## Method for pulse transformations using dispersion varying optical fibre tapers |

Optics Express, Vol. 18, Issue 23, pp. 24060-24069 (2010)

http://dx.doi.org/10.1364/OE.18.024060

Acrobat PDF (771 KB)

### 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

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]

*ψ*(

*z*,

*t*) is the slowly varying envelope of the electric field,

*β*

_{2}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.

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]

*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]

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]

## 2. Theoretical Model

*ϕ*

_{1}(

*t*) into a desired pulse shape

*ϕ*

_{2}(

*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

*β*

_{2}(

*z*). It is also worth recalling that the NLSE has two well known scaling transformations, the first for an arbitrary time

*T*

_{0}is given by

*P*

_{0}

^{2}]/[distance] and I discuss in the conclusion what the best possible scalings are for the various pulse transformations.

*ψ*(0,

*t*) =

*ϕ*

_{1}(

*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

*ϕ*

_{2}(

*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]

*L*

_{2}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.

*ϕ*

_{1,2}(

*t*) it is in general unknown whether there is a dispersion profile that will exactly transform

*ϕ*

_{1}into

*ϕ*

_{2}(clearly some transformations are impossible such as when

*ϕ*

_{2}contains more energy than

*ϕ*

_{1}). 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

*β*

_{2}(

*z*) is an arbitrary function. Hence I need to impose some constraints. Firstly I assume that the length of the taper is fixed and can be set to unity. Given the possible rescalings of the NLSE this restriction can be seen as fixing either the actual physical length of the fibre, or the pulse width or the value of the nonlinearity (or some combination of all three). Next I approximate the dispersion using a Taylor series approximation although any other set of basis functions such as Chebyshev polynomials or trigonometric functions can be used. In this way the problem is reduced from having an infinite number of degrees of freedom to a finite number

*N*. As long as

*N*is sufficiently large the precise value of

*N*is relatively unimportant as I will show later. In practice I have found that between 10 and 30 terms are sufficient (or in fact overkill).

*β*

_{2}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]

*β*

_{2}given in Eq. (5) I implemented a genetic algorithm to find the optimum coefficients

*a*. In the algorithm each individual had a genome consisting of a list of the coefficients

_{i}*a*and the initial population of size

_{i}*M*was created using a uniform distribution of random numbers. To get each subsequent generation a mixture of asexual and sexual reproduction was used. Firstly the top

*N*

_{1}individuals as ranked by the misfit function [Eq. (4)] were each cloned

*C*times and a Gaussian random variable was added. Next two individuals where chosen at random and a new individual was created with

*m*genes from the first parent and

*N*–

*m*from the second with a Gaussian distributed random number being added to each coefficient. All of the new individuals were then ranked along with the old generation via their misfit function and only the top

*M*individuals were kept. The algorithm was implemented on the IRIDIS supercomputer at the University of Southampton using the OpenMPI message passing implementation. Typically for a population size of 10000 and a maximum number of generations of 20000 the program took about 5 hours to run using 360 processors and scaled as expected with the number of processors. Finally it should be noted that the results presented here do not depend on the optimisation algorithm used and while I used a genetic algorithm other approachs should give the same result.

## 3. Results

*ϕ*

_{1}to a Gaussian pulse given by

### 3.1. Generating Parabolic Pulses

*ϕ*

_{2}(

*t*) was a parabolic pulse with the form:

*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]

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]

*γ*= 1 and

*α*= 0.2. The input in the first instance was a Gaussian pulse as given by Eq. (6). As I am only interested in determining how to create the most parabolic pulse the parameters

*a*and

*b*in Eq. (7) were found using a least squares fit that minimised the misfit function. Using the genetic algorithm for an input power of 25 we obtained the results as shown in Fig. 1(a). Here the output pulse is shown in black while the green line shows the best parabolic fit to the pulse. The misfit parameter was 7.1052 × 10

^{−7}which is an improvement by several orders of magnitude over previous results [15, 16

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]

^{−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.

*β*

_{2}(

*z*) for two other input pulse shapes and these results are shown in Fig. 2(b). The first was for a sech shaped pulse with initial intensity of 25 and a width of 1, the optimised dispersion profile for this case is shown in black. The other case was for a Gaussian pulse with a peak intensity of 50 (rather than 25 in the previous examples) and the optimised profile is shown in green. In both cases the dispersion profile follows the same pattern as previously, i.e. there is a decreasing dispersion over the first half of the taper followed by a small hump. In these examples the misfit parameter was similar to those found earlier but was higher for the sech shaped pulse. This is not surprising as a sech shaped pulse is further from a parabolic pulse to begin with and hence it is perhaps harder to transform. Comparing the results for Gaussian pulses of different amplitudes while the global misfit parameter is very similar the local differences are more striking. For the low intensity pulse there is an excellent fit at the centre of the pulse while the error near the base is higher since the nonlinearity is not sufficient to generate enough bandwidth to copy the discontinuity of the intensity at the base of the parabolic pulse. In contrast for the high intensity pulse the error near the central part of the pulse is greater but local error near the base is lower.

### 3.2. Gaussian to Sech transformations

*ϕ*

_{2}(

*t*) given by :

^{−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

*ϕ*

_{2}(

*t*) is given by:

*ϕ*

_{2}(

*t*) =

*a*exp(−(

*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.

## 4. Discussion and Conclusions

*β*

_{2}= 1500ps

^{2}/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

*β*

_{2}= 1.5ps

^{2}/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

*β*

_{2}= 15ps

^{2}/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.

*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]

*n*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.

## Acknowledgments

## References and links

1. | H. Kuehl, “Solitons on an axially nonuniform optical fiber,” J. Opt. Soc. Am. B |

2. | S. V. Chernikov, E. M. Dianov, D. J. Richardson, and D. N. Payne, “Soliton pulse-compression in dispersion-decreasing fiber,” Opt. Lett. |

3. | T. Hirooka and M. Nakazawa, “Parabolic pulse generation by use of a dispersion-decreasing fiber with normal group-velocity dispersion,” Opt. Lett. |

4. | A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers .1. anomalous dispersion,” Appl. Phys. Lett. |

5. | N. Broderick, D. Richardson, and L. Dong, “Distributed dispersion measurements and control within continuously varying dispersion tapered fibers,” IEEE Photon. Technol. Lett. |

6. | S. Chernikov and P. Mamyshev, “Femtosecond soliton propagation in fibers with slowly decreasing dispersion,” J. Opt. Soc. of Am. B |

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. |

8. | M. Sumetsky, Y. Dulashko, and S. Ghalmi, “Fabrication of miniature optical fiber and microfiber coils,” Opt. Laser Eng. |

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. |

12. | C. Billet, P. Lacourt, R. Ferriere, L. Larger, and J. Dudley, “Parabolic pulse generation in comb-like profiled dispersion decreasing fibre,” Electron. Lett. |

13. | C. Finot, J. M. Dudley, B. Kibler, D. J. Richardson, and G. Millot, “Optical parabolic pulse generation and applications,” IEEE J. Quantum Electron. |

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. |

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 |

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. |

**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

- H. Kuehl, "Solitons on an axially nonuniform optical fiber," J. Opt. Soc. Am. B 5, 709-713 (1988). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- S. Chernikov, and P. Mamyshev, "Femtosecond soliton propagation in fibers with slowly decreasing dispersion," J. Opt. Soc. Am. B 8, 1633-1641 (1991). [CrossRef]
- 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]
- M. Sumetsky, Y. Dulashko, and S. Ghalmi, "Fabrication of miniature optical fiber and microfiber coils," Opt. Lasers Eng. 48, 272-275 (2010). [CrossRef]
- 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).
- N. Vukovic, and N. Broderick, "Improved flatness of a supercontinuum at 1.55 microns in tapered microstructured optical fibres," Phys. Rev. A. submitted. [CrossRef]
- 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]
- 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]
- 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]
- 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]
- N. T. Vukovic, and N. G. R. Broderick, "Parabolic pulse generation using tapered microstructured optical fibres," Adv. Non. Opt. (2008).
- 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]
- 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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