# Off-Resonance Control and All-Optical Switching: Expanded Dimensions in Nonlinear Optics
## 3 Laser-Modified Absorption
When the same symbols carry a prime, they refer to the separately controllable throughput beam. In this formulation, the latter can accommodate arbitrary polarization. Once again, the overbar denotes  
Figure 2: Energy-level diagrams for (**a**) laser-modified absorption, (**b**) laser-modified fluorescence and (**c**) optically controlled resonance energy transfer. Light blue and red horizontal lines denote ground-state 0 and excited-state \(\alpha\) energy levels of the molecules, respectively, orange dashed lines represent the off-resonant beam, \(\hbar\alpha^{\prime}\), and blue wavy lines are the resonant beam, _ho_. Crossed and open circles are one- and two-photon interactions, respectively, the vertical black arrows denote electronic transitions and horizontal black line signifies energy transfer from A to B. Without the off-resonant beam, these diagrams simply depict absorption, fluorescence, and energy transfer. The beam that is applied concurrently (not step-wise) to these processes, as a result, suffers the annihilation and creation of one photon into the same radiation mode via virtual intermediate states (not shown). These depictions avoid the complexities of Feynman diagrams, or state-sequence diagrams, that show the whole range of interaction sequences necessarily involved [52; 53; 54]. complex conjugation (signifying a polarization state that is effectively reflected in the equator of a Poincare sphere). The quasi-hyperpolarizability \(\beta^{\alpha 0}_{ijk}(-\omega^{\prime};\omega^{\prime},\omega)\) has a similar form to Equation (6), but now involves both resonant and non-resonant beams and the real, excited state \(\alpha\). Once again, the order of indices in the tensor correlates with the order of frequency arguments--the explicit form is given in [52]. Through deployment of the explicit expression for \(M^{(1)}_{fi}\) and \(M^{(3)}_{fi}\), the rate for laser-modified one-photon absorption, \(\Gamma_{\mathrm{abs}}\), emerges as  
\[ \begin{array}{rl}\Gamma_{\mathrm{abs}}&=\left(\frac{n\beta l}{c\hbar\omega_{ 0}}\right)\!\!\epsilon_{i}\overline{\epsilon}_{j}\!\left[\mu^{(0)\alpha 0}_{i}\overline{\mu}^{(0)\alpha 0 }_{j}+(I^{\prime}/c\varepsilon_{0})\mu^{(0)\alpha 0}_{i}\overline{\mu}^{(2) \alpha 0}_{j}(\omega^{\prime})\right.\\ &\left.+\left(I^{\prime 2}/4c^{2}\varepsilon_{0}^{2}\right)\!\!\mu^{(2) \alpha 0}_{i}(\omega^{\prime})\overline{\mu}^{(2)\alpha 0}_{j}(\omega^{ \prime})\right]\ \,\end{array}  \tag{9}\]  
in which \(I=n\hbar c^{2}k/V\) is the irradiance of the absorbing resonant beam and \(I^{\prime}\) is the throughput laser irradiance. It is the second term (linear in \(I^{\prime}\)) that signifies a quantum interference of the two amplitudes. This represents the leading correction to the one-photon absorption rate. Equation (9) affords a re-interpretation of earlier expression: One-photon absorption (independent of the off-resonant beam) relates to an effective excitation dipole whose zeroth order is labeled \(\mu^{(0)\alpha 0}\equiv\mu^{\alpha 0}\), and the contribution that is quadratic in the electric field (linear in the intensity) of the off-resonant beam is \(\mu^{(2)\alpha 0}_{i}(\omega^{\prime})\equiv e^{\prime}\vec{e}^{\prime}\,k \beta^{\alpha 0}_{ijk}(-\omega^{\prime};\omega^{\prime},\omega)\). The former signifies direct resonant absorption, while the latter a nonlinear amendment comprising the off-resonant radiation of frequency \(\omega^{\prime}\) (corresponding to elastic, forward scattering). The off-resonant beam thus effects a modification of the excitation dipole moment via \(\mu^{(2)\alpha 0}(\omega^{\prime})\). The size of the correction term, which can enhance or diminish the spectral intensity of each absorption band, depends on both the intensity of the off-resonant ancillary beam and its fixed frequency.