Radiation pressure on a submerged absorptive partial reflector deduced from the Doppler shift

When a light pulse is reflected from a mirror, energy and momentum are exchanged between the electromagnetic field and the material medium. The resulting change in the energy of the reflected photons is directly related to their Doppler shifts arising from the change in the state of motion of the mirror. Similarly, the Doppler shift of photons that enter an absorber is intimately tied to the kinetic energy and momentum acquired by the absorber in its interaction with the incident light. The argument from the Doppler shift yields expressions for the exchanged energy and momentum that are identical with those obtained from Maxwell's equations and the Lorentz law of force, despite the fact that the physical bases of the two methods are fundamentally different. Here, we apply the Doppler-shift argument to a submerged partial reflector (one that absorbs a fraction of the incident light), deducing, in the process, the magnitude of the photon momentum within the submerging medium. We also discuss the case of the submerging medium having a negative refractive index and show the absence of the so-called “reversed” Doppler shift when the reflector is detached from the negative-index medium.

Figure 1

A linearly polarized monochromatic plane wave of frequency ${\omega }_{\mathrm{o}}$ enters a transparent dielectric slab of refractive index $n_{\mathrm{o}}$, whose front facet is antireflection coated. An air gap of width $d$ separates the slab from a partial reflector of refractive index $n_1$+$i{\kappa }_1$. The $E$-field amplitudes of the incident and reflected beams at the rear facet of the transparent slab ($z$ = 0${}^{-}$) are denoted by $E_{\mathrm{o}}$ and $E_r$, respectively. There reside two counterpropagating plane waves in the air gap, of which the forward-propagating wave has amplitude $E_g$ at $z$ = 0${}^{+}$. The plane wave that enters the partial reflector (and is eventually extinguished due to the nonzero absorption coefficient ${\kappa }_1$ of this medium) has amplitude $E_m$ at $z$ = $d^{+}$.

Figure 2

(a) A 40-fs Gaussian pulse having center wavelength ${\lambda }_{\mathrm{o}}$ = 633 nm is reflected from a moving mirror (velocity = $V$) immersed in a stationary dielectric medium of refractive index $n_{\mathrm{o}}$. The mirror is a dielectric slab of refractive index $n_1$ = 2.0, whose rear facet is contiguous with the perfectly matched boundary layer of the FDTD mesh. (b) Simulated spectra of the reflected pulse obtained with the FDTD method for different combinations of $V$ and $n_{\mathrm{o}}$. The solid curve labeled ($V/c$, $n_{\mathrm{o}}$) = (0, 1.0) represents a stationary mirror in vacuum. The adjacent curve (solid red) labeled ($-$0.01, 1.0) corresponds to a mirror moving in vacuum toward the source at $V$ = −0.01$c$. The rightmost curve (solid green) is the reflected spectrum from a mirror moving in vacuum toward the source at $V$ = −0.015$c$. The remaining three curves with reduced heights (solid blue, red, and green) correspond to a mirror immersed in water ($n_{\mathrm{o}}$ = 1.33), moving away from the source at $V$ = 0, +0.01$c$, and +0.015$c$, respectively. In each case, the superposed circles (red and green) are obtained by scaling the spectrum of the pulse reflected from a stationary mirror by the Doppler formula ${\omega }^{\prime }=\omega {(1-n_{\mathrm{o}}V/c)}^2/(1-V^2/c^2)$ along the frequency axis and by the Lorentz transformation of the Poynting vector along the vertical axis. The simulated spectra (solid) are seen to closely mimic the Doppler-shifted spectra (circles).

Figure 3

(a) Monochromatic plane wave is reflected from a flat mirror located behind a glass slab of refractive index $n_{\mathrm{o}}$ = 1.5. The front facet of the slab is antireflection coated, the initial air gap between the mirror and the slab is $d_{\mathrm{o}}$ = 100 nm, and the mirror moves at a constant velocity $V$ away from the slab. In these simulations, the mirror is a dielectric slab of refractive index $n_1$ = 2.0, whose rear facet is contiguous with the perfectly matched boundary layer of the FDTD mesh. (b) Time dependence of the $E$ field of the reflected beam monitored at a point slightly to the left of the light source. The green (light gray) curve corresponds to $V$ = 10${}^{-4}c$, whereas the blue (black) curve represents the case of $V$ = 0.01$c$. The fields are captured starting at time $t_{\mathrm{o}}$ (set to 90 fs in all simulations) to allow the transients initiated at the source startup to leave the computational domain.

Figure 4

Computed Doppler shift $\Delta {\omega }_r$ versus time for the reflected beam in the system of Fig. 3a. From (a) to (c), the mirror velocity is $V$ = 0.0001$c$, 0.003$c$, and 0.01$c$, respectively. In each case, the dashed curve is the theoretical estimate of $\Delta {\omega }_r$ given by Eq. (9). The theoretical estimate has been shifted forward in time to account for the delay caused by the propagation of the reflected beam from the rear facet of the glass slab to the observation point. In (c), the gap between the slab and the mirror widens by $\Delta d$ = 316.5 nm between $t$ = $t_{\mathrm{o}}$ and $t$ = $t_{\mathrm{o}}$+105.5 fs. This increase by ${\lambda }_{\mathrm{o}}$/2 in the gap width has no effect on the optical properties of the system, causing the plotted Doppler shift to repeat itself at regular intervals of $\Delta t$ = 105.5 fs.