Fourier-Domain Phase Retardation Vortex Measurement

. Optical vortices have found a wide range of applications thanks to their helical phase topology allowing to carry the orbital angular momentum. In this work, self-interfering vortex beams are utilized in a new single-shot holographic method for the circular phase retardation measurement. The vortices carrying information about the phase retardation introduced between two orthogonal circular polarization modes are generated by the spin to orbital angular momentum conversion. The phase retardation is stored in off-axis holographic records acquired in a common-path setup using a geometric-phase grating. In the proposed method, the circular phase retardation is reconstructed in the Fourier domain, surpassing the measurement precision provided by methods restoring the retardation from the rotation of a Double-Helix Point Spread Function (DH PSF). The developed method can be adapted for application to polarimetry, orientation imaging and diagnostics of nano-emitters.


Introduction
The seminal paper on the depth from diffracted rotation [1] laid the foundations of optical experiments for accurate localization of point-like scatterers.In these experiments, pairs of interfering light vortices of opposite topological charges created two-lobe diffraction image spots (DH PSFs) rotating under the image defocusing.Super-resolution localization microscopy [2] and photoactivation imaging [3] have been established with this approach, enabling 3D tracking of microparticles [4].By creating well-separated point-like emitters on smooth or rough surfaces, vortex topography was proposed and implemented [5,6].The optical capabilities of localization microscopy have further expanded thanks to new technologies for manipulating the angular momentum of light, namely the spin (SAM) to orbital angular momentum (OAM) conversion [7].Using the spin-orbit interaction, the DH PSF rotation became sensitive to the geometric phase varying in the polarization transformation.This approach led to the development of orientation imaging built on vortex beams and polarization of light scattered by plasmonic nanoparticles (sub 100 nm), allowing their 3D orientation imaging during Brownian motion or interaction with cells (frame rates up to 100 Hz) [8].
In the presented research, we aimed at the quantitative reconstruction of the circular retardation encoded into the interfering light vortices.When using the vortex imaging, the measurement accuracy was limited by determining the angular position of the lobes forming the recorded DH PSF.With this accuracy, the method was not sufficiently sensitive to variations in the circular retardation of common optically active samples.We combined the Fourier-domain vortex localization technique with geometric-phase microscopy [9,10] to enhance the measurement performance.

Results and discussion
In applications of the method, the measured circular retardation is introduced by an optically active sample or through the shape anisotropy of a nano-emitter [8].To verify the principle and assess the accuracy of the method, a controlled phase retardation was introduced between the right-hand and left-hand circular polarizations [state vectors |⟩ and |⟩] by rotating the linear polarizer RLP.The optical setup used in the calibration measurement is shown in Fig. 1a.Light with linear polarization precisely set to the states |⟩ is captured by a microscope objective MO (Nikon 10x, NA=0.45) and directed to a Q-plate (Thorlabs WPV10L-633) at its back focal plane.The polarization |⟩ incident on the Q-plate is represented by the |⟩ and |⟩ states with the circular phase retardation  2 .The SAM to OAM conversion mediated by the Q-plate reverts the handedness of the circular polarizations while creating light vortices with topological charges  = ±1.The light transformed by the Q-plate is focused by the tube lens TL (f=200 mm).When circularly polarized vortex beams are projected through a linear polarizer LP1, letting them interfere, a DH PSF is created at the back focal plane of the TL and captured by a CCD (Fig. 1b).By processing the recording, the angular orientation of the DH PSF lobes can be reconstructed, determining the circular retardation introduced by the rotating linear polarizer LPR.This measurement was used as comparative for the developed method.In the proposed measurement technique, the linear polarizer LP1 is not used, and the Fourier lens FL (f=200 mm) transforms the light coming from the TL.In the back focal plane of the FL, the Q-plate image is formed, and a geometric-phase grating GPG (Edmund Optics, 114 groves/mm) is inserted there.This grating is polarization sensitive, and |⟩ and |⟩ polarizations are deflected to +1st and -1st diffraction orders.The diffracted light is further transformed by a 4-f system composed of Fourier lenses FL1 and FL2 (f1=125 mm and f2=200 mm).After projecting the circular polarizations by the polarizer LP2, an off-axis Fourier hologram is created on the CCD in the back focal plane of the FL2 [9,10].The cross term of the Fourier hologram is separated thanks to the spatial carrier frequency, and its phase is restored quantitatively.The helical phase of light vortices with the topological charges  = ±1 is transferred to the k-space; hence, the reconstructed phase creates a spiral pattern with two arms determined by the topological charge difference |Δ| = 2 (Fig. 1c).Because the phase of optical vortices is influenced by circular retardation, the phase pattern rotates when this retardation changes.The angular setting of the RLP is evaluated by determining the angular position of the phase edges in the reconstructed phase pattern.This can be done with higher accuracy than evaluating the angular position of the DH PSF lobes.The rotation of vortex intensity and phase patterns depending on the circular retardation was proved in developed simulation models.Fig. 2a shows the DH PSFs and vortex phase maps for three different circular retardation values, set by the RLP with orientations 0°, 45°, and 90°.The related image patterns acquired using the setup in Fig. 1a are shown in Fig. 2b.The circular retardation was directly restored from the DH PSF azimuthal orientation by determining the angle given by the lobe centroids (each lobe sampled by 250 pixels) (Fig. 2c).The developed algorithm processed the phase maps obtained from the cross term of the hologram taken in k-space.Using this algorithm, the angular positions of the phase edges were located with accuracy increased compared to the DH PSF method (Fig. 2c).The accuracy of circular retardation reconstructed using numerical simulation models is shown in Fig. 3.The RMS error of the holographic method was 0.01°, while the reconstruction using the DH PSF was degraded by an RMS error of 0.2°.

Conclusion
This paper presents a new technique for measuring circular phase retardation, combining vortex localization with holographic phase reconstruction.The optical performance of the method is promising for the investigation of optically active samples and anisotropic nano-emitters.
This work has been supported by the project No. 21-01953S of the Grant Agency of the Czech Republic and the Palacký University project IGA_PrF_2023_002.

Fig. 2
Fig. 2 Evaluation of the circular retardation.a) Numerical simulation of the rotating DH PSFs and phase maps restored from Fourier holograms.b) The same as in a) but for experimental data.c) Processing of DH PSFs and phase maps.

Fig. 3
Fig. 3 Numerical accuracy assessment in the reconstruction of the circular phase retardation: -proposed holographic method, -DH PSF method.