Download The Field and Energy Distributions of the Fundamental Mode in the

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ISRN Optics
Volume 2013, Article ID 567501, 8 pages
http://dx.doi.org/10.1155/2013/567501
Research Article
The Field and Energy Distributions of the Fundamental
Mode in the Solid-Core Photonic Crystal Fibers for Different
Geometric Parameters
Halime Demir Inci and Sedat Ozsoy
Department of Physics, Faculty of Science and Arts, Erciyes University, 38039 Kayseri, Turkey
Correspondence should be addressed to Sedat Ozsoy; [email protected]
Received 16 June 2013; Accepted 12 July 2013
Academic Editors: Y. S. Kivshar and S. Ponomarenko
Copyright © 2013 H. Demir Inci and S. Ozsoy. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
For a solid core photonic crystal fiber with the triangular lattice, the field and energy distributions of fundamental mode are
considered at 1.55 𝜇m wavelength. The silica core is constituted by removing the 7 air holes. The cladding consists of the twodimensional silica-air photonic crystal with the 4 rings of air holes. The field and energy distributions were investigated for three
different values of d/Λ. Here, d and Λ represent the diameter of air holes and the pitch length, respectively. The simulations show
that, for the fixed Λ, the increase in d/Λ ratio does not cause the considerable changes in the field and energy intensity distributions,
but for the fixed d, the increase in this ratio affects the intensity distributions reasonably.
1. Introduction
In recent years, the photonic crystal fibers (PCFs) have
been of significant interest due to their unique structures
and new properties [1–6]. Generally, photonic crystal fibers
consist of an arrangement of air holes in the cladding
extending the whole length of the fiber. Photonic crystal fibers
are categorized into two groups according to light guiding
mechanism. One is the index guiding photonic crystal fiber,
and the other is the photonic band gap PCF. In the index
guiding PCFs, the core region is solid and the light is confined
in the central core as in the conventional fibers. The photonic
crystal fiber consists of the pure silica fiber with an array of the
air holes along the length of the fiber. The core is constituted
by removing the central hole from the structure. The higher
effective refractive index of the surrounding holes forms
cladding in which leading the index guiding mechanism
analogous to total internal reflection. Consequently, the light
guiding can be explained by the total internal reflection which
is also the way light is guided in step index fibers. If around
the central air hole there is the two-dimensional photonic
crystal cladding which consists of a periodic array of the silica
and air, such fibers are called the photonic band gap photonic
crystal fibers. The central hole in these fibers acts as the core,
and in this place the light is guided by the photonic band gap
effect. The frequencies within the band gap of the structure
will expose to the multiple Bragg reflection that leads to the
destructive interference of the light trying to propagate away
from the air core. The function of the air core is to provide
a defect at the periodic structure that the propagation of the
frequencies within the photonic band gap is really allowed.
Therefore, it is not needed for the core indexes of these fibers
to be bigger than the effective index of cladding.
PCFs have been shown to possess many important
properties as the single mode operation over wide range of
wavelength, the highly tunable dispersion, the propagation
of high power densities without exciting unwanted nonlinear
effects, and the high birefringence. These properties have the
practical importance in the design of sophisticated broadband optical telecommunication networks [7] and active
sensor systems [8].
In this work, the software based on the finite element
method (FEM) is used for the field and energy distribution
simulations of the fundamental mode at the solid core photonic crystal fiber structures with the different geometrical
parameters.
2
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2. Theoretical Model
The large index contrast and complex structure in the
photonic crystal fibers make them difficult to be treated
mathematically. The standard optical fiber analysis does not
help, and moreover in the majority of photonic crystal
fibers, it is practically impossible to perform modal analysis
analytically; therefore the Maxwell equations must be solved
numerically. The main idea consists of transforming this
complicated problem, which can be described by the curl-curl
equation [9]:
∇ × (𝜇𝑟−1 ∇ × 𝐸)⃗ − 𝑘02 𝜀𝑟 𝐸⃗ = 0.
d
(1)
This is an eigenvalue equation, the eigenvalues are 𝛽/𝑘0
the effective indexes, and the eigenvectors are the electric field
components (𝐸𝑥 , 𝐸𝑦 , 𝐸𝑧 ). In (1), 𝐸⃗ is the electric field, 𝑘0 is
the wavenumber in vacuum, and 𝜀𝑟 and 𝜇𝑟 are the dielectric
permittivity and magnetic permeability tensors, respectively.
Here in the analysis of the properties of PCFs, the FEM
is used. The FEM allows the photonic crystal fiber crosssection in the 𝑥-𝑦 plane to be divided into a patchwork of
triangular elements in which they can possess different sizes,
shapes, and refractive indexes. In this way, any geometry
including the PCF air hole and the medium characteristics
can be accurately described. In particular, the FEM is suited
for studying fibers with the nonperiodic arrangements of the
air holes. However, it provides a full vectorial analysis that
is necessary to model the PCFs with the large air holes and
high index variations and to accurately predict the properties
of these fibers [10].
The FEM is basically divided in to four steps [11]. The
domain discretization is the first and perhaps the most
important step in any finite element method. The first
step consists of dividing the solution domain Ω into the
subdomains of finite number in which form a patchwork
of fundamental elements that can possess different sizes,
shapes, and physical properties. The second step is choosing
the interpolation functions that provide an approximation of
the unknown solution within each element. In this specific
case, the solution domain is the transverse cross-section
of optical waveguide that is divided in triangular finite
elements, and the nodal approach is followed using secondorder polynomials as interpolating functions in each finite
element (each triangular is characterized by 6 nodes, and the
unknowns are the magnetic field components). In the next
step, the curl-curl equation is transformed into a generalized
eigenvalue problem by applying the variational formulation
as follows:
[𝐴] {Φ} − 𝜆 [𝐵] {Φ} = {0} ,
Λ
(2)
where the eigenvalue 𝜆 is the mode effective refractive index
(𝑛eff ) and the eigenvector Φ is the full vectorial electric
distribution (𝐸𝑥 , 𝐸𝑦 , 𝐸𝑧 ) in the nodal points.
Finally, the fourth and final step in the finite element analysis is to solve the eigenvalue system. The matrices [𝐴] and
[𝐵] are sparse and symmetric; therefore, the computational
time can be effectively minimized by using the sparse matrix
solver.
Figure 1: The cross-section of the solid core PCF considered. Λ is
the pitch length, and 𝑑 is the diameter of air holes.
Table 1: The effective refractive indexes of the fundamental mode
for different 𝑑/Λ ratios.
𝑑/Λ
Fixed pitch
(Λ = 4.2 𝜇m)
0.08
0.09
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
Fixed diameter
(𝑑 = 0.84 𝜇m)
0.08
0.09
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
𝜆/Λ
𝑛eff (𝜆 = 1.55 𝜇m, 𝑛silica = 1.44409)
0.37
1.442911 − 2.534351𝑒 − 8𝑖
1.442822 − 1.378536𝑒 − 8𝑖
1.442747 − 5.387208𝑒 − 9𝑖
1.442334 − 4.984438𝑒 − 12𝑖
1.442108 − 2.353948𝑒 − 14𝑖
1.441914 − 8.43704𝑒 − 17𝑖
1.441721 − 2.22923𝑒 − 19𝑖
1.44152 − 3.398141𝑒 − 19𝑖
1.441303 − 5.284044𝑒 − 19𝑖
1.441068 − 4.492512𝑒 − 20𝑖
1.440809 − 3.595418𝑒 − 19𝑖
0.15
0.17
0.19
0.37
0.56
0.71
0.91
1.11
1.25
1.48
1.67
1.443843 − 7.982422𝑒 − 11𝑖
1.443773 − 9.149398𝑒 − 11𝑖
1.443693 − 4.61459𝑒 − 11𝑖
1.442334 − 4.984438𝑒 − 12𝑖
1.439848 − 1.198296𝑒 − 12𝑖
1.436078 − 1.976832𝑒 − 13𝑖
1.43085 − 8.180793𝑒 − 14𝑖
1.423967 − 3.413292𝑒 − 14𝑖
1.41519 − 2.279162𝑒 − 14𝑖
1.404232 − 2.485407𝑒 − 14𝑖
1.390737 − 3.793315𝑒 − 14𝑖
3. Simulation Results
For the simulations, the photonic crystal fiber with crosssection shown in Figure 1 is investigated at 1.55 𝜇m wavelength. Here Λ is the pitch length, and 𝑑 is the diameter of
air holes. The fiber core is silica, and it is formed by removing
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3
Max: 2.103e − 4
×10−4
2
Max: 0.915
Max: 233.945
200
0.8
150
0.6
100
0.4
50
0.2
0
0
−50
−0.2
1
0.8
−100
−0.4
0.6
−150
−0.6
0.4
−200
−0.8
0.2
Min: −234.822
(a)
1.8
1.6
1.4
1.2
Min: −0.917
(b)
Min: 1.715e − 13
(c)
Figure 2: For Λ = 4.2 𝜇m and 𝑑/Λ = 0.2, (a) the distribution of 𝑧-component of the electric field, (b) the distribution of 𝑧-component of the
magnetic field, and (c) the total energy distribution.
Max: 4.995e − 5
×10−5
Max: 0.578
Max: 149.414
4.5
100
0.4
50
0.2
0
0
−50
−0.2
1.5
−100
−0.4
1
0.5
4
3.5
3
2.5
2
Min: −149.38
(a)
Min: 3.07e − 8
Min: −0.579
(b)
(c)
Figure 3: For Λ = 4.2 𝜇m and 𝑑/Λ = 0.08, (a) the distribution of 𝑧-component of the electric field, (b) the distribution of 𝑧-component of
the magnetic field, and (c) the total energy distribution.
Max: 202.371
200
150
0.6
8
100
0.4
7
50
0.2
6
0
0
−50
−0.2
−100
−0.4
2
−0.6
1
−150
−200
Min: −201.783
(a)
Max: 9.003 e − 5
×10−5
9
Max: 0.770
5
4
3
Min: 4.689e − 11
Min: −0.772
(b)
(c)
Figure 4: For Λ = 4.2 𝜇m and 𝑑/Λ = 0.1, (a) the distribution of z-component of the electric field, (b) the distribution of z-component of the
magnetic field, and (c) the total energy distribution.
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200
150
0.6
100
0.4
50
0.2
0
0
−50
−0.2
−100
−0.4
−150
−0.6
−200
2
1.5
1
0.5
−0.8
Min: −0.879
Min: −224.829
(a)
Max: 2.444e − 4
×10−4
Max: 0.879
0.8
Max: 225.835
(b)
Min: 5.523 e − 16
(c)
Figure 5: For Λ = 4.2 𝜇m and 𝑑/Λ = 0.3, (a) the distribution of 𝑧-component of the electric field, (b) the distribution of 𝑧-component of the
magnetic field, and (c) the total energy distribution.
Max: 0.814
0.8
Max: 221.453
200
0.6
150
100
0.4
50
0.2
0
0
−50
−0.2
−100
−0.4
−150
−0.6
−200
2.5
2
1.5
1
0.5
−0.8
Min: −0.814
Min: −222.651
(a)
Max: 2.788e − 4
×10−4
(b)
Min: 1.143e − 26
(c)
Figure 6: For Λ = 4.2 𝜇m and 𝑑/Λ = 0.6, (a) the distribution of 𝑧-component of the electric field, (b) the distribution of 𝑧-component of the
magnetic field, and (c) the total energy distribution.
Max: 217.578
200
Max: 2.607 e − 4
×10−4
2.5
Max: 0.792
150
0.6
100
0.4
50
0.2
0
0
−50
−0.2
−100
2
1.5
1
−0.4
−150
0.5
−0.6
−200
Min: −0.792
Min: −217.289
(a)
(b)
Min: 2.713e − 39
(c)
Figure 7: For Λ = 4.2 𝜇m and 𝑑/Λ = 0.9, (a) the distribution of 𝑧-component of the electric field, (b) the distribution of 𝑧-component of the
magnetic field, and (c) the total energy distribution.
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5
Max: 1.823 e − 4
×10−4
1.8
Max: 0.609
0.6
Max: 155.133
150
1.6
100
0.4
50
0.2
1.2
0
0
1
−50
−0.2
−100
−0.4
1.4
0.8
0.6
0.4
0.2
−150
Min: −154.856
(a)
−0.6
Min: −0.607
(b)
Min: 1.555e − 12
(c)
Figure 8: For 𝑑 = 0.84 and 𝑑/Λ = 0.08 𝜇m, (a) the distribution of 𝑧-component of the electric field, (b) the distribution of 𝑧-component of
the magnetic field, and (c) the total energy distribution.
150
0.6
100
0.4
50
0.2
0
0
−50
−0.2
−100
−0.4
−150
−0.6
Min: −169.009
(a)
Max: 1.888e − 4
×10−4
1.8
Max: 0.668
Max: 168.202
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
Min: 6.405e − 13
Min: −0.665
(b)
(c)
Figure 9: For 𝑑 = 0.84 and 𝑑/Λ = 0.1 𝜇m, (a) the distribution of 𝑧-component of the electric field, (b) the distribution of 𝑧-component of
the magnetic field, and (c) the total energy distribution.
Max: 1.141
Max: 284.171
Max: 2.166e − 4
×10−4
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
1
200
0.5
100
0
0
−100
−0.5
−200
−1
Min: −285.155
(a)
Min: 2.611 e − 13
Min: −1.141
(b)
(c)
Figure 10: For 𝑑 = 0.84 𝜇m and 𝑑/Λ = 0.3 𝜇m, (a) the distribution of 𝑧-component of the electric field, (b) the distribution of 𝑧-component
of the magnetic field, and (c) the total energy distribution.
6
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Max: 365.285
Max: 1.921e − 4
×10−4
1.8
Max: 1.484
300
1
200
100
0.5
0
0
1.6
1.4
1.2
1
0.8
−100
−0.5
−200
0.6
0.4
−1
0.2
−300
Min: −364.21
Min: −1.481
(a)
Min: 5.172e − 16
(b)
(c)
Figure 11: For 𝑑 = 0.84 𝜇m and 𝑑/Λ = 0.6 𝜇m, (a) the distribution of 𝑧-component of the electric field, (b) the distribution of 𝑧-component
of the magnetic field, and (c) the total energy distribution.
Max: 1.578e − 4
×10−4
Max: 1.967
Max: 610.695
600
1.5
1.4
1
1.2
200
0.5
1
0
0
0.8
400
−200
−400
−600
Min: −610.153
−0.5
0.6
−1
0.4
−1.5
0.2
Min: −1.969
(a)
Min: 5.851e − 17
(b)
(c)
1.2
1.05
1
1
0.95
0.8
Normalized w
Normalized w
Figure 12: For 𝑑 = 0.84 𝜇m and 𝑑/Λ = 0.9 𝜇m, (a) the distribution of 𝑧-component of the electric field, (b) the distribution of 𝑧-component
of the magnetic field, and (c) the total energy distribution.
0.6
0.4
0.9
0.85
0.8
0.2
0.75
0
0.7
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
d/Λ
(a)
0.6
0.8
1
d/Λ
(b)
Figure 13: The normalized total energy of the fundamental mode at the central of the core for different 𝑑/Λ ratios for (a) fixed pitch Λ = 4.2 𝜇m
and (b) fixed diameter 𝑑 = 0.84 𝜇m.
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7
Normalized z-component of the Poynting flux
1.2
Normalized z-component
of the Poynting flux
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1.05
1
0.95
0.9
0.85
0.8
0.75
0.7
1
0
0.2
0.4
0.6
0.8
1
d/Λ
d/Λ
(a)
(b)
Figure 14: The normalized z-component of the Poynting flux of the fundamental mode at the central of the core for different 𝑑/Λ ratios for
(a) fixed pitch Λ = 4.2 𝜇m and (b) fixed diameter 𝑑 = 0.84 𝜇m.
1.4435
1.45
1.443
1.44
1.43
1.4425
neff
neff
1.42
1.442
1.41
1.4415
1.4
1.441
1.39
1.38
1.4405
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
d/Λ
(a)
0.6
0.8
1
d/Λ
(b)
Figure 15: The effective refractive indexes of the fundamental mode for different 𝑑/Λ ratios for (a) fixed pitch Λ = 4.2 𝜇m and (b) fixed
diameter 𝑑 = 0.84 𝜇m.
7 air holes from the structure. The cladding is the twodimensional photonic crystal with 4 rings of the triangular
lattice air holes in the silica matrix.
Firstly, for the fixed pitch length Λ = 4.2 𝜇m, the field and
energy distributions are investigated by varying the diameters
of air holes for the 𝑑/Λ values of 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7,
0.8, and 0.9. Later, a similar investigation is executed for the
fixed diameter of air-hole 𝑑 = 0.84 𝜇m, by varying the pitch
length.
For Λ = 4.2 𝜇m and the cross-section with the radius
of 23.1 𝜇m, the electric and magnetic fields and the energy
distributions of the structure for 𝑑/Λ = 0.2, 0.08, 0.1, 0.3, 0.6,
and 0.9 are shown in Figures 2, 3, 4, 5, 6, and 7, respectively.
For a given pitch length, Λ = 4.2 𝜇m, the electric
and magnetic field distributions are more extended into the
cladding region (i.e., guiding is weakened) when the 𝑑/Λ
ratio reduces (Figures 2–9). In other words, the light is well
confined in the core when the 𝑑/Λ ratio increases. For 𝑑/Λ =
0.08, the effective index of the fundamental mode becomes
complex, and thus the mode becomes leaky. As a result of
this, the energy intensity transported in the core region is
also reduced. The reason of this behavior is that the effective
cladding index approaches the core index, and hence the
index difference decreases.
For a fixed hole diameter, 𝑑 = 0.84 𝜇m, Figures 9 and 10
show the fields and energy distributions of the structure for
𝑑/Λ = 0.1 and the cross-section with the radius of 46.2 𝜇m
and 𝑑/Λ = 0.3 and the cross-section with the radius of
15.4 𝜇m, respectively. In addition, Figures 11 and 12 show the
fields and energy distributions of the structure for 𝑑/Λ = 0.6
and the cross-section with the radius of 7.7 𝜇m and 𝑑/Λ = 0.9
and the cross-section with the radius of 5.13 𝜇m, respectively.
Figures 2, 8, and 9 show that, for a given diameter of the
air holes, the confinement and the total energy confined in
8
the core region increase when the 𝑑/Λ ratio increases up to
a certain 𝑑/Λ value, for which the effective index becomes
complex (see Table 1). However, when the 𝑑/Λ ratio increases
above this value, it can be seen that the confinement is
lower and the energy involved in the core region decreases
(see Figures 2 and 10). This behaviour of the energy can be
seen more clearly from Figure 13 and attributed to the mode
becoming leaky due to approaching its cutoff. In addition,
𝑧-component of the Poynting flux of the fundamental mode
versus 𝑑/Λ values is illustrated in Figure 14.
For the different 𝑑/Λ ratios, the effective index variation
of the fundamental mode is given in Table 1. The effective
indexes are obtained by using cylindrical PML condition.
Figure 15 shows the real part of the effective refractive index
of the fundamental mode versus 𝑑/Λ ratios for both fixed
pitch and fixed diameter.
4. Conclusion
In this study, the effect of the diameters of air holes and
the pitch length on the field and energy distributions of the
fundamental mode in a solid core PCF with the triangular
lattice of air holes in the silica matrix is investigated. For a
given pitch length, the confinement and the energy density
for the fundamental mode decrease when 𝑑/Λ decreases. For
a given diameter of the air holes, the confinement and the
total energy confined in the core region increase when the
𝑑/Λ ratio increases up to a certain 𝑑/Λ value, for which
the effective index becomes complex. However, when the
𝑑/Λ ratio increases above this value, the confinement is
lower and the energy involved in the core region decreases.
This behavior of the energy can be attributed to the mode
becoming leaky due to approaching its cutoff.
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New York, NY, USA, 2nd edition, 2002.
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