Nonlinear unitary transformations of space-variant polarized light fields from self-induced geometric-phase optical elements

We propose to couple the optical orientational nonlinearities of liquid crystals with their ability to self-organize to tailor them to control space-variant-polarized optical fields in a nonlinear manner. Experimental demonstration is made using a liquid crystal light valve that behaves like a light-driven geometric phase optical element. We also unveil two original nonlinear optical processes, namely self-induced separability and nonseparability. These results contribute to the advancement of nonlinear singular optics that is still in its infancy despite 25 years of effort, which may foster the development of nonlinear protocols to manipulate high-dimensional optical information both in the classical and quantum regimes.

Figure 1

Experimental setup. ${\mathbf{E}}_{\mathrm{in}}$: linearly polarized incident Gaussian beam; CA clear circular aperture with diameter 3 mm; ${\mathrm{PC}}_i$: polarization controllers; $L_1$: lens with focal length $f_1=100\mathrm{mm}$; LCLV: liquid crystal light valve whose enlargement depicts its structure (see text for details); $U$: applied voltage on the LCLV. $L_2$ and $L_3$: lenses with focal length $f_2=f_3=75\mathrm{mm}$; BS: beamsplitter; Cam: camera.

Figure 2

(a) Power dependence of the parameter $\eta$ that characterizes the self-induced generated scalar vortex field for an incident circularly polarized unstructured light. Markers: measurements for incident power giving extremal values of $\eta$ as well as $\eta \simeq 0.5$. Solid line: spline fit of the experimental data. Interference pattern between the output field and a reference Gaussian beam in the linear and nonlinear regimes in panels (b) and (c), respectively. The second of these refers to the fourth maximum of $\eta \left(P\right)$, which corresponds to the realization of a SIUQP. Labels A to D refer to local maxima occurring at $P=2$, 4.3, 7.7, and 16 mW, see Fig. 3 while labels ${\mathrm{A}}^{\prime }$ to ${\mathrm{C}}^{\prime }$ refer to local minima occurring at $P=3.2$, 5.8, and 10.7 mW.

Figure 3

(a) Images of the LCLV observed between crossed circular polarizers at incident optical power values that correspond to local maxima for the scalar optical vortex generation as reported in Fig. 2, see labels A to D. (b) Same as in (a) using crossed linear polarizers. The dashed white circles refer to the geometric image of CA.

Figure 4

(a) Images of the LCLV illuminated by a incident circularly polarized light, adjusting the polarization controller $P{C}_2$ as a polarizer oriented at an angle $\alpha =0^{\circ },{45}^{\circ },{90}^{\circ }$, and ${135}^{\circ }$ from the $x$ axis. (b) Experimentally reconstructed map ${\theta }_{\exp }$ of the effective optical axis orientation of the SIUQP. Dashed white circle: contour of the geometric image of the circular aperture CA. Solid white circle: region of interest used to evaluate experimentally the swirl function ${\phi }_{\exp }\left(r\right)$. (c) Determination of the swirl function ${\phi }_{\exp }\left(r\right)$ according to the protocol described in the text. The inset displays the modeled map $\theta (r,\varphi )$ of the optical axis orientation of the SIUQP over the region of interest.

Figure 5

(a) Experimental output maps ${\psi }_{\exp }$ of the electric field vector orientation in the linear regime for input optical states ${\mathbf{\Psi }}_{\mathrm{V}}\left(\ell \right)$ with $-2\le \ell \le +2$. (b) Simulated maps associated with ${\mathbf{\Psi }}_{\mathrm{V}}\left(\ell \right)$.

Figure 6

(a) Experimental output maps ${\psi }_{\exp }$ of the electric field vector orientation in the nonlinear regime for input optical states ${\mathbf{\Psi }}_{\mathrm{V}}\left(\ell \right)$ with $-2\le \ell \le +2$. (b) Simulated maps associated with ${\mathbf{\Psi }}_{\mathrm{V}}(2-\ell )$. (c) Simulated maps associated with the hybrid states ${\tilde{\mathbf{\Psi }}}_{\mathrm{V}}(2-\ell )$, see text for details.

Figure 7

(a,b) Nonlinear optical nonseparability when $\ell =0$. Interference pattern between the ${\mathbf{c}}_{{\sigma }_{\mathrm{out}}}$-polarized components of the output field with a copolarized reference Gaussian beam in the (a) linear and (b) nonlinear regimes. (c,d) Same as in (a,b) but for nonlinear optical separability when $\ell =+2$.