B-integral (nonlinear phase)

The B-integral is the nonlinear phase a beam accumulates along its path through the optical Kerr effect, evaluated at the peak of the pulse on the beam axis. Since the refractive index rises with intensity as $n = n_0 + n_2 I$, each element of length $dz$ adds a phase $(2\pi/\lambda)\,n_2 I\,dz$, and the total through a crystal, a fiber or a whole amplifier chain is

$$B = \frac{2\pi}{\lambda}\int n_2\,I(z)\,dz .$$

For fused silica, with $n_2 = 2.6 \times 10^{-20}$ m²/W, an intensity of 1 GW/cm² over 1 cm at 1064 nm gives $B$ = 0.015 rad, the value worked in the nonlinear optics entry. One radian at the same intensity would take 65 cm of glass. High-power amplifiers usually aim for a total of 1 to 3 rad or less; the exact limit depends on the beam quality and pulse fidelity required.

Why it matters

The B-integral is a single number that tracks three distinct kinds of damage to the pulse.

Self-focusing. The intensity is highest on axis, so the Kerr index change forms a positive lens across the beam. When the power exceeds the critical power, about 4.5 MW in fused silica at 1064 nm, the whole beam collapses (self-focusing). Below that, a large beam carrying many critical powers still suffers small-scale self-focusing: ripples on the beam profile grow, and in the Bespalov–Talanov analysis the fastest-growing spatial frequency is amplified by roughly $e^B$, a factor of about 20 at $B$ = 3. The ripples become filaments that damage optics. Large Nd:glass chains limit this with spatial filters between stages that strip the high spatial frequencies before they grow.

Self-phase modulation. The intensity also varies in time, so the phase does too, and the pulse acquires a chirp and a broadened spectrum (self-phase modulation). Near the pulse center the chirp is nearly linear, but the wings are not, and a pulse compressor designed for the stretcher's dispersion cannot remove it. As $B$ grows beyond about 1 rad the compressed pulse increasingly develops a pedestal and side lobes, and at a peak phase of $2.5\pi$ (7.9 rad) the spectrum shows three peaks.

Wavefront distortion. Because $B$ follows the transverse intensity profile, a beam with a nonuniform profile carries a matching phase map, which degrades focusability even when no filamentation occurs.

Chirped-pulse amplification

Chirped-pulse amplification was developed largely to keep the peak power, and with it $B$, low inside the amplifier. That entry works an example: 10 mJ at 800 nm in a Gaussian beam of 2.5 mm radius through 2 cm of sapphire gives $B \approx 45$ rad at 100 fs and about 0.015 rad once the pulse is stretched to 300 ps. A regenerative amplifier is the hardest case, since the pulse passes the crystal and the Pockels cell on each of its round trips, typically 10 to several tens and each pass adds to the total. Because $B$ scales with peak power, stretching the pulse by a factor of a thousand reduces it by the same factor.

The same quantity in fiber

In a fiber the Kerr phase is usually written with the nonlinear coefficient $\gamma = 2\pi n_2/(\lambda A_\text{eff})$:

$$\phi_\text{NL} = \gamma\,P\,L_\text{eff} ,$$

where $P$ is the peak power and $L_\text{eff}$ the effective length. This is the B-integral with the intensity replaced by power over effective area. Standard single-mode fiber at 1550 nm has $\gamma$ = 1.32 W⁻¹km⁻¹, so 1 W over 1 km gives 1.3 rad. Fiber amplifiers reach large $B$ quickly: a 10 µJ pulse stretched to 1 ns has a peak power near 10 kW, and in a 30 µm mode-field-diameter core at 1030 nm ($A_\text{eff} \approx 707$ µm², $\gamma \approx 2.2 \times 10^{-4}$ W⁻¹m⁻¹) two meters at that power would add about 4.5 rad. This is an upper bound, since the energy grows along the gain fiber and reaches its final value only near the end, but it shows why fiber chirped-pulse systems use large cores and long stretched pulses.

Pitfalls

Values of $n_2$ for the same glass vary by tens of percent among measurements and depend on pulse duration, so a computed $B$ carries the same uncertainty. Intensity must be the peak value, $2P/(\pi w^2)$ on axis for a Gaussian beam of radius $w$, rather than the average over the beam. Every transmissive element counts: windows, polarizers, Pockels cells and air paths at high intensity all contribute, and in a multipass amplifier the contributions of all passes add. Some texts give $n_2$ in electrostatic units or define it through the squared field rather than the intensity; conversion errors between these conventions are a common source of order-of-magnitude mistakes.

Common questions

What B-integral is acceptable?

For good beam quality and clean compression, designs typically keep the total below about 1 rad; some systems accept 2–3 rad with spatial filtering or spectral phase correction. Values of several radians or more without such measures generally lead to filamentation or degraded pulses.

Can the nonlinear phase be compensated?

Partly. The linear part of the SPM chirp near the pulse center can be absorbed by adjusting the compressor, and adaptive spectral phase shaping can correct more, but the spatial part (small-scale self-focusing) cannot be undone once ripples have grown.

References: W. Koechner, Solid-State Laser Engineering, 6th ed. (Springer, 2006); R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, 2008); G. P. Agrawal, Nonlinear Fiber Optics, 5th ed. (Academic Press, 2013); V. I. Bespalov and V. I. Talanov, "Filamentary structure of light beams in nonlinear liquids," JETP Lett. 3, 307 (1966); D. Strickland and G. Mourou, "Compression of amplified chirped optical pulses," Opt. Commun. 56, 219 (1985).