= Solution
A finite beam has an <intensity> gradient across its wavefront. In an instantaneous optical Kerr model, $n(r)=n_0+n_2I(r)$. A bright centre therefore sees a larger index for $n_2>0$, accumulating more phase than the wings and producing <self-focusing>. Negative $n_2$ gives <self-defocusing>. For a short medium and Gaussian $I(r)=I_0e^{-2r^2/w^2}$, expand the nonlinear phase near the axis:
$$
\Delta\varphi(r)=k_0n_2I_0L-\frac{2k_0n_2I_0L}{w^2}r^2+\cdots,
\qquad k_0=\omega/c.
$$
Compare with the in-medium thin-lens phase $-kr^2/(2f)$, where $k=n_0k_0$. The <Kerr lens> has \b[$1/f=4n_2I_0L/(n_0w^2)$]. This local quadratic fit describes the central lens, not the whole Gaussian phase or a guaranteed aberration-free focus.
Propagation combines <diffraction> with the nonlinear phase. Choose $|A|^2=I$ and a scalar paraxial envelope, giving
$$
i\partial_zA+\frac{1}{2k}\nabla_\perp^2A+\gamma|A|^2A=0,\qquad \gamma=k_0n_2.
$$
The <diffraction> length is of order $kw^2$, while the nonlinear length is $1/(|\gamma|I_0)$. Their competition depends on total beam power as well as spot size. In two transverse dimensions, balancing them gives a critical power scale \b[$P_{\rm cr}\sim\lambda_0^2/(n_0n_2)$] for positive $n_2$; the numerical constant requires a specified transverse profile and critical solution. In the ideal unsaturated cubic model, sufficiently strong focusing can collapse rather than form a stable beam. Real saturation, <absorption>, ionization or nonparaxial propagation can stop that ideal singularity; the appropriate mechanism depends on the material and pulse regime.
A broad focusing beam can also break into filaments. Perturb the uniform solution $A_0e^{i\gamma I_0z}$ by real amplitude and phase components with transverse wave number $q$. Linearization gives the <transverse modulational instability> rate
$$
\boxed{g^2=\frac{q^2}{2k}\left(2\gamma I_0-\frac{q^2}{2k}\right).}
$$
For positive $\gamma$, modes with $0<q^2<4k\gamma I_0$ grow. <diffraction> suppresses very fine perturbations. A finite beam's boundaries and available propagation length limit which filaments develop. The defocusing sign has no growing band in this simple instantaneous local model.
A <spatial optical soliton> instead has a profile whose nonlinear focusing balances <diffraction>. A one-transverse-dimensional cubic focusing model supports a stationary bright solution; the existence of that solution does not establish robust two-dimensional cubic self-trapping. A beam can also act on another finite beam through <cross-phase modulation>, producing an induced lens or steering it towards a high-index region. Unequal polarization response can rotate or reshape the profile. Temporal <self-phase modulation> is a related consequence of a time-dependent <intensity>, but should be distinguished from these transverse effects.
\Image[/past-exam-of-the-nonlinear-optics-and-quantum-states-of-light-course-of-the-university-of-cambridge/2015/paper-1-transverse-nonlinearity.png]
{title=Original Gaussian Kerr-lens phase and the focusing model's transverse instability band}
For an experimental check, vary power at fixed input waist and compare near- and far-field profiles with a low-power <diffraction> reference. Repeat with a second beam blocked to separate <cross-phase modulation> from self-action. Vary pulse duration or repetition rate to test thermal lensing against a prompt electronic response, and monitor transmission to distinguish nonlinear refraction from preferential <absorption>. A smaller output spot alone does not establish a soliton or a pure Kerr response.
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