Numerical analysis of polarization gratings using the finite-difference time-domain method

We report the first full numerical analysis of polarization gratings (PGs), and study their most general properties and limits by using the finite-difference time-domain (FDTD) method. In this way, we avoid limiting assumptions on material properties or grating dimensions (e.g., no paraxial approximations) and provide a more complete understanding of PG diffraction behavior. We identify the fundamental delineation between diffraction regimes (thin versus thick) for anisotropic gratings and determine the conditions for $\approx 100\%$ diffraction efficiency in the framework of the coupled-wave $\rho$ and $Q$ parameters. Diffraction characteristics including the efficiency, spectral response, and polarization sensitivity are investigated for the two primary types of PGs with linear and circular birefringence. The angular response and finite-grating behavior (i.e., pixelation) are also examined. Comparisons with previous analytic approximations, where applicable, show good agreement.

Figure 1

Periodic anisotropy profiles for the two primary types of PGs studied here: (a) Circular PG and (b) linear PG. The direction or handednesses of the total local anisotropy are illustrated as arrows. ($\Lambda$ is the effective optical grating period.)

Figure 2

Schematics of the 2D FDTD simulation space: (a) layout of simulation space (not drawn to scale); (b) field distribution on square-grid lattices; (c) diffraction order geometry and notation. Note that the $x$ width is determined by the grating period $\Lambda$.

Figure 3

Diffraction behavior of a circular PG—(a) diffraction efficiency spectra and (b) polarization-sensitive first-order diffraction $(\mathrm{\Delta }{n}_{l}d∕\lambda =1∕2)$—numerically calculated (curves) and analytically estimated ($엯$, $◇$, and $◆$) using Eqs. (6), where $\mathrm{\Delta }{n}_{l}d∕\lambda$ is the normalized retardation and $\chi$ is the ellipticity angle of the incident polarization. ($\Lambda =20{\lambda }_0$, $\overline{n}=1.6$, $\mathrm{\Delta }{n}_l=0.2$.)

Figure 4

(Color online) FDTD near-field images of wave propagation through a circular PG for (a) right-handed circular $(\chi =\pi ∕4)$ and (b) vertical linear $(\chi =0)$ incident polarization, with WOLFSIM. Animated movies are available at EPAPS Ref. 30. ($\Lambda =20\lambda$, $\overline{n}=1.6$, $\mathrm{\Delta }{n}_l=0.2$, $d=\lambda ∕2\mathrm{\Delta }{n}_l$.)

Figure 5

Angular response of a circular PG when illuminated at oblique incidence with right-handed circular polarization input $(\chi =45°)$: (a) First-order and (b) zero-order diffraction efficiency; (c) polarization state of the diffracted $-1$ order. PG parameters are set for half-wave retardation. ($\Lambda =20\lambda$, $d=5\lambda$, $\overline{n}=1.6$, and $\mathrm{\Delta }{n}_l=0.2$.)

Figure 6

(Color online) The behavior of circular PG diffraction with respect to grating regime (thin or thick), from the perspective of $Q$ and $\rho$ parameters [31, 32, 34]. The contour plots show the total first-order diffraction $\Sigma {\eta }_{\pm 1}$ versus normalized retardation $\mathrm{\Delta }{n}_{l}d∕\lambda$ and normalized grating period $\Lambda ∕{\lambda }_0$. A comparison between numerical FDTD calculation (black solid-line contours) and analytical estimate (grayscale levels) is shown for a high $(\mathrm{\Delta }{n}_l=0.2)$ and low $(\mathrm{\Delta }{n}_l=0.04)$ birefringence in parts (a) and (b), respectively. We note that diffraction best follows the analytic expressions [Eqs. (6)] when $\rho <1$, the thin-grating regime. The input wave is normally incident, and the crosshatch indicates $\lambda ∕\Lambda >1$, when the first-order diffraction modes become evanescent.

Figure 7

The first maximum of the first-order diffraction $\Sigma {\eta }_{\pm 1}$ of the circular PG (calculated numerically). The normalization factor ${\lambda }_{\max }$ is the wavelength at which maximum diffraction occurs. Note that the axis for ${\theta }_{+1}$ shows the angle of diffraction for the given ratio $\Lambda ∕{\lambda }_{\max }$.

Figure 8

Diffraction behavior of finite circular PGs. We show the near-field images and intensity profiles of (a) a finite circular PG and (b) a dielectric slab, when $L=2\Lambda$ and $\Lambda =10\lambda$. In part (c), the first-order efficiencies of two finite circular PGs with $\Lambda =6\lambda$ and $10\lambda$ are compared for various pixel sizes up to $L=10\Lambda$ for each case. Animated movies of the fields are available (Ref. 30). ($\overline{n}=1.6$, $\mathrm{\Delta }{n}_l=0.2$, and $d=\lambda ∕2\mathrm{\Delta }{n}_l$.)

Figure 9

Diffraction behavior of a linear PG $(\mathrm{\Delta }{n}_l=\mathrm{\Delta }{n}_c=\mathrm{\Delta }n=0.2)$—(a) diffraction efficiency spectra and (b) polarization-sensitive first-order diffraction $(\mathrm{\Delta }nd∕\lambda =1∕2)$—numerically calculated (curves) and analytically estimated ($엯$, $◇$, and $◆$) using Eqs. (10). ($\Lambda =20{\lambda }_0$, $\overline{n}=1.6$.)

Figure 10

Diffraction behavior of the linear PG $(\mathrm{\Delta }{n}_c\ne \mathrm{\Delta }{n}_l)$ for different values of circular birefringence $\mathrm{\Delta }{n}_c$; (a) diffraction efficiency spectra and (b) polarization-sensitive first-order diffraction $(\mathrm{\Delta }{n}_{l}d∕\lambda =1∕2)$. ($\Lambda =20\lambda$, $d=5\lambda$, $\overline{n}=1.6$, and $\mathrm{\Delta }{n}_l=0.2$.)