# Off-Resonance Control and All-Optical Switching: Expanded Dimensions in Nonlinear Optics
## 3 Laser-Modified Absorption  
In conventional ultraviolet/visible spectroscopy, resonant absorption occurs as molecules are promoted to allowed electronic excited states, and the resulting absorption spectrum represents an identifying characteristic of the molecular orbital structures [51]. It transpires that simultaneously passing fixed-frequency, off-resonant laser light through the sample in the course of such absorption measurements may enhance or suppress the efficiency of light absorption differentially in each absorption band [52]. This concurrent process is shown in Figure 1. As we have seen, conventional absorption involves a single photon-molecule interaction that corresponds to the first term of Equation (4). Subjecting the absorbing system to the off-resonant laser beam provides for the additional involvement of three-photon-molecule couplings, as shown in Figure 2a, involving the third term of Equation (4). Although this is a higher order term, its contribution to the process need not be insignificant compared to the one for single-photon absorption. The off-resonant beam is independent of the resonant beam and, at sufficient intensity, it may exert a comparatively substantial influence even when the resonant beam is no more intense than is common in absorption spectroscopy. For the laser-modified process, the resonant absorption couples with elastic forward-scattering of the off-resonant beam, the latter itself delivering no net absorption or emission: Off-resonant photons are annihilated and created back into the same radiation mode that, therefore, emerges unchanged. Before going further, it is important to stress that the selection rules for any three-quantum dipole transition invariably subsume those for a single-quantum transition, provided that the optical input and/or output comprises linear polarizations. The converse is, of course, not universally true, being dependent upon the relative symmetry properties of the associated ground and excited states in each case. Therefore, when off-resonant radiation becomes involved through a three-quantum transition in a single-photon absorption process, the result is that transition strengths for each absorption band may be significantly and individually modified. As we demonstrate, new bands satisfying only three-quantum selection rules may also appear when the off-resonant beam is sufficiently intense. Figure 1: Depiction of a laser-modified absorption process. The rate of laser-modified absorption, if no other mechanism were to be allowed, has a purely quadratic dependence on the intensity of the off-resonant beam. We return to the significance of this possibility under special conditions in Section 6. More commonly, however, the rate equation also includes a term representing a larger contribution to the total rate. This relates to the cross term (quantum interference) of the quantum amplitudes for conventional and laser-modified one-photon absorption. This is determined from the Fermi rate expression as \(\Gamma_{\text{abs}}\sim\left|M_{fi}^{(1)}+M_{fi}^{(3)}\right|^{2}\), where \(M_{fi}^{(1)}\) and \(M_{fi}^{(3)}\) are the quantum amplitudes for the leading first-order (single-photon) and third-order (laser-modified) interaction processes, respectively. The interference effects depend on the relative phases of the first- and third-order amplitudes. A similar phase produces an enhanced absorption rate (denoted by a common sign for \(M_{fi}^{(1)}\) and \(M_{fi}^{(3)}\)) and dissimilar phases provide a decreased rate (opposite signs). The quantum amplitude for one-photon absorption, \(M_{fi}^{(1)}\), is well-known [43], and the one for laser-modified absorption, \(M_{fi}^{(3)}\), is given by [52]  
\[M_{fi}^{(3)}=ink^{\prime}k^{2}\frac{\hbar c}{\left(\frac{\hbar c}{2\varepsilon_ {0}V}\right)}^{\frac{3}{2}}\vec{e}_{i}e^{\prime}\,je^{\prime}\,je^{\prime}\, \omega^{0}(-\omega^{\prime};\omega^{\prime},\omega) \tag{8}\]  
Here, the resonant photon energy is \(\hbar ck\equiv\hbar\omega\) and \(\mathbf{e}\) denotes the corresponding polarization vector.