Optical Fiber Technology 14 (2008) 143–148

www.elsevier.com/locate/yofte

Transient thermal loading induced optical effects in single-coated opticalfibers with interlayer thermal resistance

Yu-Ching Yang ∗, Haw-Long Lee

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

Received 25 March 2007; revised 28 May 2007

Available online 13 November 2007

Abstract

The transient microbending loss and refractive index changes in a single-coated optical fiber subjected to thermal loading with stress-dependentinterlayer thermal resistance are investigated. The effects of interlayer thermal resistance on the transient microbending loss and refractive indexchanges of the optical fiber are analyzed and discussed. Results show that the interlayer thermal resistance will increase the transient thermalloading induced lateral pressure in the single-coated optical fiber and, thus, the microbending loss. To the contrary, the interlayer thermal resistancewill decrease the transient thermal loading induced refractive index changes.© 2007 Elsevier Inc. All rights reserved.

Keywords: Interlayer thermal resistance; Optical fiber; Microbending loss; Refractive index changes

1. Introduction

The conventional optical fiber for telecommunication is usu-ally constructed of silica glass fiber coated by two or threelayers of polymeric coatings [1,2]. However, when silica fiberswith polymeric coatings are stressed in a humid environment,strength degradation occurs after a long period of time due toslow crack growth. To solve this problem of moisture attack,inorganic coatings such as oxides, carbides, and carbon are be-ing considered [3,4]. On the other hand, metallic coatings suchas aluminum, indium, copper, tin, etc. are also applied on opti-cal fibers [5–8]. Some of these fibers exhibit higher resistanceto moisture attack and show higher strength than polymeric-coated fibers.

On the other hand, thermal stresses are important in mul-tilayer structures such as optical fibers, since they cause in-creased transmission loss in the optical fibers. Thermal stressesoccur in the optical fibers at a low temperature, which are dueto the mismatches of thermal expansion properties of fiber andcoating material [9,10]. Since thermal stresses affect the per-

* Corresponding author.E-mail address: ycyang@mail.ksu.edu.tw (Y.-C. Yang).

1068-5200/$ – see front matter © 2007 Elsevier Inc. All rights reserved.doi:10.1016/j.yofte.2007.09.009

formance of the optical fibers, they have been extensively stud-ied [11–16]. Nevertheless, to the author’s knowledge, there wasno work in the literature considering the effect of interlayerthermal resistance in such layered construction as optical fibers.In this article, we investigate the effect of interlayer thermal re-sistance on the transient temperature distributions, microbend-ing loss, and refractive index changes in single-coated opticalfibers, subjected to a sudden thermal loading. In addition, inthis article, we do not assume a constant contact thermal resis-tance. Instead, a formula between contact resistance and normalstress is taken.

The stress-dependent interlayer thermal conductance willmake the analysis more complicated, since a nonlinear thermalboundary condition relevant to this contact stress at interlayer isimposed on linear heat conduction equation. Consequently, thetemperature and displacement fields are related with each other.This implies that the heat conduction equation and equationof displacement compatibility are coupled and shall be dealtwith simultaneously. After solving the coupled heat conduc-tion equation and displacement compatibility equation, then thetransient temperature distributions, microbending loss, and re-fractive index changes in the single-coated optical fibers can becalculated.

144 Y.-C. Yang, H.-L. Lee / Optical Fiber Technology 14 (2008) 143–148

Fig. 1. Temperature drop θ distributions in the carbon-coated optical fiber att = 0.01 s.

2. Analysis

2.1. Temperature distributions

Consider the transient heat conduction problem of a single-coated optical fiber as shown in Fig. 1. The optical fiber iscomposed of a glass fiber coated by a thin layer of coating,and has intermediate and outer radii, r1 and r2, respectively.Assume the single-coated optical fiber is initially at tempera-ture Ti(r,0) = T0, and for time t > 0, the fiber at its boundarysurface r = r2 is subjected to a convective thermal loading ofthe surrounding temperature T∞, T∞ < T0. Here the subscripti = 1 refers to the region of optical fiber, and i = 2 refers tothe region of coating, respectively. Define the temperature dropθi(r, t) = Ti(r, t) − T0, then the mathematical formulation ofthis transient heat conduction problem is given by [15]

∂2θi

∂r2+ 1

r

∂θi

∂r= 1

αi

∂θi

∂t, i = 1,2, (1)

∂θ1(0, t)

∂r= 0, (2)

k1∂θ1(r1, t)

∂r= k2

∂θ2(r1, t)

∂r, (3)

θ1(r1, t) − θ2(r1, t) = −R(p1)k1∂θ1(r1, t)

∂r, (4)

−k2∂θ2(r2, t)

∂r= h

[θ2(r2, t) − �T∞

], (5)

θi(r,0) = 0, i = 1,2, (6)

where α, k, and h are the thermal diffusivity, thermal con-ductivity, and convection heat transfer coefficient, respectively;�T∞ = T∞ − T0, and R is the interlayer thermal resistance.

In this article, a formula proposed by Tauchert et al. [17] isadopted, which is in the form

R(p1) = R0 + S1 exp(S2p1), (7)

where p1 is the lateral pressure at the interface of the glass fiberand the coating. In addition, in the analysis, p1 is a function oftime.

When we apply the Laplace transformation with respect totime to the problem of Eqs. (1)–(6), the results will be

∂2θ̄i

∂r2+ 1

r

∂θ̄i

∂r= 1

αi

sθ̄i , i = 1,2, (8)

∂θ̄1(0, s)

∂r= 0, (9)

k1∂θ̄1(r1, s)

∂r= k2

∂θ̄2(r1, s)

∂r, (10)

θ̄1(r1, s) − θ̄2(r1, s) = −R(p1)k1∂θ̄1(r1, s)

∂r, (11)

−k2∂θ̄2(r2, s)

∂r= h

[θ̄2(r2, s) − �T∞/s

], (12)

where θ̄i (r, s) is the Laplace transformation of θi(r, t); s is theLaplace transformation parameter. Then applying central finitedifference in Eq. (8), we obtain the following discretized equa-tions:

θ̄i,j−1 − 2θ̄i,j + θ̄i,j+1

�r2+ r−1

j

θ̄i,j+1 − θ̄i,j−1

2�r= 1

αi

sθ̄i,j , (13)

where �r is the space increment, j = 1,2, . . . ,N , and N isthe total node number. Substituting boundary conditions ofEqs. (9)–(12) into Eq. (13) and writing in matrix form, we ob-tain the following equation

([A] − s[B])[θ̄i,j ] = [Y ], (14)

where

[A] =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−4/�r2 4/�r2 0 . . .

a2 b2 c2 0

0. . .

.

.

. am + cm bm − 2cm�r/Rk1 2cm�r/Rk1. . . 2am�r/Rk2 bm − 2cm�r/Rk2

.

.

.

0

. . . 0

. . ....

. . .

am + cm . . .

. . ....

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

aN + cN bN − 2hcN�r/k2

Y.-C. Yang, H.-L. Lee / Optical Fiber Technology 14 (2008) 143–148 145

[B] =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1/α1 0 . . . . . . 0

0 1/α1 0 . . ....

.... . .

1/α1. . . 1/α2

.... . .

...

0 1/α2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

[θ̄i,j ] =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

θ̄1,1

θ̄1,2...

θ̄1,m

θ̄2,m

...

θ̄2,N

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

, and [Y ] =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

00...

00...

−2h�r�T∞/sk2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

, (15)

where m denotes the node number at the interface and

aj = 1

�r2− 1

2rj�r, bj = −2

�r2,

cj = 1

�r2+ 1

2rj�r, (16)

in which rj is the radius of the j th node. In Eq. (14), [A] isan (N + 1) × (N + 1) band matrix with real number, [B] is an(N + 1) × (N + 1) diagonal matrix, [θ̄i,j ] is an (N + 1) × 1vector representing the unknown temperature, and [Y ] is an(N + 1) × 1 complex vector representing the forcing terms.In general, we can obtain the solutions of all the N + 1 nodalpoints in the transform domain by Eq. (14). However, it isvery involved to find the inverse Laplace transformation by theresidue theorem. In this article, the inverse Laplace transforma-tion is completed by a general method, known as the Fourierseries technique [18]. On the other hand, the R values in ma-trix [A] are calculated from the equation of displacement com-patibility, which will be described later.

2.2. Lateral pressure

The single-coated optical fiber is simply subjected to thermalloading; while we assume that there is no stress in the fiber atthe initial temperature, stress will be induced after the tempera-ture drops. Since the problem is axisymmetric, the stress–strainrelation in a zero strain condition is [19]

εθ = u

r= ωθ(1 + ν) + 1 − ν2

E

(σθ − ν

1 − νσr

), (17)

where u is the radial displacement and r is the current radius.σr , σθ , εθ , ω, ν, and E are the radial stress, tangential stress,tangential strain, effective thermal expansion coefficient, Pois-son’s ratio, and Young’s modulus of the material, respectively.

The Lamé formula for the stress components in a circularthick-walled tube subjected to internal pressure pi and externalpressure pe are [19]

σr = pia2 − peb

2

b2 − a2+ a2b2(pe − pi)

(b2 − a2)r2, (18)

σθ = pia2 − peb

2

b2 − a2− a2b2(pe − pi)

(b2 − a2)r2, (19)

where a and b are the inner and outer radii of the tube. Substi-tuting Eqs. (18) and (19) into Eq. (17), we have

u = ωθ(1 + ν)r + 1 + ν

E(1 − ξ2)

[(1 − 2ν)(piξ

2 − pe)r

− a2(pe − pi)/r], (20)

where ξ = a/b is the radii ratio.Using Eq. (20), we can obtain the radial displacement u11

of the glass fiber and the radial displacement u21 of the innerboundary of carbon coating at the interface, i.e., r = r1. Theyare

u11(r1, t) = ω1θ1(r1, t)(1 + ν1)r1

− p1(t)

E1(1 + ν1)(1 − 2ν1)r1, (21)

u21(r1, t) = ω2θ2(r1, t)(1 + ν2)r1

+ p1(t)

E2(1 − ξ21 )

(1 + ν2)[(1 − 2ν1)ξ

21 + 1

]r1, (22)

where ω1, E1, and ν1 are the effective coefficient of thermalexpansion, Young’s modulus, and Poisson’s ratio of the glassfiber, respectively; ω2, E2, and ν2 are the effective coefficientof thermal expansion, Young’s modulus, and Poisson’s ratio ofthe coating; p1(t) is lateral pressure at the interface of the glassfiber and the coating; ξ1 = r1/r2. The condition of the compat-ibility of displacement (i.e., u11 = u21) result in the followingformula:

p1(t) = E1ω1ϕ

Y, (23)

where

ϕ = (1 + ν1)θ1(r1, t) − ω2

ω1(1 + ν2)θ2(r1, t), (24)

Y = (1 + ν1)(1 − 2ν1)

+ E1

E2

1

1 − ξ21

(1 + ν2)(1 − 2ν2ξ

21 + ξ2

1

). (25)

Now, Eqs. (14) and (23) can be solved simultaneously to deter-mine θi(r, t) and p1(t).

2.3. Microbending loss

The compressive lateral pressure p1(t) in the glass fiberwould produce excess microbending loss Γ (t), so it shouldbe minimized. There exists a linear relationship between p1(t)

and Γ (t), that is [20]

Γ (t) = Kp1(t), (26)

where K is a constant and its value is approximately as 0.02(dB/km)/kpsi or 0.0029 (dB/km)/MPa [20].

146 Y.-C. Yang, H.-L. Lee / Optical Fiber Technology 14 (2008) 143–148

Fig. 2. Temperature drop θ distributions in the carbon-coated optical fiber att = 0.02 s.

3. Results and discussion

The main purpose of this paper is to investigate the effectof interlayer thermal resistance on the transient temperaturedistributions, microbending loss, and refractive index changesin a single-coated optical fiber, which is subjected to a sud-den thermal loading. Therefore, an example with T0 = 21 ◦Cand T∞ = 20 ◦C will be illustrated. In other words, the single-coated optical fiber, which has initial temperature 21 ◦C, is as-sumed to be suddenly subjected to a thermal loading of constantsurrounding temperature 20 ◦C. In addition, carbon is taken asthe coating material in the numerical illustrations.

The material properties and radii of the carbon-coated opti-cal fiber are listed as follows [15,17]:

r1 = 62.5 µm, r2 = 65 µm,

ν1 = 0.17, ν2 = 0.3,

E1 = 72.5 GPa, E2 = 45 GPa,

ω1 = 0.56 × 10−6/◦C, ω2 = 22 × 10−6/◦C,

k1 = 1.1 w/m ◦C, k2 = 54 w/m ◦C,

α1 = 5.9 × 10−5 m2/s, α2 = 1.5 × 10−5 m2/s,

R0 = 0.00033 m2 K/W, S1 = 0.001175 m2 K/W,

S2 = 0.521 MPa−1.

Figures 1–3 depict the temperature drop θ(r, t) distributionsfor various times in the carbon-coated optical fiber along radialdirection. Two different dashed lines refer to the cases with con-stant thermal contact resistance (abbreviated hereafter as linearcase) and without thermal contact resistance (abbreviated here-after as no resistance), respectively; while the solid lines refer to

Fig. 3. Temperature drop θ distributions in the carbon-coated optical fiber att = 0.04 s.

the case with stress-dependent thermal contact resistance (ab-breviated hereafter as nonlinear case). For the linear case thevalue of thermal resistance R is equal to 0.00033. For the no re-sistance case the temperature drop is continuous at the interface,whereas a discontinuity in the value of the temperature drop isfound for both linear and nonlinear cases. It can be seen fromEq. (4) that, in general, the magnitude of this discontinuity intemperature increases with increasing value of interlayer ther-mal resistance and the amount of heat flux across the interlayer.Figures 1–3 also show that there exists a discrepancy in tem-perature drop θ(r, t) distributions among the three cases above,which coincides with our expectation. On the other hand, thetemperature drop increases with the increase of radius. More-over, the interlayer thermal contact resistance will increase thetemperature drop in the carbon coating and decrease the tem-perature drop in the glass fiber, since the existence of thermalcontact resistance will decrease the heat flux from the glass fiberto the carbon coating.

Figure 4 illustrates the variation of microbending loss withtime for the case with stress-dependent thermal contact resis-tance. It shows that the microbending loss of the carbon-coatedoptical fiber increases with the increasing time, which meansthat the compressive lateral pressure at the interface also in-creases with the increase of time. On the other hand, the effectof interlayer thermal contact resistance on the microbendingloss is depicted in Fig. 5, in which DIF1(t) and DIF2(t) aredefined as

DIF1(t) = Γlin(t) − Γnon(t)

Γnon(t)× 100%,

DIF2(t) = Γnor(t) − Γnon(t) × 100%,

Γnon(t)

Y.-C. Yang, H.-L. Lee / Optical Fiber Technology 14 (2008) 143–148 147

Fig. 4. Variation of microbending loss Γ with time t for the nonlinear case.

Fig. 5. Variation of DIF1 and DIF2 with time t .

respectively. Here Γnon(t), Γlin(t), and Γnor(t) denote the mi-crobending loss of nonlinear, linear, and no resistance cases,respectively. Figure 5 shows that the microbending loss of thelinear and no resistance cases is lower than that of the nonlinearcase in the beginning of the thermal loading. In other words,the existence of thermal contact resistance will increase themicrobending loss of the optical fiber in the transient state. Nev-ertheless, as the time progresses, the difference will decreasegradually.

Fig. 6. Refractive index changes �nr and �nθ distributions in the carbon-coated optical fiber at t = 0.02 s.

Fig. 7. Refractive index change �nz distributions in the carbon-coated opticalfiber at t = 0.02 s.

Figures 6–9 illustrate the refractive index changes �nr ,�nθ , and �nz of the glass fiber at various times along the radialdirection, respectively. It can be found that the refractive indexchanges increase with the increasing radius and the increasingtime. Moreover, the refractive index changes of both the linearand no resistance cases are higher than those of the nonlinearcase. In other words, the existence of thermal contact resistance

148 Y.-C. Yang, H.-L. Lee / Optical Fiber Technology 14 (2008) 143–148

Fig. 8. Refractive index changes �nr and �nθ distributions in the carbon-coated optical fiber at various times.

Fig. 9. Refractive index change �nz distributions in the carbon-coated opticalfiber at various times.

will decrease the transient refractive index changes �nr , �nθ ,and �nz.

4. Conclusion

The transient effects of interlayer thermal resistance onthe microbending loss and refractive index changes of single-coated optical fibers, subjected to a sudden thermal loading, areinvestigated. Results show that the interlayer thermal resistanceaffects the microbending loss and refractive index changes indifferent direction. In other words, it will increase the transientthermal loading induced lateral pressure in the single-coatedoptical fibers and, thus, the microbending loss. To the contrary,the interlayer thermal resistance will decrease the transient ther-mal loading induced refractive index changes.

Acknowledgment

This work was supported by the National Science Council,Taiwan, ROC, under Grant Nos. NSC 95-2221-E-168-025 andNSC 96-2221-E-168-022.

References

[1] N. Yoshizawa, Y. Katsuyama, Electron. Lett. 25 (1989) 1429.[2] N. Yoshizawa, H. Tada, Y. Katsuyama, IEEE J. Lightwave Technol. 9

(1991) 417.[3] A.H. Lettington, C. Smith, Diamond Related Mater. 1 (1992) 805.[4] D.R. Biswas, Opt. Eng. 31 (1992) 1400.[5] D.R. Biswas, S. Raychaudhuri, in: Technical Digest Optical Fiber Com-

munication Conference, 1985, p. 124.[6] T. Nozawa, D. Tanaka, A. Wada, R. Yamauchi, in: Technical Digest Opti-

cal Fiber Communication Conference, 1992, p. 217.[7] V.A. Bogatyrjov, E.M. Dianov, S.D. Rumyantsev, A.A. Sysoliatin, in:

Technical Digest Optical Fiber Communication Conference, 1993, p. 78.[8] H.S. Seo, U.C. Paek, K. Oh, C.R. Kurkjian, J. Lightwave Technol. 16

(1998) 2355.[9] S.T. Shiue, J. Appl. Phys. 78 (1995) 6384.

[10] W.W. King, C.J. Aloisio, J. Electron. Packaging 119 (1997) 133.[11] E. Suhir, IEEE J. Lightwave Technol. 6 (1988) 1321.[12] W.J. Chang, H.L. Lee, Y.C. Yang, J. Appl. Phys. 88 (2000) 616.[13] Y.C. Yang, Opt. Eng. 40 (2001) 2107.[14] U.C. Chen, W.J. Chang, Opt. Eng. 41 (2002) 1317.[15] Y.C. Yang, S.S. Chu, W.J. Chang, J. Appl. Phys. 95 (2004) 5159.[16] H.L. Lee, Y.C. Yang, W.J. Chang, Appl. Phys. B Lasers Opt. 83 (2006)

601.[17] T.R. Tauchert, D.C. Leigh, M.A. Tracy, J. Pressure Vessel Technol. 110

(1988) 335.[18] H.L. Lee, H.M. Chou, Y.C. Yang, Energy Convers. Manage. 45 (2004)

1749.[19] S.P. Timoshenko, J.N. Goodier, Theory of Elasticity, third ed., McGraw–

Hill, New York, 1970, p. 56.[20] E. Suhir, IEEE J. Lightwave Technol. 8 (1990) 863.