Waveguiding driven by the Pancharatnam-Berry phase

We theoretically and numerically investigate the properties of waveguides based on the Pancharatnam-Berry phase, obtained by a longitudinally periodic rotation of the optic axis in a transversely twisted birefringent medium. In this paper we study the case where the period of the longitudinal modulation is chosen so that a net accumulation of geometric phase in propagation occurs. First, the interplay between different contributions to the optical potential is addressed. Second, a continuous evolution of the polarization structure of the quasimodes is observed in the numerical simulations. We explain it by a combination of plane-wave-based models and gauge transformations. We discover that, beyond the longitudinal oscillations, the polarization of the quasimode also varies through its cross section. The analogies with respect to charged particles moving in a magnetic field are outlined.

Figure 1

Optical propagation in a rotated anisotropic material. (a) Definition of the local reference system $x^{\prime }{y}^{\prime }{z}^{\prime }$ rotated of an angle $\theta$ with respect to the laboratory framework $xyz$. In $x^{\prime }{y}^{\prime }{z}^{\prime }$ the dielectric tensor is always diagonal. (b) PBP delay between light propagation in $xyz$ and $x^{\prime }{y}^{\prime }{z}^{\prime }$ versus the retardation $\zeta$ in the case of circularly polarized plane waves. The rotation angle $\theta$ is ${15}^{\circ }$.

Figure 2

Polarization evolution of initially CP plane waves mapped on the Poincaré sphere for three different longitudinal modulation of the rotation angle. The profile of $\theta$ is (a) uniform along $z$, (b) a square wave, and (c) a sinusoidal modulation. The maximum rotation angle is $\pi /8$ in all the three cases. The black curve in each panel is the case for a uniform and vanishing angle $\theta$.

Figure 3

Properties of quasimodes in a sinusoidally rotated anisotropic material infinitely extended along the transverse direction. (a) Stokes parameters of the eigenmode (corresponding to the polarization assumed at the start of each longitudinal period) and (b) the corresponding geometric phase delay $\mathrm{\Delta }{\phi }_{\mathrm{geo}}$ gained across a single rotation period versus the maximum rotation angle ${\mathrm{\Gamma }}_0$. In panel (c) the evolution of the eigenmodes over the Poincaré sphere is shown for ${\mathrm{\Gamma }}_0=4^{\circ }, {24}^{\circ }, {44}^{\circ }$, and ${86}^{\circ }$, from the longest to the smallest circuit, respectively. The dashed line in panel (b) corresponds to the geometric phase computed under the small rotation approximation, providing $\mathrm{\Delta }{\phi }_{\mathrm{geo}}=\pi {\mathrm{\Gamma }}_0$. Due to the symmetry of the system, another set of eigenmodes with opposite Stokes parameters and opposite phase delay exists.

Figure 4

Longitudinal evolution of the quasimodes shown in Fig. 3. Behavior versus $z$ of the Stokes parameters (a) $S_1$, (b) $S_2$, and (c) $S_3$ versus the phase retardation $\zeta =k_0\mathrm{\Delta }nz$. The corresponding maximum rotation angles ${\mathrm{\Gamma }}_0$ are reported in the legend in panel (b). (d) Average of the Stokes parameters over one birefringence length versus the angle ${\mathrm{\Gamma }}_0$.

Figure 5

Theoretical model in the limit of purely CP quasimodes. Photonic potential (a) $V$ and (b) the corresponding fundamental mode (the intensity profile is shown) versus $x$. From shallower to deeper potential well (corresponding to a narrower fundamental mode), the maximum rotation angle ${\mathrm{\Gamma }}_0$ is ${10}^{\circ }$ (blue), ${20}^{\circ }$ (orange), ${60}^{\circ }$ (green), ${90}^{\circ }$ (red), and ${120}^{\circ }$ (magenta). In panel (a), solid and dashed lines correspond to the full potential evaluated from the entire Eq. (14) or only its first term, respectively.

Figure 6

Full numerical simulation of the light propagation for CP inputs of opposite helicity. Time-averaged intensity distribution calculated via FDTD simulations for (a)–(d) RCP and (e)–(h) LCP input polarization for $w_D=3\mu \mathrm{m}$ and increasing ${\mathrm{\Gamma }}_0$ from left to right. The scalar eigenfunctions of the potential given by Eq. (14) are used as the transverse shape of the input. The white solid lines represent the spreading that would occur in the case of a homogeneous material. Finally, the anisotropic material starts in $z=2\mu \mathrm{m}$.

Figure 7

Effects of the gauge transformation on the polarization of the guided quasimodes. Power coupled to the Berry waveguide for input (a) circular, and (b) linear vertical and horizontal polarizations versus the maximum rotation angle ${\mathrm{\Gamma }}_0$. Symbols are values extrapolated from the FDTD simulations, whereas the shaded regions are the theoretical predictions from Eq. (13). Theoretical predictions corresponds to a surface because we are considering a range for the possible width of the quasimode, see the main text. The waveguide parameters are the same as for Fig. 6. The guided power is measured in $z=200\mu \mathrm{m}$ by integrating the intensity around the origin $x=0$ on a window of overall size $10\mu \mathrm{m}$.

Figure 8

Polarization transient at the input interface. Stokes parameters nearby the input interface extracted from FDTD simulations, plotted for ${\mathrm{\Gamma }}_0=1^{\circ }, 5^{\circ }, {45}^{\circ }$, and ${90}^{\circ }$ from top to bottom, respectively. The input is a RCP mode with shape found from the potential (14).

Figure 9

Stationary distribution for the polarization state of the quasimode. Parameters are the same of Fig. 8, but at the end of the numerical grid, i.e., for $z>250\mu \mathrm{m}$.

Figure 10

(a) The Stokes parameters versus the phase retardation for a CP wave in the rotated framework rotated back to the laboratory frame. Only ${\mathit{u}}_0$ is nonvanishing in this case. (b) Stokes parameters versus phase retardation when ${\mathit{u}}_1=-0.1{\mathit{u}}_0$. In both panels the blue dashed curve is $S_3, S_2$ the green lines with symbol, $S_1$ the solid red lines. The values used for ${\mathrm{\Gamma }}_0$ are $0^{\circ }, 7.5^{\circ }, {15}^{\circ }, 22.5^{\circ }, {30}^{\circ }$

Figure 11

Long FEM simulations for small rotation angles. Maps on the plane $xz$ of the electric field components (a) $E_x$, (b) $E_y$, and (c) of the corresponding time-averaged Poynting vector component along the propagation distance $z$. A snapshot of the time-dependent electric fields for $180\mu \mathrm{m}<z<200\mu \mathrm{m}$ is shown in panels (a) and (b). The maximum rotation angle is ${\mathrm{\Gamma }}_0=1^{\circ }$ and $w_D=8\mu \mathrm{m}$. Input is a Gaussian of width $6.6\mu \mathrm{m}$.

Figure 12

Comparison of the output intensity profile computed with FDTD and FEM simulations. The normalized intensity cross section in $z=50\mu \mathrm{m}$ is plotted versus $x$ for three different values of ${\mathrm{\Gamma }}_0$, the latter being labeled at the top of each panel; the width of the twisting distribution is $w_D=8\mu \mathrm{m}$. Solid red and green dashed lines correspond to FEM and FDTD, respectively. In both the simulators, the input is a circularly polarized Gaussian beam with a width equal to the effective fundamental mode.

Figure 13

Comparison of light evolution computed via FDTD and FEM simulations. (a) Intensity distribution for a Gaussian input of waist $2.4\mu \mathrm{m}$ over the plane $x$ for ${\mathrm{\Gamma }}_0={45}^{\circ }$ and $w_D=8\mu \mathrm{m}$, computed with FEM (left side) and FDTD (right side). (b) Corresponding evolution versus $z$ of the Stokes parameters on the beam axis $x=0\mu \mathrm{m}$; blue and red lines correspond to FDTD and FEM simulations, respectively.