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JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 6, MARCH 15, 2009 799
Orthogonal Hybrid Waveguides: An Approach to
Low Crosstalk and Wideband Photonic Crystal
Intersections Design
Kiazand Fasihi and Shahram Mohammadnejad
Abstract—A low crosstalk and wideband photonic crystal (PC)
waveguide intersection design based on two orthogonal hybrid
waveguides in a crossbar configuration is proposed. The finite-dif-
ference time-domain (FDTD) and coupled-mode theory (CMT)
methods are used to simulate the hybrid waveguides of square
lattice. The bandwidth (BW) and crosstalk of the intersection are
investigated for various radii of the coupled cavities. It is shown
that simultaneous crossing of the lightwave signals through the
intersection with negligible interference is possible. The transmis-
sion of a 200-fs pulse at 1550 nm is simulated by using the FDTD
method, and the transmitted pulse shows negligible crosstalk and
very little distortion.
Index Terms—Crosstalk, coupled-mode theory (CMT), finite-
difference time-domain (FDTD), orthogonal hybrid waveguides,
photonic crystal (PC) intersections.
I. INTRODUCTION
RECENTLY, photonic crystals (PCs) have attracted great
interests due to their potential ability of controlling light
propagation with the existence of photonic bandgap (PBG), and
the possibilities of implementing compact optical integrated cir-
cuits [1]–[6]. Waveguide intersections with low crosstalk and
high BW are the key element for implementation of integrated
photonic circuits. In 1998, Johnson et al. proposed a scheme
to eliminate crosstalk for a waveguide intersection based on a
two-dimensional (2-D) PC of square lattice by using a single de-
fect with doubly degenerate modes [7]. They also presented gen-
eral criteria for designing such waveguide intersections based on
symmetry consideration. Lan and Ishikawa presented another
mechanism where the defect coupling is highly dependent on
the field patterns in the defects and the alignment of the de-
fects (i.e., the coupling angle) [8]. They asserted that their de-
sign leads to a 10-nm wide region at the central wavelength of
1310 nm with crosstalk as low as to dB, while in [7]
the width of the transmission band with comparable crosstalk
is only 7.8 nm. In the aforementioned design, the central wave-
length value of the low crosstalk transmission band is related to
the air-holes radii of PC structure and therefore, adjusting the
Manuscript received March 15, 2008; revised July 21, 2008. Current version
published April 17, 2009.
The authors are with the Department of Electrical Engineering, Iran Univer-
sity of Science and Technology, Nanoptronics Research Center, Tehran, Iran
(e-mail: kfasihi@iust.ac.ir; shahramm@iust.ac.ir).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/JLT.2008.929422
wavelength domain of transmission band is a challenge. Fur-
thermore, Liu et al. proposed another waveguide intersection for
lightwaves with no crosstalk and excellent transmission which
was based on nonidentical PC coupled resonator optical wave-
guide (CROW), without transmission band overlap [9].
Zhaofeng et al. proposed a different approach that utilizes a
vanishing overlap of the propagation modes in the waveguides
created by line defects which support dipole-like defect modes
[10]. They claimed that in their design, over a BW of 30 nm
with the central wavelength at 1300 nm, transmission efficiency
above 90% with crosstalk below dB can be obtained. It
is obvious that in that proposal—and also in [9], simultaneous
propagation of lightwaves with equal frequencies through the
intersection is impossible and due to using of taper structure to
solve the mode mismatch problem, total length of the intersec-
tion is increased. In our solution, an approach to design of low
crosstalk and wideband PC waveguide intersections based on
two orthogonal hybrid waveguides in a crossbar configuration,
is proposed. The paper is organized as follows: In Section II,
the hybrid waveguides are introduced and analyzed using
coupled-mode theory (CMT) method. Fundamental approach
to low crosstalk and wideband intersections design is proposed
in Section III. In Section VI, the orthogonal hybrid waveguide
intersections are simulated using the FDTD method. Also,
simultaneous crossing of lightwave signals and transmission
of ultrashort pulses through the proposed intersection are
investigated.
II. THEORETICAL MODEL FOR HYBRID WAVEGUIDES:
CMT APPROXIMATION
A. Hybrid Waveguides
The PC-based coupled cavity waveguides (CCW) are formed
by placing a series of high-Q optical cavities close together. In
this case, due to weak coupling of the cavities, light will be
transferred from one cavity to its neighbors and a waveguide
can be created [11]. By combining the CCWs and the conven-
tional line defect waveguides a new waveguide can be created,
which is referred to as hybrid waveguide. Fig. 1 shows the struc-
tures of a hybrid waveguide and an orthogonal hybrid wave-
guide intersection which are implemented in a square lattice PC.
Usually, for various applications such as ultrashort pulse trans-
mission, there is a need to have a large BW and a quasi-flat trans-
mission spectrum within the transmission band. If the confine-
ment of the coupled cavities is increased, the continuous trans-
mission band will be converted to a series of discrete bands,
0733-8724/$25.00 © 2009 IEEE
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800 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 6, MARCH 15, 2009
Fig. 1. Schematic structures of square lattice PC components studied in this
paper. (a) A hybrid waveguide (
a; r
and
r
are the lattice constant, the radius
of the dielectric rods, and the radius of the coupled cavities, respectively) and
(b) an orthogonal hybrid waveguide intersection.
which are useful for implementation of some optical devices,
such as filters [12]. Generally, there are two types of PC lat-
tice structures, air-hole-type and rod-type. Despite easier fabri-
cation of PC waveguide based on air-hole-type structures than
rod-type waveguides, there are limitations on frequency BW of
the single mode region and the group velocity [13]. Moreover
in PC waveguides based on rod-type structure the wide BW and
large group velocity can be achieved, and recently such wave-
guides have been used for fabrication of photonic devices [14].
B. Modeling of Hybrid Waveguides by CMT Method
Here, we consider the CCWs that are formed by periodically
introducing defects along one direction. The corresponding
model which contains identical defects is schematically
shown in Fig. 2. According to CMT, the equations describing
the energy amplitude of th PC defect, , can
be written as [15]
(1a)
(1b)
where is the resonant frequency of the PC defects and
denotes the external decay rate of into one of its two adja-
cent defects. Here, the internal loss of energy in PC defects is
ignored. As shown in Fig. 2, and are the electromag-
netic waves entering into the th PC defect from its left and right
sides, respectively. Also, and represent the electromag-
netic waves emerging from the left and right sides of the th PC
defect, respectively. The coupling between two PC cavities de-
pends on the leakage rate of energy amplitude into the adjacent
cavity which defines the quality factor of the cavity, and
the phase-shift of the electromagnetic wave traveling between
two adjacent cavities . It can be shown that in a straight CCW
which contains PC cavities the transmission spec-
trum is given as [15]
(2)
where
(3)
In the above equation, and are the frequency of inci-
dent input, the resonant frequency and the quality factor of PC
cavities, respectively. In (2), A is a series function of
that satisfies and
. As shown in (2), apart from the
transmission spectrum of a CCW depends on three parameters
and . The and can be extracted from a simple nu-
merical simulation on a PC molecule, composed of two coupled
cavities. For , i.e., a PC molecule, the transmission spec-
trum is given as
(4)
where
(5)
In the above equations, is the minimum in transmission
band of a PC molecule. The peaks of the (4), which are equal
to unity, appear at and .
Hence, using (5) and the simulated transmission spectrum of
one PC molecule, and can be extracted. It must be noted
that the analytical results of (2) can be extended to CCWs of any
dimensions [15]. Now, we consider the hybrid waveguides that
contain identical cavities in 2-D-PCs and generalize CMT an-
alytical method to obtain the transmission spectrum. According
to (2), it can be seen that for a given the transmission spec-
trum curve has number of extremums and the min-
imum in transmission spectrum is independent of and
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FASIHI AND MOHAMMADNEJAD: ORTHOGONAL HYBRID WAVEGUIDES 801
Fig. 2. Schematic diagram for CCWs based on CMT [15].
TABLE I
VALUES OF THE MINIMUM IN THE HW3 TRANSMISSION SPECTRUM FOR
VARIOUS RADII OF THE COUPLED CAVITIES
. Therefore, we can obtain as a function of the radius of the
coupled cavities as follows.
•The relationship between and can be calculated by
repeating a numerical simulation, such as FDTD method,
for different values of .
Here we consider a hybrid waveguide which contains three
coupled cavities, [see Fig. 1(a)], in the 2-D-PC of
square lattice composed of dielectric rods in air. Now, we
have chosen to name this hybrid waveguide HW3 and ex-
tend this naming to other hybrid waveguides.The rods have
refractive index and radius where
is the lattice constant. By normalizing every parameter
with respect to the lattice constant , we can scale the wave-
guide structure to any length scale simply by scaling .
The radius of the coupled cavities are varied from
to . The grid size parameter in the FDTD simula-
tion is set to and the excitations are electromag-
netic pulses with Gaussian envelope, which are applied to
the input port from the left side. All the FDTD simulations
below are for TM (i.e., with electric field parallel to the rod
axis) polarization. The field amplitude is monitored at suit-
able location at the right side of the HW3. Table I shows
the relationship between and for the HW3 which
are obtained from the FDTD simulations.
•The relationship between and for the HW3 can be
calculated from (2).
Fig. 3 shows this relationship over one-half period of (2).
Therefore, the relationship between and of the HW3 can
be demonstrated in Fig. 4. In order to compare the results of
CMT and FDTD methods, we consider a HW2 under the same
conditions as mentioned previously and utilize the FDTD sim-
ulation results to compute and . The radius of the cou-
pled cavities are set to . The transmission spec-
trum of HW2 computed by the FDTD is shown in Fig. 5. Ac-
cording to this figure, the parameters and are equal
Fig. 3. The relationship between
T
and
'
in the HW3.
Fig. 4. The phase-shift between two adjacent cavities as a function of radius of
the coupled cavities in the HW3.
to , 130.3, and , respectively. Hence, the
CMT transmission spectrum can be calculated from (4) (see
Fig. 5). It is observed that the transmission spectrum calculated
by CMT is in good agreement with that simulated by FDTD.
As another example, we take a HW3 under the same condition
as mentioned previously, with which corresponds to
. The transmission spectra of the above HW3 sim-
ulated by FDTD and CMT are shown in Fig. 6 for comparison.
Although there is a difference in the minimum transmission
spectrum between the first and second peaks, it is observed that
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802 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 6, MARCH 15, 2009
Fig. 5. The simulation results of transmission spectrum of the HW2 obtained
by the FDTD method (the dotted curve) and the CMT method (the solid curve).
Fig. 6. The simulation results of transmission spectrum of the HW3 obtained
by (a) the FDTD method (the dotted curve) and (b) the CMT method (the solid
curve).
the spectrum calculated by analytical method is nearly in good
agreement with that simulated by the numerical simulation.
III. FUNDAMENTAL APPROAC H TOLOW CROSSTALK AND
WIDEBAND INTERSECTIONS DESIGN IN SQUARE LATTICE PCS
In this section, we first consider the basic criteria proposed in
[7] to eliminate crosstalk in waveguide intersections, and then
generalize it for two orthogonal hybrid waveguides in a crossbar
configuration. Johnson et al. presented general criteria based on
symmetry for the intersection with high throughput and very low
crosstalk. The criteria are as shown here.
•Each waveguide must have a mirror symmetry plane
through its axis and perpendicular to the other waveguide
and have a single guided mode in the frequency range of
interest. This mode will be either even or odd with respect
to the mirror plane.
•The center of the intersection must be occupied by a res-
onant cavity that is symmetric with respect to the mirror
planes of both waveguides.
•Two resonant modes must exist in the cavity, each of which
is even with respect to one waveguide’s mirror plane and
odd with respect to the other. These modes should be the
only resonant modes in the frequency range of interest.
The fundamental idea is to consider coupling of the four
branches of the intersection in terms of a single resonant cavity
at the center. If the above conditions are satisfied, then each
resonant state of the cavity will couple to modes in just one
waveguide and be orthogonal to modes in the other waveguide.
So, assuming that the branches only couple to one another
through the resonant cavity, crosstalk will be eliminated. But
the FDTD calculated transmission spectra of the intersections,
which are based on a single cavity at the center of the intersec-
tion of two line-defect waveguides, show that under the best
circumstances for several structural variations, only a narrow
low crosstalk wavelength region can be obtained [7].
We can generalize these criteria to the structure of Fig. 1(b),
which is based on two orthogonal HW3s in a crossbar config-
uration. In this structure, by photon hopping through the cou-
pled cavities, light can propagate from one branch to the other
branches. Since the structure of the proposed intersection satis-
fies the criteria of the [7], we can find wavelength region with
very low crosstalk. In this structure the BW is determined by in-
teraction of the individual resonant frequencies of the coupled
cavities. It will be shown that in the case of
where is an integer including zero, a quasi-flat impurity band
can be achieved. Consequently, a wideband and low crosstalk
intersection with high transmission efficiency can be obtained.
IV. SIMULATION AND RESULTS
A. Simulation of the Orthogonal Hybrid Waveguide
Intersections by FDTD Method
Without losing generality, once again we consider a 2-D
square lattice of infinitely long dielectric rods in air. The rods
have refractive index and radius .
These parameters lead to a PBG for the TM mode from
to here is the free-space wave-
length. To determine the PBG regions of the PC structure,
the MIT Photonic-Bands package [16] is used. To evaluate
the performance of the proposed device, the FDTD method is
used for simulation, under the same conditions as mentioned
previously. The excitations are electromagnetic pulses with
Gaussian envelope, which are launched to the input port from
the left side. The field amplitudes are monitored at suitable
locations around the intersection in horizontal and perpendic-
ular waveguides. Fig. 7(a) and (b) shows the transmission and
crosstalk characteristics of the orthogonal HW3 intersection,
where the radius of the coupled cavities are set to
and , respectively. As can be seen from Fig. 7(a)
and (b), there exists around and regions in
which the transmission is over 50%. Also, it must be noted that
the transmission properties of the proposed intersection are the
same as transmission properties of the corresponding hybrid
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FASIHI AND MOHAMMADNEJAD: ORTHOGONAL HYBRID WAVEGUIDES 803
Fig. 7. The transmission and crosstalk characteristics of the orthogonal HW3
intersection when the radius of the coupled cavities are set to (a)
r
=0
:
28
a
and (b)
r
=0
:
32
a
.
waveguide. Furthermore, by varying the radius of the coupled
cavities of the hybrid waveguides, a wide frequency domain of
transmission band will be obtained which proves the flexibility
of the proposed design. Table II, shows the transmission region,
dB BW and the crosstalk of the proposed intersection for
different values of the coupled cavities radii. Assuming the
lattice constant m, considering that in this case the
center wavelength of transmission band is equal to 1550 nm
when , the intersection BWs for different radius
of the coupled cavities at working wavelength of 1550 nm
can be obtained and is shown in the column 4 of Table II. By
comparing the results of Fig. 4 and Table II, it can be seen that
the optimum values of BW and crosstalk are obtained when
. In this case, the transmission spectra of the
intersection is quasi-flat (see Fig. 8).
TABLE II
VALUES OF THE TRANSMISSION REGION,
0
3
DB BW AND CROSSTALK IN
ORTHOGONAL HW3 INTERSECTION FOR VARIOUS RADII OF
THE COUPLED CAVITIES
Fig. 8. The transmission behavior of the orthogonal HW3 intersection when
'
(
k
+1
=
2)
.
B. Simultaneous Crossing of Lightwave Signals and
Transmission of Ultrashort Pulses Through the Orthogonal
Hybrid Waveguide Intersections
In the implementation of PC-based integrated circuits, such
as those which used in wavelength division multiplexing
(WDM) systems, it is necessary to have intersections in which
simultaneous crossing of lightwaves is possible. In the orthog-
onal hybrid waveguide intersections, lightwave signals can
cross through the intersection simultaneously because each res-
onant state of the intersection will couple to modes in just one
waveguide and be orthogonal to modes in the other waveguide.
We consider the structure shown in Fig. 1(b) and verify this idea
by using the FDTD technique. In this simulation, the radius
of the coupled cavities of the orthogonal HW3 are chosen to
be where m. During simulation,
two input pulses with Gaussian envelope are applied to input
ports from the top and the left sides. The monitors are placed
at right and bottom output ports at suitable locations. The
intensities of 500-fs pulses are adjusted to unity and 0.5, while
their central wavelengths are set at 1550 nm and the phase
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804 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 27, NO. 6, MARCH 15, 2009
Fig. 9. The simultaneous crossing of two lightwave signals through the orthog-
onal HW3 intersection with
r
=0
:
3075
a
and
a
=0
:
55
m. (a) Calculated
transmission spectra. (b) Calculated field distribution. The intensities of 500-fs
pulses are adjusted to unity and 0.5, while their central wavelengths are set at
1550 nm and the phase difference between them is 180 .
difference between them is 180 . Fig. 9 shows the transmission
behavior of simultaneous crossing of lightwave signals through
the orthogonal HW3 intersection. It can be seen that the input
pulses are transmitted through the intersection with negligible
interference effect. In a separate assessment, we again consider
the structure shown in Fig. 1(b) with where
m, and investigate the transmission property of the
intersection for ultrashort pulses by using the FDTD method.
Fig. 10 shows the transmission behavior of a 200-fs pulse
whose central wavelength is 1550 nm. We can see that not only
the crosstalk is negligible, but also the distortion of the pulse
shape is very small.
Fig. 10. The transmission behavior of a 200-fs pulse whose central wavelength
is 1550 nm through the orthogonal HW3 intersection with
r
=0
:
3075
a
and
a
=0
:
55
m.
V. CONCLUSION
In this paper, a low crosstalk and wideband waveguide in-
tersection design based on two orthogonal hybrid waveguides
in crossbar configuration was proposed and modeled by using
the FDTD and CMT methods. It has been demonstrated that the
theoretical results derived by CMT for simulation of the hybrid
waveguides are in good agreement with FDTD simulation re-
sults. Also, it has been shown that when the phase-shift of the
electromagnetic waves traveling between two adjacent PC cou-
pled cavities is approximately equal to , i.e., quasi-flat
condition, optimum performance results for the intersection can
be achieved. In addition, it has been clearly proved that simul-
taneous crossing of ultrashort pulses through the intersection is
possible with negligible interference. The proposed solution can
be easily generalized to other 2-D square as well as 3-D cubic
PCs.
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Kiazand Fasihi was born in 1977. He received the
B.Sc. degree from Razi University and the M.Sc. de-
gree from Iran University of Science and Technology
(IUST).
He is currently working toward the Ph.D. degree at
IUST where his research interests include photonic
crystal devices and optical integrated circuits.
Shahram Mohammadnejad received the B.Sc. de-
gree in electrical engineering from the University of
Houston, Houston, TX, in 1981 and the M.Sc. and
Ph.D. degrees in semiconductor material growth and
lasers from Shizuoka University, Shizuoka, Japan, in
1990 and 1993, respectively.
He invented the PdSrS laser for the first time in
1992. He has published more than 80 scientific papers
and books. His research interests include semicon-
ductor material growth, quantum electronics, semi-
conductor devices, optoelectronics, and lasers.
Dr. Mohammadnejad is a scientific committee member of the Iranian Con-
ference of Electrical Engineering (ICEE), a member of Institute of Engineering
and Technology (IET), and a member of IET- CEng.
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