# Parabolic Pulse Generation Using Tapered Microstructured Optical Fibres

Authors: Natasha Vukovic; Neil G. R. Broderick; Francesco Poletti
Journal: Advances in Nonlinear Optics
 (2008)

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## Abstract

This paper presents a numerical study of parabolic pulse generation in tapered microstructured optical fibres (MOFs). Based on our results and the algorithms presented, one can determine the linear taper profile (starting and finishing pitch values and taper length) needed to achieve parabolic pulse shaping of an initial Gaussian pulse shape with different widths and powers. We quantify the evolution of the parabolic pulse using the misfit parameter and show that it is possible to reach values significantly better than those obtained by a step index fibre.


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## Body

## 1. Introduction

The generation of parabolic pulse has been widely investigated and is still an active area of research due to
its unique properties and numerous applications. In the
normal dispersion regime, due to the interaction between dispersive and
nonlinear effects, a high intensity pulse suffers wave breaking, manifesting
itself as oscillations on the pulse's tail [1]. However, for a linearly chirped
parabolic pulse, this effect can be avoided if the pulse's chirp is such that
the rate of change of the time separation is the same for all parts of the
pulse, resulting in the pulse evolving in a self-similar fashion [1].
Importantly, linearly chirped parabolic pulses can be efficiently compressed
and are used in many telecommunications applications such as optical
regeneration [2] or temporal Fourier transforms [3]. Such pulses are useful for
applications such as pulse compression, and indeed compression of picosecond
pulses down to femtosecond range has been experimentally demonstrated in [4, 5].Different approaches to
the generation of parabolic pulses have been studied and include a fibre Bragg
grating [4], dispersion decreasing fibre in the normal dispersion regime [6],
normally dispersive two-segment fibre device [7–9], optical amplifiers [10], or
similariton lasers [11]. Applications requiring high-power parabolic pulses
often rely on an optical amplifier as parabolic pulses are automatically
generated in these amplifiers. On the other hand, there is a range of
applications, particularly related to optical signal processing, where a high
signal power is undesirable, since it requires a high-power pump source and
adds costs and complexity to the system [7]. Another drawback of using an
active system is that it introduces amplified spontaneous emission noise. Hence, the
need for other ways to generate parabolic pulses using passive fibres is required.The passive means to
generate parabolic pulses described above all require relative long lengths of
fibre (>1 km) and are suitable for picosecond pulse widths (>1picosecond). In [12],
however, it was shown that by using a comb-like dispersion a decreasing profile
also leads to parabolic pulse generation, in a few metres of fibre. Such
comb-like profiles are complicated to produce (relying on multiple splices),
and hence there is a need for a different approach for efficient generation of
parabolic pulses in relative short lengths and with the flexibility of coping
with different input pulse widths and energies. Due to the exceptional properties
of microstructured optical fibres such as control of optical properties
(dispersion, nonlinearity, and birefringence), “endlessly” single-mode guidance,
extreme mode sizes, low bend loss [13], we chose to examine microstructured
optical fibre-based tapers for parabolic pulse generation. Tapered MOFs have
found many other applications, such as pulse compression [14, 15],
supercontinuum generation [16], generating soliton frequency shift [17], and
SBS threshold enhancement [18].Our aim here is to
investigate whether tapered microstructured fibres can be used to efficiently
generate parabolic pulses in short lengths (<10 m). We choose to taper the
fibres, since, as it was pointed out in [6, 19–21], that in a system
described by the ideal lossless nonlinear Schrodinger equation (NLSE) with
decreasing dispersion, optical pulse evolution is formally similar to that of a
medium with a constant gain. As a consequence, an asymptotic self-similar
parabolic pulse solution is found to exist in a dispersion decreasing fibre
with normal group velocity dispersion, that is, a taper [6]. Using linear dispersion
profiles, we achieved parabolic pulses after a finite propagation length, with
the parabolic pulse remaining stable for some length before being distorted.
Formally, this approach does not correspond to an asymptotic self-similar parabolic
pulse, since we used a linear taper profile and not a hyperbolic dispersion
profile. However, our approach is still based on the observation that the
longitudinal decrease of the normal dispersion is formally equivalent to
optical gain.Section2 presents results of modelling and
simulation of normally dispersive microstructured optical fibres. We have
identified two normal dispersion regions, and outlined a procedure for choosing
the parameters of the taper profile (starting and finishing values of the
pitch) and the air-filling factor, for efficient pulse generation. Then in Section 3, using the NLSE, we show how initial
Gaussian pulses of different input powers and widths evolve into parabolic
pulses with a linear chirp for two different tapers. Results are quantified
using the misfit parameter, which measures the root mean square error between
the propagated pulse and its parabolic fit in the time domain, and presented so
that the optimum taper length that corresponds to the best
parabolic pulse can be determined. The best misfit distances for both regions and
a range of pulse power and initial pulse widths are given, allowing one to
choose the best taper profiles for a wide range of initial pulse parameters.


## 2. MOF and Taper Modelling

A microstructured fibre consists of a periodic array of air-holes with a central
defect acting as the core with the optical parameters of the fibre being
completely determined by the period Λ and the relative air-hole sized/Λ. We used the finite element method (FEM), implemented
in the commercial software package Comsol Multiphysics, to solve for
dispersion, effective area, and confinement loss of the lowest-order mode. The
FEM elements are flexible from a geometric point of view (easy treatments of
any shape of structure) and from the material point of view (it allows
inhomogeneous, anisotropic, or nonlinear characteristics to be incorporated) [22].
The refractive index of silica n(ω) is approximated by the standard Sellmeier
equation [23]. The geometry of the MOF used was a hexagonal pattern with 11
layers of holes, (see Figure 1). Group theory analysis allows only a quarter of
the structure to be simulated with no loss of information about the modes
saving considerable amount
of the computational time [24]. A typical fundamental mode of a MOF, for
illustrative purposes, is shown in Figure 2. Note that the light is well
confined to the core region and penetrates only slightly into the cladding
region.Figure 1
Cross-section of the MOF with 11 rings of holes (d–hole diameter, Λ–pitch).Figure 2
The magnitude of the Poynting vector.For this study, the
crucial result of the fibre simulations is the propagation constantβ and associated effective index neff which are related by(1)β=2πneffλ.Note that
for a finite structure,neff is a complex number with the imaginary part giving the confinement loss of the
mode. In our FEM analysis, we used a perfectly matched layer (PML) to treat the
open boundaries. Using the complex effective index, the confinement loss is
calculated as(2)CL[dB/m]=20×106ln102πλ[μm]Im(neff).For
useful fibres, a loss of less than 1 dB/m is needed and this forces us to reduce
the parameter space of possible fibres in our study.The
chromatic dispersion,d, of the fibre
is given by(3)D=−2πcλ2β2,where βn is a group velocity dispersion defined as follows:(4)β2=d2βdω2.Note that as pure silica
at 1.55 microns has positive dispersion, and so fibres with large negative
waveguide dispersion are needed in this study.Since ideal MOFs depend on
only two parameters, their optical properties can be best displayed using an
optical map such as that shown in Figure3 which shows the dispersion of
various MOFs at 1.55 μm
(wavelength mostly used in telecommunications applications). This plot enables
us to find the range of values for Λ and d/Λ that would provide normal dispersion regime (D<0) of
the fibre. It can be seen that almost any value of d/Λ can be
used assuming that the correct value of Λ is
chosen and so we need to choose a range that can be easily fabricated.
Importantly, the dispersion characteristics of any fibre taper can be
immediately read off the contour graph as each fibre taper corresponds to a
path in (Λ, d/Λ) space. However, for ease of taper fabrication, only vertical paths are
considered here as they correspond to fibre tapers with a constant d/Λ which can be made by simply heating and
stretching the fibre. More complicated profiles which can be made by pressuring
the taper during fabrication are not considered here.Figure 3
Dispersion (D) contour plot atλ = 1.55 μm.Since our fibre tapers
have a fixedd/Λ, we can compress the 2D map in Figure 3 into a
series of overlapping curves as shown in Figure 4.Figure 4
Dispersion versusd/Λ for
different values of Λ.In Figure4, the dispersion is calculated versus air-filling factor d/Λ, for different values of
parameter Λ. From Figure 4, it can be
seen that there are several regions of d/Λ that can be chosen so
that a fibre operates in the normal dispersion regime. We chose to examine
three regions with d/Λ = 0.2, 0.3, and 0.8,
respectively, as these regions have a low absolute value of dispersion. Both
the nonlinearity and the third-order dispersion differ by an order of magnitude
in the different regions. Thus, these regions allow us to
observe the effects of different fibre parameters on the pulse shaping.For these
regions, we determined the maximum and minimum values of Λ, in accordance with the
limits that absolute value of the dispersion is lower than −40 ps/nm/km (as
this level of dispersion can be acceptable in communications applications),
according to Figure3. We next examined the effect of the confinement loss for
each region. Figure 5 shows plots of the dispersion and confinement loss versus
pitch (Λ) for a fixed d/Λ = 0.2. Similarly, Figures 6 and 7 show the results
for d/Λ = 0.3 and d/Λ = 0.8, respectively.Dispersion and confinement loss versus pitch, whend/Λ = 0.2.
(a)(b)Dispersion and  confinement loss versus pitch, whend/Λ = 0.3.
(a)(b)Dispersion and confinement loss versus pitch, whend/Λ = 0.8.
(a)(b)From Figures5–7, we can see that although the dispersion is limited to the absolute value
of 40 ps/nm/km, the confinement loss changes by several orders of magnitude for
different d/Λ values, being the lowest for d/Λ = 0.8 and the highest for d/Λ = 0.2 (as expected). The
results obtained for d/Λ = 0.2 show that although the
dispersion is low, the confinement loss of up to 100 dB/m is the crucial factor
making this region useless for practical experiments. Therefore, we will
concentrate on the regions of d/Λ = 0.3 and d/Λ = 0.8, as both the
dispersion and the confinement loss lie within acceptable limits.From Figure6, we choose
the parameters of the first taper, with starting pitch (1.85 μm) and finishing pitch (2.2 μm) In the same manner, from Figure 7, we chose
parameters of the second taper, the starting pitch (1.18 μm) and finishing pitch (1.22 μm). Even knowing the starting and finishing
values of the dispersion, there is still an infinite range of possible tapers
to consider. For the sake of simplicity, we choose a linear profile for the
taper, determined by fibre pitch at the beginning and the end, from the equation
(5)Λ(z)=(zmax−z)Λ(0)+zΛ(zmax)zmax,
where zmax is
the taper length, Λ(0) = 1.85 μm, and  Λ(zmax) = 2.2 μm. In the normal dispersion regime, smaller
absolute values of dispersion correspond to larger core diameters, and therefore
taper should be used from the narrow end. Other taper profiles are possible
(such as exponentially decreasing profiles that will be studied in a later
paper).The parameters of the MOF that
change with tapering are the chromatic dispersionβ2,
third-order dispersion β3, and nonlinear coefficient γ. The nonlinear coefficient γ is
given by [20](6)γ=2πn2λAeff,
where n2 = 2.2×10−20 m2/W is the nonlinear refractive index of the silica fibre,
Aeff is the effective mode
area, and λ is the
optical wavelength. Note that it is the effective area that changes along the
taper and hence γ does as well. Tables 1 and 2 summarize taper
parameters for regions d/Λ = 0.3 and d/Λ = 0.8,
respectively.Table 1
Parameters of the tapered fibre (d/Λ =  0.3).
Λ [μm]β2 [ps2/m]β3 [ps3/m]γ
[W−1km−11]Start1.850.050367−0.00018864.135End2.20.001167−0.00007614.6Table 2
Parameters
of the tapered fibre (d/Λ =  0.8).
Λ
[μm]β2 [ps2/m]β3 [ps3/m]γ [W−11km−11]Start1.180.0436−0.00113145.1End1.220.00031−0.000911844.1The final linear taper profile and change of
parametersβ2, β3,
and γ with
the distance are shown in Figures 8 and 9 for d/Λ =  0.3 and d/Λ =  0.8, respectively.(a)β2 and β3 (- -) versus distance, (b) γ versus distance,
and (c) pitch (Λ) versus distance, when d/Λ =  0.3.
(a)(b)(c)(a)β2 and β3 (- -) versus distance, (b) γ versus
distance, and (c) pitch (Λ) versus distance, when d/Λ =  0.8.
(a)(b)(c)


## 3. Simulations and Discussion

Having chosen the fibre tapers, the next step is to examine optical propagation
through them. The propagation of pulses in an optical fibre with variable
dispersion and nonlinearity is described by nonlinear Schrödinger equation (NLSE) [23]:(7)∂u∂z+α2u+iβ2(z)2!∂2u∂t2−β3(z)3!∂3u∂t3−iγ(z)|u|2u=0,
where u is the complex electric field envelope, z is the distance along the fibre, t is
retarded time and is defined such that for any distance z along the fibre, the centre of the pulse is at t = 0, α is
fibre loss, β2 is group velocity dispersion, β3 is third-order dispersion, and γ is the nonlinear coefficient. We solved the
NLSE (7) numerically, using the symmetrized split-step Fourier method [23, 25],
for our two tapers. In order to quantify the evolution towards parabolic pulse,
we computed the evolution of the misfit parameter M2 between the pulse intensity profile |u|2 and the parabolic fit |p|2, using [7](8)M2=∫[|u|2−|p|2]2dτ∫|u|4dτ,
where p(t) is the generalized
expression for the parabolic pulse(9)p(t)={Pp1−2t2Tp2exp(−iC2t2),if|t|≤Tp2,0,elsewhere,
where Pp is the peak power of the pulse, Tp is the temporal full-width at half maximum (FWHM), and C is the linear chirp coefficient. We also use a local misfit
parameter which looks at the difference between the pulse and an ideal
parabolic pulse at a particular instant in time.The initial pulse launched into the fibre was a Gaussian pulse with various peak
powers and temporal widths. We choose a taper lengthzmax = 10 m
(this value determines slope of the linear taper profile in (5)), as this slope
of the linear profile gives the best parabolic fit. We propagated the pulse
along the taper from the narrow end, with Λ = 1.85 μm to Λ = 2.2 μm, in the first case, when d/Λ =  0.3.
The range of input pulse peak power was 2 KW to 20 KW, and the range of input
pulse FWHM was 200 femtoseconds to 5 picosecods. The range of input pulse peak
powers and widths ensured nonlinear propagation in all cases and prevented
pulse distortion at higher pulse energies.It has been shown in [7]
that pulse reshaping is possible using nonlinear propagation in a length of a
normally dispersive step index fibre. The best misfit parameter obtained in
that study was 0.033, for a pulse energy of 31 pJ. Thus, for our method to be
useful, we need to improve upon this figure. In fact, we find that for a range
of parameters the minimum misfit parameter for different input energies is less
than 0.002—for the example chosen, when d/Λ =  0.8, it is 0.0015. By carefully adjusting input pulse peak power and width, it is possible
to reach values corresponding to M2 < 0.001,
that is, significantly better than that obtained by a step index fibre.A typical result is presented in Figure10 which shows a plot of the intensity
profile of the output pulse for a taper length of 2.2 m (peak power 2.8 KW and
the pulse FWHM width 4.5 picoseconds) and the best parabolic fit, as well as
plot of the local misfit. The pulse envelope is almost perfectly parabolic,
apart from the pulse edges, as expected. The local misfit parameter is of the
order of 10−14 which should be acceptable for most applications.(a) Pulse envelope (normalize|u|2) against parabolic fit (normalized |p|2), and (b) local misfit parameter, d/Λ =  0.3.
(a)(b)The minimum values ofM2 along the length of the
taper (for d/Λ =  0.3) are shown in Figure 11 assuming no
loss for different initial conditions. Figure 12 shows distance along the taper
where the misfit parameter has its minimum value. Similarly, Figure 13 shows
the result obtained for the same taper with an artificially high-fibre loss
taken into account as α = 0.5 dB/m. Also, Figure 14 shows distances
that correspond to the minimum misfit parameter from Figure 13. Comparing these
two sets of graphs, only minor differences can be seen showing that realistic
fibre losses will not significantly affect the parabolic pulse generation. From Figures 11
and 12, or from Figures
13 and 14, we can extract the taper length that will provide the best
misfit parameter, that is, where the best parabolic pulse is generated. From
the contour plots given, it can be concluded that there is a range of values of
input pulse widths and powers that will enable generation of parabolic pulses.
Minimum value of misfit parameter, from Figure 11, is 0.002, obtained for input
powers of around 3 KW, for different input pulse widths. From Figure 11, it can
be seen that misfit parameter is a function of the peak power whilst it is
mostly independent on the value of the pulse width. The taper length that
corresponds to the chosen pulse peak power and width can be determined from Figure
12. Therefore, from Figures 11 and 12, it is possible to choose values
for input pulse power, width, and the taper length (this choice depends on the
choice of the equipment used in the particular application). For the
illustration purposes, we chose input pulse power, width, and the taper length: P0 = 2.8 KW,
T0 = 4.5 picoseconds, and Zmax = 2.2 m, respectively. Figure 15
shows parabolic pulse evolution for the chosen parameters of the taper, when d/Λ =  0.3
and peak power P0 = 2.8 KW,
FWHM width T0 = 4.5 picoseconds
and the taper length Zmax = 2.2 m.Figure 11
Misfit parameter (minimum)
for input Gaussian pulse:α = 0, d/Λ =  0.3.Figure 12
Taper length where the best (minimum)
misfit parameter is obtained for input Gaussian pulse:α = 0, d/Λ =  0.3.Figure 13
Misfit parameter (minimum) for input Gaussian pulse:α = 0.5 dB/m, d/Λ =  0.3.Figure 14
Taper length where the
best (minimum) misfit parameter is obtained for input Gaussian pulse:α = 0.5 dB/m, d/Λ =  0.3.Figure 15
Parabolic pulse evolution,d/Λ =  0.3.Figure16 shows
the pulse envelope, spectrum, phase, and chirp, for the chosen values. The
spectra broadening factor is calculated as B/B0 = 28 (where B is bandwidth at zmax = 2.2 m,
and B0 is bandwidth, when z = 0 m). From the bottom plot in Figure 16,
which represents instantaneous frequency, or chirp, it can be seen that the generated
parabolic pulse has a linear chirp across it and therefore can be useful for
practical applications. Figure 17 shows the result of the pulse compression
performed numerically (using ideal pulse compression), where initial pulse
full-width at half maximum (FWHM) is 4.5 picoseconds and after compression 310 femtoseconds,
so that a compression ratio (calculated as the ratio
of the FWHM of the input pulse to that of the compressed pulse) of 14.5
is obtained.Pulse envelope, spectrum, phase, and chirp coefficients,d/Λ =  0.3.
(a)(b)(c)(d)Figure 17
Initial pulse envelope,
compressed pulse envelope (•), and parabolic pulse (- -).For the second taper, whend/Λ =  0.8,
as the nonlinearity is much higher than when d/Λ =  0.3, the range of input pulse peak power was 10 W to 1 KW and the range of input
pulse FWHM was 200 femtoseconds to 5 picoseconds. As for the first taper, we
plotted contour plots of the minimum misfit parameter along the taper length
(see Figure 18) and the corresponding distances (Figure 19) versus pulse peak
powers and FWHM widths. From
Figures 18 and 19 (as in the previous case), we could extract the
information about the taper length which will provide the best misfit
parameter. Minimum value of misfit parameter, from Figure 18, is 0.002,
obtained for input powers of around 0.1 KW, for different input pulse widths.
The taper length that corresponds to the chosen pulse peak power and width can
be determined from Figure 19. Therefore, from Figures 18
and 19, it is
possible to choose values for input pulse peak power, width, and the taper
length: P0 = 0.1 KW, T0 = 2 picoseconds, and zmax = 2.3 m, respectively. Similarly to the taper, when d/Λ =  0.3, from the contour plots given, it can be concluded that there is a range
of values of input pulse widths and powers that will enable generation of
parabolic pulses.Figure 18
Misfit
parameter (minimum) for input Gaussian pulse,d/Λ =  0.8.Figure 19
Taper length
where the best (minimum) misfit parameter is obtained for input Gaussian
pulse,d/Λ =  0.8.Figure20 shows parabolic pulse evolution for the chosen parameters of the taper, when d/Λ =  0.8 and peak
power P0 = 0.1 KW, FWHM width T0 = 2
picoseconds, and the taper length Z = 2.3 m.Figure 20
Parabolic
pulse evolution,d/Λ =  0.8.Figure21 shows the pulse envelope, spectrum, phase and chirp, for the taper, when d/Λ =  0.8,
and the chosen values for the peak power, FWHM
width, and the taper length. The spectra broadening factor is calculated as B/B0 = 9.25, when d/Λ =  0.8. Result of the numerical pulse
compression is shown in Figure 22. Initial pulse FWHM is 2 picoseconds and pulse is
compressed to 332 femtoseconds, so that compression factor of 6 is obtained for
this case. The sidelobes observed in Figures 17 and 22 are the
theoretical wings of a recompressed parabolic pulse.Pulse envelope, spectrum, phase
and chirp coefficients,d/Λ =  0.8.
(a)(b)(c)(d)Figure 22
Initial pulse envelope, compressed pulse
envelope (•), and parabolic pulse (- -).In Figure23, for d/Λ =  0.8, we present the pulse envelope at the
taper length 2.3 m, P0 = 0.1 KW,
and T0 = 2 picoseconds and
at the bottom plot, the misfit parameter or the error between the pulse
intensity profile and the parabolic fit calculated using (8). It can be seen
that the pulse exhibits the parabolic intensity profile, but with the small
discrepancy in the wings of the pulse. By comparison of Figures 23 and 10,
it can be observed that the minimum misfit parameter has a lower value when d/Λ =  0.8,
so the intensity profile of the pulse for the taper when d/Λ =  0.8
deviates less from the parabolic shape in comparison to the case, when d/Λ =  0.3,
although it deviates slightly more at the top of the pulse. We anticipate that
this asymmetric deviation from the parabolic shape is due to the value of the
third-order dispersion, which is nearly one order of magnitude higher, when d/Λ =  0.8,
in comparison to the case d/Λ =  0.3. Also, the higher power of the pulse,
when d/Λ =  0.3, is another reason for this greater
misfit.(a) Pulse envelope (normalized|u|2) against
parabolic fit (normalized |p|2), and (b) local misfit parameter, d/Λ =  0.8.
(a)(b)


## 4. Conclusion

We have presented results of
modelling and simulation of a microstructured optical fibre taper, where the
parameters of the pitch and the air-filling factor are determined so that the
normal dispersion regime is obtained. For two different values of the
air-filling factor 0.3 and 0.8, two different tapers with a linear profile are
proposed. In the case whend/Λ = 0.3, the confinement loss is non-negligible, so it has been taken into
consideration. Results show that a Gaussian input pulse of different peak powers and widths can
evolve into the parabolic shape with a linear chirp coefficient in both cases. The
two different spectra broadening factors are calculated for two different
cases, 28 when d/Λ =  0.3 and 9.25 when d/Λ = 0.8. Also, results show that using the
first taper, when d/Λ = 0.3, linearly chirped
parabolic pulse of FWHM width of 4.5 picoseconds can be compressed down to 310 femtoseconds,
so that compression factor of 14.5 is achieved, and for the second taper, when d/Λ =  0.8, initial pulse FWHM width of 2 picoseconds is
compressed to 332 femtoseconds, so that compression factor 6 is obtained. The optimal taper length, to
achieve the best parabolic characteristic for a range of input powers and pulse
widths, has also been determined. Results show that due to the different nonlinearity parameters for the two tapers under consideration,
the best parabolic fit can be achieved for different pulse energies. When d/Λ =  0.3, minimum misfit
parameter is achieved for pulse energy in range from 10 nJ to 24 nJ, while when d/Λ =  0.8, due to the higher
nonlinearity coefficient, the pulse energy range that gives minimum misfit
parameter is from 177 pJ to 440 pJ. Therefore, we can conclude that these results
might be of interest where pulse reshaping is conditioned by the initial pulse
energy availability. Based on the presented procedure, optimization can be
performed to find the best possible taper profile and length for the parabolic
pulse generation.



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*Source: 480362-2008-02-26.xml*