Tunable High-Resolution Macroscopic Self-Engineered Geometric Phase Optical Elements

Artificially engineered geometric phase optical elements may have tunable photonic functionalities owing to their sensitivity to external fields, as is the case for liquid crystal based devices. However, liquid crystal technology combining high-resolution topological ordering with tunable spectral behavior remains elusive. Here, by using a magnetoelectric external stimulus, we create robust and efficient self-engineered liquid crystal geometric phase vortex masks with a broadly tunable operating wavelength, centimeter-size clear aperture, and high-quality topological ordering.

Figure 1

(a) Geometry of the liquid crystal film under magnetoelectric external field. The magnetization of the nickel-plated neodynium (grade N42) ring magnet is directed along the $z$ axis. The magnet parameters are $R_{\mathrm{in}}=4\mathrm{mm}$, $R_{\mathrm{out}}=7.5\mathrm{mm}$, and $H=6\mathrm{mm}$ and are associated with a pull force of $\sim 50N$ (manufacturer data sheet). (b),(c) Images of the sample observed between crossed linear polarizers (oriented along the $x$ and $y$ axes) using white light incoherent illumination 10 min after a square-waveform voltage $U=2.15\mathrm{V}$ at 2 kHz frequency is applied without (b) and with (c) the presence of the ring magnet. The magnet is placed at a distance $d=2\mathrm{mm}$ from the liquid crystal layer in panel (c), while $d=1\mathrm{mm}$ in the inset. The dark cross reveals an umbilical defect with topological charge $+1$. (d)–(f) Side view of the lines of the director field—the local-averaged orientation of the liquid crystal molecules—of the sample at rest (d), under the magnetic field alone (e), and under the combined action of the magnetic and electric fields (f).

Figure 2

Experimental spectral dependence of the reduced voltage values $U_n/U_{\mathrm{th}}$, where $U_{\mathrm{th}}\simeq 1.8\mathrm{V}$ refers to the threshold below which the liquid crystal remains unperturbed in the absence of the magnet, which fulfill the half-wave retardance condition ${\mathrm{\Delta }}_{\infty }=n\pi$ with an $n$ odd integer. Color markers refer to different values for $n$, while the black curves are only a guide for the eyes.

Figure 3

Optical characterization of the generation of spectrally agile optical vortex beams from a 2-mm-diameter collimated supercontinuum laser beam filtered with a set of nine wavelengths from 400 to 800 nm in steps of 50 nm, in the case of an incident linear polarization state along the $x$ axis. The study is performed for $n=1$, hence, adjusting the applied voltage according the specifications reported in Fig. 2. (a) Far-field total intensity profiles. (b) Far-field intensity patterns when a linear polarizer oriented along the $x$ axis is placed at the output of the device.

Figure 4

(a) Temporal evolution of the defect location given by its coordinates $(x_d,y_d)$ in the plane of the film over almost 20 h, where $⟨·⟩$ denotes the time average over the whole duration of the observation. (b)–(d) Imaging of the device placed between crossed linear polarizers using white light incoherent illumination spectrally filtered at 633 nm wavelength when a circularly polarized Gaussian beam at 532 nm wavelength with total optical power $P$ impinges at normal incidence onto the sample. Shown data recorded at $U/U_{\mathrm{th}}=1.2$.

Figure 5

(a) Observation of the swirl in the neighborhood of the liquid crystal topological defect by placing the device between crossed linear polarizers using white light incoherent illumination. (b) Observation of the umbilic core by placing the sample between crossed circular polarizers under white light incoherent illumination spectrally filtered at 532 nm wavelength. The data refer to applied voltage values that correspond to $n=1$ and $n=7$ for 532 nm wavelength.

Figure 6

Experimental setup used for the demonstration of spiral phase Fourier filtering at 700 nm wavelength using a so-called $4f$ optical system made of a pair of lenses $L$ with focal length 200 mm. This corresponds to an Airy spot radius $r_0\simeq 40\mu \mathrm{m}$ (taken at the first zero of intensity) in the Fourier plane of the telescope. The input circular aperture has a diameter $D=4\mathrm{mm}$. The self-engineered magnetoelectric liquid crystal $q$ plate is placed in the Fourier plane. The input pupil is reimaged on a camera (using a lens with 75 mm focal length) when the defect is displaced of axis at a distance $\simeq 10r_0$ (b) and when it is placed on axis (c), for $n=5$.