## Self-similar pulses in coherent linear amplifiers |

Optics Express, Vol. 19, Issue 10, pp. 9750-9758 (2011)

http://dx.doi.org/10.1364/OE.19.009750

Acrobat PDF (866 KB)

### Abstract

We discover and analytically describe self-similar pulses existing in homogeneously broadened amplifying linear media in a vicinity of an optical resonance. We demonstrate numerically that the discovered pulses serve as universal self-similar asymptotics of any near-resonant short pulses with sharp leading edges, propagating in coherent linear amplifiers. We show that broadening of any low-intensity seed pulse in the amplifier has a diffusive nature: Asymptotically the pulse width growth is governed by the simple diffusion law. We also compare the energy gain factors of short and long self-similar pulses supported by such media.

© 2011 OSA

## 1. Introduction

1. J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. **3**, 597–603 (2007). [CrossRef]

2. S. An and J. E. Sipe, “Universality in the dynamics of phase grating formation in optical fibers,” Opt. Lett. **16**, 1478–1480 (1991). [CrossRef] [PubMed]

3. C. R. Menyuk, D. Levi, and P. Winternitz, “Self-similarity in transient stimulated Raman scattering,” Phys. Rev. Lett. **69**, 3048–3051 (1992). [CrossRef] [PubMed]

4. T. M. Monroe, P. D. Millar, P. L. Poladian, and C. M. de Sterke, “Self-similar evolution of self-written waveguides,” Opt. Lett. **23**, 268–270 (1998). [CrossRef]

5. M. Soljacic, M. Segev, and C. R. Menyuk, “Self-similarity and fractals in soliton-supporting systems,” Phys. Rev. E **61**, R1048–R1051 (2000). [CrossRef]

6. 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]

8. M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. D. 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]

9. F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. **92**, 213902 (2004). [CrossRef] [PubMed]

10. B. Oktem, F. O. Ulgudur, and F. O. Ilday, “Soliton-similariton fibre laser,” Nat. Photon. **4**, 307–311 (2010). [CrossRef]

11. W. H. Renninger, A. Chong, and F. W. Wise, “Self-similar pulse evolution in an all-normal-dispersion laser,” Phys. Rev. A **82**, 021805(R) (2010). [CrossRef]

12. S. A. Ponomarenko and G. P. Agrawal, “Do soliton-like self-similar waves exist in nonlinear optical media?” Phys. Rev. Lett. **97**, 013901 (2006). [CrossRef] [PubMed]

16. K. D. Moll, A. Gaeta, and G. Fibich, “Self-similar optical wave collapse: observation of the townes profile,” Phys. Rev. Lett. **90**, 203902 (2003). [CrossRef] [PubMed]

18. I. R. Gabitov and S. V. Manakov, “Propagation of ultrashort pulses in degenerate laser amplifiers,” Phys. Rev. Lett. **50**, 495–498 (1983). [CrossRef]

15. S. A. Ponomarenko and S. Haghgoo, “Spatial optical similaritons in conservative nonintegrable systems,” Phys. Rev. A **81**, 051801(R) (2010). [CrossRef]

20. S. A. Ponomarenko and S. Haghgoo, “Self-similarity and optical kinks in resonant nonlinear media,” Phys. Rev. A **82**, 051801(R) (2010). [CrossRef]

15. S. A. Ponomarenko and S. Haghgoo, “Spatial optical similaritons in conservative nonintegrable systems,” Phys. Rev. A **81**, 051801(R) (2010). [CrossRef]

20. S. A. Ponomarenko and S. Haghgoo, “Self-similarity and optical kinks in resonant nonlinear media,” Phys. Rev. A **82**, 051801(R) (2010). [CrossRef]

*resonant linear*media. At first glance, the very proximity to optical resonance(s), coupled with the system linearity, appears to preclude self-similarity of a sufficiently short pulse: Strong dispersion at resonance(s) would, in general, seem to cause severe pulse reshaping. One would then also wonder whether self-similar pulses in such media, if any, would be universal asymptotics of very weak seed pulses. The affirmative answer to the last question would augur well for the experimental realization of such similaritons.

## 2. Physical model and mathematical preliminaries

*ω*

_{0}, thereby limiting our consideration to the case of one internal resonance. We assume here-after that the pulse spectrum is mainly affected by homogeneous broadening, implying that

*γ*

_{⊥}≫

*δ*,

*γ*

_{⊥}and

*δ*being transverse (dipole moment) relaxation rate and a characteristic spectral width of inhomogeneous broadening, respectively. Under these conditions, the evolution of a pulse with the carrier frequency

*ω*in the medium is governed by a reduced wave equation, subject to the slowly-varying envelope approximation (SVEA): Here Ω = 2

*d*

*ℰ*

_{eg}*/h̄*is the Rabi frequency associated with the pulse amplitude ℰ,

*N*is an atom density,

*d*

*is a dipole matrix element between the ground and excited states of any atom, labeled with the indices*

_{eg}*g*and

*e*, respectively;

*κ*=

*ωN*|

*d*

*|*

_{eg}^{2}/

*cε*

_{0}

*h̄*is a coupling constant,

*k*=

*ω*/

*c*, and Eq. (1) is written in terms of the transformed coordinate and time,

*ζ*=

*z*and

*τ*=

*t*–

*z/c*. The dipole moment matrix element

*σ*and one-atom inversion

*w*obey the Bloch equations [21] and In Eqs. (3) and (4)

*γ*

_{||}is a longitudinal relaxation rates associated with one-atom inversion damping, Δ =

*ω*–

*ω*

_{0}is a detuning from the resonance and

*w*

*is a value of the one-atom inversion in the absence of the pulse (equilibrium).*

_{eq}*P*and the upper level relaxation rate must be large enough,

*P*≫

*γ*

_{⊥}and Γ ≫

*γ*

_{⊥}, to achieve population inversion between levels “e” and “g”.

*t*

*≫*

_{p}*T*

_{⊥}and “short” ones,

*t*

*≤*

_{p}*T*

_{⊥}–in the case

*t*

*≪*

_{p}*T*

_{⊥}, the pulses may be called “ultrashort”–where

*t*

*is a characteristic pulse width.*

_{p}### Long pulses

*σ*to its quasi-steady-state value with respect to Ω–which will result in the pulse evolution equation in the form Here we introduced an inverse Beer’s gain/absorption length

*α*and an overall phase accumulation rate

*β*by the expressions, It follows at once from Eq. (7) that for sufficiently long pulses,

*any*pulse grows/decays exponentially in such a medium, maintaining its overall shape, where Ω

_{0}(

*t*) describes an initial pulse profile, and Eq. (9) is well-known Beer’s amplification/absorption law.

### Short pulses

*α*

_{0}= 2

*κ*/

*γ*

_{⊥}. Our treatment to this point is equally applicable to amplifying and absorbing media. Hereafter we focus on short pulse propagation in amplifiers.

## 3. Short self-similar pulses

*θ*(

*τ*) is a unit step function describing a sharp leading edge of the pulse,

*F*(

*ζ*) and

*G*(

*ζ*) are arbitrary at the moment and

*are dimensionless functions. Substituting the Ansatz (12) and (13) into Eqs. (10) and (11), we can show that self-similarity is sustained provided that Further, the dimensionless pulse envelope*σ ¯

*η*by and the prime denotes a derivative with respect to

*η*.

_{1}

*F*

_{1}(

*a, c, x*) is a confluent hypergeometric function, and we dropped an arbitrary (small) initial pulse amplitude. Eq. (17) can be expressed in a more compact form as where

*I*

_{0}(

*x*) is a modified Bessel function of zero order. We note that for sufficiently long propagation distances,

*α*

_{0}

*ζ*≪

*T*

_{⊥}/

*t*

*, the self-similar pulse profile no longer depends on*

_{p}*t*

*, yielding a universal self-similar profile*

_{p}6. 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]

8. M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. D. 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]

23. V. V. Kozlov and S. Wabnitz, “Quasi-parabolic pulses in a coherent nonlinear optical amplifier,” Opt. Lett. **35**, 2058–2060 (2010). [CrossRef] [PubMed]

*T*=

*τ*/

*T*

_{⊥}for several values of the dimensionless propagation distance

*Z*= α

_{0}

*ζ*. The self-similar character of the pulse dynamics is clearly discernable in the figure. To demonstrate the universal nature of the discovered self-similar regime, we numerically simulate the evolution of a generic asymmetric Gaussian pulse,

*Z*= 0:

*t*

*=*

_{p}*T*

_{⊥}/2. In the inset to the figure, we compare short-distance pulse dynamics of the two pulses. We see in the figure that although the Gaussian pulse profile deviates from the self-similar asymptotics over short distances–at least over first few Beer’s amplification lengths as is seen in the inset–it quickly converges to the universal asymptotics over longer distances. It then is seen to coincide with the self-similar asymptotics profile to within numerical round-off errors [24]. We obtained qualitatively similar results for hyperbolic secant and exponential profiles with cut off leading edges: Ω

_{2}(

*t,*0) ∝

*θ*(

*t*)sech(

*t*/

*t*

*), and Ω*

_{p}_{3}(

*t,*0) ∝

*θ*(

*t*)exp(–

*t*/

*t*

*).*

_{p}*T*

_{⊥}–of the universal self-similar asymptotics on pulse propagation in the amplifier. The averaging is taken over the pulse intensity distribution, for instance, With the help of the asymptotic expansion of

*I*

_{0}(

*x*) [25], an analytical expression for the rms width can be derived and presented in an exceptionally simple form In other words, the pulse rms width grows with the distance in a diffusive manner with the effective diffusion coefficient equal to

*t*

*describing the fast rise of its leading edge. Thus for a short pulse,*

_{sw}*t*

_{p}*~ T*⊥, the pulse duration (rise time of the pulse front edge) has to be much shorter than the pulse width, yet much longer than an optical cycle for the SVEA–see Eq. (2)–to hold: The complimentary conditions (22) can be realized in a laboratory for picosecond pulses in dilute atomic vapors, say, for which, typically

*T*

_{⊥}~ 1 ÷ 10 ps [21] by choosing, for example,

*t*

*~ 10 ÷ 100 fs. Mathematically, the leading front step function can then be approximated, for instance, as with the excellent approximation attainable for*

_{sw}*t*

*= 0.01*

_{sw}*T*

_{⊥}.

## References and links

1. | J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. |

2. | S. An and J. E. Sipe, “Universality in the dynamics of phase grating formation in optical fibers,” Opt. Lett. |

3. | C. R. Menyuk, D. Levi, and P. Winternitz, “Self-similarity in transient stimulated Raman scattering,” Phys. Rev. Lett. |

4. | T. M. Monroe, P. D. Millar, P. L. Poladian, and C. M. de Sterke, “Self-similar evolution of self-written waveguides,” Opt. Lett. |

5. | M. Soljacic, M. Segev, and C. R. Menyuk, “Self-similarity and fractals in soliton-supporting systems,” Phys. Rev. E |

6. | 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 |

7. | K. Tamura and M. Nakazawa, “Pulse compression by nonlinear pulse evolution with reduced optical wave breaking in erbium-doped fiber amplifiers,” Opt. Lett. |

8. | M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. D. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. |

9. | F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. |

10. | B. Oktem, F. O. Ulgudur, and F. O. Ilday, “Soliton-similariton fibre laser,” Nat. Photon. |

11. | W. H. Renninger, A. Chong, and F. W. Wise, “Self-similar pulse evolution in an all-normal-dispersion laser,” Phys. Rev. A |

12. | S. A. Ponomarenko and G. P. Agrawal, “Do soliton-like self-similar waves exist in nonlinear optical media?” Phys. Rev. Lett. |

13. | S. A. Ponomarenko and G. P. Agrawal, “Optical similaritons in nonlinear waveguides,” Opt. Lett. |

14. | L. Wu, J.-F. Zhang, L. Li, and Q. Tian, “Similaritons in nonlinear optical systems,” Opt. Express |

15. | S. A. Ponomarenko and S. Haghgoo, “Spatial optical similaritons in conservative nonintegrable systems,” Phys. Rev. A |

16. | K. D. Moll, A. Gaeta, and G. Fibich, “Self-similar optical wave collapse: observation of the townes profile,” Phys. Rev. Lett. |

17. | S. V. Manakov, “Propagation of an ultrashort optical pulse in a two-level laser amplifier,” Sov. Phys. JETP |

18. | I. R. Gabitov and S. V. Manakov, “Propagation of ultrashort pulses in degenerate laser amplifiers,” Phys. Rev. Lett. |

19. | I. R. Gabitov, V. E. Zakharov, and A. V. Mikhailov, “Nonlinear theory of superfluorescence,” Sov. Phys. JETP |

20. | S. A. Ponomarenko and S. Haghgoo, “Self-similarity and optical kinks in resonant nonlinear media,” Phys. Rev. A |

21. | L. Allen and J. H. Eberly, |

22. | P. W. Milonni and J. H. Eberly, |

23. | V. V. Kozlov and S. Wabnitz, “Quasi-parabolic pulses in a coherent nonlinear optical amplifier,” Opt. Lett. |

24. | A small deviation from the universal asymptotics in the pulse tails can be explained by limited accuracy of our sharp leading edge approximation. |

25. | M. Abramowitz and I. A. Stegan |

**OCIS Codes**

(260.0260) Physical optics : Physical optics

(260.2030) Physical optics : Dispersion

(320.0320) Ultrafast optics : Ultrafast optics

(320.5540) Ultrafast optics : Pulse shaping

(320.5550) Ultrafast optics : Pulses

**ToC Category:**

Ultrafast Optics

**History**

Original Manuscript: March 18, 2011

Revised Manuscript: April 21, 2011

Manuscript Accepted: April 21, 2011

Published: May 4, 2011

**Citation**

Soodeh Haghgoo and Sergey A. Ponomarenko, "Self-similar pulses in coherent linear amplifiers," Opt. Express **19**, 9750-9758 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-10-9750

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### References

- J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. 3, 597–603 (2007). [CrossRef]
- S. An and J. E. Sipe, “Universality in the dynamics of phase grating formation in optical fibers,” Opt. Lett. 16, 1478–1480 (1991). [CrossRef] [PubMed]
- C. R. Menyuk, D. Levi, and P. Winternitz, “Self-similarity in transient stimulated Raman scattering,” Phys. Rev. Lett. 69, 3048–3051 (1992). [CrossRef] [PubMed]
- T. M. Monroe, P. D. Millar, P. L. Poladian, and C. M. de Sterke, “Self-similar evolution of self-written waveguides,” Opt. Lett. 23, 268–270 (1998). [CrossRef]
- M. Soljacic, M. Segev, and C. R. Menyuk, “Self-similarity and fractals in soliton-supporting systems,” Phys. Rev. E 61, R1048–R1051 (2000). [CrossRef]
- 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]
- K. Tamura and M. Nakazawa, “Pulse compression by nonlinear pulse evolution with reduced optical wave breaking in erbium-doped fiber amplifiers,” Opt. Lett. 21, 68–70 (1996). [CrossRef] [PubMed]
- M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. D. 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]
- F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92, 213902 (2004). [CrossRef] [PubMed]
- B. Oktem, F. O. Ulgudur, and F. O. Ilday, “Soliton-similariton fibre laser,” Nat. Photon. 4, 307–311 (2010). [CrossRef]
- W. H. Renninger, A. Chong, and F. W. Wise, “Self-similar pulse evolution in an all-normal-dispersion laser,” Phys. Rev. A 82, 021805(R) (2010). [CrossRef]
- S. A. Ponomarenko and G. P. Agrawal, “Do soliton-like self-similar waves exist in nonlinear optical media?” Phys. Rev. Lett. 97, 013901 (2006). [CrossRef] [PubMed]
- S. A. Ponomarenko and G. P. Agrawal, “Optical similaritons in nonlinear waveguides,” Opt. Lett. 32, 1659–1661 (2007). [CrossRef] [PubMed]
- L. Wu, J.-F. Zhang, L. Li, and Q. Tian, “Similaritons in nonlinear optical systems,” Opt. Express 16, 6352–6360 (2008). [CrossRef] [PubMed]
- S. A. Ponomarenko and S. Haghgoo, “Spatial optical similaritons in conservative nonintegrable systems,” Phys. Rev. A 81, 051801(R) (2010). [CrossRef]
- K. D. Moll, A. Gaeta, and G. Fibich, “Self-similar optical wave collapse: observation of the townes profile,” Phys. Rev. Lett. 90, 203902 (2003). [CrossRef] [PubMed]
- S. V. Manakov, “Propagation of an ultrashort optical pulse in a two-level laser amplifier,” Sov. Phys. JETP 56, 37–44 (1982).
- I. R. Gabitov and S. V. Manakov, “Propagation of ultrashort pulses in degenerate laser amplifiers,” Phys. Rev. Lett. 50, 495–498 (1983). [CrossRef]
- I. R. Gabitov, V. E. Zakharov, and A. V. Mikhailov, “Nonlinear theory of superfluorescence,” Sov. Phys. JETP 59, 703–709 (1984).
- S. A. Ponomarenko and S. Haghgoo, “Self-similarity and optical kinks in resonant nonlinear media,” Phys. Rev. A 82, 051801(R) (2010). [CrossRef]
- L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover Publications Inc., 1975).
- P. W. Milonni and J. H. Eberly, Lasers (Wiley, 1985).
- V. V. Kozlov and S. Wabnitz, “Quasi-parabolic pulses in a coherent nonlinear optical amplifier,” Opt. Lett. 35, 2058–2060 (2010). [CrossRef] [PubMed]
- A small deviation from the universal asymptotics in the pulse tails can be explained by limited accuracy of our sharp leading edge approximation.
- M. Abramowitz and I. A. SteganHandbook of Mathematical Functions (Dover, 1972).

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