## Abstract

A frequency chirped continuous wave laser beam incident upon a resonant, two-level atomic absorber is seen to evolve into a Jacobi elliptic pulse-train solution to the Maxwell-Bloch equations. Experimental pulse-train envelopes are found in good agreement with numerical and analytical predictions.

© Optical Society of America

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### Equations (11)

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(1)
$$\mathbf{E}(t,z)={E}_{0}(t,z)\left[{e}^{i\left\{\omega t-Kz+\varphi \left(t\right)\right\}}+\mathrm{c.}c.\right]$$
(2)
$$\omega \left(t\right)=\omega +\frac{d\varphi}{dt}$$
(3)
$$\frac{\partial u}{\partial t}=-\left(\Delta \omega -\dot{\varphi}\right)v-{E}_{2}w-\frac{u}{{T}_{2}^{\prime}}$$
(4)
$$\frac{\partial v}{\partial t}=\left(\Delta \omega -\dot{\varphi}\right)u+{E}_{1}w-\frac{v}{{T}_{2}^{\prime}}$$
(5)
$$\frac{\partial w}{\partial t}=-{E}_{1}v+{E}_{2}u-\frac{\left(w+1\right)}{{T}_{1}}$$
(6)
$$\frac{\partial {E}_{1}}{\partial z}=\frac{\alpha}{2\pi g\left(0\right)}\underset{-\infty}{\overset{\infty}{\int}}v(t,z,\Delta \omega )g\left(\Delta \omega \right)d\Delta \omega $$
(7)
$$\frac{\partial {E}_{2}}{\partial z}=-\frac{\alpha}{2\pi g\left(0\right)}\underset{-\infty}{\overset{\infty}{\int}}u(t,z,\Delta \omega )g\left(\Delta \omega \right)d\Delta \omega $$
(8)
$${E}_{0}=\frac{2k}{\tau l}{\left[1-{l}^{2}{\mathrm{sn}}^{2}(t\u2044\tau ;k)\right]}^{\frac{1}{2}}$$
(9)
$$\frac{d\phi}{dt}=\frac{{\left(1-{l}^{2}\right)}^{\frac{1}{2}}{\left({l}^{2}-{k}^{2}\right)}^{\frac{1}{2}}}{l\tau \left[1-{l}^{2}{\mathrm{sn}}^{2}(t\u2044\tau ;k)\right]}$$
(10)
$$\frac{d\mathbf{P}}{d\mathbf{t}}=\mathbf{\Omega}\times \mathbf{P}$$
(11)
$$\frac{d\varphi \left(t\right)}{dt}={\varphi}_{0}\mathrm{sin}\phantom{\rule{.2em}{0ex}}\delta t,\mathrm{or}\phantom{\rule{.2em}{0ex}}\varphi \left(t\right)=-\frac{{\varphi}_{0}}{\delta}\mathrm{cos}\delta t,$$