# Pulse Propagation Modeling

Author: the photonics expert Dr. Rüdiger Paschotta

Definition: working with physical models describing the propagation of ultrashort pulses e.g. in lasers or optical fibers

**Summary:** This in-depth article on pulse propagation model answers the following questions:

- Which kinds of physical effects typically affect the propagation of ultrashort pulses?
- In which typical situations is it important to analyze pulse propagation effects?
- What kinds of physical models exist for describing and stimulating ultrashort pulse propagation, and what are their specific advantages and limitations?

When propagating in transparent optical media, the properties of ultrashort light pulses can undergo complicated changes. Typical physical effects influencing pulses are:

*Chromatic dispersion*can lead to dispersive pulse broadening, but also to pulse compression, generation of a chirp, etc.- Various
*nonlinearities*can become relevant at high peak powers. For example, the Kerr effect can cause self-phase modulation, and Raman scattering may e.g. induce Raman gain within the pulse spectrum (*Raman self-frequency shift*). - Optical
*gain*and*losses*can modify the pulse energy and the spectral shape. - The spatial properties can change due to linear effects such as
*diffraction*and*waveguiding*, but also due to nonlinear effects such as*self-focusing*. In highly nonlinear interactions,*filamentation*may occur.

Of course, different effects can act simultaneously, and often interact in surprising ways. For example, chromatic dispersion and Kerr nonlinearity can lead to soliton effects.

## Case Study: Raman Scattering in a Fiber Amplifier

We investigate the effects of stimulated Raman scattering in an ytterbium-doped fiber amplifier for ultrashort pulses, considering three very different input pulse duration regimes. Surprisingly, the effect of Raman scattering always gets substantial only on the last meter, although the input peak powers vary by two orders of magnitude.

For longer pulse durations (nanosecond durations or longer), the situation is normally much simpler: chromatic dispersion is quite irrelevant, and nonlinearities can substantially change the pulse spectrum, but not substantially affect the temporal profile – unless extreme processes occur at very high optical intensity, such as laser-induced damage with a dielectric breakdown.

## Case Study: Nonlinear Pulse Compression in a Fiber

We explore how we can spectrally broaden light pulses by self-phase modulation in a fiber and subsequently compress the pulses using a dispersive element. A substantial reduction in pulse duration by more than an order of magnitude is easily achieved, while the pulse quality is often not ideal.

## Relevance of Pulse Propagation Effects

Pulse propagation effects as mentioned above are relevant in various kinds of situations. Some examples are:

- Details of the propagation of ultrashort pulses in a mode-locked laser determine the steady-state pulse properties such as pulse duration, bandwidth and chirp, and the stability of pulse generation, multiple pulsing, etc.
- The propagation in fibers is relevant e.g. for pulse amplification, pulse compression and supercontinuum generation, and in optical fiber communications.
- Nonlinear frequency conversion of ultrashort pulses can lead to complicated changes of pulse shapes. In addition to the nonlinear interaction, there can be influences from effects such as temporal spatial walk-off and dispersive broadening.

A detailed understanding of pulse propagation is thus important, for example, for the development of ultrafast laser and amplifier systems, of supercontinuum sources and other kinds of nonlinear frequency conversion devices.

For gaining such an understanding, the modeling and simulation of pulse propagation is usually indispensable, as experimental exploration is far more cumbersome and limited. For example, there are sophisticated (and expensive) instruments for pulse characterization, but a complete characterization remains difficult, and can usually be applied only to freely accessible pulses – for example, to the output pulses of a laser system, but not to pulses at any location within a laser resonator or a fiber. Further, a computer simulation model can be used to very quickly and easily find out how the pulse propagation would react to various changes. That way, optimal performance can be realized far more efficiently.

Note that the article on laser modeling and simulation explains in detail various general aspects of modeling and simulations, which also apply to pulse propagation modeling. In the following, we treat only aspects which are specifically relevant for ultrashort pulse propagation.

## Case Study: Numerical Experiments With Soliton Pulses in Fibers

We investigate various details of soliton pulse propagation in passive fibers, using numerical simulations.

## Case Study: Soliton Pulses in a Fiber Amplifier

We investigate to which extent soliton pulses could be amplified in a fiber amplifier, preserving the soliton shape and compressing the pulses temporally.

## Simulation Models for Ultrashort Pulse Propagation

Depending on the situation, different kinds of physical modeling techniques are required. Some of the most important ones are described in the following:

### Haus Master Equation

The Haus Master equation is an analytical tool (a differential equation) mainly for calculating the steady-state pulse properties obtained in mode-locked lasers. It can be seen as a generalization of the *nonlinear Schrödinger equation*.

This equation is sometimes used for simulating the evolution of pulses over the resonator round trips of an ultrashort pulse in a mode-locked lasers. It does *not* deal with details of the intracavity evolution (i.e., the evolution within a single round trip), but only describes the *total* effects in one round trip. It is thus mostly applicable to cases where the nonlinear and dispersive effects within a single resonator round trip are relatively weak. That is often the case in mode-locked bulk lasers, but not in mode-locked fiber lasers.

In not too complicated cases, some analytical results can directly be derived from Haus Master equation, but it is also common to solve in numerically.

### Soliton Perturbation Theory

Soliton perturbation theory describes the propagation of soliton pulses which can be subject to gain or loss, spectral filtering, or additional details of nonlinearities such as the delayed nonlinear response. A number of dynamic equations describe the evolution of the basic parameters of solitons (e.g., pulse energy and duration, center wavelength and chirp) under the influence of various effects. Also, the so-called *continuum* is included, i.e. a temporally broad background radiation with which a soliton can interact. Soliton perturbation theory can be used, e.g., to describe the generation of Kelly sidebands in a mode-locked fiber laser, or propagation effects in optical fiber communications.

### Other Dynamic Models

Some models based on second-order moments of the complex electric field of a pulse [5] can also greatly reduce the number of dynamic variables. However, they are applicable only as long as the pulse shapes remain relatively simple.

A difficulty is that it is not always obvious where the parameter region with a reasonable accuracy ends.

The advantage of a significantly faster computation (compared with a full numerical simulation) becomes less important as the power of computers is increasing.

### General Numerical Pulse Propagation

Numerical techniques are available for simulating pulse propagation in more general cases. The common approach to describe a short pulse in such models is briefly explained in the following:

- Analytically, one uses a complex amplitude <$A(t)$> in the time domain, or alternatively an amplitude <$A(\nu)$> in the frequency domain. Typically, this is a “slowly varying” amplitude, not containing the fast optical oscillation; the electric field strength is proportional to <$A(t) \exp(-i \omega_0 t)$> with the optical (angular) center frequency <$\omega_0$>. The frequency-dependent amplitudes then have their maximum around <$\nu = 0$>, rather than around the center frequency. The time- and frequency-dependent functions are related to each other by a Fourier transform.

### Simulations on Pulse Propagation

Ultrashort pulses change in complicated ways when propagating through a fiber, for example. A suitable simulator is essential for getting complete insight – not only on the resulting output pulses, but also on the pulses at any location in your system. The RP Fiber Power software is an ideal tool for such work.

- Numerically, we represent the pulse with an array of discrete complex amplitudes in the time or frequency domain. Normally, the number of amplitudes is an integer power of 2, because fast Fourier transform algorithms work best with that.
- In some cases, a moderate number of amplitudes (e.g., 2
^{8}= 256) are well sufficient to represent a pulse. A substantially higher number of amplitudes is required in cases where, for example, pulses can be relatively long in time and broadband at the same time. A typical case is supercontinuum generation, where one sometimes uses 2^{15}or more amplitudes. - The chosen numerical parameters (for example, width of the considered temporal range and number of amplitudes) determine the temporal resolution, which is related to the width of the optical frequency range, and the spectral resolution. Inappropriately chosen parameters can lead to numerical artifacts.

## Case Study: Parabolic Pulses in a Fiber Amplifier

We explore the regime of parabolic pulse amplification in an Yb-doped single-mode fiber. We find suitable system parameters and investigate limiting effects.

Various aspects need to be considered for the numerical propagation of pulses:

- Linear effects such as chromatic dispersion or frequency-dependent propagation losses are easily treated in the frequency domain, whereas nonlinear interactions are often (but not always) more conveniently handled in the time domain. As required, switching between both domains can be done with a fast Fourier transform algorithm (
*FFT techniques*). - There are some more challenging cases where both time and frequency dependencies must be considered. An example is the saturation of frequency-dependent gain [9].
- One frequently uses the
*symmetrized split-step Fourier method*, used particularly for pulse propagation in fibers [12]. The (weak) dispersive and nonlinear effects corresponding to short fiber pieces are alternately applied. The numerical errors associated with the finite longitudinal step size can be minimized with a special symmetrization technique, which allows for higher accuracies without excessively increased computation times. - Automatic control of the step size (in the propagation direction) can be very important for ensuring numerical accuracy while also maintaining a high computational efficiency.

One distinguishes between simpler 1D models and more complex 3D models:

- In many cases, it is sufficient to ignore the transverse dimensions. For example, for propagation of pulses in a single-mode fiber, the transverse profile can usually be regarded as fixed (although strictly speaking the mode radius is frequency-dependent, and at extremely high intensities there are self-focusing effects). A pulse at one longitudinal position with a fiber, for example, can then be represented by a single time- or frequency-dependent complex amplitude. A model can propagate such a pulse along a fiber, for example, by using a differential equation containing influences of chromatic dispersion and nonlinearities. In the simplest case, only the simple Kerr effect is considered, but one can also include self-steepening and the delayed nonlinear response for stimulated Raman scattering. Further, in an active fiber or a laser crystal there is gain (often with relevant gain saturation).
- In special situations, such as the detailed investigation of Kerr lens mode locking or filamentation phenomena of terawatt pulses propagating in gases, one requires a full 3D model, also considering the transverse dimensions. This can be realized with sophisticated numerical beam propagation methods, including all the above-mentioned details for ultrashort pulse propagation. Such models are difficult to set up and slow to execute on a computer, but essential in some situations.
- A kind of intermediate approach, for example for pulse propagation in multimode waveguides, is based on a description of the optical field as a superposition of propagation modes, which can be coupled e.g. via nonlinearities. Similarly, one can describe interactions between light components with orthogonal polarization states.

By applying statistical techniques, pulse propagation models can also be used to investigate noise phenomena [7].

Although the construction even of 1D pulse propagation models involves various non-trivial challenges, it is relatively easy to simulate pulse propagation with sufficiently powerful and flexible software such as RP Fiber Power. It makes sense to have such models and software developed by few people and used by many others, who do not need to have programming skills and experience with numerical algorithms. A substantially less detailed understanding of the underlying physics is also sufficient.

## Modeling of Pulse Amplification

This tutorial explains the amplification of light pulses, with particular focus on fiber amplifiers, but also some content on bulk laser amplifiers.

## More to Learn

- Case Study: Raman Scattering in a Fiber Amplifier
- Case Study: Nonlinear Pulse Compression in a Fiber
- Case Study: Numerical Experiments With Soliton Pulses in Fibers
- Case Study: Soliton Pulses in a Fiber Amplifier
- Case Study: Parabolic Pulses in a Fiber Amplifier
- Case Study: Collision of Soliton Pulses in a Fiber
- Case Study: Erbium-doped Fiber Amplifier for Rectangular Nanosecond Pulses

Encyclopedia articles:

## Suppliers

The **RP Photonics Buyer's Guide** contains five suppliers for pulse propagation modeling software. Among them:

### RP Photonics

Our RP ProPulse and RP Fiber Power software are suitable for modeling mode-locked solid-state lasers and fiber lasers, ultrafast amplifier systems etc., and provide particularly flexible solutions.

### Bibliography

[1] | P. V. Mamyshev and S. V. Chernikov, “Ultrashort-pulse propagation in optical fibers”, Opt. Lett. 15 (19), 1076 (1990); https://doi.org/10.1364/OL.15.001076 |

[2] | G. P. Agrawal, “Optical pulse propagation in doped fiber amplifiers”, Phys. Rev. A 44 (11), 7493 (1991); https://doi.org/10.1103/PhysRevA.44.7493 |

[3] | H. A. Haus et al., “Structures for additive pulse mode locking”, J. Opt. Soc. Am. B 8 (10), 2068 (1991); https://doi.org/10.1364/JOSAB.8.002068 |

[4] | P. L. François, “Nonlinear propagation of ultrashort pulses in optical fibers: total field formulation in the frequency domain”, J. Opt. Soc. Am. B 8 (2), 276 (1991); https://doi.org/10.1364/JOSAB.8.000276 |

[5] | M. Potasek et al., “Analytic and numerical study of pulse broadening in nonlinear dispersive fibers”, J. Opt. Soc. Am. B 3 (2), 205 (1992); https://doi.org/10.1364/JOSAB.3.000205 |

[6] | D. Marcuse, “RMS width of pulses in nonlinear dispersive fibers”, IEEE J. Lightwave Technol. 10 (1), 17 (1992); https://doi.org/10.1109/50.108730 |

[7] | R. Paschotta, “Noise of mode-locked lasers. Part I: numerical model”, Appl. Phys. B 79, 153 (2004)“,”http://link.springer.com/article/10.1007%2Fs00340-004-1547-x; R. Paschotta, “Noise of mode-locked lasers. Part II: timing jitter and other fluctuations”, Appl. Phys. B 79, 163 (2004); https://doi.org/10.1007/s00340-004-1548-9 |

[8] | B. Burgoyne et al., “Nonlinear pulse propagation in optical fibers using second order moments”, Opt. Express 15 (16), 10075 (2007); https://doi.org/10.1364/OE.15.010075 |

[9] | R. Paschotta, “Modeling of ultrashort pulse amplification with gain saturation”, Opt. Express 25 (16), 19112 (2017); https://doi.org/10.1364/OE.25.019112 |

[10] | Y.-H. Chen et al., “Accurate modeling of ultrafast nonlinear pulse propagation in multimode gain fiber”, J. Opt. Soc. Am. B 40 (10), 2633 (2023); https://doi.org/10.1364/JOSAB.500586 |

[11] | C. R. Phillips, M. Jankowski, N. Flemens and M. M. Fejer, “General framework for ultrafast nonlinear photonics: unifying single and multi-envelope treatments [Invited]”, Opt. Express 32 (5), 8284 (2024); https://doi.org/10.1364/OE.513856 |

[12] | G. P. Agrawal, Nonlinear Fiber Optics, 4th edn., Academic Press, New York (2007) |

[13] | R. Paschotta, tutorial “Passive Fiber Optics” |

[14] | R. Paschotta, tutorial “Passive Fiber Optics”, Part 12: Ultrashort Pulses and Signals in Fibers |

[15] | R. Paschotta, tutorial “Modeling of Fiber Amplifiers and Lasers”, part 7 |

[16] | R. Paschotta, tutorial “Modeling of Pulse Amplification” |

This encyclopedia is authored by Dr. Rüdiger Paschotta, the founder and executive of RP Photonics AG. How about a tailored training course from this distinguished expert at your location? Contact RP Photonics to find out how his technical consulting services (e.g. product designs, problem solving, independent evaluations, training) and software could become very valuable for your business!

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