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Coupling of a laser diode to single-mode fiber with anupside-down tapered lens end

Samir Kumar Mondal and Somenath Sarkar

We present in detail a simple analysis of the coupling efficiency and possible transverse and angularmisalignment losses of a laser diode to single-mode, step-index fiber excitation with an upside-downtapered lens ~UDTL! end, drawn by molding the end of a step-index fiber. The analysis employs ourrecently formulated ABCD matrix for an UDTL and has the advantage of simplicity compared withcomplicated methods involving cumbersome numerical integrations. Both the analysis and the resultsshould be useful in designing coupling optics that use such lenses in the context of probable misalign-ments. © 1999 Optical Society of America

OCIS codes: 060.2430, 350.3950, 140.2020, 060.0060, 060.2340.

t

1. Introduction

To achieve efficient coupling between the laser sourceand the single-mode fiber for optimal performance inoptical communication systems, various lensingschemes are still being proposed, designed, and fab-ricated. From these proliferating schemes the for-mation of different integral microlenses on the fibertip is the most recent state of the art.1–3 These in-tegral microlenses that replace intermediate large-lens systems placed disjointly4 between source andfiber have hemispherical and hyperboloidal inter-faces. Although a hemispherical microlens can beeasily drawn, its aperture is limited, and the hyper-bolic lens with greater light-gathering power involvesan intricate construction technique.

A recent prospective candidate for such integrallensing schemes is the upside-down tapered lens5–7

~UDTL! drawn from the fiber end where the fiber tips tapered into a large hemispherical shape. Such aens can be drawn easily and can be used to gather aarge amount of light from the source.5 Thus itsensing properties can be exploited for coupling to aource or a detector and used in other micro-opticalystems. Although the techniques used to draw

The authors are with the Fiber Optics Research Group, Depart-ment of Electronic Science, University of Calcutta, 92 AcharyaPrafulla Chandra Road, Calcutta 700 009, India. The e-mail ad-dress for S. N. Sarkar is sns@cucc.ernet.in.

Received 4 January 1999; revised manuscript received 16 June1999.

0003-6935y99y306272-06$15.00y0© 1999 Optical Society of America

6272 APPLIED OPTICS y Vol. 38, No. 30 y 20 October 1999

such a tapered end are highlighted in Ref. 5, thenature of the refractive-index distribution in the ta-pered region is not predicted experimentally. Thisfield of study should attract more and more theoret-ical and experimental attention to the analysis of therefractive-index profile and the coupling optics of theUDTL. Owing to the wide availability of the step-index fiber for significant application in communica-tion, the derivation of the possible refractive-indexdistribution and the corresponding ABCD matrixransformation7 for a tapered lens end drawn mono-

lithically from a step-index fiber has been reportedrecently.

An extensive study has been reported on the cou-pling performance of conical microlenses, includingtheir possible misalignments with cumbersome nu-merical methods,3 and the recently derived simpleABCD matrix methods for hyperbolic8,9 integral lenssystems. However, to the best of our knowledge, nosuch study is available on the UDTL lens. Althoughonce the simple ABCD matrix is derived for a system,the coupling efficiency calculation is conventionaland simple in terms of the well-known overlap inte-gral.8

It may be relevant in this connection to mentionthat a considerable amount of research has been re-ported on deriving the geometrical optics of taperedgradient-index rods,10 the associated ABCD matrix inarbitrary tapered quadratic-index waveguides,11 andsingle-mode coupling by tapered gradient-index-fiberlenses.12 But this research needs more specific ori-entation for one to incorporate the emerging UDTLdrawn monolithically from a step-index fiber into aready reference in coupling optics. In this context

idUepi

mfimscgss

ss

the simple ABCD matrix derived recently and specif-ically for such an UDTL predicts the coupling opticsinvolving UDTL in the usual way so that one canhave direct access to this formulation. Thereforenow is the time to use the prescribed ABCD matrix tonvestigate coupling efficiencies in the case of a laseriode to single-mode fiber excitation through anDTL drawn on the fiber tip in the absence or pres-

nce of misalignments. The current formalism isrecisely the form to deal with coupling optics involv-ng an UDTL.

In this paper we use the above-mentioned ABCDatrix derived for the UDTL end from a step-index

ber to estimate the theoretical coupling perfor-ance between the laser source and the single-mode,

tep-index fiber through the UDTL. Coupling effi-iency in the presence of possible transverse and an-ular misalignments has also been studied. Such atudy might be useful to the designer in molding andhaping the desired UDTL from the fiber end.

2. Theory

A. Coupling Efficiency

The coupling arrangement to be studied is shown inFig. 1, where u is the separation between the laserource and the UDTL end fiber. As in earlier re-earch,3 we consider the usual approximations, such

as the Gaussian distribution of fiber and source fieldsand the perfect matching of the polarization of thefield at the lens end and fiber face. However, inconnection with the approximation of the Gaussiansource field, it would be appropriate to point out thatthe field in the direction perpendicular to the epi-taxial layers departs from Gaussian, given that thedimension of the junction of most sources in this di-rection is much less than that in the other direction,the latter being close to the wavelength of propaga-tion. But for simplicity a Gaussian beam has beenused in the literature to estimate coupling optics in-volving such sources and various kinds of lens.1–3,8,13

An even simpler assumption has been made of aspherical wave from a laser source1 with a Gaussianfield to investigate the performance of the novel hy-perbolic lens on the fiber tip. Furthermore, becausewe are employing the parameters of Ref. 3 in which

Fig. 1. Schematic diagram of a laser beam emitted from input plsingle-mode, step-index fiber.

the dimensions of the parallel and the perpendicularjunctions of the laser source are comparable, we, as inRef. 3, use Gaussian-field distributions of the sourcewith elliptical waist spot sizes with a view to explor-ing the applicability of the recently formulated ma-trix for the UDTL.

Accordingly the field cu at the output of the lasersource at distance u from the UDTL end is expressedas14

cu 5 expF2S X2

W1X2 1

Y2

W1Y2DGexpF2jk1SX2 1 Y2

2R1DG , (1)

where R1 is the radius of curvature of the wave frontfrom the laser diode, W1X and W1Y are the spot sizesalong two perpendicular directions, X and Y, withk1~52pyl1! as the wave number in the incident me-dium.

We also consider the Gaussian approximation14 ofthe fundamental mode field in the step-index, single-mode fiber and express it as

cf 5 expS2X2 1 Y2

Wf2 D , (2)

where Wf is the fundamental spot size and is approx-imated by15

Wf 5 aS0.65 11.619V1.5 1

2.879V6 D , (3)

where a is the core radius of the fiber and V is thenormalized frequency given by k0a@~nco

2 2 ncl2!#1y2,

where nco and ncl are axial and cladding refractiveindices, respectively, and k0 is the free-space wavenumber.

The laser field cu is transformed to field cv by theUDTL on fiber plane 2 where it matches the fiber’sfundamental mode. The UDTL-transformed laserfield is given by14

cv 5 expF2S X2

W2X2 1

Y2

W2Y2DGexpF2j

k2

2 S X2

R2X1

Y2

R2YDG ,

(4)

and refracted through an UDTL onto plane 2, the endface of the

ane 1

20 October 1999 y Vol. 38, No. 30 y APPLIED OPTICS 6273

l

e

Tt

sat

a

6

where k2 is the wave number of the lens medium andW2X and W2Y correspond to the transformed spotsizes and R2X and R2Y to the radii of curvature in theX and Y directions, respectively. It is well knownthat the coupling efficiency is expressed by the over-lap integral4 as

h0 5

U** cv*cfdXdYU2

** ucvu2dXdY ** ucf u2dXdY

, (5)

which, after substitution of cf and cv from Eqs. ~2!and ~4! into Eq. ~5! and integration, reduces to

where transformed spot sizes W2X and W2Y andtransformed radii of curvature R2X and R2Y are re-ated to W1X, W1Y, and R1 according to the prescribed

relations8 in Appendix A. In our computation theffective UDTL length, say ZL, which is equal to @~D 2

a!yD#L, is the length from the front refracting surfaceto the fiber input face with D and L having the usualmeanings,7 as shown in Fig. 1.

B. Misalignments

1. Transverse MisalignmentThe schematic diagrams for transverse and angularmisalignments are in Figs. 2~a! and 2~b!, respectively.

o obtain coupling efficiency in the presence of aransverse offset in the X–Y plane, we show that the

center of the fiber is shifted to a point with coordi-nates ~d1, d2!.

The fundamental mode in the fiber can then bewritten as

cf 5 expH2F~X 2 d1!2 1 ~Y 2 d2!

2

Wf2 GJ . (7)

Employing Eqs. ~4! and ~7! in Eq. ~5!, we obtain

ht 5 ho expH2d12

Wf2

3 F W2X2~W2X

2 1 Wf2!

~Wf2 1 W2X

2!2 1 ~k22W2X

4Wf4!y4R2X

2 2 1GJ3 expH2d2

2

Wf2

3 F W2Y2~W2Y

2 1 Wf2!

~Wf2 1 W2Y

2!2 1 ~k22W2Y

4Wf4!y4R2Y

2 2 1GJ .

(8)

h0 54W

F~Wf2 1 W2X

2!2 1k2

2W2X4Wf

4

4R2X2

274 APPLIED OPTICS y Vol. 38, No. 30 y 20 October 1999

2. Angular MisalignmentIn Fig. 2~b! we show the angular misalignment of themall angle between the UDTL-transformed endfacend the fiber endface. Now we can express the lens-ransformed laser field on the fiber as16

cv 5 expF2S X92

W2X2 1

Y92

W2Y2DGexpF2j

k2

2 SX92

R2X

1Y92

R2YDGexp~ jk2 X9u!, (9)

whereas the fundamental mode of the fiber becomes

cf 5 expF2SX92 1 Y92

Wf2 DG , (10)

where X9 and Y9 represent the new coordinate systems shown in Fig. 2~b!. Using Eqs. ~9! and ~10! in Eq.

2YWf2

~Wf2 1 W2Y

2!2 1k2

2W2Y4Wf

4

4R2Y2 G1y2

, (6)

Fig. 2. ~a! Transverse misalignment between the centers of thefiber and the imaged laser spot. ~b! Angular mismatch betweenthe endface of the fiber and the UDTL-transformed input face.

2XW

G1y2F

ciffidrFe

ss

co

~5!, we obtain coupling efficiency in the presence of asmall angular mismatch u as in Eq. ~11! ~Ref. 16!:

ha 5 h0 expH2k2

2u2

2

3 F ~Wf2 1 W2X

2!W2X2Wf

2

~Wf2 1 W2X

2!2 1 ~k22W2X

4Wf4y4R2X

2!GJ . (11)

The analytical expressions for ht and ha thus ob-tained are used to calculate the coupling losses forrespective misalignments.

3. Results and Discussion

To estimate the excitation efficiency of the single-mode, step-index fiber by the laser source with anUDTL, we consider laser-diode-emitting radiation ofwavelength 1.3 mm with spot sizes of W1X 5 1.081 mmand W1Y 5 1.161 mm. These parameters correspondto the typical laser used in an experiment3 of therecent past. We choose a typical step-index fiberwith core and cladding refractive indices of 1.46 and1.45, respectively, a spot size Wf of 4.794 mm, and aore diameter 2a of 4 mm. In addition, we take thenput beam from the laser facet to be a plane waveront. Two UDTL’s to be drawn from this typicalber were chosen each with 2D 5 6 mm and twoifferent UDTL lengths, ZL as 26.6 and 23.3 mm cor-esponding to L of 80 and 70 mm, respectively, as inig. 1. The radius of curvature R0 of the sphericalnd of the UDTL is taken to be 90 mm in each case.We depict a variation in coupling efficiency with

ource distance by curves ~a! and ~b! in Fig. 3 corre-ponding to the above-mentioned UDTL’s with ZL 5

26.6 and 23.3 mm, respectively. The curves showhigh peak values of more than 90% but also a gener-ally flat nature in contrast with the sharp variationusually seen.1 The maximum coupling efficiency isobserved in the case of curve ~a! for ZL 5 26.6 mm,apparently favoring the higher UDTL length. Fur-thermore, from the flat nature of the curve, the sys-tem appears to be more tolerant with respect to smallfluctuations in the source positions. Thus a small

Fig. 3. Variation of coupling efficiency against the source positionfor two UDTL lengths: curve ~a!, ZL 5 26.6 mm; curve ~b!, ZL 523.3 mm.

displacement from the actual position of the sourcewith maximum coupling efficiency does not causemuch change in efficiency. It is clearly evident fromFig. 3 that a displacement of the source position byapproximately 1.5 times the source position corre-sponding to maximum efficiency changes the effi-ciency by only ;2%. Similarly a variation in taperlength does not cause much change in maximum ef-ficiency. Figure 3 reveals that a 12% change inUDTL length causes only an ;2% shortfall in effi-iency. Again a similar result is observed in the casef variation in the radius of curvature R0 of the spher-

ical end where no appreciable change in efficiencyoccurs owing to the change in the radius of the cur-vature. Thus, in view of this advantage of flexibilityin lens parameters, with respect to optimal couplingperformance, the UDTL system should definitely beeffective in designing suitable coupling optics.

Next we study angular and transverse misalign-ment losses, considering the X direction only for thelens where ZL 5 26.6 mm and 2D 5 6 mm. Figure4~a! shows a variation in coupling loss for differenttransverse misalignments. It shows a loss of 0.74dB for 2-mm transverse misalignment. Figure 4~b!shows losses for different angular misalignments.For an angular misalignment of 2°, coupling loss is at1.96 dB. In both cases, loss without misalignment is

Fig. 4. Variation of coupling loss against ~a! transverse misalign-ment along the X direction and ~b! angular misalignment for theUDTL length ZL 5 26.6 mm.

20 October 1999 y Vol. 38, No. 30 y APPLIED OPTICS 6275

eowpcasTtesa

fbaU

b

U

6

at 0.1 dB, which corresponds to 97% efficiency incurve ~a! of Fig. 3. Such loss observation in the pres-nce of misalignments is also consistent with the lossbtained for the hyperbolic lens system.9 Thus,hen choosing an UDTL lens, one should not onlyrefer a higher UDTL length but should also be moreareful to avoid angular mismatch, since, as is seen,slight angular misalignment causes a pronounced

etback compared with a slight transverse offset.his theoretical study is, to the best of our knowledge,

he first loss analysis of UDTL coupling in the pres-nce and absence of misalignment. This studyhould also, from its confirmation, generate interestmong researchers who have proposed5 that the

UDTL be an alternative and prospective state of theart.

In all, our investigation should be useful for explor-ing the tolerance limit of the UDTL lens system andfor designing suitable coupling optics.

4. Conclusion

A theoretical investigation has been carried out onthe coupling efficiency of a laser source to a single-mode, step-index fiber with an upside-down taperedlens on a fiber tip. We have also studied coupling inthe presence of angular and transverse misalign-ments. All the calculations have been carried outwith the simple and currently derived ABCD matrixor the UDTL region. Suitable parameters for theest mode matching can also be determined from ournalysis. Such a study will be useful in designing anDTL on a fiber tip by laser molding.

Appendix A

Taking into consideration the distance u of the laserdiode from the UDTL end, the q parameters of theGaussian beams at the input laser facet and outputlens fiber interface can be related by the ABCD ma-trix:

q2 5 ~Aq1 1 Au 1 B!y~Cq1 1 Cu 1 D!, (A1)

where

1q1,2

51

R1,22

jl0

~pW1,22n1,2!

, (A2)

and A, B, C, and D are the matrix elements of the lensmatrix7:

A 5 r2~z! 2~1 2 n!

nR0r1~z!, B 5

r1~z!

n,

C 5dr2~z!

dz2

~1 2 n!

nR0

dr1~z!

dz,

D 51n

dr1~z!

dz

n 5n2

n1. (A3)

276 APPLIED OPTICS y Vol. 38, No. 30 y 20 October 1999

The z dependence of the above matrix elements cane explicitly expressed by substituting7

r1~z! 5 2La S1 2

zLD

1y2

sin K~z!,

dr1

dz5

1

S1 2zLD

1y2 Fcos K~z! 11

2asin k~z!G ,

r2~z! 5 S1 2zLD

1y2Fcos K~z! 21

2asin K~z!G ,

dr2

dz5

A02L

aS1 2zLD

1y2 sin K~z!, (A4)

where K~z! 5 a ln@12~zyL!# and a 5 ~A02L2 2 1⁄4!1y2,

L is the length of the tapered cone, and A0 is a con-stant given by A0 5 ~1yD!@2 ln~ncoyncl!#

1y2 for anDTL with aperture 2D. To obtain W2X,2Y, we eval-

uate the matrix for ZL 5 L@~D 2 a!yD#.The transformed spot sizes and radii of curvature

are related through the matrix elements with theinput spot size and radius of curvature:

W2X,2Y2 5

A12W1X,1Y

2 1 ~l12B1

2!y~p2W1X,1Y2!

n~A1 D1 2 B1 C1!,

1R2X,2Y

5A1 C1 W1X,1Y

2 1 ~l12B1 D1!y~p2W1X,1Y

2!

A12W1X,1Y

2 1 ~l1 B1!2y~p2W1X,1Y

2!, (A5)

where B1 5 Au 1 B, D1 5 Cu 1 D, A1 5 A 1 ~B1yR!,and C1 5 C 1 ~D1yR!, where R is the radius of cur-vature of the wave front from the laser facet, and inour plane wave-front model R 5 ` and l1 5 ~l0yn1!.

The financial support of the All India Council forTechnical Education, India, is gratefully acknowl-edged. S. K. Mondal acknowledges UniversityGrants Commission, India, for the award of a seniorresearch fellowship and both authors thank S.Gangopadhyay for helpful discussions. The authorsare also grateful to the anonymous reviewer for hisconstructive and helpful suggestions.

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