INTRODUCTION
Photonic crystals have inspired great interest recently because
of their potential ability to control the propagation of light. They can
modify and even eliminate the density of electromagnetic (EM) states inside
the crystal (Yablonovitch, 1987; John, 1987). Such periodic dielectric
structures with complete band gaps can find many applications, including
the fabrication of lossless dielectric mirrors and resonant cavities for
optical light (Joannopoulos et al., 1995).
The realization of efficient sharp and compact waveguide bends is still
a challenging task in micro optics. With the introduction of Photonics
Crystals (PCs) major interest has also focused on the issue of efficient
waveguide bends embedded in PCs. There are various proposals for bend
design in order to minimize losses. Examples are smoothening the sharp
bends (Moosburger et al., 2001), introducing cavities or intermediate
straight sections (Olivier et al., 2002), or placing smaller holes
around the bend (Talneau et al., 2002). To inhibit the modal mismatch
at a Y splitter (Boscolo et al., 2002) an additional hole has been
added at the bend. On the other hand, it has been shown that a CoupledDefect
Waveguide (CDW) composed of a chain of point defects in a PC can also
have a very high transmission for a certain wavelength range at sharps
bends (Johnson and Joannopoulos, 2001).
In this study, we demonstrate a novel method for guiding light around
sharp corners, using photonic crystal waveguides. This method is based
on the one hand on the modification in the geometry at the corner and
the other hand on the use of the absorbing boundary conditions proposed
by Mekis et al. (1999) which reduce reflection from PBG waveguide
ends to under a few percent.
The FDTD method (Kunz and Luebbers, 1993) has been widely used to study
EM properties of arbitrary dielectric structures.
In this method, one simulates a space of theoretically infinite extent
with a finite computational cell. To accomplish this, a number of boundary
conditions such as Berenger`s Perfectly Matched Layer (PML) (Berenger,
1994), have been proposed that absorb outgoing waves at the computational
cell boundaries. Applications of the FDTD method are to simulate photonic
crystal waveguides, however, pose unique difficulties. While reflection
from a PML boundary is minute for a traditional dielectric waveguide substantial
reflection from the boundary is observed if a PBG waveguide is terminated
so, on the order of 2030% in amplitude (Mekis et al., 1996). Such
reflection introduces unphysical reflected (parasite) pulses which may
significantly compromise the accuracy of the simulated response. Reflected
waves introduce interference and result in large errors in transmission
measurements.
FINITE DIFFERENCE TIME DOMAIN
ALGORITHM (FDTD)
For a linear isotropic material in a sourcefree region, the timedependent
Maxwell`s equations can be written in the following form,
where, ε(r), μ(r) and σ(r) are the position dependent
permittivity, permeability and conductivity of the material, respectively.
In the two dimensional case, the fields can be decoupled into two transversely
polarized modes, namely, the E polarization and the H polarization. These
equations can be discretized in space and time by a socalled Yeecell
technique (Yee, 1966). The following FDTD time stepping formulas are spatial
and time discretizations of Eq. 1 and
2 on a discrete twodimensional mesh within the xy coordinate system
for the E polarization (Yee, 1966),
where the index n denotes the discrete time step, indices i and j denote
the discretized grid point in the xy plane respectively. Δt is the
time increment and Δx and Δy are the intervals between two neighbouring
grid points along the x and y directions, respectively. Similar equations
for the H polarization can be easily obtained.
It can be easily see that for a fixed total number of times steps the
computational time is proportional to the number of discretization points
in the computation domain, i.e., the FDTD algorithm is of order N.
The FDTD timestepping formulas are stable numerically if the following
conditions are satisfied (Taflove, 1995).
where, c is the fastest peed of the light in all the materials involved
in the simulation (In FDTD program, we always choose c be the speed of
the light in vacuum). Thus, smaller Δt, this means even longer calculation
time.
The number of total time steps should be chosen carefully, too. For pulse
propagation, the total time steps should be larger enough, in order to
allow the pulse passes the detectors. In particular, when the simulations
involve cavities, the number should be sufficiently large.
BOUNDARY CONDITIONS
One approach to eliminate errors due to reflected pulses has been
to increase the cell size such that the useful and the parasite pulses
can be separated (Mekis et al., 1996). This approach, however significantly
increases the computational cost in terms of memory and time. Special
care must be taken to separate well the pulses since due to interference
the error is proportional to the amplitude and not to the power, of the
reflected pulse. In addition, in the case of steady state simulations,
or when a highQ resonance is involved, it becomes impractical or even
impossible to separate the reflected signal amplitude from the useful
one. In the earlier research Mekis et al. (1999) demonstrate that
it is possible to reduce the reflection amplitude from photonic crystal
waveguide ends to a few percent by using a kmatched Distributed Bragg
Reflector (DBR) waveguide. This provides a simple means to reduce the
computational costs associated with simulating PBG waveguides. In our
work, we use this concept of absorbing boundary conditions to obtain an
efficient 90 ° bend structure.
ROLES OF RESONATORS
In this section, we evaluate the relative importance of cavity resonance
on the bend performance of the resonator. The idea is based on the principle
of a symmetric resonator with two parts. The notations employed here are
those used by Manolatou et al. (1999). At resonance, the transmission
is complete with no reflection if the resonator is lossless. The effects
of radiations can be counteracted by making the external Q of the resonator
very small. This is achieved by strong coupling of the waveguide modes
to the resonator mode. This concept is simply explained using coupling
of modes in time (Manolatou et al., 1999). Because this analysis
is based on perturbation theory, it can only provide a qualitative prediction
in the case of strong coupling between the cavity and the waveguide modes.

Fig. 1: 
Schematic of a twoport resonator connected to the waveguides
of the 60 ° bend (Manolatou et al., 1999) 
Following the approach of Manolatou et al. (1999), the amplitude of the mode in
the cavity is denoted by u and is normalized to the energy in the mode. The
decay rates of the mode amplitude due to the coupling to the waveguides are
1/τ_{e1} and τ_{e2}, respectively, related to the external
Q`s by Q_{e1} = ω_{0}τ_{e1}/2 and Q_{e2} =
ω_{0}τ_{e2}/2, where ω_{0} is the resonance frequency.
The decay rate due to radiation loss is 1/τ_{0} = ω_{0}/2Q_{0}.
The incoming (outgoing) waves at the two parts are denoted by S_{+1}
(S_{1}) and S_{+2} (S_{2}) (Fig.
1) and are normalized to the power carried by the waveguide mode.
If the excitation is S_{+1} with exp(jωt) time dependence
and S_{+2} = 0 then at steady state we have (Manolatou et al.,
1999):
and
Which, due to (7) finally give:
At ω = ω_{0} the reflection is zero and the transmission
maximized if:
Thus asymmetric system (1/τ_{e1} = 1/τ_{e2}
= 1/τ_{e} = ω_{0}/2Q_{e}) allows complete
transmission provided that it is lossless. The width of the frequency
response is determined by 1/τ_{e} if the loss is present
the ratio τ_{e}/τ_{0} = Q_{e}/Q_{0}
determines the peak transmission and minimum reflection as:
DESCRIPTION OF THE PROPOSAL STRUCTURE
The specific structure that we investigated is a 90 ° sharp bend
formed by the intersection of two PC channel waveguides at 90 ° Fig.
2a in an otherwise uniform photonic lattice. We assume a square lattice
of air holes etched in a dielectric substrate, with refractive index n
= 3.24, having filling factor of 39 %. Since we want to use this device
around 1550 nm we calculate the lattice constant to be 430 nm and obtain
therefore a hole radius of 141.9 nm, respectively. The structure is assumed
to be bidimensional; i.e., the air holes are infinitely long, the 2D PC
supports a photonic band gap in the region 0.203 < c/a <0.35 for
TE polarized light.
In the design process, we use 2D FDTD simulation. This technique is powerful
and versatile and has been introduced and adapted to optical waveguide
devices (Chutinan and Noda, 2000; Chutinan et al., 2002). In FDTD
very small time step size must be used because both the carrier and the
modulated envelope are included in the wave propagator.
We start our optimization by first looking at a wavelength scan in the
non optimized case, wherein the original position of the holes remains
unchanged (inset of Fig. 2).
SIMULATION RESULTS AND DISCUSSION
The cartography of the magnetic field Hz in the non optimize structure
and the transmission and reflection spectra are shown in Fig.
2b and c, respectively. We note that the transmission reached a value
of only 10% (reflection 73%). The reason for this very poor transmission
is twofold.

Fig. 2: 
(a) Non optimised 90 ° bend structure, (b) Magnetic field
amplitude distribution in a plain 90 ° bend and (c) Transmission
and reflection spectra of the plain 90 ° bend 
First, a large fraction of the power is lost to radiation or reflected backward due to mode mismatch at the
corner of the sharp bend. Moreover, the poor transmission originates from
modal mismatch at the junction. In fact, if the incoming mode has space
to expand in the junction area, it excites a higher mode with odd parity
that is either very lossy or cannot propagate in the output waveguide,
so most of the incoming light is reflected and transmission is poor. Therefore,
the excitation of modes with odd parity would act as a loss mechanism
for the 90 ° sharp bend. Our conclusion is therefore as follows: the
transmission through a junction depends strongly on the relationship between
the modes of that may propagate in the PC waveguides and the modes of
the junction are not compatible with those of the waveguide, transmission
will be poor. The resulting cartography is a mixture of the stationary
mode on the level of the bend and propagative wave (Fig.
2b).
To improve matters, the obvious choice is to modify the junction region.
By removing hole at the center of the junction, we reduce the optical
size of the cavity, thus eliminating multimode effects.
TOWARDS A COMPLETELY ACHROMATIC TRANSMISSION
The research of the compromise zero reflectionbroad bandwidth
was based up to now on the research of optimal topology, in terms of modal
agreement between the mode of the guide and that of the bend. Insinuation,
the possible improvements made to the properties of transmission, relate
always to a resonant cavity and thus by definition always chromatic. We
have just shown, with simple considerations, results translating of the
notable improvements of the answer related to a corner: the width of the
spectral range of high transmission for a bend with 60 ° more than
was doubled compared to the improvements reported in the literature (Chutinan
and Noda, 2000; Chutinan et al., 2002). We would however wish to
use all the monomode range including that which would be located above
the line of light in the 3D case. The only possible exit to transmit on
all the monomode range consists in, consequently, returning to a specular
approach, such as it is practised conventionally in integrated optics, or in other words to kill the resonance all while
preserving reflections with the bend.

Fig. 3: 
Losses by curve in a conventional approach (Grillet,
2003) 
We go, before considering the use
of a specular approach, to point out the limitations of the conventional
approach (case of guides ribbons for example) for the realization of turns.
We can distinguish three sources from losses in a conventional approach.
The losses by curve: Certain guided rays, when the guide is right,
will see their angle of reflection to pass below the limiting angle of
total reflection when the guide is curved and a part of the luminous power
is thus refracted outside the guide with each reflection (Fig.
3). In the case of turn to CP, these losses do not have obviously
course.
The losses by transition: When we couples a mode of right guide
in a curved guide, it will have an effectiveness of coupling lower than
the unit because the mode of the curved guide is shifted towards the outside
of the curve (Fig. 4a). Two strategies are often used
in optics guided to reduce these losses: the first consists in shifting
the input of the curved guide compared to the right guide (Fig.
4b). When the right guide is shifted towards the outside of the curve,
it anticipates the shift which the wave will undergo while entering the
guide curves and makes thus the coupling optimal between right guide and
curved guide. the other strategy consists in producing guides with continuously
variable curve (Fig. 4c) so as to pass in an adiabatic
way of the right guide to a curved guide of given curve. The advantage
of this method is that it can also be optimized to reduce the losses by
curve. But it appears clearly that these strategies are not easily transposable
with a realization of turns in a CP2D and besides do not constitute quite
simply a satisfactory solution for compact applications in integrated
optics.
The losses by roughness with the turns: Let us consider the light
suitably installed in the curved guide and observing the conditions of
total reflection interns in this portion: this one is not saved for as much! Roughness will induce a coupling
between the guided modes and the radiative modes of the structure.

Fig. 4: 
(a): Losses by transitions in a conventional curved guide, (b)
Configuration planned to reduce these losses by shift and (c) Curve
continuement variable (Grillet, 2003) 
That
result in losses, all the more important as the fraction of luminous energy
conveyed outside the guide is high.
An approach combining mirror with 45 ° and cavity slightly resonant,
developed with MIT, rises in remarkable performances (transmission 98%,
on a bandwidth de 10% around 1.55 mm) for the turn with 90 ° in a
configuration with high contrast from index (Manolatou et al.,
1999). We took as a starting point this research, by adapting the use
of the mirror to our own configuration.
The optimised structure: The essential function of W1 PC 90 °
sharp bend is to convert a single mode train in the input waveguide into
a single mode train in the output waveguide.
In order to improve the transmission of the 90 ° sharp bend and to
avoid the losses at the 90 ° bend we insert a mirror in the bend of
reference, the mirror is obtained by doing small displacement for the
most critical hole around the proper 90 ° bending region. The optimized
structure resulting is showing in Fig. 5a.

Fig. 5: 
(a) Optimised 90 ° bend structure resulting, (b)
Magnetic Field amplitude distribution in the optimized 90 ° bend
and (c) The transmission spectra of the optimise 90 ° bend structure 
RESULTS AND DISCUSSION
Using the novel numerical scheme for the reduction of spurious reflections
from photonic crystal waveguide ends and the reflections induces et the
corner of the sharply bend, then clearly increase the bandwidth and power
transmission, as directly observed in our, comparative, simulations of
a 90 ° sharp bend with and without modifications. This solution has
been compared to a several previous independent works.
Although, the bend`s radius of curvature is less than the light`s wavelength,
nearly all the light is transmitted through the bend over a wide range
of frequencies through the gap. The small fraction of light that is not
transmitted is reflected. For specific wavelengths we can achieve 100%
transmission (Fig. 5c) efficiency is that the photonic
crystal waveguide be single mode in the frequency range of interest. The
Fig. 5b shows clearly that the light is confined around the sharp
bend and it can be seen that the radiation has been vanished compared
with Fig. 2b.
CONCLUSION
While performing a simple sensitivity analysis based on FDTD code,
we have obtained an efficient method for improving the frequency response
of 2DPC devices. In particular we have applied this technique to a simple
90 ° sharp bend waveguide emerging from an underlying 2DPC with square
lattice symmetry. A single optimization step has already obtained nearly
zero reflection over almost the entire PBG. This technique can easily
be extended to other 2DPC properties for optimization purposes, because
key problem in the design of future integrated optical devices is how
to balance ease of fabrication with the reduction of radiation losses.