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16 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. LT-2, NO. 2, APRIL 1984 On the Theory of Backscattering in Single-Mode Optical Fibers ARTHUR H. HARTOG AND MARTIN P. GOLD Abstract-A new theory of backscattering i n s ingle-mode f ibers is In contrast, the theory of backscattering in multimode fibers is presented. It allows backscatter waveforms to be predicted for fibers well established [ 181 - [ 191. of any r efractive-index p rofile or scattering-loss d istribution. T he r e- The p urpose o f t he present contribution is, therefore, t o of an sults agree with experimental d ata a nd p rovide c onfirmation resolve the disagreement in the literature. To this end, we pre- earlier, more restricted theory. T I. INTRODUCTION HE BACKSCATTERING METHOD [1]-[3] has been used increasingly in recent years to characterize loss and imperfections in single-mode fibers. The t echnique involves the launching of a pulse of light into the fiber and examining the temporal behavior of the return signal. The latter consists of energy which, having been scattered from the guided wave, is recaptured by the fiber in the reverse direction. A theoretical analysis of backscattering in single-mode fibers appeared in 1980 [ 4] a nd was subsequently verified experi- mentally [5]. A series of experimental results has been pub- lished covering such aspects as long-range fault location [6]- [ 121, determination of structural parameter variations [ 131, measurements of splicing loss [7], [8], and the examination [ 141 - [ 161. of polarization effects The b ackscatter f actor, i.e., the ratio of the backscattered power to the energy launched into the fiber, is important in long-range fault location since it determines the magnitude of the signal and hence the range which the apparatus can cover. Moreover, for m ore q uantitative m easurements, s uch as the evaluation of splicing losses, or the determination of variations of fiber parameters along the length, it is essential to have an accurate m odel of the d ependence of the b ackscatter signal on the parameters of the fiber. Without such a model, valid and a ccurate i nterpretations o f t he results cannot be m ade since one is usually interested in quite subtle changes i n the received power. It is, therefore, of some concern that doubts have been ex- 121 as to the validity of the presently available pressed [6], [ wave-optics theory, due to Brinkmeyer [4]. U nfortunately, the only alternatives to the latter are either approximate [l 11 171, an approach clearly inap- or based on geometric optics [ propriate to the analysis of single-mode fibers. The resulting impression is one of a confusion which requires clarification. sent a new theory of backscattering in single-mode fibers which is derived using a different and more general approach from those proposed previously. The result is a simple expres- sion for the backscatter factor which may be used in circularly symmetric fibers having arbitrary refractive-index profiles and an a rbitrary d istribution o f s cattering loss. Comparisons a re made with previously published results. It is found that if the near-field distribution is approximated by a Gaussian function, our results agree exactly with those of Brinkmeyer [4]. 11. TIME-DEPENDENCE OF THE BACKSCATTER SIGNAL For the purpose of the present argument, we assume that the pulse launched i nto t he f iber is a Dirac function of en- width. The e ffect of a ergy E(0j and of vanishingly narrow finite pulsewidth W is to limit the distance resolution of the ugW/2 where ug is the g roup velocity measurement t o 6x11 a constant i nput e nergy, varying the pulse- in t he f iber; f or width will not alter the backscatter signal level. The f orward pulse energy is attenuated a t a rate a(np/m) and its dependence on position z is thus E(z) (1) (-azj. = E(0) exp The energy scattered while the pulse travels a distance element dz situated at z is dE, (z, z t dz) = E(z) a,(zj dz (21 where as is the Rayleigh scattering-loss coefficient. We define the c apture f raction B(z) as the p roportion o f the total energy scattered at z which is recaptured by the fiber returning in the return direction. The energy dEBs(z, z + dz) from the length element dz to the launching end is, therefore, dEss(z, z t dz) = E(0) a,(z)B(z) exp (-2 azj dz. This energy arrives spread over a time interval given by dt = - dz ug (3) L (4) Manuscript r eceived June 6, 1983, revised October 26, 1983. This which is simply the time taken by the impulse to travel the dis- work was supported in part by the UK Science and Engineering R e- tance dz in both directions. In the case of a Dirac pulse (which search Council and the Pirelli General Cable Company. The authors are with the Department of Electronics, The University we have assumed), no energy will arrive from any other part of Southampton, Southampton, England SO9 5NH. the fiber during this time interval. The power received at the 0733-8724/84/0400-0076$01.00 0 1984 IEEE HARTOG AND GOLD: THEORY OF BACKSCATTERING IN OPTICAL FIBERS launch end of the fiber at time t = 22/ ug is, therefore, Pss(t) = - - - E(0) cr,(z)B(zj exp (-2az). dt 2 dEBS - ug (5) A similar expression has been put forward recently [I21 in which the factor of is omitted. Note that the result given above, in (5), and the derivation are virtually the same as for 181. multimode fibers [ 4 111. EVALUATION OF THE BACKSCATTER CAPTURE [F lJ FRACTION Rayleigh scattering is described by classical theory in terms i I of electric dipoles driven by an electromagnetic wave traveling through the material [20]. In a homogeneous medium, inter- ference between the radiation patterns of the dipoles results in cancellation of the secondary waves in all but the forward di- rection. A localized inhomogeneity of the refractive index, however, results in a dipole moment whose radiation is not canceled by adjacent dipoles. A portion of the incident wave is then radiated in all directions and power is lost to Rayleigh scattering. T he s cattering process may, t herefore, be repre- sented by a large number of dipoles o scillating with a fixed phase relative to the incident wave, but whose amplitude is proportional to the random local deviation z. DmX of the electric susceptibility from its mean value With coherent illumina- tion, t he phase relationship b etween t he light scattered b y separate dipoles is fixed. The scattered light, therefore, suffers interference, a p henomenon a kin t o laser speckle. S ome as- pects of these coherence e ffects have been discussed in the context of frequency-domain reflectometry by Eickhoff and Ulrich [21]. T hey can also be observed in time-domain re- flectometry [I31 with pulsed lasers of sufficiently narrow spectral width. In the present analysis, we assume that the source used is sufficiently incoherent that such interference effects are elimi- nated [13] . The contribution of all dipoles to the scattered light can, therefore, be added in intensity. The calculation of the backscatter capture fraction B is then reduced to the evalu- ation of the power coupling between the electric dipoles and the fundamental mode of the fiber. The power coupled into the HE1, mode is calculated with thesaid of an overlap integral in the far field following an ap- proach used by Marcuse and Marcatili [22] for slab waveguides and, more recently, by Wagner and Tomlinson [23] to study components for single-mode fibers. For simplicity, we assume linearly polarized mode fields. For weakly g uiding f ibers there is no loss of generality s ince an arbitrary state of polarization may be described as a combina- tion of linearly polarized waves. The orientation of the electric- field vector of the backscattered wave at the point of capture is that o f t he dipole moment. F or i sotropic materials this orientation coincides with that of the incident electric field, and the state of polarization is thus preserved by the scatter- ing process. Practical fiber materials are locally anisotropic which results in a small degree (-5 percent) of depolarization of the scattered light. It is interesting to note that this depo- larized component o f t he s cattered light recaptured in the / Fig. 1. Coordinates used in the near and far field. forward d irection imposes a f undamental l imitation o n t he ability of high-birefringence fibers to maintain a linear state of polarization. In order to calculate the power coupled to the fiber mode, we consider one direction of propagation o nly a nd p erform the coupling o verlap i ntegral over a hemisphere centered on the f iber axis. The coupling e fficiency b, of a s catter p oint situated at a distance R, from the fiber axis is b(R,) = - 1 2 where $F and $s are the far-field distributions of the HEll mode and of the dipole, respectively, and dS1 is a solid angle element. Distance R, has been normalized to the core radius a. The dipole, in fact, radiates the same amount of power into the complementary hemisphere, which couples to the HEll mode in the f orward d irection. T he f actor o f a ppearing i n (6) is, therefore, r equired t o give the overall power coupling e ffi- ciency between the dipole and the backscattered guided wave. The far field $F(Y, 0, @of the HEIl mode may ) be obtained from the near-field distribution J/N (R, Go) using the Fraun- hofer diffraction formula. The spherical polar coordinates system used is represented schematically in Fig. 1. $F is given by [241 ~51 . exp (jkanR sin 0 cos (@ - $o)) dG0 R dR (7) where C is a normalization constant and R is the radial coordi- nate normalized to the core radius a. X is the wavelength of the incident light measured in free space and k = 27r/h. On the assumption of weakly guiding fibers, we make the approxima- tion: n1 E nz = n. 78 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. LT-2, NO. 2, APRIL 1984 From the definition of the Hankel transform [27], 111 l2 sim- Assuming circular symmetry and using the well-known inte- gral representation of Jo [26] the Bessel function of the first plifies to kind, (7) may be rewritten in the form $,&, 6) =i y Ckla e-jkflrL- $N(R)Jo(klR sin O)R dR Similarly, the first term in the denominator of (6) is given by (8) where kl = kun. Similarly, the f ar field of t he dipole m ay be expressed as [201 $s(r, 0, $1 = $,,e -j[knr+G(R,)] (1 - sin2 e cos2 @)1’2 (9) where $So is a normalization constant. 6(RS) is a phase shift resulting from the displacement R, of the scatter point from the fiber a xis. I t carries the only information which remains in the far field as to the value of R,. Using a simple geometric argument, 6 may be shown to be 6(RS) = kl R, sin 0 sin I#J. (10) The numerator 1I1I2 of (6) then becomes .ln’2 $N(R)Jo(klR sin O)R dR For the values of refractive-index difference normally used in single-mode fibers, the far-field distribution only has signifi- cant intensity for small values of 8. We can, therefore, make the approximation 1 - sin2 e cos2 $ 1. For the same reason, the upper limit of the integral over 6 in (1 1) may be taken to infinity and sin 0 replaced b y 8. The accuracy of these approximations has been checked numerically and found to be better than 0.5 percent for numerical a per- tures of 0.2 or less and for V-values of 1 or more. Using these approximations and the integral representation of Jo, (1 1) may be rewritten as 1I1l2 = j - 27rGl m kl $,oe-”knr~ xJo(xR,) .lw r $~~(R)Jo(xR)R dR dx I: (1 2) where x = kl 8. . sin B[Lm GN(R)Jo(klR sin 0)R dR I’ dB (14) and, making the same approximation as previously, n amely sin 19 2: 6 and 7r/2 -+ 00, (1 4) becomes From Parseval’s equation, as applied to Hankel transform pairs [27], it follows that the integration over x can be per- formed in the near field, (1 5) is the equivalent to Finally, it is found that Substituting (1 3), (1 6) and (1 7) into (6), the following expres- sion for the capture fraction at R, (the local capture fraction) is obtained In order to derive the overall capture fraction B it is now only necessary to average b(R,) over the entire near field, weighted by the intensity distribution of the scattered light. For a uni- form distribution of the scattering loss, B is, therefore, HARTOG AND GOLD: THEORY OF BACKSCATTERING IN OPTICAL FIBERS 79 i.e., B= 3 4k2a2n2 [l- 2 R $id?) dR] or, in terms of the normalized frequency V 'lY.- 0 1.0 12 1.4 1.6 1.8 2.0 2 2 2.4 2 6 2 8 Narmallsed Frewency V In the case of nonuniform scatterifig-loss distributions, B is Fig.,2. Calculated values of the normalized capture fraction B' = B X given by (NA/n)-' for a .step-index fiber having a uniform scattering-loss distribution. Solid l ine: P resent theory (27). Dotted l ine: G aussian approximation (25). proximation, the validity of [4]. However, in those cases where the detailed behavior of the function B(V) is of inter- est, (21) and (22) are more useful since the accuracy of the JO Gaussian approximation is itself sensitive to the V-value. (22) It is then necessary to replace, i n (5), a, with the intensity- B. Step-Index Fibers Having Uniform Distributions weighted mean value $ defined by of Scattering Loss In the case of the step-index profile, the field distribution /.- is the well-known Bessel function expression q= - JO r- where U is the eigenvalue and W2 = Vi - U2. B is then given by Equation (22) is a general expression for the capture fraction for arbitrary refractive-ifidex profiles and scattering-loss distri- butions. Hence the backscatter factor may be evaluated from the near field of the HEll mode and the radial scattering-loss distribution without resorting to the equivalent-step approach. IV. RESULTS Hence, as is the case for the Gaussian approximation, it is pos- A. Gaussian Approximation sible to .separate the dependence of B on V-value and on nu- Bririkmeyer's theory of backscattering in single-mode fibers merical a pe'rture. A general curve of B' = B(NA/n)-' which [4] is based on the Gaussian approximation, i.e., it is assumed depends only on V may, therefore, be produced. that the near-field distribution is of the form In Fig. 2, the values of the function B'(V j have been com- pared for the two above r esults. The solid curve is obtained from (27); it has a peak of 0.258 at V= 1.75 and decreases gradually for larger V-values. At low V-values, the fall of B' where oo is the spot size. Substituting GG into (21) leads to is more dramatic and reflects the spreading of the power into (for a uniform distrihtion of the scattering loss) the cladding as the f requency decreases. The d otted line is calculated with the aid of the Gaussian approximation, (25), 1 B= - 3 (NAj2 and using the expression for oo/a due to Marcuse [28] n2 . od/a = 0.65 + 1.619V-1'5 + 2.879V-6. (28) The peak value bf B' in this case is some 10 percent below that 80 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. LT-2, NO. 2, APRIL 1984 I 10 1.2 1.4 16 2 18 2.8 2.4 2 0 2 2 8 V. DISCUSSION A. Interpretation of Backscatter Measurements The interpretation of features in backscatter waveforms re- lies on the availability of a theory of backscattering applicable to any refractive-index profile or distribution of scattering loss. If the backscatter waveform of a fiber exhibits departures from the simple exponential decay, one or more of the struc- tural parameters must be a function of position. It would be useful to d etermine which of t he fiber properties are non- uniform in order t o provide feedback t o t he m anufacturing process. Experimentally, the local backscatter factor [5 J defined as Norma1is.d Frequency v Fig. 3 . C alculated v alues o f t he n ormalized c apture f raction B' for various distributions of scattering loss in a step-indes fiber. Labeling parameter corresponds to the fiber numbers of Table I. TABLE I Scattermc loss (dB km-1wn41 Fibre NO. core claddlng 1.4 0.7 1.05 1.05 0.7 0.7 0.7 1.05 1.05 1.4 i Note: Fibers 1 and 5 have unusually large indes differences for single- mode fibers. fibers 2 and 4 have a more typical numerical aperture, and the scattering loss of fiber 3 is uniform for comparison. the Gaussian approximation and the detailed s hapes of t he two curves are thus significantly different. T he difference is scarcely important in the prediction of the range of backscatter apparatus; use of the Gaussian approximation could, however, lead to errors in the interpretation of features in backscatter waveforms. C Step-Index Fibers Having Nonuniform Distributions of Scattering Loss In most fibers the scattering loss is a function of radial posi- tion. In particular, the admixture of dopants such as Ge02 or B203 to silica in order to alter the refractive index is known to modify the scattering level. In general, such additives lead to increased s cattering loss owing to small-scale fluctuations of the glass composition a nd h ence o f i ts refractive index. In order to evaluate the effect of nonuniform scattering loss on has been calculated the backscatter factor, the function B'( V) s tep-index fibers having the s cattering losses from (22) for given in Table I. T he r ight-hand c olumn in the t able shows approximate c orresponding values of refractive-index differ- and assuming that either ence, obtained from [29] and [30], the core or the cladding is made from pure silica. curves. It is immediately Fig. 3 shows the resulting B'( V) clear from the figure that the capture fraction increases as the proportion of the scattering loss occurring in the core increases. We note also that the peak values of the capture fraction oc- cur at d ifferent V-values. The curves come t ogether a t large V-values, as the power confinement improves and t he e ffect of the cladding loss is less important. is obtained by combining backscatter waveforms measured J, [31]. In general, it is not from each end of the fiber [ 13 possible to interpret this information unambiguously in terms of the fiber properties, since the b ackscatter f actor d epends on all of the major fiber parameters, even under the simplify- ing assumption of a Gaussian near-field distribution. In the case of a step-index fiber, the parameters which affect the backscatter factor are the core radius, the index difference the core and cladding. The impor- and the scattering loss in tance of the distribution of the scattering loss, as well as its mean value has been emphasized by the present theory. Since all of these parameters are potentially length-dependent, it is clearly not possible to distinguish their individual contribution without additional information. However, we propose a method whereby the origin of the observed nonuniformities may be determined under certain conditions, as follows. In order to calculate the backscatter factor. it is necessary first to obtain the near-field distribution. This may either be or calculated from the refractive- measured directly on the fiber, index profile (by numerical solution of the wave equation). A third possibility is to adopt one of the equivalent step-index approaches. For the sake of clarity, the remainder of the pres- ent discussion is restricted t o t he case of a s tep-index fiber whose core scattering loss is allowed to be different from that of the cladding. We assume that the nonuniformity of interest is a perturba- tion of one or more fiber parameters a bout t heir nominal values. The sensitivity of q to changes in fiber parameters has been calculated and the results are given in Figs. 4 and 5. Fig. 4 shows the relative sensitivity (a/q * &/cia) of q to core radius variations, for the same distributions of scattering loss as used in Fig. 3. For variations of the numerical aperture, the curves are the same except for a constant offset of 2 which re- sults from the (NA)2 factor in (27). The exact value depends to some extent on the distribution of scattering and varies sub- stantially with V-value? even over the limited range where low microbending-loss single-mode operation is feasible. From (29), the relative sensitivity of 77 to variations of the mean scattering loss is unity. This is not the case if a$ varies differently in the core or cladding r egions as might b e ex- pected for defects in MCVD fibers. The relative sensitivity of q to core scattering variations is shown in Fig. 5. The effect complementary of variations of as in the cladding is exactly i s? t herefore, f ound t hat, if V is to the curves of Fig. 5. It greater than -1.5, q is sensitive to variation of the scattering HARTOG AND GOLD: THEORY OF BACKSCATTERING IN OPTICAL FIBERS 81 5 ,,, 17 -2 1.0 12 14 t.8 IO 2.0 2.2 2.4 2.8 2.8 Nern.ll*.d Fr.qu.noy V Fig. 4. Relative sensitivity of the backscatter factor q to variations of the core radius (left-hand axes) and of the numerical aperture (right- hand axes) for the distributions of 01, given in Table I. t -I 1.0 1.2 14 18 16 20 22 2.4 26 2.8 Norrnallssd Frequency v Fig. 5. Relative sensitivity of the backscatter factor to the scatter loss in the core, for the distributions of 01, given in Table I. loss in the core but largely unaffected by changes of a, occur- ring in the cladding region. The interesting point to note from Figs. 4 and 5 is that the relative sensitivities of the backscatter factor to the different fiber parameters vary in markedly different ways as the I/-value ch‘anges. This characteristic can be exploited b y p erforming backscatter measurements at 2 or 3 well-spaced wavelengths. Comparison of the behavior of the measured features at the different I/-values with t he p arameter sensitivities predicted by the theory for the nominal fiber characteristics will then reveal which parameters are changing at the observed back- scatter features. B. Comparison of the Present Theory with Experiment Measurements of the backscatter factor in single-mode fibers have been p ublished by the p resent a uthors 151. The results are reproduced i n Table 11. The m easurements were shown to agree with the predictions of the Gaussian-approxi- mation theory of Brinkmeyer. Comparison of the experi- mental results with t he present theory shows an agreement slightly worse than was found with the Gaussianrapproximation theory but nevertheless within the accuracy to which the fiber parameters are known. If, h owever, t he f actor o f h ad been omitted f rom (5), as [12] suggests it ought, then there is no longer agreement be- tween theory and experiment (see Table 11). In [12], this lack TABLE I1 ‘I Backscatter factor dB/km W/J 347 1.88 * obtained from published scattering loss dataL2T or the basls of measured f ibre p arameters. of agreement is put down to uncertainty in the estimates of the scattering loss in the fibers. We note, h owever, t hat in order for our previous experimental results to agree with the theory of [ 121, the scattering loss at X = 1.06 pm in our fibers would have to’ be about 0.3 dB/km. Such a low scattering loss is, sadly, unrealistic in silica-based fibers a t t his wavelength. The present theory and, by extension, that of [4] are thus the only ones of those presented to date to agree accurately with the experimental evidence. VI. CONCLUSIONS A new approach to the theory of backscattering in single- mode fibers has been presented. A simple expression has been derived for t he b ackscatter c apture f raction whch in- volves only the near-field distribution of the HEll mode. This result e nables the b ackscatter power to be calculated f or a fiber of arbitrary refractive-index p rofile and scattering-loss distribution. The predictions of the theory were found to agree with pre- viously published experimental data and also to confirm the validity of an earlier, less general, result derived by a separate argument. Finally, the theory presented in this paper allows the effect of fiber imperfections to be modeled so that the features ob- served in backscatter waveforms may be interpreted more accurately. ACKNOWLEDGMENT The a uthors would like to t hank D. N. Payne a nd C. M. Ragdale for h elpful discussions, and W. A. Gambling for his guidance. REFERENCES F. P. Kapron, R. D. Maurer, and M. P. Teter, “Theory of back- scattering effects in waveguides,” Appl. Opt., vol. 11, no. 6, pp. M. 1352-1356, 1972. K. 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Quantum Electron., vol. 12, no. 1, pp. 17-22, 1980. * Arthur H. Hartog was born in Toronto, Canada in 1955. He received the B.Sc. degree and Ph.D. degrees f rom t he U niversity of Southampton, Southampton, England. He has worked in the Optical Fiber Group at Southampton University since 1976, initially as a Research Student and presently as a Research Fellow. His research interests a re in propaga- tion studies, and measurement techniquqs in optical fibers and fiber sensors. IC Martin P. Gold was born in London, England in 1959. He received the B.A. degree i n p hysics from Oxford University, England in 1980. In 1980 he joined the Optical Fiber Group in the D epartment of Electronics, University of Southampton, Southampton, England as a Re- search Student working for t he Ph.D. degree on reflectometry techniques for optical fibers. In October 1983 he was appointed as a Research Fellow at the University of Southampton.