## Dispersion-to-spectrum mapping in nonlinear fibers based on optical wave-breaking |

Optics Express, Vol. 21, Issue 23, pp. 28550-28558 (2013)

http://dx.doi.org/10.1364/OE.21.028550

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### Abstract

In this work we recognize new strategies involving optical wave-breaking for controlling the output pulse spectrum in nonlinear fibers. To this end, first we obtain a constant of motion for nonlinear pulse propagation in waveguides derived from the generalized nonlinear Schrödinger equation. In a second phase, using the above conservation law we theoretically analyze how to transfer in a simple manner the group-velocity-dispersion curve of the waveguide to the output spectral profile of pulsed light. Finally, the computation of several output spectra corroborates our proposition.

© 2013 Optical Society of America

## 1. Introduction

1. W. J. Tomlinson, R. H. Stolen, and A. M. Johnson, “Optical wave-breaking in nonlinear optical fibers,” Opt. Lett. **10**, 457–459 (1985). [CrossRef] [PubMed]

2. D. Anderson, M. Desaix, M. Lisak, and M. L. Quiroga-Teixeiro, “Wave breaking in nonlinear-optical fibers,” J. Opt. Soc. Am. B **9**, 1358–1361 (1992). [CrossRef]

*χ*

^{3}susceptibility. Indeed OWB can be understood in the spectral domain as a four-wave mixing (FWM) process that produces two spectral sidelobes [3]. At the same time, the interference of such frequencies results in some temporal ripples near the pulse edges.

4. D. Anderson, M. Desaix, M. Karlsson, M. Lisak, and M. L. Quiroga-Teixeiro, “Wave-breaking-free pulses in nonlinear-optical fibers,” J. Opt. Soc. Am. B **10**, 1185–1190 (1993). [CrossRef]

5. C. Finot, B. Kibler, L. Provost, and S. Wabnitz, “Beneficial impact of wave-breaking for coherent continuum formation in normally dispersive nonlinear fibers,” J. Opt. Soc. Am. B **25**, 1938–1948 (2008). [CrossRef]

6. A. M. Heidt, A. Hartung, G. W. Bosman, P. Krok, E. G. Rohwer, H. Schwoerer, and H. Bartelt, “Coherent octave spanning near-infrared and visible supercontinuum generation in all-normal dispersion photonic crystal fibers,” Opt. Express **19**, 3775–3787 (2011). [CrossRef] [PubMed]

7. Y. Liu, H. Tu, and S. A. Boppart, “Wave-breaking-extended fiber supercontinuum generation for high compression ratio transform-limited pulse compression,” Opt. Lett. **37**, 2172–2174 (2012). [CrossRef] [PubMed]

6. A. M. Heidt, A. Hartung, G. W. Bosman, P. Krok, E. G. Rohwer, H. Schwoerer, and H. Bartelt, “Coherent octave spanning near-infrared and visible supercontinuum generation in all-normal dispersion photonic crystal fibers,” Opt. Express **19**, 3775–3787 (2011). [CrossRef] [PubMed]

1. W. J. Tomlinson, R. H. Stolen, and A. M. Johnson, “Optical wave-breaking in nonlinear optical fibers,” Opt. Lett. **10**, 457–459 (1985). [CrossRef] [PubMed]

8. J. E. Rothenberg, “Femtosecond optical shocks and wave breaking in fiber propagation,” J. Opt. Soc. Am. B **6**, 2392–2401 (1989). [CrossRef]

2. D. Anderson, M. Desaix, M. Lisak, and M. L. Quiroga-Teixeiro, “Wave breaking in nonlinear-optical fibers,” J. Opt. Soc. Am. B **9**, 1358–1361 (1992). [CrossRef]

5. C. Finot, B. Kibler, L. Provost, and S. Wabnitz, “Beneficial impact of wave-breaking for coherent continuum formation in normally dispersive nonlinear fibers,” J. Opt. Soc. Am. B **25**, 1938–1948 (2008). [CrossRef]

5. C. Finot, B. Kibler, L. Provost, and S. Wabnitz, “Beneficial impact of wave-breaking for coherent continuum formation in normally dispersive nonlinear fibers,” J. Opt. Soc. Am. B **25**, 1938–1948 (2008). [CrossRef]

## 2. Nonlinear propagation in optical fibers: a constant of motion

*A*is the complex envelope of the electric field,

*β*=

_{k}*∂*(

^{k}β*ω*)/

*∂ω*|

^{k}_{ω=ω0}, being

*β*(

*ω*) the propagation constant of the mode supported by the waveguide and

*ω*

_{0}the carrier frequency, and

*γ*

_{0}represents the waveguide nonlinear coefficient. Within this framework, and without any additional assumption, in the appendix A we deduce a conservation law that generalizes a previous expression derived by Zakharov and Shabat [9], where

*β*(

_{p}*ω*) =

*β*(

*ω*) −

*β*

_{0}−

*β*

_{1}(

*ω*−

*ω*

_{0}) and

*Ã*is the Fourier transform of

*A*.

*β*(

_{p}*ω*) in Taylor series around

*ω*

_{0}, the second fraction in Eq. (2) can be rewritten as

*μ*is the normalized

_{k}*k*th moment of the pulse spectrum at the baseband. This expression includes the

*β*coefficients, which account for the dispersive effects in Eq. (1). For a fiber far from the zero-dispersion wavelength and assuming a smooth pulse profile during the propagation, the quantity

_{k}*T*

_{0}denotes the temporal width of the input pulse) traditionally estimates the impact of dispersion. Therefore, we also define the function that generalizes the standard amount

*z*.

*C*can be calculated only taking into account the initial conditions. From this point of view, nonlinear pulse propagation can be understood as a competition between the activities of SPM and dispersion.

10. F. Shimizu, “Frequency broadening in liquid by a short light pulse,” Phys. Rev. Lett. **19**, 1097–1100 (1967). [CrossRef]

*z*according to Eq. (5). Assuming that

*β*

_{2}

*μ*

_{2}/2 is the dominant contribution in

*β*

_{2}> 0 since the pulse spectrum broadens through propagation along the fiber (i.e.,

*∂*

_{z}μ_{2}> 0). Therefore, as is well known, it is crucial to pump at the normal dispersion regime to achieve a smooth output spectrum in conventional fibers [5

**25**, 1938–1948 (2008). [CrossRef]

*z*distance (say,

*z*

_{out}, the output distance). So, based in Eq. (5), in a first-order approximation we can write

2. D. Anderson, M. Desaix, M. Lisak, and M. L. Quiroga-Teixeiro, “Wave breaking in nonlinear-optical fibers,” J. Opt. Soc. Am. B **9**, 1358–1361 (1992). [CrossRef]

*z*for which the new nonlinear and dispersive functions of

_{c}*z*intersect each other. In this case we write

*z*we estimate

_{c}*μ*at this distance using the SPM-induced chirp with an equivalent peak pulse power of

_{k}*β*= 0 for

_{k}*k*> 2, and a Gaussian input pulse, we obtain

*z*≈ 9.0 m, which is in close agreement with the abscissa of the intersection point of the curves in this figure. In addition, the above distance is greater than the OWB distance derived in [2

_{c}**9**, 1358–1361 (1992). [CrossRef]

*z*as the OWB distance at which OWB is the dominant nonlinear process at the second stage of the pulse propagation. Unlike the procedure for calculating the OWB distance in [2

_{c}**9**, 1358–1361 (1992). [CrossRef]

*z*for both any dispersion curve and any input pulse profile. In this way, following our criterion, the OWB distance of the system corresponding to Fig. 1(b), with a non constant dispersion, turns to be

_{c}*z*= 5.5 m.

_{c}## 3. Dispersion-to-spectrum mapping: direct spectral shaping through dispersion engineering

*z*we cannot ignore the nonlinear processes involving instantaneous frequencies in the central region of the pulse. In other words, the pulse evolves in the spectral domain through a set of intrapulse FWM processes [11

_{c}11. A. V. Gorbach, D. V. Skryabin, J. M. Stone, and J. C. Knight, “Four-wave mixing of solitons with radiation and quasi-nondispersive wave packets at the short-wavelength edge of a supercontinuum,” Opt. Express **14**, 9854–9863 (2006). [CrossRef] [PubMed]

*t*as where we consider both a linear approximation of the phase

_{k}*φ*and a slow variation of the envelope modulus |

*A*| compared with the phase. Equation (8) defines locally monochromatic waves with angular frequency

*δω*=

*ω*−

*ω*

_{0}= −

*∂*|

_{t}φ_{tk}, power |

*A*(

*t*)|

_{k}^{2}, and phase

*φ*(

*t*). The above statement is on the basis of the physical meaning of the instantaneous frequency [12].

_{k}*z*<

*z*, we consider that only SPM rules the pulse evolution. In this way, we can use the SPM-induced chirp to define the frequency of locally monocromatic waves at

_{c}*z*. Note that the instantaneous power and frequency of every monochromatic wave can be worked out at

_{c}*z*. In particular, for a Gaussian input pulse the power associated at a certain instantaneous frequency is given by where

_{c}*𝒲*is the Lambert function of order

_{l}*l*[13

13. R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert 𝒲 function,” Adv. Comput. Math. **5**, 329–359 (1996). [CrossRef]

*l*= −1 for the tails and

*l*= 0 for the central region of the pulse), and

*z*[see Fig. 2].

_{c}*z*>

*z*, each locally monochromatic wave that is present at

_{c}*z*acts as the pump in multiple degenerate FWM processes with the nearby waves such that 2

_{c}*ω*=

_{p}*ω*+

_{s}*ω*, where the subscripts

_{i}*p*,

*s*and

*i*refer to pump, signal, and idler, respectively. Note that the above frequency mixing occurs at both the central part and the tails of the pulse. This panorama is graphically sketched in Fig. 2. In order to simplify the analysis, we only take into acount processes for which the pump power is much greater than the signal and idler powers (

*P*≫

_{p}*P*,

_{s}*P*) and

_{i}*P*(

_{i}*z*) = 0. In any such case, the production of idler photons for any input pulse shape is given by [14

_{c}14. J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P.-O. Hedekvist, “Fiber based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. **8**, 506–520 (2002). [CrossRef]

*g*|

^{2}= Δ

*β*[Δ

*β*/4 +

*γ*

_{0}

*P*(

_{p}*z*)] is the squared modulus of the gain of a generic elementary process and Δ

_{c}*β*=

*β*(

*ω*) +

_{s}*β*(

*ω*) − 2

_{i}*β*(

*ω*) is the low power propagation mismatch. We point out that the boundary values of the pump frequencies are given by the frequencies at which the pulse chirp reaches its minimum or maximum at

_{p}*z*, as one can see in Fig. 2.

_{c}*z*. This fact is due to the high value that the gain shows around

*z*,

_{c}*z*, we obtain the trend for the power corresponding to the production of idler frequencies, i.e., Finally the total flow from a pump frequency to the rest can be estimated by integrating for all the possible values of

*ω*satisfying the above power requirements (i.e.,

_{s}*P*≫

_{p}*P*,

_{s}*P*), The above expression seems to diverge when

_{i}*ω*≈

_{s}*ω*since Δ

_{p}*β*≈ 0. However the description of those frequencies are, in fact, out of the model depicted by Eq. (10). In addition, it is easy to check, bearing in mind [14

14. J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P.-O. Hedekvist, “Fiber based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. **8**, 506–520 (2002). [CrossRef]

*β*≈ (

*ω*−

_{s}*ω*)

_{p}^{2}

*β*

_{2}(

*ω*), in the lowest-order approximation. In this way, the integrand of the right-hand side in Eq. (12), excluding

_{p}*β*

_{2}(

*ω*) and

_{p}*P*(

_{p}*z*), shows a small variation on

_{c}*ω*and

_{s}*ω*since it is far away from the poles. Actually, Eq. (9) can be used to verify this issue when we deal with a Gaussian input pulse. So, the above statements indicate that the power spreading rate roughly goes as the inverse of the group-velocity dispersion. This fact can be expressed in mathematical terms as This conclusion is in agreement with the fact that supercontinuum spectrum becomes nearly flat when

_{p}*β*

_{2}(

*ω*) is constant [15

15. T. Hori, J. Takayanagi, N. Nishizawa, and T. Goto, “Flatly broadened, wideband and low noise supercontinuum generation in highly nonlinear hybrid fiber,” Opt. Express **12**, 317–324 (2004). [CrossRef] [PubMed]

16. J. J. Miret, E. Silvestre, and P. Andrés, “Octave-spanning ultraflat supercontinuum with soft-glass photonic crystal fibers,” Opt. Express **17**, 9197–9203 (2009). [CrossRef] [PubMed]

*δP*(

_{p}*z*)|, is higher in the regions in which

*P*is greater. So, the power leakage shifts the spectrum power from the high values to the low ones. In this way, if

_{p}*z*

_{out}is large enough, the rapid and nearly-regular oscillations of the spectrum at the OWB distance [3] are mitigated and the output power spectrum,

*S*(

*ω*,

*z*

_{out}), becomes uniform.

*β*

_{2}(

*ω*) is not constant, the above process still operates locally, in such a way that oscillations are also damped. However, the spectral power spreading is stronger when 1/

*β*

_{2}is larger. So, now

*S*(

*ω*,

*z*

_{out}) should adopt the (1/

*β*

_{2})-profile around the carrier frequency. At this point it is important to recognize that the variation of −1/

*β*

_{2}(

*ω*) around the central frequency of the pulse (

*δω*= 0) approximately agrees with that of the function

_{p}*β*

_{2}(

*ω*) itself, except by a negative additive constant. This plausible conclusion is, in addition, consistent to Eq. (7) and can be mathematically expressed as where

_{p}*ℳ*and

*𝒩*are in principle nearly flattened functions of

*ω*and consequently they only account for the fine detail of the spectral shape. Despite of the approximations considered in the derivation of Eq. (14), it retains enough information about the physical processes governing the nonlinear pulse propagation and predicts a clear spectral trend within the regime where it is derived (namely, high nonlinearity and normal dispersion), as it is verified numerically in the next section.

## 4. Numerical results

*z*

_{out}= 20 mm. The rest of fiber and pulse parameters are the same as in Fig. 3(b). In this case we have included higher order effects as self-steepening and intrapulse Raman scattering in the GNLSE for the computation of the nonlinear propagation of such a pulse. The evolution of the functions

## 5. Conclusions

*β*

_{2}(

*ω*), to the power spectrum profile of the output pulse,

*S*(

*ω*), around the carrier frequency. This result has been computationally checked even under conditions that overpass the initial input pulse requirements. We point out that this mapping permits in a very simple way to manipulate the emerging spectrum by dispersion engineering of any nonlinear waveguide in which pulse propagation is described by means of a GNLSE-type equation.

## Appendix: Derivation of the conservation law

*β*(

_{p}*ω*) both sides of Eq. (16) and taking into account Eq. (15), we obtain At this point, if we consider

*Ã*

^{*}(

*ω*) =

*ℱ*(

*A*

^{*}(−

*t*)) and apply the convolution theorem of Fourier theory, we achieve Now, integrating over

*ω*and taking into account

*δ*is the Dirac delta function, we derive Finally, Eq. (2) is obtained considering the conservation energy of these systems. Note that in the particular case

*β*(

_{p}*ω*) = (

*ω*−

*ω*

_{0})

^{2}

*β*

_{2}, we recover one of the conservation laws derived in [9].

## Acknowledgments

## References and links

1. | W. J. Tomlinson, R. H. Stolen, and A. M. Johnson, “Optical wave-breaking in nonlinear optical fibers,” Opt. Lett. |

2. | D. Anderson, M. Desaix, M. Lisak, and M. L. Quiroga-Teixeiro, “Wave breaking in nonlinear-optical fibers,” J. Opt. Soc. Am. B |

3. | G. P. Agrawal, |

4. | D. Anderson, M. Desaix, M. Karlsson, M. Lisak, and M. L. Quiroga-Teixeiro, “Wave-breaking-free pulses in nonlinear-optical fibers,” J. Opt. Soc. Am. B |

5. | C. Finot, B. Kibler, L. Provost, and S. Wabnitz, “Beneficial impact of wave-breaking for coherent continuum formation in normally dispersive nonlinear fibers,” J. Opt. Soc. Am. B |

6. | A. M. Heidt, A. Hartung, G. W. Bosman, P. Krok, E. G. Rohwer, H. Schwoerer, and H. Bartelt, “Coherent octave spanning near-infrared and visible supercontinuum generation in all-normal dispersion photonic crystal fibers,” Opt. Express |

7. | Y. Liu, H. Tu, and S. A. Boppart, “Wave-breaking-extended fiber supercontinuum generation for high compression ratio transform-limited pulse compression,” Opt. Lett. |

8. | J. E. Rothenberg, “Femtosecond optical shocks and wave breaking in fiber propagation,” J. Opt. Soc. Am. B |

9. | V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP |

10. | F. Shimizu, “Frequency broadening in liquid by a short light pulse,” Phys. Rev. Lett. |

11. | A. V. Gorbach, D. V. Skryabin, J. M. Stone, and J. C. Knight, “Four-wave mixing of solitons with radiation and quasi-nondispersive wave packets at the short-wavelength edge of a supercontinuum,” Opt. Express |

12. | L. Cohen, |

13. | R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert 𝒲 function,” Adv. Comput. Math. |

14. | J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P.-O. Hedekvist, “Fiber based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. |

15. | T. Hori, J. Takayanagi, N. Nishizawa, and T. Goto, “Flatly broadened, wideband and low noise supercontinuum generation in highly nonlinear hybrid fiber,” Opt. Express |

16. | J. J. Miret, E. Silvestre, and P. Andrés, “Octave-spanning ultraflat supercontinuum with soft-glass photonic crystal fibers,” Opt. Express |

**OCIS Codes**

(190.4370) Nonlinear optics : Nonlinear optics, fibers

(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

(320.7110) Ultrafast optics : Ultrafast nonlinear optics

(060.4005) Fiber optics and optical communications : Microstructured fibers

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: July 10, 2013

Revised Manuscript: November 2, 2013

Manuscript Accepted: November 5, 2013

Published: November 13, 2013

**Citation**

David Castelló-Lurbe, Pedro Andrés, and Enrique Silvestre, "Dispersion-to-spectrum mapping in nonlinear fibers based on optical wave-breaking," Opt. Express **21**, 28550-28558 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-23-28550

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### References

- W. J. Tomlinson, R. H. Stolen, and A. M. Johnson, “Optical wave-breaking in nonlinear optical fibers,” Opt. Lett.10, 457–459 (1985). [CrossRef] [PubMed]
- D. Anderson, M. Desaix, M. Lisak, and M. L. Quiroga-Teixeiro, “Wave breaking in nonlinear-optical fibers,” J. Opt. Soc. Am. B9, 1358–1361 (1992). [CrossRef]
- G. P. Agrawal, Nonlinear Fiber Optics, 4 (Academic, 2007).
- D. Anderson, M. Desaix, M. Karlsson, M. Lisak, and M. L. Quiroga-Teixeiro, “Wave-breaking-free pulses in nonlinear-optical fibers,” J. Opt. Soc. Am. B10, 1185–1190 (1993). [CrossRef]
- C. Finot, B. Kibler, L. Provost, and S. Wabnitz, “Beneficial impact of wave-breaking for coherent continuum formation in normally dispersive nonlinear fibers,” J. Opt. Soc. Am. B25, 1938–1948 (2008). [CrossRef]
- A. M. Heidt, A. Hartung, G. W. Bosman, P. Krok, E. G. Rohwer, H. Schwoerer, and H. Bartelt, “Coherent octave spanning near-infrared and visible supercontinuum generation in all-normal dispersion photonic crystal fibers,” Opt. Express19, 3775–3787 (2011). [CrossRef] [PubMed]
- Y. Liu, H. Tu, and S. A. Boppart, “Wave-breaking-extended fiber supercontinuum generation for high compression ratio transform-limited pulse compression,” Opt. Lett.37, 2172–2174 (2012). [CrossRef] [PubMed]
- J. E. Rothenberg, “Femtosecond optical shocks and wave breaking in fiber propagation,” J. Opt. Soc. Am. B6, 2392–2401 (1989). [CrossRef]
- V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP34, 62–70 (1972).
- F. Shimizu, “Frequency broadening in liquid by a short light pulse,” Phys. Rev. Lett.19, 1097–1100 (1967). [CrossRef]
- A. V. Gorbach, D. V. Skryabin, J. M. Stone, and J. C. Knight, “Four-wave mixing of solitons with radiation and quasi-nondispersive wave packets at the short-wavelength edge of a supercontinuum,” Opt. Express14, 9854–9863 (2006). [CrossRef] [PubMed]
- L. Cohen, Time-Frequency Analysis (Prentice Hall, 1995).
- R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert 𝒲 function,” Adv. Comput. Math.5, 329–359 (1996). [CrossRef]
- J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P.-O. Hedekvist, “Fiber based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron.8, 506–520 (2002). [CrossRef]
- T. Hori, J. Takayanagi, N. Nishizawa, and T. Goto, “Flatly broadened, wideband and low noise supercontinuum generation in highly nonlinear hybrid fiber,” Opt. Express12, 317–324 (2004). [CrossRef] [PubMed]
- J. J. Miret, E. Silvestre, and P. Andrés, “Octave-spanning ultraflat supercontinuum with soft-glass photonic crystal fibers,” Opt. Express17, 9197–9203 (2009). [CrossRef] [PubMed]

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