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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 23 — Nov. 8, 2010
  • pp: 24060–24069
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Method for pulse transformations using dispersion varying optical fibre tapers

N. G. R. Broderick  »View Author Affiliations


Optics Express, Vol. 18, Issue 23, pp. 24060-24069 (2010)
http://dx.doi.org/10.1364/OE.18.024060


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Abstract

I introduce the problem of transforming one optical pulse into another via nonlinear propagation in a length of dispersion varying optical fibre. Then using a genetic algorithm to design the dispersion profiles, I show that the problem can be solved leading to high quality pulse transforms that are significantly better than what has been published previously. Finally I suggestion further work and other applications for this method.

© 2010 Optical Society of America

1. Introduction

Optical pulse propagation in an optical fibre can be well described by the Nonlinear Schrödinger equation (NLSE) given by [4

4. A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers .1. anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973). [CrossRef]

]:
iψzβ222ψt2+iα2ψ+γ|ψ|2ψ=0,
(1)
where ψ(z, t) is the slowly varying envelope of the electric field, β2 represents the dispersion of the fibre, α is the loss and γ represents the usual Kerr nonlinearity. In a standard optical fibre the coefficients of the NLSE are all constant, however it is easy to vary the the size of the dispersion through fibre tapering where the diameter of the fibre changes by a relatively small amount (typically less than 10%). Using tapered fibres can dramatically alter the optical properties of pulses propagating through them in the nonlinear regime as is well known from numerous experimental and theoretical studies.

There are in the literature several methods for making shallow tapers for controlling the dispersion of optical fibres depending on the required length of the taper. The controlled fabrication of kilometre lengths of dispersion varying fibres is possible by varying the exit speed of the fibre on the draw tower itself [5

5. N. Broderick, D. Richardson, and L. Dong, “Distributed dispersion measurements and control within continuously varying dispersion tapered fibers,” IEEE Photon. Technol. Lett. 9, 1511–1513 (1997). [CrossRef]

, 6

6. S. Chernikov and P. Mamyshev, “Femtosecond soliton propagation in fibers with slowly decreasing dispersion,” J. Opt. Soc. of Am. B 8, 1633–1641 (1991). [CrossRef]

] while more recently Vukovic et al. [7

7. N. Vukovic, N. G. R. Broderick, M. Petrovich, and G. Brambilla, “Novel method for the fabrication of long optical tapers,” IEEE Photon. Technol. Lett. 20, 1264–1266 (2008). [CrossRef]

] described a novel taper rig capable of producing metre length fibre tapers with slowly varying diameters. Similarly, Sumetsky et al. [8

8. M. Sumetsky, Y. Dulashko, and S. Ghalmi, “Fabrication of miniature optical fiber and microfiber coils,” Opt. Laser Eng. 48, 272–275 (2010). [CrossRef]

] published a different design for making metre length fibre tapers. The experiments and modelling by Vukovic et al. [9

9. N. Vukovic, F. Parmigiani, A. Camerlingo, M. Petrovich, P. Petropoulos, and N. G. R. Broderick, “Experimental investigation of a parabolic pulse generation using tapered microstructured optical fibres,” Proc. SPIE, Photonics Europe 2010 (2010).

, 10

10. N. Vukovic and N. Broderick, “Improved flatness of a supercontinuum at 1.55 microns in tapered microstructured optical fibres,” Phys. Rev. A. (submitted) (2010). [CrossRef]

] suggest that it is possible to vary the dispersion of a highly nonlinear photonic crystal fibre between ±150ps/nm/km and one of the goals of this work is to consider what pulse transformations are possible within such limitations.

2. Theoretical Model

The aim of this work is to demonstrate that a carefully designed fibre taper can transform an initial pulse shape ϕ1(t) into a desired pulse shape ϕ2(t). In order to model propagation along an optical fibre taper the standard NLSE [Eq. (1)] needs to be modified to include a position dependent dispersion function β2(z). It is also worth recalling that the NLSE has two well known scaling transformations, the first for an arbitrary time T0 is given by
ttT0,andβ2(z)β2(z)T02.
(2)

While the amplitude and nonlinearity can be scaled by an arbitrary power factor P0
γγP02,andψP0ψ.
(3)

These two scalings means that I can set the initial pulse width to unity and set gamma to unity without loss of generality. Importantly the physical dimensions of dispersion are [time2]/[distance] and I discuss in the conclusion what the best possible scalings are for the various pulse transformations.

For the normalised NLSE the initial pulse shape is given by ψ(0,t) = ϕ1(t) while the output from the fibre taper is ψ(L,t) where L is the length of the taper. The difference between the taper output and the desired pulse shape ϕ2(t) is given by the misfit parameter [11

11. A. Peacock, N. Broderick, and T. Monro, “Numerical study of parabolic pulse generation in microstructured fibre Raman amplifiers,” Opt. Commun. 218, 167–172 (2003). [CrossRef]

]:
M=(|ψ(L,t)|2|ϕ2(t)|2)2dt|ψ(L,t)|4dt.
(4)

This is not the usual L2 norm for functions but rather it is usual norm weighted by the square of the pulse energy at the output. The primary reason for choosing this misfit function is that it was used previously by C. Finot et al. to characterise the evolution of a pulse into a parabolic shape and so I can compare my results with the earlier published results. The main effect of the weighting is to reduce the misfit for larger pulses since they contain more energy. Note that the smaller the misfit the better the global match between the output pulse and the desired pulse shape.

For arbitrary pulse shapes ϕ1,2(t) it is in general unknown whether there is a dispersion profile that will exactly transform ϕ1 into ϕ2 (clearly some transformations are impossible such as when ϕ2 contains more energy than ϕ1). Furthermore an analytic approach is impossible and so numerical techniques must be used. In the subsequent section I discuss the numerical method used and then proceed to discuss the results for a variety of different pulse shapes.

2.1. Genetic Algorithms for Pulse Shape Optimisation

An advantage of writing the dispersion as a sum of polynomials is that the resulting profile is relatively smooth and so can be fabricated using standard tapering techniques. An alternative approach would be to discretise the function β2 into N discrete values along the length. This disadvantage of this is that at the interface between any two values the dispersion will change discontinuously which is hard to implement except through splicing bits of fibre together which is time-consuming and impractical for small lengths of fibre. This approach would however model the use of comb-like dispersion profiles used by some researchers [12

12. C. Billet, P. Lacourt, R. Ferriere, L. Larger, and J. Dudley, “Parabolic pulse generation in comb-like profiled dispersion decreasing fibre,” Electron. Lett. 42, 965–966 (2006). [CrossRef]

] for parabolic pulse generation.

3. Results

In the cases studied below I usually restricted the input pulse ϕ1 to a Gaussian pulse given by
ϕ1(t)=5et2.
(6)

The choice of a fixed input pulse corresponds to a common situation in many labs, namely that the number of choices of mode-locked lasers is limited and it is desirable to transform the output of the fixed laser source to an arbitrary pulse shape. I have however included a couple of different input pulse shapes in the results to show that the success of this method does not depend critically on the input pulse shape. In the subsequent sections I discuss the results of transforming the input Gaussian pulse into a Parabolic, Sech or square shaped pulse.

3.1. Generating Parabolic Pulses

The first problem I consider is the generation of parabolic pulses using a Gaussian input. The desired output pulse ϕ2(t) was a parabolic pulse with the form:
ϕ2(t)={a(1(t/b)2)|t|<b0otherwise.
(7)

The reason for considering this problem is that parabolic pulse generation is well studied (see the recent review by C. Finot et al. [13

13. C. Finot, J. M. Dudley, B. Kibler, D. J. Richardson, and G. Millot, “Optical parabolic pulse generation and applications,” IEEE J. Quantum Electron. 45, 1482–1489 (2009). [CrossRef]

] for more details) and hence there are a number of different solutions available in the literature. Another useful feature of this problem is that in fibres with a constant gain, parabolic pulses are asymptotic solutions to the NLSE [14

14. M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84, 6010–6013 (2000). [CrossRef] [PubMed]

] and all pulses will evolve towards a parabolic shape. As the NLSE with decreasing dispersion and no loss is equivalent to a fibre with gain the solution to the pulse transformation problem for an infinite fibre is thus known. Hence I can compare the numerical results for a fibre of finite length to that of the known dispersion profile for an infinite fibre.

Fig. 1 (a) Intensity profile of the output pulse (black line) alone with the best parabolic fit (green line). (b) Dispersion profile (green line) and evolution of the misfit parameter (red dotted line) for the optimum fibre taper.

The optimised dispersion profile processes some interesting features. Firstly compared to the expected monotonically decreasing profile it actually increases in the second half of the fibre before decreasing again towards the end. Surprisingly the misfit parameter also increases slightly in the second half of the taper before decreasing dramatically towards the end. In order to test the genetic algorithm I ran it several times (with different random seeds) and with different numbers of polynomials with the results as shown in Fig. 2(a). Here three optimised dispersion profiles are shown and in all cases the resulting misfit parameter is ≈ 7.2 × 10−7 and does not vary significantly. From Fig. 2(a) it can be seen that in the three cases shown the initial dispersion varies considerably more than the dispersion in the last half of the fibre. This suggests that in order to get the best parabolic pulse it is the dispersion profile in the later stages that is critical while the initial dispersion profile is less important.

Fig. 2 (a) Differing dispersion profiles for different runs of the genetic algorithm. The green and red line are the results when there are 21 different polynomials but with different random seeds. The black line is the case for 31 polynomials. (b) The black line shows the optimised profile for converting a sech shaped pulse into a parabolic pulse while the green line shows the profile for a high intensity Gaussian pulse.

Importantly in all the cases examined the resulting dispersion profile is slowly varying and does not change too much in absolute terms over the length of the taper. This is important for a number of reason. Firstly it means that the dispersion profiles can be fabricated using existing taper rigs and so such dispersion profiles can be practically realised. Secondly from a numerical point of view it shows that the number of terms in the Taylor series expansion [Eq. (5)] can be reduced without altering the result. This would allow the program to run faster or alternatively tackle problems of greater complexity (which I will return to in the discussion).

3.2. Gaussian to Sech transformations

The reason behind looking at this transformation is that it is well known that in the lossless regime a Gaussian pulse will evolve into a soliton provided that the energy is sufficient. Hence again we can compare the optimised solution for a fibre of finite length with the constant dispersion profile expected for the lossless infinitely long case. The best results for this transformation are shown in Fig. 3.

Fig. 3 (a) Intensity profile of the output pulse (black line) alone with the best sech shaped fit (green line). (b) Dispersion profile (green line) and evolution of the misfit parameter (red dotted line) for the optimum fibre taper.

In this case the input pulse was again a Gaussian pulse with unit width and a peak intensity of 25. After optimisation the best misfit parameter was 2.6 × 10−6 resulting in a extremely good match between desired and actual pulse shapes. It can be seen from Fig. 3(a) that the resulting pulse is well match down to the −20dB level. Figure 3(b) shows the best dispersion profile (in green) and surprisingly the dispersion decreases monotonically from −1.8 to −8.8. This increase in the magnitude of the dispersion during propagation is what drives the increase in the pulse width which nearly doubles during propagation. The evolution of the misfit parameter is also interesting since in undergoes three large oscillations before reaching the final value.

3.3. Generating Square Pulses

The last problem I consider is generating square pulses from Gaussian pulses. This problem is chosen as it illustrates some of the limitations of using tapers for pulse transformations. The desired output pulse ϕ2(t) is given by:
ϕ2(t)={a|t|<b0otherwise.
(9)

As with the parabolic pulse a notable feature of a square pulse is that it has an extremely broad spectrum due to the discontinuities in the intensity. These discontinuities are however more significant since they occur at points of maximum amplitude rather than at points of zero intensity for the parabolic pulse. As before the input was a Gaussian pulse given by Eq. (6) and the results are shown in Fig. 4.

Fig. 4 (a) Intensity profile of the output pulse (black line) alone with the best square shaped fit (green line). (b) Dispersion profile (green line) and evolution of the misfit parameter (red dotted line) for the optimum fibre taper.

In order to further examine the effects of the discontinuities on the performance of the pulse transformation I examined the evolution towards a super-Gaussian pulse given by ϕ2(t) = aexp(−(t/b)12). These results are shown in Fig. 5 and show a much better fit than the for the square pulse. Note that in the wings the resulting pulse does not decay as fast as the super-Gaussian and this is again due to the limited bandwidth that can be created during propagation. However it does suggest that the smoother the desired output pulse the smaller the optimum misfit for a given input pulse.

Fig. 5 (a) Intensity profile of the output pulse (black line) alone with the best super-Gaussian shaped fit (green line). (b) Dispersion profile (green line) and evolution of the misfit parameter (red dotted line) for the optimum fibre taper.

4. Discussion and Conclusions

In this work I have used a normalised version of the nonlinear Schrödinger equation and it is worth examining how the results scale for realistic parameters. For the parabolic pulse generation the dispersion decreases from ≈ 1.5 to zero. Assuming a one metre long fibre taper and a picosecond pulse this corresponds to a value of β2 = 1500ps2/km which is significantly higher than what can be easily achieved using either conventional or even highly dispersive fibres (such as small core photonic crystal fibres). However there are two ways around this problem, firstly if we assume that instead of a metre long fibre taper we have a kilometre long fibre taper the starting dispersion becomes β2 = 1.5ps2/km which is similar to standard commercial dispersion shifted fibres and can easily be fabricated. Here the loss used becomes 0.2km−1 which is a realistic value for commercial step index fibres. In the past similar kilometre long tapers have been fabricated by changing the speed while drawing the fibres and so this is a practical method for generating parabolic pulses. Alternatively if the initial pulse width is 100fs rather than 1ps the dispersion scaling of the NLSE means that the required starting dispersion for a one metre long fibre taper becomes β2 = 15ps2/km which can be easily achieved using either photonic crystal fibres or standard step index fibres. These values are typical of the dispersion profiles for the other pulse shapes and suggest that this method can be used either with picosecond pulses and kilometre long fibre tapers or with 100 femtosecond pulses and metre length fibre tapers. An additional attraction of using 100 femtosecond pulses over picosecond pulses is that the scale of the deliberate dispersion variations are significantly higher than the intrinsic flucations in the fibre dispersion and also over a one metre length the uniformity of the fibre’s dispersion is much better than it would be for a kilometre length.

In this work I have ignored any higher order corrections to the NLSE such as third order dispersion or the Raman effect. Such effects are unlikely to be important for picosecond pulses however for femtosecond pulses such effects will be more important and will be considered in future work. In addition trying to optimised tapers for effects such as super-continuum generation will require inclusion of higher order dispersion as well as the Raman effect and self-steepening. Future work will also look at modelling the geometric fibre parameters rather than the dispersion profile using the genetic algorithm. For example a photonic crystal fibre can be described by two parameters, the hole size d and the hole to hole spacing Λ [17

17. T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, “Holey Optical Fibres: An efficient Modal Model,” J. Lightwave Technol. 17(6), 1093 (1999). [CrossRef]

], while a step index fibre is characterised by the refractive index difference Δn and the core diameter d. In a more realistic genetic algorithm both parameters would be separately optimised and the algorithm would calculate the dispersion (to all orders) which would be used in the generalised NLSE. While this doubles the number of dimensions of the problem using 15 terms in the Taylor series for each parameter rather than 30 as done in this work would keep the size of the simulation the same without reducing the accuracy of the results. An additional advantage of this approach would be that it would be possible to include the variation of the effective area in the simulations as this will change the effective nonlinearity along the length of the fibre. For the cases studied here such effects are unlikely to be significant but for effects such as super-continuum generation the change in effective area would need to be considered in the model. Finally in this work I have only looked at the intensity of the output pulse rather than it’s phase profile and clearly in the future I could look at the optimising the phase profile as well as the intensity profile.

In conclusion I shown various examples of how one pulse shape can be transformed into another during propagation down an optical fibre taper. In all cases considered the misfit parameter that described the success of the transform was extremely low showing that this method is extremely flexible. The major area where this method fails is where the desired pulse shape changes discontinuously such as in the case of the square pulse or the wings of the parabolic pulse. In all cases examined the genetic algorithm produced slowly varying tapers that can be fabricated using existing technology. Compared to linear filtering techniques for pulse shaping nonlinear pulse transformations can increase the pulse’s frequency bandwidth allowing pulse compression simultaneously with pulse transformations which is an attractive feature of this method that could be exploited further in the future.

Acknowledgments

The author gratefully acknowledges Dr. Peter Horak, for many helpful conversations. The computing resources were provided by iSolutions at the University of Southampton.

References and links

1.

H. Kuehl, “Solitons on an axially nonuniform optical fiber,” J. Opt. Soc. Am. B 5, 709–713 (1988). [CrossRef]

2.

S. V. Chernikov, E. M. Dianov, D. J. Richardson, and D. N. Payne, “Soliton pulse-compression in dispersion-decreasing fiber,” Opt. Lett. 18, 476–478 (1993). [CrossRef] [PubMed]

3.

T. Hirooka and M. Nakazawa, “Parabolic pulse generation by use of a dispersion-decreasing fiber with normal group-velocity dispersion,” Opt. Lett. 29, 498–500 (2004). [CrossRef] [PubMed]

4.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers .1. anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973). [CrossRef]

5.

N. Broderick, D. Richardson, and L. Dong, “Distributed dispersion measurements and control within continuously varying dispersion tapered fibers,” IEEE Photon. Technol. Lett. 9, 1511–1513 (1997). [CrossRef]

6.

S. Chernikov and P. Mamyshev, “Femtosecond soliton propagation in fibers with slowly decreasing dispersion,” J. Opt. Soc. of Am. B 8, 1633–1641 (1991). [CrossRef]

7.

N. Vukovic, N. G. R. Broderick, M. Petrovich, and G. Brambilla, “Novel method for the fabrication of long optical tapers,” IEEE Photon. Technol. Lett. 20, 1264–1266 (2008). [CrossRef]

8.

M. Sumetsky, Y. Dulashko, and S. Ghalmi, “Fabrication of miniature optical fiber and microfiber coils,” Opt. Laser Eng. 48, 272–275 (2010). [CrossRef]

9.

N. Vukovic, F. Parmigiani, A. Camerlingo, M. Petrovich, P. Petropoulos, and N. G. R. Broderick, “Experimental investigation of a parabolic pulse generation using tapered microstructured optical fibres,” Proc. SPIE, Photonics Europe 2010 (2010).

10.

N. Vukovic and N. Broderick, “Improved flatness of a supercontinuum at 1.55 microns in tapered microstructured optical fibres,” Phys. Rev. A. (submitted) (2010). [CrossRef]

11.

A. Peacock, N. Broderick, and T. Monro, “Numerical study of parabolic pulse generation in microstructured fibre Raman amplifiers,” Opt. Commun. 218, 167–172 (2003). [CrossRef]

12.

C. Billet, P. Lacourt, R. Ferriere, L. Larger, and J. Dudley, “Parabolic pulse generation in comb-like profiled dispersion decreasing fibre,” Electron. Lett. 42, 965–966 (2006). [CrossRef]

13.

C. Finot, J. M. Dudley, B. Kibler, D. J. Richardson, and G. Millot, “Optical parabolic pulse generation and applications,” IEEE J. Quantum Electron. 45, 1482–1489 (2009). [CrossRef]

14.

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84, 6010–6013 (2000). [CrossRef] [PubMed]

15.

N. T. Vukovic and N. G. R. Broderick, “Parabolic pulse generation using tapered microstructured optical fibres,” Adv. Non. Opt. (2008).

16.

C. Finot, L. Provost, P. Petropoulos, and D. Richardson, “Parabolic pulse generation through passive nonlinear pulse reshaping in a normally dispersive two segment fiber device,” Opt. Express 15, 852–864 (2007). [CrossRef] [PubMed]

17.

T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, “Holey Optical Fibres: An efficient Modal Model,” J. Lightwave Technol. 17(6), 1093 (1999). [CrossRef]

OCIS Codes
(060.2280) Fiber optics and optical communications : Fiber design and fabrication
(190.4370) Nonlinear optics : Nonlinear optics, fibers

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: July 12, 2010
Revised Manuscript: September 8, 2010
Manuscript Accepted: September 17, 2010
Published: November 3, 2010

Citation
N. G. R. Broderick, "Method for pulse transformations using dispersion varying optical fibre tapers," Opt. Express 18, 24060-24069 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-23-24060


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References

  1. H. Kuehl, "Solitons on an axially nonuniform optical fiber," J. Opt. Soc. Am. B 5, 709-713 (1988). [CrossRef]
  2. S. V. Chernikov, E. M. Dianov, D. J. Richardson, and D. N. Payne, "Soliton pulse-compression in dispersion-decreasing fiber," Opt. Lett. 18, 476-478 (1993). [CrossRef] [PubMed]
  3. T. Hirooka, and M. Nakazawa, "Parabolic pulse generation by use of a dispersion-decreasing fiber with normal group-velocity dispersion," Opt. Lett. 29, 498-500 (2004). [CrossRef] [PubMed]
  4. A. Hasegawa, and F. Tappert, "Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. 1. anomalous dispersion," Appl. Phys. Lett. 23, 142-144 (1973). [CrossRef]
  5. N. Broderick, D. Richardson, and L. Dong, "Distributed dispersion measurements and control within continuously varying dispersion tapered fibers," IEEE Photon. Technol. Lett. 9, 1511-1513 (1997). [CrossRef]
  6. S. Chernikov, and P. Mamyshev, "Femtosecond soliton propagation in fibers with slowly decreasing dispersion," J. Opt. Soc. Am. B 8, 1633-1641 (1991). [CrossRef]
  7. N. Vukovic, N. G. R. Broderick, M. Petrovich, and G. Brambilla, "Novel method for the fabrication of long optical tapers," IEEE Photon. Technol. Lett. 20, 1264-1266 (2008). [CrossRef]
  8. M. Sumetsky, Y. Dulashko, and S. Ghalmi, "Fabrication of miniature optical fiber and microfiber coils," Opt. Lasers Eng. 48, 272-275 (2010). [CrossRef]
  9. N. Vukovic, F. Parmigiani, A. Camerlingo, M. Petrovich, P. Petropoulos, and N. G. R. Broderick, "Experimental investigation of a parabolic pulse generation using tapered microstructured optical fibres," Proc. SPIE, Photonics Europe 2010 (2010).
  10. N. Vukovic, and N. Broderick, "Improved flatness of a supercontinuum at 1.55 microns in tapered microstructured optical fibres," Phys. Rev. A. submitted. [CrossRef]
  11. A. Peacock, N. Broderick, and T. Monro, "Numerical study of parabolic pulse generation in microstructured fibre Raman amplifiers," Opt. Commun. 218, 167-172 (2003). [CrossRef]
  12. C. Billet, P. Lacourt, R. Ferriere, L. Larger, and J. Dudley, "Parabolic pulse generation in comb-like profiled dispersion decreasing fibre," Electron. Lett. 42, 965-966 (2006). [CrossRef]
  13. C. Finot, J. M. Dudley, B. Kibler, D. J. Richardson, and G. Millot, "Optical parabolic pulse generation and applications," IEEE J. Quantum Electron. 45, 1482-1489 (2009). [CrossRef]
  14. M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, "Self-similar propagation and amplification of parabolic pulses in optical fibers," Phys. Rev. Lett. 84, 6010-6013 (2000). [CrossRef] [PubMed]
  15. N. T. Vukovic, and N. G. R. Broderick, "Parabolic pulse generation using tapered microstructured optical fibres," Adv. Non. Opt. (2008).
  16. C. Finot, L. Provost, P. Petropoulos, and D. Richardson, "Parabolic pulse generation through passive nonlinear pulse reshaping in a normally dispersive two segment fiber device," Opt. Express 15, 852-864 (2007). [CrossRef] [PubMed]
  17. T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, "Holey Optical Fibres: An efficient Modal Model," J. Lightwave Technol. 17(6), 1093 (1999). [CrossRef]

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