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Unformatted text preview: 5'20 (E)ES determination from modefield diameter and
refractive index profile measurements on singlemode 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 singlemode ﬁbres, the (E)ESI and MFD methods are interrelated in a
selfconsistent model with the theoretical cutoff wavelength playing a pivotal role. Three indepen
dent measurement approaches are examined: modeﬁeld 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 modeﬁeld diameter (MFD) measurements.* Between these two there exist equivalent step index (ESI)
methods. The two ESI parameters (equivalent coreradius
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 singlemode 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 singlemode 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 singlemode
ﬁbres. Measurements of refractive index proﬁle and
modeﬁeld 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 singlemode
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 singlemode 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 stepindex profile l. The problem is then to determine the propagation
characteristics of a singlemode 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 singlemode
fibres look similar. Since the analytical solutions for the
stepindex ﬁbre are already available and well known, it
is then very convenient to have a method which has as its
reference a stepindex 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. corecladding boundary) from the
proﬁle measurements. Because of these problems, the
alternative of characterising singlemode ﬁbres from
MFD measurements has received a lot of attention. 1.2 Modefield diameter measurement (MFD)
techniques The MF D is the width of the fundamental mode, guided
by a singlemode 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 singlemodeﬁbre at V = 2.1, and its Gaussian approximation
MFD, is the Gaussian Modeﬁeld diameter actual ﬁeld amplide distribution
       gaussian approximation for the ﬁrst generation of singlemode 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, singlemode 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 nearﬁeld, 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
singlemode ﬁbres, but some problems arise because the
secondmode 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 singlemode
fibres Millar’s method is perhaps the most standard technique
for determining the two ESI parameters and the second
mode cutoff wavelength for singlemode ﬁbres. One basic
assumption in the Millar method is that the MFD for
arbitrary proﬁles behaves simply as a scaled version of
the stepindex 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 triangulartype 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 singlemode regime) to vagaries
in the refractive index proﬁle. Moreover, the Millar
method has no builtin 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 farﬁeld
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 cusplike proﬁles, its value can
approach 0.3. In terms of our three parameters, the normalised near
ﬁeld Petermann modeﬁeld 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 modeﬁeld diameter with that of some
stepindex ﬁbre. Additionally, the term in square brackets
in eqn. 1 is a monotonically increasing function of V, so
that by measuring the modeﬁeld diameter at cutoff, we
are choosing the worst sampling point for ﬁtting the
simple stepindex 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 triangularproﬁle ﬁbre. 2.0 U1 'o O
U" normalised modefield radius, a O
1.0 1,5 2.0 2 5 3.0 3.5 effective normalised trequencyW ' Fig. 3 Modeﬁeld 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+AQ4q (4)
where
_ 1
“(Dim
,_ I73’_b,_
qZﬂVH f(V).(V) 4 The quantities p and q are speciﬁed once I7 is known.
Similarly, the farﬁeld 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 singlemode ﬁbre measurements.
Millar’s method determines 11m from observation of
the nearﬁeld, 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 singlemode
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+AQ441)
= 0°in + IAQ442) (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 nearﬁeld mode ﬁeld radius at AL 2. If we use the farﬁeld 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 twoparameter 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 ﬁbrepreform is reported here,
one other nominally stepindex ﬁbrepreform 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 nearﬁeld 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 1217111
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 Modefield diameter measurements The nearﬁeld Petermann’s MFD was obtained from the inverse of the RMS farﬁeld 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: Nearfield 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 powerlaw 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
slammu» — N b 00 a) 8 responds approximately to a powerlaw proﬁle with
a = 8, which is very steplike. The enhancement para
meter in the ﬁbre (IAQ4I = 0.08) again, according to
Table 6, corresponds approximately to a powerlaw
proﬁle with a z 4 which is also very steplike. 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
singlemode 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 selfconsistent
model for characterising singlemode 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 modeﬁeld diameter measure
ments. Mr. Martinez would also like to thank the
Mexican Institutions CONACYT and IIE for their
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fibre determination‘. Electron. Lett., 1983, 19, pp. 643—644 19 FOX, M.: 'Calculation of equivalentstepindex parameters for
singlemode ﬁbres”, Opt. and Quanl. Electron, 1983, 15, pp. 451—455 20 HUSSEY. CD, and MARTINEZ, F.: ‘Approximate analytical
forms for the propagation characteristics of singlemode 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'hrirhtvmech. Z., 1976, 29, pp. 328329 22 HUSSEY, CD, and PASK, C: ‘Theory of proﬁle moments descrip
tion of singlemode 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 singlemode 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 singlemode 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 modefield radiusw o 1 .L._ __ _1
1.0 1.5 2.0 2.5 3.0
normalised frequency V
Fig. 5 Petermann modeﬁeld radius (DP and Marcuse modeﬁeld 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 wellknown 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 stepindex
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 lowestorder 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 powerlaw proﬁle with a = 8. V“ indicates the limit of
singlemode operation exact       EESI _.
05 ix 'N 7‘
C3 normalised modefield 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 singlemode 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
Vvalues 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 Vvalues. 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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