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| '''Self-phase modulation''' (SPM) is a [[Nonlinear optics|nonlinear optical]] effect of [[light]]-[[matter]] interaction.
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| An [[ultrashort pulse]] of light, when travelling in a medium, will induce a varying [[refractive index]] of the medium due to the [[optical Kerr effect]]. This variation in refractive index will produce a [[phase (waves)|phase]] shift in the pulse, leading to a change of the pulse's [[frequency spectrum]].
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| Self-phase modulation is an important effect in [[optics|optical]] systems that use short, intense pulses of light, such as [[laser]]s and [[optical fiber|optical fibre communications]] systems.<ref>{{cite doi|10.1103/PhysRevA.17.1448}}</ref>
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| ==Theory==
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| [[File:Self-phase-modulation-en.svg|thumb|right|325px|A pulse (top curve) propagating through a nonlinear medium undergoes a self-frequency shift (bottom curve) due to self-phase modulation. The front of the pulse is shifted to lower frequencies, the back to higher frequencies. In the centre of the pulse the frequency shift is approximately linear.]]
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| For an ultrashort pulse with a [[Gaussian]] shape and constant phase, the intensity at time ''t'' is given by ''I''(''t'')'': | |
| :<math>I(t) = I_0 \exp \left(- \frac{t^2}{\tau^2} \right)</math>
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| where ''I''<sub>0</sub> is the peak intensity, and τ is half the pulse duration.
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| If the pulse is travelling in a medium, the [[optical Kerr effect]] produces a refractive index change with intensity:
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| :<math>n(I) = n_0 + n_2 \cdot I</math>
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| where ''n''<sub>0</sub> is the linear refractive index, and ''n''<sub>2</sub> is the second-order nonlinear refractive index of the medium.
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| As the pulse propagates, the intensity at any one point in the medium rises and then falls as the pulse goes past. This will produce a time-varying refractive index:
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| :<math>\frac{dn(I)}{dt} = n_2 \frac{dI}{dt} = n_2 \cdot I_0 \cdot \frac{-2 t}{\tau^2} \cdot \exp\left(\frac{-t^2}{\tau^2} \right).</math>
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| This variation in refractive index produces a shift in the instantaneous phase of the pulse:
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| :<math>\phi(t) = \omega_0 t - kx = \omega_0 t - \frac{2 \pi}{\lambda_0} \cdot n(I) L</math>
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| where <math>\omega_0</math> and <math>\lambda_0</math> are the carrier frequency and (vacuum) [[wavelength]] of the pulse, and <math>L</math> is the distance the pulse has propagated.
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| The phase shift results in a frequency shift of the pulse. The instantaneous frequency ω(''t'') is given by:
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| :<math>\omega(t) = \frac{d \phi(t)}{dt} = \omega_0 - \frac{2 \pi L}{\lambda_0} \frac{dn(I)}{dt},</math>
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| and from the equation for ''dn''/''dt'' above, this is:
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| :<math>\omega(t) = \omega_0 + \frac{4 \pi L n_2 I_0}{\lambda_0 \tau^2} \cdot t \cdot \exp\left(\frac{-t^2}{\tau^2}\right).</math>
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| Plotting ω(''t'') shows the frequency shift of each part of the pulse. The leading edge shifts to lower frequencies ("redder" wavelengths), trailing edge to higher frequencies ("bluer") and the very peak of the pulse is not shifted. For the centre portion of the pulse (between ''t'' = ±τ/2), there is an approximately linear frequency shift ([[chirp]]) given by:
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| :<math>\omega(t) = \omega_0 + \alpha \cdot t</math> | |
| where α is:
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| :<math>\alpha = \left. \frac{d\omega}{dt} \right |_0 = \frac{4 \pi L n_2 I_0}{\lambda_0 \tau^2}.</math>
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| It is clear that the extra frequencies generated through SPM broaden the frequency spectrum of the pulse symmetrically. In the time domain, the envelope of the pulse is not changed, however in any real medium the effects of [[dispersion (optics)|dispersion]] will simultaneously act on the pulse.<ref>{{cite doi|10.1364/JOSAB.9.001358}}</ref><ref>{{cite doi|10.1364/ON.15.1.000007}}</ref> In regions of normal dispersion, the "redder" portions of the pulse have a higher velocity than the "blue" portions, and thus the front of the pulse moves faster than the back, broadening the pulse in time. In regions of [[anomalous dispersion]], the opposite is true, and the pulse is compressed temporally and becomes shorter. This effect can be exploited to some degree (until it digs holes into the spectrum) to produce ultrashort pulse compression.
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| A similar analysis can be carried out for any pulse shape, such as the [[Hyperbolic function|hyperbolic secant]]-squared (sech<sup>2</sub>) pulse profile generated by most [[ultrashort pulse]] lasers.
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| If the pulse is of sufficient intensity, the spectral broadening process of SPM can balance with the temporal compression due to anomalous dispersion and reach an equilibrium state. The resulting pulse is called an optical [[soliton]].
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| == Applications of SPM ==
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| Self-phase modulation has stimulated many applications in the field of ultrashort pulse including to cite a few:
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| * spectral broadening<ref name="OE Parmigiani">{{cite doi|10.1364/OE.14.007617}}</ref> and [[supercontinuum]]
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| * temporal pulse compression<ref>{{cite doi|10.1109/JQE.1969.1081928}}</ref>
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| * spectral pulse compression<ref>{{cite doi|10.1364/OL.18.000699}}</ref>
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| The nonlinear properties of Kerr nonlinearity has also been beneficial for various optical pulse processing techniques such as optical regeneration<ref name="ECOC Mamyshev">{{cite doi|10.1109/ECOC.1998.732666}}</ref> or wavelength conversion.<ref name="PTL Parmigiani">{{cite doi|10.1109/LPT.2008.927887}}</ref>
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| ==Mitigation strategies in DWDM systems==
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| In long-haul single-channel and [[DWDM]] systems SPM is one of the most important reach limiting nonlinear effects. It can be reduced by:<ref>{{cite book |last1=Ramaswami |first1=Rajiv |last2=Sivarajan |first2=Kumar N. |title=Optical Networks: A Practical Perspective |year=1998 |edition=5th |publisher=[[Morgan Kaufmann Publishers]] |isbn=1-55860-445-6}}</ref>
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| * Lowering the optical power at the expense of increased noise
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| * Dispersion management, because dispersion can partly mitigate the SPM effect
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| ==See also==
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| Other non-linear effects:
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| * [[Cross-phase modulation]] — XPM
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| * [[Four wave mixing]] — FWM
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| * [[Modulational instability]]— MI
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| * [[Raman scattering|Stimulated Raman scattering]] — SRS
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| Applications of SPM:
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| * [[Supercontinuum]]
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| * [[Mamyshev 2R regenerator]]
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| ==Notes and references==
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| {{reflist}}
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| {{DEFAULTSORT:Self-Phase Modulation}}
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| [[Category:Optics]]
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| [[Category:Nonlinear optics]]
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