# Self-phase Modulation

Acronym: SPM

Definition: nonlinear phase modulation of a beam, caused by its own intensity via the Kerr effect

More general term: nonlinear optical effects

Opposite term: cross-phase modulation

German: Selbstphasenmodulation

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

Author: Dr. Rüdiger Paschotta

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

Get citation code: Endnote (RIS) BibTex plain textHTML

Due to the Kerr effect, high optical intensity in a medium (e.g. an optical fiber) causes a nonlinear phase delay of the propagating light. That phase delay which has the same temporal shape as the optical intensity. In simple models, the effect is described as a nonlinear change in the refractive index:

$$\Delta n = {n_2}\;I$$with the nonlinear index <$n_2$> and the optical intensity <$I$>. In the context of self-phase modulation, the emphasis is on the *temporal* dependence of the phase shift, whereas the transverse dependence for some beam profile leads to the phenomenon of self-focusing.

Of course, the nonlinear phase delay comes *in addition* to the linear phase delay.

Note that the description of self-phase modulation with a time-dependent refractive index is somewhat simplified, and accurate only for not too short pulses. In more extreme situations, extended models are required, which also take into account the effect of self-steepening – a change of the temporal pulse shape even in the absence of chromatic dispersion. That may be interpreted as a nonlinear modification of the group velocity. There can also be higher-order nonlinear effects, leading to more complicated propagation phenomena.

## Effects on Optical Pulses

### Time-dependent Phase Shift

If an optical pulse is transmitted through a medium, the Kerr effect causes a time-dependent phase shift according to the time-dependent pulse intensity. In this way, an initial unchirped optical pulse acquires a so-called chirp, i.e., a temporally varying instantaneous frequency.

The central part of the pulse exhibits an up-chirp.

For a Gaussian beam with beam radius <$w$> in a homogeneous medium with length <$L$>, the on-axis phase change per unit optical power is described by the proportionality constant

$${\gamma _{{\rm{SPM}}}} = \frac{{2\pi }}{\lambda } n_2 L\;{\left( {\frac{\pi }{2}{w^2}} \right)^{ - 1}} = \frac{4 \: n_2 \: L}{{\lambda \;{w^2}}}$$(In some cases, it may be more convenient to omit the factor <$L$>, obtaining the phase change per unit optical power and unit length.)

Note that 2 times smaller coefficients sometimes occur in the literature, which can have two different reasons:

- An incorrect (2 times too low) equation for the peak intensity of a Gaussian beam is sometimes used. With that, one underestimates the on-axis phase change.
- One may calculate the overall nonlinear phase shift of a Gaussian-shaped waveguide mode, where one gets a kind of averaging between higher- and lower-intensity parts. That reduces the nonlinear phase shift by a factor of 2.

### Spectral Changes

The time-dependent phase change caused by SPM is associated with a modification of the optical spectrum:

- If the pulse is initially unchirped or up-chirped, SPM leads to spectral broadening (an increase in optical bandwidth).
- Spectral compression can result if the initial pulse is downchirped (always assuming a positive nonlinear index).

For strong SPM, the optical spectrum can exhibit strong oscillations (see Figure 2). The reason for the oscillatory character is essentially that the instantaneous frequency undergoes strong excursions, so that in general there are contributions from two different times to the Fourier integral for a given frequency component. Depending on the exact frequency, these contributions may constructively add up or cancel each other (see also Ref. [7]).

The simulation has been done with the software RP Fiber Power.

### Spatial Effects

For a propagating laser beam, the optical intensity is not only time-dependent, but also spatially dependent. The nonlinear phase changes are then also spatially dependent, and that leads to nonlinear self-focusing.

## Self-phase Modulation in Optical Fibers

In optical fibers, SPM can be the dominant effect on an ultrashort pulse if the peak power is high (leading to strong SPM) while the chromatic dispersion is weak, so that the pulse duration remains approximately constant. Figure 3 shows an example case where that assumption is well fulfilled within the first 30 mm of fiber; here, the overall spectral width rises about linearly with the propagation distance. Thereafter, it grows faster because anomalous dispersion leads to pulse compression and thus to an increased peak power and an enhanced nonlinear interaction.

In optical fibers with anomalous chromatic dispersion, the chirp from self-phase modulation may be compensated by dispersion; this can lead to the formation of solitons. In the case of fundamental solitons in a lossless fiber, the spectral width of the pulses stays constant during propagation, despite the SPM effect.

In optical fibers with normal dispersion, a modulational instability can occur. That can also contribute to pulse break-up in supercontinuum generation.

## Self-phase Modulation in Semiconductors via Carrier Density Changes

The term *self-phase modulation* is occasionally used outside the context of the Kerr effect, when other effects cause intensity-dependent phase changes. In particular, this is the case in semiconductor lasers and semiconductor optical amplifiers, where a high signal intensity can reduce the carrier densities, which in turn lead to a modification of the refractive index and thus the phase change per unit length during propagation. Comparing that effect with SPM via the Kerr effect, there is an important difference: such carrier-related phase changes do not simply follow the temporal intensity profile because the carrier densities do not instantly adjust to modified intensities. That aspect is pronounced for pulse durations below the relaxation time of the carriers, which is typically in the range of picoseconds to a few nanoseconds.

## Self-phase Modulation in Mode-locked Lasers

Self-phase modulation has important effects in mode-locked femtosecond lasers. It results mainly from the Kerr nonlinearity of the gain medium, although for very long laser resonators even the Kerr nonlinearity of air can be relevant [5]. Without chromatic dispersion, the nonlinear phase shifts can be so strong that stable operation is no longer possible: the laser would not reach a steady state with well-defined pulses. In that case, (quasi-)soliton mode locking [4] is a good solution, where a balance of self-phase modulation and dispersion is utilized, similar to the situation of solitons in fibers.

## Self-phase Modulation via Cascaded Nonlinearities

Strong self-phase modulation can also arise from cascaded <$\chi^{(2)}$> nonlinearities. Basically this means that a not phase-matched nonlinear interaction leads to frequency doubling, but with subsequent backconversion. In effect, there is little power conversion to other wavelengths, but the phase changes on the original wave can be substantial. This effect may also be used to compensate self-phase modulation from other origins [6].

### Bibliography

[1] | F. Shimizu, “Frequency broadening in liquids by a short light pulse”, Phys. Rev. Lett. 19 (19), 1097 (1967); https://doi.org/10.1103/PhysRevLett.19.1097 (first demonstration of self-phase modulation) |

[2] | R. R. Alfano and S. L. Shapiro, “Observation of self-phase modulation and small-scale filaments in crystals and glasses”, Phys. Rev. Lett. 24 (11), 592 (1970); https://doi.org/10.1103/PhysRevLett.24.592 |

[3] | R. H. Stolen and C. Lin, “Self-phase-modulation in silica optical fibers”, Phys. Rev. A 17 (4), 1448 (1978); https://doi.org/10.1103/PhysRevA.17.1448 |

[4] | F. X. Kärtner et al., “Stabilization of solitonlike pulses with a slow saturable absorber”, Opt. Lett. 20 (1), 16 (1995); https://doi.org/10.1364/OL.20.000016 |

[5] | S. V. Marchese et al., “Pulse energy scaling to 5 μJ from a femtosecond thin-disk laser”, Opt. Lett. 31 (18), 2728 (2006); https://doi.org/10.1364/OL.31.002728 |

[6] | F. Saltarelli et al., “Self-phase modulation cancellation in a high-power ultrafast thin-disk laser oscillator”, Optica 5 (12), 1603 (2018); https://doi.org/10.1364/OPTICA.5.001603 |

[7] | R. Paschotta, "Effect of self-phase modulation on the pulse bandwidth" |

[8] | R. Paschotta, tutorial on "Passive Fiber Optics", Part 11: Nonlinearities of Fibers |

See also: Kerr effect, Kerr lens, solitons, cross-phase modulation, self-steepening, self-focusing, modulational instability, spotlight 2015-07-01

## Questions and Comments from Users

2021-07-12

How can we account for the ray trajectories of a Gaussian beam for the above-mentioned equation, rather than solely considering on-axis phase delay?

The author's answer:

In the simplest case, we still consider unidirectional propagation, just taking into account the spatially dependent nonlinear phase shifts. That leads to nonlinear self-focusing, as explained in the article.

If you consider ray trajectories in different directions, things become substantially more complicated. Quite sophisticated models are required for accurately describing for example the nonlinear effect on tightly focused light pulses, exhibiting strong beam divergence.

2023-03-16

Pure self-phase modulation leads to symmetrical broadening of the spectrum, but why do I see that self-phase modulation has a higher peak at shortwave in some articles？

The author's answer:

Indeed, you see that effect e.g. in Figure 2 on this page. The curve would be symmetrical when shown on a frequency axis, but if you calculate a power spectral density with wavelength units, the conversion makes it asymmetric. This is basically because the frequency range corresponding to a 1-nm wavelength interval, for example, gets larger as the wavelength gets shorter.

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2021-07-08

Is there a reference in which I can find the derivation of the self-phase modulation proportionality constant?

The author's answer:

I don't have a reference for that, but it is a fairly simple calculation which you can quickly do yourself.