Testing the reliability of a velocity definition in a dispersive medium

We introduce a method to test if a given velocity definition corresponds to an actual physical flow in a dispersive medium. We utilize the equivalence of the pulse dynamics in the real-$\omega$ and real-$k$ Fourier expansion approaches as a test tool. To demonstrate our method, we take the definition introduced by Peatross et al. [Phys. Rev. Lett. 84, 2370 (2000)] and calculate the velocity in two different ways. We calculate (i) the mean arrival time between two positions in space, using the real-$\omega$ Fourier expansion for the fields and (ii) the mean spatial displacement between two points in time, using the Fourier expansion in real-$k$ space. We compare the velocities calculated in the two approaches. If the velocity definition truly corresponds to an actual flow, the two velocities must be the same. However, we show that the two velocities differ significantly (3$\%$) in the region of superluminal propagation even for the successful definition of Peatross et al.

Figure 1

Real-$\omega$ ($C_1$: red thin ${\omega }_I=0$ line) and real-$k$ ($C_2$: thick contour) integration paths corresponding to the Lorentzian dielectric constant (1) with parameters ${\omega }_p=0.1{\omega }_0$ and $\gamma =0.12{\omega }_0$. $C_2$ contour is deduced by solving the nonlinear index equation $c\overline{k}=n\left(\omega \right)\omega$ for real-$\overline{k}$ values. The two lines $L_1$ and $L_2$ are the branch cuts of the index [24]. Both branch cuts are below the real-$k$ integration path. Length of the branch cuts are exaggerated only for visual purposes.

Figure 2

The same reflection-transmission problem considered (a) in the real-$\omega$ and (b) in the real-$k$ Fourier spaces. Incident light penetrates from a vacuum on the left-hand side to an absorbing dielectric slab of complex index $n\left(\omega \right)$ on right-hand side (RHS). (a) Incident wave of Fourier coefficients $A_1\left(\omega \right)$ results in a reflected wave of coefficient $B_1\left(\omega \right)$ and a transmitted wave of coefficients $D_1\left(\omega \right)$. (b) Incident wave of Fourier coefficient $A_2\left(k\right)$ results in a reflected wave of coefficient $B_2\left(k\right)$ and a transmitted wave of coefficient $D_2\left(k\right)$. The two solutions on the RHS must match at the origin $x=0$ for all times. This results in the relation of Eq. (6) between $D_1\left(\omega \right)$ and $D_2\left(k\right)$.

Figure 3

(a) Comparison of the two velocities (in units of $c$) deduced from definition of Peatross et al. [8]. The velocities $v_1=\Delta x/({\langle t\rangle }_{\Delta x}-{\langle t\rangle }_0)$ and $v_2=({\langle x\rangle }_{\Delta t}-{\langle x\rangle }_0)/\Delta t$ are calculated using the same definition but performing real-$\omega$ (solid line) and real-$k$ (dashed line) Fourier expansions for the fields, respectively. For a consistent definition, the two results must be identical. However, 3$\%$ discrepancy between $v_1$ and $v_2$ is observed in the superluminal region. Thus, this description is not so reliable in the superluminal regime. We use a Gaussian pulse of carrier frequency ${\omega }_c$ and temporal width ${\omega }_0\tau =20$. Propagation distance is $\Delta x_1=150c/{\omega }_0$. (b) Real ($n_R$) and imaginary ($n_I$) parts of the index of refraction. Parameters for the index are the same as described in the legend of Fig. 1.