Full text of DOI:10.1109/50.848400

JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 18, NO. 6, JUNE 2000
869
Theoretical Investigation on the Dispersion of
Graded-Index Polymer Optical Fibers
Gnitabouré Yabre, Member, IEEE
Abstract—Various experiments on polymer optical fibers of 100 m [1], [2]. In more recent experiments [3], [4], we suc-
(POF’s) suggest that they present a large transmission capacity,
but this is not confirmed by the standard theory. In this paper we
present a full dispersion model based on evaluating the frequency
response of the fiber instead of the direct calculation of the pulse
cessfully reached the record transmission length of 200 m. A
full description of the state of the art regarding POF’s and re-
lated technologies is to appear in [5]. However, the achieve-
ments have been accomplished so far without real reference to
broadening from the moments of the impulse response. The
mathematical formalism is firstly displayed and subsequently prediction tools. The existing simulation models have been de-
applied to the graded-index POF made with polymethylmethacry-
late (PMMA). The simulation results show that the baseband
bandwidth is indeed enhanced due to the strong differential mode
attenuation (DMA).
veloped for GOF systems some decades ago and involve some
approximations that appear invalid in consideration of polymer
optical fibers. On the basis of the standard theory as it will be
clarified further, the error-free transmission experiment at 2.5
Index Terms—Bandwidth, differential mode attenuation, disper-
Gb/s over a 200-m-long fiber that we successfully carried out in
sion, optical communications, polymer optical fibers.
[3], [4] would have been rather problematic.
In the present work, we solve the above discrepancy and show
I. INTRODUCTION
that the factors that have until now been given little attention
can have a large impact on the data rate transmission perfor-
mance. This model presents a full description of the dispersion
which incorporates all the parameters involved in the determi-
nation of the total bandwidth. It is based on analytically eval-
uating the frequency response of the fiber instead of the direct
calculation of the pulse broadening from the moments of the
impulse response. We show that the baseband bandwidth is in-
deed increased in GIPOF’s mainly due to the strong differential
mode attenuation (DMA). Actually, this illustrates the classical
conflicting relation between dispersion and loss in multimode
fibers (MMF’s) in general. As a matter of fact, the large DMA
of high-order modes necessarilly causes a large power penalty
during light propagation, but at the same time it yields a band-
width enhancement as a result of the mode stripping effect.
To make it easy to apply the present analysis to any
multimode optical fiber link, the mathematical formalism
is explicitly displayed before specific computer simulations
applied to the PMMA-type GIPOF doped with benzyl benzoate
(BEN) are presented and discussed through illustrative curves.
We achieved a more than fourfold enhancement in the baseband
bandwidth in the presence of DMA even if a complete mode
filling condition is imposed at the input end of the fiber.
OLYMER optical fibers (POF’s) are being developed as
important high-speed communication media for short-
P
and medium-distance applications. Polymer fibers including
the step-index as well as the graded-index versions have a large
core-diameter (500–1500 m) and a large numerical aperture
(0.2–0.9), which allows for easy connectorization of systems
and efficient launching from semiconductor lasers or light
emitting diodes. Furthermore, they show a complete immunity
to EMI/EMR (electromagnetic interference/electromagnetic
radiation) and a great resistance to mechanical damage as a
consequence of the intrinsic flexibility of the polymer material.
On the other hand, the graded-index type polymer optical fiber
(GIPOF) can combine these properties with a large capacity
for digital transmission. Accordingly, the GIPOF as a physical
transmission medium is foreseen to be the best candidate
for the replacement of the metallic cables traditionally used
in customer premises networks (CPN’s), interconnect and
access links. The utilization of conventional glass optical fibers
(GOF’s) could also be envisaged in these applications, but
the corresponding connectors are expensive. In addition, the
fabrication and termination of GOF’s are more difficult than
those of POF’s and also the glass material is more expensive
than polymer, so the cost of the overall GOF-based system
would be inevitably high.
II. THEORY
A number of researchers has previously demonstrated that
undistorted bit streams of 2.5 Gb/s could be transmitted through
the polymethylmethacrylate (PMMA) GIPOF over a distance
We consider the class of circular-symmetric fibers described
by the refractive index profile
for
for
(1)
Manuscript received August 17, 2000; revised February 16, 2000. This
work was supported and conducted within the framework of the Dutch
research project IOP Electro-Optics ongoing in the COBRA research institute, where is an offset distance from the core center, is the
Eindhoven University of Technology.
core index exponent
is the refractive index in the center of the core,
is the refractive-index contrast, is the core radius (i.e., the
,
is the free-space wavelength,
The author is with COBRA, Interuniversity Research Institute, Eindhoven
University of Technology, EH 12.33, Eindhoven 5600 MB, The Netherlands.
Publisher Item Identifier S 0733-8724(00)05077-5.
0733–8724/00$10.00 © 2000 IEEE

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radius at which the index
reaches the cladding value where is the transmission distance and is the so-called cou-
. For the following, we will pling length,
being the corresponding bandwidth. But this
simply use the identifications
,
and model uses the uniform launching assumption and also ignores
, and let the quantities dependence on wavelength be the DMA [9]. Furthermore, it no longer associates the contri-
bution of the material dispersion which is known to be strong
implicit.
Multimode optical fibers described by (1) potentially support in POF’s. On the other hand, the right hand side of formula (8)
a large but finite number of modes which are particular solutions contains two unknowns, that is,
and that are dependent
of the Maxwell’s equations. Each guided mode has its own prop- on the launching conditions and therefore difficult to put on ac-
agation constant and therefore propagates at its own particular curate numerical values. Any attempt to measure
and
velocity. From the WKB analysis it was shown that propagated should be altered by a large error because the effect of DMA and
modes are actually clustered within groups in which each mode that of chromatic dispersion cannot be prevented from affecting
has nearly the same propagation constant derived to be [6]
the results. The coupling length was estimated in [8] to be 2 m,
but this value seems too small for ordinary GIPOF’s. This view
is supported by the fact that the suggested coupling-length of 2
m is not confirmed by measurements. In the reality, the effect
of mode-coupling in GIPOF should be negligible for the fol-
lowing reason. The length-dependent mode-coupling is a diffu-
sion process resulting from small power exchanges on imper-
fections along the fiber. These exchanges can cause the modal
delay times to be averaged [10] and consequently a bandwidth
enhancement if the transmission distance is long enough for
each propagated mode to undergo a statistically large number
of interactions before detection. This requirement appears to be
unsatisfied if the transmission length is limited to a few hundred
meters as it is for instant the case. This explanation can be best
understood by reference to numerical values which compare the
graded-index POF with a graded-index GOF.
(2)
where is the principal mode group number (or principal mode
group order) and is the total number of mode groups that can
potentially be excited. is analytically given by
(3)
The modal delay per unit length can be derived from (2) using
the definition where the prime denotes
the derivation with respect to wavelength and is the speed of
light in vacuum. The calculation leads to
Propagated modes belonging to different groups most easily
interchange energy when they have field distribution that
strongly overlap, and fiber imperfections are most effective
in causing this when they present a periodicity equal to the
beat period of the two modes in consideration. If this criterion
(4)
in which
parameter, respectively, given by
is the group index and is the profile dispersion
(5)
(6)
is fulfilled, the coupling between level
and level
adds in phase from one interaction to the next and becomes
considerable. In other words, if we assume that the perturba-
tions are periodic with “wavelength” , the mode-mixing is
strongest when
For a parabolic standard glass fiber with
, we obtain mm. By adopting a
coupling length of 1 km (which should be a strict minimum
for classical GOF’s under normal utilization [11]), this yields a
minimum of 706 515 modal interactions required to impact the
dispersion. Let us now consider a parabolic-index POF with
[with
being given by (7)].
For further use, let us also express the difference in prop-
agation constant between adjacent mode groups. Considering
m,
level and level
, the spacing is approximately given by
, which yields
(7)
m and
. The strongest perturbation
mm. This will cause
wavelength is calculated to be
A. Background
202 modal interactions over 2 m, which is insufficient to impact
the dispersion when compared to the result of glass fiber. For a
longer sample of 400 m in length, for example, the number of
interactions increases to 40 360, but this number is still 17 times
less than that required for the mode-coupling to be efficient.
Obviously, the mixing phenomenon is not the reason behind the
bandwidth enhancement in GIPOF’s. Additional length-depen-
dent power mixing may occur between noncontiguous modes
(with smaller , but this is not expected to greatly affect the
propagation [9]. Instead of the mode-coupling, we show here
that the strong differential mode attenuation which has been
given little attention until now is in fact the main cause of the
improvement in the fiber dispersion. This trend was recently
So far two different analytic approaches have been used to
evaluate the pulse dispersion in polymer optical fibers [7], [8].
The approach taken in [7] is based on Olshansky and Keck’s
theory [6] which assumes a uniform excitation and neglects both
distributed loss and mode-coupling. Moreover, these parame-
ters, in particular the distributed loss, may have a significant im-
pact as we show further.
The approach taken in [8] is an attempt to incorporate the
mode-coupling effect in the dispersion model by using the fol-
lowing bandwidth formula derived in [9]:
(8)

YABRE: THEORETICAL INVESTIGATION ON THE DISPERSION OF GRADED-INDEX POLYMER OPTICAL FIBERS
871
shown by some measurements that are to be published in [12],
but a proper pulse broadening model which would incorporate
the DMA has not been given. This is the topic of this paper that
will be discussed in the following sections.
B. Refractive Index and Material Dispersion
We assume that the core and cladding materials of the GIPOF
follow a three-term Sellmeier function of wavelength [7], i.e.
with
(9)
where
and
are, respectively, the oscillator strength
and
(a)
and the oscillator wavelength. The parameters
are often gathered under the term of Sellmeier constants. For
the ben-doped PMMA GIPOF, these constants are given in [7].
Once the refractive index is known, the material dispersion
and the corresponding dispersion slope can be obtained,
respectively, from the definitions
(10)
(11)
The top part of Fig. 1 reports the refractive index of the
cladding and that of the central core region, whilst the bottom
part plots the material dispersion as a function of wavelength.
In the 650-nm transmission window the previous parameters
(b)
Fig. 1. Refractive index and material dispersion for the ben-doped PMMA
GIPOF as a function of wavelength. These results are based on the Sellmeier
parameters of [7]: (a) Refractive index of the cladding material and that of the
central core region and (b) dispersion of the central core region.
can be derived as
, and
,
,
,
. These values allow to theo-
retically determine the numerical aperture (NA) of the fiber
as NA . On the other hand, the
material dispersion at the 650-nm wavelength and the corre-
sponding dispersion slope are calculated approximately to be
403 ps/(nm km) and 2 ps/(nm km), respectively.
where
with
(13)
C. Simplified Dispersion Model
Using the above modal delay expression together with the
calculation process of [6] (based on evaluating the first three
moments of the full impulse response), produces the following
expression for the pulse broadening due to chromatic dispersion
Toallowforcleardemonstration,wefounditconvenienttofirst
presentthedispersionanalysiswithinthesimplifiedtheorywhich
assumes a uniform excitation and neglects both the effects of dis-
tributedlossandmode-coupling.Althoughtheseassumptionsare
invalid in the general case, this analysis should allow for a better
understanding of the improved bandwidth model.
Within the weak guidance rule, formula (4) can be approxi-
mated by a Taylor series in powers of
of a fifth-order expansion is given by
. The result
(12)
(14)

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JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 18, NO. 6, JUNE 2000
In the same way, the intermodal pulse broadening is calcu- did Olshansky and Keck [6]. For a system operated within the
lated to be
standard low-loss transmission windows, the contribution of the
extra terms may not be significant as long as the weak guidance
rule remains valid. Their effect will become important if the
numerical aperture (NA) of the fiber in consideration is fairly
high and if a nearly zero material dispersion wavelength is used.
Therefore, dispersion formula [14] and [15] can be regarded as
refinements to Olshansky and Keck’s expressions.
D. Full Dispersion Analysis
We consider a multimode fiber characterized by the -class
refractive index profile defined in (1) as linear in its input-output
relationship [14]. In this case, we have recently shown that
within some conditions that are expected to be satisfied in
practice, the complex transfer function of such MMF’s can be
modeled by a product of two filter functions as follows [15]
(15)
The minimum of the modal pulse broadening can be obtained
from (15) by solving = 0. For this, we will
retain only the first two terms in (15) as the most relevant. With
(19)
this approximation it can be shown that the minimum occurs for
[6]
where
and
represent chromatic
dispersion and modal dispersion, respectively. The parameters
appearing in argument of both functions are the baseband an-
gular frequency , and the transmission length .
(16)
Equation (19) expresses the latent idea that chromatic and
modal dispersions are independent effects and can be evaluated
separately. Under the assumption that the power spectral den-
sity of the driving source has a Gaussian lineshape with rms
linewidth , the chromatic transfer function can be calculated
exactly, yielding [15], [16]
The total rms width of the impulse response,
resultant 3-dBo bandwidth can be determined from (14) and
(15) as
, and the
(17)
(18)
(20)
The above relations show that the rms width increases
in direct proportion to the transmission distance, whilst the
bandwidth reduces inversely, i.e., the product of the bandwidth
times the length is constant for a given type of fiber. This is the
reason why pulse broadening in MMF’s has traditionally been
described in terms of a constant bandwidth-distance product.
But, it should be mentioned in passing that the inclusion of
the nonlinear parameters (nonuniform excitation, distributed
loss, eventual mode-coupling) will cause the bandwidth-dis-
in which
and
have been introduced as abreviations for
(21)
(22)
tance product to depend on length to a certain degree. So this where
concept will no longer be meaningful in characterizing the guided modes and
is the modal velocity dispersion averaged over all
is the averaged dispersion slope. It is im-
fiber dispersion behavior. Instead, the determination of the portant to realize that for a system operated around a zero ma-
bandwidth-versus-length characteristic will be relevant for link terial dispersion wavelength, the chromatic effect is not neces-
designers.
sarily negligible because of the presence of the dispersion slope.
It is of interest to compare formula (14) and (15) with equiv- Therefore, this term cannot be systematically ignored even in
alent expressions derived by Olshansky and Keck. Firstly, we the zero dispersion region. The fact that averaged values are
note that the third term in (46) of [6] involves a multiplying used for the dispersion and dispersion slope may cause the chro-
factor
. Instead, the present intermodulation disper- matic transfer function as defined in (20) to slightly depend on
, which is quite all parameters that determine the propagation, i.e., launch condi-
sion formula (14) involves
in agreement with the correction already reported by Soudagar tions, distributed loss and mode-coupling. We have to mention,
and Wali [13]. Secondly, it is noteworthy that our calculation in- however, that, after computation at the 650-nm wavelength, we
corporates more terms in the expressions of the chromatic and have not recorded any significant difference between these av-
intermodal pulse broadenings compared to those displayed in eraged values compared to those calculated previously by con-
[6]. This follows from the fact that we used a fith order expan- sidering only the material contribution, i.e.,
sion of the modal delay instead of a second order expansion as km) and ps/(nm km).
ps/(nm

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873
The modal transfer function is given by
expressed in cartesian coordinate (with the core center taken as
the origin of axes) as
(25)
(23)
where stands for the beam spot radius.
We also adopt the simplified approximation that the
launching field given above excites solely the Hermite–Gauss
fields of the fiber given by [18]
where
,
and
is the modal
power in Fourier domain. The numerator in the right hand side
of (23) is a normalization factor representing the total power
at position . As a general rule,
is a solution of a
partial differential equation (referred to as power flow equation
or diffusion equation) which incorporates not only the modal
delay but also the distributed loss and mode-coupling effects.
The power flow equation has no analytical solutions and can be
solved only numerically. Such an analysis was previously car-
ried out in [15] concerning glass fibers and could readily be ex-
tended to GIPOF’s. As we explained in Section II-A, the contri-
bution of the mode-coupling phenomenon should be negligible
over the lengths of GIPOF samples usually considered. In this
case the modal power can simply be calculated as
(26)
where
is the Hermite polynomial of degree and
is
referred to as the beam radius of the fundamental modes given
by
(27)
The overlap integral of (25) and (26) is calculated from the
following definition where the integration is taken over the input
plane
(24)
where
is the mode excitation coefficient (or cou-
is the mode-dependent at-
tenuation. The fact that it is the difference in the modal param-
eters that causes the intermodal dispersion can readily be veri-
fied from the expressions (23) and (24). Indeed, it is clear that
the shape of the frequeny response will not be affected by mul-
pling/launching efficiency), and
(28)
This integral can be solved exactly, yielding [19]
for or odd
tiplying (24) by any overall constant factor. Thus,
be replaced by which represents the differen-
tial mode attenuation. In the same way, could be replaced
which represents the differential mode delay
could
(29)
for and even.
by
(DMD).
in which
To enable further frequency response simulations to be done
incorporating all the parameters involved, the mode excitation
coefficients as well as the differential mode attenuation must be
specified. Let us discuss these aspects in two separate Sections.
1) Excitation Coefficients: The presence of the excitation
coefficient in (24) permits to use the present model under any
(30)
(31)
launching condition.
can be evaluated as the overlap
integral of the electrical field of each fiber mode with the elec-
trical field of the incident light. If we consider that the fiber
launching is uniform as did Olshansky and Keck, the initial
power distribution is by assumption the same for each mode and
can be formally set to unity. This approximation is
where
hypergeometrical function.
The mean power excitation coefficient of modes in group
that we denote by , can be derived from (29) as
is the gamma function and
is the
,
realistic if a light emitting diode is being used as a launching
source. But most of experiments so far carried out on the PMMA
GIPOF’s employ visible light beams from semiconductor lasers.
Therefore, it is closest to reality to adopt the assumption of a
Gaussian-type shape for the input beam spot [17].
(32)
The excitation coefficient as a function of the normalized
We will additionally assume that the fiber is excited centrally group number can be calculated as follows. During the nu-
with a laser beam that forms a circularly symmetric spot on merical evaluation of (23), using for example Simpson rule, the
the input surface. It is equally considered that the fiber axis is variable is handled as discrete values . From each of these
well aligned with the beam axis, and that its end surface is well discrete values , the corresponding mode group number
positioned in the focal plane. The field of such a beam can be is derived as the integer nearest to the product
and the

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JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 18, NO. 6, JUNE 2000
-dependent excitation coefficient is simply computed by iden-
tifying to
.
2) Differential Mode Attenuation: The other parameter re-
quired in the computation of the modal transfer function is the
differential mode attenuation. The modal attenuation originates
from conventional loss mechanisms that are present in a usual
fiber, that is, absorption [20], Rayleigh scattering [20] and loss
on reflection at the core-cladding interface [21]. These different
loss mechanisms act on each mode in a different manner, which
causes the attenuation coefficient to vary from mode to mode.
For example the fiber boundary has a strong effect on modes
near cutoff but little on the fundamental ones. Because of the
multiple origins of the DMA, the modeling of this phenomenon
is not an easy task. Even though we may succeed through sim-
plifying assumptions, the resulting model will enevitably intro-
duce a number of parameters for which it may be difficult to
find correct numerical values. We believe that as it is a common
practice to systematically mesure a fiber attenuation in the usual
transmission windows, it should be the same for the DMA. Such
measured data could readily be incorporated in the dispersion
model by fitting an appropriate function. For this, we suggest to
use the following functional expression for the distributed loss
Fig. 2. Mode-dependent attenuation (MDA) as a function of the normalized
mode order : The bullets represent the measurements reported in [22,
Fig. 7(c)], whilst the solid curve is the fitted MDA formula (33) with
(33)
and
. Note that the curve is properly shifted in order to take the
minimum attenuation of 124 dB/km into account.
where
is the attenuation of low-order modes,
is the
th-order modified Bessel function of the first kind, and is a
weighting constant.
III. SIMULATION RESULTS
The question now is to put proper numerical values on the
parameters appearing in formula (33), for the computation to be
possible. Unfortunately, no measured DMA data for POF’s exist
in the literature yet. However, various measurements on glass
optical fibers show that, more than anything, the DMA in deci-
bels as a function of the principal mode group number depends
on the wavelength and on the numerical aperture (NA) [20],
[22]. In other words, the fiber materials seem to have no sig-
nificant effect on the DMA shape as long as the refractive index
conforms to a circular-symmetric profile law in accordance with
We have carried out computer simulations for the graded-
index type POF having a PMMA uniform cladding and a gradu-
ally varying concentration of benzyl benzoate dopant in the core
[7]. It is assumed that the exciting light source emits a visible
light near 650 nm and has an rms linewidth of 0.5 nm. These
are the characteristics of the MQW laser diode specially devel-
oped by NEC Co. for high-performance POF systems, and that
we used in previous transmission experiments [3], [4].
Fig. 3 reports the chromatic, intermodal and total bandwidth
the definition (1). Therefore, we will adopt the profile of [22, of a 100 m-long PMMA GIPOF as a function of the core re-
Fig. 7(c)], but we will vertically shift the curve to accommo- fractive index exponent for the 650-nm wavelength. These plots
date the estimated minimum attenuation of 124 dB/km of POF’s are based on the analysis of Section II-C which assumes a uni-
[23]. Although a modification of the data is proposed for POF’s, form excitation and neglects both the distributed attenuation
this approximation that the DMA profile is the same as that of and mode-coupling. The chromatic bandwidth is seen to show
GOF’s may seem rough at first glance, in particular if account is little dependence on , which means that the material disper-
taken of a possible influence of the size of the core. Because, the sion is the dominant contribution in the transmission window
numerical results obtained here are quite in line with previous considered. On the other hand, the modal bandwidth shows a
transmission experiments, as we explain further, we believe that highly peaked resonance with . This is the well-known char-
this assumption should present a large degree of validity. The re- acteristic feature of the grading. With the present choice of pa-
sults are reported in Fig. 2. The fitting of measurements of [22] rameter-values, the maximum turns out to be located around
to the empirical expression (33) leads to
and . It is worth noticing from these values that culated from expression (16). Another result that can be noted
the strength of the DMA in POF’s follows mainly from the fact from Fig. 2 is the presence of crossover points at
2.8 10 km
,
an -factor of 2.33, which quite corresponds to the value cal-
that the minimum attenuation is extremely large. For compar- and
between the chromatic and modal characteris-
ison, the minimum attenuation of single cladded fluoride glass tics. This shows that the total bandwidth may mainly be limited
fibers in the 1300 nm transmission window is less than 1 dB/km, either by the modal dispersion or the chromatic dispersion de-
that is,
km . Future experiments on POF samples pending on the value of the index exponent. As shown by the
could be done for a better understanding of the DMA behavior. solid curve in Fig. 2, the chromatic dispersion will essentially

YABRE: THEORETICAL INVESTIGATION ON THE DISPERSION OF GRADED-INDEX POLYMER OPTICAL FIBERS
875
Fig. 4. Frequency responses of a 100-m-long PMMA GIPOF for a wavelength
of 650 nm and varying index exponent . These plots result
from formula (19) within the assumption of a uniform excitation in the absence
of distributed loss and mode-coupling.
Fig. 3. 3-dBo baseband bandwidth of a 100-m-long fiber as a function of
refractive index exponent for a 650-nm wavelength. All three curves are based
on the results of analysis presented in Section II-C: (——) Total bandwidth, (–
–) Modal bandwidth,
Chromatic bandwidth.
limit the total bandwidth for
, whilst for
or the modal dispersion will cause the main limitation.
But these comments are valid only if the approximations related
to this model are likely to be satisfied. We will show further that
the modal response of the GIPOF greatly enhances due to the
strong differential mode attenuation.
Before discussing the full dispersion analysis through illus-
trations, it is of interest to firstly verify its consistency with Ol-
shansky and Keck’s theory when the same approximations are
imposed. For this, a series of frequency response simulations has
been done under the assumption of uniform excitation, constant
mode attenuation and absence of mode-coupling. Some of these
results are displayed in Fig. 4 where plots of the fiber frequency
response for a sample length of 100 m and a varying index ex-
ponent of 2.1, 2.3, and 2.4 are given. The 3-dB baseband band-
widths corresponding to the three values of are found approx-
imately to be 2.0, 8.3, and 5.6 GHz, respectively. These values Fig. 5. Modal frequency responses of a 100-m-long PMMA GIPOF for an
index exponent of 2.1 and a wavelength of 650 nm: (——) Uniform excitation
exactly match with those obtained from formula (18). We have
also separately verified through various numerical simulations,
without DMA, (– –) Gaussian beam excitation
m) without DMA, (–
–) Uniform excitation with DMA; The dotted curve represents the chromatic
not reported here, that the chromatic and modal bandwidths cal- frequency response given for comparison.
culated from expressions (20) and (23) coincide well with the
values calculated from formula (14) and (15), respectively. Let proving the modal response of the GIPOF. It can be observed, in
us mention a posteriori that the bandwidth is defined here as particular, that the bandwidth enhancement caused by the DMA
the half-power frequency of the modulus of the transfer func- is dramatically large. The modal response with DMA is even
tion over the distance considered.
seen to become superior to the response due to chromatic dis-
The modal frequency response simulations are displayed in persion. Obviously, this result shows that the DMA is a deter-
Fig. 5, where the effects of the launching condition and that of mining factor for accurate assessment of the baseband band-
the differential mode attenuation are shown. For comparison a width in POF’s, and consequently it cannot be excluded from
dotted curve is reported in the same figure representing the fre- the model as it has been the case for glass fibers.
quency response due to chromatic dispersion. These results in-
Fig. 6 plots the total fiber bandwidth as a function of trans-
dicate that both the Gaussian beam excitation (with overfilled mission length, showing a comparison between Olshansky and
core diameter) and the presence of DMA are favorable for im- Keck’s theory and the improved model including the effect of

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JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 18, NO. 6, JUNE 2000
It is also of interest to mention that because the bandwidth
of the GIPOF is essentially limited by material dispersion (as a
consequence of the DMA effect), its operation under a restricted
mode excitation using the techniques described in [24] to sup-
port increased transmission rates in GOF’s should not result in
a significant bandwidth improvement. Nevertheless, a selective
mode launching scheme could be implemented in the view of
benefitting from the minimum loss advantage. In that case, in-
deed, low-order modes are involved that propagate near fiber
axis with the slowest attenuation, which should cause a dramatic
reduction of the inline power penalty. A limited mode launch of
the GIPOF in the center of the core may reduce the mean at-
tenuation of 164 dB/km [3], [4] to the minimum of 124 dB/km
[23] or less. In other words, the 32.8-dB power loss over the
200-m-long sample may drop to 24.8 dB or less. In this way
one could release at least 8 dB in the power budget, thereby al-
lowing for a less sensitive and certainly less expensive receiver
to be employed.
In summary, since the DMA in GIPOF’s affects both disper-
sion and loss to a considerable degree, it becomes a determining
factor in the choice of system parameters. Accordingly, a good
knowledge of the DMA will be of prime importance for link de-
signers and users. Also, always owing to the large effect of this
parameter, its clear understanding can no longer be bypassed if
appropriate network standards are to be developed. Although a
distributed loss curve that would directly be measured using a
GIPOF sample is not expected to depart as much from the shape
of Fig. 2, this experiment needs to be done in future.
Fig. 6. Total 3-dB bandwidth of the PMMA GIPOF as a function of
transmission length for an index exponent of 2.1 and a wavelength of 650 nm:
(——) Result of formula (18), (– –) Result of formula (23) including DMA.
DMA. It is clear again that the DMA enhances the total band-
width to a considerable extent. Indeed, it can be seen from the
figure that at least a fourfold bandwidth enhancement may be
gained. For example, the 3-dBo bandwidth corresponding to
a 200-m-long sample is found from the simplified dispersion
model to be 1.0 GHz but it increases to 4.4 GHz in the pres-
ence of DMA. This result is more consistent with the error-free
transmission experiment at 2.5 Gb/s that we achieved in [3], [4].
Most important, the dash-dot curve of Fig. 6 indicates that the
PMMA GIPOF has greater potential than suggested by previous
experiments. It appears that the data rate can be increased up
to 8.7 and 4.4 Gb/s through the 100- and 200-m-long samples,
respectively, if the laser source is capable of such modulation
speeds. An error-free transmission at a bit rate of 2.5 Gb/s via
a 300-m GIPOF can even be realized. These predicted figures
should be verified in future experiments. Consequently, it has
become a reality that the PMMA GIPOF can span longer dis-
tances than those foreseen so far. As an example, one could en-
visage the coverage of medium-size building backbones with
this type of GIPOF provided that a sufficient power budget can
be made available.
V. CONCLUSION
We have described the dispersion behavior of laser-based
graded-index polymer optical fiber systems. This model in-
corporates all parameters involved in the determination of the
baseband bandwidth, in particular, the differential mode attenu-
ation that was ignored in the standard theory. The mathematical
development is based on analytically evaluating the frequency
response of the fiber instead of the direct calculation of the
pulse broadening from the moments of the impulse response.
It can be used at any wavelength under any input condition
including the fiber excitation by a Gaussian-shape beam spot
which is closest to most practical applications.
We have shown that the 3-dBo bandwidth is enhanced in
GIPOF’s mainly due to the strong DMA. The numerical results
are quite in line with the transmission performances obtained
previously. Since this simulation tool can be used to establish
more likely performance limits, it should be useful to the design
of more reliable GIPOF systems. We believe that this analysis
is an important milestone for the development of gigabit links
in graded-index polymer optical fibers.
IV. DISCUSSION
As mentioned before, the fact that the GIPOF bandwidth in-
creases with a strong differential mode attenuation is unsur-
prising. During the propagation of light through the fiber, high-
order modes that have the fastest attenuation gradually lose their
energy and only a small fraction of the number of guided modes
supported by the waveguide have significant power to be de-
ACKNOWLEDGMENT
The author would like to thank Prof. G. D. Khoe, Dr. H. de
tected. In other words, the bandwidth enhancement follows from Waardt, and Ir. H. P. A. van den Boom for continuous and helpful
the filtering effect of the DMA. This in fact reflects the clas- discussions. He equally acknowledges useful discussions with
sical trade-off relation between dispersion and loss in multi- Dr. T. Ishigure from Keio University, Japan, about the parame-
mode fibers in general.
ters of the PMMA material.

YABRE: THEORETICAL INVESTIGATION ON THE DISPERSION OF GRADED-INDEX POLYMER OPTICAL FIBERS
877
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Gnitabouré Yabre (M’97) was born in
Boutaya-P/Zabré, Burkina Faso, in 1962. He
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From 1989 to 1995, he worked in the RESO
laboratory at the “Ecole Nationale d’Ingénieurs de
Brest,” France. He is now with COBRA, Interuni-
versity Research Institute, Eindhoven University of
Technology, The Netherlands. During his previous
post, his research activities were focused on semiconductor laser nonlin-
earities, linearization alternatives, injection-locking, subcarrier-multiplexing
techniques, fiber-to-wireless communication systems, broad-band optical com-
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multimode optical fibers and related systems.
Dr. Yabre has served as a reviewer for the JOURNAL OF LIGHTWAVE
TECHNOLOGY, IEEE TRANSACTIONS ON MICROWAVE THEORY AND
TECHNIQUES, and Optics Communications.