Equilibrium mode distribution and steady-statedistribution in 100–400 μm core step-index

silica optical fibers

Svetislav Savović,1,2,* Alexandar Djordjevich,1 Ana Simović,2 and Branko Drljača2

1City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong, China2University of Kragujevac, Faculty of Science, R. Domanovića 12, 34000 Kragujevac, Serbia

*Corresponding author: savovic@kg.ac.rs

Received 4 April 2011; accepted 20 June 2011;posted 22 June 2011 (Doc. ID 145334); published 19 July 2011

Using the power flow equation, the state of mode coupling in 100–400 μm core step-index silicaoptical fibers is investigated in this article. Results show the coupling length Lc at which the equilibriummode distribution is achieved and the length zs of the fiber required for achieving the steady-state modedistribution. Functional dependences of these lengths on the core radius and wavelength are also given.Results agree well with those obtained using a long-established calculation method. Since large coresilica optical fibers are used at short distances (usually at lengths of up to 10m), the light they transmitis at the stage of coupling that is far from the equilibrium and steady-state mode distributions. © 2011Optical Society of AmericaOCIS codes: 060.2310, 060.2400.

1. Introduction

For decades, glass optical fibers (GOFs) have beenthe preferred transmission medium in high-capacitycommunications networks and long-distance com-munications systems [1]. Graded index multimodeGOFs are used for 0–300m 10Gb Ethernet links or0–100m 40–100Gb Ethernet links. Step-index (SI)multimode GOFs are often used for laser beam deliv-ery, sensing systems, as part of lane control signalequipment, etc. For laser delivery, it is desirable touse relatively large core (200–500 μm core radius) si-lica optical fibers for transmission of high-power la-ser pulses with high beam quality [2]. On the otherhand, multimode plastic optical fibers are usuallyconsidered for short data links (<100m). Local net-working with plastic optical fibers benefits from therapid (less laborious) interconnectivity with low-precision and low-cost components as plastic opticalfibers couple light efficiently due to their large

diameter (∼1mm) and high numerical aperture.However, plastic optical fiber performance is clearlyattenuation limited. A typical attenuation level forSI plastic optical fibers is ∼100dB=km—comparedwith ∼0:5dB=km for SI GOFs [3]. This limits plasticoptical fiber data links to lengths shorter than 100m.

Transmission characteristics of SI optical fibersdepend heavily upon the differential mode attenua-tion and rate of mode coupling. The latter representspower transfer from lower- to higher-order modescaused by fiber impurities and inhomogeneities in-troduced during the fiber manufacturing process(such as microscopic bends, irregularity of the core–cladding boundary, and refractive index distributionfluctuations). When installing an optical-fiber-basedlink, the cable has to be bent repeatedly, thus in-creasing radiation losses [4,5].

Output angular power distribution in the nearand far fields of an optical fiber end has been studiedextensively. Work has been reported using geometricoptics (ray approximation) to investigate mode cou-pling and predict output-field patterns [6,7]. By em-ploying the power flow equation [8–12] as well as the

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Fokker–Planck and Langevin equations [13], thesepatterns have been predicted as a function of thelaunch conditions and fiber length. A key prerequi-site for achieving this is the knowledge of the rate ofmode coupling expressed in the form of the couplingcoefficient D [8–10], which has been shown to cor-rectly predict coupling effects observed in practice(e.g., [14]).

The method of determining the coupling coefficientD proposed by Gambling et al. [8] required that thefar-field output pattern be observed for various fiberlengths and at different launch angles. Only one fiberlength and two launch angles must be consideredin the method by Zubía et al. [15]. It determinesthe coupling coefficient D from the intersection pointbetween two far-field output patterns that corre-spond to the two launch angles. A further alternativeis Savović–Djordjevich method [16], which deter-mines the mode coupling coefficient D from just onefar-field output pattern. This single pattern is for theinput Gaussian beam launched along the fiber axis.The variance of the launch-beam distribution has tobe known, which is usually the case. Should it not beknown, variances of the far-field output patterns attwo fiber lengths have to be measured.

Hurand et al. [2] have employed Savović–Djordjevich method of determining the coupling coef-ficient D [16] in their investigation of mode couplingin SI silica optical fibers with core diameters of 100–400 μm and lengths of 2m for central launch beamθ0 ¼ 0. In this work, using the power flow equation,we extend their work and examine the state of modecoupling in SI silica optical fibers with core diam-eters of 100 μm, 200 μm, and 400 μm and lengths ofkilometers, both for central launch beam θ0 ¼ 0 aswell as for a launch beam with θ0 > 0. Furthermore,for 200 μm core SI silica optical fiber, a dependence ofthe output angular power distribution on fiber lengthis investigated for three wavelengths. As a result, thecoupling length Lc at which the equilibrium modedistribution (EMD) is achieved and the length zs offiber required for achieving the steady-state modedistribution (SSD) are obtained. We compare our re-sults with those obtained using a long-establishedcalculation method [2,9].

2. Power Flow Equation

Gloge’s power flow equation is [9]

∂Pðθ; zÞ∂z

¼ −αðθÞPðθ; zÞ þDθ

∂

∂θ

�θ ∂Pðθ; zÞ

∂θ

�; ð1Þ

where Pðθ; zÞ is the angular power distribution, z isthe distance from the input end of the fiber, θ is thepropagation angle with respect to the core axis, D isthe coupling coefficient assumed constant [8,9,12],and αðθÞ is the modal attenuation. The boundary con-ditions are Pðθc; zÞ ¼ 0, where θc is the critical angleof the fiber and Dð∂P=∂θÞ ¼ 0 at θ ¼ 0. ConditionPðθc; zÞ ¼ 0 implies that modes with infinitely highloss do not carry power. Condition Dð∂P=∂θÞ ¼ 0 at

θ ¼ 0 indicates that the coupling is limited to themodes propagating with θ > 0. Except near cutoff,the attenuation remains uniform αðθÞ ¼ α0 through-out the region of guided modes 0 ≤ θ ≤ θc [10,11] [itappears in the solution as the multiplication factorexpð−α0zÞ that also does not depend on θ]. Therefore,αðθÞ need not be accounted for when solving Eq. (1)for mode coupling, and this equation reduces to [12]

∂Pðθ; zÞ∂z

¼ Dθ∂Pðθ; zÞ

∂θ þD∂2Pðθ; zÞ

∂θ2 : ð2Þ

The solution of Eq. (2) for the steady-state powerdistribution is given by [10]

Pðθ; zÞ ¼ J0

�2:405

θθc

�expð−γ0zÞ; ð3Þ

where J0 is the Bessel function of the first kindand zero order and γ0½m−1� ¼ 2:4052D=θ2c is the at-tenuation coefficient. We used this solution to testour numerical results for the case of the fiber lengthat which the power distribution becomes indepen-dent of the launch conditions. This length at which asteady-state distribution is achieved can be obtainedusing Eq. (4) [2,9]:

zs ¼0:2D

�NAn

�2; ð4Þ

where n is the refractive index of the core and NA isnumerical aperture of the fiber.

In order to obtain numerical solution of the powerflow Eq. (2) we have used the explicit finite-differencemethod (EFDM) employed in our earlier works[12,17]. To start the calculations, we used Gaussianlaunch-beam distribution of the form

Pðθ; zÞ ¼ exp�−ðθ − θ0Þ2σ2

�; ð5Þ

with 0 ≤ θ ≤ θc, where θ0 is the mean value of theincidence angle distribution, with the full width athalf-maximum FWHM ¼ 2σ

ffiffiffiffiffiffiffiffiffiffiffiffi2 ln 2

p¼ 2:355σ (σ is

standard deviation). This distribution is suitableboth for LED and laser beams (LED distribution canalso be described by a Lambertian source).

3. Results and Discussion

In this paper, we analyze mode coupling in relativelylarge core SI silica optical fibers used in the experi-ment reported recently [2]. The first fiber has corediameter dcore ¼ 100 μm and clad diameter dclad ¼660 μm (100=660 fiber), the second fiber has dcore ¼200 μm and clad diameter dclad ¼ 745 μm (200=745fiber), while the third fiber has core diameter dcore ¼400 μm and clad diameter dclad ¼ 720 μm (400=720fiber). All fibers have NA ¼ 0:22, core refractive in-dex n ¼ 1:4570 at λ ¼ 633nm, and critical angle θc ¼8:8°. Fiber samples with a length of 2m were tested

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to obtain their mode coupling properties under theexcitation of only a small number of modes, which isrealized by a narrow (with standard deviation σ0 ¼0:054°) centrally launched beam [2]. By measuringthe standard deviation of the output angular powerdistribution at the end of the fiber and using the re-lation for mode coupling coefficient D ¼ ðσ2z − σ20Þ=2z[16], Hurand et al. [2] obtainedD ¼ 4:9 × 10−7 rad2=mfor 100=600 fiber, D ¼ 1:9 × 10−6 rad2=m for 200=745fiber, and D ¼ 6:4 × 10−6 rad2=m for 400=720 fiber atλ ¼ 633nm—which we have adopted in this work.

In Fig. 1, our numerical solution of the power flowequation is presented by showing the evolution ofthe normalized output power distribution with fiberlength for 200=745 fiber at λ ¼ 633nm. We showresults for three different input angles θ0 ¼ 0°, 4°,and 8°. We selected Gaussian launch-beam distribu-tion with ðFWHM0Þ ¼ 0:127° by setting σ0 ¼ 0:054°in Eq. (5). Using the step lengths of Δθ ¼ 0:1° andΔz ¼ 0:02m, we have achieved the stability of ourfinite-difference scheme. Since a truncation errorfor our explicit finite-difference scheme is OðΔz;Δθ2Þ[18], using a small enough value ofΔz, the truncationerrors were reduced until the accuracy achieved waswithin the error tolerance.

Radiation patterns in the short fiber (z ¼ 500m)in Fig. 1(a) indicate that the coupling is stronger forthe low-order modes: their distributions have shiftedtoward θ ¼ 0°. Coupling of higher-order modes canbe observed better only after longer fiber lengths[Fig. 1(b)]. It is not until the fiber’s coupling lengthLc that all the mode distributions shift their mid-points to 0° (from the initial value of θ0 at the inputfiber end), producing the EMD in Fig. 1(c): Lc is1380m. The coupling continues further along the fi-ber beyond the Lc mark until all distributions’widthsequalize and SSD is reached at length zs in Fig. 1(d):zs ¼ 2470m. For the 200=745 fiber, Fig. 1(d) showsnormalized curves of the output angular distributionobtained by solving the power flow equation usingthe EFDM (solid curve) as well as the steady-stateanalytical solution of Eq. (2) (solid squares), whereγ0 ¼ 0:00047m−1. The two are in good agreement,with the relative error below 0.8%. In the same man-ner, we obtained Lc ¼ 5400mand zs ¼ 9600m for the100=600 fiber and Lc ¼ 430m and zs ¼ 730m for the400=720 fiber. In order to test the accuracy of results,we compared zs with theoretical values determinedby function (4), which are zs ¼ 9306m for the100=600 fiber, zs ¼ 2400m for the 200=745 fiber, andzs ¼ 712m for the 400=720 fiber. The relevant valuesare summarized in Table 1 to facilitate easier com-parisons. Good agreement is apparent between ournumerically obtained values for zs and theoreticalpredictions by Gloge’s function. One can observe thatmode coupling coefficients varied as ∼dcore

1:85. Theincrease of mode coupling coefficient (rate of modecoupling) with core diameter is due to a simultaneousdecrease of angular separation between adjacentmodes Δθ ¼ λ=ð2dcorenÞ. As a consequence, the cou-pling length Lc where the equilibriummode distribu-tion is achieved and length zs where steady-statedistribution is established decrease with increasingcore diameter.

Hurand et al. [2] studied 200=745 fiber for twoother wavelengths (λ ¼ 403 and 1064nm). Themeasured coupling coefficients are D ¼ 2:4 × 10−6

and 1:5 × 10−6 rad2=m at λ ¼ 403 and 1064nm, re-spectively. By adopting these two values for D, wesolved the power flow equation numerically in orderto investigate how wavelength influences character-istic fiber lengths, Lc and zs. We obtained that Lc ¼1090m, zs ¼ 1910m at λ ¼ 403nm and Lc ¼ 1800m,zs ¼ 3170m at λ ¼ 1064nm. Using Gloge’s func-tion, we obtain zs ¼ 1866m at λ ¼ 403nm andzs ¼ 3069m at λ ¼ 1064nm (core refractive index

Table 1. Core Diameter, Clad Diameter, Coupling Coefficient D , Coupling Length Lc , and Length zs for Silica Fibers at λ � 633 nma

CoreDiameter (μm)

CladDiameter (μm) D ðrad2=mÞ

Lc ðmÞ (NumericalResults)

zs ðmÞ (NumericalResults)

zs ðmÞ [AnalyticalResults, Eq. (4)]

100 600 4:9 × 10−7 5400 9600 9306200 745 1:9 × 10−6 1380 2470 2400400 720 6:4 × 10−6 430 730 712

aValues for D are those determined by Hurand et al. [2].

Fig. 1. Normalized output angular power distribution at differ-ent locations along the 200=745 silica fiber calculated for threeGaussian input angles θ0 ¼ 0° (solid curve), 4° (dashed curve), and8° (dotted–dashed curve) with ðFWHMÞz¼0 ¼ 0:127° for (a) z ¼500m, (b) z ¼ 900m, (c) z ¼ 1380m, and (d) z ¼ 2470m (solidsquares represent the analytical steady-state solution).

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n ¼ 1:4703 at λ ¼ 403nm and n ¼ 1:4502 at λ ¼1064nm are assumed in the calculations). For threewavelengths mentioned, the relevant numericalvalues are summarized in Table 2 to facilitate easiercomparisons. A good agreement is again apparentbetween our numerically obtained values for zsand theoretical predictions obtained using Gloge’sfunction [4]. One can observe that mode couplingcoefficients varied as ∼λ−1=2. The decrease of themode coupling coefficient with increasing wave-length is due to the simultaneous increase ofangular separation between adjacent modes Δθ ¼λ=ð2dcorenÞ. As a consequence, the coupling lengthLc where the equilibrium mode distribution isachieved and length zs where steady-state distribu-tion is established increase with increasing wave-length. Similar wavelength dependence ∼λ−1=3 isobtained for chalcogenide-glass SI optical fiber, forwhich coupling length is even larger (Lc ≈ 13 to20km) [19]. The coupling coefficients of the SI silicaoptical fibers that were analyzed are 2 orders ofmagnitude lower than for SI plastic optical fibers(typically ≅ 10−4 rad2=m). Consequently, the cou-pling length Lc and length zs are much shorter (Lc ≅

15 to 35m, zs ¼ 45 to 100m) [20]. This is attributedto strong intrinsic perturbation effects in plastic op-tical fibers.

Finally, we conclude that, since large core silica op-tical fibers are used at short distances (usually atlengths of up to ten meters), the light they transmitis at the stage of coupling that is far from the equili-brium and steady-state mode distributions.

4. Conclusion

A solution is reported of the power flow equationemployed to investigate the state of mode couplingalong large core SI silica optical fibers. Results havebeen verified against the analytical solution for thesteady-state coupling condition. Coupling lengthsand lengths for achieving the steady-state distribu-tion are shown to decrease with increasing the coreradius. On the other hand, they increase with in-creasing the wavelength. Since these fibers are usedat short distances (usually at lengths of up to ten me-ters), the light they transmit is at a stage of couplingthat is far from the equilibrium or steady-state modedistributions. These results are of interest in predict-ing the transmission properties of large core SI silicaoptical fibers used for power delivery and sensingsystems.

The work described in this paper was supportedby a grant from the Serbian Ministry of Science andTechnological Development (project 171011).

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Table 2. Coupling Coefficient D, Coupling Length Lc , and Length zs

at Different Wavelengths λ for 200=745 Silica Fibera

λðnmÞ

Dðrad2=mÞ

Lc ðmÞ(NumericalResults)

zs ðmÞ(NumericalResults)

zs ðmÞ[AnalyticalResults,Eq. (4)]

403 2:4 × 10−6 1090 1910 1866633 1:9 × 10−6 1380 2470 2400

1064 1:5 × 10−6 1800 3170 3069aValues for D are those determined by Hurand et al. [2].

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