# Splice Loss of Graded-Index Fibers: Accurate Semianalytical Descriptions Using Nelder-Mead Nonlinear Unconstrained Optimization with Three-Parameter Fundamental Modal Field

Authors: Raja Roy Choudhury; Arundhati Roy Choudhury; Mrinal Kanti Ghose
Journal: Advances in Optical Technologies
 (2013)

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## Abstract

A faster and accurate semianalytical formulation with a robust optimization solution for estimating the splice loss of graded-index fibers has been proposed. The semianalytical optimization of modal parameters has been carried out by Nelder-Mead method of nonlinear unconstrained minimization suitable for functions which are uncertain, noisy, or even discontinuous. Instead of normally used Gaussian function, as the trial field for the fundamental mode of graded-index optical fiber a novel sinc function with exponentially andR-3/2 (R is the normalized radius of the optical fiber) decaying trailing edge has been used. Due to inclusion of three parameters in the optimization of fundamental modal solution and application of an efficient optimization technique with simple analytical expressions for various modal parameters, the results are found to be accurate and computationally easier to find than the standard numerical method solution.


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## Body

## 1. Introduction

Single mode fiber is considered as the most important broadband transmission media for optical communication system. Achieving accurate values of modal field distribution in such fiber is very essential, as it can provide basic solutions for wave equation and many useful properties like splice loss, microbending loss, fiber coupling, and the prediction of intramodal dispersion [1]. However, the various expressions for the fundamental modal field that have been reported so far are not able to predict propagation constant and modal parameters exactly in all regions of single mode operation [2]. The Gaussian approximation shows poor accuracy for lower normalized frequency region although this region may involve single mode fiber operation [2]; however, it can perform satisfactorily only for higher normalized frequency region and give good result near the cut-off frequency of next higher mode [3]. Besides, it is also equally important that the approximation should describe the field in the cladding accurately, as it is useful in the study of evanescent coupling problem. To overcome these inefficiencies, an exponentially and R-3/2 decaying trailing edge fundamental modal field solution in core-cladding interface region has been considered.To achieve higher accuracy compared to Gaussian function, the Gaussian-Hankel [2], the generalized Gaussian [4], the extended Gaussian [5], and the Laguerre-Gauss/Bessel expansion approximation [6, 7] have been proposed so far. An approximate analytical description with no requirement for optimization has also been presented [8]. But such analytical expression may not work for all specifications of an optical fiber. In the proposed formulation, Nelder-Mead method of nonlinear unconstrained minimization and the process of minimization of core parameter (U) for all specific requirements have been used to achieve an accurate and computationally appropriate result.Unlike the existing reported fundamental modal solution with one or two parameters [2–8], an attempt has been made to propose a three-parameter fundamental modal field solution for graded-index fiber to introduce more flexibility to solve the fundamental modal solution more accurately, especially in core-cladding interface region wherein the solution has different form (exponentially and R-3/2 decaying trailing edge). Ghatak et al. [9] had arrived at simple analytical expressions to describe different optical fiber characteristics by implementing variational technique. Again, the optimization process requires expressions for propagation constant β and core parameter U. The analytical expressions for β and U used for the present study involve many fiber parameters, such as core radius (a), refractive indices of core and cladding (nco and ncl), aspect ratio (S0), and wavelength (λ=2π/k), where k is the free space wave number. Hence, any desired specification can be incorporated by varying these parameters. Now, the task of optimization can be carried out by using Nelder-Mead method of nonlinear unconstrained minimization, to meet a particular design.For graded-index optical fiber at the splices, the power transmission coefficients with transverse and angular mismatch have been estimated by using the methods given by Meunier and Hosain [10] and Hosain et al. [11]. For arbitrarily graded-index fiber, the Gaussian approximation does not give accurate result at lower normalized frequency or in cases where the power law profile deviates from its simplest form [12]. Further, the numerical solution requires rigorous computations and specialized numerical techniques [13]. However, using the proposed three-parameter fundamental modal solution coupled with Nelder-Mead method of nonlinear unconstrained minimization, the algorithm becomes comparatively easier to be implemented on an ordinary personal computer, which provides computationally more efficient result [14, 15] than standard numerical method and yields excellent agreement with exact solutions. This is achieved due to the fact that requisite analytical formulae are deduced beforehand and then parameters of those analytical expressions are found by optimization using Nelder-Mead simplex method for nonlinear unconstrained minimization. Furthermore, Nelder-Mead simplex method for nonlinear unconstrained minimization is a direct search method [16, 17] which does not require any derivative information, so it can optimize nonstationary functions, as needed for the problems under study [18–20]. The proposed semianalytical model can also be used in the study of nonlinear fiber [21].


## 2. Formulation of the Problem

### 2.1. Theory

Splice loss can be evaluated analytically with the help of the following equations [22]:
(1)∫|ψ1|2RdR=log(αR/R0)-Ci(2αR/R0)2,(2)∫|ψ2|2RdR=sin2αe2μR(-R0e-2μR/R0+2μREi(1,2μRR0)),(3)∫nf22R|ψ1|2dR=1(-4α+4αS0)×[2n22αS0log(αRR0)-2n22αS0Ci(2αRR0)-2n12αlog(αRR0)+2n12αCi(αRR0)-n12R0sin(2αRR0)+2n12Rα+n22R0sin(2αRR0)-2n22αR],(4)∫nf22R|ψ2|2dR=e2μsin2αR(S0-1)×[-n22R0S0e-(2μR/R0)+2n22μRS0Ei(1,2μRR0)+n12R0e-(2μR/R0)-2n12μREi(1,2μRR0)-n12R0REi(1,2μRR0)+n22R0REi(1,2μRR0)],(5)∫|dψ1dR|2RdR=[12R02log(αRR0)-12R02Ci(2αRR0)]α2+α2RR0sin(2αRR0)+14R2cos(2αRR0)-14R2,(6)∫|dψ2dR|2RdR=-sin2αe2μ4R02R3e2μR/R0((1,2μRR0)3R03+3R02μR-2R0μ2R2+4Ei(1,2μRR0)μ3R3e2μR/R0),(7)∫|ψ1|2R3dR=18α2[-2αRR0sin(2αRR0)+2α2R2+R02-R02cos(2αRR0)],(8)∫|ψ2|2R3dR=-sinα2e2μR022μe2μR/R0,(9)∫|ψ1|2RqdR=α2R02(q+1)Rq+1×Hypergeom([1,q2+12],[2,32,32+q2],-α2R2R02),(10)∫|ψ2|2RqdR=2(2-q)(μR0)-qμ2R0e2μ×sinα2[1(q-2)(q-1)q(q+1)(μRR0)-q/2×e-μR/R0Rq2q/2(μR0)q×WM(q2,q2+12,2μRR0)+2q/2-3(q-2)(q-1)qμ3(μRR0)-q/2×e-μR/R0R03Rq-3(μR0)q×(4μ2R2R02+2qμRR0-q+q2)×WM(q2+1,q2+12,2μRR0)(μRR0)-q/2],(11)∫1R|dψ1dR|2dR=18R02R4[-2α2R2+2R0αRsin(2αRR0)-R02+R02cos(2αRR0)],(12)∫1R|dψ2dR|2dR=sin2α120R04R5[(1,2μRR0)63R04μRe(-2μ(-R0+R)/R0)-2R03μ2R2e(-2μ(-R0+R)/R0)+2R02μ3R3e(-2μ(-R0+R)/R0)-4R0μ4R4e(-2μ(R0+R)/R0)+54R05e(-2μ(-R0+R)/R0)+8μ5R5e2μEi(1,2μRR0)],(13)∫|d2ψ1dR2|2RdR=α42R04[log(αRR0)-Ci(2αRR0)]-α2R02R2cos(2αRR0)+αR0R3sin(2αRR0)-12R4+12R4cos(2αRR0),(14)∫|d2ψ2dR2|2RdR=sin2α32R04R5e2μ[(1,2μRR0)-90R05e(-2μR/R0)-135R04μRe(-2μR/R0)-86R03μ2R2e(-2μR/R0)-10R02μ3R3e(-2μR/R0)-12R0μ4R4e(-2μR/R0)+24μ5R5Ei(1,2μRR0)],(15)∫|d3ψ1dR3|2RdR=α6[12R06log(αRR0)-12R06Ci(2αRR0)]+α52R05Rsin(2αRR0)+α4[54R04R2cos(2αRR0)+34R04R2]-4α3R03R3sin(2αRR0)-6α2R02R4cos(2αRR0)+6αR0R5sin(2αRR0)+3R6cos(2αRR0)-3R6,(16)∫|d3ψ2dR3|2RdRR=sin2α64R06R7e2μ×[40μ7R7Ei(1,2μRR0)+1575R07e-(2μR/R0)+2625R06μRe-(2μR/R0)+2082R05μ2R2e-(2μR/R0)+999R04μ3R3e-(2μR/R0)+246R03μ4R4e-(2μR/R0)+42R02μ5R5e-(2μR/R0)-20R0μ6R6e-(2μR/R0)(1,2μRR0)],(17)∫1R|dψ1dR||d3ψ1dR3|dR=18R04R6[(2αRR0)-4R04-3α2R02R2+2α4R4-2α3R0R3sin(2αRR0)+8αR03Rsin(2αRR0)-5α2R02R2cos(2αRR0)+4R04cos(2αRR0)],(18)∫1R|dψ2dR||d3ψ2dR3|dR=sin2α240R06R7e(2μR/R0)e2μ×[(1,2μRR0)975R06μR+474R05μ2R2+123R04μ3R3-2R03μ4R4+2R02μ5R5-4R0μ6R6+8μ7R7e(2μR/R0)Ei(1,2μRR0)+675R07(1,2μRR0)],(19)|d2ψdR2|=sinα4R5(R0/R)3/2eμe-(μR/R0)×(15R02+12R0μR+4μ2R2),
where ψ1 and ψ2 are given in (26). α, R0, and μ are the three variational parameters present in the fundamental modal solution.E
i
(
z
) is exponential integral given by [23] as follows:
(20)Ei(1,z)=∫z∞e-ttdt.Ci(z) is the cosine integral function, defined by [23], as follows:
(21)Ci(z)=χ+ln(z)+∫0zcost-1tdt,
where χ is Euler's constant 0.5772.W
M
(
k
,
m
,
z
) are the Whittaker functions which are solutions to the Whittaker differential equation [23].Hypergeom
(
n
,
d
,
z
) is the generalized hypergeometric function F(n,d,z), where [23]
(22)F(n,d,z)=∑k=0∞Cn,kCd,k·zkk!,
with,
(23)Cv,k=∏j=1vΓ(vj+k)Γ(vj),
where Γ(a) is the gamma function [23].


### 2.2. Basic Formulations

The refractive index profile for a weakly guiding fiber is given by(24)nf12=n22+(n12-n22)f1,for0≤R≤S0,nf22=n22+(n12-n22)f2,forS0≤R≤1,nf32=n22+(n12-n22)f3,forR>1,
where the normalized profile functions for the trapezoidal and triangular index profiles fi (i=1,2,3) are given by
(25)f1=1,f2=1-R1-S0,f3=0.
Here, S0 is the aspect ratio, R is the normalized radius (=r/a), a is the core radius, r is the actual radius of the optical fiber, and n1 and n2 are, respectively, the refractive indices of the core axis and cladding.For the present study, the following approximations for the fundamental mode as the trial field have been proposed:(26)ψ1=sin(αR/R0)RforR≤R0,ψ2=(sin(α)R)eμ(1-(R/R0))R0RforR>R0,
where α, R0, and μ are the three variational parameters present in the fundamental modal solution.To employ variational technique, first the scalar variational expression for the propagation constantβ as given by (27) has been considered and is shown in equations through (28) to (30) as follows:
(27)β2=k2∫0∞n2(R)|ψ(R)|2RdR-(1/a2)〈ψ′2〉〈ψ2〉,(28)β2=1〈ψ2〉[∫0R0k2nf12|ψ1|2RdR+∫R0S0k2nf12|ψ2|2RdR+∫S01k2nf22|ψ2|2RdR+∫1∞k2nf32|ψ2|2RdR-1a2〈ψ′2〉∫0R0k2nf12|ψ1|2RdR],
for R0<1 and R0<S0,
(29)β2=1〈ψ2〉[∫0S0k2nf12|ψ1|2RdR+∫S0R0k2nf22|ψ1|2RdR+∫R01k2nf22|ψ2|2RdR+∫1∞k2nf32|ψ2|2RdR-1a2〈ψ′2〉∫0S0k2nf12|ψ1|2RdR],
for R0<1 and R0>S0,
(30)β2=1〈ψ2〉[∫0S0k2nf12|ψ1|2RdR+∫S01k2nf22|ψ1|2RdR+∫1R0k2nf32|ψ1|2RdR+∫R0∞k2nf32|ψ2|2RdR-1a2〈ψ′2〉],
for R0>1, where
(31)〈ψ2〉=∫0R0|ψ1|2RdR+∫R0∞|ψ2|2RdR,〈ψ′2〉=∫0R0|dψ1dR|2RdR+∫R0∞|dψ2dR|2RdR.
Now, the core parameter U is given by
(32)U2=a2(k2n12-β2).Now for a fixed value of normalized frequency, the core parameterU is minimized with respect to the variational parameters α, R0, and μ. Once the optimized values of these three parameters are obtained, the propagation constant and other design parameters can be obtained as explained in the next section.


### 2.3. Splice Loss

For small angular misalignment (θ) at the splice of two optical fibers, following Hosain et al. [11], the well-known overlap integral can be represented as
(33)Ca(p)=∫02π∫0∞dϕRdR|ψ(R)|2exp(ipRcosϕ),
where p=aknθ, n being refractive index of the index matching fluid joining the fibers and θ being the angular misalignment.The transmission coefficientTa(p) at the splice with angular mismatch can then be expressed as
(34)Ta(p)=|Ca(p)Ca(0)|2.
Expanding the exponential term, (33) can be written as(35a)Ca(p)=2π∑n=0n=∞(-p2/4)n(n!)2∫0∞|ψ(R)|2R2n+1dR,
and from (33), Ca(0) is given by
(35b)Ca(0)=2π∫0∞RdR|ψ(R)|2.According to Hosain et al. [11], only the first four terms in (35a) are enough to obtain sufficient accuracy for misalignment up to 10, which corresponds to p≈0.8 for an optical fiber with n=1.5 and a=4 μm working at a wavelength λ=0.8 μm. Here, up to the fifth term of (35a) have been calculated and the required expressions are given in (9)-(10).The transmission coefficientTt(Δ) at the splice for a transverse offset d is expressed as follows:
(36)Tt(Δ)=|Ct(Δ)Ct(0)|2,
where Δ=d/a is the normalized transverse offset, and in practice for Δ ≤ 0.8, one can approximately write [11]
(37)Ct(Δ)Ct(0)=1-B1B0(Δ2)2+B2B0(Δ2)4-B3B0(Δ2)6,
where
(38)B0=〈ψ2〉,(39)B1=〈ψ′2〉,(40)B2=14(∫0∞|d2ψdR2|2RdR+∫0∞|dψdR|2dRR),(41)B3=136∫0∞|d3ψdR3|2RdR-112∫0∞dψdRd3ψdR3dRR-124(d2ψdR2)R=0.
Integrals given in (38)–(41) can be evaluated by using the expression of fundamental modal field given by (26). Hence, (34) and (36) can be evaluated with the help of (35a), (35b), and (37).


### 2.4. Evaluation of Integrals

Evaluation of integrals to determine propagation constant and splice loss is presented in (1)–(23). Substituting (1)–(6) into (28), (29), and (30), analytical expression of propagation constant β can be obtained with the help of (26) and (27). The transmission coefficient Ta(p) (34) at the splice with angular mismatch can be obtained by substituting (9)-(10) into (35a) and (35b). Using (31), (1), (2), (5), and (6), (38) can be obtained. Equations (40) and (41) can be evaluated using (11). Once (38)–(41) are evaluated, Ct(Δ)/Ct(0) (see (37)) can be calculated. Then, the analytical expression of transmission coefficient Tt(Δ) at the splice between two identical optical fibers having a transverse offset can be evaluated using (36).


## 2.1. Theory

Splice loss can be evaluated analytically with the help of the following equations [22]:
(1)∫|ψ1|2RdR=log(αR/R0)-Ci(2αR/R0)2,(2)∫|ψ2|2RdR=sin2αe2μR(-R0e-2μR/R0+2μREi(1,2μRR0)),(3)∫nf22R|ψ1|2dR=1(-4α+4αS0)×[2n22αS0log(αRR0)-2n22αS0Ci(2αRR0)-2n12αlog(αRR0)+2n12αCi(αRR0)-n12R0sin(2αRR0)+2n12Rα+n22R0sin(2αRR0)-2n22αR],(4)∫nf22R|ψ2|2dR=e2μsin2αR(S0-1)×[-n22R0S0e-(2μR/R0)+2n22μRS0Ei(1,2μRR0)+n12R0e-(2μR/R0)-2n12μREi(1,2μRR0)-n12R0REi(1,2μRR0)+n22R0REi(1,2μRR0)],(5)∫|dψ1dR|2RdR=[12R02log(αRR0)-12R02Ci(2αRR0)]α2+α2RR0sin(2αRR0)+14R2cos(2αRR0)-14R2,(6)∫|dψ2dR|2RdR=-sin2αe2μ4R02R3e2μR/R0((1,2μRR0)3R03+3R02μR-2R0μ2R2+4Ei(1,2μRR0)μ3R3e2μR/R0),(7)∫|ψ1|2R3dR=18α2[-2αRR0sin(2αRR0)+2α2R2+R02-R02cos(2αRR0)],(8)∫|ψ2|2R3dR=-sinα2e2μR022μe2μR/R0,(9)∫|ψ1|2RqdR=α2R02(q+1)Rq+1×Hypergeom([1,q2+12],[2,32,32+q2],-α2R2R02),(10)∫|ψ2|2RqdR=2(2-q)(μR0)-qμ2R0e2μ×sinα2[1(q-2)(q-1)q(q+1)(μRR0)-q/2×e-μR/R0Rq2q/2(μR0)q×WM(q2,q2+12,2μRR0)+2q/2-3(q-2)(q-1)qμ3(μRR0)-q/2×e-μR/R0R03Rq-3(μR0)q×(4μ2R2R02+2qμRR0-q+q2)×WM(q2+1,q2+12,2μRR0)(μRR0)-q/2],(11)∫1R|dψ1dR|2dR=18R02R4[-2α2R2+2R0αRsin(2αRR0)-R02+R02cos(2αRR0)],(12)∫1R|dψ2dR|2dR=sin2α120R04R5[(1,2μRR0)63R04μRe(-2μ(-R0+R)/R0)-2R03μ2R2e(-2μ(-R0+R)/R0)+2R02μ3R3e(-2μ(-R0+R)/R0)-4R0μ4R4e(-2μ(R0+R)/R0)+54R05e(-2μ(-R0+R)/R0)+8μ5R5e2μEi(1,2μRR0)],(13)∫|d2ψ1dR2|2RdR=α42R04[log(αRR0)-Ci(2αRR0)]-α2R02R2cos(2αRR0)+αR0R3sin(2αRR0)-12R4+12R4cos(2αRR0),(14)∫|d2ψ2dR2|2RdR=sin2α32R04R5e2μ[(1,2μRR0)-90R05e(-2μR/R0)-135R04μRe(-2μR/R0)-86R03μ2R2e(-2μR/R0)-10R02μ3R3e(-2μR/R0)-12R0μ4R4e(-2μR/R0)+24μ5R5Ei(1,2μRR0)],(15)∫|d3ψ1dR3|2RdR=α6[12R06log(αRR0)-12R06Ci(2αRR0)]+α52R05Rsin(2αRR0)+α4[54R04R2cos(2αRR0)+34R04R2]-4α3R03R3sin(2αRR0)-6α2R02R4cos(2αRR0)+6αR0R5sin(2αRR0)+3R6cos(2αRR0)-3R6,(16)∫|d3ψ2dR3|2RdRR=sin2α64R06R7e2μ×[40μ7R7Ei(1,2μRR0)+1575R07e-(2μR/R0)+2625R06μRe-(2μR/R0)+2082R05μ2R2e-(2μR/R0)+999R04μ3R3e-(2μR/R0)+246R03μ4R4e-(2μR/R0)+42R02μ5R5e-(2μR/R0)-20R0μ6R6e-(2μR/R0)(1,2μRR0)],(17)∫1R|dψ1dR||d3ψ1dR3|dR=18R04R6[(2αRR0)-4R04-3α2R02R2+2α4R4-2α3R0R3sin(2αRR0)+8αR03Rsin(2αRR0)-5α2R02R2cos(2αRR0)+4R04cos(2αRR0)],(18)∫1R|dψ2dR||d3ψ2dR3|dR=sin2α240R06R7e(2μR/R0)e2μ×[(1,2μRR0)975R06μR+474R05μ2R2+123R04μ3R3-2R03μ4R4+2R02μ5R5-4R0μ6R6+8μ7R7e(2μR/R0)Ei(1,2μRR0)+675R07(1,2μRR0)],(19)|d2ψdR2|=sinα4R5(R0/R)3/2eμe-(μR/R0)×(15R02+12R0μR+4μ2R2),
where ψ1 and ψ2 are given in (26). α, R0, and μ are the three variational parameters present in the fundamental modal solution.E
i
(
z
) is exponential integral given by [23] as follows:
(20)Ei(1,z)=∫z∞e-ttdt.Ci(z) is the cosine integral function, defined by [23], as follows:
(21)Ci(z)=χ+ln(z)+∫0zcost-1tdt,
where χ is Euler's constant 0.5772.W
M
(
k
,
m
,
z
) are the Whittaker functions which are solutions to the Whittaker differential equation [23].Hypergeom
(
n
,
d
,
z
) is the generalized hypergeometric function F(n,d,z), where [23]
(22)F(n,d,z)=∑k=0∞Cn,kCd,k·zkk!,
with,
(23)Cv,k=∏j=1vΓ(vj+k)Γ(vj),
where Γ(a) is the gamma function [23].


## 2.2. Basic Formulations

The refractive index profile for a weakly guiding fiber is given by(24)nf12=n22+(n12-n22)f1,for0≤R≤S0,nf22=n22+(n12-n22)f2,forS0≤R≤1,nf32=n22+(n12-n22)f3,forR>1,
where the normalized profile functions for the trapezoidal and triangular index profiles fi (i=1,2,3) are given by
(25)f1=1,f2=1-R1-S0,f3=0.
Here, S0 is the aspect ratio, R is the normalized radius (=r/a), a is the core radius, r is the actual radius of the optical fiber, and n1 and n2 are, respectively, the refractive indices of the core axis and cladding.For the present study, the following approximations for the fundamental mode as the trial field have been proposed:(26)ψ1=sin(αR/R0)RforR≤R0,ψ2=(sin(α)R)eμ(1-(R/R0))R0RforR>R0,
where α, R0, and μ are the three variational parameters present in the fundamental modal solution.To employ variational technique, first the scalar variational expression for the propagation constantβ as given by (27) has been considered and is shown in equations through (28) to (30) as follows:
(27)β2=k2∫0∞n2(R)|ψ(R)|2RdR-(1/a2)〈ψ′2〉〈ψ2〉,(28)β2=1〈ψ2〉[∫0R0k2nf12|ψ1|2RdR+∫R0S0k2nf12|ψ2|2RdR+∫S01k2nf22|ψ2|2RdR+∫1∞k2nf32|ψ2|2RdR-1a2〈ψ′2〉∫0R0k2nf12|ψ1|2RdR],
for R0<1 and R0<S0,
(29)β2=1〈ψ2〉[∫0S0k2nf12|ψ1|2RdR+∫S0R0k2nf22|ψ1|2RdR+∫R01k2nf22|ψ2|2RdR+∫1∞k2nf32|ψ2|2RdR-1a2〈ψ′2〉∫0S0k2nf12|ψ1|2RdR],
for R0<1 and R0>S0,
(30)β2=1〈ψ2〉[∫0S0k2nf12|ψ1|2RdR+∫S01k2nf22|ψ1|2RdR+∫1R0k2nf32|ψ1|2RdR+∫R0∞k2nf32|ψ2|2RdR-1a2〈ψ′2〉],
for R0>1, where
(31)〈ψ2〉=∫0R0|ψ1|2RdR+∫R0∞|ψ2|2RdR,〈ψ′2〉=∫0R0|dψ1dR|2RdR+∫R0∞|dψ2dR|2RdR.
Now, the core parameter U is given by
(32)U2=a2(k2n12-β2).Now for a fixed value of normalized frequency, the core parameterU is minimized with respect to the variational parameters α, R0, and μ. Once the optimized values of these three parameters are obtained, the propagation constant and other design parameters can be obtained as explained in the next section.


## 2.3. Splice Loss

For small angular misalignment (θ) at the splice of two optical fibers, following Hosain et al. [11], the well-known overlap integral can be represented as
(33)Ca(p)=∫02π∫0∞dϕRdR|ψ(R)|2exp(ipRcosϕ),
where p=aknθ, n being refractive index of the index matching fluid joining the fibers and θ being the angular misalignment.The transmission coefficientTa(p) at the splice with angular mismatch can then be expressed as
(34)Ta(p)=|Ca(p)Ca(0)|2.
Expanding the exponential term, (33) can be written as(35a)Ca(p)=2π∑n=0n=∞(-p2/4)n(n!)2∫0∞|ψ(R)|2R2n+1dR,
and from (33), Ca(0) is given by
(35b)Ca(0)=2π∫0∞RdR|ψ(R)|2.According to Hosain et al. [11], only the first four terms in (35a) are enough to obtain sufficient accuracy for misalignment up to 10, which corresponds to p≈0.8 for an optical fiber with n=1.5 and a=4 μm working at a wavelength λ=0.8 μm. Here, up to the fifth term of (35a) have been calculated and the required expressions are given in (9)-(10).The transmission coefficientTt(Δ) at the splice for a transverse offset d is expressed as follows:
(36)Tt(Δ)=|Ct(Δ)Ct(0)|2,
where Δ=d/a is the normalized transverse offset, and in practice for Δ ≤ 0.8, one can approximately write [11]
(37)Ct(Δ)Ct(0)=1-B1B0(Δ2)2+B2B0(Δ2)4-B3B0(Δ2)6,
where
(38)B0=〈ψ2〉,(39)B1=〈ψ′2〉,(40)B2=14(∫0∞|d2ψdR2|2RdR+∫0∞|dψdR|2dRR),(41)B3=136∫0∞|d3ψdR3|2RdR-112∫0∞dψdRd3ψdR3dRR-124(d2ψdR2)R=0.
Integrals given in (38)–(41) can be evaluated by using the expression of fundamental modal field given by (26). Hence, (34) and (36) can be evaluated with the help of (35a), (35b), and (37).


## 2.4. Evaluation of Integrals

Evaluation of integrals to determine propagation constant and splice loss is presented in (1)–(23). Substituting (1)–(6) into (28), (29), and (30), analytical expression of propagation constant β can be obtained with the help of (26) and (27). The transmission coefficient Ta(p) (34) at the splice with angular mismatch can be obtained by substituting (9)-(10) into (35a) and (35b). Using (31), (1), (2), (5), and (6), (38) can be obtained. Equations (40) and (41) can be evaluated using (11). Once (38)–(41) are evaluated, Ct(Δ)/Ct(0) (see (37)) can be calculated. Then, the analytical expression of transmission coefficient Tt(Δ) at the splice between two identical optical fibers having a transverse offset can be evaluated using (36).


## 3. Results and Discussions

Detailed comparison between the proposed formulation and available exact numerical results [10, 13] has been carried out in terms of accuracy assessment. It has been justified by many authors [1–3] that two-parameter approximations are more accurate than single-parameter approximation. The proposed approximation of fundamental field involving three optimizing parameters incorporates more flexibility to modify the fundamental modal solution of optical fibers having different specifications. Optimized values of these parameters for different normalized frequencies are given in Tables 1 and 2 for a particular specification of optical fiber having trapezoidal and triangular index profiles, respectively. Values for other normalized frequencies having different specification of optical fiber can also be obtained by using Nelder-Mead method of nonlinear unconstrained minimization.Table 1
Values of optimizing parameters with different normalized frequencies for trapezoidal index profile.
α
R
0
μ
V
1.839506
1.487480
0.006634
1.6000
1.854451
1.152568
0.040590
1.8000
1.878781
1.006541
0.097612
2.0000
1.908874
0.923490
0.171137
2.2000
1.941083
0.867514
0.253638
2.4000
1.954907
0.814689
0.289126
2.6000
1.987346
0.781444
0.277328
2.8000
1.967303
0.722072
0.256257
3.0000
2.207612
0.797867
0.283156
3.2000
1.930533
0.637208
0.226140
3.4000Table 2
Values of optimizing parameters with different normalized frequencies for triangular index profile.
α
R
0
μ
V
1.836461
1.962401
−0.000309
1.7000
1.843169
1.323589
0.014877
1.9000
1.858852
1.102224
0.050720
2.1000
1.881018
0.986778
0.103042
2.3000
1.907224
0.914502
0.167036
2.5000
1.934891
0.862840
0.237307
2.7000
1.954757
0.817735
0.290207
2.9000
1.947828
0.764272
0.271233
3.1000
1.941593
0.718887
0.255127
3.3000
2.150373
0.784010
0.278234
3.5000In order to verify the feasibility of the proposed approximation, the outcomes of the proposed study have been compared with the earlier reported numerical results [10, 13]. In the present study, S0=0.25 and V=2.4 are considered for trapezoidal index profile, which corresponds to a typical dispersion shifted silica fiber with a=3.2 μm, δ=(n12-n22)/2n12=0.008, and zero dispersion wavelength at 1.55 μm [12]. For triangular index profile, V=2.7 and S0=0 have been chosen, taking a=3.5 μm, δ=0.008, and zero dispersion wavelength at 1.5 μm [12].For evaluation of splice loss, the applicability of the proposed formulations in case of power transmission coefficientsTa(p) and Tt(Δ) at splices between two identical optical fibers has been considered. Gaussian approximation gives accurate result for the evaluation of transmission coefficient only in the region near the cutoff of single mode operation, but it leads to considerable error throughout the single mode region [11]. The variation of Ta with p and Tt with Δ, in case of splicing of two identical triangular index fibers, has been plotted in Figures 1 and 2, respectively. Similarly, the variations of these power transmission coefficients for the case of splicing of two identical trapezoidal index fibers are illustrated in Figures 3 and 4. For the practical range of p and Δ, the results obtained by the proposed approximation are identically matching with the exact available and numerical results [10, 13].Figure 1
Variation of power transmission coefficientsTa with the normalized angular offset p for splicing of two identical single mode triangular index fibers with V=2.7 (exact numerical results [10, 13]; results by our approximation; results based on Gaussian approximation [10, 13]).Figure 2
Variation of power transmission coefficientsTt with the normalized transverse offset Δ for splicing of two identical single mode triangular index fibers with V=2.7 (exact numerical results [10, 13]; results by our approximation; results based on Gaussian approximation [10, 13]).Figure 3
Variation of power transmission coefficientsTa with the normalized angular offset p for splicing of two identical single mode trapezoidal index fibers with V=2.4 (exact numerical  results [10, 13]; results by our approximation; results based on Gaussian approximation [10, 13]).Figure 4
Variation of power transmission coefficientsTt with the normalized transverse offset Δ for splicing of two identical single mode trapezoidal index fibers with V=2.4 (exact numerical results [10, 13]; results by our approximation; results based on Gaussian approximation [10, 13]).


## 4. Conclusions

An accurate three-parameter approximation of fundamental modal field solution of an optical fiber has been presented, which can effectively be used to estimate the power transmission coefficients in case of splicing of two identical single mode graded-index fibers in presence of both transverse and angular misalignments. Taking trapezoidal and triangular index fibers as examples, it has been shown that the results obtained with our function are excellently matching with the exact available and numerical results [10, 13]. Besides providing values of optimizing parameters involved in the approximate field obtained by Nelder-Mead method of nonlinear unconstrained minimization, all related simplified analytical expressions have also been presented, which can be used directly by optical fiber designer while predicting splice losses of an optical fiber, having triangular and trapezoidal index profiles for a wide range of normalized frequencies. The salient features of the proposed solution are easy computation on an ordinary personal computer and a robust algorithm for nonlinear unconstrained optimization being applied in an optical fiber having triangular and trapezoidal index profiles.



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*Source: 168923-2013-07-08.xml*