Left-handed electromagnetic waves in materials with induced polarization and magnetization

We analyze the properties of electromagnetic waves inside materials with induced polarization and magnetization. We show that if the polarization and magnetization of the material are sufficiently large and appropriately phased, then the system supports the formation of left-handed waves. In some respects, such a system behaves similarly to materials with a negative index of refraction, yet there is one important advantage: Left-handed waves in materials with induced polarization and magnetization do not require as stringent material properties (such as the strength of resonances and the density of radiators). We numerically investigate the formation and propagation of such left-handed waves using finite-difference approximation to Maxwell's equations. We also discuss possible experimental observation of these ideas in a rare-earth-doped crystal.

Figure 1

Left-handed waves inside a material with induced polarization and magnetization. If the medium is polarized and magnetized at the same frequency $\omega$ with exactly the same $k$ vector, then there are forced-wave solutions with the vectors $\vec{E},\vec{H}$, and $\vec{k}$ forming a left-handed triad. The solutions are such that the field quantities $\vec{E}$ and $\vec{H}$ are $\pi$ out of phase with the induced polarization and magnetization.

Figure 2

Beam refraction at normal incidence from a right-handed material into a material with induced polarization and magnetization. The boundary conditions at the interface dictate that ${\vec{E}}_2={\vec{E}}_1$ and ${\vec{H}}_2={\vec{H}}_1$. If the induced polarization and magnetization in medium 2 are $\vec{M}=-2\overrightarrow{H_2}$ and $\vec{P}=-2{\epsilon }_0\overrightarrow{E_2}$, then forced left-handed waves are excited. These waves satisfy both the boundary conditions at the interface and Maxwell's equations in the material.

Figure 3

Inducing polarization and magnetization using second-order atomic nonlinearity ${\chi }^{\left(2\right)}$. Three intense laser beams with fields $E_a,E_b$ (electric), and $B_c$ (magnetic), two-photon excite an electric-dipole $\left(|g\rangle \to |e\rangle \right)$ and a magnetic-dipole $\left(|g\rangle \to |m\rangle \right)$ transition through an intermediate level $|r\rangle$. The nonlinear response produces polarization and magnetization at the sum frequency of $\omega ={\omega }_a+{\omega }_b={\omega }_a+{\omega }_c$.

Figure 4

The calculated intensity of the left-handed waves that can be formed as the atomic density is varied. For atomic densities of ${10}^{12}/{\mathrm{cm}}^3$ (which is achievable with most vapor cells or laser-cooled ultracold clouds), left-handed waves with intensities of about 1 pW/${\mathrm{cm}}^2$ can be formed. In contrast, all negative-index proposals in atomic systems require densities exceeding ${10}^{16}/{\mathrm{cm}}^3$.

Figure 5

Finite-difference numerical simulations of Maxwell's equations that demonstrate left-handed waves inside a material with induced polarization and magnetization. The plots show the snapshots for the electric field of a wave packet at three different instants in time, $t=0,t=1$, and $t=60$ fs. Both the phase and group velocities are oriented along the $+x$ direction. The electric and magnetic field components are oriented along $+z$ and $+y$ directions, respectively.

Figure 6

Wave propagation with the initial field values chosen to be ${\mathcal{E}}_z(x,y,t=0)={\mathcal{H}}_y(x,y,t=0)=0$. In addition to a left-handed wave, a right-handed wave with equal amplitude and opposite initial phase (phase at $t=0)$ is formed.

Figure 7

(a) Beam refraction from a “normal,” right-handed material into a material with induced polarization and magnetization. The wave refracts with a negative angle of refraction. (b) The time-reversed case where the wave refracts from the material with induced polarization and magnetization into right-handed material.

Figure 8

Wave refraction from a material with induced polarization and magnetization into free space. Here, we assume polarization and magnetization waves in the material propagating along the $k$ vector as shown in Fig. 4 and start with zero initial field values, ${\mathcal{E}}_z(x,y,t=0)={\mathcal{H}}_y(x,y,t=0)=0$. Before the polarization and magnetization exist in the region on the right, there is no generated EM wave (a). Once the induced polarization and magnetization appear at the interface, a refracted right-handed wave appears in the region of free space (b). As the polarization and magnetization generate the left-handed wave in the region of $x>0$ a refracted wave is generated along with it in the region of free space. The direction of the polarization, magnetization, electric field, and magnetic field are indicated (c). After the pulse of polarization and magnetization has completely entered the region of $x>0$ the left-handed wave inside the material and the refracted right-handed wave outside the material become two separate pulses and propagate as expected (d).

Figure 9

Proposed scheme for exciting left-handed waves in a Eu-doped crystal. The magnetization $\left(M\right)$ is induced using the ${}^{7}F{}_0\to {}^{5}D{}_1$ strong magnetic-dipole transition of the ${\mathrm{Eu}}^{3+}$ ions. Two-photon excitation with infrared light at a wavelength of 1054 nm is used to generate the coherence between the two levels. The two-photon excitation is through the $4f5d$ configuration as the intermediate level. The polarization can be induced using a number of processes. One approach would be to use the second-order nonlinear response of the host crystal. The second-order susceptibility $\left({\chi }^{\left(2\right)}\right)$ of the crystal can produce a nonlinear polarization $P={\epsilon }_0{\chi }^{\left(2\right)}{E}_c^2$ using a separate infrared laser with field amplitude $E_c$.

Figure 10

The predicted intensity of left-handed waves that can be excited inside a 0.1% doped Eu:YSO crystal. Here we take the power of the infrared laser beam $(\lambda =1054$ nm) to be 1 W and calculate the magnetization that can be induced in the material as we vary the focused beam size. The induced magnetization in turn determines the field values and therefore the intensity for the left-handed waves.