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Our empirical formula for the dispersion is based on the capillary model derived by Marcatili and Schmeltzer [8]. The capillary radius is corrected to account for the

HC-ARF’s core-cladding structure, as well as the operating wavelength. Moreover, the formula may incorporate the structural resonances in accordance with the

ARROW model, which arise due to the glass-web thickness of the antiresonant tubes. Our formula for the effective indexneff ofHC-ARFs has the form:

neff = v u u u t 1− _u 01λ 2πReff 2 +X m σmλ2 λ2− 2t_g/mqn2 g−1 2. (3.2)

Here, the second term accounts for the effect of the core-cladding arrangement, whereu01 ≈2.405is the first zero of the Bessel function of the first kind andReffis

the effective radius which we discuss below. The last term represents the effect of theARROWmodel and accounts for the structural resonances. ngis the refractive

index of the waveguide material,mis the resonance order, andσ_mdescribes its corresponding strength.

The empirical formula for the effective radiusR_eff is given by:

R_eff = f1R 1−

f2λ2

Rtg

!

, (3.3)

where f1 and f2 are two dimensionless parameters that need to be determined

to provide fitting in the short and long wavelength regions within the spectral range of our interest, respectively. As mostHC-ARFapplications are carried out at wavelengths far from the structural resonances, we may drop the last term in Eq. (3.2) for simplicity. Hence, the formula then becomes:

neff ≈1− ₁ 8 u₀₁λ πReff 2 . (3.4)

Figure3.2presentsGVDcalculated forR=12µm,tg=0.2µm,g=1µm, andn=6

3.2 Dispersion and fitting parameters 39 are also plotted in Fig.3.2. For the best fit, f1=1.1039and f2 =0.0480were used for

both Eqs. (3.2) and (3.4), while the effect of the first structural resonance was well represented by settingσ1 ≈2×10−6in Eq. (3.2). Moreover, for comparison, we have

also plottedGVDfollowing the approximate equation presented in Ref. [80] and perturbative extension of the Marcatili-Schmeltzer model presented in Poor-man’s model [81] for the same fiber structure. It clearly shows that the use of the effective radius significantly improves the fitting in the long wavelength region.

Figure 3.2 GVDcalculated usingFEM, Eq. (3.2), and Eq. (3.4). HereR = 12µm,t_g = 0.20µm, g = 1µm, andn = 6. The fitting parameters f1 = 1.1039and f2 = 0.0480in both Eq. (3.2) and

Eq. (3.4), andσ1≈2×10−6in Eq. (3.2). The amplitude and damping of the first structural resonance

for Ref. [80] are1 THzand4 THz, respectively. The same structural parameters are also used for

the perturbative extension of the Marcatili–Schmeltzer model (cyan color curve) presented in Poor-man’s model [81].

Figure3.3showsGVDcalculated numerically usingFEM, the empirical formula given in Eq. (3.4), and the Marcatili and Schmeltzer’s capillary model [8], for the threeHC-ARFstructures given in Fig.3.1. The other design parameters are fixed at R=12µm,tg =0.20µm, andg= 1µm. As for the fitting parameters, f1= 1.1039

was used for all three structures, while different f2values were required for different

Figure 3.3 TheGVDcalculated usingFEM, Eq. (3.4), and the Marcatili and Schmeltzer’s capillary

model forHC-ARFs with (a)n =6, (b)n=8, and (c)n=10. Here,R=12µm,tg =0.2µm, and g=1µm. The fitting parameters f1 =1.1039for all three structures and f2 =0.0480,0.0608, and 0.0729for (a), (b), and (c), respectively.

(b), and (c), respectively. This indicates that the dispersion is not strongly influenced by the cladding arrangement in the short-wavelength side, whereas it becomes important at longer wavelengths. Also, Fig.3.3clearly shows that Eq. (3.4), with the right choice of fitting parameters, accurately approximatesGVDover a wide spectral region, while for the simple capillary model, the agreement is poor in the long-wavelength region. We find that forHC-ARFs,GVDdiscrepancy between the Marcatili and Schmeltzer’s capillary model andFEMis small (<5%) at wavelengths

3.2 Dispersion and fitting parameters 41 below∼0.6p

Rt_gand far from the structural resonances. The deviation becomes more significant (>5%) at wavelengths longer than∼0.6p_Rtg, meaning that using

R_effwith the right choice of f1and f2is crucial for the accurate approximation of

GVDin this region.

In this section we systematically investigate the effect of changing each of the fibre structural parameters, to acheive the equation of the best fitting parameters f1and

f2. Those equations will provide us the values of f1and f2without need ofFEM

calculation.

3.2.1.

Core radius, perimeter gap, glass-web thickness, and num-

ber of antiresonant tubes

To see the effect of core radius, perimeter gap, and glass-web thickness, we present

GVD calculated usingFEM(solid-lines), as well as empirical formulae (dotted- lines) for various sets of the structural parameters as shown in Fig.3.4. In Fig.3.4(a), we maintain the same ratio between the core radius and the perimeter gapR/ g, while changing their absolute values plotted in different colored curves. It clearly indicates the long-wavelength-side drop in the dispersion slope becomes sharper as the core size is decreased. This is an evidence that, the smaller core confines light more tightly, therefore light experiences stronger effect of the waveguide. Another important observation is that the fitting parameters, f1 = 1.1061 and

f2=0.05remain the same for all curves in Fig.3.4(a), suggesting that these values

are determined by the relative parameterR/gand not by their absolute values. Figure 3.4 (b) shows the effect of changing the glass-web thickness while all other parameters are kept constant. The thickness alone dictates the resonance wavelengths in accord with the ARROWmodel [16]. Similar to Fig. 3.4(a), the fitting parameters f1 =1.1061and f2 = 0.05do not vary with the change intg. It

is worth noting that the bandwidth of the resonances widens as the glass-web thickness is increased. Also, the glass-web resonance becomes stronger in fibers with smaller cores due to stronger effect of the cladding on light. Hence, we can conclude thatRandt_gare the dominating factors in determining the resonances strength and its bandwidth. We have found that they are well approximated by

Figure 3.4 GVDcalculated usingFEM(solid-lines) and empirical formulae—Eq. (3.4) in (a) and

Eq. (3.2) in (b)—(dotted-lines) to investigate the role of various fiber structural parameters. (a) ChangeRandgwhile maintaining the ratio between the twoR/g=9.6154. The glass-web thickness tg=0.4167µmandn=6for all curves. (b) Changetgwhile keepingR=40µm,g=4.16µm, and n=6. usingσm ≈ tg ngR 2.303_nga m+2 3m 3.57a

, wherea=1.83+(2.3t_g/R),mis the resonance order, andngis the waveguide material index. We have checked the validity of the above

expression forσm up tom=4.

In fact, our analysis reveals that the fitting parameters f1 and f2 depend on the

dimensionless ratio between the core radius and the perimeter gap. Figures3.5(a) and (b) present the variations in f1and f2as a function of the ratioR/gforHC-ARF

with differentnshown in Fig.3.1. They clearly indicate that f1curve does not vary

with the change inn, while f2curve shifts up with increasingn. Figures3.5(c) and

(d) also present how f1 and f2 vary with increasingnfrom six to twelve for three

differentR/ gratios.

3.2 Dispersion and fitting parameters 43

Figure 3.5 The fitting parameters (a) f1and (b) f2determined using the best fit as a function of R/gforHC-ARFs. Note that in (a), f1does not change withnand the three lines coincide. (c)

and (d) show the changes in f1and f2, respectively, for different number of antiresonant tubes, as

indicated in the right-hand side panel.

ratioR/g, which are given by:

f1=A1exp

A0

R/ g

!

f2 =B1nexp

B0

R/g

!

−_B2n+_B3, (3.5b)

hereA0,A1,B0,B1,B2, andB3are empirical constants. We found that the choice of

A0 =0.097041,A1 =1.095,B0 =0.76246,B1= 0.007584,B2 =0.002, andB3=0.012

provides the best agreement seen when Eq. (3.5a) and (3.5b) are compared with curves in Fig.3.5.

3.2.2.

Number of nested elements

Past studies have shown that having nested elements in the antiresonant tubes improves the guiding properties [26,31]. Hence, we study the effect of having the nested elements on the group-velocity dispersion formula in Fig. 3.6. We

Figure 3.6 The fitting parameters (a) f1and (b) f2determined using the best fit as a function of R/gforHC-ARFwith three different nested structures in the antiresonant tubes. Note that in (a),

f1does not change withNand the three lines coincide.

3.3 Effective mode area 45