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Put $g=\gamma P_0$. For a temporally uniform envelope, $A_z=igA$ and $A_{TT}=0$, so the propagation equation reduces to $-gA+\gamma P_{0}A=0$. Thus the power stays $P_0$ and the accumulated nonlinear wave phase is $gz$. The optical Kerr effect gives an intensity-dependent refractive index, and hence self-phase modulation. For the usual power normalization $\gamma \simeq n_2{\omega }_0/(c{A}_{\mathrm{eff}})$, with effective guided-mode area $A_{\mathrm{eff}}$. Losslessness preserves the background power; it does not remove the self-phase modulation. The carrier wave phase has already been factored out.

Write $A=(\sqrt{P_0}+ϵ)e^{igz}$. To first order,Cancelling the background and its phase derivative leavesThe last term couples a perturbation to its complex conjugate: the pump can transfer energy into a pair of sidebands. It is not an independent wave phase of each Fourier component. This sideband coupling is the linear origin of fibre modulational instability.

For real $K,\Omega ,c_1,c_2$, put $d={\beta }_2{\Omega }^2/2$. Matching the two exponential coefficients givesA nonzero pair requires a zero determinant, henceFor unstable complex $K$, the real-coefficient assumption is too restrictive. The consistent pair is $ϵ=u{e}^{iKz-i\Omega T}+v^{\ast }{e}^{-i{K}^{\ast }z+i\Omega T}$, with complex $u,v$. Its equations give the same determinant and allow the growing and decaying branches. Merely inserting imaginary $K$ while treating the two original coefficients as real would not respect complex conjugation.

For $P_0>0$ and nonzero ${\beta }_2,\gamma$, exponential growth requires ${\beta }_2\gamma <0$ andFor ordinary positive Kerr nonlinearity this means anomalous group velocity dispersion, ${\beta }_2<0$. Define ${\Omega }_c=\sqrt{4g/|{\beta }_2|}$. The growing branch is $K=-iG$, with amplitude growth rateSideband power grows at rate $2G$, rather than $G$. The cutoff and zero-frequency modes have no exponential growth; in a degenerate zero-frequency sector a secular phase response can remain. Positive dispersion with positive $\gamma$ gives real $K$ and oscillatory perturbations.

Initially small ripples grow into pronounced intensity peaks and deplete the uniform background. Dispersion and self-phase modulation can then support localized bright pulses. A fundamental temporal bright soliton obeys $\gamma P_s{T}_0^2=|{\beta }_2|$, so its width is comparable to $1/{\Omega }_c$ when its peak power is comparable to the original power. The dominant ripple spacing is $2\pi /|\Omega {|}_{\max }$, not simply $1/{\Omega }_c$. Propagation distance $z$ is the evolution variable; $T$ measures time within the envelope. An ideal conservative fibre need not settle permanently into a unique pulse train: nonlinear compression, interactions and recurrence depend on the seed and boundary conditions.

A detector too slow to resolve a femtosecond field can measure its integrated nonlinear response instead. Split the pulse, delay one replica, overlap both in a thin phase-matched crystal, and select the sum-frequency signal. For negligible depletion and sufficiently broad phase matching, the cross-generated field is proportional to $A(t)A(t-\tau )$; integrated signal gives the second-harmonic intensity autocorrelationA noncollinear geometry can reject each replica's individual second harmonic. The optical response supplies the fast gate while the detector integrates slowly. Crystal thickness, group-velocity mismatch, spectral acceptance, beam overlap and delay calibration limit resolution.

For a Gaussian pulse specified by its intensity profile, the autocorrelation's full width at half maximum is $\sqrt{2}$ times the intensity full width at half maximum. For a sech-squared intensity profile the ratio is about $1.543$. These are model-dependent conversion factors. An intensity autocorrelation is symmetric in delay and loses phase information; different pulses can share it. An interferometric autocorrelation retains rapid carrier fringes but still does not generally reconstruct an arbitrary complex field uniquely.

frequency-resolved optical gating also resolves the gated signal's spectrum, giving a two-dimensional delay-frequency record. For second-harmonic frequency-resolved optical gating,where $\omega$ is measured relative to the signal's carrier. An iterative retrieval seeks an envelope whose predicted record matches the measurement. Alternatively, polarization-gating frequency-resolved optical gating uses the optical Kerr effect to gate the probe by a delayed replica's intensity, with signal envelope proportional to $A(t)|A(t-\tau ){|}^2$. The original FROG work introduced nonlinear optical gating plus spectral resolution to characterize arbitrary femtosecond pulses.^[ref]

Absolute wave phase and absolute time origin remain undetermined by these intensity records. The SHG record also cannot distinguish $A(t)$ from $A^{\ast }(-t)$, so its time-direction ambiguity remains. Retrieval agreement, an independently measured spectrum, adequate signal dynamic range and gate-response calibration are practical consistency checks. A pulse train that changes between shots can yield an averaged record that no single pulse faithfully represents. Thus nonlinear gating solves the detector-speed problem but does not remove inverse problem ambiguities or experimental limitations.

The optical Kerr effect gives an intensity-dependent refractive index, $n=n_0+n_{2}I$, in the simplest instantaneous isotropic model. A beam whose intensity varies transversely therefore writes a transverse index profile into the medium. Over length $L$ its nonlinear phase is approximately ${\varphi }_{\mathrm{NL}}({\mathbf{r}}_{\perp })=k_0{n}_{2}I({\mathbf{r}}_{\perp })L$, with $k_0=\omega /c$.

For a bright-centered beam, positive $n_2$ raises the index at the center and produces self-focusing; negative $n_2$ produces self-defocusing. The medium acts as an intensity-dependent lens. This is a transverse effect, distinct from temporal self-phase modulation broadening a pulse's spectrum, though both originate in the same nonlinear index. Kerr lenses can provide intensity-dependent mode selection in a laser, and a beam's index profile can guide or deflect a second beam through cross-phase modulation.

The competition with diffraction appears in a paraxial model,In one transverse dimension, focusing can balance diffraction to support a bright spatial optical soliton under suitable boundary conditions. Transverse perturbations can also grow through modulational instability, breaking a broad beam into localized structures. In an ideal two-transverse-dimensional cubic model, sufficiently strong focusing permits collapse; the critical-power scale is proportional to ${\lambda }^2/(n_0{n}_2)$, with its numerical coefficient depending on the convention and profile used.

A real medium does not develop an actual infinite intensity. Saturation, multiphoton absorption, ionization, nonparaxial propagation and damage can arrest or invalidate the simple model. Filament formation therefore requires more physics than inserting a real constant $n_2$ into a lens formula. The useful applications exploit a controllable index profile; the same positive feedback can be a serious limitation on transporting high-power beams.