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Self-steepening

Definition: a nonlinear effect in pulse propagation, leading to an increasingly steep trailing slope of the temporal pulse shape

Categories: fiber optics and waveguides, nonlinear optics, light pulses

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Cite the article using its DOI: https://doi.org/10.61835/t5d

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When ultrashort pulses of light propagate in a nonlinear medium (e.g., an optical fibers), they may experience self-steepening, meaning a distortion of the temporal pulse shape such that the trailing slope (i.e., the decay of optical power after the pulse maximum) gets increasingly steep.

For simplicity, in Figure 1 we illustrate this effect under the assumption of zero chromatic dispersion in the medium, although that is unrealistic for optical fibers.

self-steepening
Figure 1: Self-steepening of a 1-nJ 100-fs pulse in a hypothetical optical single-mode fiber without chromatic dispersion. Results for increasing propagation distances (up to 1.3 m) are shown with colors ranging from light blue to red.

The phenomenon can be interpreted as a reduction in group velocity proportional to the optical intensity, although the original concept of group velocity does not include nonlinear effects.

Under the additional influence of chromatic dispersion, the actual pulse dynamics can be substantially different, as discussed below.

Spectral Changes

Self-steepening is also associated with pronounced changes of the optical spectrum of the pulses, particularly when a very steep trailing slope (an optical shock) results.

self-steepening spectrum
Figure 2: Evolution of the optical spectra in the same situation as for Fig. 1. Again, results for increasing propagation distances are shown with colors ranging from light blue to red.

Typical observations are:

  • The spectral broadening is stronger on the high-frequency (short-wavelength) side, because the higher frequencies are created by the falling slope, which gets steeper.
  • The power spectral density is higher on the low-frequency side, since the energy is distributed over a narrower range of optical frequencies.

Self-steepening becomes important if light pulses in a fiber are (a) sufficiently intense to experience strong self-phase modulation over the given length of fiber, and (b) sufficiently broadband. Often, these conditions are fulfilled for pulse durations below 100 fs.

Self-steepening can also occur in other types of nonlinear media, including solids, liquids and gases, although it may then be less common than in fibers, where high optical intensities can easily be maintained over a long length due to the waveguiding.

Numerical Simulation of Self-steepening

Self-steepening is included in simulations which are based on the following differential equation [13], which also includes propagation losses and chromatic dispersion up to third order:

$$\frac{{\partial A}}{{\partial z}} = - \frac{\alpha }{2}A - i\frac{{{\beta _2}}}{2}\frac{{{\partial ^2}A}}{{\partial {t^2}}} + \frac{{{\beta _3}}}{6}\frac{{{\partial ^2}A}}{{\partial {t^3}}} + i\gamma \: \left(1 + \frac{i}{\omega_0} \frac{\partial}{\partial t}\right) \: \left(A(z,t)\int\limits_0^\infty {R(\tau )\;{{\left| {A(z,t - \tau )} \right|}^2}{\rm{d}}\tau } \right) $$

Specifically, the term with the temporal derivative is responsible for self-steepening. The equation also describes a delayed nonlinear response, causing stimulated Raman scattering (SRS); if that aspect can be neglected (for not too broadband pulses), the equation simplifies to

$$\frac{{\partial A}}{{\partial z}} = - \frac{\alpha }{2}A - i\frac{{{\beta _2}}}{2}\frac{{{\partial ^2}A}}{{\partial {t^2}}} + \frac{{{\beta _3}}}{6}\frac{{{\partial ^2}A}}{{\partial {t^3}}} + i\gamma \: \left(1 + \frac{i}{\omega_0} \frac{\partial}{\partial t}\right) \: \left( A(z,t) \: \left|{A(z,t)}\right|^2 \right) $$

Note that the time derivative in these time-domain equations essentially corresponds to an <$\omega$> factor in the frequency domain (disregarding the imaginary unit), where <$\omega$> is not the absolute optical frequency, but rather the deviation of that frequency from the center frequency <$\omega_0$>. Therefore, the parenthesis part containing the time derivative corresponds to a factor proportional to the optical frequency. Numerically, that factor can be conveniently applied after applying a Fast Fourier Transform (FFT). Further frequency dependencies from the nonlinear index and from a frequency-dependent effective mode area of the fiber can also be treated in that context.

Influence of Chromatic Dispersion

Pulse propagation in optical fibers is usually also substantially influenced by chromatic dispersion, resulting in pulse dynamics which may substantially differ from a hypothetical situation without chromatic dispersion.

We illustrated this with an example case, again assuming 1-nJ input pulses with 100 fs duration in a single-mode fiber, but now with a group velocity dispersion of −10000 fs2/m. We first neglect self-steepening, and the two following diagrams show the resulting pulse evolution in the time and frequency domain:

pulse propagation with positive GVD and nonlinearity
Figure 3: Propagation of an optical pulse in a fiber with negative GVD and nonlinearity, shown in the time domain. Self-steepening is neglected.
pulse propagation with positive GVD and nonlinearity
Figure 4: Pulse evolution in the frequency domain for the same situation as in the previous diagram.

The result is somewhat similar as for the propagation of a higher-order soliton pulse, although the pulse parameters do not accurate fit to those of such a soliton, so that a non-period evolution results.

The following diagrams show the same with the self-steepening term included:

pulse propagation with positive GVD and nonlinearity
Figure 5: Time domain results, now with self-steepening.
pulse propagation with positive GVD and nonlinearity
Figure 6: Frequency domain results with self-steepening.

We see that while the propagation initially looks similar, self-steepening soon leads to a splitting of the pulse into two, with dramatically different results for longer propagation. For a fiber with positive GVD, however, the impact of self-steepening would be weak.

Influence of Self-steepening Term in Cases with Raman Scattering

In cases with substantial stimulated Raman scattering, simulations with omitted self-steepening term lead to results where the pulse energy is conserved – although in reality Raman scattering preserves the photon number but reduces the pulse energy (inelastic scattering). The self-steepening term corrects for that.

It has also been shown that self-steepening can reduce the amount of the soliton self-frequency shift [8].

Bibliography

[1]F. DeMartini et al., “Self-steepening of light pulses”, Phys. Rev. 164, 312 (1967); https://doi.org/10.1103/PhysRev.164.312
[2]D. Grischkowsky, Eric Courtens and J. A. Armstrong, “Observation of self-steepening of optical pulses with possible shock formation”, Phys. Rev. Lett. 31, 422 (1973); https://doi.org/10.1103/PhysRevLett.31.422
[3]N. Tzoar and M. Jain, “Self-phase modulation in long-geometry optical waveguides”, Phys. Rev. A 23, 1266 (1981); https://doi.org/10.1103/PhysRevA.23.1266
[4]D. Anderson and M. Lisak, “Nonlinear asymmetric self-phase modulation and self-steepening of pulses in long optical waveguides”, Phys. Rev. A 27, 1393 (1983); https://doi.org/10.1103/PhysRevA.27.1393
[5]G. Yang and Y. R. Shen, “Spectral broadening of ultrashort pulses in a nonlinear medium”, Opt. Lett. 9 (11), 510 (1984); https://doi.org/10.1364/OL.9.000510
[6]J. Manassah et al., “Spectral extent and pulse shape of the supercontinuum for ultrashort laser pulse”, IEEE J. Quantum Electron. 22 (1), 197 (1986); https://doi.org/10.1109/JQE.1986.1072850
[7]J. R. de Oliveira et al., “Self-steepening of optical pulses in dispersive media”, J. Opt. Soc. Am. B 9 (11), 2025 (1992); https://doi.org/10.1364/JOSAB.9.002025
[8]A. A. Voronin and A. M. Zheltikov, “Soliton self-frequency shift decelerated by self-steepening”, Opt. Lett. 33 (15), 1723 (2008); https://doi.org/10.1364/OL.33.001723
[9]J. Kasparian, P. Béjot and J. Wolf, “Arbitrary-order nonlinear contribution to self-steepening”, Opt. Lett. 35 (16), 2795 (2010); https://doi.org/10.1364/OL.35.002795
[10]L. Zhang et al., “On-chip two-octave supercontinuum generation by enhancing self-steepening of optical pulses”, Opt. Express 19 (12), 11584 (2011); https://doi.org/10.1364/OE.19.011584
[11]N. Linale et al., “Measuring self-steepening with the photon-conserving nonlinear Schrödinger equation”, Opt. Lett. 45 (16), 4535 (2020); https://doi.org/10.1364/OL.401096
[12]R. Kormokar, Md H. M. Shamim and M. Rochette, “High-order analytical formulation of soliton self-frequency shift”, J. Opt. Soc. Am. B 38 (2), 466 (2021); https://doi.org/10.1364/JOSAB.409240
[13]G. P. Agrawal, Nonlinear Fiber Optics, 4th edn., Academic Press, New York (2007)
[14]R. Paschotta, tutorial on "Passive Fiber Optics", Part 11: Nonlinearities of Fibers

(Suggest additional literature!)

See also: nonlinearities, delayed nonlinear response, ultrashort pulses

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