The normalized PCF parameters \(v_\) and \(u_\) in the infinite cladding region of that fiber are defined as [1]:

$$v_ = k\Lambda \left( ^ - 1} \right)^$$

(11)

and

$$u_ = k\Lambda \left( ^ - \frac }} }}} \right)^$$

(12)

with

where \(w_\) is the third normalized parameter in the same region of the PCF.

To compute the \(\beta_\) and hence the \(n_\), the approximation of a hexagonal unit cell in a regular photonic crystal by a circle is considered [2, 25]. Then, using the boundary conditions for the field at the interface of air-hole and silica and its first derivative at the outer boundary of the circular unit cell, and making use of Bessel functions, the following Eigen value Eq. (1) can be derived, from which corresponding to a fixed value of \(v_\), obtained from Eq. (11) for fixed \(\lambda\) and \(\Lambda\) values, the concerned \(u_\) can be determined:

$$\begin w_ I_ (a_ w_ )[J_ (bu_ )Y_ (a_ u_ ) - J_ (a_ u_ )Y_ (bu_ )] \hfill \\ \hfill \\ + u_ I_ (a_ w_ )[J_ (bu_ )Y_ (a_ u_ ) - J_ (a_ u_ )Y_ (bu_ )] = 0 \hfill \\ \end$$

(14)

With.

$$a_ = \frac\,b = \left( }} \right)^$$

To develop the empirical relation for the index \(n_\) of the PCF, a specific value of \(n_\) is assumed and the roots of Eq. (6), i.e., the \(u_\) values are computed for different \(d/\Lambda\) and \(\lambda /\Lambda\) values. From the \(u_\) values, it is possible to determine the values of \(n_\), by replacing \(\beta /k\) in Eq. (4) by its highest value \(n_\). This has led to propose the simple relation as follows:

$$n_ = A + B\left( } \right) + C\left( } \right)^$$

(15)

where \(A\), \(B\) and \(C\) are three different optimization parameters, dependent only on the relative air-hole diameter \(d/\Lambda\). Only up to the quadratic term in Eq. (7) is tested as sufficient to provide results with tolerable accuracy.

Here, for each value of \(d/\Lambda\), \(\lambda /\Lambda\) is varied, and the \(n_\) values are obtained from the corresponding \(u_\) values, computed from Eq. (14) for fixed \(v_\) values. By least-squares fitting of \(n_\) in terms of \(\lambda /\Lambda\) to Eq. (15) for a particular \(d/\Lambda\), the values of \(A\), \(B\) and \(C\) are generated. Following the same process, various \(A\), \(B\) and \(C\) values are simulated for various \(d/\Lambda\) in the ESM region of the PCF, from which empirical relations of \(A\), \(B\) and \(C\) in Eq. (7) are written in terms of \(d/\Lambda\), as given in the following:

$$A = a_ + a_ \left( } \right) + a_ \left( } \right)^ = \sum\limits_^ } \left( } \right)^$$

(16)

$$B = b_ + b_ \left( } \right) + b_ \left( } \right)^ = \sum\limits_^ } \left( } \right)^$$

(17)

and

$$C = c_ + c_ \left( } \right) + c_ \left( } \right)^ = \sum\limits_^ } \left( } \right)^$$

(18)

where ai, bi, ci, and (i = 0, 1 and 2) are the optimization parameters for \(A\), \(B\) and \(C\), respectively. The values of these constant coefficients are presented in Table 1 and 2 of Sect. "Evaluation of Splice Loss between two fibers without any misalignment or offset".

After knowing \(A\), \(B\) and \(C\) from Eqs. (16) and (18), those can be substituted into Eq. (15) to find out the index \(n_\), directly, for any \(\lambda /\Lambda\) and \(d/\Lambda\) in the ESM region of PCFs.

See Eqs. 11, 12, 13, 14, 15, 16, 17, 18.