Engineering & Science
Optical Kerr Effect Nonlinear Refractive Index & Laser B-Integral
Calculate intensity-dependent Kerr refractive index shift delta_n = n_2 * I and accumulated ultrafast laser amplifier B-integral = (2 * PI() / lambda_0) * n_2 * I * L_eff (radians) to prevent catastrophic filamentation breakup.
Keep ultrafast chirped-pulse amplifiers below the B-integral < 2–3 radian breakup limit!
At gigawatt-per-cm^2 ultrafast laser peak intensities, every transparent optical glass exhibits the third-order chi^(3) Optical Kerr Effect: local refractive index rises linearly with irradiance I (W/m^2) via n(I) = n_0 + n_2 * I! Because a Gaussian laser beam is brightest at its center, Kerr delta_n acts as a self-induced focusing lens (Kerr self-focusing) and imprints self-phase modulation across optical path length L_eff (m).
High-energy laser engineers track total accumulated on-axis nonlinear phase shift via the dimensionless B-Integral (in radians): B = (2 * PI() / lambda_0) * n_2 * I * L_eff! Passing a lambda_0 = 1.053 um Nd:glass pulse at peak intensity I = 15 GW/cm^2 (1.5e14 W/m^2) through L_eff = 8.0 cm (0.08 m) of fused silica (n_2 = 2.7e-20 m^2/W) accumulates B = 1.933 rad ("Safe < 2.0 rad CPA Window"); doubling peak intensity to 30 GW/cm^2 pushes B to 3.867 rad—triggering small-scale Bespalov-Talanov filamentation that shatters expensive amplifier slabs!
| Ultrafast Pulse Intensity (lambda = 1.053 um, n2 = 2.7e-20 m^2/W, L = 8 cm) | B-Integral (radians) | Laser Damage & Beam-Quality Regime |
|---|---|---|
| 15 GW/cm^2 Stretched Pulse (1.5e14 W/m^2) | 1.933 rad | Safe Amplifier Window (B <= 2.0 rad) |
| 30 GW/cm^2 Unstretched Spike (3.0e14 W/m^2) | 3.867 rad | Filamentation Danger (B > 3.0 rad)! |
| Invalid Wavelength or Optical Path <= 0 | Guarded Optical Error | "Invalid Kerr Inputs" |
Accumulated Nonlinear Phase Shift B-Integral (radians)
Inputs n_2 in units of 10^-20 m^2/W (B3) and peak intensity I in GW/cm^2 (B4, where 1 GW/cm^2 = 10^13 W/m^2, so n_2 * I = B3 * B4 * 10^-7!).
=IF(OR(B2 <= 0, B3 <= 0, B4 < 0, B5 <= 0), "Invalid Kerr Inputs", ROUND((2 * PI() / (B2 * 0.000001)) * (B3 * B4 * 0.0000001) * B5, 3))CPA Small-Scale Self-Focusing Breakup Safety Status
Classifies B <= 2.0 rad as Safe, 2.0 < B <= 3.0 rad as Caution, and B > 3.0 rad as High Filamentation Damage Risk.
=IF(OR(B2 <= 0, B3 <= 0, B4 < 0, B5 <= 0), "Invalid Kerr Inputs", IF(((2 * PI() / (B2 * 0.000001)) * (B3 * B4 * 0.0000001) * B5) > 3, "Danger: Filamentation Risk (B > 3 rad)", IF(((2 * PI() / (B2 * 0.000001)) * (B3 * B4 * 0.0000001) * B5) > 2, "Caution: Marginal CPA Phase (2-3 rad)", "Safe CPA Regime (B <= 2 rad)")))Try the example
Use the original values or change them to test your understanding.