# All-optical switching, bistability, and slow-light transmission in photonic crystal waveguide-resonator structures.

**ABSTRACT** We analyze the resonant linear and nonlinear transmission through a photonic crystal waveguide side-coupled to a Kerr-nonlinear photonic crystal resonator. First, we extend the standard coupled-mode 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 coupled-mode 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 geometry-driven 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 all-optical devices based on slow-light photonic crystal waveguides.

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- Journal of Modern Optics 06/2014; 61(11):904-909. · 1.17 Impact Factor
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**ABSTRACT:**We study the dynamics of the nonlinear Fano resonance in the 2D photonic-crystal waveguides with a side-coupled cavity. The photonic crystal is created by a lattice of dielectric rods in air. The cavity is assumed to be nonlinear, which can be achieved by replacing a normal dielectric rod by a nonlinear rod. The stationary regime of the light transmission through such a structure can be studied by using the Green's function approach. In particular, such a system shows a possibility of the nonlinear Fano resonance and bistable transmission. In order to study the full dynamical problem, here we generalize our approach by utilizing the time-dependent Green's function formalism. By using the proper boundary conditions, we are able to obtain fast and accurate results which then verified by the numerical FDTD simulations. We perform two types of simulations for the Gaussian pulses and continuous waves.Lasers and Electro-Optics, 2007 and the International Quantum Electronics Conference. CLEOE-IQEC 2007. European Conference on; 01/2007 - SourceAvailable from: David Anthony Powell[Show abstract] [Hide abstract]

**ABSTRACT:**We demonstrate an analogue of electromagnetically-induced transparency (EIT) in a magnetoelastic metamaterial system and experimentally realize nonlinear electromagnetic tuning of this EIT-like transmission. We study a single meta-molecule, consisting of two split-ring resonators (SRRs) and one closed-ring resonator, with one SRR free to rotate about the common axis of the structure in response to torques induced by the microwave pump. We observe EIT-like narrow-band resonant transmission in the carefully optimized device, which we characterize in a microwave waveguide, and verify that the resonance is due to the hybridized mode of all three resonators. We demonstrate nonlinear spectral narrowing and an increase of the group delay upon increasing the pump power and show the significant role of the intrinsic rotation of the freely rotatable SRR in this process.Applied Physics Letters 03/2014; 104(16). · 3.52 Impact Factor

Page 1

arXiv:physics/0605156v1 [physics.optics] 18 May 2006

All-optical switching, bistability, and slow-light transmission

in photonic crystal waveguide-resonator 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 Ultra-high 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 side-coupled to a Kerr-nonlinear photonic crystal resonator. Firstly, we extend the standard

coupled-mode 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 coupled-mode 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, geometry-driven 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 all-optical devices based on slow-light 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 all-optical 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 all-optical 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 side-coupling

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 two-port systems of the first type, with direct-

coupled resonator, can be realized in one-dimensional

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 side-coupled res-

onators, can only be realized in higher-dimensional 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 side-coupled to resonators as shown in Fig. 1.

Moreover, we assume that the waveguide and resonator

are created in two- or three-dimensional 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 ultra-compact

photonic crystal devices which are very promising for ap-

plications in photonic integrated circuits. As an illustra-

tion, side-coupled waveguide-resonator 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

two-dimensional 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 three-line PhC waveg-

uide with a large-area 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

all-optical bistability in direct-coupled PhC waveguide-

resonator systems.

Photonic-crystal based devices offer two major advan-

tages over corresponding ridge-waveguide 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 photonic-crystal waveguide side-coupled to a non-

linear optical resonator, Aα.

ory is based on the geometry (a) which does not account

for discreteness-induced effects in the photonic-crystal waveg-

uides. For instance, light transmission and bistability are

qualitatively different for (b) on-site and (c) inter-site loca-

tions of the resonator along waveguide and this cannot be

distinguished within the conceptual framework of structure

of type (a).

Standard coupled-mode 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 ultra-compact

high-Q resonators, which are essential for the further

miniaturization of all-optical 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 coupled-mode theory for

continuous systems (see Refs. [11–21] and the discussion

in Sec. II).

However, an inspection of Figs. 1(a-c) 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 small-volume 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 α-resonator to the waveguide cavities along the

waveguide. As a matter of fact, we may (laterally) place

the α-resonator 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 high-intensity light in one direction only. This

is an intriguing peculiarity of photonic-crystal 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 on-site coupling of

the α-resonator to the PhC waveguide, shown schemat-

ically in Figs. 1(b), or inter-site coupling, as shown in

Fig. 1(c).

To address these issues, we employ a recently devel-

oped approach [22–24] and describe the photonic-crystal

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 α-resonator with

respect to the edges of the waveguide passing band. If

ωαlies deep inside the passing band, all devices shown in

Figs. 1(a-c) are qualitatively similar, and can adequately

be described by the conventional coupled-mode theory.

However, if the resonator’s frequency ωα moves closer

to the edge of the passing band, standard coupled-mode

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 all-optical switching in

the slow-light regime.

The paper is organized as follows. In Sec. II we sum-

marize and extend the results of standard coupled-mode

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 waveguide-resonator 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 two-dimensional

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 waveguide-resonator couplings are briefly

summarized in Appendix B.

II. COUPLED-MODE THEORY

In this Section, we first summarize the results of stan-

dard coupled-mode theory and other similar approaches

developed for the analysis of continuous-waveguide 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 waveguide-resonator systems

is usually studied in the linear limit using a coupled-mode

theory based on a Hamiltonian approach. This approach

has been pioneered by Haus and co-workers [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 photonic-crystal 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 single-moded as well as that the resonator α is

non-degenerate 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 real-valued and frequency-dependent func-

tion σ(ω) and the reflection phase ϕr(ω). Accordingly,

the absolute values of the transmission coefficient T =

|t|2and reflection coefficient R = |r|2are

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, Fano-like 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 coupled-mode theory analysis

(for instance, see Ref. [16]) indicate that in the vicinity of

a high-quality (or high-Q) 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 α-resonator.

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 Kerr-nonlinear 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 α-resonator,

?E(? r) ≃ Aα?Eα(? r), is equal to |Aα|2; (iii) The α-resonator

is made of a Kerr-nonlinear 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)

α|Aα|2, (7)

where κ is the dimensionless and scale-invariant 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 = |Ak|2and 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 high-contrast 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 small-volume 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 waveguide-resonator 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 low-index materials into a host photonic crystal

made of a high-index 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 photonic-crystal waveguide.

C. Limitations of the coupled-mode theory

Standard coupled-mode 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 coupled-mode 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 non-Lorentzian transmission spectra in cou-

pled waveguide-resonator 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 waveguide-resonator 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

continuous-waveguide 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 on-site 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 photonic-crystal 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 n-th 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 third-order suscep-

tibility of the n-th cavity.

Similar to Sec. II, we describe the electric field of the

n-th 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)

n|An|2An, (21)

where Dn(ω) = 1−Vn,n(ω) is the dimensionless frequency

detuning from the resonance frequency, ωn, of the n-th

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 n-th and

the m-th 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 scale-invariant nonlinear feed-

back parameter which should be compared with the

analogous parameter (8) introduced in the conventional

coupled-mode 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

higher-order 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)

α|Aα|2Aα, (24)

where we assume that all cavities of the photonic-crystal

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 side-coupled 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.On-site 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 α-resonator (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) side-coupled to a one-site 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) (dot-dashed line) and Eq. (B1) (dashed line).

the detuning parameter, Eq. (26), for a linear α-resonator

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 coupled-mode

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 all-optical switching devices based on slow-

light photonic crystal waveguides.

We support this conclusion by another observation.

First, the light intensity at the 0-th cavity, |A0|2=

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 α-resonator

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 α-resonator

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 α-resonator 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 non-local 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 inter-coupling between waveguide cavities

as well as a nonlocality of cross-coupling 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, non-local 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 non-local couplings between the cavities and

resonator and novel effects originating solely from the

non-locality may be expected when the non-local 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 local-coupling 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

α|Aα|2, (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 α-resonatorwith 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. Inter-site resonator

In the system where the α-resonator 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) side-coupled to an inter-site 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 real-valued function (see also the discussion following

Eq. (1) above).

In the case of the linear α-resonator (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 high-quality 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 α-resonator 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 on-site coupled structure, Eq. (42) does not

vanish at the edge of the transmission band k = ±π/s.

Therefore, we can expect that for inter-site 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 α-resonator

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 |Aα|2at the α-resonator do not vanish at the

band edge k = ±π/s since the quality factor Q at this

band edge grows to infinity for the inter-site structure of

Fig. 1(c). Therefore, this structure may be utilized for

realizing efficient all-optical switching devices based on

slow-light 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 photonic-crystal 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 two-dimensional 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) side-coupled to a single on-site (a)

or inter-site (b) polymer-rod 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 polymer-rod 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 on-site and inter-site positions

of the side-coupled resonator for three different values of

resonator dielectric constant εα. We notice that in the

case of the on-site 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 on-site system cannot

exhibit bistability.

On the other hand, bistability may be realized for the

inter-site position of the side-coupled 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 inter-site structure at k(ω) close to

±π/s leads to very low bistability thresholds as compared

to the cases of on-site 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

side-coupled waveguide-resonator 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 inter-site loca-

tion of the α-resonator whose resonance frequency lies close

to the edge k = ±π/s of the passing band; example B rep-

resents a sub-optimal structure with an inter-site location of

the α-resonator whose resonance frequency lies near the cen-

ter of the passing band; example C represents a sub-optimal

but commonly used structure with an on-site location of the

α-resonator 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 high-contrast

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 inter-site cou-

pled example B still exhibits a much smaller bistability

threshold than the on-site coupled system with the same

waveguide design in example C. This is caused by (usu-

ally) much smaller waveguide-resonatorcoupling and, ac-

cordingly, much larger Q in the inter-site structures as

compared to the on-site structures.

Summarizing, the inter-site structure of the resonant

waveguide-resonator 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 continuous-waveguide and on-site structures

depicted in Figs. 1(a,b), provides us with a possibility to

create low-threshold all-optical switching devices based

on slow-light photonic crystal waveguides.

V.CONCLUSIONS

We have presented a detailed analysis of PhC waveg-

uides side-coupled to Kerr nonlinear resonators which

may serve as a basic element of active photonic-crystal

circuitry. First, we have extended the familiar approach

based on standard coupled-mode 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 semi-analytical 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 semi-quantitative description

of photonic-crystal devices that involve a discrete set of

small-volume cavities. We have shown that this novel

approach allows to adequately describe light transmis-

sion in the waveguide-resonator structures near the band

edges. Specifically, we have demonstrated that while the

transmission coefficient vanishes at both spectral edges

for the on-site coupled structure (see Fig. 1(b)), light

transmission remains perfect at one band edge for the

inter-site 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 photonic-crystal devices in integrated

Page 12

12

photonic circuits.

In addition, we would like to emphasize that the engi-

neering of the geometry of photonic-crystal based devices

such as that presented in Fig. 1(c) becomes extremely

useful for developing novel concepts of all-optical switch-

ing in the slow-light 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 photonic-crystal waveguides will be useful in the

study of more complicated devices and circuits and, in

particular, for various slow-light applications.

stance, this concept may be applied to the transmission

of a side-coupled resonator placed between two partially

reflecting elements embedded into the photonic-crystal

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 three-port channel-drop 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 PECS-VI 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 two-dimensional

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

two-dimensional 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

[7-16] 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

(E-polarized 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 on-site side-coupled waveguide-resonator

system shown in Fig. 1-b. 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 on-site 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 polymer-rod

resonator, we find νn≈ 0.295.

When the n-th 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 |An|2Anwith the new important coefficient —

κn(ω) ≈ −0.111 is constant and can rewrite Eq. (21)

according to

|An|2=

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 |An|2as

?

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)

n|An|2?

. (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 photonic-crystal 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 coupled-resonator 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 photonic-crystal 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 low-frequency 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 super-cell plane-wave 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 Floquet-Bloch 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 photonic-crystal waveguide and

compare it with exact numerical results calculated by the

super-cell plane-wave method [47]. For this case, even

the simplest tight-binding approximation (i.e., at L = 1)

gives quite satisfactory results.

In the tight-binding 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 half-bandwidth 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 LONG-RANGE

INTERACTIONS

1. Effects of nonlocal dispersion

As follows from the results of Sec. IIIB above, the

local-coupling 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 photonic-crystal

devices.

In Fig. 5, we provide a comparison of T(ω) calculated

from Eq. (29) in the local-coupling 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 local-coupling

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 long-range

coupling between cavities inside the waveguide and (ii) a

long-range 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 side-coupled resonator α. Here, we dis-

cuss this effect in the framework of the tight-binding ap-

proximation for the waveguide dispersion (i.e., L = 1)

to distinguish it from the other type of nonlocal effects

discussed in the previous sub-section. 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 long-range 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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