Input/output fiber configuration in a laser package design: optimization for lower stresses

Input/output fiber configuration in a laserpackage design: optimization for lower stresses

Ephraim Suhir

Mechanically and thermally induced stresses in the input/output (I/O) fiber of a laser package design areevaluated for different configurations of this fiber. It is found that if the fiber exhibits bendingdeformations, mechanical stresses can be minimized if a proper end offset is applied and thermal stressescan be reduced if the fiber is mechanically prestressed. It is found also that if the optical device can berotated by a small angle around the transverse axis, this rotation can be used effectively to minimize thestresses in both categories. It is shown that the smallest fiber span can be obtained if one makes the endplanes of the device perpendicular to the package axis, i.e., by making the fiber straight. Clearly in thiscase the fiber should be made short enough to avoid buckling under the compressive action of thermallyinduced stresses. Such a configuration is the most feasible because it results in the shortest fiber span(length) and in minimal optical losses. Such a configuration should be employed in all cases whenrotation of the optical device is possible, when the fiber ends can be easily aligned, and if the supportstructures are strong enough to withstand the higher thermally induced forces from the compressedfiber. Thermal stresses can be brought down by the use of low-expansion materials such as Kovar orInvar for the package enclosure. It should be pointed out that although the results of this analysisprovide designers with a useful theoretical guide for optimizing the I/O fiber configuration, the finalconfiguration can be selected only after the allowable stress and the achievable alignment (in the case ofstraight fiber) are evaluated experimentally.

Introduction

In a laser package design the inner ends of theinput/output (I/O) fibers (the ends at the opticaldevice) form acute angles (determined by the Snelllaw) with the axis of the package, while their outerends should be perpendicular to the package walls(simply for esthetic considerations). This configura-tion results in bending of the I/O fibers.

Typically, the fiber ends are bonded with an adhe-sive or soldered to the supports. Because such bond-ing (soldering) is done at a high temperature andbecause the coefficient of thermal expansion (contrac-tion) of the glass is considerably smaller than that ofthe package enclosure, the I/O fiber experiences bothmechanical bending stresses and thermally inducedstresses, when the package is cooled down to lowtemperatures (e.g., during operating, testing, ship-ment, or storage). Clearly, the fiber configurationshould be chosen in such a way that the total stresscaused by the combined actions of mechanical and

The author is with AT&T Bell Laboratories, Murray Hill, NewJersey 07974.

Received 2 February 1993; revision received 7 September 1993.0003-6935/94/12230007$06.00/0.c 1994 Optical Society of America.

thermal loading be low enough so as to not compro-mise fiber strength.

In the analysis that follows the mechanical andthermal stresses for different I/O fiber configurationsare evaluated. These configurations are character-ized by the fiber span (distance from the package wallto the fiber's inner end), the off-set of the ends, andthe angle that the inner end of the fiber forms withthe package axis (Fig. 1). The purpose of this analy-sis is to provide designers with a useful guide thatpermits evaluation of the induced stresses and selec-tion of the most feasible fiber configuration, depend-ing on the materials under consideration and thetechnology available.

Analysis

Curved Fibers

The elastic curve (deflection function) w(x) of an I/Ofiber with a given span 1, an end offset , and an angle1 at its inner end (Fig. 1) can be written in the form ofan equation with initial parameters of

Mox 2

w(x) = 8 - 2EINox 3

6EI (1)

(see, e.g., Refs. 1 and 2). Here MO and No are the

2300 APPLIED OPTICS / Vol. 33, No. 12 / 20 April 1994

Package Wall

1/0 Fiber Optical Device

1 £ -?

Fig. 1. Mechanical bending of the I/O fiber.

bending moment and the lateral force at the origin, Eis the Young modulus of the glass, I = (/4)ro 4 is themoment of inertia of the fiber cross section, and r isthe fiber radius. The origin of the coordinate x is atthe outer end of the fiber (at the package wall).

With Eq. (1) the conditions w(0) = and w'(0) = 0at the origin are fulfilled automatically, and theconditions w(l) = 0 and w'(I) = - at end x = I resultin the following equations for the initial parametersMO and No:

6EI3Mo - No = 12 8

2EI2M - No= - 13.

From these equations we find

2EI -Mo= - (3S -13),

6Eo -No = - (2 - ), (2)

where 8 = S/1 is the ratio of the end offset to the fiberspan. Substituting Eq. (2) in Eq. (1) we obtain theelastic curve of the fiber in the form

In this case the fiber is subjected to the bendingmoment

EIMP= PI (8)

which is uniformly distributed over its length, andthe elastic curve given by Eq. (3) is a parabola where

I~)=l X) 2g

When the value is larger than 13/2 (Fig. 2) themoment MO at the origin is greater than the momentMl at the end where x = 1, and it exceeds the momentMp, which corresponds to pure bending, by a factor of2(39/ - 1). For instance, if = 3, this factor isequal to 4. When the value is smaller than /2moment Ml becomes greater than MO and exceedsmoment Mp by a factor of 2(2 - 3/). If forinstance = 0, moment Ml is 4 times greater thanmoment Mp. Note that moments MO and Ml at thefiber ends change their signs for = /3 and =21/3, respectively.

The length of the curved fiber can be evaluated ifwe assume small deflections as in

l IlS = + [w'(x)]2dx

2

= I + - [-2(38 - P)g + 3(28 - 2]2d,

= I + -(23 + 183 - 38). (10)30

w(x) = 8 - ax2 + bx3,where the following notation is used:

38 - 3I

23 -b=

(3)

(4)5

The bending moment can be evaluated with Eq. (3)by differentiation:

M(x) = -EIw"(x)

2EI 3 -1=1 - 3(29 - 13)

The bending moments at the fiber ends are

2EI -MO =M(0)= -I(36 P)

2EI -Ml =M(l) = - -I (3 -213).

(5)

0

(6)

As is evident from Eq. (5), the fiber undergoes purebending if

2

-5

(7)

Mo=-Mo f =2E13 1

Mt =M

Fig. 2. End bending moments in an I/O fiber.

20 April 1994 / Vol. 33, No. 12 / APPLIED OPTICS 2301

5/3

=2-3 p

The minimum difference between the length of thecurved fiber and its span is

AS = S - = 1/1 21, (11)

and it takes place when 6 = 1/12. Note that in thecase of pure bending this difference is substantiallygreater:

As = /6121. (12)

Let the laser package be heated for bonding (solder-ing) to a temperature At. Then the span of the I/Ofiber increases by the amount Al = lAaAt, whereAaAt is the thermal strain, and Aa is the differencebetween the coefficients of thermal expansion of thematerials of the package enclosure and the glass.When after bonding (soldering) of the fibers thepackage is cooled down, the fiber experiences ther-mally induced bending in addition to mechanicalbowing. The thermally induced bending moment(Fig. 3) can be expressed as

m(x) = mO - nox + T[8 - w(x)]

= mO - nox + T(ax 2- bx3), (13)

where mO, no, and T are the initial bending moment,the initial lateral force, and the initial axial force,respectively. These variables can be determined withthe Castigliano theorem from the formula for thestrain energy",2

m 2(x)dx.

In accordance with the Castigliano theorem, thepartial derivative of the strain energy V with respectto the given generalized force Q (in the case inquestion, with respect to mO, no, or T) is equal to thecorresponding generalized displacement A:

av 1 I' am(x)aQ =EI mx jQ dx = A.

The generalized displacements that correspond to themoment mo and the force no are zero, and the

displacement corresponding to the force T, is equal toAl. Then the application of the above formula re-sults in the following system of algebraic equationsfor the unknowns mO, no, and T:

aV 1 am() I I m(x) am dx = -I m(x)dx = 0,amo El am0 Elj

aV 1 g' am(x)

ano EI jx dno= - EI m(x)xdx = 0,

av 1 C' am(x)

aT EI J AT

E=-J m(x)(ax 2 - bx3)dx = Al,EI

or if we consider Eq. (13)

12mo - 61no + 1(68 - )T = 0,

30mO - 201no + 31(78 - )T = 0,

35(63 - )mo - 211(78 - )no

_ _ ~AlEI+ 41(12 + 3952 - 111i3)T = 420 l2

12

These equations have the following solutions:

AlEI 33+MO = 210 _3 +2912 11p2 + 92 - '

AlEI 123 - 1nO= 630_ _

13 112 + 932 - 918

AlEI 1T= 6300 _13 11l 9-2 -9p8

(14)

Note that as can be obtained from the second formulain Eqs. (14), the lateral thermally induced force nobecomes zero if S = 1/12, which is the case for theshortest length of the bent fiber. The thermallyinduced bending moment ml at end x = can bedetermined from Eq. (13) as

ml = mO - nol + T(al2 - bl3)

A IEI 3S - 413= -210 2 112 + 92 - 98 (15)

Optical D1/0 Fiber

Fig. 3. Thermally induced bending of the I/O fiber.

eviceAs is evident from Eqs. (14) and (15), the end

moments mO and ml become equal when S = 1/2, i.e.,x in the case of pure bending of the fiber. In this case

AlEI EI AaAtmp = mO = ml = 60 - = 60 - -I2 13 1 13 (16)

It should be pointed out that unlike the mechanicalmoment the thermally induced moment increases


T

with a decrease in the angle 13 at the inner end of theI/O fiber.

Using Eqs. (8) and (16), we obtain the followingformula for the total maximum-bending momentresulting from the combined actions of the mechani-cal and the thermal loading

M=Mp+mp=- Au-At+60 i .

13The section modulus of the glass fiber is I/ro, where rois the radius of the fiber. Then moment M results inthe following bending stress:

M ro AoLAt)or = I r = E 1 + 60 Z .

The critical (buckling) force T and the postbucklingelastic curve of the fiber can be determined from thefollowing equation of the bending of a buckled beam1 2:

EIwN(x) + Tw"(x) = 0. (20)

The elastic curve w (x) of a buckled fiber can be soughtin the form

w(x) = CO + Cl cos kx, (21)

where

T E-El (22)

(17)

As is evident from this formula, the maximum totalstress in the I/O fiber is inversely proportional to thefiber span 1. As to the angle , its increase results inlarger mechanical stresses and smaller thermalstresses. Therefore there exists an angle 13* thatminimizes the total stress. This angle can be foundfrom Eq. (17) as

A* = 60AaAt. (18)

In this case the mechanical and the thermal stressesbecome equal, and the total stress is

or* = 2E I *. (19)

Initially Straight Fibers

Examine now a situation in which optical device canbe easily rotated in such a way that its end planesbecome perpendicular to the package axis. In thiscase the I/O fiber becomes straight (Fig. 4) and doesnot undergo bending deformations. However, afterthe fiber is bonded (soldered), it undergoes uniaxialcompression (we are assuming of course that its endsare well aligned). If the fiber is made short, thiscompression will not cause it to buckle and will have afavorable effect on the brittle strength and long-termreliability of the fiber. Clearly, a shorter fiber also isdesirable when we consider package size and transmis-sion lasers.

is the eigenvalue of the problem, and C0 and C1 arethe constants of integration. The boundary condi-tion w(112) 0 (there is no vertical displacement atend x = 1/2) yields

CC= - C .cos U

Here the parameter u is expressed as

k1 1 TU =-= -V-

Then Eq. (21) results in the followingconfiguration of the fiber:

w(x) = (l -cos kx

(23)

postbuckling

(24)

Using the boundary condition w'(l/2) = 0 (there is norotation of the fiber axis at end x = 1/2), we obtainsin u = 0. Therefore, as follows from Eq. (23), thecritical force is expressed by the formula

4rr 2 EIC- = 12 (25)

which is a well-known formula for the critical forcefor a beam clamped at the ends.12 The correspond-ing compressive stress is

=E(Jc = 2 =E 0o, (26)

Optical Device where

(rro)2s = I

X is the nominal critical strain.It has been found that the stress-strain relation-

ship in silica materials subjected to uniaxial loading isnonlinear.3 4 For compressive strains not exceeding5% this relationship is

Fig. 4. I/O fiber with a straight axis. u C = E0(E - 1/20t2).


(27)

(28)

Here E0 is the Young modulus of the material in thearea of low stresses and ao is the parameter ofnonlinearity. For silica materials typically employedin fiber optics E0 = 10.5 x 106 psi (= XX Torr or XXPa), and a = 6.4

From Eq. (28) we find

E = d = E0(1 - OF),

and Eq. (26), with consideration of Eq. (28), results inthe following equation for the critical strain E,:

2 2-, -(1 + ato)E, + -o = 0. (29)

exponent n in this formula can be assumed to be n 18 - 20.6 Then we conclude that the short-term(high-level) stress a, applied to the fiber during solder-ing is related to the long-term (low-level) stress UT bythe formula

t -=/n(t = UST - (33)

where t is the shorter time during which the stress utis applied and T is the longer expected life-time of thefiber material at room or operating temperature.Obviously, if the calculated ut value predicted on thebasis of Eq. (33) exceeds the expected thermallyinduced stress

This equation has the solution

1sIC =-(1 + o - 1 + 2s 0

2).

r0 AaAtth = 60E 13,

(30)

Clearly no buckling could occur if the critical strain E,were greater than the actual thermal strain Et =Au-At. Using Eq. (27) we obtain the formula for themaximum allowable length * of the fiber (i.e., thelength that will not lead to buckling)

1* = wrro/2(1 - ast) (31)<\£t (2 ~- at)

If a low-expansion enclosure is used so that the strainEt is small, the nonlinear stress-strain relationshipneed not be considered, and a simplified formula

* wo

(32)

can be applied.Because the use of a straight I/O fiber enables one

to achieve the shortest laser package and lowertransmission losses, we view such a configuration asthe most feasible and suggest that it be used in allcases when appropriate rotation of the optical deviceis possible, when the fiber ends can be easily aligned,and if the fiber's support structures can withstandthe increased forces from the thermally compressedfiber. It should be pointed out that these forces canbe brought down if necessary by the use of low-expansion materials for the package enclosure (e.g.,Kovar or Invar).

Mechanical Prestressing for Lower Thermal Stress

Thermal stresses at low (room or testing) tempera-tures can be reduced for improved reliability if thefiber is mechanically stressed. When choosing aprestress level, one can proceed from the tentativerelationship between the time to failure Tf of the fiberand the applied stress r5,6

Tf= ar-'.

For the silica materials employed in fiber optics, the

(34)

then it is the latter value that should be taken as theshort-term stress.

It goes without saying that mechanical prestress-ing also can be helpful in the case of an initiallystraight fiber if there is a need to increase the lengthof the fiber (e.g., to bring down the stresses resultingfrom an inevitable misalignment) while ensuring itselastic stability.

Numerical Examples

Let the package enclosure be made of phosphorbronze with the coefficient of thermal expansion18.5 x 10-6 1/0 C. If the I/O fiber is fixed bysoldering at the temperature 185°C, then with acoefficient of thermal expansion of the glass of 0.5 x10-6 1/0C, the thermal contrarction mismatch strainat room temperature ( 25 0C) is E = Au-At = 0.00288.

Let the angle that the inner end of the fiber formswith the axis of the package be 30. This angle is verysmall and in accordance with Eq. (16) can result inrather high thermally induced moments for a suffi-ciently small span 1. These moments can be reducedby an appropriate rotation of the optical device, sothat the above angle is increased, e.g., 1 = 10°. Letthe allowable strain based on the long-term reliabilityrequirement for the fiber be [E] = 0.5% = 0.005.5Formula (17) then results in the expression for theallowable span

ro AaIAt\=-61 + 6013).- (40)

With ro = 0.0625 mm, we obtain 1 = 14.6 mm.The span can be reduced with mechanical prestress-

ing of the fiber. If the desired life-time of the fiber atroom temperature is T = 20 yr = 10 483 200 min, andthe soldering time is t = 30 sec = 0.5 min, then Eq.(33) with n = 20 predicts that the corresponding(allowable) short-term (high-level) strain can be afactor of (T/t)l/n = 2.3 greater than the long-term(low-level) strain. Using a (conservative) assump-tion that the safety margins can be accepted as thesame in both cases, we conclude that the allowable


short-term strain is approximately 0.011. Assumingfurther that short-term prestressing has a negligibleeffect on the allowable long-term strain, one cansimply reduce the above calculated span by a factor of2.3, i.e., make it only I = 6.3 mm. The elongation ofthe fiber corresponding to such prestress is Al =0.063 mm = 63 Am. Note that the optimal angle 1,calculated with Eq. (18) is for this case * = 0.4157 =23.80, i.e., it significantly exceeds 100.

Now examine a situation in which the packageenclosure is made of a low-expansion material such asInvar with a coefficient of thermal expansion of only2.0 x 10-6 1/0C and the package geometry is suchthat the angle at the inner end of the fiber cannot bemade larger than 10 C. The calculated thermallyinduced strain is S = AaoAt = 0.00024, and theoptimal angle of rotation in accordance with Eq. (18)is * = 0.12 = 6.90. Clearly such a small angle canbe achieved easily. The calculated fiber span in thiscase is only

= 2 o" = 3mm.

If a Kovar package (= 5.0 x 10-6 1/0C) is em-ployed, the predicted thermally induced strain is St =AuLAt = 0.00072, and the optimal angle at the innerend of the fiber is * = 0.2078 = 11.9°. This issomewhat larger than the maximum angle of 100.Note that with the 100 angle, Eq. (40) yields 1 = 5.3mm, which is only slightly greater than the minimumvalue 1 = 5.2 mm in the case of an optimal angle of13= 11.90.

In an initially straight fiber, it is the maximum notthe minimum span that should be restricted to avoidbuckling. The maximum allowable span, calculatedin accordance with Eq. (31) is 3.3 mm for a phosphorbronze enclosure, 12.7 mm for an Invar enclosure,and 7.3 mm for a Kovar enclosure. Clearly as far asthe elastic stability of the fiber is concerned, theactual span can be made smaller than these values.However application of very short fibers might im-pose requirements too stringent for the allowablemisalignment of the opening in the package wall andthe aperture in the optical device.

For instance let the package enclosure be made ofphosphor bronze and the change from the soldering(bonding) temperature to the low (testing) tempera-ture be 220 'C. Using Eq. (31) we conclude that thefiber should not be longer than 3 mm or its elasticstability might be compromised. The supports inthis case should be able to withstand a compressiveforce of approximately 360 gf. If a Kovar enclosureis used, the calculated maximum allowable length ofthe fiber is approximately 7 mm and the predictedcompressive thermally induced force is 90 gf. Clearly,application of low expansion enclosures enables oneto apply larger fiber spans and hence to accommodatehigher end misalignments.

Conclusions

The following major conclusions can be drawn fromthe performed analysis:

* Simple, easy-to-use formulas have been ob-tained for the evaluation of mechanical and thermalstresses in I/O fibers of laser packages.

* The smallest span of an I/O fiber can be achievedif the optical device can be rotated within the packagein such a way that its end planes become perpendicu-lar to the axis of the package. Because in this casethe fiber will remain straight, it will not experiencebending deformations. However, at low (room ortesting) temperatures it will be subjected to thermallyinduced compression. It is imperative that the fiberbe short enough to withstand such compressionwithout buckling. Because an initially straight I/Ofiber enables one to design the shortest package withminimum transmission losses, such a configuration isviewed the most feasible and it should be employed inall cases when the appropriate rotation of the opticaldevice is possible, when good alignment between theopening in the package wall and the aperture in theoptical device can be easily achieved, and if the fibersupports can be made strong enough to withstandhigh thermally induced forces from the compressedfiber. One can reduce these forces if necessary byemploying low-expansion materials for the packageenclosure or by mechanically prestressing the fiber.

* If the optical device cannot be rotated or if asufficiently good alignment between the I/O fiberends cannot be achieved, bending stresses in the fiberbecome inevitable. In this case the configuration ofthe fiber should be chosen so that the bendingstresses are as low as possible. This can be achievedwith an appropriate fiber end offset so that at roomtemperature (prior to heating the package for solder-ing) the fiber experiences pure bending. Thermalbending stresses can be reduced by employment oflow-expansion package enclosures, appropriate rota-tion of the optical device (so as to increase the acuteangle at the inner end of the fiber), and with prelimi-nary mechanical stressing of the fiber.

* The execution of the above analysis enables adesigner to understand the mechanical behavior ofI/O fibers of different configurations and provides auseful theoretical guide for optimizing the fiber con-figuration and achieving lower stresses. However,the final selection of such a configuration should bemade only after the allowable stress and the achiev-able alignment (which might be crucial in the case of astraight fiber) are evaluated experimentally for thegiven materials and the available technology.

The author thanks C. Paola, W. M. MacDonald, andG. Bubel of AT&T Bell Laboratories for valuablediscussions.

References

1. S. P. Timoshenko and D. H. Young, Theory of Structures, 2nded. (McGraw-Hill, New York, 1965).


2. E. Suhir, Structural Analysis in Microelectronic and Fiber-Optic Systems, Vol. 1 of Basic Principles of Engineering Elastic-ity and Fundamentals of Structural Analysis (Van NostrandReinhold, New York, 1991).

3. F. P. Mallinder and B. A. Proctor, "Elastic constants of fusedsilica as a function of large tensile strain," Phys. Chem. Glasses5, 91-103 (1964).

4. J. T. Krause, L. R. Testardi, and R. N. Thurston, "Deviations

from linearity in the dependence of elongation upon free fibersof simple glass formers and of glass optical lightguides," Phys.Chem. Glasses 20, 135-139 (1979).

5. W. R. Wagner, "Extensive fiber damage and its effect on thereliability of optical fiber connectors and splices," in Fiber OpticComponents and Reliability, P. M. Kapera, ed., Proc. Soc.Photo-Opt. Instrum. Eng. 1580, 5-7 (1992).

6. G. Bubel, AT&T Bell Laboratories, Murray Hill, N.J. 07974(personal communication, 1992).


Input/output fiber configuration in a laser package design: optimization for lower stresses

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