Practical additional degrees of freedom which can be easily

Practical existence of
photonic crystal fibers (PCFs) as waveguides with a transverse air-silica
matrix around a solid-silica core has been found since early 1996 9. The
geometrical properties of PCFs like air-hole diameter (d) and the distance between the centers of two adjacent air-holes
i.e. pitch (?), give us additional degrees of freedom which can be easily
tailored to achieve the novel light properties of PCFs. PCFs are become more
popular from the user’s view point of interest due to its ability to increase
the non-linearity compared to the conventional fibers by orders of magnitude
4. Because of few popular features of PCFs, they are promising candidates in
the wide arena of interesting applications, like surface plasmon generation,
super-continuum generation etc. PCF based surface plasmonic resonance sensors are
our prior focusing arena for this thesis work which consists of selectively
metal-filled air-hole(s), enhances the phase matching between the plasmonic
mode and the core-guided mode (ref. Chapter 5 for more details).

            One of the attractive features of PCF is the ease of
manipulation in its core size by suitable tuning of the geometrical parameters d and
?. We can easily achieve very small effective mode area with the well-confined
propagating mode within the core by controlling those two above mentioned
parameters 4. Such PCFs are better known as suspended core fibers, in which
the first ring of air-holes of the cladding are elongated and tapered towards
the center of the core 20. This amount of elongation and tapering of
air-holes can be controlled by a geometrical parameter, called suspension
factor (SF), will be discussed in details in the Section 4.6.1. Overlapping
between the core guided mode and surface plasmon polariton (SPP) mode can be
appreciably controlled by this suspension factor.

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            The main advantages of such fibers manifests the
extremely small core size, the core mode extends substantially into the
surrounding air-holes in the cladding. Such evanescent fields of the
propagating core modes can be used to detect various bio-fluids and gases. This
suspended core PCFs offer the higher sensitivity due to the potential for long
interaction wavelengths. Proper focusing and launching of light are really
challenging in a sub-micron scaled non-circular core of PCFs.

 

 

4.6.1  
Suspension Factor

We have already
discussed in the earlier section that the first ring of air-holes of the
cladding of a particular PCF are elongated and tapered towards the center of
the core and this amount of elongation and tapering can be controlled by a geometrical
parameter known as the suspension factor (SF). The limitation or range of SF
for a specific PCF structure is restricted by the geometrical parameters d and
?. We have taken the relation between the suspension factor with d and
? as reference 20 for our further implementation into our thesis works. We
must mention here that the ratio d/?
is the standard parameter for describing any PCF geometry containing a perfectly
hexagonal structure and it gives a measure of the air-filling fraction of the
PCF 21. In such PCFs, the air-filling fraction is indirectly controlled by
the suspension factor. In a suspended core PCF, the d/? values are applicable to the second and third rings of
air-holes of the microstructured cladding only, which retain the hexagonal
geometry.

            Each air-hole presents in the first ring is duly formed
by a combination of a circle and a second order Bézier curve 22. Bézier curve
is basically the path traced by the function B(t), is defined by the eqn.
(4.9) where P1, P2 and P3 are the mentioned points in the Fig. 4.7
20, 23 where t is the varying
parameter. The curve starts to move from the point P1
in the direction of P2 and then bends to move back to the point P3 in the direction from P2. In other words, the
tangents in both P1 and P3
intersect each other at the point P2.

 

                  (4.9)

 

            In the context of solid core PCF, the formation of such a
Bézier curve and a circle is schematically depicted in Fig. 4.7 (b), where O is
the centre of the circular air-hole in the first ring, C is the centre of the
entire PCF geometry and the control points of the Bézier curve are P1,
P2 and P3 are
already mentioned in the eqn. (4.9).  denotes the different position of P2.
The suspension
factor (SF) of the solid core is defined as OP2/OC.
By manipulating the control point P2
of the Bézier curve, air-holes with different suspension conditions can be
achieved. It is worth mentioning that the SF cannot contain
any arbitrary value and it is delimited by an upper and lower bound based on
the structural integrity.

Figure 4.7: Schematic diagram of the (a) Second order Bézier curve, and (b) Elongated air-hole formed by Bézier curve.

            Different Bézier curves like P1Q2P3
(indicated by red dashed line) and P1P3
(indicated by blue dashed line) with the control points P2
and
 have been shown in the above Fig. 4.7 (b),
which denote two two different suspension condition of air-hole. The
geometrical formation of each suspension factor applied air-hole in the first
ring is also depicted in the Fig. 4.8 20, 23.

Figure 4.8: Schematic representation of the ideal circular and egg-shaped air-holes formed by using Bézier curves.

            From the Fig. 4.8, the difference between an ideal
circular structure and suspended core structure is clearly observed. A
perfectly circular air-hole indicates the minimum value of SF while the maximum
value of SF is limited by the width of silica presents in between two adjacent
air-holes. Suspension factor can be increased up to the point where the
individuality of the air-holes is strictly maintained and do not merge or collapse
with each other. D. Ghosh et al. 20
suggested the following relation mentioned in the eqn.
(4.10.a) and eqn. (4.10.b) to determine the maximum and minimum values of SF,
based on numerical calculations within the limit, 0.75 ? d/? ? 0.95.

                                               (4.10.a)

                                     (4.10.b)

x

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