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

TL;DR: It is found 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.

Abstract: We analyze the resonant linear and nonlinear transmission through a photonic crystal waveguide sidecoupled 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.

Summary

Introduction

• Paste shows great promise to achieve the denser integration schemes that are required for the application of high resolution ultrasonic imaging.
• A design of experiments has been carried out to characterize and optimize a flip-chip bonding technology that utilizes a novel, magnetically aligned anisotropic conductive paste.
• This optimized process has the potential to implement more reliable and electrically conductive, fine pitch bonding for the production of high density ultrasound transducer arrays in needle devices.
• The resulting low penetration depth of the image requires that the transducer be operated close to the tissue of interest, which is commonly achieved by packaging the high-frequency transducers directly into surgical tools.
• ACPs are composed of fine electrically conductive particles uniformly dispersed in an adhesive matrix.

A. Bonding Experiments

• Initial experiments consisted of bonding two rigid PCB substrates, an example of which is shown in Fig. 1, forming a copper daisy chain electrical test structure with 200 μm track width and 200 μm pitch.
• ZTACHTM low temperature, thermally cured ACP (Sunray Scientific, USA) [4] was manually dispensed at room temperature onto a 50 m thick stencil before being forced through to the PCB surface.
• During curing of the epoxy, the beads align themselves along a uniform magnetic field applied perpendicularly to the PCB surface, forming thereby conductive tracks between the PCBs.
• The bonded substrates were placed in a Gallenkamp Plus II oven (Gallenkamp, UK) and left to cure at 150oC for 15 minutes 24 hours after curing the resistance of the daisy chain test structures was measured via two-probe measurement with a handheld multimeter (Fluke, USA).

B. Design of Experiments

• The use of a full factorial Design of Experiments (DoE) enables the identification of the sensitivity of the bonding process to alterations of various parameters and the effect of interactions between these parameters on the result.
• If the bonding force is too high the particles will fracture, leading to the disruption of the formation of a continuous electrical path.
• The main effects (1st order effects) and interaction matrix (2nd order effects) were calculated for both the average and standard deviation of the resistance as shown in Fig 2.
• Initial analysis of the results show that width of the bonding duration, then the stencil slit and bonding duration have a more dominant effect than the bonding force.
• These parameters act independently of one another.

ACKNOWLEDGMENT

• This work was supported by the UK Engineering and Physical Sciences Research Council through the Programme Grant entitled “Sonopill: minimally invasive gastrointestinal diagnosis and therapy”, grant no.
• The authors also acknowledge the financial contribution of the UK Medical Research Council through its Confidence in Concept programme (MRC-CiC3/036) and the support of the Edinburgh & Lothian Health Foundation.

Figures

FIG. 9. Color online Frequency dependence of the detuning coefficient Dn for the 2D photonic crystal described in Appendix A 1, for two types of resonators: Removing a single rod n=1.0; solid line leads to a localized mode at n=0.392, while replacing a single rod by a geometrically identical rod made of polymer n =2.56, dashed line leads to a localized mode at n=0.374.

FIG. 1. Color online Three types of the geometries of a straight photonic-crystal waveguide side coupled to a nonlinear optical resonator, A . Standard coupled-mode theory is based on the geometry a which does not account for discreteness-induced effects in the photonic-crystal waveguides. For instance, light transmission and bistability are qualitatively different for b on-site and c inter-site locations of the resonator along waveguide and this cannot be distinguished within the conceptual framework of structure of type a .

FIG. 6. Color online Liner transmission through a photoniccrystal waveguide created by removing every second rod in a row s =2a 1 side coupled to an inter-site resonator created by removing a single rod. The underlying 2D photonic crystal described in Appendix A 1. We compare exact numerical results solid line with the analytical results based on Eq. 39 dashed line .

FIG. 8. Color online a Linear transmission spectrum 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 location of the resonator whose resonance frequency lies close to the edge k= ± /s of the passing band; example B represents a suboptimal structure with an inter-site location of the resonator whose resonance frequency lies near the center of the passing band; example C represents a suboptimal but commonly used structure with an on-site location of the resonator whose resonance frequency lies near the center of the passing band. Closed circles in a indicate frequencies with T =80% that are used for achieving high-contrast bistability in b . Red circles in c indicate positions of the nonlinear polymer rods with =2.56. Other parameters of the 2D photonic crystal are described in Appendix A 1.

FIG. 7. Color online Linear transmission spectrum for a photonic-crystal waveguide created by removing every second rod in a row s =2a 1 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 described in Appendix A 1 and results for three different values of the resonator dielectric constant are shown.

FIG. 12. Color online Dispersion relation for a photoniccrystal waveguide created by removing every second rod in a row s =2a 1 in the 2D photonic crystal described in Appendix A 1. Numerical exact results solid line are calculated with the supercell plane-wave method 47 and the approximate results are obtained from Eq. A5 with L=1 dotted line and L=2 dashed line , using the coupling coefficients from Fig. 11.

FIG. 5. Color online Linear transmission through a photoniccrystal waveguide that is created by removing every second rod in a row s =2a 1 side coupled to a one-site resonator created by removing a single rod. The underlying 2D photonic crystal is described in Appendix A 1. We compare exact numerical results solid line with the analytical results based on Eq. 29 dotted-dashed line and Eq. B1 dashed line .

FIG. 3. Color online Dependencies of the transmission coefficient, T, the outgoing light intensity, Jout, and the resonator’s mode intensity, J , on the incoming light intensity, Jin, for several different values of 2 and positive product J .

FIG. 2. Color online Dependencies of the transmission coefficient, T, the outgoing light intensity, Jout, and the resonator’s mode intensity, J , on the incoming light intensity, Jin, for 2 =1 and negative product J .

FIG. 11. Color online Frequency dependence of the coupling coefficients Vn,m for the 2D photonic crystal described in Appendix A 1 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 R n=2a 1n whereas the on-site resonator is created by replacing a single rod at R =−2a 2 with a polymer rod n=2.56 of the same radius. The dispersion relation for such a waveguide is displayed in Fig. 12.

FIG. 10. Frequency dependence of the nonlinear feedback parameter n for the 2D photonic crystal described in Appendix A 1. The nonlinear resonator is created by replacing a single rod with a polymer rod of the same radius which supports at n=2.56 a localized mode with frequency n=0.374.

FIG. 4. Color online Dependencies of the threshold incoming light intensity and the corresponding transmission coefficients on 2 for the four critical points 1 – 4 indicated by circles in Fig. 3. Here, we assume that J 0.

Full text

All-optical switching, bistability, and slow-light transmission
in photonic crystal waveguide-resonator structures
Sergei F. Mingaleev,
1,2
Andrey E. Miroshnichenko,
3
Yuri S. Kivshar,
3
and Kurt Busch
1
1
Institut für Theoretische Festkörperphysik, Universität Karlsruhe, Karlsruhe 76128, Germany
2
Bogolyubov Institute for Theoretical Physics of the National Academy of Sciences of Ukraine, 03143 Kiev, Ukraine
3
Nonlinear 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
共Received 17 May 2006; published 5 October 2006
兲
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 associ-
ated 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 enhance-
ment 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 impor-
tance for the design of all-optical devices based on slow-light photonic crystal waveguides.
DOI: 10.1103/PhysRevE.74.046603 PACS number共s兲: 42.70.Qs, 42.65.Pc, 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 physics and designs of
such all-optical devices have attracted significant research
efforts during the last two decades, and most of these studies
utilize the concepts of optical switching and bistability 关1兴.
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 transmission 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 ob-
tain a system with resonant transmission in a narrow fre-
quency range, while in the second case, we obtain a system
with resonant reflection. 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 resonant systems of the second type, with side-coupled
resonators, can only be realized in higher-dimensional struc-
tures, and their functionalities are not yet completely under-
stood.
Our goal in this paper is to study in detail the second 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 共PhC兲关2兴. Due to a
periodic modulation of the refractive index of PhCs, such
structures may possess complete photonic 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 applications in photonic integrated
circuits. As an illustration, side-coupled waveguide-resonator
FIG. 1. 共Color online兲 Three types of the geometries of a
straight photonic-crystal waveguide side coupled to a nonlinear op-
tical resonator, A
␣
. Standard coupled-mode theory is based on the
geometry 共a兲 which does not account for discreteness-induced ef-
fects in the photonic-crystal waveguides. For instance, light trans-
mission and bistability are qualitatively different for 共b兲 on-site and
共c兲 inter-site locations of the resonator along waveguide and this
cannot be distinguished within the conceptual framework of struc-
ture of type 共a兲.
PHYSICAL REVIEW E 74, 046603 共2006兲
1539-3755/2006/74共4兲/046603共15兲 ©2006 The American Physical Society046603-1

systems created in PhCs through arrays of cavities are sche-
matically depicted in Fig. 1共b兲 and Fig. 1共c兲.
Practical applications of such PhC devices are becoming 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 in-
creased radius 关3–6兴; Smith et al. demonstrated coupling of a
three-line PhC waveguide with a large-area hexagonal reso-
nator 关7兴; Seassal et al. have investigated the mutual cou-
pling of a PhC waveguide with a rectangular microresonator
关8兴; Notomi et al. 关9兴 and Barclay et al. 关10兴 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: 共i兲 the
PhC waveguides may have very low group velocities and, as
a result, may significantly enhance the effective coupling be-
tween short pulse and resonators, and 共ii兲 photonic crystals
allow the creation of ultracompact high-Q resonators, which
are essential for the further miniaturization of all-optical
nanophotonic devices. Despite 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 共see Refs. 关11–21兴 and the
discussion in Sec. II兲.
However, an inspection of Figs. 1共a兲–1共c兲 reveals, that a
major difference between the ridge waveguide in 共a兲 and
PhC waveguides in 共b, c兲 is that a PhC waveguide is always
created by an array of coupled small-volume cavities and,
therefore, exhibits an inherently discrete nature. This sug-
gests that in these systems an additional coupling parameter
appears which relates the position of the
␣
resonator to the
waveguide cavities along the waveguide. As a matter of fact,
we may 共laterally兲 place the
␣
resonator at any point relative
to two successive waveguide cavities, thus creating a gener-
ally asymmetric device which 共in the nonlinear transmission
regime兲 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 devices
which we will analyze in a future presentation. In this paper,
however, we restrict our analysis to symmetric structures and
study the cases of either on-site coupling of the
␣
resonator
to the PhC waveguide, shown schematically in Figs. 1共b兲,or
inter-site coupling, as shown in Fig. 1共c兲.
To address these issues, we employ a recently developed
approach 关22–24兴 and describe the photonic-crystal devices
via effective discrete equations that are derived by means of
a Green’s function formalism 关25–29兴. This approach 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 fre-
quency
␻
␣
of the
␣
resonator with respect to the edges of the
waveguide passing band. If
␻
␣
lies deep inside the passing
band, all devices shown in Figs. 1共a兲–1共c兲 are qualitatively
similar, and can adequately be described by the conventional
coupled-mode theory. However, if the resonator’s frequency
␻
␣
moves closer to the edge of the passing band, standard
coupled-mode theory fails 关30兴. More importantly, we show
that in this latter case the properties of the devices shown in
Figs. 1共b兲 and Fig. 1共c兲 become qualitatively different: light
transmission vanishes at both edges of the passing band, for
the cases shown in Fig. 1共a兲 and Fig. 1共b兲, but for the case
shown in Fig. 1共c兲 it remains perfect at one of the edges.
Moreover, the resonance quality factor for the structure 共c兲
grows indefinitely as
␻
␣
approaches this latter band edge,
accordingly reducing the threshold intensity 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 summa-
rize and extend the results of standard coupled-mode theory
which accurately describes the system shown in Fig. 1共a兲.
Then, in Sec. III A we derive a system of effective discrete
equations 关25,26兴 and utilize a recently developed approach
for its analysis 关22,23兴. Specifically, in Sec. III B and Sec.
III C, respectively, we study the two geometries of the
waveguide-resonator coupled systems schematically de-
picted in Fig. 1共b兲 and Fig. 1共c兲. 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. Finally, in Sec. V we summarize and dis-
cuss 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 cavi-
ties and waveguides. The effects of nonlocal waveguide dis-
persion and nonlocal waveguide-resonator couplings are
briefly summarized in Appendix B.
II. COUPLED-MODE THEORY
In this section, we first summarize the results of standard
coupled-mode theory and other similar approaches devel-
oped for the analysis of continuous-waveguide structures
similar to those displayed in Fig. 1共a兲. Then, we extend these
results in order to obtain analytical formulas for the descrip-
tion of bistable nonlinear transmission in such devices.
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 关11,12兴 and is simi-
lar to that used by Fano 关13兴 and Anderson 关14兴 for describ-
ing the interaction between localized resonances and con-
tinuum states in the context of an effect which is generally
referred to as “Fano resonance.” For the analysis of the trans-
mission of photonic-crystal devices, this approach has been
employed first by Fan et al. 关15兴 and has been elaborated on
by Xu et al. 关16兴.
Throughout this paper we consider the propagation of a
monochromatic wave with the frequency
␻
lying inside the
waveguide passing band; we assume that the waveguide is a
single moded as well as that the resonator
␣
is nondegenerate
and losses can be neglected. In this case, the complex trans-
mission and reflection amplitudes, t共
␻
兲 and r共
␻
兲, can be
written in the form
MINGALEEV et al. PHYSICAL REVIEW E 74, 046603 共2006兲
046603-2

t共
␻
兲 =
␴
共
␻
兲
␴
共
␻
兲 − i
, r共
␻
兲 =
e
i
␸
r
共
␻
兲
␴
共
␻
兲 − i
, 共1兲
with a certain real-valued and frequency-dependent function
␴
共
␻
兲 and the reflection phase
␸
r
共
␻
兲. Accordingly, the abso-
lute values of the transmission coefficient T=兩t兩
2
and reflec-
tion coefficient R=兩r兩
2
are
T共
␻
兲 =
␴
2
共
␻
兲
␴
2
共
␻
兲 +1
and R共
␻
兲 =
1
␴
2
共
␻
兲 +1
, 共2兲
and it is easy to see that T + R = 1 for any
␴
共
␻
兲.
If the frequency
␻
␣
of the resonator
␣
lies inside the
waveguide passing band, Fano-like resonant scattering with
zero transmission at the resonance frequency
␻
res
, lying in
the vicinity of the resonator’s frequency,
␻
␣
, should be ob-
served 关13,22兴. This corresponds to the condition
␴
共
␻
res
兲=0
and, based on the terminology developed in Refs. 关17,18兴,
␴
共
␻
兲 may be interpreted as the detuning of the incident fre-
quency from resonance.
The results of standard coupled-mode theory analysis 共for
instance, see Ref. 关16兴兲 indicate that in the vicinity of a high-
quality 共or high-Q兲 resonance, the detuning function
␴
共
␻
兲
can be accurately described through the linear function
␴
共
␻
兲⯝
␻
res
−
␻
␥
, where
␥
=
␻
res
2Q
, 共3兲
which leads to a Lorentzian spectrum. Here, Q is the quality
factor of the resonance mode of the
␣
resonator. From the
Hamiltonian approach 关16兴, we find that the resonance fre-
quency
␻
res
almost coincides with the resonator frequency
␻
␣
共see, however, Appendix A in Ref. 关19兴 for a more accu-
rate estimate of
␻
res
兲, the reflection phase is
␸
r
=
␲
/2, and the
resonance width
␥
is determined by the overlap of the mode
profiles of waveguide and resonator:
␥
⬇
L
v
gr
␻
res
2
4W
k
W
␣
冉
冕
dr
ជ
␦
␧共r
ជ
兲E
ជ
k
*
共r
ជ
兲E
ជ
␣
共r
ជ
兲
冊
2
. 共4兲
Here, E
ជ
␣
共r
ជ
兲 is the normalized dimensionless electric field of
the resonator mode, E
ជ
k
共r
ជ
兲 is the corresponding field of the
waveguide mode at wave vector k=k共
␻
res
兲,
v
gr
=共d
␻
/dk兲 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
冕
wg section
dr
ជ
␧
wg
共r
ជ
兲兩E
ជ
k
共r
ជ
兲兩
2
= W
k
,
冕
all space
dr
ជ
␧
␣
共r
ជ
兲兩E
ជ
␣
共r
ជ
兲兩
2
= W
␣
. 共5兲
Furthermore, ␧
␣
共r
ជ
兲 and ␧
wg
共r
ជ
兲 are the dielectric functions
that describe the resonator
␣
and waveguide, respectively.
From Eqs. 共4兲 and 共5兲 it is easy to see that the resonance
width
␥
does not depend on the length L.
However, within the Hamiltonian approach, the function
␦
␧共r
ជ
兲 in Eq. 共4兲 remains undetermined. Generally, it is as-
sumed to be a difference between the total dielectric function
and the dielectric function ␧
0
共r
ជ
兲 “associated with the unper-
turbed Hamiltonian” 关16兴 which is an ill-defined quantity. A
different approach based on a perturbative solution of the
wave equation for the electric field 关20兴 sheds some light on
the resolution of this ambiguity and shows explicitly that
␧
0
共r
ជ
兲 can be taken as either ␧
wg
共r
ជ
兲 or ␧
␣
共r
ជ
兲.
B. Nonlinear transmission
If the resonator
␣
is made of a Kerr-nonlinear material,
increasing the intensity of the localized mode of the resona-
tor 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. 共1兲 and 共2兲, with the only difference that the
frequency detuning parameter
␴
共
␻
兲 should be replaced by
the generalized intensity-dependent frequency detuning pa-
rameter 关
␴
共
␻
兲− J
␣
兴. Here, J
␣
is a new dimensionless param-
eter which is, as we show below, proportional to the intensity
of the resonator’s localized mode. In particular, Eqs. 共2兲 take
the form
T =
关
␴
共
␻
兲 − J
␣
兴
2
关
␴
共
␻
兲 − J
␣
兴
2
+1
, R =
1
关
␴
共
␻
兲 − J
␣
兴
2
+1
. 共6兲
In order to find an explicit expression for J
␣
, we assume the
following: 共i兲 The dimensionless mode profiles E
ជ
␣
共r
ជ
兲 and
E
ជ
k
共r
ជ
兲 introduced in Eqs. 共4兲 and 共5兲 are normalized to their
maximal values 共as functions in real space兲, i.e., 兩E
ជ
␣
共r
ជ
兲兩
max
2
=兩E
ជ
k
共r
ជ
兲兩
max
2
=1; 共ii兲 the physical electric fields are described
by amplitudes, A
␣
and A
k
, multiplying the field profiles. Con-
sequently, the maximum intensity of the electric field in the
vicinity of the
␣
resonator, E
ជ
共r
ជ
兲⯝A
␣
E
ជ
␣
共r
ជ
兲, is equal to 兩A
␣
兩
2
;
共iii兲 the
␣
resonator is made of a Kerr-nonlinear material
with the nonlinear susceptibility
␹
␣
共3兲
and it covers the area
described by the function
␪
␣
共r
ជ
兲. This function is equal to
unity for all r
ជ
inside the cavities which form the resonator
structure and vanishes outside. In this case, J
␣
takes the
form
J
␣
=
12
␲
Q
␬
W
␣
2
␹
␣
共3兲
兩A
␣
兩
2
, 共7兲
where
␬
is the dimensionless and scale-invariant nonlinear
feedback parameter 共first introduced in similar form in Refs.
关17,18兴兲 which measures the geometric nonlinear feedback of
the system. It depends on the overlap of the resonator’s mode
profile with spatial distribution
␪
␣
共r
ជ
兲 of nonlinear material
according to
␬
=
3
W
␣
2
冉
c
␻
res
冊
d
冕
all space
dr
ជ
␪
␣
共r
ជ
兲␧
␣
共r
ជ
兲兩E
ជ
␣
共r
ជ
兲兩
4
, 共8兲
where d is the system dimensionality.
The dependence of J
␣
on the power of the incoming light
has already been studied analytically in Refs. 关18,20,21兴.
Here, we suggest a simpler form for this dependence
ALL-OPTICAL SWITCHING, BISTABILITY, AND… PHYSICAL REVIEW E 74, 046603 共2006兲
046603-3

J
in
= J
␣
兵关
␴
共
␻
兲 − J
␣
兴
2
+1其, 共9兲
where we have introduced the dimensionless intensity J
in
which is proportional to the experimentally measured power
of the incoming light
P
in
=
c
2
k共
␻
兲
2
␲␻
I
in
= P
0
J
in
. 共10兲
In this expression, we have abbreviated the incoming light
intensity as I
in
=兩A
k
兩
2
and introduced the characteristic power
P
0
of the waveguide defined as 共see Refs. 关17,18,20,21兴 for
derivation兲:
P
0
=
冉
c
␻
res
冊
d−1
冑
␧
␣
Q
2
␬
␣
␹
␣
共3兲
. 共11兲
Finally, the outgoing light power P
out
= P
0
J
out
can be deter-
mined through the dimensionless intensity of the outgoing
light J
out
=TJ
in
with the transmission coefficient T defined
by Eq. 共6兲.
It follows from Eqs. 共6兲 and 共9兲 that the nonlinear trans-
mission problem is completely determined by the value of
␴
共
␻
兲 and the sign of the product
␴
共
␻
兲· J
␣
. As is illustrated in
Fig. 2, for frequencies where 关
␴
共
␻
兲J
␣
兴⬍0, the transmission
coefficient T and the outgoing light intensity J
out
grow
monotonically with J
in
for all values of
␴
共
␻
兲.
The situation becomes more interesting for frequencies
lying on the other side of the resonance where 关
␴
共
␻
兲J
␣
兴
⬎0. In this case T and, therefore, J
out
become nonmonotonic
functions of J
in
, as is illustrated in Fig. 3. Moreover, for
␴
2
共
␻
兲⬎3 these functions become multivalued functions of
J
in
in the interval J
in
共3,4兲
艋J
in
艋J
in
共1,2兲
, where
J
in
共1,2兲
=
2
27
关
␴
3
+9
␴
+ 共
␴
2
−3兲
3/2
兴,
J
in
共3,4兲
=
2
27
关
␴
3
+9
␴
− 共
␴
2
−3兲
3/2
兴, 共12兲
which are also shown in Fig. 4. In this interval the nonlinear
light transmission becomes bistable: low- and high-
transmission regimes coexist at the same value of the incom-
ing light intensity J
in
, as can be seen in Fig. 3 for
␴
2
⬎3
共intermediate parts of the curves correspond to unstable
transmission兲. Therefore, by increasing an initially low inten-
sity J
in
we obtain a hysteresis where we jump from the point
共1兲 to 共2兲, and then upon decreasing J
in
, we jump from the
point 共3兲 to 共4兲. The transmission coefficients at these char-
acteristic points are
T
共1,3兲
=
1
2
␴
2
共1 ⫿
冑
1−3/
␴
2
兲 −2
,
T
共2,4兲
=
共1 ⫿ 2
冑
1−3/
␴
2
兲
2
5−3/
␴
2
⫿ 4
冑
1−3/
␴
2
, 共13兲
and they are depicted in Fig. 4. For completeness, we also
present the expressions for the resonator’s mode intensity at
these points
J
␣
共1,3兲
=
2
␴
3
⫿
1
3
冑
␴
2
−3,
FIG. 2. 共Color online兲 Dependencies of the transmission coeffi-
cient, T, the outgoing light intensity, J
out
, and the resonator’s mode
intensity, J
␣
, on the incoming light intensity, J
in
, for
␴
2
共
␻
兲= 1 and
negative product 关
␴
共
␻
兲J
␣
兴.
FIG. 3. 共Color online兲 Dependencies of the transmission coeffi-
cient, T, the outgoing light intensity, J
out
, and the resonator’s mode
intensity, J
␣
, on the incoming light intensity, J
in
, for several differ-
ent values of
␴
2
共
␻
兲 and positive product 关
␴
共
␻
兲J
␣
兴.
MINGALEEV et al. PHYSICAL REVIEW E 74, 046603 共2006兲
046603-4

J
␣
共2,4兲
=
2
␴
3
±
2
3
冑
␴
2
−3. 共14兲
From a practical point of view, these solutions have impor-
tant consequences. First, the bistability condition
␴
2
⬎3 cor-
responds to a linear transmission T ⬎ 3/4 . That is, the
bistable transmission becomes possible only for frequencies
where 关
␴
共
␻
兲J
␣
兴 is positive and linear transmission exceeds
75%. As demonstrated in Fig. 4 and Eq. 共12兲, when
␴
2
grows, all threshold intensities grow, too, starting with the
minimum threshold intensity J
in
共1,2,3,4兲
=8/3
1.5
⬇1.54 at
␴
2
=3.
For ideal nonlinear switching the coefficients T
共1兲
and T
共4兲
should be close to unity while T
共2兲
and T
共3兲
should vanish.
However, as can be seen from Fig. 4 and the asymptotic 共for
large
␴
2
兲 expressions
T
共1兲
⬇ 1−
9
␴
2
, T
共2兲
⬇ 1−
9
4
␴
2
,
T
共3兲
⬇ 1−
1
␴
2
, T
共4兲
⬇
1
4
␴
2
, 共15兲
of Eqs. 共13兲, these conditions cannot be satisfied simulta-
neously. In particular, the transmission coefficient T
共2兲
does
not vanish but approaches unity for large
␴
2
. Moreover, there
exists no condition under which T
共2兲
and T
共3兲
vanish simul-
taneously. Therefore, it is impossible to create ideal nonlin-
ear switches in these systems.
A reasonable compromise for realistic nonlinear switching
schemes of this type could be the usage of the frequency
with
␴
2
⯝5, for which the linear light transmission is close to
83%. For this case, the critical transmission coefficients
T
共2兲
⯝3.7% and T
共3兲
⯝7% are sufficiently small, while T
共1兲
⯝60% and T
共4兲
⯝74% are large enough for practical pur-
poses. The threshold intensities J
in
共1,2兲
⯝2.53 and J
in
共3,4兲
⯝2.11 differ about 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 dimension-
less threshold intensities are fixed around J
in
共i兲
⬃2.5 so that
the real threshold power of the incoming light, P
in
共i兲
= P
0
J
in
共i兲
,
can only be minimized by minimizing the characteristic
power, P
0
, of the system. An inspection of Eq. 共11兲 shows
that this can be facilitated by increasing the resonator non-
linear feedback parameter,
␬
␣
, the material nonlinearity,
␹
␣
共3兲
,
or the resonator quality factor, Q. For small-volume photonic
crystal resonators, it has been established that
␬
⬃0.2 共see
Refs. 关17,18兴兲, 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 material properties:
We should create the resonator
␣
from a material with the
largest possible value of
␹
␣
共3兲
. In high-index semiconductors,
nearly instantaneous Kerr nonlinearity reaches values of n
2
⬃1.5⫻ 10
−13
cm
2
/W 关31兴, where n
2
⬀
␹
共3兲
/n
0
and n
0
is the
linear refractive index of the material. Even such relatively
weak nonlinearity is already sufficient for many experimen-
tal observations of the bistability effect in the waveguide-
resonator systems 关9,10兴. However, using polymers with
nearly instantaneous Kerr nonlinearity of the order of n
2
⬎10
−11
cm
2
/W and, at the same time, sufficiently weak two-
photon absorption 关32兴, one could potentially decrease the
value of P
0
by at least two orders of magnitude. Polymers,
however, have a low refractive index which is insufficient for
creating a 共linear兲 photonic band gap required to obtain good
waveguiding and low losses. The solution to this could be
the embedding of such highly nonlinear but low-index mate-
rials into a host photonic crystal made of a high-index semi-
conductor. Optimized waveguiding designs for the basic
functional devices of this kind are available 关33–35兴 and re-
cent experimental progress 关36,37兴 may soon allow a realiza-
tion of corresponding linear and nonlinear devices.
The second approach is based on designing waveguide-
resonator structures with the largest possible quality factor,
Q. Potentially, one can increase Q indefinitely by mere in-
crease of the distance between the waveguide and the reso-
nator. However, this leads to a corresponding increase in the
size of the nonlinear photonic devices. A very attractive al-
ternative possibility for increasing Q is based on the adjust-
ment of the resonator geometry 关38兴.
In what follows, we suggest yet another possibility to dra-
matically increase Q through an optimal choice of the reso-
nator 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 limi-
tations. First, it gives analytical expression for the detuning
parameter
␴
共
␻
兲 only near the resonator frequency
␻
␣
. And
this immediately highlights the second limitation: standard
coupled-mode theory 关16–21兴 cannot analytically describe
resonant effects near waveguide band edges. However, nu-
merical studies 关30兴 have recently demonstrated that the ef-
fects of the waveguide dispersion become very important at
the band edges and may lead to non-Lorentzian transmission
spectra in coupled waveguide-resonator systems.
As a matter of fact, the question “what happens if the
resonator frequency
␻
␣
lies near the edge of the waveguide
passing band or even outside it?” may be of a great practical
importance due to two reasons. First, in realistic structures it
is not always possible to appropriately tune the frequency
FIG. 4. 共Color online兲 Dependencies of the threshold incoming
light intensity and the corresponding transmission coefficients on
␴
2
共
␻
兲 for the four critical points 共1兲–共4兲 indicated by circles in Fig.
3. Here, we assume that 关
␴
共
␻
兲J
␣
兴⬎ 0.
ALL-OPTICAL SWITCHING, BISTABILITY, AND… PHYSICAL REVIEW E 74, 046603 共2006兲
046603-5

Frequently asked questions

Q1. What are the contributions mentioned in the paper "All-optical switching, bistability, and slow-light transmission in photonic crystal waveguide-resonator structures" ?

In this paper, Mingaleev et al. presented a detailed analysis of PhC waveguides side coupled to Kerr-nonlinear resonators which may serve as a basic element of active photonic-crystal circuitry. 

Q2. What are the future works mentioned in the paper "All-optical switching, bistability, and slow-light transmission in photonic crystal waveguide-resonator structures" ?

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 authors study light propagation in the plane of periodicity, assuming that the rods have a radius r=0. As a matter of fact, their studies indicate that sufficiently accurate results can be obtained already for L 4a /s. In Fig. 12 the authors plot the dispersion relation for a 2D model photonic-crystal waveguide and compare it with exact numerical results calculated by the supercell plane-wave method 47. First, the authors explore the former possibility. 

Q3. What is the effect of Eq. 21 on the waveguide?

If the authors neglect nonlinear effects assuming that either the waveguide cavities are linear, n3 =0, or the light intensity in the waveguide remains sufficiently small , Eq. 21 reduces toDw An = j=1 Vjw An+j + 

Q4. How can the authors tune the resonant frequency of the polymer rod?

The resonant frequency of the polymer-rod resonator lies very close to the edge k= ± /s of the waveguide passing band, and can be tuned by changing the linear dielectric constant of the rod.046603-9 

Q5. What is the possibility of nonlinearity 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. 

Q6. What is the sensitivity of the resonator at the site n?

As the authors see, the nonlinear sensitivity of the resonator at the site n is a product of its nonlinear feedback parameter, n, the sensitivity to a change of the dielectric constant, n, and the Kerr susceptibility of material, n3 . 

Q7. What is the main argument for the nonlinearity enhancement of photonic crystals?

The authors believe that the basic concept of the geometric enhancement of nonlinear effects based on the discrete nature of photonic-crystal waveguides will be useful in the study of more complicated devices and circuits and, in particular, for various slow-light applications. 

Q8. What is the transmission coefficient for the inter-site coupled structure?

the authors 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 . 

Q9. What is the effect of vanishing transmission at the waveguide band edges?

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. 

Q10. What is the simplest explanation for the bistability of the PhC waveguide?

The authors have presented a detailed analysis of PhC waveguides side coupled to Kerr-nonlinear resonators which may serve as a basic element of active photonic-crystal circuitry. 

Q11. What is the ef-fect at the band edge?

the authors can expect that for inter-site coupled structure nonlinear ef-fects at the band edge k= ± /s should be sufficiently strong to allow bistable transmission and switching. 

Q12. What is the effect of vanishing transmission at the edge of the waveguide?

a nonlocality of the intercoupling between waveguide cavities as well as a nonlocality of cross coupling with the resonator lead to a small shift in the resonance frequency, res, but do not change the main result about the suppression of the detuning and transmission T at both edges of the waveguide passing band. 

Q13. What is the effect of the assumption of linear waveguide cavities on the resonator frequency?

Below the authors show that the assumption of linear waveguide cavities may be relaxed for frequencies near the resonator resonance frequency because then the amplitudes 

Q14. What is the dimensionless linear coupling between the nth and the mth cavity?

Furthermore,Vn,m = m Wn c 2 dr dr E n* r ̂n rm r Ĝ r + R n − R m,r E m r , 22is the dimensionless linear coupling between the nth and the mth cavity. 

Q15. How does the light intensity at the resonator at the spectral band edges differ?

In contrast, the light intensity at the resonator reaches its maximum value at ,A 2 4 V1wV0, 2 sin2 k s Iin vgrs w wV0, 2 Iin2Q V ,0 2Iin, 34which may significantly exceed the incoming light intensity Iin when the coupling V0, between the resonator and waveguide becomes small enough relative to the coupling V1w between the cavities in the waveguide. 

Q16. What is the main argument for the nonlinearity enhancement of the photonic crystal waveguide?

In addition, the authors would like to emphasize that the engineering of the geometry of photonic-crystal-based devices such as that presented in Fig. 1 c becomes extremely useful for developing concepts of all-optical switching in the slow-light regime of PhC waveguides which may have much wider applications in nanophotonics and is currently under active experimental research 40 .