Why are laser pulses Sech Squared in temporal shape?

Question

Ultrashort pulses from mode-locked lasers often have a temporal shape which can be described with a squared hyperbolic secant ($\mathrm{sech}^2$) function:

$$ P(t)=P_0 \mathrm{sech}^2 \left( \frac{t}{\tau} \right) = \frac{P_0}{\mathrm{cosh}^2\left( \frac{t}{\tau} \right)} $$

This fucntion looks similar to but is subtly different from the Gaussian (normal distribution) function. The Gaussian function shows up in many different physical phenomenon and its appearance can be explained by the Central Limit Theorem.

Is there a similar theorem or theory to explain the appearance of the $\mathrm{sech}^2$ fuction in pulsed laser dynamics.

Answer 1

The ${\rm sech}$ pulse is, in Kerr effect nonlinear optical mediums, an Optical Soliton.

This means that it is the particular time variation such that the tendency of the pulse to spread out in time owing to linear dispersion is exactly counterbalanced by the nonlinear effect that tends to confine pulses in time. This balance is a stable one in a Kerr medium, meaning that small perturbations of the ${\rm sech}$ pulse tend to decay. Alternatively, a pulse that looks vaguely like a ${\rm sech}$ pulse will evolve towards the latter. This means that, at high power, the nonlinear lasing medium will tend to produce ${\rm sech}$ pulses. The Kerr model, where the refractive index varies like $n_0 + \kappa |\vec{E}|^2$ (where $\vec{E}$ is the electric field envelope) is a good first approximation to many nonlinear mediums.

As you can see, this has nothing to do with the central limit theorem, which explains the emergence of Gaussian probability distributions from the summing, or general linear operations, on a large number of identically distributed random variables. The other way that Gaussian shapes arise in optics is as a transverse spatial variation in the Gaussian Beam because Gaussian and related transverse spatial variations are modal solutions to the paraxial wave equation, or, equivalently, they are "like" eigenfunctions of the Fresnel diffraction integral insofar that a diffracted Gaussian beam is also a Gaussian beam (with different parameters, so we're not quite talking eigenfunctions here) and, to first approximation, a paraxial Gaussian beam passing through a thin lens or reflected from a large radius spherical mirror is also a Gaussian beam. So Gaussian beams are the eigenfunctions of a laser cavity: they are the ones left invariant by a round trip through the cavity.

Answer 2

The hyperbolic secant pulse envelope for the electric field (which gives $sech^2(t/t_0)$ pulse envelope for intensity) is obtained from the solution of the pendulum equation ($d^2 \theta/dt^2 - \sin(\theta)/t_0^2 = 0$ ).

The pendulum equation describes 2-level atoms interacting with a monochromatic pulse with slowly varying envelope (varying slowly compared to the optical frequency, but still possibly "ultrafast"). Here, theta is the area of the pulse. So if the laser pulse is shorter than the dephasing time of the atoms/molecules that produce the light, then the interaction producing the light is coherent and is well described by the Optical Bloch equations, which give the pendulum equation.

References:

• Eqn. 4.19 in Allen and Eberly, Optical Resonance and Two-Level Atoms;
• SL McCall and EL Hahn, Phys Rev Lett 18, 908 (1967).