Lowest-order relativistic corrections to the fundamental limits of nonlinear-optical coefficients

The effects of small relativistic corrections to the off-resonant polarizability, hyperpolarizability, and second hyperpolarizability are investigated. Corrections to linear- and nonlinear-optical coefficients are demonstrated in the three-level ansatz, which includes corrections to the Kuzyk limits when scaled to semirelativistic energies. It is also shown that the maximum value of the hyperpolarizability is more sensitive than the maximum polarizability or second hyperpolarizability to lowest-order relativistic corrections. These corrections illustrate how the intrinsic nonlinear-optical response is affected at semirelativistic energies.

Figure 1

The intrinsic hyperpolarizability as a function of ${\lambda }_{00}$ and ${\lambda }_{11}$ when $X=1/\sqrt[4]{3}$ and $E=0$. The relativistic behavior of the Kuzyk limit is illustrated for (a) ${\lambda }_{10}=0$, (b) ${\lambda }_{10}=0.1$, and (c) ${\lambda }_{10}=0.2$. Values outside the magnitude for the nonrelativistic limit are given by the solid (red) region.

Figure 2

The maximum intrinsic value of the second hyperpolarizability as a function of ${\lambda }_{00}$ and ${\lambda }_{11}$ from the relativistic corrections to the TRK sum rules. The corrected maximum is shown for (a) ${\lambda }_{10}=0$, (b) ${\lambda }_{10}=0.1$, and (c) ${\lambda }_{10}=0.2$.

Figure 3

The minimum intrinsic value of the second hyperpolarizability as a function of ${\lambda }_{00}$ and ${\lambda }_{10}$ from the relativistic corrections to the TRK sum rules.

Figure 4

The second hyperpolarizability as a function of atomic number for an H-like ion using the three-level model with lowest-order relativistic corrections to the TRK sum rules.