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OpticsOptikOptikOptik 120 (2009) 647–651

0030-4026/$ - se

doi:10.1016/j.ijl

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Power transfer characteristics among N parallel single-mode optical fibers

Ye Wang, Dajian Xue, Xuanhui Lu�

State Key Laboratory of Modern Optical Instrumentation, Department of Physics, Optics Institute, Zhejiang University,

Hangzhou 310027, PR China

Received 22 August 2007; received in revised form 10 January 2008; accepted 11 February 2008

Abstract

Based on the coupled-mode theory, the power transfer among ‘‘- - -’’ arranged parallel single-mode optical fibers hasbeen investigated. The analysis shows that the distances between each two of the N fibers centers have effects on thecoupling coefficient and power transfer. The solution of the coupled equations for three parallel single-mode opticalfibers is given, and is studied for different initial conditions comparatively. Numerical simulations show that powertransfer will be periodical during coupling among parallel single-mode optical fibers. These results can be extended tomulti-parallel single-mode optical fibers.r 2008 Elsevier GmbH. All rights reserved.

Keywords: Optical fiber array; Modes coupling; Power transfer

1. Introduction

It is a well-known fact that if the optical fiber coresare brought close enough to each other, wave couplingand power transfer among optical fibers happens due tothe interaction of the extended evanescent optical fieldsright outside of the cores. The coupling is important todivide and mix optical signals for various optical fibercomponents, such as fiber couplers [1,2] and powerdividers [3]. For example, the well-known phenomenonof evanescent wave coupling has wide applications suchas design of waveguide switches [4], an importantpassive device in optical communications. However, insome other cases, an avoidance of wave coupling effectsamong ‘‘- - -’’ arranged optical fibers is needed. Arepresentative example is pivotal components in a newscanning system for imaging.

e front matter r 2008 Elsevier GmbH. All rights reserved.

eo.2008.02.011

ing author. Tel./fax: +86 571 87953232.

ess: xhlu@zju.edu.cn (X. Lu).

In previous studies, Snyder [5] presented a generalcoupled-mode theory and obtained conclusive resultsthat could be used in a number of different wavepropagation problems. In his study, the solution of thecoupled equations for two parallel optical fibers wasfurther presented. He also studied the coupling effectson the situation of a special fiber surrounded by other N

identical fibers. However, it is only a special case amongthe generic two-body coupling problems. McIntyre [6] etal. extended Snyder’s results to the coupling of multi-mode fibers, and discussed the power transfer between aspecial fiber and six other non-identical fibers, which isalso a generic two-mode coupling problem. Chang et al.[7] analyzed numerically the cases of power transferbetween a signal-mode fiber and a multi-mode fiber. Ingeneral cases, two-mode coupling of many ‘‘- - -’’arranged parallel optical fibers cannot be easily solved.To determine the details of the power transfer amongthe ‘‘- - -’’ arranged N fibers, it is required to solve theequations for N systems. In this paper, we derived thesolution of the coupled equations of three parallel

ARTICLE IN PRESSY. Wang et al. / Optik 120 (2009) 647–651648

single-mode (SM) optical fibers. The solution can beextended to N parallel SM optical fibers.

2. The coupled-mode method

We consider N infinitely long and parallel identicalcores embedded in the cladding background, as shownin Fig. 1. The refractive index of the SM core and thecladding are denoted as n1 and n2, respectively. Theradius of the SM core and the distance between twocenters of the neighboring cores are shown as r and d,respectively.

We adopt the coupled-mode equations derived in Ref.[5], in which only the closest coupling is taken intoaccount. For the present system of N (N43) parallelfibers of no loss, the equations are as follows:

dA1ðzÞ

dzþ ib1A1ðzÞ ¼ �iA2ðzÞC12 (1)

dAmðzÞ

dzþ ibmAmðzÞ ¼ � iAm�1ðzÞCmðm�1Þ

� iAmþ1ðzÞCmðmþ1Þ

�ðm ¼ 2; 3; . . . ;N � 1Þ (2)

dAN ðzÞ

dzþ ibNANðzÞ ¼ �iAN�1ðzÞCNðN�1Þ (3)

The z-axis in Cartesian coordinates is taken to beparallel to the fiber axes. In Eq. (1)–(3), the subscripts m

and N refer to the mth fiber and Nth fiber, respectively. brepresents the propagation constant of mode, and A(z)represents the coefficient of mode on z ¼ constant. Thesymbols Cab is the coupling coefficient between the athand the bth fibers, and is defined as

Cab ¼o2

ZAb

ðn21 � n2

2Þ e*

a � e*

b ds (4)

where o is the wave frequency, Ab represents the crosssection of the bth cores, e

*is the electric field vector of

mode. All modes are assumed to travel in the positive z

direction. All the propagation constants of mode areequal because the N SM fibers are identical, andall coupling coefficients are equal as well according to

dd

n1n1 n1

n2 z

1 2 3ρρ

N

Fig. 1. N parallel single-mode optical fibers system.

Eq. (4). Thus all the propagation constants of mode andall coupling coefficients can be shown as b and C,respectively. It has been shown that the power transferbetween the forward and backward modes is verysmall and can be neglected for the case where thecoupling coefficients are much smaller than the propa-gation constants of mode, that is, the case satisfyingthe weak-coupling condition [5]. In the followingexamples, C is found to be much smaller than theaccompanying value b. Thus, we only consider weak-coupling cases.

b and C were given [5,6] in the following formsby introducing three dimensionless parameters U, V

and W:

b ¼2pn1

l

� �2

�U2

r2

" #1=2(5)

C ¼

ffiffiffidp

U2K0½W ðd=rÞ�rV 3K2

1ðW Þ(6)

where l is the wavelength of light in vacuum, Ks aremodified Hankel functions, and d is defined as

d ¼ 1�n2

n1

� �2

(7)

A dimensionless frequency V is defined to be

V ¼2prn1

ffiffiffidp

l¼ U2 þW 2 (8)

U is given approximately as

U ffi 2:405e�ð1�d=2ÞV . (9)

3. Results and discussion

3.1. Solutions for two different initial conditions and

discussion

When N ¼ 3, the solutions to Eqs. (1)–(3) are given as

A1ðzÞ

A2ðzÞ

A3ðzÞ

0BB@

1CCA ¼ q1

1

0

�1

0BB@

1CCAe�ibz þ q2

1ffiffiffi2p

1

0BB@

1CCAe�iðbþ

ffiffi2p

CÞz

þ q3

1

�ffiffiffi2p

1

0BB@

1CCAe�iðb�

ffiffi2p

CÞz (10)

where q1 ,q2 ,q3, are determined by initial conditions.Two cases of unit power coupling are considered.

ARTICLE IN PRESS

0.00

0.2

0.4

0.6

0.8

1.0

POW

ER

DISTANCE z / mm

123

00.0

0.2

0.4

0.6

0.8

1.0

POW

ER

DISTANCE z / mm

123

00.0

0.2

0.4

0.6

0.8

1.0

POW

ER

DISTANCE z / m

123

25

2 4 6 8 10

50 75 100 125

200 400 600 800 1000 1200

Fig. 2. Power variations versus propagation distance of three

optical fibers according to the initial condition 1. Three cases

with N ¼ 3, d ¼ 12, 20, and 50mm are shown. Curves 1, 2, 3

represent the power variations of optical fibers 1, 2, 3

respectively. The unit power is defined as the original value

of power when Z ¼ 0, and it is the same as below.

Y. Wang et al. / Optik 120 (2009) 647–651 649

3.1.1. Excitation of the 1st fiber with unit power A1ð0Þ ¼0 with A2ð0Þ ¼ 1,?A3ð0Þ ¼ 0?

By combining this condition with Eq. (10), we obtainq1 ¼ 1/2, q2 ¼ q3 ¼ 1/2. The power of each fiber can beshown as PiðzÞ ¼ jAiðzÞj

2 (i ¼ 1,2,3). If l ¼ 1.31 mm, therefractive index of the core is n1 ¼ 1.458, the refractiveindex of the cladding is n2 ¼ 1.455, and the radiuses ofthe cores are r ¼ 4 mm. The results of the powercoupling among the three fibers are shown in Fig. 2for d ¼ 12, 20, and 50 mm, respectively. In all three cases,the three fibers appear to undergo significant powercoupling and variations, as indicated by the power vs.distance curves shown in the figures. It is noticed inFig. 2(a) that at the distance of 4.48mm, almost allthe power of the 1st fiber has been transferred to that ofthe 3rd fiber. In the cases of Fig. 2(b) and (c), the powerof the 1st fiber drops to zero at z ¼ 56.5mm and482.8m, respectively, and the power of the 3rd fiberreaches the peak of unit power simultaneously. At thebeginning of the propagation, the distance at which thepower of the 1st fiber drops to half of its initial value isfound to be 1.63mm for the case of d ¼ 12 mm. Andas the coupling strength decreases for the case ofd ¼ 50 mm, the distance increases monotonously to175.5m. Initially, the power of the 2nd fiber is observedto gain most of its power from the 1st fiber in all threecases. The maximum power point of the 2nd fiberdepends on the coupling strength and the spacingbetween the two fibers, and reaches half of themaximum power value of the 1st and the 3rd fibers inall three cases. It is found that the power of the 2nd fiberdecreases to zero at nearly the same position as that ofthe 1st fiber. As propagation distance goes on, the powerof the 1st fiber begins to increase, while the power of the3rd fiber decreases. The power of the 2nd fiber goes upinitially and then drops. The Fig. 2(a) shows that thepowers of the 2nd and the 3rd fibers drops to zero andthe power of the 1st fiber returns to unit power at thedistance of 8.965mm at d ¼ 12 mm. The cases of d ¼ 20and 50 mm show the same variation trend of power. Allthree fibers have the periodical power transfer along thepropagation distance.

It can be seen by analyzing Fig. 2(a)–(c) that thecoupling period increases as the space d increases. Thecause is that the coupling coefficient goes downobservably with the increasing space between the twoneighboring fibers, and the coupling strength amongoptical fibers falls dramatically accordingly. Thus theoutward transfer velocity of the power of the 1st fiberalso decreases. So the wave propagation distance islonger as the initial power of the 1st power drops tozero. Fig. 3 does show the relationship between thecoupling coefficient and the space between the twoneighboring fibers. When d432 mm (correspondinglyC ¼ 0.999), the coupling is so weak that it can beneglected in this case.

3.1.2. Excitation of the 2nd fiber with unit power A2ð0Þ ¼1 with A1ð0Þ ¼ 0, A3ð0Þ ¼ 0

The power coupling of another case is presented in Fig. 4.Now the initial condition is A2ð0Þ ¼ 1, A1ð0Þ ¼ 0, and

ARTICLE IN PRESSY. Wang et al. / Optik 120 (2009) 647–651650

A3ð0Þ ¼ 0. Using Eq. (10), we have q1 ¼ 0, q2 ¼ffiffiffiffiffiffiffiffi2=4

p,

q3 ¼ �ffiffiffiffiffiffiffiffi2=4

p, for d ¼ 20mm. It can be observed that the

variations are similar to those shown in Fig. 2(b). However,the power transfer of the 1st and the 3rd fibers issynchronous and their transfer velocities are also same.The maximal power of the 1st and the 3rd fibers reacheshalf of the total power. The Figs. 4 and 2(b) show that thepower transfer velocities of the 1st and the 3rd fibers inSection 2 are faster than those in Section 1, and thecoupling period in Fig. 4 is shorter than that in Fig. 2(b).

3.2. Coupling analysis when N43

If the initial condition is A1(0) ¼ 1 and zero for otherfibers, the Eqs. (1)–(3) with N ¼ 4 and 5 can be solvednumerically. Figs. 5 and 6 show the power variationsversus propagation distance with d ¼ 12 mm in the

12

0

200

400

600

800

1000

d / um

Cou

plin

g co

effi

cien

t C

16 20 24 28 32

Fig. 3. Coupling coefficient as a function of the distance

between two optical fiber centers, derived from Eq. (6).

00.0

0.2

0.4

0.6

0.8

1.0

POW

ER

DISTAN

15 30

Fig. 4. Power variations versus propagation distance according to th

are overlapping.

4-fiber and 5-fiber systems, respectively. The couplingof 4 or 5 fibers is complicated. The power of each fiberoscillates in the propagation period, and the peak doesnot vary monotonously. In Fig. 5, the power peak valueof the 1st fiber is unit power initially. After coupling andoscillation, it drops and then increases, until it reachesits secondary peak finally. The variations of the powerof the 2nd and the 3rd fibers are similar to each other,increasing initially and decreasing later. However, theyare different from the 1st one, which has the same trendas the 4th fiber. The coupling velocity of the 2nd fiber islarger than that of the 3rd fiber, and the 1st power peakof the 2nd fiber is higher than that of the 3rd one, whichis around 0.55. The 4th fiber has the slowest couplingvelocity, while it has higher power peak than the 2nd

CE z / mm

123

45 60 75

e initial conditions 2 when N ¼ 3, d ¼ 20 mm. Now curves 1, 3

00.0

0.2

0.4

0.6

0.8

1.0

POW

ER

DISTANCE z / mm

1234

30 60 90 120 150

Fig. 5. Power variations versus propagation distance when

N ¼ 4, d ¼ 20mm, with initial conditions A1(0) ¼ 1 and zero

for others.

ARTICLE IN PRESS

00.0

0.2

0.4

0.6

0.8

1.0

POW

ER

DISTANCE z / mm

12345

40 80 120 160 200

Fig. 6. Power variations versus propagation distance when

N ¼ 5, d ¼ 20 mm, with initial conditions A1(0) ¼ 1 and zero

for others.

Y. Wang et al. / Optik 120 (2009) 647–651 651

and the 3rd fibers. The power of the 4th fiber occupiesalmost 97 percent of total power when it is at the peak ofits power, with the powers of the 2nd and the 3rd fiberssimultaneously transferred out, except the 1st one. Sothis case is different from that of N ¼ 3. In other words,the fringe Nth fiber cannot get all the power when thepower of middle fibers is zero.

By comparing Figs. 5 and 6, it is found that there islittle difference between the power variation of N ¼ 4and that of N ¼ 5. The coupling is strong between the1st and last fibers no matter. The maximal power ofthe Nth fiber exceeds 90 percent of the total power. Asthe number of fibers increases, every oscillating peakvalue of the fringe fibers in a coupling period decreases,and the wave propagation distance gets short from theinitial to the zero-power point.

4. Conclusion

In this paper, we have calculated the power transferamong many ‘‘- - -’’ arranged parallel single-mode (SM)optical fibers based on a coupled-mode analysis.Different initial coupling conditions and different

numbers of optical fibers are studied comparatively.Numerical simulations and theoretical curves show thatthe power transfer varies regularly among N ‘‘- - -’’arranged parallel SM optical fibers. The couplingbecomes stronger when the space between the twoneighboring fibers is smaller. When using N ‘‘- - -’’arranged parallel optical fibers, the power coupling canbe controlled in different applications if the spacebetween two neighboring fibers and the propagationdistance are adjusted properly. On the other hand, afterinvestigating these results, the coupling can be avoidedefficiently in the scan imaging system, where it isnecessary to keep uniformity for every spot.

Acknowledgments

This work is supported by National Science Founda-tion of China (Grant no. 10334050) and ScientificProject of Zhejiang Province (Grant no. 2005c21003).

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