Radiation pressure in stratified moving media

A general theory of optical forces on moving bodies is here developed in terms of generalized $4\times 4$ transfer and scattering matrices. Results are presented for a planar dielectric of arbitrary refractive index placed in an otherwise empty space and moving parallel and perpendicular to the slab-vacuum interface. In both regimes of motion the resulting force comprises lateral and normal velocity-dependent components, which may depend in a subtle way on the Doppler effect and $s$-$p$-polarization mixing. For lateral displacements in particular, polarization mixing, which is here interpreted as an effective magnetoelectric effect due to the reduced symmetry induced by the motion of the slab, gives rise to a velocity-dependent force contribution that is sensitive to the phase difference between the two polarization amplitudes. This term gives rise to a rather peculiar optical response on the moving body, and specific cases are illustrated for incident radiation of arbitrarily directed linear polarization. The additional force due to polarization mixing may cancel to first order in $V/c$ with the first order Doppler contribution yielding an overall vanishing of the velocity-dependent component of the force on the body. The above findings bear some relevance to modern developments of nano-optomechanics and to the problem of the frictional component of the Casimir force.

Figure 1

Geometry of the slab interfaces showing the input (red) field amplitudes ${\alpha }_{\text{L}}^{(+)}$ and ${\alpha }_{\text{R}}^{(-)}$ and the output (blue) amplitudes ${\alpha }_{\text{L}}^{(-)}$ and ${\alpha }_{\text{R}}^{(+)}$. The slab surface normal is along $\pm \widehat{\mathbf{x}}$ while the orientation of the two polarizations ($1,2$) are indicated in the input amplitudes. The inset shows the unit polarization vectors (6, 7) and their relative orientation in terms of the angle defined in (8).

Figure 2

For certain angles of incidence, a laterally moving medium mixes polarizations in transmission and reflection. For instance, an $s$-polarized input will generate $p$-polarized reflected and transmitted amplitudes so long as $k_z\ne 0$. In all cases the Doppler shift makes the reflection and transmission coefficients dependent upon $k_y$ in such a way that, for example, ${\mathcal{R}}_{qr}\left(k_y\right)\ne {\mathcal{R}}_{qr}(-k_y)$.

Figure 3

Contribution to the normal force (36) from the mixing of polarizations due to the motion of the medium. Here we plot the quantity, $\langle d{\tilde{P}}_M^{\text{(PM)}}/dt\rangle ={\zeta }_x^2{\sum }_{q}2{(-1)}^{\overline{q}}\eta \left(\right|{\mathcal{R}}_{qq}{|}^2-|{\mathcal{T}}_{qq}{|}^2)$. This represents the contribution to the normal force due to polarization mixing when ${\alpha }_{1L}^{(+)}={\alpha }_{2L}^{(+)}$, normalized in units of the incident momentum flux times $V_y/c$. In panel (a) we plot $\langle d{P}_M^{\text{(PM)}}/dt\rangle$ as a function of $\theta$ for various $\varphi$. In the main plot the radiation is incident onto a slab of thickness $d=10c/\omega$ with $\epsilon =6.0+0.01i$ and $\mu =1.0$. The inset shows the case when the thickness is $d=40c/\omega$. Panel (b) illustrates the increase in magnitude of $\langle d{P}_M^{\text{(PM)}}/dt\rangle$ when the refractive index drops below unity: $\epsilon =0.1+0.01i$ and $\mu =1.0$, where the medium becomes sensitive to the difference between $s$ and $p$ polarization at small angles of incidence [in this regime $\eta =\cot \left(\theta \right)\sin \left(\varphi \right)$ is large]. Within both plots we used the reflection and transmission coefficients for a slab of dielectric of arbitrary thickness as can be found in Ref. [39].

Figure 4

A slab in motion along the $x$ axis does not mix polarizations, but mixes frequencies. Incidence from the left, when $V_x>0$ yields a reflected amplitude at a frequency, ${\omega }^{(-)}<\omega$, while incidence from the right yields a reflected amplitude at ${\omega }^{(+)}>\omega$, which breaks reciprocity.