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Optics Communications 278 (2007) 81–89

Estimation of thermal contact resistance and thermally inducedoptical effects in single-coated optical fibers

Yu-Ching Yang *

Department of Mechanical Engineering, Kun Shan University, Tainan 710-03, Taiwan, ROC

Received 26 January 2007; received in revised form 4 June 2007; accepted 4 June 2007

Abstract

In this study, a conjugate gradient method based on an inverse algorithm is applied to estimate the unknown time-dependent thermalcontact resistance in a single-coated optical fiber, which is subjected to transient thermal loading. While knowing the temperature historyat the measuring position, no prior information is needed on the functional form of the unknown contact resistance. The temperaturedata obtained from the direct problem are used to simulate the temperature measurement. The influence of measurement errors, initialguess values, and measurement locations upon the precision of the estimated results is also investigated. Results show that an excellentestimation on the time-dependent thermal contact resistance, temperature distributions, thermally induced microbending loss, andrefractive index changes can be obtained for the case considered in this study.� 2007 Elsevier B.V. All rights reserved.

Keywords: Inverse problem; Thermal contact resistance; Conjugate gradient method; Optical fibers; Microbending loss; Refractive index changes

1. Introduction

The conventional optical fiber for telecommunication isusually constructed of silica glass fiber coated by two orthree layers of polymeric coatings [1,2]. However, whensilica fibers with polymeric coatings are stressed in ahumid environment, strength degradation occurs after along period of time due to slow crack growth. To solvethis problem of moisture attack, inorganic coatings suchas oxides, carbides, and carbon are being considered[3,4]. On the other hand, metallic coatings such as alumi-num, indium, copper, tin, etc. are also applied on opticalfibers [5–7]. Some of these fibers exhibit higher resistanceto moisture attack and show higher strength than poly-meric-coated fibers.

On the other hand, thermal stresses are important inmulti-layer structures such as optical fibers, since theycause increased transmission loss in the optical fibers. Ther-

0030-4018/$ - see front matter � 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.optcom.2007.06.013

* Tel.: +886 6 2050496; fax: +886 6 2050509.E-mail address: ycyang@mail.ksu.edu.tw

mal stresses occur in the optical fibers at a low temperature,which are due to the mismatches of thermal expansionproperties of fiber and coating material [8,9]. Since thermalstresses affect the performance of the optical fibers, theyhave been extensively studied [10–15]. Nevertheless, tothe author’s knowledge, there was no work in the literatureinvestigating the effect of interlayer thermal contact resis-tance in such layered construction as optical fibers. Inaddition, the thermal contact resistance can neither bemeasured nor calculated directly. However this problemcan be solved by the technique of Inverse Heat Conduction

Problem (IHCP).A direct heat transfer problem is concerned with the

determination of temperature at interior points of aregion when thermophysical properties and initial andboundary conditions are specified. In contrast, an inverseheat transfer problem considered in this study involves thedetermination of the unknown time-dependent thermalcontact resistance from the knowledge of the temperaturemeasurements taken within the body. Many methods ofsolving the inverse heat transfer problem have been

Nomenclature

E Young’s modulus (GPa)h convection coefficient (Wm�2 K�1)J functionalJ 0 gradient of functionalk thermal conductivity (Wm�1 K�1)q direction of descentR thermal contact resistance (m2 KW�1)r radius (lm)T temperature (�C)T0 initial temperature (�C)T1 surrounding temperature (�C)t time (s)u displacement radial component (lm)Y measurement temperature (�C)C microbending loss (dB/km)D small variation qualityDnr, Dnz, Dn1 refractive index changesa thermal diffusivity (m2 s�1)

b step sizeC conjugate coefficienteh strain tangential componentg very small valueh temperature drop (�C)k variable used in adjoint problemm Poisson’s ratior standard deviationrr stress radial component (MPa)rz stress axial component (MPa)rh stress tangential component (MPa)s transformed time (s)x thermal expansion coefficient (K�1)- random variable

Superscript

K iterative number

++

++

++

++

++

++

++

++

++

+ +

t, sec

R(t

),m

2K

/W

0 0.02 0.04 0.06 0.083

4

5

6

7

8

9

Exact SolutionInverse Solution+

× 10-3

Glass FiberCoating

rr 12

Fig. 1. Estimated thermal contact resistance at 20th iteration with initialguesses eR0 ¼ 0:003, measurement error r = 0.0, and measurement loca-tion rm = 63 lm.

82 Y.-C. Yang / Optics Communications 278 (2007) 81–89

developed. In this article, we present the conjugate gradi-ent method (CGM) [16–23], which converges very rapidlyand is not sensitive to the measurement errors, to estimatethe time-dependent thermal contact resistance of a single-coated optical fiber by using the simulated temperaturemeasurements. Subsequently, the temperature distribu-tions, thermally induced transient microbending loss,and refractive index changes of the optical fiber are alsocalculated.

The conjugate gradient method derives basis from theperturbational principles and transforms the inverse prob-lem to the solution of three problems, namely, the directproblem, the sensitivity problem, and the adjoint problem,which will be discussed in detail in text.

2. Direct problem

To illustrate the methodology of developing expressionsfor use in determining the unknown time-dependent ther-mal contact resistance, R(t), in a single-coated optical fiber,we consider the following transient heat conductionproblem.

Fig. 1 shows the geometry of a single-coated opticalfiber, which is composed of a silica glass fiber coated by athin layer of coating, and has intermediate and outer radii,r1 and r2, respectively. Assume the single-coated opticalfiber is initially at temperature Ti(r, 0) = T0, and for timet > 0, the fiber at its boundary surface r = r2 is subjectedto a convective thermal loading of the surrounding temper-ature T1, T1 < T0. Here the subscript i = 1 refers to theregion of glass fiber, and i = 2 refers to the region of coat-ing, respectively. Then the mathematical formulation of thistransient heat conduction problem is given by [20]:

o2T i

or2þ 1

roT i

or¼ 1

ai

oT i

ot; i ¼ 1; 2; ð1Þ

oT 1ð0; tÞor

¼ 0; ð2Þ

k1

oT 1ðr1; tÞor

¼ k2

oT 2ðr1; tÞor

; ð3Þ

Y.-C. Yang / Optics Communications 278 (2007) 81–89 83

T 1ðr1; tÞ � T 2ðr1; tÞ ¼ �RðtÞk1

oT 1ðr1; tÞor

; ð4Þ

� k2

oT 2ðr2; tÞor

¼ h½T 2ðr2; tÞ � T1�; ð5Þ

T iðr; 0Þ ¼ T 0; i ¼ 1; 2; ð6Þ

where R(t) is the interlayer thermal contact resistance; k

and h are the thermal conductivity and the convection heattransfer coefficient, respectively. The direct problem con-sidered here is concerned with the determination of themedium temperature when the thermal contact resistance,thermal properties, and initial and boundary conditionsare known. A hybrid numerical method of Laplace trans-formation and finite difference used in our previous workcan be applied to solve the direct problem [24].

3. Inverse problem

For the inverse problem, the thermal contact resistance,R(t), is regarded as being unknown, while everything else inEqs. (1)–(6) is known. In addition, temperature readingstaken at r = rm are considered available. The objective ofthe inverse analysis is to predict the unknown time-depen-dent thermal contact resistance R(t) from knowledge ofthese temperature readings. Let the measured temperatureat position r = rm and time t be denoted by Y(rm, t). Thenthis inverse problem can be stated as follows: by utilizingthe above mentioned measured temperature data Y(rm, t),the unknown R(t) is to be estimated over the specified timedomain.

The solution of the present inverse problem is to beobtained in such a way that the following functional isminimized:

J ½RðtÞ� ¼Z tf

t¼0

½T 2ðrm; tÞ � Y ðrm; tÞ�2 dt; ð7Þ

here T2(rm, t) is the estimated (or computed) temperature atthe measurement location r = rm. These quantities aredetermined from the solution of the direct problem givenpreviously by using an estimated eRKðtÞ for the exact R(t).Here eRKðtÞ denotes the estimated quantities at the Kth iter-ation. tf is the final time of the measurement. In addition, inorder to develop expressions for the determination of theunknown R(t), a ‘‘sensitivity problem’’ and an ‘‘adjointproblem’’ are constructed as described below.

4. Sensitivity problem

The sensitivity problem is obtained from the originaldirect problem defined by Eqs. (1)–(6) in the followingmanner: It is assumed that when R(t) undergoes a variationDR(t), Ti(r, t) is perturbed by Ti + DTi. Then replacing inthe direct problem R by R + DR and Ti by Ti + DTi, sub-tracting from the resulting expressions the direct problem,and neglecting the second-order terms, the following sensi-tivity problem for the sensitivity function DTi can beobtained.

o2DT i

or2þ 1

roDT i

or¼ 1

ai

oDT i

ot; i¼ 1;2; ð8Þ

oDT 1ð0; tÞor

¼ 0; ð9Þ

k1oDT 1ðr1; tÞ

or¼ k2

oDT 2ðr1; tÞor

; ð10Þ

DT 1ðr1; tÞ�DT 2ðr1; tÞ ¼�RðtÞk1

oDT 1ðr1; tÞor

�DRðtÞk1

oT 1ðr1; tÞor

;

ð11Þ

� k2

oDT 2ðr2; tÞor

¼ hDT 2ðr2; tÞ; ð12Þ

DT iðr;0Þ ¼ 0; i¼ 1;2: ð13Þ

The sensitivity problem of Eqs. (8)–(13) can be solved bythe same method as the direct problem of Eqs. (1)–(6).

5. Adjoint problem and gradient equation

To obtain the adjoint problem, Eq. (1) is multiplied bythe Lagrange multipliers (or adjoint functions) k1(r, t) andk2(r, t), respectively, and the resulting expression is inte-grated over the time and correspondent space domains.Then the result is added to the right hand side of Eq. (7)to yield the following expression for the functional J[R(t)]:

J ½RðtÞ� ¼Z tf

0

½T 2ðrm; tÞ � Y ðrm; tÞ�2 dt

þZ r1

0

Z tf

0

r � k1ðr; tÞ �o2T 1

or2þ 1

roT 1

or� 1

a1

oT 1

ot

� �dtdr

þZ r2

r1

Z tf

0

r � k2ðr; tÞ �o

2T 2

or2þ 1

roT 2

or� 1

a2

oT 2

ot

� �dtdr:

ð14Þ

The variation DJ is obtained by perturbing R by DR and Ti

by DTi in Eq. (14). Subtracting from the resulting expres-sion the original Eq. (14) and neglecting the second-orderterms, we thus find:

DJ ½RðtÞ� ¼Z r2

r1

Z tf

0

2r � ½T 2ðrm; tÞ� Y ðrm; tÞ�rm

� dðr� rmÞdtdr

þZ r1

0

Z tf

0

r � k1ðr; tÞ �o2DT 1

or2þ 1

roDT 1

or� 1

a1

oDT 1

ot

� �dtdr

þZ r2

r1

Z tf

0

r � k2ðr; tÞ �o2DT 2

or2þ 1

roDT 2

or� 1

a2

oDT 2

ot

� �dtdr;

ð15Þwhere d(r � rm) is the Dirac function. We can integrate thesecond and third double integral terms in Eq. (15) by parts.Utilizing the initial and boundary conditions of the sensi-tivity problem, then DJ is allowed to go to zero. The van-ishing of the integrands containing DTi leads to thefollowing adjoint problem for the determination of k1(r, t)and k2(r, t):

o2ki

or2þ 1

roki

orþ 1

ai

oki

otþ 2½T 2ðrm; tÞ� Y ðrm; tÞ�

rdðr� rmÞ ¼ 0; i¼ 1;2;

ð16Þ

84 Y.-C. Yang / Optics Communications 278 (2007) 81–89

ok1ð0; tÞor

¼ 0; at r ¼ 0; ð17Þ

ok1ðr1; tÞor

¼ ok2ðr1; tÞor

; at r ¼ r1; ð18Þ

ok1ðr1; tÞor

¼ 1

RðtÞk2ðr1; tÞ

k2

� k1ðr1; tÞk1

� �; at r ¼ r1; ð19Þ

� k2

ok2ðr2; tÞor

¼ hk2ðr2; tÞ; at r ¼ r2; ð20Þ

kiðr; tfÞ ¼ 0; i ¼ 1; 2; for t ¼ tf : ð21Þ

The adjoint problem is different from the standard initialvalue problem in that the final time condition at time t = tf

is specified instead of customary initial condition. How-ever, this problem can be transformed to an initial valueproblem by the transformation of the time variable ass = tf � t. Then the adjoint problem can be solved by thesame method as the direct problem.

Finally the following integral term is left

DJ ¼Z tf

0

r1

½T 2ðr1; tÞ�T 1ðr1; tÞ�R2ðtÞ

k2ðr1; tÞk2

�k1ðr1; tÞk1

� ��DRðtÞdt:

ð22ÞFrom the definition used in the Ref. [25], we have

DJ ¼Z tf

0

J 0ðtÞDRðtÞdt; ð23Þ

where J 0(t) is the gradient of the functional J[R(t)], a com-parison of Eqs. (22) and (23) leads to the following form:

J 0ðtÞ ¼ r1

½T 2ðr1; tÞ � T 1ðr1; tÞ�R2ðtÞ

k2ðr1; tÞk2

� k1ðr1; tÞk1

� �: ð24Þ

6. Conjugate gradient method for minimization

Assuming the functions of Ti(r, t), DTi(r, t), ki(r, t), andJ 0(t) are available at the Kth iteration, the iteration processbased on the conjugate gradient method is now used for theestimation of R(t). By minimizing the above functionalJ[R(t)], the function R(t) can be evaluated at the(K + 1)th step by

eRKþ1ðtÞ ¼ eRKðtÞ � bKqKðtÞ; K ¼ 0; 1; 2; . . . ; ð25Þwhere bK is the search step size in going from iteration K toiteration K + 1, and qK is the direction of descent (i.e.,search direction) given by:

qKðtÞ ¼ J 0KðtÞ þ CKqK�1ðtÞ; ð26Þwhich is conjugation of the gradient direction J 0K(t) at iter-ation K and the direction of descent qK�1(t) at iterationK � 1. The conjugate coefficient CK is determined from:

CK ¼R tf

0½J 0KðtÞ�2 dtR tf

0 ½J0K�1ðtÞ�2 dt

with C0 ¼ 0: ð27Þ

The convergence of the above iterative procedure in mini-mizing the functional J is proved in the reference [26]. Toperform the iterations according to Eq. (25), we need to

compute the step size bK and the gradient of the functionalJ 0K(t).

The functional J ½eRKþ1ðtÞ� for iteration K + 1 is obtainedby rewriting Eq. (7) as:

J ½eRKþ1ðtÞ� ¼Z tf

0

½T 2ðeRK � bKqKÞ � Y ðrm; tÞ�2 dt; ð28Þ

where we replace eRKþ1 by the expression given by Eq. (25).If temperature T 2ðeRK � bKqKÞ is linearized by a Taylorexpansion, Eq. (28) takes the form:

J ½eRKþ1ðtÞ� ¼Z tf

0

½T 2ðeRKÞ � bKDT 2ðqKÞ

� Y ðrm; tÞ�2 dt; ð29Þ

where T 2ðeRKÞ is the solution of the direct problem at r = rm

by using estimate eRKðtÞ for exact R(t) at time t. Thesensitivity function DT2(qK) are taken as the solution ofEqs. (8)–(13) at the measured position r = rm by lettingDR = qK [27]. The search step size bK is determined byminimizing the functional given by Eq. (29) with respectto bK. After rearrangement, the following expression isobtained:

bK ¼R tf

0 DT 2ðqKÞ½T 2ðeRKÞ � Y �dtR tf

0½DT 2ðqKÞ�2 dt

: ð30Þ

7. Stopping criterion

If the problem contains no measurement errors, the con-vergence condition for the minimization of the criterion is:

JðRKþ1Þ < g; ð31Þwhere g is related to the accuracy of the direct problemsolution.

For measured temperature Y perturbed by an additiverandom error with a standard deviation r, the followingexpression is obtained for g:

g ¼ r2tf : ð32ÞThen, the stopping criterion is given by Eq. (31) with gdetermined from Eq. (32).

8. Computational procedures

The computational procedure for the solution of thisinverse problem may be summarizes as follows:

Suppose eRKðtÞ is available at iteration K.

Step 1. Solve the direct problem given by Eqs. (1)–(6) forT1(r, t) and T2(r, t), respectively.

Step 2. Examine the stopping criterion given by Eq. (31)with g given by Eq. (32). Continue if not satisfied.

Step 3. Solve the adjoint problem given by Eqs. (16)–(21)for k1(r, t) and k2(r, t), respectively.

Step 4. Compute the gradient of the functional J 0 fromEq. (24).

Y.-C. Yang / Optics Communications 278 (2007) 81–89 85

Step 5. Compute the conjugate coefficient CK and direc-tion of decent qK from Eqs. (27) and (26),respectively.

Step 6. Set DR(t) = qK(t) and solve the sensitivity problemgiven by Eqs. (8)–(13) for DT1(r, t) and DT2(r, t),respectively.

Step 7. Compute the search step size bK from Eq. (30).Step 8. Compute the new estimation for eRKþ1ðtÞfrom Eq.

(25) and return to Step 1.

9. Lateral pressure

The single-coated optical fiber is simply subjected tothermal loading; while we assume that there is no stressin the fiber at the initial temperature, stress will be inducedafter the temperature drops. Since the problem is axisym-metric, the stress–strain relation in a zero strain conditionis [28]:

eh ¼ur¼ xhð1þ mÞ þ 1� m2

Erh �

m1� m

rr

� �; ð33Þ

where u is the radial displacement and r is the current ra-dius; h = T � T0 is the temperature drop. rr, rh, eh, x, m,and E are the radial stress, tangential stress, tangentialstrain, effective thermal expansion coefficient, Poisson’s ra-tio, and Young’s modulus of the material, respectively.

The Lame formula for the stress components in a circu-lar thick-walled tube subjected to internal pressure pi andexternal pressure pe are [28]:

rr ¼pia

2 � peb2

b2 � a2þ a2b2ðpe � piÞðb2 � a2Þr2

; ð34Þ

rh ¼pia

2 � peb2

b2 � a2� a2b2ðpe � piÞðb2 � a2Þr2

; ð35Þ

where a and b are the inner and outer radii of the tube.Substituting Eqs. (34) and (35) into Eq. (33) , we have

u¼xhð1þ mÞrþ 1þ m

Eð1�n2Þ½ð1�2mÞðpin

2�peÞr�a2ðpe�piÞ=r�;

ð36Þ

where n = a/b is the radii ratio.Using Eq. (36), we can obtain the radial displacement

u11 of the glass fiber and the radial displacement u21 ofthe inner boundary of coating at the interface, i.e., r = r1.They are

u11ðr1; tÞ ¼ x1h1ðr1; tÞð1þ m1Þr1 �p1ðtÞE1

ð1þ m1Þð1� 2m1Þr1;

ð37Þ

u21ðr1; tÞ ¼ x1h2ðr1; tÞð1þ m2Þr1 þp1ðtÞ

E2ð1� n21Þð1þ m2Þ

� ½ð1� 2m1Þn21 þ 1�r1; ð38Þ

where h1(r1, t) = T1(r1, t) � T0 and h2(r1, t) = T2(r1, t) � T0.x1, E1, and m1 are the effective coefficient of thermal expan-sion, Young’s modulus, and Poisson’s ratio of the glass fi-

ber, respectively; x2, E2, and m2 are the effective coefficientof thermal expansion, Young’s modulus, and Poisson’s ra-tio of the coating, respectively; p1(t) is lateral pressure atthe interface of the glass fiber and the coating; n1 = r1/r2.The condition of the compatibility of displacement (i.e.,u11 = u21) results in the following formula:

p1ðtÞ ¼ E1x1

uY; ð39Þ

where

u ¼ ð1þ m1Þh1ðr1; tÞ �x2

x1

ð1þ m2Þh2ðr1; tÞ; ð40Þ

Y ¼ ð1þ m1Þð1� 2m1Þ þE1

E2

1

1� n21

ð1þ m2Þð1� 2m2n21 þ n2

1Þ:

ð41ÞThe lateral pressure p1(t) at r = r1 is a function of time, anddepends on the thickness and material properties of thecoating layer.

10. Normal stresses

The radial stress rr and tangential stress rh in the glassfiber can be obtained by setting a = 0 and pe = p1 in Eqs.(34) and (35); and they are

rrðtÞ ¼ rhðtÞ ¼ �p1ðtÞ: ð42ÞThe axial strain is zero for the case of plain strain; and thenthe axial stress from the stress–strain relation is given by

rz ¼ Exhþ mðrr þ rhÞ: ð43ÞHence, the axial stress in the glass fiber is

rzðr; tÞ ¼ E1x1h1ðr; tÞ � 2m1p1ðtÞ; r 6 r1; ð44Þ

11. Microbending loss

The compressive lateral pressure p1(t) in the glass fiberwould produce excess microbending loss C(t) [29], so itshould be minimized. There exists a linear relationshipbetween p1(t) and C(t), that is

CðtÞ ¼ Lp1ðtÞ; ð45Þwhere L is a constant and its value is approximately as0.02 (dB/km)/kpsi or 0.0029 (dB/km)/Mpa [29].

12. Refractive index changes

The refractive index would be changed when glass fiberis subjected to stress. The changes are given by [30]:

Dnrðr; tÞ ¼ nr � n ¼ �B2rrðtÞ � B1½rhðtÞ þ rzðr; tÞ�; ð46ÞDnhðr; tÞ ¼ nh � n ¼ �B2rhðtÞ � B1½rrðtÞ þ rzðr; tÞ�; ð47ÞDnzðr; tÞ ¼ nz � n ¼ �B2rzðr; tÞ � B1½rrðtÞ þ rhðtÞ�; ð48Þ

where n is the refractive index of the unstressed glass fiber;nr, nh, and nz are the refractive indices of light rays. B1 andB2 are the stress–optical coefficients for the ordinary and

++

++

++

++

++

++

++

++

++

+ +

t, sec

R(t

),m

2K

/W

0 0.02 0.04 0.06 0.083

4

5

6

7

8

9

Exact SolutionInverse Solution+

× 10-3

Fig. 2. Estimated thermal contact resistance at 20th iteration with initialguesses eR0 ¼ 0:003, measurement error r = 0.01, and measurementlocation rm = 63 lm.

86 Y.-C. Yang / Optics Communications 278 (2007) 81–89

extraordinary rays, respectively; the values are B1 =4.2 · 10�6/Mpa and B2 = 6.5 · 10�7/Mpa [31].

13. Results and discussion

The objective of this article is to show the validity of thepresent approach in estimating R(t) accurately with noprior information on the functional form of the unknownquantities, which is the so-called function estimation.Therefore, an example with T0 = 21 �C and T1 = 20 �Cwill be illustrated. In other words, the single-coated opticalfiber, which has initial temperature 21 �C, is assumed to besuddenly subjected to a thermal loading of constant sur-rounding temperature 20 �C, while the thermal contactresistance at the interface is time varying. In addition, car-bon is taken as the coating material in the numericalillustrations.

In order to illustrate the accuracy of the present inverseanalysis, we consider the simulated exact value of the time-dependent thermal contact resistance R(t) as

RðtÞ ¼ 0:0091= expðt=0:08Þ m2 K=W: ð49ÞThe material properties and radii of the carbon-coatedoptical fiber are listed as follows [14]:

r1 ¼ 62:5 lm; r2 ¼ 65 lm;

m1 ¼ 0:17; m2 ¼ 0:3;

E1 ¼ 72:5 GPa; E2 ¼ 45 GPa;

x1 ¼ 0:56� 10�6=K; x2 ¼ 22� 10�6=K;

k1 ¼ 1:1 W=m K; k2 ¼ 54 W=m K;

a1 ¼ 5:9� 10�5 m2=s; a2 ¼ 1:5� 10�5 m2=s;

h ¼ 120 W=m2 K:

The temperature measurement is located at the positionrm = 63 lm. The total measurement time is chosen astf = 0.08 sec and measurement time step is taken 0.004 s.

In practice, the temperature measurements always con-tain some degree of error, whose magnitude depends uponthe particular measuring method employed. Therefore, thesimulated temperature measurements adopted in the cur-rent inverse problem are also considered to include mea-surement errors. For reasons of practicality, the presentstudy adds a random error noise to the exact temperaturevalues computed from the direct problem. Hence, the mea-sured temperature Y is expressed as

Y ¼ Y exact þ -r; ð50Þwhere Yexact is the temperature at position r = 63 lm of thedirect problem with the exact thermal contact resistanceR(t), - is a random variable within �2.576 to 2.576 for a99% confidence bounds, and r is the standard deviationof the measurement.

The estimated values of thermal contact resistance R(t),obtained at the 20th iteration with initial guesseseR0 ¼ 0:003 and measurement error r = 0.0 are shown inFig. 1. These results confirm that the estimated results

are in very good agreement with those of the exact thermalcontact resistance. Meanwhile, Fig. 2 shows that the pro-posed inverse method is still capable of yielding satisfac-tory results even when a measurement error (r = 0.01) isintroduced. For a thermal contact resistance of unity and99% confidence, that standard deviation corresponds tomeasurement error of 2.58%. Figs. 1, 2show that, for thecases considered in this paper, increases in the measure-ment error do not cause obvious decreases in the accuracyof the inverse solution.

Fig. 3 demonstrates the estimated values of thermal con-tact resistance R(t), obtained at the 20th iteration with ini-tial guesses eR0 ¼ 0:00003 and measurement error r = 0.0.It can be found that the variation of the initial guess valueeR0 has little effect on the accuracy of the estimation. Toinvestigate the influence of measurement location uponthe estimated results, Fig. 4 illustrates the estimated con-tact resistance, with temperature measurement taken atr = r2. In other words, the thermocouples are located atthe outer surface of the carbon-coated optical fiber. Mean-while, the results in Fig. 4 are obtained at the 20th iterationwith initial guesses eR0 ¼ 0:003 and measurement errorr = 0.0. Fig. 4 shows that satisfactory results are still avail-able even with a different measurement location.

Fig. 5 depicts the prediction of the temperature distribu-tions T(r, t) along radial direction at various times in thecarbon-coated optical fiber, which are obtained at the20th iteration with initial guesses eR0 ¼ 0:003 and measure-ment error r = 0.0. From the results of Fig. 5, it can befound that the predicted values for the temperature distri-butions are in excellent agreement with the exact results.

++

++

++

++

++

++

++

++

++

+ +

t, sec

R(t

),m

2K

/W

0 0.02 0.04 0.06 0.083

4

5

6

7

8

9

Exact SolutionInverse Solution+

× 10 -3

Fig. 4. Estimated thermal contact resistance at 20th iteration with initialguesses eR0 ¼ 0:003, measurement error r = 0.0, and measurement loca-tion rm = 65 lm.

+ + + + + +

+ + + + + +

+ + + + + +

62.5 65 20.3

20.4

20.5

20.6

20.7

t = 0.02 sec

t = 0.04 sec

t = 0.06 sec

+ + + + + + + + + + + + +

+ + + + + + + + + + + + +

+ + + + + + + + + + + + +

r, m

T,o C

0 10 20 30 40 5020.3

20.4

20.5

20.6

20.7

Exact SolutionInverse Solution+

t = 0.02 sec

t = 0.04 sec

t = 0.06 sec

Fig. 5. Temperature distributions in the carbon-coated optical fiber atvarious times.

++

++

++

++

++

++

++

++

++

+ +

t, sec

R(t

),m

2K

/ W

0 0.02 0.04 0.06 0.083

4

5

6

7

8

9

Exact SolutionInverse Solution+

× 10-3

Fig. 3. Estimated thermal contact resistance at 20th iteration with initialguesses eR0 ¼ 0:00003, measurement error r = 0.0, and measurementlocation rm = 63 lm.

++

++

++

++

++

++

++

++

++

++

t, sec

(t),

dB

/ km

0 0.02 0.04 0.06 0.080.4

0.6

0.8

1

1.2

Exact SolutionInverse Solution+

× 10-4

Fig. 6. Variation of microbending loss C with time t in the carbon-coatedoptical fiber.

Y.-C. Yang / Optics Communications 278 (2007) 81–89 87

In addition, Fig. 5 shows that there exists a discrepancy intemperature at the interface, which is the result of the ther-mal contact resistance. In addition, this discrepancy in tem-perature will decrease with the increasing time.

Furthermore, the temperature distributions decrease withthe increase of time and with the increasing radial distancein both the glass fiber and the carbon coating, which coin-cides with our expectation.

The estimated values of microbending loss are illus-trated in Fig. 6, which are obtained at the 20th iterationwith initial guesses eR0 ¼ 0:003 and measurement error

88 Y.-C. Yang / Optics Communications 278 (2007) 81–89

r = 0.0. It shows that the microbending loss of the car-bon-coated optical fiber increases with the increasingtime, which means that the compressive lateral pressureat the interface also increases with the increase of time.Finally, Figs. 7–10 demonstrate the prediction of therefractive index changes Dnr, Dnh, and Dnz in the glass

+ + + + + + + + + + ++

++

++

++

++

++

++

++

r, μm

Δn , r

Δnθ

0 12.5 25 37.5 50 62.5

2.03

330

2.03

335

2.03

340

2.03

345

2.03

350

2.03

355

Exact SolutionInverse Solution+

× 10-7

Fig. 7. Refractive index changes Dnr and Dnh distributions in the glassfiber at t = 0.02 s.

+ + + + + + + + + + + + +

+ + + + + + + + + + + + +

+ + + + + + + + + + + + +

r, μm

Δnr,

Δnθ

0 12.5 25 37.5 50 62.51.5

2.0

2.5

3.0

3.5

Exact SolutionInverse Solution+

× 10

t = 0.02 sec

t = 0.04 sec

t = 0.06 sec

-7

Fig. 8. Refractive index changes Dnr and Dnh distributions in the glassfiber at various times.

fiber at various times along the radial direction, respec-tively. The results in Figs. 7–10 are also obtained at the20th iteration with initial guesses eR0 ¼ 0:003 and mea-surement error r = 0.0. It can be found that the refractiveindex changes increase with the increasing radius and theincreasing time.

+ + + ++

++

+

+

+

+

+

+

r, μm

Δnz

0 12.5 25 37.5 50 62.5

1.98

792

1.98

793

1.98

794

1.98

795

1.98

796

Exact SolutionInverseS olution+

× 10-7

Fig. 9. Refractive index changes Dnz distributions in the glass fiber att = 0.02 s.

+ + + + + + + + + + + + +

+ + + + + + + + + + + + +

+ + + + + + + + + + + + +

r, μm

Δnz

0 12.5 25 37.5 50 62.51.5

2.0

2.5

3.0

3.5

Exact SolutionInverse Solution+

t = 0.02 sec

t = 0.04 sec

t = 0.06 sec

× 10-7

Fig. 10. Refractive index changes Dnz distributions in the glass fiber atvarious times.

Y.-C. Yang / Optics Communications 278 (2007) 81–89 89

14. Conclusion

The conjugate gradient method was successfully appliedfor the solution of the inverse problem to determine theunknown time-dependent thermal contact resistance of acarbon-coated optical fiber, while knowing the temperaturehistory at the measurement location in the optical fiber.Subsequently, the temperature distributions, thermallyinduced transient microbending loss, and refractive indexchanges of the optical fiber are also calculated. The influ-ence of measurement errors, initial guess values, and mea-surement locations upon the precision of the estimatedresults is also investigated. Numerical results have con-firmed that the method proposed herein can accurately esti-mate the time-dependent thermal contact resistance,temperature distributions, microbending loss, and refrac-tive index changes for the considered problem. In addition,the conjugate gradient method does not require prior infor-mation for the functional form of the unknown quantitiesto perform the inverse calculation and excellent estimatedvalues can be obtained.

Acknowledgement

This work was supported by the National Science Coun-cil, Taiwan, Republic of China, under the Grant No. NSC95-2221-E-168-025.

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