# Delayed Nonlinear Response

Author: the photonics expert Dr. Rüdiger Paschotta

Definition: the phenomenon that the nonlinear polarization of a medium does not instantly follow the electric field strength

Opposite term: instantaneous nonlinear response

In nonlinear optics, one frequently assumes nonlinearities in transparent optical materials to be described by a nonlinear polarization which instantly follows the driving electric field, and thus does not depend on the electric field strength at earlier times. For example, the <$\chi^{(3)}$> nonlinearity, on which we focus in this article because it is most relevant in this context, results in a nonlinear polarization proportional to the third power of the electric field strength (ignoring the tensor nature of the nonlinearity):

$${P_{{\rm{nl}}}}(t) = {\varepsilon _0}{\chi ^{(3)}}E(t)\;{\left| {E(t)} \right|^2}$$where the square modulus of the field strength is related to the optical intensity. This is a reasonable approximation in many situations, where e.g. the optical Kerr effect is observed. However, the nonlinear response actually has different contributions which behave differently in that respect:

- The response of the electrons of the medium is normally assumed to be virtually instantaneous.
- There are also contributions from vibrations of the crystal lattice, which can be excited by intense electric fields and influence the polarization of the medium. Such contributions to the nonlinear polarization occur on rather short timescales, but still long enough to be substantially non-instantaneous e.g. in the context of ultrashort pulses of light.

Such a non-instantaneous response means that the induced nonlinear polarization at a certain time <$t$> depends not only on the electric field intensity at that time, but rather on that intensity during some time interval before <$t$>. This can be described with a response function <$R(t)$>:

$${P_{{\rm{nl}}}}(t) = {\varepsilon _0}{\chi ^{(3)}}E(t)\int\limits_0^\infty {R(\tau )} \;{\left| {E(t - \tau )} \right|^2}\;{\rm{d}}\tau $$The response function is in principle defined for arbitrarily large time delays, but essentially vanishes within a certain time, during which the system “forgets” any influences from that distant past. There are no contributions from negative time delays because those would violate the principle of causality.

An instantaneous response would simply be described with a response function which is a delta function. The above-mentioned combination of a virtually instantaneous electronic response and a delayed response related to lattice vibrations leads to a combination of a delta function with an oscillatory (but decaying) function <$h(\tau )$>:

$$R(\tau ) = (1 - {f_{\rm{R}}})\;\delta (\tau ) + {f_{\rm{R}}}\;h(\tau )$$Here, the factor <$f_\textrm{R}$> quantifies how strong the contribution of the oscillatory <$h$> function is; that function is normalized such that its integral over all non-negative time arguments is unity. In a situation where the nonlinear polarization is driven by a relatively long light pulse, the result will be approximately the same as if one would set <$f_\textrm{R}$> = 0. For shorter light pulses, however, the oscillatory term makes a difference.

### Simulations on Pulse Propagation

Stimulated Raman scattering comes together with many other effects, making ultrashort pulse propagation rather complicated. Only with a suitable simulator, you can get 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.

## Brillouin and Raman Scattering

Lattice vibrations of solid state media are quantized as phonons. One distinguishes two types of those:

*Acoustic phonons*are related to long-wavelength vibrations, where neighbored ions oscillate nearly in phase. They have relatively low frequencies in the gigahertz region and are related to Brillouin scattering.*Optical phonons*involve the oscillation of neighbored ions with a phase difference around 180°. They have much higher frequencies in the terahertz region and are related to Raman scattering.

In the context of ultrashort pulses, one usually needs to consider only Raman scattering. One then uses a *Raman response function* <$h_\textrm{R}(\tau )$>, which typically exhibits oscillations with terahertz frequencies, vanishing within a couple of picoseconds.

The imaginary part of the Fourier transform of the Raman response function <$h_{\rm R}(t)$> is related to the nonlinear gain spectrum (e.g., the Raman gain spectrum). From a measured Raman gain spectrum, one can thus obtain that imaginary part, from that calculate the real part using Kramers–Kronig relations and finally obtain the Raman response function.

## Delayed Nonlinear Response in Ultrashort Pulse Simulations

Ultrashort pulses are often described with a complex amplitude <$A(z, t)$>, if propagation only in one spatial dimension is relevant, for example if the pulses propagate in a single-mode fiber. One then describes the propagation with a partial differential equation for that amplitude. For example, if linear absorption with an absorption coefficient <$\alpha$>, second-order chromatic dispersion with a coefficient <$\beta_2$> and the Kerr nonlinearity with a nonlinear coefficient <$\gamma$> (related to the nonlinear index) and the response function <$R(\tau )$> are relevant, that equation is [6] (with no self-steepening term included):

$$\frac{{\partial A}}{{\partial z}} = - \frac{\alpha }{2}A - i\frac{{{\beta _2}}}{2}\frac{{{\partial ^2}A}}{{\partial {t^2}}} + i\gamma \;A(z,t)\int\limits_0^\infty {R(\tau )\;{{\left| {A(z,t - \tau )} \right|}^2}{\rm{d}}\tau } $$Higher-order chromatic dispersion terms can also be easily added.

The amplitudes are assumed to be normalized such that <$|A(z,t)|^2$> is the optical power, and the nonlinear coefficient <$\gamma$> is calculated such that it takes into account the effective mode area.

The Raman response function in the time domain is related to the Raman gain spectrum: the latter is proportional to the imaginary part of the Fourier transform of the Raman response function. If the latter is experimentally measured, the Raman response function can be derived from it.

In simple models, one often uses a Raman response function of the form

$${h_\rm{R}}(t) = \frac{{\tau _1^2 + \tau _2^2}}{{{\tau _1}\;\tau _2^2}}\;\sin (t/{\tau _1})\;\exp ( - t/{\tau _2})$$which describes a simple damped sinusoidal oscillation, related to a single vibrational mode and characterized by only two parameters <$\tau_1$> and <$\tau_2$>. (For fused silica, one may use the values <$\tau_1$> = 12.2 fs and <$\tau_2$> = 32 fs, and <$f_\rm{R}$> = 0.18 [1].) This is sufficient in some cases, but for more accurate simulations one requires more detailed Raman response data [1, 2, 3], taking into account multiple vibrational modes of the medium. Figure 1 shows an example for fused silica.

An instantaneous nonlinear response could be described with the response function <$R(\tau ) = \delta (\tau )$>. That would let the integral simply be <$|A(z,t)|^2$>, and the equation turns into the well-known nonlinear Schrödinger equation.

## More to Learn

Encyclopedia articles:

### Bibliography

[1] | K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers”, IEEE J. Quantum Electron. 25 (12), 2665 (1989); https://doi.org/10.1109/3.40655 |

[2] | R. H. Stolen et al., “Raman response function of silica-core fibers”, J. Opt. Soc. Am. B 6 (6), 1159 (1989); https://doi.org/10.1364/JOSAB.6.001159 |

[3] | D. Hollenbeck and C. D. Cantrell, “Multiple-vibrational-mode model for fiber-optic Raman gain spectrum and response function”, J. Opt. Soc. Am. B 19 (12), 2886 (2002); https://doi.org/10.1364/JOSAB.19.002886 |

[4] | Q. Lin and G. P. Agrawal, “Raman response function for silica fibers”, Opt. Lett. 31 (21), 3086 (2006); https://doi.org/10.1364/OL.31.003086 |

[5] | G. P. Agrawal, “Nonlinear fiber optics: its history and recent progress”, J. Opt. Soc. Am. B 28 (12), A1 (2011); https://doi.org/10.1364/JOSAB.28.0000A1 |

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

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