# Off-Resonance Control and All-Optical Switching: Expanded Dimensions in Nonlinear Optics
## 5 Optically Controlled Resonance Energy Transfer  
Resonance energy transfer (RET) is the transport of the electronic excitation between electronically distinct molecular components, commonly designated donor A and acceptor B [65; 66; 67; 68; 69; 70]. Analogous to the effects of laser-modified absorption and fluorescence, an off-resonant beam of sufficient intensity  
Figure 4: Contour maps of the fluorescence rate, against \(x\) and \(y\) (both over the range \(-100\) nm, \(100\) nm) with \(z=40\) nm, from a light-emitting molecule at the source to a detector positioned in the \(x\),\(y\)-plane. Here, the input intensity of the off-resonant beam is: (**a**) \(3\times 10^{11}\) W cm\({}^{-2}\), (**b**) \(1\times 10^{12}\) W cm\({}^{-2}\), (**c**) \(2\times 10^{12}\) W cm\({}^{-2}\), and (**d**) \(3\times 10^{12}\) W cm\({}^{-2}\). The darker the shade of blue, the higher the rate. The right-hand diagram shows the model geometry, which involves the source A and detector B at a separation \(R\), while \(r\) is the distance of the detector from the \(z\)-axis. The input beam, with a Gaussian profile, propagates upward along the \(z\)-axis. The beam waist parameter \(w=500\) nm, transition dipole magnitude is 5 D, and both the source and detector bandgap = 0.3 eV. Modified from the original image in [60]. is able to increase or reduce the rate of such energy transfer. Again, the key equations are derived in the form of a Fermi rate expression, but this time via \(\Gamma_{\text{ret}}\sim\left|M_{fi}^{(2)}+M_{fi}^{(4)}\right|^{2}\) where \(M_{fi}^{(2)}\) and \(M_{fi}^{(4)}\) are the quantum amplitudes for the second-order (single-photon emission at A and single-photon absorption at B, i.e., RET) and fourth-order interaction processes (optically controlled RET, also called laser-assisted RET, or LARET), respectively. In detail, optically controlled RET is the process whereby energy transfer proceeds with concurrent absorption of an off-resonant beam at A and re-emission at B (and vice-versa). RET intrinsically involves the emission and annihilation of a virtual photon [71]. Thus, four photon-molecule interactions are involved overall, as shown in Figure 2c, and the precise form of the associated quantum amplitude is accordingly secured from the fourth term in Equation (4). In the near field, where RET is most commonly measured, the quantum amplitude of the laser-modified process is given by [72; 73]  
\[M_{fi}^{(4)}=\left(\frac{I^{\prime}}{2\varepsilon_{0}c}\right)e\vec{\varepsilon} _{i}V_{jk}(0,R)\bigg{(}\alpha_{ij}^{0\alpha(\Lambda)}(-\omega^{\prime})\alpha_ {lk}^{0\beta\;(\text{B})}(\omega^{\prime})+\alpha_{ij}^{0\beta\;(\text{B})}(- \omega^{\prime})\alpha_{lk}^{0\alpha(\Lambda)}(\omega^{\prime})\bigg{)} \tag{12}\]  
where -\(\alpha_{ij}\) is a transition polarizability, \(\beta\) is an excited state of B (for simplicity, denoted \(\alpha\) in Figure 2c) and \(V_{jk}(0,R)\) is the dipole-dipole coupling tensor, which acquires the form of its static limit in the near-field. The rate of optically controlled resonance energy transfer is found from Equation (12) and the matrix element, \(M_{fi}^{(2)}\), for conventional RET [43], to give  
\[ \begin{array}{ll}\Gamma_{\text{ret}}&=V_{jk}(0,R)V_{mm}(0,R)\bigg{[}\mu_{j}^{ \alpha\alpha(\Lambda)}\mu_{k}^{0\beta\;(\text{B})}\mu_{m}^{0\alpha(\Lambda)}\mu _{n}^{0\beta\;(\text{B})}+(I^{\prime}/c\varepsilon_{0})e\vec{\varepsilon}_{i} \mu_{m}^{0\alpha(\Lambda)}\mu_{n}^{0\beta\;(\text{B})}\\ &\times\bigg{(}\alpha_{ij}^{0\alpha(\Lambda)}(-\omega^{\prime})\alpha_{lk}^{0 \beta\;(\text{B})}(\omega^{\prime})+\alpha_{ij}^{0\beta\;(\text{B})}(-\omega ^{\prime})\alpha_{lk}^{0\alpha(\Lambda)}(\omega^{\prime})\bigg{)}+\ldots\big{]} \ \.\end{array}  \tag{13}\]  
Here, the first term is the rate of RET as it occurs without the input beam, and the second term is the quantum interference between RET and optically controlled RET. The latter term, linear in the irradiance of the off-resonant laser beam, is the leading correction to the transfer rate when the laser beam is present.