January 15, 2008 / Vol. 33, No. 2 / OPTICS LETTERS
119
Bending loss analyses of photonic crystal fibers
based on the finite-difference time-domain method
Ngoc Hai Vu,1 In-Kag Hwang,1,* and Yong-Hee Lee2
1
Department of Physics, Chonnam National University, 300 Yongbong-dong, Buk-gu, Gwangju 500-757, South Korea
2
Department of Physics, Korea Advanced Institute of Science and Technology, 373-1 Guseong-dong,
Yuseong-gu, Daejeon 305-701, South Korea
*Corresponding author: [email protected]
Received October 8, 2007; revised November 29, 2007; accepted December 6, 2007;
posted December 11, 2007 (Doc. ID 88351); published January 8, 2008
This is a report on an effective simulation method for the bending loss analyses of photonic crystal fibers.
This method is based on the two-dimensional finite-difference time-domain algorithm and a conformal transformation of the refractive index profile. We observed the temporal dynamics of light waves in a bent fiber in
a simulation and obtained the bending loss as a function of bend radius and optical wavelength for the commercial photonic crystal fibers. The accuracy of this method was verified by good agreement between the
simulation and experimental data. © 2008 Optical Society of America
OCIS codes: 060.5295, 060.2280, 060.2310.
First proposed in 1995, photonic crystal fibers
(PCFs), with silica–air microstructures, attract many
researchers as these fibers have unique applications
in ultrawide-band transmission, supercontinuum
generation, high power delivery, optical amplifiers,
and other functional devices [1]. One of the important
issues regarding the practical development of these
PCFs concerns their bending loss properties. When a
fiber is bent, the modal field distorts outwards in the
direction of the bend and a radiation loss occurs. The
bending loss is usually regarded as an adverse effect
in the context of optical transmission. However, the
bent fibers can also be used as a new and unique optical component employable in optical communications or optical sensing [2]. It is important to be able
to accurately estimate the bending loss of a given fiber structure for the design and characterization of
various PCFs.
The complicated microstructure in a PCF makes
the calculation a challenging problem. Most of the
analytical methods such as antenna theory developed
for conventional fibers with circularly symmetric index profiles cannot be directly applied to PCFs. Here,
we, for the first time to our knowledge, adopted a
two-dimensional finite-difference time-domain (2DFDTD) algorithm [3] for the simulation of optical
propagation in a bent PCF. The three-dimensional
(3D) structure of the bent fiber was transformed into
a two-dimensional straight fiber by using conformal
mapping of the refractive index profile of the PCF.
The temporal evolution of the optical field was properly interpreted to yield the bending loss per unit
length. We compared the simulation results with the
experimental results to validate the accuracy of our
method.
The FDTD method has some distinct features compared with other previous methods such as the efficient modal model [4,5] or the finite-element method
[6] used for the calculation of bending loss of the PCF.
The FDTD algorithm is a very general tool and is applicable to a wide range of electromagnetic problems.
It directly solves Maxwell’s equations with minimal
0146-9592/08/020119-3/$15.00
assumptions and approximations and thus provides
fairly reliable results as long as the spatial and temporal resolution are high enough. Recent advancement of computer technology allows the use of high
spatial and temporal resolution, making this technique more useful and popular. This method enables
full access to electromagnetic waves at an arbitrary
time and position, so that one can collect any desired
information from these waves. Therefore it is distinguishable from other numerical methods that provide
specific information only.
The direct simulation of optical propagation in
bent fiber may be performed by the FDTD method in
a complete 3D structure of fiber loops with an input
of an optical wave from one end of the fiber loops [2].
In this case, we record the optical powers at several
locations of the fiber loops to obtain the bending loss
as a function of the propagation length. However, this
approach requires huge memory sizes and long computation time, which is practically unacceptable.
Here, we implemented a time-domain approach for
the calculation of the bending loss, instead of the
space-domain approach, for computation efficiency.
First we imagined an infinitely long fiber with a bend
radius of Rb as shown in Fig. 1(a). At t = 0, the fiber is
filled with an optical wave with a propagation constant ␤ and a uniform intensity along the length. After t = 0, the optical wave starts to attenuate with a
rate of ␣⬘ due to the bending loss. Note that the optical intensity is always uniform over the whole fiber
length during the attenuation. Finally the loss coefficient per unit time 共␣⬘兲 is converted to a loss coefficient per unit length 共␣兲 using the equation ␣ = ␣⬘ / v,
where v is the velocity of light in the fiber.
Since the optical wave has no variation along the
length except the phase in the above situation, the
computation structure for the FDTD method can be
reduced to one slice of the bent fiber with an arbitrarily small thickness 共⌬z兲 as shown in Fig. 1(b).
Then, the 3D structure of the bent piece is again simplified to a flat one [Fig. 1(c)] by employing an equivalent index profile given by
© 2008 Optical Society of America
120
OPTICS LETTERS / Vol. 33, No. 2 / January 15, 2008
Fig. 1. (Color online) Illustration of the simulation
scheme: (a) infinitely-long bent fiber model; (b) one sliced
piece of the bent fiber and its index profile along x; (c) same
as (b) after conformal transformation of the index profile;
(d) E2 at the center of the fiber as a function of time, showing the attenuation of optical intensity.
冉 冊
2
neq
共x,y兲 = n2共x,y兲 1 +
2x
Rb
,
where Rb is the radius of curvature and n共x , y兲 is the
refractive index profile of the straight fiber [6–8]. The
bottom part of Fig. 1(c) shows the transformed refractive index profile of a PCF with a bend radius of
5.5 mm. It clearly shows that the transformation superimposes a gradient onto the refractive index of the
straight fiber in the direction of the bend. Therefore
the final computation structure is effectively a 2D
one containing only one grid along the z. The sizes of
the computation grids were set to ⌬x = ⌬y = ⌳ / 20 and
⌬z = ⌳ / 100, where ⌬ is the hole pitch. Those parameters were optimized to maximize the numerical accuracy within a reasonable computation time.
Figure 1(d) shows the recorded optical intensity E2
at the center of the fiber as a function of time. The
optical intensity corresponds to I共t兲 = 具E2共t兲典. Here we
obtained the loss factor per unit time, ␣⬘, from curve
fitting with the function I共t兲 = I0 exp共−␣⬘t兲. The velocity of the light was calculated from v = ␻ / ␤, where ␻
was the oscillation frequency of Fig. 1(d), to get the
loss factor per unit length, ␣ = ␣⬘ / v.
The simulation was performed for the ESM-5 PCF
(hole pitch, ⌳ = 8 ␮m; normalized hole diameter, d / ⌳
= 0.46) from Crystal Fibre A/S. The parameters ⌳ and
d were extracted directly from a scanning electron
microscope image of the real fiber. The refractive index of the silica was set as 1.444, and no material
dispersion was assumed. Figure 2(a) shows the intensity distributions of the fundamental mode at ␭
= 1550 nm for the bend in the x direction of the radius
Fig. 2. (Color online) (a) Optical intensity distribution in
the cross section of a bent fiber with a radius of 5.5 mm (log
scale) and (b) central regions of intensity profiles taken at
successive times 共⌬t = 1 fs兲 in one period of oscillation.
of 3.5 mm. It clearly shows that the mode of the bent
fiber was asymmetric in shape and shifted towards
the outside of the bend [4]. We generated multiple intensity profiles at successive time frames in one period of the optical oscillation to observe the dynamics
of the optical field, which are shown in Fig. 2(b). Here
we could observe a “propagating” field radiated from
the center toward the outside of the bend while the
“stationary” core mode was blinking at its optical frequency. This energy propagation across the fiber was
the cause of the bending loss and resulted in the dissipation of the optical power in the core mode. The direct observation of this phenomenon could not be
made by other calculation methods.
We calculated the bending losses of the ESM-5
PCF for several different bending radii. The plots are
denoted with triangles in Fig. 3(a). The typical computation time was about 2 – 3 h for each data point in
our case, although it depends largely on the spatial
and temporal resolutions of FDTD. For comparison,
the bending losses were experimentally measured using a narrow-linewidth laser and an optical powermeter. Two turns of fiber loops were made for a bend
radius in the range of 3.0 to 7.0 mm in increments of
Fig. 3. (Color online) Dependence of bending loss on bending radius for (a) ESM-5 PCF and (b) LMA-8 PCF. The
squares and triangles denote the experimental and simulation data, respectively.
January 15, 2008 / Vol. 33, No. 2 / OPTICS LETTERS
0.3 mm. For a bend radius smaller than 3.0 mm, the
bent fiber was easily broken; while for a radius larger
than 7 mm, the loss was too low for reliable and repeatable measurements. Each measurement was repeated three times, and its average value and deviation are shown with squares and error bars,
respectively, in Fig. 3(a). A comparison was carried
out for another PCF, LMA-8 (⌳ = 5.6 ␮m, d / ⌳ = 0.49)
also from Crystal Fibre A/S. The results are shown in
Fig. 3(b). There was reasonably good agreement between the simulation and experimental results for
both fibers. It is interesting to see the small bumps in
the experimental curves at the bending radii of
⬃7.8 mm for ESM-5 PCF and ⬃4.2 mm for LMA-8,
which did not appear in the simulation results. These
loss peaks seemed to come from resonant coupling
from the core mode to a cladding mode in the
multilayer structure of the cladding. Note that the
full cladding structure of PCF should be included in
the computation structure to investigate this phenomenon [9]. A detailed study of its origin is underway.
The wavelength dependence of the bending loss
was also calculated and measured in this report for
ESM-5. In FDTD, the optical wavelength was redetermined by simultaneous changes of the propagation constant, ␤, and the optical frequency, ␻. For the
experiment, we used an optical spectrum analyzer
and a superluminescent diode with a bandwidth of
⬎50 nm. The simulation and experimental data are
shown in Fig. 4 as dashed and solid curves, respectively. The strong wavelength dependence was observed for a small bending radius, while it was relatively flat for a large bending radius. The fine
structures in the experimental data seemed to result
from the reflection of the radiated light at the
Fig. 4. (Color online) Loss spectrum of ESM-5 PCF with
different bend radii. Experimental and simulation data are
shown by the solid and dashed curves, respectively.
121
cladding–jacket or jacket–air boundaries back to the
core mode. Note that the exceptionally large bending
loss at R ⬃ 7.7 mm in Fig. 3(a) was observed again in
Fig. 4. We also found good agreement between the
simulation and experiment, which again verified the
accuracy of our method.
We proposed an efficient numerical method for
bending loss analyses of PCF, based on the 2D-FDTD
method and conformal transformation of the index
profile. The time-domain simulation of the optical
propagation in a bent fiber provided a view of the
temporal dynamics of the optical field as well as the
mode profile, dispersion and the bending loss. The accuracy of the method was verified by good agreement
between a simulation and experimental data. The
technique outlined here is directly applicable to not
only PCFs but also any kind of waveguides with arbitrary index profiles. It is important to note that this
FDTD method can be easily extended by adding new
functions to include nonlinear or strain effects in the
simulation, which may not be available in other
methods. We believe it is a useful tool for analyses
and design of various microstructured fibers [10–12].
This work was supported by IT R&D program of
Ministry of Information and Communication and Institute for Information Technology Advancement
(2005-S099-03, Development of photonic crystal fibers and their application technology for high-speed
optical communication system).
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