The dimensionless zero-frequency electronic first hyperpolarizability $3^{-1/4}\beta E_{10}^{7/2}{m}^{3/2}{(e\hslash )}^{-3}$ of an electron in one dimension was maximized by adjusting the shape of a piecewise linear potential. Careful maximizations converged quickly to 0.708951 with increasing numbers of parameters. The Hessian shows that $\beta$ is strongly sensitive to only two parameters in the potential: sensitivity to additional parameters decreases rapidly. With more than two parameters, a wide range of potentials and an apparently narrower range of wavefunctions have nearly optimal hyperpolarizability. Modulations of the potential to which the unique maximum is insensitive were characterized. Prospects for concise description of the two important constraints on near-optimum potentials are discussed.

© 2012 Optical Society of America

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