Ultrasmall Photonic Crystal Coupler for Wave Division
Multiplexing Networks
R Chatta1, C Ben Neïla2 and M Zghal2
1.
Institut Supérieur des Etudes Technologiques en Communications (Iset’Com), Cité
Technologique des Communications 2083 Ariana, Tunisia. Tel : +21671857000, Fax :
+21671856829.
2.
Ecole Supérieure des Communications de Tunis (Sup’Com), Cité Technologique
des Communications 2083 Ariana, Tunisia.
rihab@isetcom.rnu.tn
Abstract. Several designs of photonic crystal (PC) directional coupler, composed by two
nearby linear defect waveguides in two-dimensional (2-D) photonic crystal of GaAs rods, are
numerically analysed. Shown that the coupling lengths are on a wavelength scale, we study the
coupling efficiency by fitting the geometry of the interaction region lying between the two PC
defects. We show that reducing the radius of rods separating both waveguides implies a higher
coupling coefficient for a 2-D PC directional coupler. By reducing the coupling length, we
realize an efficient wavelength selective coupler that acts as ultrasmall channel interleaver and
as a micrometer sized wavelength demultiplexer
1. Introduction
Two-Dimensional (2-D) photonic crystal (PC) waveguides are considered as attractive candidates in
the implementation of compact optical devices guaranteeing the miniaturization of integrated optical
circuits [1]. They are realized by inserting linear defects in a 2-D perfect crystal. Proposed in 1987, PC
are dielectric structures with a refractive index periodicity in one, two or three dimensions, which
prohibit the propagation of electromagnetic waves in specific frequencies ranges, known as photonic
band gaps (PBGs). In particular, two-dimensional photonic crystals have been studied extensively
since they are relatively easier to fabricate for the optical regime. However, if the periodicity of
structure is broken by the introduction of some defects into the crystal such as linear ones, the light of
certain frequencies will be forced to propagate along them. Henceforth, it is possible to provide an inplane confinement for in-plane propagation. Thanks to their relevant features, PC waveguides, as well
as planar waveguides [2] or coupled-cavity waveguides [3], may serve as a basic building block of the
miniaturized optical circuits. Therefore, new component designs are no longer proposed such as PC
filters realized by coupling defect cavity to a linear waveguide [4], for wavelength selectivity. Like the
conventional optical coupler which is commonly composed of two parallel waveguides, a directional
coupler can be constructed by setting two 2-D PC waveguides in close proximity. Such a coupler
exhibits highly strong coupling due to the narrow separation between both linear defects. Thus, PC
couplers are used to implement wavelength demultiplexers [5], [6]. They can act also as channel
interleavers or wavelength multiplexers for wave-division multiplexing (WDM) networks provided
that their total length is considerably reduced to control propagation losses, relatively important in 2-D
PC waveguides.
In the present paper, we study numerically the behavior of photonic directional couplers formed by
two single-row defect waveguides in a square lattice of dielectric rods in the air, using both 2-D plane
wave expansion (PME) and finite difference time domain (FDTD) computations. We propose to
decrease the coupling length in PC coupler by diminishing the high-index volume between
waveguides, reducing the radius of the rods of the interaction region. Applications as channel
interleaver and wavelength demultiplexer are demonstrated.
This paper is organized as follows. In Section 2, we introduce the photonic coupler structure. The
propagation constants of the guided modes and consequently the coupling length are numerically
evaluated. Then, we stress the significant reduction in length as the radius of rods separating both
waveguides is reduced. The coupler performance as channel interleaver is studied. Finally, concluding
remarks are presented in Section 3.
2. PC Waveguide Coupler Structure
The PC structure that we considered is a square lattice of infinitely dielectric rods in air. The design
parameters are as follows: the rods have a dielectric constant and a radius r  0.2 a , where a is the
rod-to-rod pitch. The rods are parallel to the z axis, with a refractive index equals to 3.4. This
important dielectric contrast is recommended since it enables PBG. The structure is analyzed using
PWE [1], [7]. Therefore, from the modal dispersion diagram for the eigenmodes of this perfect square
lattice, we find that the structure has a complete PBG in the spectral range 0.28  a   0.42 for the
transverse magnetic (TM) modes (magnetic field in the plane), but not for transverse electric (TE)
modes,  being the free-space wavelength. To create the directional coupler, two parallel photonic
waveguides are set along the ΓK direction by inserting two line defects in the perfect PC, separated by
two dielectric rods, as shown in figure.1.
Figure 1 Schematic view of the proposed 2-D photonic crystal coupler. The shadowed rectangle shows the
supercell used to obtain dispersion relation.
In order to evaluate numerically the supermodes of the directional coupler, we use the PWE which
computes the Bloch phase shift kz (expressed also as the propagation constant β) that a wave acquires
when it travels through the supercell shadowed in figure 1. It includes the two linear defects
corresponding to the two coupled waveguides. In order to separate the two output waveguides, lowloss 90° bend is introduced since it demonstrates highly efficient transmission [8]. The dispersion
curves of the supermodes propagating in the PC waveguides are illustrated in figure 2. The shaded
regions correspond to the extended modes lying in the infinitely square lattice, in absence of any
defect. The dashed line refers to the isolated single waveguide. The band structure diagram shows that
the even mode of the single the waveguide is split into two supermodes of the directional photonic
coupler. Regarding the symmetry plane between the two waveguides orthogonal to the z axis, there are
both even (symmetric) and odd (antisymmetric) modes at every given frequency. To verify the
different parities of both the single waveguide mode and the photonic coupler modes, we investigate
the corresponding field patterns for a   0.35 , as shown in figure 3. The two supermodes assume
distinguished behaviors through their propagation in the photonic device. In the spectral range
0.295  a   0.36 , the even and odd modes exhibit a significant splitting, enhancing the coupling
strength. However, for higher frequencies, the dispersion curves of the two supermodes become
almost linear, revealing their degeneracy with a weak coupling. A comparison between figure 3.b and
3.c demonstrates the important difference in the field distributions over high and low refractive index
material for the even and odd modes, due to the PBG confinement. This property leads to an increased
frequency difference at equal β between them.
Figure 2 Dispersion diagram for TM guided modes of the coupler of Figure. 1. The solid line is the
dispersion relation for a mode with even symmetry. The dashed line is the dispersion relation of a mode
with odd symmetry.
Although the coupler structure is composed of two nearby waveguides, it can be considered as a
single mode device, such the fundamental mode is defined as a superposition of the even and the odd
supermodes existing in the directional coupler [9]. After a certain propagation length, an
electromagnetic wave launched into one of the waveguides can couple entirely into the other if the
phase shift achieved by both guided modes differs by an odd multiple of π. However, complete
exchange of optical power doesn’t occur until the modes have equal propagation constants, for each
isolated guide.
Figure 3 (a) Field pattern for single waveguide mode. Field patterns for (b) even and (c) odd modes in the
PC coupler of Figure 1.
This criterion, known as phase synchronization, is naturally achieved since the waveguides are
identical, being designed in the same way in the square PC. Thus, they can couple to each other at all
wavelengths. In this way, we define the coupling length L c as the required interaction length for an
input wave to switch completely to the output waveguide. In order to calculate L c necessary for a
certain wavelength, we apply the normal mode theory [6]. Considering the photonic coupler design,
when a mode with frequency ω is fed into of the guides, it excites an electromagnetic mode of the
form:
Ez x,z   Ψ x , z   c e Ψ e x , z  exp (-j e z )  c o Ψ o x , z  exp (-j o z )
(1)
where c i is the field coefficient, Ψ i x , z  exp (-j i z ) is the localized Bloch wave function with a
propagation constant  i , and the index i stands for even or odd. The two supermodes will be in the
same phase if only if exp (-j e z )  exp (-j o z ) . This specific length is defined as the beat length
LB  2 e  o . Applying the switch condition, expressed as follows:
 e   o  2m  1  π , m  0,1,2,...
Thus, the minimum coupler length can be written:
Lc 

e  o

(2)
1
LB
2
(3)
From the dispersion curves of the supermodes, we can derive L c necessary to provide a complete
power transfer from one waveguide into the other, using the following procedure: we find the
corresponding values of the normalized modal Bloch phase constants for both even and odd
eigenmodes referring to the selected wavelength  . Then, we compute the coupling length according
to the expression (3).
The channel spacing may be reduced by increasing the photonic coupler length or the wavelength
dependence of the coupling coefficient [11]. In this paper, we propose to diminish the coupling length
by reducing the radius of the rods which separate the two waveguides, defined as r d such rd  r .
figure 4 depicts the dispersion relation of the modes of the structure inside the BPG for different rd a
values.
Figure 4 Dispersion relation of the TM guided modes of photonic coupler for different
rd a values: 0.2
(filled circles), 0.18 (open squares), 0.16 (filled triangles), 0.14 (open circles), 0.12 (filled squares).
We observe that the odd mode shifts to higher frequencies as the radius of the rods, lying between
both photonic waveguides, decreases. Meanwhile, the even mode is almost unaffected. In fact, as
shown in figure 3.a, the even mode has an electric field node in the region separating the linear
defects, whereas the odd mode has maximum in this region, as plotted in figure 3.b. Therefore, this
implies that the propagation constants difference rises, leading to a smaller coupling coefficient. To
investigate this feature, we compute the variation of the coupling length against optical frequency,
using PWE calculations. Figure. 5 illustrates the normalized coupling length L c a as function of the
normalized frequency for different rd a values.
Figure 5 Frequency dependence of the
normalized coupling length for different rd a
values: rd a = 0.2 (dots), 0.18 (open squares),
0.16 (filled squares), 0.14 (open circles), 0.12
(filled circles).
Figure 6 Phase shift of supermodes
through 280 cells of the directional coupler
with rd a  0.12 .
In figure 5, for the same frequency range, the coupling length diminishes clearly as r d decreases, as
expected. Since the main goal is to reduce the wavelength spacing between adjacent channels at both
outputs, we consider the photonic coupler, with rd a  0.12 . To appreciate the efficiency of such
device as WDM interleavers, we compute the frequency dependence of the total relative phase shift
 TOT accumulated after propagation along 280 lattice cells (corresponds to L  150 μm for
a  536.5 nm ) for the coupler of Fig.1 and the coupler with rd a  0.12 , as shown in figure. 6.
In the frequency range 0.34  a   0.42 ,   scales between 8 and 99 for the photonic coupler
with rd a  0.12 . Thus, 46 different frequencies(wavelengths) experience an overall phase shift equal
to an odd multiple of π, which means that they are completely transferred to the other waveguide (port
2). However, for the other 45 frequencies, their path length is an integer multiple of the beat length:
they come out the input waveguide (port 3). Then, reducing r d yields a smaller spacing. For example,
considering a photonic coupler with rd a  0.12 and a  536.5 nm in order to center its response in
the 1550 nm range (wavelengths 1541 to 1561.5 nm), we obtain a 0.8-nm channel spacing, which
corresponds to 100 GHz with a device length as short as L  280 a  150.22 m , in the frequency range
0.343 to 0.348. To illustrate this previous example, Figure 7 depicts the normalized power at port 2
and 3 as function of the wavelength obtained from FDTD calculations.
However, as shown in figure 6, it can be seen that  TOT sharply bends upwards at shorter
frequencies. Then, the interleaved channels are not equally spaced. To by-pass this drawback, a
tradeoff between the photonic coupler length and the channel uniformity is recommended: the spectral
range should be limited to a range where the deviation of the phase shift can be approximated by a
linear function of frequency.
Figure 7 Normalized response of PC coupler with
rd  0.12a and L  280 a at both outputs: Port2:
solid line and Port 3: dashed line.
3. Conclusion
In this paper, we propose a simple method to improve the coupling coefficient and to reduce the
coupling length in 2-D photonic coupler consisting of nearby PC defect waveguides designed in a
square lattice of dielectric rods in air. By reducing the radius of dielectric rods composing the
interaction region, a 150 µm directional coupler can serve as a channel interleaver with 0.8-nm
channel spacing at 1.55 µm wavelength.
References
[1] J. D. Joannopoulos, R. D. Meade, and J. N. Winn, “Photonic Crystals: Molding the flow of
light,” Princeton Univ. Press, Princeton, 1995.
[2] R. Stoffer, H. J. W. M. Hoekstra, R. M. D. Ridder, E. V. Groesen and F. P. H. V. Beckum,
“Numerical Studies of 2D Photonic Crystals: Waveguides, Coupling Between Waveguides
and Filters,” Opt. Quantum Electron, vol. 32, pp. 947–961, 2000.
[3] M. Bayindir and E. Ozbay, “Heavy photons at coupled-cavity waveguide band edges in a threedimensional photonic crystal,” Phys. Rev. B, vol. 62, R2247–R2250, 2000.
[4]
S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, “Channel drop filters in photonic
crystals,” Opt. Express., vol. 3, no. 1, pp. 960–963, Feb. 1998.
[5] M. Koshiba, “Wavelength Division Multiplexing and Demultiplexing with Photonic Crystal
Waveguide couplers,” J. Lightwave Technol., vol. 19, no. 12, pp. 1970–1975, Dec. 2001.
[6] S. Boscolo., M. Midrio, and C. G. Someda, “Coupling and decoupling of electromagnetic waves
in parallel 2-D photonic crystal waveguides,” IEEE J. of Quantum Electron., vol. 38, no. 1,
pp. 47–53, Jan. 2002.
[7]
R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O.L. Alerhand, “Accurate
theoretical analysis of photonic band gap materials,” Phys. Rev. B, vol. 48, pp. 8434-8437,
1993.
[8] A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D.Joannopoulos, “High
transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett., vol. 77,
pp. 3787–3790, Oct. 1996.
[9] R. März, Integrated Optics: Design and Modeling, Artech House, Norwood, 1995.
[10] A. Taflove, S.C. Hagness, Computational Electrodynamics: The Finite-Difference TimeDomain Method, second ed., Artech House, Boston, 2000.
[11] A. Martinez, F. Cuesta, and J. Marti, “Ultrashort 2-D photonic crystal directional couplers,”
IEEE Photonics Tech. Lett., vol. 15, pp. 694–696, 2003.