# Forces in the Interaction of Light with Matter
## 2 Optical Forces--Fundamentals
### Optical Forces in Dielectric Media  
Given their mode of operation on particles in free space, it is not surprising that optical forces may also have an effect on condensed phase matter. They can, for example, lead to electric phenomena such as the photon drag effect, in which a current is induced due to light forces acting on quasi-free electrons in semiconductors and other dielectric media [47; 51]. Optical forces in some dielectric media can be regarded as arising from the response of a distribution of microscopic atomic electric dipoles, representing the source of polarisation \(\mathcal{P}\), subject to electric and magnetic fields \(\mathbf{E}\) and \(\mathbf{B}\). We can then associate the following optical force density with the effect on an individual dipole \(\mathbf{d}\) upon being subjected to electromagnetic fields:  
\[\mathbf{f}=\delta(\mathbf{r}-\mathbf{R})(\mathbf{d}\cdot\
abla)\mathbf{E}+ \dot{\mathbf{d}}\times\mathbf{B}\delta(\mathbf{r}-\mathbf{R}). \tag{8}\]  
The individual polarisation field of a single dipole is \(\mathbf{p}=\mathbf{d}\delta(\mathbf{r}-\mathbf{R})\). This leads to the generalisation of the Lorentz force acting on a homogeneously distributed ensemble of such dipoles forming a dielectric medium, in which case the force density acting on an individual dipole of the ensemble is  
\[\mathbf{f}_{d}=(\mathbf{p}\cdot\
abla)\mathbf{E}+\dot{\mathbf{p}}\times \mathbf{B}. \tag{9}\]  
The dielectric medium is modelled as a set of \(N\) elementary dipoles per unit volume such that, aside from Lorentz local field corrections, the medium polarisation density is \(\mathcal{P}=N\mathbf{p}\); \(\mathbf{p}\) is proportional to the electric field acting on the dipole \(\mathbf{p}=\alpha\mathbf{E}\), where \(\alpha\) is the dipole polarisability. So, we now have for the medium polarisation  
\[\mathcal{P}=N\alpha\mathbf{E}=\chi\mathbf{E}. \tag{10}\]  
where we have defined the linear susceptibility \(\chi=N\alpha\). The optical force in the dielectric medium is  
\[\mathbf{F}_{dielectric}=(\mathcal{P}\cdot\
abla)\mathbf{E}+\dot{\mathcal{P}} \times\mathbf{B}. \tag{11}\]  
The polarisability and its bulk counterpart susceptibility are, in general, complex. On substituting for \(\mathcal{P}\), we have, assuming that \(\alpha=\alpha^{\prime}\) is real,  
\[\mathbf{F}_{dielectric}=\frac{1}{2}\alpha^{\prime}\
abla|\mathbf{E}(\mathbf{r },t)|^{2}+\alpha^{\prime}\frac{\partial}{\partial t}\mathbf{E}(\mathbf{r},t) \times\mathbf{B}(\mathbf{r},t). \tag{12}\]  
In practice, we require the cycle average of this force, which we denote as \(<\mathbf{F}_{dielectric}>\) and which is given by  
\[<\mathbf{F}_{dielectric}>=\frac{1}{2}\alpha^{\prime}\
abla|\mathbf{E}|^{2}, \tag{13}\]  
where we have set the cycle average of the second term in Equation (12) to zero. On the other hand, if the polarisability is considered to be complex, as befits near-resonant radiation, then \(\alpha=\alpha^{\prime}+i\alpha^{\prime\prime}\), and the force expression in Equation (13) becomes  
\[<\mathbf{F}_{dielectric}>=\frac{1}{2}\Re\big{\{}[(\alpha^{\prime}+i\alpha^{ \prime\prime})\mathbf{E}\cdot\
abla]\mathbf{E}^{*}\big{\}}. \tag{14}\]  
This expression contains both gradient and dissipative forces.