou
pled example B still exhibits a much smaller bistability
threshold than the onsite coupled system with the same
waveguide design in example C. This is caused by (usu
ally) much smaller waveguideresonatorcoupling and, ac
cordingly, much larger Q in the intersite structures as
compared to the onsite structures.
Summarizing, the intersite structure of the resonant
waveguideresonator interaction schematically shown in
This is illustrated in examples B
Fig. 1(c) allows to achieve much higher values for the
linear quality factor Q. As a consequence, much smaller
bistability threshold intensities for the nonlinear trans
mission are obtained. To employ these advantages, the
wavevector k(ωres) of the guided mode at the resonance
frequency ωres, Eq. (39), should be as close as possible to
π/s. This requirement coincides with the condition of a
very small group velocity in the waveguide and, in con
trast to the continuouswaveguide and onsite structures
depicted in Figs. 1(a,b), provides us with a possibility to
create lowthreshold alloptical switching devices based
on slowlight photonic crystal waveguides.
V.CONCLUSIONS
We have presented a detailed analysis of PhC waveg
uides sidecoupled to Kerr nonlinear resonators which
may serve as a basic element of active photoniccrystal
circuitry. First, we have extended the familiar approach
based on standard coupledmode theory and derived ex
plicit analytical expressions for the bistability thresholds
and transmission coefficients related to light switching in
such structures. Our results reveal that, from the point
of view of bistability contrast (a small difference between
two threshold intensities and robustness of switching) the
best conditions for bistability are realized for those pa
rameter values for which the dimensionless detuning pa
rameter σ(ω) is close to√5. Practically, this corresponds
to the choice of operation frequencies for which the linear
light transmission is close to 83%.
We have pointed out that the conventional coupled
mode theory does not allow to describe the light trans
mission near the band edges, and we have developed an
improved semianalytical approach based on the effective
discrete equations derived in the framework of a consis
tent Green’s function formalism. This approach is ideally
suited for a qualitative and semiquantitative description
of photoniccrystal devices that involve a discrete set of
smallvolume cavities. We have shown that this novel
approach allows to adequately describe light transmis
sion in the waveguideresonator structures near the band
edges. Specifically, we have demonstrated that while the
transmission coefficient vanishes at both spectral edges
for the onsite coupled structure (see Fig. 1(b)), light
transmission remains perfect at one band edge for the
intersite coupled structure (see Fig. 1(c)). These fea
tures allow a significant enhancement of the resonator
quality factor and, accordingly, a substantial reduction
of the bistability threshold. As a consequence, we refer
to this type of nonlinearity enhancement as a geomet
ric enhancement. The possibility of such enhancement is
a direct consequence of the discreteness of the photonic
crystal waveguide and is in a sharp contrast to similar
resonant systems based on ridge waveguides. The poten
tial of this novel type of the nonlinearity enhancement
may be regarded as an additional argument to support
the application of photoniccrystal devices in integrated
Page 12
12
photonic circuits.
In addition, we would like to emphasize that the engi
neering of the geometry of photoniccrystal based devices
such as that presented in Fig. 1(c) becomes extremely
useful for developing novel concepts of alloptical switch
ing in the slowlight regime of PhC waveguides which may
have much wider applications in nanophotonics and is
currently under active experimental research [40].
We believe that the basic concept of the geometric en
hancement of nonlinear effects based on the discrete na
ture of photoniccrystal waveguides will be useful in the
study of more complicated devices and circuits and, in
particular, for various slowlight applications.
stance, this concept may be applied to the transmission
of a sidecoupled resonator placed between two partially
reflecting elements embedded into the photoniccrystal
waveguide where sharp and asymmetric line shapes have
been predicted with associated variations of the transmis
sion from 0% to 100% over narrow frequency ranges [41].
Similarly, the concept can be extended to a system of
cascaded cavities [42] and threeport channeldrop fil
ters [43], optical delay lines [44], systems of two nonlinear
resonators with a very low bistability threshold [45], etc.
For in
Acknowledgments
S.F.M.
the
the
project A1.1.
from the Organizers of the PECSVI Symposium
(http://cmp.ameslab.gov/PECSVI/),
these results have been presented for the first time.
The work of Y.K. and A.E.M. was supported by the
Australian Research Council through the Center of
Excellence Program.
and K.B.
for
Forschungsgemeinschaft
S.F.M. also acknowledges a support
acknowledge
Functional
a support from
Center
Deutsche
Nanostructures of
withinthe
where some of
APPENDIX A: CALCULATION OF THE MODEL
PARAMETERS AND EXAMPLES
1.Coupling coefficients for twodimensional
photonic crystals
To obtain deeper insight into the basic properties of
the effective discrete equations (21), we should know how
the coupling coefficients Dn(ω), κn(ω), and Vn,m(ω) de
pend on frequency ω. As an illustration, we consider a
twodimensional model of a photonic crystal consisting
of a square lattice (lattice spacing a) of infinitely long
dielectric rods (see Refs. [15, 17, 18] and also references
[716] in Ref. [25]). We study light propagation in the
plane of periodicity, assuming that the rods have a radius
r = 0.18a and a dielectric constant of εrod= 11.56 (GaAs
or Si at the telecommunication wavelength λ ∼ 1.55 µm).
For light with the electric field polarized along the rods
(Epolarized light), this photonic crystal exhibits a large
0,32 0,34
0,36
0,380,4 0,420,44
Frequency, a/λ = ωa/2πc
2
1,5
1
0,5
0
0,5
Dn(ω)
εn=1.0 (removed rod)
εn=2.56 (polymer rod)
FIG. 9: (Color online) Frquency dependence of the detun
ing coefficient Dn(ω) for the 2D photonic crystal described
in Appendix A1, for two types of resonators: Removing a
single rod (εn = 1.0; solid line) leads to a localized mode at
ωn = 0.392, while replacing a single rod by a geometrically
identical rod made of polymer (εn = 2.56, dashed line) leads
to a localized mode at ωn = 0.374.
(38% of the center frequency) photonic bandgap that ex
tends from ω = 0.303(2πc/a) to ω = 0.444(2πc/a)
Our task is to evaluate the coupling coefficients Dn(ω),
κn(ω), and Vn,m(ω) using Eqs. (22)–(23) with the
Green’s functionˆG(? r,? r′ω) calculated by the method de
scribed earlier in Refs. [28, 29]. The results of these cal
culations are displayed in Figs. 9–11.
2. Isolated optical resonators
For the case of an isolated (Vn,m= 0) linear (χ(3)
optical resonator at the site n, Eq. (21) takes a sim
plest possible form, Dn(ω)An = 0.
only need to know the dimensionless frequency detun
ing coefficient, Dn(ω). In Fig. 9 we plot Dn(ω) for
two types of resonators: a resonator created by re
moving a single rod and a resonator created by replac
ing a single rod with a polymer rod of the same ra
dius and εn = 2.56. Introducing the dimensionless fre
quency ˜ ω = a/λ ≡ (ωa/2πc), we can express these
coefficients, with a very good accuracy in the range
0.36 ≤ ˜ ω ≤ 0.41, by the following cubic dependencies:
Dn(ω) = 9.426(˜ ω−˜ ωn)−10.889(˜ ω−˜ ωn)2+840.36(˜ ω−˜ ωn)3
with ˜ ωn = 0.3919, for the removed rod, and Dn(ω) =
9.047(˜ ω − ˜ ωn) − 49.555(˜ ω − ˜ ωn)2+770.14(˜ ω − ˜ ωn)3with
˜ ωn= 0.3744, for the replaced rod.
The resonator mode can only be excited at the res
onator frequency ωn, which is determined by the equation
Dn(ωn) = 0. Fig. 9 suggests that changing the dielectric
constant of the resonator εnallows to tune the frequency
ωn. In all cases, in the vicinity of the resonator frequency
ωn, the coupling coefficient Dn(ω) can be approximately
expanded into the Taylor series with a linear dependence
n = 0)
In this case, we
Dn(ω) ≃ω − ωn
νnωn
,νn=
1
ωnD′n(ωn),(A1)
Page 13
13
0,320,34
Frequency, a/λ = ωa/2πc
0,36
0,380,40,42 0,44
0,3
0,2
0,1
0
κn(ω)
FIG. 10: Frequency dependence of the nonlinear feedback
parameter κn(ω) for the 2D photonic crystal described in
Appendix A1. The nonlinear resonator is created by re
placing a single rod with a polymer rod of the same radius
which supports at εn = 2.56 a localized mode with frequency
ωn = 0.374.
0,320,34
Frequency, a/λ = ωa/2πc
0,36
0,380,4 0,420,44
0
0,1
0,2
0,3
Vn,m(ω)
Vα,0=V1w
V0,α
V2w
Vα,1
V1,α
FIG. 11: (Color online) Frequency dependence of the coupling
coefficients Vn,m(ω) for the 2D photonic crystal described in
Appendix A1 with a onsite sidecoupled waveguideresonator
system shown in Fig. 1b. The notations are the same as those
in Eq. (24) and we assume that the waveguide is created by
removing every second rod in a row (located at?Rn = 2? a1n)
whereas the onsite resonator is created by replacing a single
rod at?Rα = −2? a2 with a polymer rod (εn = 2.56) of the
same radius. The dispersion relation for such a waveguide is
displayed in Fig. 12.
where we have introduced a dimensionless parameter νn
which describes the resonator sensitivity to a change of
the dielectric constant. For our example of a polymerrod
resonator, we find νn≈ 0.295.
When the nth resonator is nonlinear (i.e., χ(3)
0), Eq. (21) reduces to the equation Dn(ω)An
κn(ω)χ(3)
nonlinear feedback parameter κn(ω). In Fig. 10, we de
pict the frequency dependence of κn(ω) for the case of
a nonlinear polymer resonator. In the frequency range
0.36 ≤ ˜ ω ≤ 0.41, this behavior can be approximated
as κn(ω) = −0.111 + 1.005(˜ ω − ˜ ωn) − 5.501(˜ ω − ˜ ωn)2+
85.57(˜ ω−˜ ωn)3with ˜ ωn= 0.3744. Therefore, in the vicin
ity of the resonator’s frequency, ωn, we may assume that
n
?=
=
n An2Anwith the new important coefficient —
κn(ω) ≈ −0.111 is constant and can rewrite Eq. (21)
according to
An2=
Dn(ω)
κn(ω)χ(3)
n
≈
D′
n(ωn)
κn(ωn)χ(3)
n
(ω − ωn). (A2)
The solution of the above equation gives us the depen
dence of the resonator frequency ωreson the resonator’s
mode intensity An2as
?
Here, we have used the notation κn= κn(ωn). As we see,
the nonlinear sensitivity of the resonator at the site n is
a product of its nonlinear feedback parameter, κn, the
sensitivity to a change of the dielectric constant, νn, and
the Kerr susceptibility of material, χ(3)
product defines the direction of the resonator frequency
shift. In particular, for the polymer resonator used in
Figs. 9–10, we obtain a rather small shift, κnνn≈ −0.033
which indicates that for χ(3)
n > 0 the resonator frequency
decreases as the light intensity grows. Designing optical
resonators with larger κn or νn, may allow to enhance
their nonlinear properties for a given material with Kerr
nonlinearity χ(3)
n .
ωres≈ ωn
1 + κnνnχ(3)
nAn2?
. (A3)
n . The sign of this
3.Straight waveguides
Now let us consider an array of identical coupled
cavities separated by the distance s = ? s which cre
ate a straight photoniccrystal waveguide depicted in
Figs. 1(b,c). Before proceeding, we would like to em
phasize that our analysis can equally well be applied to
the coupledresonator optical waveguides (CROWs) sug
gested in Ref. [46]. If we neglect nonlinear effects (assum
ing that either the waveguide cavities are linear, χ(3)
or the light intensity in the waveguide remains sufficiently
small), Eq. (21) reduces to
n = 0,
Dw(ω)An=
∞
?
j=1
Vjw(ω)(An+j+ An−j) .(A4)
Here we have defined, similar to Eq. (24), Dw(ω) ≡
Dn(ω) and Vjw(ω) ≡ Vn,n±j(ω) which are identical for
all n.
In Fig. 11 we plot the frequency dependencies of
V1w(ω) and V2w(ω) for a photoniccrystal waveguide cre
ated by removing every second rod in a row, either with
? s = 2? a1or with ? s = 2? a2. In the vicinity of the polymer
rod resonator frequency, the coupling coefficients are to
lowest order constant: V1w≈ 0.096 and V2w≈ 0.0086 .
In the general case, our calculations show that the co
efficients Vjw(ω) decay nearly exponentially with j. In
terms of frequency, they take on a constant value at the
central passing band frequency and grow rapidly towards
the lowfrequency bandgap edge.
Page 14
14
0 0,10,2 0,30,4
0,5
Wave vector (ks/2π)
0,37
0,38
0,39
0,4
0,41
Frequency, a/λ = ωa/2πc
accurate
L=1
L=2
k
FIG. 12: (Color online) Dispersion relation for a photonic
crystal waveguide created by removing every second rod in
a row (? s = 2? a1) in the 2D photonic crystal described in Ap
pendix A1. Numerical exact results (solid line) are calculated
with the supercell planewave method [47] and the approxi
mate results are obtained from Eq. (A5) with L = 1 (dotted
line) and L = 2 (dashed line), using the coupling coefficients
from Fig. 11.
According to the FloquetBloch theorem, Eq. (A4) has
a solution An= A0exp[±ik(ω)sn] with an arbitrary com
plex amplitude A0. The corresponding dispersion k(ω)
is determined by the equation
Dw(ω) =
L
?
j=1
2Vjw(ω)cos[k(ω)sj] ,(A5)
where we assume that the coupling coefficients Vjw(ω)
vanish for all j above L. As a matter of fact, our studies
indicate that sufficiently accurate results can be obtained
already for L ∼ 4a/s. In Fig. 12 we plot the dispersion
relation for a 2D model photoniccrystal waveguide and
compare it with exact numerical results calculated by the
supercell planewave method [47]. For this case, even
the simplest tightbinding approximation (i.e., at L = 1)
gives quite satisfactory results.
In the tightbinding approximation (L = 1) the dis
persion relation can be described by the following simple
expression
cos[k(ω)s] =
Dw(ω)
2V1w(ω)≃ω − ωw
ωw∆w
, (A6)
where ωw is the resonance frequency of the waveguide
cavities. Furthermore, we have the dimensionless param
eter
∆w=
2V1w(ωw)
ωwD′w(ωw)= 2V1wνw, (A7)
with V1w ≡ V1w(ωw) and νw defined by Eq. (A1), that
equals halfbandwidth of the waveguide transmission
band. This band extends from ωw(1−∆w) to ωw(1+∆w).
For our example of photonic crystal waveguide, we find
∆w≈ 0.052, i.e., its bandwidth is about 10%.
APPENDIX B: EFFECT OF LONGRANGE
INTERACTIONS
1. Effects of nonlocal dispersion
As follows from the results of Sec. IIIB above, the
localcoupling approximation provides us with an ex
cellent qualitative analysis of the structure shown in
Fig. 1(b). However, certain physically important effects
may be missed in this approximation. A detailed anal
ysis of the effects of nonlocal coupling was performed in
Ref. [22], so that here we may discuss this issue very
briefly, and may specify it directly to photoniccrystal
devices.
In Fig. 5, we provide a comparison of T(ω) calculated
from Eq. (29) in the localcoupling approximation with
the exact numerical results for the structure shown in
Fig. 1(b) for the model photonic crystal described in Ap
pendix A1. The results suggest that the localcoupling
approximation introduces a frequency shift for the band
edges which agrees well with the corresponding frequency
shift in the dispersion relation shown in Fig. 12.
In addition, the resonance frequency is also shifted; it
is not equal to ωαbut is slightly larger. In principle, this
shift can be produced by two effects: (i) a longrange
coupling between cavities inside the waveguide and (ii) a
longrange coupling between the waveguide and the side
coupled resonator. First, we explore the former possibil
ity.
Solving Eqs. (24)–(25) for L = 2, we obtain the trans
mission and reflection coefficients (1)–(2) with the detun
ing parameter
σ(ω) = 2sin(ks) ·Dα
?V3
1w+ 3DwV1wV2w+ 3V1wV2
2w+ 2V3
1w− V2
2wcos(3ks)?− V0,αVα,0V1wV2w
V0,αVα,0(V2
2w+ DwV2w)
, (B1)
where all the coefficients are assumed to be frequency
dependent analogous to Eqs. (28)(29). The waveguide
dispersion k(ω) is now calculated from Eq. (A5) with
L = 2.
Fig. 5 shows that the transmission calculated from
Eq. (B1) is much closer to the exact numerical results.
In fact, the nominator of Eq. (B1) indicates that, indeed,
the resonance frequency is slightly shifted from the value
Page 15
15
ωα, and that this shift is proportional to V2w. Since V2wis
always much smaller than V1w(see Fig. 11), we can safely
neglect all the terms proportional to Vn
obtain the resonance frequency according to
2wwith n ≥ 2, and
ωres≈ ωα
?
1 +
V0,αVα,0V1wV2w
(V3
1w+ 3DwV1wV2w)να
?
. (B2)
Here, the values of all coefficients are calculated at the
frequency ωα, and ναis defined by Eq. (A1).
In addition to the shift of the resonance frequency,
a perfect transmission may occur at the frequencies for
which the denominator in Eq. (B1) vanishes:
V2w= V1w
?
cos(ks) ±?1 + cos2(ks) − 2cos(2ks)
1 − 2cos(2ks)
?
. (B3)
However, an analysis reveal that Eq. (B3) has solutions
only when V2w(ω) exceeds V1w(ω)/2, a condition that
appears to be impossible to realize in realistic photonic
crystals.
2. Effects of nonlocal coupling
Another possible reason for a shift of the resonance
frequency is a nonlocal coupling between the waveguide
cavities and the sidecoupled resonator α. Here, we dis
cuss this effect in the framework of the tightbinding ap
proximation for the waveguide dispersion (i.e., L = 1)
to distinguish it from the other type of nonlocal effects
discussed in the previous subsection. We assume that
Vj,α(ω) = Vα,j(ω) = 0 for all j ≥ 2, and take into account
that, for the symmetric structure shown in Fig. 1(b),
the coupling coefficients are: V−1,α(ω) ≡ V1,α(ω) and
Vα,−1(ω) ≡ Vα,1(ω).
Eqs. (24)–(25) in the form of Eqs. (1)–(2) with the de
tuning parameter defined as
Then, we obtain a solution of
σ(ω) = 2sin[ks] ·DαV1w+ V0,αVα,1+ Vα,0V1,α+ 2Vα,1V1,αcos(ks)
?Vα,0+ 2Vα,1cos(ks)??V0,α+ 2V1,αcos(ks)?
. (B4)
Here, all coefficients are assumed to be frequency
dependent, similar to Eqs. (28)(29).
the waveguide dispersion k(ω) is calculated again from
Eq. (A6).
Eq. (B4) suggests that in this case the resonance
frequency becomes slightly shifted from the value ωα,
and this shift is proportional to the values of V1,α and
Vα,1, which for our example (see Fig. 5) are equal to
Vα,1 = −0.0026 and V1,α = −0.0022. Assuming that
these coupling coefficients are always much smaller than
V0,α and Vα,0 (cf. Vα,0 = 0.096 and V0,α = 0.082), we
obtain for the resonance frequency
Furthermore,
ωres≈ ωα
?
1 −V0,αVα,1+ Vα,0V1,α
V1w
να
?
. (B5)
Here, the coefficients are calculated at the resonance fre
quency ωα, and ναis defined by Eq. (A1). For the exam
ple shown in Fig. 5, this frequency shift is much smaller
than that described by Eq. (B2) because in this case the
values of Vα,1 and V1,α are 3.3 times smaller than the
value of V2w.
Due to this longrange coupling, there appears a pos
sibility of perfect light transmission, as discussed in
Ref. [22], but only in the case when Vα,1(ω) exceeds
Vα,0(ω)/2 or V1,α(ω) exceeds V0,α(ω)/2. Again, such
a scenario appears to be impossible to realize in realistic
photonic crystal structures.
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