Goos-Hänchen shifts of the reflected waves from a cold, inhomogeneous, and magnetized plasma slab

We discuss theoretically the Goos-Hänchen (GH) shifts of the reflected waves from a cold, inhomogeneous, and magnetized plasma slab by using the invariant imbedding approach. Aiming at the linear and parabolic electron-density profiles, we demonstrate numerically the dependences of the co- and cross-polarized GH shifts on the angle of incidence, external static magnetic field, and the thickness of the plasma slab. The results show that the different electron-density profiles of plasma can result in the very different dependences of the GH shifts on the angle of incidence, external magnetic field, and the slab’s thickness; the GH shifts can be switched between the considerably large positive and negative values under certain conditions. Particularly, without altering the structure of the plasma slab, the GH shifts can be manipulated by modifying the angle of incident or the external static magnetic field.

Figure 1

Schematic of the cold, inhomogeneous, and magnetized plasma slab. External static magnetic field is directed along the $x$ axis. Dashed line is the path of reflection usually expected from geometrical optics. Inset in the upper left shows the linear and parabolic electron-density profiles.

Figure 2

Dependences of the GH shifts and reflectivity on the angle of incidence for $L=2\lambda$ and ${\omega }_c=0.3{\omega }_0$ in the case of the linear electron-density profile. Dotted and solid lines correspond to the GH shifts and reflectivity, respectively. (a) and (c) for the copolarized GH shifts and reflectivity and (b) for the cross-polarized GH shifts and reflectivity.

Figure 3

Variation of the lateral component of the time-averaged Poynting vector ${\eta }_0⟨S_x\left(z\right)⟩$ inside the plasma slab with the coordinate $z$ for $\theta =35.1°$ and $L=2\lambda$. A giant negative-energy flow occurs at $z/\lambda =1.64$ (corresponding to $z/L=0.82$). Insets (a) and (b) show the distributions of ${\eta }_0⟨S_x\left(z\right)⟩$ on both the sides of $z/\lambda =1.64$.

Figure 4

Variations of the GH shifts with ${\omega }_c$ for $L=2\lambda$ and $\theta =35.1°$, $43.9°$, and $75°$. The electron-density profile is the same as in Fig. 2. Dashed, solid, and dotted lines correspond to the GH shifts for $\theta =35.1°$, $43.9°$, and $75°$, respectively. (a) and (c) for the copolarized GH shifts and (b) for the cross-polarized GH shifts.

Figure 5

Variations of the GH shifts with $L$ for ${\omega }_c=0.3{\omega }_0$ and $\theta =35.1°$, $43.9°$, and $75°$. The electron-density profile is the same as in Fig. 2. Dashed, solid, and dotted lines correspond to the GH shifts for $\theta =35.1°$, $43.9°$, and $75°$, respectively. (a) and (c) for the copolarized GH shifts and (b) for the cross-polarized GH shifts.

Figure 6

Dependences of the GH shifts and reflectivity on the angle of incidence for the same parameters as in Fig. 2, but for the parabolic electron-density profile. Dotted and solid lines correspond to the GH shifts and reflectivity, respectively. (a) and (c) for the copolarized GH shifts and reflectivity and (b) for the cross-polarized GH shifts and reflectivity.

Figure 7

Dependences of the GH shifts on ${\omega }_c$ for $L=2\lambda$ and $\theta =31.3°$, $37.1°$, and $75°$. The electron-density profile is the same as in Fig. 6. Dashed, solid, and dotted lines correspond to the GH shifts for $\theta =31.3°$, $37.1°$, and $75°$, respectively. (a) and (c) for the copolarized GH shifts and (b) for the cross-polarized GH shifts.

Figure 8

Dependences of the GH shifts on $L$ for ${\omega }_c=0.3{\omega }_0$ and $\theta =31.3°$, $37.1°$, and $75°$. The electron-density profile is the same as in Fig. 6. Dashed, solid, and dotted lines correspond to the GH shifts for $\theta =31.3°$, $37.1°$, and $75°$, respectively. (a) and (c) for the copolarized GH shifts and (b) for the cross-polarized GH shifts. The magnified inset in (c) is the GH shifts for $\theta =37.1°$.