Pure and achromatic spin-orbit shaping of light from Fresnel reflection off space-variant anisotropic media

We propose an approach to achieve achromatic pure geometric phase shaping of light from anisotropic media by exploiting the fundamental laws of electromagnetic waves. Its practical implementation requires a cover layer made of a dispersive isotropic medium. The approach applies to any anisotropic material and any geometric phase spatial distribution and preserves the geometric phase reversal upon the reversal of the incident polarization handedness. An experimental demonstration is made over the whole visible range but can be extended to any wavelength range without conceptual issues.

Figure 1

Purity $\xi$ as a function of the refractive index $n$ of the isotropic cover layer. Markers show experimental data and the solid curve denotes theory [see Eq. (4)]. The values of $n$ at 532 nm are extrapolated from the known values at ${\lambda }_{\mathrm{F}}=486.1$ nm, ${\lambda }_{\mathrm{D}}=589.3$ nm, and ${\lambda }_{\mathrm{C}}=656.3$ nm by using a Cauchy law of the form $n=c_1+c_2/{\lambda }^2$.

Figure 2

(a) Dispersion of the refractive indices $n_{\parallel }$ (blue curve), $n_{\perp }$ (red curve), and $\overline{n}$ (black curve) for the calcite slab in the wavelength range $400\mathrm{nm}<\lambda <800\mathrm{nm}$, from Ref. [20]. The dispersion for the oil that gives the maximal value of $\xi$ in Fig. 1 is shown in magenta. (b) Dispersion of the fraction $\xi$ of the reflected light exhibiting spin angular momentum reversal. The markers show the experimental data. The magenta curve shows the theoretical behavior from Eq. (4) and the black curve the ideally pure behavior $\xi =1$ when $n\left(\lambda \right)=\overline{n}\left(\lambda \right)$. The inset shows the calculated value up to $\lambda =2\mu \mathrm{m}$ using the Cauchy extrapolation for the oil and the dispersion equations from Ref. [20].

Figure 3

Product $\sigma {\mathrm{\Phi }}_g$ as a function of the optical axis orientation angle $\psi$. The markers show the experimental data and the solid line denotes theory, $\sigma {\mathrm{\Phi }}_g=2\psi$.

Figure 4

Rotational Doppler experiment at a 532-nm wavelength as described in the text, where the sample is rotated at constant angular frequency $\mathrm{\Omega }$ and the reflected light is recorded after a linear polarizer. (a) Snapshot collection of the optical intensity distribution over one oscillation period of duration $\pi /\mathrm{\Omega }$. (b) Normalized time-dependent power $p\left(t\right)$ as a function of the normalized time $t/T$, with $T=2\pi /\mathrm{\Omega }$. The thick gray curve denotes the experimental data and the thin black curve theory, $p\left(t\right)={cos}^2\left(\mathrm{\Omega }t\right)$.

Figure 5

Map of the reflectance $R^{*}$ in the plane of parameters $(\langle n\rangle ,\delta n)$, where $\langle n\rangle =(n_{\parallel }+n_{\perp })/2$. The plus markers refer to the following practical cases: quartz at $\lambda ={\lambda }_{\mathrm{D}}$ (1), calcite at $\lambda ={\lambda }_{\mathrm{D}}$ (2), common nematic liquid crystal E7 at $\lambda ={\lambda }_{\mathrm{D}}$ (3), very high birefringence nematic liquid crystal at $\lambda =598$ nm [24] (4), diamond subwavelength grating patterns at $\lambda =1550$ nm with pitch $\mathrm{\Lambda }=\lambda /4$ (5), and amorphous silicon subwavelength grating patterns at $\lambda =1550$ nm with pitch $\mathrm{\Lambda }=\lambda /4$ (6). Cases 5 and 6 are evaluated using the second-order effective-medium theory [25].