Optics 100 · Kerr self-trapping in a planar guide

Nonlinear Light Shapes Its Own Medium

When can a bright beam raise the refractive index enough to guide itself and stop spreading? When the Kerr effect, an index rise proportional to intensity, is strong enough to cancel diffraction. The bright centre of the beam raises the index most, so the beam writes its own waveguide and travels without spreading, as a spatial soliton. In a thin planar guide, where light spreads only sideways, the balance is stable; in bulk glass it is unstable and a stronger beam collapses.

Send 100 fs pulses into a planar glass guide. Near soliton power the beam writes its own waveguide; reverse the Kerr effect and it spreads, widen it and it breaks up.

Output width at the pulse peak (brightest line) $\text{—}$

Output width of weak light, which only diffracts $\text{—}$

Extra phase at the beam centre at the pulse peak $\text{—}$

Spectrum at the output centre: width over the input’s $\text{—}$

Output width on a slow camera, summed over the pulse $\text{—}$

Peak power over the fundamental soliton’s $\text{—}$

Guide length in soliton periods of 3.73 mm $\text{—}$

Bright lines in the wide beam’s output $\text{—}$

Mean spacing of those lines $\text{—}$

Period of the fastest-growing ripple, plane-wave theory $\text{—}$

Light that left the computed window through its sides $\text{—}$

Instants of the pulse computed $\text{—}$

Model of the planar glass waveguide of Aitchison et al. (1991): 620 nm pulses 100 fs long, a 22 µm wide beam (full width at half maximum) and a linear index of 1.5. The thin guiding layer holds the light across its thickness, so the beam can spread only sideways, and its envelope follows the one-dimensional nonlinear Schrödinger equation that Zakharov and Shabat (1972) solved exactly. Widths are measured in units of 12.5 µm, the width parameter of a fundamental soliton 22 µm wide; lengths in the diffraction length of 2.38 mm; the soliton period is 3.73 mm. Peak intensity is given in units of the peak intensity of that soliton. The equation is integrated by the symmetric split-step Fourier method on 512 points across ±600 µm, with an absorbing band in the outer 100 µm on each side and a step limited so the Kerr phase changes by at most 0.1 rad. Each instant of the pulse travels through the guide on its own: the pulse is resolved into 24 instantaneous intensities from zero to the peak, and the camera image and the spectrum at the centre of the output are built from them. A random ripple of 1 % root-mean-square amplitude, smooth on the scale of 2 µm, is added to the input. Not included: dispersion, two-photon absorption, Raman scattering, self-steepening, saturation of the Kerr effect and the time the glass takes to respond, so the model keeps narrowing at powers where the 1991 experiment stopped. Widths across the guide are drawn about 17 times larger than lengths in the scene.

Physics tutorial

Let a beam write its own waveguide

BackgroundA narrow beam spreads as it travels: the narrower it is, the faster it spreads. That is diffraction, and a 22 µm beam of red light doubles its width in about 5 mm of glass.

Why it mattersIn glass the refractive index rises slightly where light is bright. A bright beam therefore raises the index most at its own centre, like a lens that only exists while the light is there. Aitchison and colleagues sent 100 fs pulses into a thin planar glass guide, where the light can spread only sideways, and watched the output on a camera.

Start with the essentials

Focus question: At what intensity does the self-made lens exactly cancel diffraction, and what happens above and below it?
One-sentence intuition: Diffraction and the Kerr lens both act over the diffraction length $L_D=kw^2$ , 2.38 mm here. When the peak intensity is near the soliton intensity $I_{\mathrm{s}}$ they balance: the beam keeps a fixed width, a spatial soliton. Each instant of a pulse has its own intensity, so only the bright middle of the pulse is trapped; its weak front and tail still spread.

Core mathematical model

Index that follows the light

n=n_0+n_2I

The Kerr effect: the index rises in proportion to the intensity $I$ . In glass $n_2>0$ , so bright light focuses itself; the reversed setting uses $n_2<0$ .

Envelope of the beam

i\frac{\partial u}{\partial\zeta}+\frac{1}{2}\frac{\partial^2u}{\partial\xi^2}+\sigma|u|^2u=0

Here $\zeta=z/L_D$ is distance along the guide, $\xi=x/w$ position across it with $w=12.5\,\mu\mathrm{m}$ , $|u|^2$ is intensity in units of $I_{\mathrm{s}}$ and $\sigma=\pm1$ is the sign of $n_2$ . The second term is diffraction, the third the Kerr lens (Zakharov and Shabat 1972, eq. 3).

The fundamental soliton

\begin{gathered}u=\mathrm{sech}(\xi)\,e^{i\zeta/2}\\z_0=\frac{\pi}{2}L_D=\frac{\pi^2n_0w^2}{\lambda}\end{gathered}

This beam keeps its shape forever. A beam with four times its intensity returns to its input shape after each soliton period $z_0$ , 3.73 mm for the 22 µm beam (Aitchison et al. 1991).

Self-phase modulation

\begin{gathered}\delta\omega(t)=-\frac{\partial\phi_{\mathrm{NL}}}{\partial t}\\\frac{\Delta\omega_{\mathrm{rms}}}{\Delta\omega_0}=\sqrt{1+\frac{4}{3\sqrt3}\,\phi_{\max}^2}\end{gathered}

The bright middle of the pulse picks up more phase $\phi_{\mathrm{NL}}$ than its edges, so the frequency shifts while the pulse passes. For a Gaussian pulse whose phase simply follows its intensity, the root-mean-square spectral width grows as the second line shows; in the guide the phase also changes as the beam reshapes.

Ripples that grow

\begin{gathered}g(\kappa)=|\kappa|\sqrt{I_0-\kappa^2/4}\\\Lambda_{\mathrm{fast}}=\frac{2\pi w}{\sqrt{2I_0}}\end{gathered}

On a wide beam of intensity $I_0$ (in units of $I_{\mathrm{s}}$ ), a ripple across it with wavenumber $\kappa$ (in units of $1/w$ ) grows by the factor $e^{g}$ per diffraction length. The fastest ripple has period $\Lambda_{\mathrm{fast}}$ , 45 µm at $I_0=1.5$ .

Common difficulties

There is no lens in the glass

Typical misconceptionThe guide has a built-in lens or index channel that holds the beam.

Better mental modelAcross the guide the glass is uniform. The index channel exists only where and while the light is bright, which is why the weak front and tail of each pulse still spread.

More power is not a steadier beam

Typical misconceptionAbove the soliton intensity the beam just gets narrower and stays there.

Better mental modelIn a planar guide extra power makes the width oscillate along the guide, and a wide beam breaks into several. In bulk glass, where light can spread in two directions, the same balance is unstable and the beam collapses (Aitchison et al. 1991).

The spectrum broadens without a new frequency source

Typical misconceptionSelf-phase modulation adds new light at other colours from somewhere else.

Better mental modelThe pulse only changes its own phase in time. A phase that changes in time is a frequency shift, so the same light is spread over a wider band.

Run the experiment

01
Watch the beam hold its width
Keep the default preset, Light writes its own waveguide, and watch the top view as the pulse passes.
What to observe: At the pulse peak the light runs as one straight line and leaves 10 mm of glass 24 µm wide, close to its 22 µm input. Weak light, which only diffracts, comes out 86 µm wide. The front and tail of the pulse spread.
02
Compare the camera with the peak
Look at Output across the guide below the scene.
What to observe: The dashed camera curve is 32 µm wide, wider than the 24 µm at the pulse peak, because a camera adds up the whole pulse. The 1991 experiment also recorded whole pulses, and its narrowest camera image came at about 1.5 times the soliton power.
03
Turn the power down
Drag the attenuator wheel down, or load Too weak: the beam spreads.
What to observe: At a quarter of the soliton intensity the beam spreads to 71 µm, not much less than the 86 µm of weak light.
04
Four times the intensity
Load Four times the intensity: the beam breathes, then open Width along the guide and phase at the centre.
What to observe: The beam narrows to about 5 µm, widens again to 16 µm and keeps doing so every 3.6 mm, close to the 3.73 mm soliton period. The centre picks up 29 rad of extra phase and the spectrum becomes 32 times wider.
05
Reverse the Kerr effect
Load Reversed Kerr effect: the beam defocuses itself.
What to observe: Now bright light lowers the index. At three times the soliton intensity the beam spreads to 210 µm, far more than the 86 µm of weak light.
06
Break up a wide beam
Load A wide beam breaks into filaments and look at Angles leaving the output.
What to observe: The 300 µm beam breaks into four bright lines about 59 µm apart. The output gains side angles near 7 mrad; plane-wave theory puts the fastest-growing ripple at 45 µm and 9 mrad.

Nonlinear Light Shapes Its Own Medium

Let a beam write its own waveguide

Core mathematical model

Index that follows the light

Envelope of the beam

The fundamental soliton

Self-phase modulation

Ripples that grow

Common difficulties

There is no lens in the glass

More power is not a steadier beam

The spectrum broadens without a new frequency source

Run the experiment

Watch the beam hold its width

Compare the camera with the peak

Turn the power down

Four times the intensity

Reverse the Kerr effect

Break up a wide beam

Related experiments and articles

Controlled light and precision measurement