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Unformatted text preview: 5'20 (E)ES| determination from mode-field diameter and refractive index profile measurements on single-mode fibres F. Martinez, MSc C.D. Hussey, PhD Indexing terms: Measurements and measuring, Optical fibres, Waveguides and waveguide components Abstract: We show, both experimentally and theoretically, that for single-mode fibres, the (E)ESI and MFD methods are interrelated in a self-consistent model with the theoretical cutoff wavelength playing a pivotal role. Three indepen- dent measurement approaches are examined: mode-field diameter measurements, preform profile measurements and fibre profile measure- ments. 1 Introduction Currently, there are two trends in characterising single- mode fibres. They are: (a) from refractive index profile measurements and (b) from mode-field diameter (MFD) measurements.* Between these two there exist equivalent step index (ESI) methods. The two ESI parameters (equivalent core-radius and equivalent numerical aperture) can be obtained from either the refractive index profile or the MFD. Having the ESI parameters and the MFD, we can predict most of the essential characteristics of single-mode fibres, such as bending loss, splice loss, microbending loss, waveguide dispersion (and therefore total dispersion). However, there is still confusion because there are several ESI defi- nitions and measurement techniques, and several MFD definitions and measurement techniques, which have been proposed and none of them has been recognised as a standard method for characterising single-mode fibres. In this paper, we propose a theory, which links the ESI parameters obtained from measurements of the ref- ractive index profile to the same parameters derived from measurements of the MFD. We can therefore, in prin- ciple, unify both trends in characterising single-mode fibres. Measurements of refractive index profile and mode-field diameter are performed on MCVD fibres, to provide experimental support for this theory. 1.1 Equivalent step index (ESI) techniques In general, the refractive index profile of a single-mode fibre deviates from the ideal step index as shown in Fig. * Note: In addition to the term ‘mode field diameter’ MFD, we will use the term ‘mode field radius’ to (i.e. MFD = 20)). at has also been referred as ‘spot size” in the literature. Paper 5954] (E13), received 19th October 1987 The authors are with the Optical Fibre Group, Department of Elec- tronics and Computer Science, The University, Southampton, Hamp- shire SO9 SNH, United Kingdom 202 refractive index difference ¢hm°x radius , pm Fig. 1 Typical refractive index profile of a MC VD single-mode fibre and a possible equivalent step index representation at is the equivalent core radius and he the equivalent index difference actual refractive index profile v v 7 7 equivalent step-index profile l. The problem is then to determine the propagation characteristics of a single-mode fibre with an arbitrary refractive index profile. Exact solutions are difficult to implement, and good approximate methods are preferred. It has been observed that the fields of all single-mode fibres look similar. Since the analytical solutions for the step-index fibre are already available and well known, it is then very convenient to have a method which has as its reference a step-index fibre. This gives rise to equivalent step index methods. The ESI fibre should have a second mode cutofi wave- length, fundamental mode propagation constant, MFD, and evanescent field as close as possible to the corre- sponding parameters of the actual fibre. If this is the case, the bending losses, microbending losses, and splice losses can be predicted from the ESI fibre [1]. There exist, however, some problems including [2]: (a) Several ESI techniques are available, each of which gives different values for the two ESI parameters. (b) The accuracy of the predicted propagation charac- teristics is not always good, particularly with the predic- tion of waveguide dispersion. Referring to the latter problem, we have shown that one particular ESI method can be readily enhanced from the accurate prediction of waveguide dispersion [3], we IEE PROCEEDINGS. Vol. 135, Pt. J, No.3, JUNE I988 call this the enhanced ESI, or (E)ESI model. This model is based on the moments of the refractive index profile, and it proposes the use of an additional third parameter called the enhancement parameter. Unfortunately, per- forming refractive index profile measurements on the fibre is difficult and, occasionally, it is difficult to estimate the core diameter (i.e. core-cladding boundary) from the profile measurements. Because of these problems, the alternative of characterising single-mode fibres from MFD measurements has received a lot of attention. 1.2 Mode-field diameter measurement (MFD) techniques The MF D is the width of the fundamental mode, guided by a single-mode fibre above its cutoff wavelength. MFD is in principle a very useful parameter, as it allows the prediction of splicing and microbending losses. In addi- tion to that, the wavelength dependence of the MFD allows us to predict the bending losses (through the E81 parameters), and the waveguide dispersion. However, quoting Reference 4, ‘MFD appears to be a parameter in turmoil. Its importance is well understood, but no consensus on its fundamental definition or mea- surement method has yet emerged. Several measurement techniques and definitions have been proposed during the past few years. None of these methods is universally accepted as a reference test method, nor is any method more or less fundamentally correct than the others.’ The degree of consistency between the different tech- niques is a direct result of the choice of a definition for the MF D. The Gaussian approximation, as produced by Marcuse [5, 6] and illustrated in Fig. 2, so widely used rudius,pm Fig. 2 Field amplitude distribution of the fundamental mode, in a step- index single-modefibre at V = 2.1, and its Gaussian approximation MFD, is the Gaussian Mode-field diameter actual field amplide distribution - - - - - - - gaussian approximation for the first generation of single-mode fibres, is now severely questioned. Small systematic errors result even for quasistep index fibres when the Gaussian approx- imation is used, especially at longer wavelengths [7]. Also, the second generation, single-mode fibres with nonstep refractive indices for dispersion shifting and dis- persion flattening do not exhibit Gaussian fields at any wavelength. An alternative definition of MFD based, on the moments of the near-field, which is frequently called the Petermann 2 definition, can, in principle, solve the problem of consistency, and reduce the systematic errors [8]. The primary difference with respect to the Gaussian definition is that the Petermann definition uses the mea- sured field directly, making no assumption about the IEE PROCEEDINGS, Vol. 135, Pt. J, No.3, JUNE 1988 shape of the field. Because of this, the Petermann 2 defini- tion is becoming more widely accepted. Cutoff wavelength is also essential in characterising single-mode fibres, but some problems arise because the second-mode becomes highly attenuated at wavelengths lower than the theoretical cutoff wavelength. This effec~ tive cutoff is dependent on the length and the bending of the fibre. There is some agreement on the effective cutoff definition and measurement techniques [9], but the rela- tion between the effective and theoretical cutoff is pre- sumably very difficult to determine. This suggests that one has to be careful when using the measured cutoff wavelength, as an input in the determination of the E81 parameters from the spectral variation of MFD, for example. 2 - Unifying E51 and MFD methods in single-mode fibres Millar’s method is perhaps the most standard technique for determining the two ESI parameters and the second- mode cutoff wavelength for single-mode fibres. One basic assumption in the Millar method is that the MFD for arbitrary profiles behaves simply as a scaled version of the step-index MFD [10]. While this is true for most fibres used for telecommunications, it is not true in general, and in particular the method breaks down when applied to the triangular-type profile fibres and dual shape profile fibres [11] which are receiving a lot of attention for dispersion shifting. In its simplest form, the Millar method measures the cutoff wavelength, together with the MFD. Unfor- tunately it is at the cutoff wavelength where the MFD is most sensitive (over the single-mode regime) to vagaries in the refractive index profile. Moreover, the Millar method has no built-in alarm to show that it breaks down for a specific fibre under measurement. Here we show how the Millar method can be adapted to cater for a more complex MFD structure. Our approach uses the Petermann MF D [12, 13] (or far-field RMS width) to find the two parameters for the E81 or the three parameters necessary to specify the enhanced ESI (or (E)ESI). Having determined the three parameters, we can then decide whether to use all three parameters in the (E)ESI, or only two in the simple ESI approximation, but now we have a greater appreciation of the errors involved, if we choose the latter. The implementation of our approach is aided by our analytic approximation to the Petermann MFD, which is outlined in Appendix 8. 2.2 Petermann MFD and (E)ESI approximation The (E)ESI approximation is outlined in Appendix 8, and is specified by three parameters. For our purpose we will find it convenient to use the parameters 7, ac, and |A§i4 |, where l7 = (Zn/1M, NA, = \/(290)V, a, = a\/2.(-22 , and NA, = NAJQo/Qz; a is the core radius, NA is the numerical aperture, 90 is the guidance factor and Q" is the moment of profile. We can see from Appendix 7 that AQ. is the enhancement parameter; its magnitude gives an indication of the deviation_of the refractive index from the step index (for which A9,, = 0). From Table 6, for example, we find that for the parabolic profile fibre, the enhancement parameter is 0.125, for the triangular profile fibre it is 0.190, and for cusp-like profiles, its value can approach 0.3. In terms of our three parameters, the normalised near- field Petermann mode-field radius for fibres of arbitrary 203 profile can be expressed as [14] 1 - ft?) V3J"(V)bsitl7l]}’ =__+AQ[__+--——- (l) imam ' " mm 4 where bs,(l7)=(1.1428l7—0.996)2/l72 is the Rudolph— Neumann approximation, m7) =0.0313 —0.013i72 ' (2) (from Appendix 8.2), and (2),, = 0.65 + 1619le2 + 2.87917"6 — (0.016 + 1.56117‘7) (3) (from Appendix 8.1). Since the first term in eqn. 1 represents the simple ESl, the full form of this equation illustrates dramatically the limitations of any procedure which arbitrarily attempts to fit a measured mode-field diameter with that of some step-index fibre. Additionally, the term in square brackets in eqn. 1 is a monotonically increasing function of V, so that by measuring the mode-field diameter at cutoff, we are choosing the worst sampling point for fitting the simple step-index mode field diameter for fibres which have a large enhancement parameter. These points are also illustrated in Fig. 3, where we plot d) as a function of I7 for the triangular-profile fibre. 2.0 U1 'o O U" normalised mode-field radius, a O 1.0 1,5 2.0 2 5 3.0 3.5 effective normalised trequencyW ' Fig. 3 Mode-field radius (2) against Vfor the triangular pro/ilefibre exact r 7 ~ ES] - ~ ~ ~ r ~ enhanced ESI It is convenient to rewrite eqn. 1 as follows: 2 2__ ac w _p+|AQ4|q (4) where _ 1 “(Dim ,_ I73’_b,_ qZflVH f(V).(V) 4 The quantities p and q are specified once I7 is known. Similarly, the far-field mode radius can be expressed as wff=g=;§(P+lA§—24i€1) (5) 2.3 Adapted Mil/ar procedure The starting point of the Millar method [10] is the fact that the theoretical normalised cutoff frequency I7“, is unlike any other fibre characteristic, because it is extremely insensitive to profile shape. The cutofi‘ wave- length ,im should therefore, in principle, be a xed refer- ence point for all single-mode fibre measurements. Millar’s method determines 11m from observation of the near-field, MFD behaviour. Further refinements have produced more accurate determinations of A“, [15]. The possibility of using the far—field MFD behaviour in deter- mining Am has also been suggested [16]. Such a tech— nique could depend critically on the detectOr sensitivity. By combining such a determination of the cutoff wave- length, with the measurement of the Petermann MFD at two different wavelengths (A, and 12) in the single-mode region, i.e. A, # 1.2 2 Am , the three necessary parameters for specifying the (E)ESI are obtained as follows: 17,, 2 = 2.405 A” (6a) 11.2 — p1 — sz Um I = ———— (6h) 4 R‘Iz “ ‘11 03 = wi(P1+|AQ4|41) = 0°in + IAQ4|42) (6") where R is the ratio of the two mode field radius mea- surements, (mg/(of), pl. 2 and ql. 2 are the values ofp and q of eqn. 2 evaluated at 17L 2, and w,“ 2 are the measured values of the near-field mode field radius at AL 2. If we use the far-field RMS width measurements at the two different wavelengths, then the parameters 17,. 2 and Mill are still given by eqns. 6a and h, while the equiva— lent core radius is now: 2 _ a? 2T”): + IAQ4iql) (DH! 2 _ = 2 (P2 + IAQ4I‘12) (6d) (Of/2 while R in eqn. 6b is now defined as (will/(of). The two-parameter ESl can be obtained from mea- surements of the Petermann MFD at one wavelength. together with the cutoff wavelength. Neglecting the enhancement parameter (i.e. |AQ4| = 0 in eqn. (it, then the two parameters are given by: 17,. = 2.405 (7) and (U- a. = _ '— (8) (USN/r) and (2)54 17,-) can be given by eqn. 3. The subscript i now corresponds to any wavelength about the cutoff wave- length. The equivalent numerical aperture NA‘, is now obtained simply from NA? = 2.405}.m/27ra,.. 3 Measurements In this paper we implement and critically assess our approach using three independent measurement pro- cedures. The (E)ESI parameters as obtained from MFD measurements, preform profile measurements, and fibre IEE PROCEEDINGS. Vol. 135, Pt. J. No. 3, JUNE 1988 profile measurements, are compared for a nominally step index fibre fabricated by the MCVD process. A statistical approach is adopted in analysing the measurement data, thereby reducing the influence of measurement error and other inherent uncertainties. Results show that while the preform and the fibre profile measurements produce essentially the same equivalent core radius and the same equivalent numerical aperture (with allowance for diffusion effects), the E81 parameters produced by the MFD are substantially different. The discrepancy arises from the use of the effective cutoff wavelength, as derived from the MF D measure- ments, instead of the theoretical cutoff wavelength (corresponding to a I7 value of 2.405) in the evaluation of the E51 parameters. However, it is found that if the theoretical cutoff wave- length, as derived from the profile measurements, is used in the interpretation of the MFD data, then the three measurement approaches predict very similar ESI para- meters. While only one fibre-preform is reported here, one other nominally step-index fibre-preform exhibited similar trends. 3.1 Profile measurements Figs. 4a and 4b show the refractive index profiles of the preform and the fibre, respectively, as measured using commercial equipment (spatial filtering technique and refracted near-field technique, respectively). The (E)ESI parameters were derived from the moment analysis of the refractive index profile for the preform and the fibre. Problems in implementing this procedure arise because both the core radius and the refractive index levels of the core and the cladding are not well defined. Because of this, six limits were proposed, as sketched in the insets of 8 0.01 C ‘1' 2 6 0.005 K 11.: ‘D E 0 l .L_._J_.L. I I I I l .L .1, _A___l J l I -5 -6 J. —2 0 2 A 6 8 radius,mm a , Is" 0 m E, 0.01 E 3% 3 0.005 E j 0 —60 —40 -20 0 20 1.0 60 radius, pm b Fig. 4 Refractive index profile 0f(a) preform and (b) fibre The insets show the six different core sizes and cladding levels used IEE PROCEEDINGS, Vol. I35, Pt. J, No.3, JUNE I988 Fig. 4. When using the preform data, the same measured core radii as used for points 1, 2, 3, 4, 5 and 6 in Fig. 4b were assigned for the corresponding points in Fig. 4a. Assuming circular symmetry only, the right hand half of each profile has been used. The (E)ESI parameters resulting from the profile mea- surements also differ slightly from one choice of profile to another. Table 1 shows the average (E)ESI parameters Table 1: Average (E)ESI parameters as obtained from ref- ractive index profile measurements and moments theory Parameter Profile measurements Fibre Preform 1100(nm) 1226i25 12171-11 a,(um) 2.86i0.12 2.71 i0.1 NA, 0.164 t 0.005 0.1722 1 0.0007 IAQ‘l 0.08 i 0.02 0.03 :1: 0.01 (Average theoretical cutoff is Aw=1222 nm). The uncertainty quoted is the standard deviation resulting from the six profiles, both for the fibre and the preform. 3.2 Mode-field diameter measurements The near-field Petermann’s MFD was obtained from the inverse of the RMS far-field width measured by the vari- able aperture method using commercial equipment. The cutoff wavelength was measured using two techniques: (a) the Millar procedure from the variation of MFD as a function of wavelength (b) the transmitted power technique (as recommended by CCITT) [17]. Table 2 shows the values measured for the Peter- mann’s MFD at four wavelengths. In theory, for the determination of the (E)ESI parameters, the cutoff wave- length together with the MF D at any other two wave- lengths are sufficient. In practice, if we try different Table 2: Near-field Petermann's MFD at four wavelengths for the fibre used in the comparative analysis A (nm) Petermann's MFD (pm) 1300 6.47 i1"o 1350 6.64 :1: 1% 1400 6.85 :t 1% 1450 7.07 i 1% wavelength pairs we obtain different results. Our approach is to take the average value of the (E)ESI parameters predicted from different pairs of the four wavelengths in Table 2. Table 3 shows the average value Table 3: Average (E)ESI parameters as obtained from MFD measurements Parameter Aw, = 1275 :t 25 (nm) A”, = 1295 i 25 (nm) a,(pm) 3.16 :t 0.12 3.22 i 0.14 NA, 0.155 t 0.006 0.154 d: 0.007 |AO.| 0.13:0.08 0.14i0.09 A”, was measured using the transmitted power technique. A”, was measured using the variation of MFD as a function of wavelength of these (E)ESI parameters for the two different measured cutoff wavelengths. If we choose to ignore the enhancement parameter in determining the two parameters of the simple ESI, we require the cutoff wavelength and the MFD at only one wavelength. We again determine the average values from 205 Table 4: Average (E)ESI parameters derived from MFD mea- surements assuming |AO‘| = 0 Parameter A”, = 1275 + 25 (nm) A“, = 1295 :t 25 (nm) a,(pm) 2.94 i 0.005 2.97 a 0.008 NAe 0.1662 a 0.0003 0.1668 1 0.0004 AM, and A502 as in Table 3 the measurements in Table 1. These are shown in Table 4 for the two different measured cutoff wavelengths. 4 Discussion of results Taking the results for the profile measurements in Table 1, it is found that the two parameters a, and N A, are in good agreement between the fibre and the preform. However, there is a slight increase in a, and a slight decrease in NA, in the transition from preform to fibre, this is consistent with the occurrence of diffusion during fibre drawing. The fact that there is only a slight varia- tion in these parameters from preform to fibre is inter- esting, since these two parameters are derived from the first two even moments of the profile shape function, and as such they rely on the average properties of the refrac- tive index profile which are not expected to change very much in the fibre draw. It should be pointed out that the poor resolution in the fibre profile measurement will also appear as diffusion, however, even within the error bound there is an overall true diffusion effect. In addition, the theoretical cutofl' wavelengths, as derived from: 1,, = ZnaeNAe/ZAOS (9) are in good agreement between fibre and preform. The enhancement parameter IASLI is small in both cases of preform and fibre profile measurements. The average value of the enhancement parameter of the preform is |AQ4| = 0.03, and according to Table 6, it cor- Table 5: Average (E)ESI parameters as obtained from MFD measurements, but using now the average theoretical cutoff wavelength (1.“, = 1222 nm) derived from the refrac- tive index profile and moments theory Parameter (E)ESI ESI a,(pm) 3.0 t 0.07 2.84 :t 0.01 NA, 0.156 :t 0.003 0.1649 :i: 0.0004 |AO4| 0.09 i 0.05 — Table 6: Data for the first two moments (Q0 and 9,) and for the enhancement parameter m, for several values of the exponent a of the power-law profiles a (20 02 AO‘ 0.500 0.500 0.000 0.444 0.450 0.01 0 0.400 0.417 0.029 0.333 0.375 0.067 0.250 0.333 0.125 0.167 0.300 0.190 0.136 0.289 0.215 0.100 0.278 0.246 0.056 0.265 0.285 —I slam-mu» —- N b 00 a) 8 responds approximately to a power-law profile with a = 8, which is very step-like. The enhancement para- meter in the fibre (IAQ4I = 0.08) again, according to Table 6, corresponds approximately to a power-law profile with a z 4 which is also very step-like. The enhancement parameter is therefore small enough to be neglected, because its effect on the dispersion parameters b, b1 and b2 is small (see Appendix 8), and therefore its 206 influence on the behaviour of the MFD is also very small. It is also interesting to note that [Afi4l in the fibre is slightly larger than in the preform. This gives a measure- ment of the slight diffusion that occurs in the profile of a single-mode fibre when it is drawn from the preform. The comparatively larger standard deviation in the determi- nation of the (E)ESI parameters for the fibre is an indica- tion of the intrinsic difficulty when working with the small dimensions involved. On examining the results from the MFD measure- ments in Table 3, it is found that the results are com- pletely at odds with those from the profiles. The discrepancy is due to the use of the measured cutoff wavelength (1mm) in determining the fibre parameters from eqn. 9. On substituting the average theoretical cutoff value A“, deduced from the profiles (i.e. Aw = 1222 nm) into the MF D analysis, the parameter values as shown in Table 5 are obtained. These parameters are now in good agreement with those derived from the fibre profile. From Table 5 we can conclude, as we have done for the profiles, that the simple two parameter ESI should be adequate for this fibre. Indeed the simple two parameter ESI results in this Table are found to be in remarkably good agreement with those in Table 1, once we use the theoretical cutoff wavelength in the analysis. 5 Conclusions We show that the theoretical cutoff wavelength, as defined by eqn. 9, is the key to obtaining a self-consistent model for characterising single-mode fibres which unites both the E81 and MFD models. The measured cutoff wavelength, which for a given fibre remains consistent between different reference measurement techniques, does not have a definite relationship to the theoretical cutoff wavelength when comparing different fibres. As such, the measured cutoff wavelength can lead only to confusion when used as a reference wavelength in determining the E81 parameters. This conclusion is drawn not only from the results from a single fibre as presented here, but also from other experimental results in our laboratory, and from results quoted by other workers (such as in Refer- ences 18 and 19). One of the limitations of the MFD measurements pre- sented here is the restricted set of data. Ideally, a larger number of measurements would improve the results for the average and the standard deviation of the (E)ESI parameters. This, however, would increase the time spent performing the measurements. 6 Acknowledgments This work was supported by the SERC. The authors would like to thank Dr. N. McFarlane from York Tech- nology for performing the mode-field diameter measure- ments. Mr. Martinez would also like to thank the Mexican Institutions CONACYT and IIE for their support. 7 References 1 JEUNHOMME, L.B.: “Single-mode fibre optics: Principles and applications’ (Marcel Dekkr. lng, New York, 1983) 2 SAMSON, P.J.: ‘Usage-based comparison of BS] techniques‘, J. Lightwave Tech, 1985, LT-3, pp. 165—175 3 MARTINEZ, F., and HUSSEY, C.D.: ‘Enhanced ESI for the pre- diction of waveguide dispersion in single-mode optical fibres”, Elec- tron. Lett., 1984, 24, pp. 1019—1021 IEE PROCEEDINGS, Vol. 135, Pt. J, No.3, JUNE 1988 4 DICK. J.M.. and SHAAR, C: ‘Mode field diameter: toward a stan- dard definition”, Laser and Appl, 1986, 5, pp. 91 V94 5 MARCUSli. 1).: ‘Loss analysis of single-mode fibre splices”, Bell Syst. Tcrh. J.. 1977, 56, pp. 703718 6 MARCUSE, D.: ‘Gaussian approximation for single-mode fibres”, J. Opt. Sor‘. Am., 1978. 68, pp. 103 7-109 7 SARAVANOS, C, and LOWE, R.S.: ‘The measurement of non- gaussian mode fields by the far-field axial scanning technique’, J. Lightwavc T(‘(‘l’l., 1986, LT-4, pp. 15634566 8 ANDERSON, W.T.: “Status of single-mode fibre measurements”. Technical Digest Symposium on Optical Fibre Measurements, National Bureau of Standards, US. Department of Commerce, 1986, pp. 77 90 9 FRANZEN. 0.(‘.: 'Determining the effective cutofl‘ wavelength of single~mode fibres: An interlaboratory comparison”, J. Lightwave Twit, 1985. LT-3, pp. 128 134 10 MILLAR, C: ‘Direct method for determining equivalent step index profiles for monomode libres‘. Electron. Lett., 1981, 17, pp. 4584160 11 OGAI, M., KlNOSHlTA, E.. TAMURA, 1., NAKAMURA, S., and HlGASHlMOTO, M.: ‘Low-loss dispersion shifted fibre with dual shape refractive index profile‘, Tech. Dig. ECOC '87, 1, pp. 171—174 12 PETERMANN, K.: ‘(‘onstraints for fundamental modal spot size for broadband dispersion compensated single-mode fibres”, Elertron. 1.011.. 1984.19, pp. 712* 714 13 HUSSEY. (‘.D.: ‘l-‘ield to dispersion relationships in single-mode fibres‘. ibid., 1984. 20, pp. 105171052 14 HUSSEY. (‘.D.. and MARTINEZ, F.: ‘New interpretation of spot- size measurements on singly-clad single-mode fibres”, ihid., 1986, 22, pp. 28-30 15 CAMPOS, A.(‘.. SRlVASTAVA, R., and ROVERSI, .l.A.: ‘Charac- terization of single—mode fibres from wavelength dependence of modal field and far field”, .1. Lightwuvc Tt‘t'h., 1984. LT—2, pp. 334 7340 16 PASK, C, and RUHL, 1’.: ‘New method for equivalent-step index fibre determination', Elm-tron. 1.011., 1983, 19, pp. 6587659 17 C(‘lTT: Recommendation (3.652: ‘Characterization of single-mode optical fibre cable‘, Section 111: ‘Test methods for the cutoff wave- length” 18 PASK, C, and RUHL, F.: Effects of loss on equivalent-step index fibre determination‘. Electron. Lett., 1983, 19, pp. 643—644 19 FOX, M.: 'Calculation of equivalent-step-index parameters for single-mode fibres”, Opt. and Quanl. Electron, 1983, 15, pp. 451—455 20 HUSSEY. CD, and MARTINEZ, F.: ‘Approximate analytical forms for the propagation characteristics of single-mode optical fibres”, Electron. Lett., 1985, 21, pp. 110371104 21 RUDOLPH, H.D., and NEUMANN, E.G.: “Approximation for the eigenvalues of the fundamental mode of a step index glass fibre waveguide”, Nar'hrir-htvmech. Z., 1976, 29, pp. 328329 22 HUSSEY, CD, and PASK, C: ‘Theory of profile moments descrip- tion of single-mode fibres”, IEE Proc. 11., 1982, 129, pp. 123—134 23 GAMBLING, W.A.. MATSUMURA, H., and RAGDALE, C.M.: ‘Mode dispersion, material dispersion and profile dispersion in graded index single-mode fibres”, Microwave, Optics and Acoustics, 1979, 3, pp. 239 7245 8 Appendix 8.1 Approximate analytical forms for the propagation characteristics of the single -mode fibres We present an approximate analytic form for the Peter- mann mode field radius, as required in the paper. The new equation adds two extra terms to the old Marcuse mode field radius expression [7]. 8.1.1 Mode fie/d radius and eigenvalue: The exact analytical formula for the normalised Petermann 2 mode field radius, 6)”, (\/2/W)(J,(U)/J0(U)) can be expressed approximately as [20]: a; = a,“ — (0.016 +1.561V‘7) (10) P where (DM is the Marcuse formula for the normalised mode field radius of an optimally exciting Gaussian beam, and which we repeat here for convenience [5]: LDM:O.650+1.619V’3’2+2.879V_° (11) For our purposes we have optimised eqn. 10 to be accu- rate to within 1% in the region 1.5 S V S 2.5, since this is the range of most practical interest in single-mode fibre IEE PROCEEDINGS, Vol. I35. P1. J. No. 3, JUNE 1988 transmission. Like the Marcuse formula, eqn. 10 was determined empirically. Increased accuracy over a more extended range would require additional terms, and would be of questionable advantage. Fig. 5 shows 6)”, and both the exact and approximate curves for dip. The very good accuracy of eqn. 10 for larger V values (V > 2.5), and the good qualitative La.) 0 2.0 1.0 normalised mode-field radiusw o 1 .L._ _|_ _1 1.0 1.5 2.0 2.5 3.0 normalised frequency V Fig. 5 Petermann mode-field radius (DP and Marcuse mode-field radius (7),” plotted as a function of normalised frequency V r Petermann (exact) Petermann (approximate) w r r Marcuse behaviour for small V values (V < 1.5) are additional bonuses. For completeness and easy reference we also include here the Rudolph—Neumann approximation for the modal eigenvalue, which is given by (see reference 21): W =1.1428V — 0.996 1.5 < V S 2.5 (12) This is accurate to within 0.1% for our chosen range. 8.1.2 Other propagation characteristics: The dispersion parameters b, b1, and b2 and the fraction of power propa- gation in the core, 11, have been shown to be intimately related to the normalized Petermann mode field radius (1),, [13], namely d 1 = 4 — 1 b2 dV (Wei) ( 3a) 4 b1: MD: + b (13b) 2 We propose to use eqns. 10 and 12 in eqns. (13a—c), thereby eliminating the need for numerical methods in evaluating eigenvalue or Bessel function terms. The accuracy of this approach is illustrated in Fig. 6 for our parameters b, b1, b2, and 11. The curve for b is simply the well-known Rudolph—Neumann approx- imation. The curves for b1 and n are of remarkable accu- racy: they are within 1% for our chosen range, and do not deteriorate appreciably beyond this range. For low V values, we are clearly obtaining a compensating effect of a poor eigenvalue combined with a poor mode field radius to give good results. The parameter b2 has always been the severest test of any approximation, since it relies on derivatives for its determination. In this case, however, the result is clearly impressive; the accuracy at V = 1.5 is 1.6%, and is well within this over the most of the range 207 (1.5g V<2.5). The percentage accuracy at large V values becomes meaningless, since the function is approaching zero; however, a good indication of its 1.8 normalised mode—field radiusw .0 1.5 2.0 2.5 30 normalised frequency V Fig. 6 Approximate and exact dispersion and (are power curves plotted as functions of normalised frequency V —- exact - - - - - ~ - approximate validity is that the approximation predicts the zero dis- persion point (Vzd z ) to within 2%. 8.2 ESI and (E) 58/ based on the moments of the refractive index profile The model we use is based on the moments of the refrac- tive index shape function, and is much easier to use than a similar model proposed in Reference 22. 8.2.1 Moments definition: The profile moments are defined as 1 QM =f s(R)RM"l dR (14) 0 where s(R) is the profile shape function n2 r — n3 s(R) = % (15) n0 .— "c! which can be rewritten as s(R) z m (16) no _ "c1 Since the fibres are weakly guiding, R = r/p (r is the usual radial variable), p is the core radius, n(r) is the refractive index distribution, n0 and no, are the maximum core and uniform cladding refractive indices, respectively. The shape function s(R) has a maximum of 1, i.e. 0 S s(R) S 1, in the region 0 < R < 1 and s(R) = 0 for R > 1. The normalised moments are defined by n —— (17) and it has been established that only the even moments ((20, 92, Q4, ...) are required to specify any refractive index distribution [22]. The ESI requires that the first two even moments ((20, ()2) of both the actual profile and the equivalent step profile are equal. This condition renders an equivalent step with a core radius of : pe = 208 (18a) and a profile height of: = .— (18b) The average waveguide parameter, V, is related to the usual waveguide parameter V by r = «mow (19) Using the average waveguide parameter and the E81, defined by eqn. 18, the dispersion expressions take the following forms: W2 no b(V) = 7 ~ bsr(V) (2011) d Vb Q b1(V)= W) z bum (zob) Vd b Q bzm = (M z bzm (20c) where b,,(V) is evaluated for the step index fibre at V = V. Any error introduced in using this 1381, is caused by neglecting t_he difference between the higher moments of the profile QM and the higher moments of BS], QM“. We define the normalised difference as follows: AQM = w (21) QMsr where M 2 4. M is even and _ (ZS—22W” 9”" ’ M/2 + 1 (22) In enhancing the simple ESI model, we introduce the effects of adding only one extra parameter, this is the enhancement parameter A94. Using eqns. 21 and 22, A94 can be expressed more specifically as: (3/094 - 9% 9% We have, therefore adopted the following functional form for the dispersion parameter b(V): A0,, = (23) Q _ _ _ b( V) = bmrl + mm mm (24) 2 This is an enhancement of the simpler expression of eqn. 20a where f(V) is our enhancement function. We will refer to eqn. 24 as the Enhanced ESI, or (E)ESI approx- imation for b( V). _ Table 6 shows A94 for several :1 values of the power law profiles. A94 decreases as we approach the step-index fibre; the correction term in eqn. 24 decreases accord- ingly. The size of A04 can be used to decide whether or not the function f (V) is required. 8.2.2 The (E)ESI and its prediction of disper- sion: From the exact calculation of b(V) for a range of profiles, we found the lowest-order polynomial approx- imated for f (V) to be the quadratic f(l7)=0.313I7—0.013I72 (25) The power law profiles with exponent a have been used for testing the (E)ESI model, and the main results are described as follows: IEE PROCEEDINGS, Vol. 135, Pt, J. No. 3, JUNE 1988 normalised mode—field radiusw normalised frequency V Fig. 7 Comparison between the dispersion curves b, bl, and b2 obtained by using: a An exact numerical technique b The Enhanced ESl c The simple ESI approximation The results are for the power-law profile with a = 8. V“ indicates the limit of single-mode operation exact - - - - - -- EESI _. 05 ix 'N 7‘ C3 normalised mode-field radiusjii Q 0 .0 b 'm on .0 N '-o.. 2 6 normalised frequency V Fig. 8 As Fig. 7, with at = 2 exact - - - - - - - EESI — — — — ESl 60 t‘ Ln (3 C) to C3 dispersion, pslnm km 8 3 1.0 1.2 wavelength, pm —20 Fig. 9 Total dispersion for the parabolic index profile fibre (a = 2) exact - r - - - -- EESI — A 7 — ESI IEE PROCEEDINGS, Vol.135, Pt. J, No. 3, JUNE 1988 (a) Profiles with a 2 2. For profiles which range from the clad parabola (a = 2) to the step index (a = 00), the error in predicting the dispersion parameters is negligible in the single-mode region. Figs. 7 and 8 show the exact E81 and (E)ESI dispersion curves for a = 8 and a = 2, respectively. Fig. 9 shows the ESL (E)ESI and exact total dispersion curves for the clad parabolic core profile (a = 2). The curves were obtained using the equation of the com- ponents of total dispersion, and data of Reference 23, and the E81 and (E)ESI dispersion parameters b, b1, and b2. It is seen that the total dispersion curve predicted by the EESI and the exact curve are, for all practical purposes, the same. The wavelength of zero total dispersion (10",) is predicted exactly by the (E)ESI, while the 1.0", is in error by 5.4% for the simple ESI. (b) Profiles with at < 2. For these extreme profiles it has been found that the (E)ESI provides the bulk of the correction term for waveguide dispersion in the range of V-values for which the simple ESI breaks down. Fig. 10 —0.2 normalised frequency V Fig. 10 As Fig. 7 with a = 1 exact v - - v - -- EESI — — — — ESI dispersion, pslnm km 1.4 wavelength , urn -20 Fig. 11 4444*7 exact Total dispersion for the triangular index profile fibre (a = l) ----- -- EESI ~——— 1381 209 illustrates the triangular profile (at = 1) case. For the tri- angular profile, the (E)ESI provides an estimate of the waveguide dispersion paramater, b2(V), of 92% of its exact value at the cutoff of the second mode, and improves rapidly for smaller V-values. In this case, the estimate of b and 171 are accurate. 210 Fig. 11 shows the E81, (E)ESI, and exact total disper- sion curves for the triangular core profile fibre. The (E)ESI total dispersion curve is in good agreement with the exact calculation, and 10“, is the same for both the exact and (E)ESI curves. The ESI approximation predicts the zero of total dispersion with an error of 6.7%. IEE PROCEEDINGS, Vol.135, Pt. J, No. 3, JUNE 1988 ...
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