Spano, FC; Warren, WS, *Preparation of constant-bandwidth total inversion, independent of optical density, with phase-modulated laser pulses*,
Physical Review A - Atomic, Molecular, and Optical Physics, vol. 37 no. 3
(1988),
pp. 1013-1016 [doi] .
**Abstract:**

*We show that pulses of electromagnetic radiation with a hyperbolic-secant envelope and a hyperbolic-tangent frequency sweep are capable of exciting an inversion line shape in a collection of two-level absorbers that is highly insensitive to optical density. This is not true of the hyperbolic-secant pulse (with no frequency modulation) which undergoes significant reshaping as it propagates. The analysis is based on numerical solutions of the coupled Maxwell-Bloch equations in both the plane-wave and paraxial approximations. We also show an additional insensitivity of the inversion profile to finite relaxation times (T2) that approach the pulse width. © 1988 The American Physical Society.*