Interplay between diffraction and the Pancharatnam-Berry phase in inhomogeneously twisted anisotropic media

We discuss the propagation of an electromagnetic field in an inhomogeneously anisotropic material where the optic axis is rotated in the transverse plane but is invariant along the propagation direction. In such a configuration, the evolution of an electromagnetic wave packet is governed by the Pancharatnam-Berry phase (PBP), which is responsible for the appearance of an effective photonic potential. In a recent paper [ACS Photon. 3, 2249 (2016)] we demonstrated that the effective potential supports transverse confinement. Here we find the profile of the quasimodes and show that the photonic potential arises from the Kapitza effect of light. The theoretical results are confirmed by numerical simulations, accounting for the medium birefringence. Finally, we analyze in detail a configuration able to support nonleaky guided modes.

Figure 1

(a) Definition of the rotation angle $\theta$. (b) Gaussian distribution of the rotation angle $\theta$ on the plane $xy$ when maximum rotation is ${90}^{\circ }$: the arrows correspond to the local optic axis (i.e., the extraordinary axis), superposed to the spatial distribution of $\theta$ represented as a color map. When $\delta =\pi$, the distribution plotted in panel (b) corresponds to a polarization-dependent lens [3, 21].

Figure 2

Pancharatnam-Berry phase difference $\mathrm{\Delta }{\varphi }_{\mathrm{geo}}$ versus retardation $\delta$ between two regions differing only by a relative rotation $\theta$ when spatial dispersion is neglected, i.e., in the plane-wave limit. The blue solid lines are the exact results computed from Eq. (23); the red dashed lines are a sinusoidal approximation for $\mathrm{\Delta }{\varphi }_{\mathrm{geo}}$ sharing the same peak $2\theta$ of the exact solutions.

Figure 3

(a) Rotation angle $\theta$ (blue solid line) and corresponding photonic potential (dashed green line) versus $x$ for a Gaussian-shaped rotation of principal axes. Here ${\theta }_0=\pi /2$ and $w_{\theta }=5\mu \mathrm{m}$. (b) As in panel (a) but when $\theta$ has a hyperbolic tangent distribution. Here $L_{\theta }=5\mu \mathrm{m}$. When $\delta =\pi$, panels (a) and (b) behave like a polarization-dependent lens and deflector, respectively.

Figure 4

(a) Intensity profile for a Gaussian distribution of $\theta$ and $w_{\theta }=5\mu \mathrm{m}$; each curve is labeled with the corresponding maximum rotation ${\theta }_0$. (b) Corresponding width of the quasimode versus maximum rotation angle ${\theta }_0$ for three different $w_{\theta }$, as labeled (solid lines). The dashed lines graph the half width of the angular distribution: above them the model is not accurate due to a finite overlap of the mode with the edges of the photonic potential [37]. Here we solved Eqs. (14) and (15) by considering only the term given by Eq. (16) when computing the quasimode; we assumed $n_{\perp }=1.5$ and $n_{\parallel }=1.7$.

Figure 5

Light propagation in the presence of (a) a focusing or (b) a defocusing effective potential. The first and second rows correspond to input wave packets polarized along $y$ and $x$, respectively. In panel (a) and (b) the transverse distribution of $\theta$ is given by Eqs. (25) and (26), respectively. Here ${\theta }_0={360}^{\circ },w_{\theta }=L_{\theta }=5\mu \mathrm{m}, n_{\perp }=1.5$, and $n_{\parallel }=1.7$, the wavelength is 1064 nm. The input is a Gaussian beam of width $3\mu \mathrm{m}$ and with a flat phase front.

Figure 6

Intensity distribution in the plane $xz$ versus maximum rotation angle ${\theta }_0$, computed with BPM simulations when $\theta$ is Gaussian with $w_{\theta }=5\mu \mathrm{m}$, see Eq. (25); here the refractive indices are $n_{\perp }=1.5$ and $n_{\parallel }=1.7$ and a $y$-polarized Gaussian beam of waist $3\mu \mathrm{m}$ is launched at the input. In the first and second rows $|{\psi }_e{|}^2$ and $|{\psi }_o{|}^2$ are plotted, respectively. In the third and fourth rows the corresponding fields $|E_y{|}^2$ and $|E_x{|}^2$ are graphed in the laboratory coordinate system, respectively.

Figure 7

Columns 1 to 5: Intensity evolution in the plane $xz$ when the mode plotted in Fig. 4 is launched at the input $z=0$, for different values of $n_{\parallel }$ as labeled (the corresponding birefringence $\mathrm{\Delta }n$ is 0.001, 0.01, 0.05, 0.2, and 0.4, from left to right, respectively). Top and bottom rows correspond to local ordinary and extraordinary polarizations, respectively. Column 6: modulus of the field $z=500\mu \mathrm{m}$ versus $x$ and the refractive index $n_{\parallel }$. Here $\theta$ is Gaussian with ${\theta }_0={360}^{\circ }$ and $w_{\theta }=5\mu \mathrm{m}, n_{\perp }=1.5$.

Figure 8

FDTD simulations of the intensity distribution for (a) defocusing [$\theta$ obeying Eq. (25)] and (b) focusing [$\theta$ distributions as provided by Eq. (26)] effective potential. The input beam is linearly polarized along $y,n_{\parallel }=1.7,w_{\theta }=L_{\theta }=5\mu \mathrm{m}$, and ${\theta }_0={360}^{\circ }$. The white dashed line indicates the input interface, the red solid lines show beam diffraction for ${\theta }_0=0^{\circ }$.

Figure 9

FDTD simulations versus ${\theta }_0$. Evolution of $|E_x{|}^2$ (first row), $|E_y{|}^2$ (second row) and the overall intensity $|E_x{|}^2+{\left|E_y\right|}^2$ (third row) in the plane $xz$. The input beam is linearly polarized along $y$.

Figure 10

Direct comparison between analytic theory and FDTD simulations when the input is linearly polarized along $y,\theta$ has is Gaussian and $w_{\theta }=5\mu \mathrm{m}$. Upper panels: overall intensity cross-sections calculated at $z=120\mu \mathrm{m}$ for a Gaussian wave packet undergoing diffraction when ${\theta }_0=0^{\circ }$ (solid black lines) and the dielectric tensor is transversely rotated (solid blue lines). The red dashed lines are the corresponding quasimodes from Eqs. (14) and (15). Lower panels: wave packet width versus propagation distance $z$ using ${\mathrm{\Pi }}_{E_y}$ (blue solid line) and ${\mathrm{\Pi }}_I$ (red solid lines) for the probability densities. The black dashed line corresponds to a diffracting Gaussian beam in the limit ${\theta }_0=0^{\circ }$. Here $L=10\mu \mathrm{m}$.

Figure 11

FDTD simulations versus birefringence for a fixed $\theta$ profile. The beam is linearly polarized along $x$ (left column) and $y$ (right column). Birefringence $\mathrm{\Delta }n$ is 0.001, 0.01, 0.2, and 0.4, from the top to bottom row, respectively. The red solid lines show the diffracting case, the dashed white lines correspond to the initial section of the anisotropic medium. Here $n_{\perp }=1.5,{\theta }_0={360}^{\circ }$, and $w_{\theta }=5\mu \mathrm{m}$.

Figure 12

V-shaped PBP waveguide. (a) Rotation angle $\theta$ (blue solid line) and corresponding potential $V$ from Eq. (27) (green dashed line) for $x_0=10\mu \mathrm{m}$ and $a=4\times {10}^{10}{\mathrm{m}}^{-2}$. (b) Mode width versus $a$ for (top to bottom curves) $x_0=5\mu \mathrm{m}$ (blue line), $10\mu \mathrm{m}$ (green), and $20\mu \mathrm{m}$ (red). The stars graph the width $w_G$ of the Gaussian fundamental mode of the parabolic potential in the limit $x_0\to \infty$. Inset: mode intensity profile when $a=2\times {10}^{10}{\mathrm{m}}^{-2}$ for $x_0=5\mu \mathrm{m}$ (blue solid line) and $x_0=20\mu \mathrm{m}$ (red dashed line). Blue stars represent the Gaussian mode when $x_0\to \infty$.

Figure 13

V-shaped PBP waveguide. FDTD simulations in the plane $xz$ for $a=0,3\times {10}^{10}$, and $3\times {10}^{11}{\mathrm{m}}^{-2}$, respectively (slices from bottom to top) when (a) $x_0=2\mu \mathrm{m}$ and (b) $x_0=10\mu \mathrm{m}$. The red dashed lines correspond to the straight lines $x=\pm |x_0|$, i.e., the waveguide edges.