A Simple but Accurate Method for Analytical Estimation of Splice Loss in Single-Mode Triangular Index Fibers for Different V Numbers Including the Low Ones

Rahul Debnath; Sankar Gangopadhyay

doi:10.1515/joc-2015-0062

Published by De Gruyter October 13, 2015

A Simple but Accurate Method for Analytical Estimation of Splice Loss in Single-Mode Triangular Index Fibers for Different V Numbers Including the Low Ones

Rahul Debnath and Sankar Gangopadhyay

From the journal Journal of Optical Communications

https://doi.org/10.1515/joc-2015-0062

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Abstract

We report formulation of simple but accurate analytical expressions at the splice for both angular and transverse mismatches in case of single mode triangular index fibers. Here, we employ the simple series expression for fundamental modal field of such fibers. The analysis takes care of large V values appropriate for single-moded guidance in such fibers as well as low V values. As regards evaluation of the said parameters by our formalism, little computation will be involved. We also show that our predictions match excellently with the available exact numerical results. Moreover, splices show high tolerance with respect to longitudinal separation and accordingly we restrict our analysis to the practical cases of transverse and angular offsets only. The present investigation should be of immense importance to the packagers and system engineers in the field of optical technology involving such kind of fibers.

Keywords: single-mode triangular index fibers; angular and transverse mismatches; splice loss

PACS: 42.81.-i; 42.82.-m

Acknowledgement

The authors are thankful to the anonymous reviewer for the constructive and helpful suggestions.

Appendix A

For weakly guiding single-mode fiber, the fundamental modal field ψ(R) in the fiber core is expressed by the following scalar wave equation [14]

(15)d2ψdR2+1RdψdR+V2(1−f(R))−W2ψ=0, R≤1

along with the boundary condition

(16)1ψdψdRR=1=−WK1(W)K0(W)

where V[=k0a(n12−n22)1/2] and W[=a(β2−n22k02)1/2] are the normalized frequency and cladding decay parameter respectively with k₀ and β representing the free space wave number and propagation constant respectively.

The fundamental modal field in the cladding of the fiber is given as

(17)ψ(R) K0(WR),R>1

Taking care of the fact that the fundamental modal field ψ(R) is an even function of R with ψ′(0) being zero and ψ(0) nonzero, one can approximate ψ(R) in the following form of Chebyshev power series [15, 16]

(18)Ψ(R)=∑j=0j=M−1a2jR2j

For the sake of simplicity without sacrifice of accuracy, it is sufficient to retain terms up to j=3 in eq. (18) [5–10] whereby one gets

(19)ψ(R)=a0+a2R2+a4R4+a6R6

The Chebyshev points are given as follows [15]

(20)Rm=cos(2m−12M−1π2); m=1,2,…,(M-1)

Clearly, we will get Chebyshev points appropriate for eq. (19) by putting M=4 and accordingly, the relevant three values of R are as follows

(21)R1=0.9749,R2=0.7818andR3=0.4338

Using eq. (19) in eq. (15), one gets the following three equations corresponding to three values of R given in eq. (21).

(22)a0(V21−f(Ri)−W2)+a2[4+Ri2(V21−f(Ri)−W2)]+a4[16Ri2+Ri4(V21−f(Ri)−W2)]+a6[36Ri4+Ri6(V21−f(Ri)−W2)]=0

where i=1, 2 and 3.

By applying least square fitting in the region W≤2.5, one can formulate the following linear relationship

(23)K1(W)K0(W)=α+βW

It deserves mentioning in this connection that over the range 0.60≤W≤2.5, the above linear relationship corresponds to α=1.0364623; β=0.3890323.ButforW≤0.60, the validity of (23) demands that least square fitting technique should be applied for some short intervals in order to find out the values of α and β. The relevant values of α and β for such W numbers in the low V region are presented in appendix Table 1 (below) for ready reference.

Interval of W	α	β
0.1–0.2	1.36850086	0.27248694
0.2–0.3	1.22199127	0.30182093
0.3–0.4	1.15243299	0.32309927
0.4–0.5	1.11884710	0.33673703
0.5–0.6	1.09220070	0.35008770
0.6–2.3	1.03462500	0.38903230
2.3–2.4	1.01280706	0.43094821

Using eqs (19) and (23) in eq. (16), one obtains

(24)a0(αW+β)+a2(αW+2+β)+a4(αW+4+β)+a6(αW+6+β)=0

The nontrivial solution of a₂, a₄, a₆ in terms a₀ from three equations in eq. (22) and one equation in eq. (24) requires the following condition

(25)α1β1γ1δ1α2β2γ2δ2α3β3γ3δ3α4β4γ4δ4=0

here,

(26)αi=V2(1−f(Ri))−W2βi=4+Ri2(V2(1−f(Ri))−W2)γi=16Ri2+Ri4(V2(1−f(Ri))−W2)δi=36Ri4+Ri6(V2(1−f(Ri))−W2)

where i = 1, 2, 3 and

α4=αW+ββ4=2+α4;γ4=4+α4;δ4=6+α4;

Employing eq. (25), one can obtain W for a given value of V. Further, from the knowledge of W for a particular V, one can easily find a₂, a₄, a₆ in terms of a₀ by using any three of four equations given by eqs (22) and (24). Therefore, the normalized fundamental modal field with respect to a₀ for a particular value of V is found by this simple method and following eq. (19), those are presented below

ψ(R)=a0+a2R2+a4R4+a6R6

ψ(R)=a01+a2a0R2+a4a0R4+a6a0R6

(27)ψ(R)=a01+A2R2+A4R4+A6R6,R≤1=a0(1+A2+A4+A6)K0(WR)K0(W),R>1

where A2j=a2j/a0;j=1, 2, 3.

Appendix B

We present the following important relations involving Bessel functions [17–20]

(28)(ρ+μ+ν)∫xρ−1Zμ(x)Zv(x)dx±(ρ−μ−ν−2)∫xρ−1Zμ+1(x)Zv+1(x)dx=xρ(Zμ(x)Zν(x)±Zμ+1(x)Zν+1(x))

(μ+2)∫Zν2(x)xμ+2dx=±(μ+1)[ν2−14(μ+1)2]∫Zν2(x)xμdx

(29)(±12xμ+1[{xddxZν(x)−(μ+1)2Zν(x)}2+{±x2−ν2+(μ+1)24}Zν2(x)])

(30)∫x−μ−ν−1Zμ+1(x)Zν+1(x)dx=∓x−μ−ν2(μ+ν+1)[Zμ(x)Zν(x)±Zμ+1(x)Zν+1(x)]

(31)∫xμ+ν+1Zμ(x)Zν(x)dx=xμ+ν+22(μ+ν+1)[Zμ(x)Zν(x)±Zμ+1(x)Zν+1(x)]

∫xZ02(x)dx=x22Z02(x)±Z12(x)

∫xZ12(x)dx=x22Z12(x)−Z0(x)Z2(x)

(32)∫x5Z12(x)dx=x6405Z12(x)±4Z22(x)−Z32(x)∫x3Z12(x)dx=x46Z12(x)±Z22(x)

(33)1xddxmZv(x)xν=±1mZν+m(x)xν+m

(34)Zν−1(x)−Zν+1(x)=±2νxZν(x)

Here, Zν(x) represents either the Bessel function Jν(x) or the modified Bessel function Kν(x) with the upper and lower signs standing for J-type and K-type Bessel functions respectively.

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Received: 2015-7-13

Accepted: 2015-9-23

Published Online: 2015-10-13

Published in Print: 2016-9-1

A Simple but Accurate Method for Analytical Estimation of Splice Loss in Single-Mode Triangular Index Fibers for Different V Numbers Including the Low Ones

Abstract

Acknowledgement

Appendix A

Appendix B

References

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