During recent years, lots of studies have been done about PCFs or holy fibers.
This is due to their capabilities of these fibers in handling propagation modes
through themselves (Saitoh and Koshiba, 2005). This aspect has introduced these
devices as the most popular and applicable optical instruments such as channel
allocation in the wavelength division multiplexing transmission systems and
Pressure Sensor Applications (Kim, 2003).
PCFs are categorized as mono-material fibers which have a central light guiding
area surrounded by rods in a triangular lattice (Li et
al., 2004). These rods are filled by air and their diameters and hole
pitches are almost the same as the amount of wavelength. This novel structure
of PCF causes new properties such as wide single-mode wavelength range, unusual
chromatic dispersion and high or low non-linearity (Saitoh
and Koshiba, 2005). There are several methods to analyze these fibers including:
Effective Index Method, (EIM), Localized Basis Function Method, Finite Element
Method (FEM), Finite Difference Method (FDM), Plane Wave Expansion Method (PWM)
and Multi-Pole Method (Saitoh and Koshiba, 2005; Li
et al., 2006; Sinha and Varshney, 2003).
Numerical methods consume too long time consuming and need large amount of iterative computations (Saitoh and Koshiba, 2005). Usually these methods are too mighty and their broad capabilities are not required for studying of PCFs. Despite of limitations and inaccuracies, other analytic methods are introduced to replace these ones (Saitoh and Koshiba, 2005). In the present study, Modified Fully Vectorial Effective Index Method (MFVEIM) and Empirical Relations Method (ERM) are studied among them.
Here, via fully vectorial effective index method (FVEIM), the effective cladding
index of a hexagonal unit cell which consists of a fiber rod, is calculated
with respect to the rod diameter and pitch (Λ). Then the effective index
of PCF is obtained by using the effective cladding index (Li
et al., 2004). However, comparing with an accurate method like as
full vector finite element method (FVFEM), the effective index obtained by FVEIM
is not accurate for values of d/Λ. In order to correct this problem, Yong-Zhao
et al. (2006) suggested a method so called Modified FVEIM, which
efficiently improved FVEIM. In fact, MFVEIM applies an effective core radius
(rc) which changes by hole diameter and hole pitch; while FVEIM uses
a constant rc (Yong-Zhao et al., 2006).
In Empirical Relations Method (ERM), empirical relations for parameters of V
(Normalized Frequency) and W (Normalized Transverse Attenuation Constant) of
PCFs with respect to the basic geometrical parameters (i.e., the air hole diameter
and the hole pitch) are formed (Saitoh and Koshiba, 2005).
Then V and W are computed and used to calculate PCF's basic parameters (Saitoh
and Koshiba, 2005). Hereafter, the obtained results of these two methods
are compared and we show that the accuracy of the methods changes by Λ
We will also present the calculations of the second and third order dispersions for chromatic dispersion in PCFs with well known properties. The Sellmeier relation has been used to calculate material dispersion.
MODIFIED FULLY VECTORIAL EFFECTIVE INDEX METHOD
Both the effective cladding index and the effective index of the guided mode
of the PCF are calculated using fully vectorial equations (Li
et al., 2004).
Solving Maxwell equations in the infinite two-dimensional photonic crystal
structure, we will get the modal index of fundamental space filling mode (nFSM)
(Bjarklev et al., 2003; Birks
et al., 1997).
In order to calculate nFSM for the PCF, the hexagonal unit cell
(Fig. 1) is approximated by a circular one of radius R. In
calculating nFSM using FVEIM, the boundary conditions at point P
should be perfect electric and perfect magnetic conductor (Li
et al., 2007). After applying the boundary condition to the characteristic
equation, we will have the following equation:
where, l = 1,
and Il is Bessel function. One should note that Ul and
Il should play a similar role as Jl and Kl,
respectively, in the characteristic equation of step index fiber (Li
et al., 2004). K(w) and T(w) are defined as
In all above equations, the derivatives are taken with respect to the function
arguments. Note that the optimal radius for FVEIM is R = Λ/2 (Midrio
et al., 2000) and we use the same radius for MFVEIM.
By solving Eq. 1 for β(w), we can calculate effective cladding index using nFSM(w) = β(w)c/w.
Afterwards, βc(w) is achieved via solving Eq. 2:
|| The hexagonal unit cell and its circular equivalent
where, l = 1,
and, with nc(w) being the refractive index of the core material.
Note that both nsilica(w) in Eq. 1 and nc(w)
in Eq. 2 are 1.45 as a fixed value in the current method.
In FVEIM, the parameter of rc takes a fixed value and different values are suggested for that in the references. But in MFVEIM, rc changes when PCF has different relative hole diameters. In fact rc is calculated by following formula:
where, b = 0.6962, a0 = 0.0236, a2 = 0.0056 and a3 = 0.1302 (Saitoh and Koshiba, 2005).
Afterwards, by using neff(w) = βc(w)c/w, the effective index of PCF is obtained.
Now, it is possible to calculate the total dispersion using the following formula:
where Dm is the material dispersion obtained from the Sellmeier relation.
EMPIRICAL RELATIONS METHOD
In this method, the refractive index of silica is considered constant as ncore = 1.45 and the effective core radius is defined as (Saitoh and Koshiba, 2005).
Recently, it has been claimed that the triangular PCFs can be well parameterized
in terms of the V parameter (Koshiba and Saitoh, 2004)
that is given by:
First, from the study by Saitoh and Koshiba (2005),
we calculate V by using
(Saitoh and Koshiba, 2005), where,
Subsequently, the effective cladding index nFSM is obtained from
Eq. 5. Then referring to Table by Saitoh
and Koshiba (2005) and from:
(Saitoh and Koshiba, 2005), where
we can calculate W. From Eq. 6 for given W and nFSM,
can be obtained and finally one can calculate the total dispersion using Eq.
Figure 2 shows nFSM calculated as a function of
d/Λ by ERM, MFVEIM and FVFEM (Koshiba and Saitoh, 2002).
The accuracy of our calculations is proved by Fig. 2.
Figure 3a shows that for d/Λ = 0.2, 0.3, 0.4, 0.7 and 0.8, the relative difference between by ERM and MFVEIM is almost high, while for d/Λ = 0.5 and 0.6 this difference is smooth and low. So, we can conclude that as d/Λ either increases or decreases more, two methods result in more different amounts for neff.
Hereby, the comparison between the accuracies of two above mentioned methods
(ERM and MFVEIM) with respect to the method of Fully Vectorial Finite Element
(FVFEM) will be made. Referring to (Saitoh and Koshiba,2005),
it has been shown that achieved by ERM deviates less than 15% from that of FVFEM,
while it is calculated in restricted range (Saitoh and Koshiba,
||neff as a function of λ/Λ, obtained by
ERM, MFVEIM and FVFEM
||Relative difference between neff obtained by different
methods. (a) ERM and MFVEIM for several s. (b) FVFEM and MFVEIM for several
d/Λs. (b) FVFEM and MFVEIM for d/Λ=0.4
Moreover,Fig. 3b shows that the relative difference between
neff obtained by MFVEIM and FVFEM can exceed 15%.
||The second order dispersion obtained by ERM, MFVEIM and FVFEM.
(a) Λ = 2 μm (b) Λ = 3 μm
So, it can be concluded that ERM is preferable from the accuracy view point.
Next, we show the accuracy of MFVEIM and ERM via comparing the results of second
order dispersion obtained by them with the results of second order dispersion
obtained by FVFEM. Figure 4a and b illustrate
this comparison for Λ = 2 and 3 μm for same d/Λs.
The evaluation of neff via ERM causes the parameter of second order dispersion being closer to that was achieved by FVFEM.
It is interesting to observe that for Λ = μm, not only the does the
second order dispersion from ERM and MFVEIM become closer to FVFEM, but also
both methods agree better in results. And something else can be obtained, is
that the MFVEIM and ERM can be used for big Λ. Because we have seen in
our studying that the increasing the error in small pitch in both methods due
to they can not be able to have good accuracy for calculating of effective interaction
between mutual rods and between core and rods.
||The third order dispersion obtained by ERM, MFVEIM (a) Λ
= 2 μm (b) Λ = 3 μm
Figure 5a and b show the third order dispersion
obtained by MFVEIM and ERM for Λ = 2 and 3 μm. Our previous claim
for the second order dispersion is accurate for the third order dispersion too.
It means that as pitch increases, both methods agree more.
We have seen that ERM has less error than MFVEIM in defined range. On the other
hand, ERM is faster and simpler than MFVEIM. According to Li
et al. (2004, 2006, 2007)
show that FVEIM is more accurate than scalar effective index method (SEIM).
Meanwhile, referring to (Yong-Zhao et al., 2006), one can see
that MFVEIM is more accurate than FVEIM and it is shown that ERM is better than
SEIM (Pourkazemi and Mansourabadi, 2008). As the result,
so we can claim that ERM is more accurate, simpler and faster than three other
methods (i.e., SEIM, FVEIM and MFVEIM) in its defined range.