Superluminal group velocity in a birefringent crystal

We examine the effect of superluminal signal propagation through a birefringent crystal, where the effect is not due to absorption or reflection, but to the filtration of a special polarization component. We first examine the effect by a stationary phase analysis, with results consistent with those of an earlier analysis of the system. We supplement this analysis by considering the transit of a Gaussian wave and find bounds for the validity of the stationary phase result. The propagation of the Gaussian wave is illustrated by figures.

Figure 1

(Color online) Schematic illustration of the experimental situation considered in the text. A plane polarized electromagnetic wave is sent through a birefringent crystal $C$ of width $d$. After leaving the crystal the wave passes a polarization filter $F$ with a polarization angle $\beta$ that can be varied, and the amplitude of the filtered signal is registered in the detector $D$.

Figure 2

(Color online) Amplitude and phase of the filtered monochromatic plane wave as functions of the polarization angle $\beta$ and the wave number $k$. Three singular points with vanishing amplitude are shown, where the lower one corresponds to the half waveplate condition. The curves that circulate the singular points are contour lines of constant amplitude and the lines that interconnect these points are contour lines of constant phase. The numbers along the horizontal axis gives $\Delta kd$ measured from the closest point with an integer number of wavelengths $\left(2\pi N\right)$.

Figure 3

(Color online) Lines of constant time shift for the outgoing signal in the $\beta$, $k$ plane. The lower (red) curves correspond to advanced signals and the upper (blue) curves correspond to retarded signals. Close to the point of vanishing amplitude the time shift increases without bound. The region of superluminal transit, for parameter values given in the text, is indicated by shaded red.

Figure 4

(Color online) Advanced and retarded signals. The figure shows the Gaussian envelopes (the squared relative amplitudes $f$) of the fast component (green curve to the right) and slow component (blue curve to the left) after leaving the crystal. The (red) two-peaked curve shows the filtered signal for polarization angle $\beta =\pi ∕4$. For this angle destructive interference between the fast and slow components gives rise to a symmetric signal with an advanced and a retarded maximum.

Figure 5

(Color online) Outgoing signals for Gaussian input signals of different widths. The three curves show the squared amplitude $f$ as a function of the dimensionless $x$-coordinate $\xi$ for polarization angle $\beta =0.21\pi$. For the (blue) curve with the highest peak the width $\ell$ of the Gaussian is given by $d∕\ell =2.6$, with $d$ as the width of the crystal. The corresponding values are $d∕\ell =1.6$ for the (green) curve of medium height and $d∕\ell =0.6$ for the broadest (red) curve. The position ${\xi }_0$ indicates where the peak position would be if the Gaussian signal moved with the speed of light, and ${\xi }_1$ if the signal moved with the speed determined by the stationary phase argument. These positions depend on the indexes of refraction which have here been chosen as $\overline{n}=1.30$ and $\Delta n=0.15$.

Figure 6

(Color online) Propagation of the $\beta$ component of the Gaussian wave through the birefringent crystal. The figure shows the envelope of the component at four subsequent times, $t_1$, $t_2$, $t_3$, and $t_4$. The polarization angle is $\beta =0.21\pi$ and the width is $\ell =4d$. The region where the crystal is located is indicated by the shaded (blue) area. For comparison the propagating wave (red curve) is shown together with an undistorted Gaussian wave (green curve) which propagates with the speed of light. The peak position of the outgoing signal, which is indicated by the dotted vertical line, clearly is advanced relative to the peak position of the signal propagating with the speed of light. In this plot the following values have been chosen for the indexes of refraction, $\overline{n}=1.35$ and $\Delta n=0.5$.