Anomalous phase in one-dimensional, multilayer, periodic structures with birefringent materials

We have studied biaxial, birefringent, one-dimensional, multilayer structures and found a wavelength region where the phase of one specific polarization component of the transmitted field increases with wavelength, giving rise to unusual polarization dependent dispersive effects of the input beam. We discuss the conditions that lead to these effects, and examine possible ways to enhance them.

Figure 1

(Color) Representation of the field inside and outside the anisotropic layer. The two electric fields inside the layer are the fields in the two channels of the anisotropic material. The input semispace is isotropic while the output semispace is anisotropic.

Figure 2

Representation in the complex plane of the sum of the electric fields of the two channels of the anisotropic semispace in output. The sum is referred to the first ray in output only.

Figure 3

(Color) Representation of the field that is inside and outside two anisotropic layers.

Figure 4

The phase of the wave in output from channel ${\epsilon }_{\mathrm{o}\mathrm{x}}$ for two anisotropic layers with $n_{\mathrm{i}}=1$, ${\epsilon }_{\mathrm{L}\mathrm{x}}=1.8$, ${\epsilon }_{\mathrm{L}\mathrm{y}}=3.7$, ${\epsilon }_{\mathrm{L}\mathrm{z}}=2.0$, ${\epsilon }_{\mathrm{o}\mathrm{x}}=3.0$, ${\epsilon }_{\mathrm{o}\mathrm{y}}=1.7$, ${\epsilon }_{\mathrm{o}\mathrm{z}}=2.0$. The crystal of the first layer is rotated by 40° around the $z$ axis while the crystal of the material in output has the principal axis directed along the directions of the Cartesian axis. (a) gives the phase of the first ray in output. (b) represents the phase of the wave obtained as a sum of the first and the second traversal rays at output. (c) is represents the phase for the wave obtained as a sum of the first three rays at the output. (d) represents the phase of the output field with all the rays included for the structure.

Figure 5

Simple system that uses an anisotropic multilayer to create an anomalous phase in the output field. The periodic structure has ten periods, formed by two birifrangent layers with dielectric constants: ${\epsilon }_{1x}=3$, ${\epsilon }_{1y}=6$, ${\epsilon }_{1z}=4$, ${\epsilon }_{2x}=7$, ${\epsilon }_{2y}=5$, ${\epsilon }_{2z}=4$, the thickness of the two layers are both $0.5\mu \mathrm{m}$, the wave in input is a beam with cylindrical section (on the $xy$ plane) Gaussian, the axis of the cylinder is the $y$ axis. The beam propagates in a transversal way on the $xz$ plane; the polarization will be linear, normal to the beam and to the $y$ axis. The input and output medium is vacuum. The incident angle is 30° with respect to the $z$ axis. While the second crystal is rotated with respect to the first one by 30° around the $z$ axis.

Figure 6

(Color) The phase of the output wave in channel ${\epsilon }_{\mathrm{o}\mathrm{x}}$, for the system of Fig. 5.

Figure 7

(Color) Field behavior in channel ${\epsilon }_{\mathrm{o}\mathrm{x}}$ along the $x$ axis for the case of Fig. 5 with a wavelength $\lambda =0.5207\mu \mathrm{m}$. (a) input Gaussian beam, (b) output Gaussian beam, (c) transmission spectrum of the multilayer, (d) intensity of incident, reflected, and transmitted beam: the output transmitted beam has been normalized to the transmission value in order to be visualized.

Figure 8

(Color) Gaussian input wave in a multilayer structure. (a) Phase of the output. (b) Contour plot (amplitude) of the input and output wave in the propagation $y\text{−}z$ plane within the middle the space for the multilayer, the output has been normalized to the transmission, it has been plotted only the input pulse and not the reflected one. (c) Transmission spectrum.

Figure 9

(a) Graphical representation of the later spatial shift in the $x\text{−}z$ plane in the case of anomalous phase. (b) Graphical representation of the refraction of an extended and focalized for an anomalous phase of the transmission of the geometry (named system $s$), equivalent to the one expected for a “negative refractive index” material.