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Abstract
Recent advances in ptychography have extended to anisotropic specimens, but vectorial reconstruction of probes owing to polarization aliasing remains a challenge. A polarization-sensitive ptychography that enables full optical property measurement of vector light is proposed. An optimized reconstruction strategy, first calibrating the propagation direction and then performing faithful retrieval, is established. This method avoids multiple image acquisitions with various polarizer configurations and significantly improves the measurement accuracy by correlating the intensity and position of different polarization components. The capability of the proposed method to quantify anisotropic parameters of optical materials and polarization properties of vector probe is demonstrated by experiment.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Ptychography is a high-precision phase retrieval technique that exploits a set of partially overlapping diffraction patterns to simultaneously reconstruct the complex amplitudes of both the sample and probe beam [1–4]. Owing to its high data redundancy, ptychography has several advantages over conventional phase-retrieval techniques, including strong anti-interference ability, no requirement for prior information, and more reliable reconstruction accuracy [5–9]. Currently, ptychography is not only widely applied in biomedical research, materials science, chemistry, and other sample imaging fields, but also serves as a new competitive method of beam characterization and optical component measurement because of its capability to quantitatively obtain the phase of the probe [6,10–15].
However, ptychography traditionally models the probe-object interaction in a scalar approximation and thus ignores polarization information. Recently, polarization-sensitive ptychography techniques, such as vectorial ptychography and vectorial Fourier ptychography, have been developed to reconstruct the anisotropic properties of objects and extend the application range of ptychography to the birefringence evaluation of biological specimens [16–25]. Ptychography needs to be leveraged to reconstruct vector probes in several cases, for example, to quantitatively evaluate the birefringence properties of slightly larger scale materials (such as liquid crystals, crystals, stressed polymers, etc.) as well as optical elements under stress [12,26]. The processes are not suitable for scanning as diffractive objects, but can be placed in the probe-formed path so that their birefringence properties can be derived from the reconstructed probe. Additionally, the accurate measurement of the polarization distribution of the beam is a key problem in studying vector beams, which has significant application value in the fields of microscopic imaging, particle acceleration, optical capture, and near-field optics. Polarization ptychography aimed at vector probes provides a useful means to access this information [27–31]. Baroni et al. employed multiple ptychography measurements with a rotating polarizer to investigate the vector properties of the probe, where the angle error and the influence on beam propagation direction of polarizer rotation are unavoidable [32].
In another recent development, the concept of mixed-state ptychography was proposed and demonstrated, which efficiently leverages the inherent data redundancy to reconstruct multiple modes of the probe beam and object. The primary application of multi-probe modes is beam coherence and stability obtained by solving for transverse modes. However, polarization decomposition remains a significant challenge because of reconstruction ambiguities from polarization aliasing [33–37].
In this paper, we report for the first time mixed-state ptychography aimed at full optical property measurement of vector light. Two different measurement methods are described, and an advanced iterative algorithm suitable for both methods is presented to significantly improve the reconstruction accuracy. The relevance of this approach for studying the birefringent properties of optical materials and polarization properties of vector beams was experimentally confirmed.
2. Method
A Wollaston prism was introduced into a traditional ptychography measurement setup to solve the polarization-aliasing problem of vector light measurement. The Wollaston prism can be placed either in front of the diffractive object, that is, beam splitting before modulation [ Fig. 1(a)], or between the diffractive object and the charge-coupled device (CCD), that is, beam splitting after modulation [Fig. 1(b)]. In both cases, the object O only needs to be isotropic, thereby reducing the selection difficulty. Although the physical processes of the two methods are different, the core idea is the same, and we can adopt the same iterative algorithm for accurate oblique wavefront reconstruction.
Fig. 1. Schematic of mixed-state ptychography for vector light reconstruction. (a) Beam splitting before modulation; (b) beam splitting after modulation.
2.1 Beam splitting before modulation
As shown in Fig. 1(a), the vector light E first passes through the Wollaston prism and is divided into two orthogonal polarization components with a small separation angle. The exit field from the Wollaston prism can be written as
Then, EW propagates forward over a distance d and forms two probe components Pm (m = 1,2) on the diffractive object plane. There is a small lateral separation between the two probe components, causing them to undergo different but mostly overlapping modulations on the diffractive object. Pm is derived from the diffraction propagation (considering P1 as an example):
2.2 Iterative algorithm
A mixed-state reconstruction algorithm can be used; however, the propagation direction after passing through a Wollaston prism must be accurate, otherwise the reconstruction will be fuzzy or even fail. This is shown in detail in section 3. In the general ptychography reconstruction algorithm, the tilt factor in the probe phase has no practical physical significance, because there is no constraint on the position of the diffractive object. However, in the proposed method, horizontal position relationships exist between different parts of the object illuminated by the two probe components in the object plane and between two diffraction spots in the CCD plane, which makes the propagation direction of the two probe components an important parameter that must be accurate. From the relationship $\tan \alpha = x/z$, where z is the distance from the object to the CCD, it can be estimated that the angles α and β need to be accurate to 0.01° to achieve pixel-level lateral accuracy; otherwise, the reconstructed object will appear fuzzy or will not converge. In addition, the intensities of the reconstructed probe and object are relative values and the two can complement each other in general ptychography reconstruction. To address this problem, we propose a new reconstruction method based on a mixed-state reconstruction algorithm. A flowchart is shown in Fig. 2, and the entire process is described in the following sequence.
- (1) The initial estimates of the two probe components were set according to Eq. (3), including a constant amplitude, phase guess of ${P^{\prime}_m}$, and separate oblique wavefront guess of $\varphi = 2\pi (x\cos \alpha + y\cos \beta )/\lambda$,where the initial values of α and β are the separation angles of the Wollaston prism. The two objects were initialized with random amplitude and phase distributions, and each object interacted individually with a single probe component.
- (2) The reconstruction begins with five iterations of the common mixed-state algorithm, updating the two objects and probe components with the constraint of Ij, where Ij represents the jth recorded intensity in the object scanning process. Leveraging the fast convergence of the early iterations, the general shapes of the two objects are produced within a few iterations. The offset $({\Delta x,\Delta y} )$ of the two reconstructed objects is determined by a cross-correlation operation, and used as feedback to adjust the guessed values of α and β in step (1), where $\mathrm{\Delta }\alpha = {\tan ^{ - 1}}(\mathrm{\Delta }x/z)$, $\mathrm{\Delta }\beta = {\tan ^{ - 1}}(\mathrm{\Delta }y/z)$.
- (3) Repeat the above steps (1) and (2) until the offset of two reconstructed objects is 0, and then output the accurate oblique wavefront φ0.
- (4) After φ0 is determined, the initial guesses, which are consistent with the actual situation, are given including an object with random amplitude and phase distributions, and two probe components ${P^{\prime}_m}$ with constant amplitude and phase distributions. The exit wave function is ${U_m} = {P_m}O = {P^{\prime}_m}O\textrm{exp} ({i{\varphi_0}} )$. Subsequently, a mixed-state algorithm is applied to update both the object and two probe components, where the object and probe components are updated using
- (5) Finally, the iterations continue until a termination condition is satisfied or until a fixed number of iterations have been completed.
In fact, just adopting the process in the dashed box in Fig. 2 to complete the iterations, that is, to reconstruct with two objects from start to finish, is also convergent, but it does not conform to the actual physical process. More importantly, it is impossible to establish the spatial and intensity correlation of the two reconstructed probe components, which makes the subsequent vector synthesis highly erroneous or impossible to implement.
After the iterative computations are completed, by propagating the reconstructed probe components Pm back to the plane of the Wollaston prism and removing the influence W introduced by the Wollaston prism, the vector light E can be obtained, from which the polarization information of interest, such as the birefringence, polarization direction, and vector characteristics, can be extracted.
2.3 Beam splitting after modulation
In Fig. 1(b), the vector light E is the probe P incident on a diffractive object, where different polarization states undergo the same modulation. After the exit-wave ${\boldsymbol U} = {\boldsymbol P}O = {\left[ {\begin{array}{{cc}} {{U_1}}&{{U_2}} \end{array}} \right]^T} = {\left[ {\begin{array}{{cc}} {{P_1}O}&{{P_2}O} \end{array}} \right]^\textrm{T}}$ propagates over the distance z1 forward, it passes through the Wollaston prism and is divided into two orthogonal-polarization states. The exit field from the Wollaston prism is expressed as
Similar to the method shown in Fig. 1(a), this method also has two horizontal constraints: the same object is illuminated by the two probe components at the object plane and position relationship between the two diffraction spots at the CCD plane. This makes the propagation direction introduced by the Wollaston prism an important parameter that must be accurate. Therefore, we can also use the above iterative algorithm to first determine the oblique wavefront with two objects, and then reconstruct it precisely with a single object. Unlike the flowchart in Fig. 2, the entire propagation process comprises two parts: Eq. (6) and Eq. (7), and the oblique wavefront φ interacts with the intermediate wave function UW,m, rather than with the probe. Moreover, the reconstructed probe was the exact measured light, and no additional calculations were required.
Compared with the measurement method of multiple analyzers, the above two methods not only reduce the number of measurements, but also establish the correlation of the intensity and spatial position between two reconstructed polarization components, thus improving the measurement accuracy significantly. To validate the proposed methods, we conducted experiments for two main applications: the quantitative evaluation of the anisotropic properties of optical materials and measurement of vector beam properties.
3. Measurement of crystal uniformity
Crystals are widely used in various laser systems owing to their frequency-conversion and phase-modulation properties. During crystal growth, owing to gravity, temperature, impurities, and other factors, the optical uniformity of the crystal inevitably decreases, leading to a local phase mismatch of the laser beam and reducing the conversion efficiency of the laser system. When using crystalline materials, it is essential to test the inhomogeneity of the refractive index and select the appropriate area for processing [39–41]. Light incident on a crystal in directions other than the optical axis was decomposed into two light waves with mutually perpendicular polarization directions and different refractive indices (fast and slow axes). Therefore, the refractive-index inhomogeneity of crystals is generally obtained by measuring the transmitted wavefronts under different polarization states. Polarization interferometry is the most commonly used method with a high measurement accuracy; however, it requires a stable environment and high-quality optical components [41]. There are also some measurement methods based on the Hartmann wavefront sensor, but they have low resolution, which affects the judgment of crystal uniformity.
The proposed method, which avoids the problems of poor robustness and low resolution of traditional measurement methods, can be used to simultaneously obtain two transmitted wavefronts with orthogonal polarizations and further determine the refractive index inhomogeneity of the crystal. The setup is shown in Fig. 3, where the wavelength of the laser was 632.8 nm. The incident laser was transformed into linearly polarized light by the polarizer set at an angle π/4 with respect to the fast axis of the tested crystal. After passing through the spatial filter and lens L1, a parallel light with a diameter of 24 mm was incident onto the crystal. We employed a 10 mm thick Er0.12Y0.88COB crystal as the test sample. A 4F optical system comprising two lenses, L2 and L3, was used to convert the beam diameter to 9 mm. Here, we selected the approach illustrated in Fig. 1(a). A Wollaston prism with a separation angle of 1° was placed in front of the diffractive object, a resolution test target (USAF, 1951) placed on a two-dimensional translation stage. Rotating the prism makes the polarization directions of the two probe components correspond to the fast and slow axes. The CCD (Pike F421b) had 2048 × 2048 pixels (pixel size 7.4 µm) and was placed 75-mm downstream from the diffractive object. According to the system parameters, the pixel sizes at the diffractive object plane and at the tested crystal plane were 7.4 µm and 19.7 µm, respectively. The diffraction patterns Ij are generated from a 10 × 10 regular grid with random offset to avoid the raster grid pathology. The grid interval and the maximum of random offset were respectively 740 µm and 37 µm.
Figure 4(a) shows one of the recorded diffraction patterns; the two cross-polarized spots have an overlap rate of approximately 70% at the CCD plane. Without correcting the propagation direction, and using the mixed-state algorithm directly to reconstruct a single object O and two probe components P1 and P2, the result is as shown in Figs. 4(b)-(d). Object O is very fuzzy and aliased, and the two probe components, P1 and P2 do not converge.
Fig. 4. (a) One of recorded diffraction patterns; (b)-(d) reconstructed amplitude distributions without propagation direction correction. (b) O; (c) P1; (d) P2.
The reconstruction results obtained using the proposed iterative algorithm in Section 2.2 are shown in Figs. 5(a)-(d). The complex amplitudes of object O and probes P1 and P2 were accurately reconstructed with a reconstruction error of 0.8%. The resolution achieved is approximately 13.92 µm [see the zoomed-in image in Fig. 5(b)]. To observe the wavefront change introduced by the crystal, Figs. 5(c) and (d) show the complex amplitudes of ${P^{\prime}_m}$, which removed the oblique wavefront φ0 from Pm. The reconstructed amplitudes of ${P^{\prime}_m}$ are clear, and the obvious horizontal separation between ${P^{\prime}_1}$ and ${P^{\prime}_2}$ is consistent with Eq. (3), which verified the accuracy of the proposed algorithm. The vector light E was obtained by propagating Pm back to the plane of the Wollaston prism whose components E1 and E2 were in the polarization directions of the fast and slow axes, respectively. The propagation distance d is initially measured and accurately determined from the reconstructed angles α and β and horizontal separation between ${P^{\prime}_1}$ and ${P^{\prime}_2}$. The phases of the two polarized lights Em can be written as
where $C({x,y} )$ represents the influence of the measurement system on the wavefront. The local refractive index and thickness are respectively expressed as ${n_m}(x,y) = {\bar{n}_m} + \delta {n_m}(x,y)$ and $l(x,y) = \bar{l} + \delta l(x,y)$, where ${\bar{n}_m}$ is the average refractive index of the tested crystal under the polarization direction m, $\bar{l}$ is the average thickness of the crystal, and δn and δl represent the changes in refractive index and thickness, respectively.Fig. 5. Reconstruction results with the proposed iterative algorithm. (a) Amplitude of object; (b) zoomed-in image of the red box in (a); (c) ${P^{\prime}_1}$; (d) ${P^{\prime}_2}$ (left: amplitude, right: phase); (e) phase of background light C1; (f) phase of C2.
To eliminate the effect of wavefront distortion introduced by other optical elements in the measurement optical path, we removed the tested crystal and measured the background light Cm [see Figs. 5(e) and (f)]. The measurement system has different effects on the wavefronts of different polarization states, indicating the necessity for background-light measurement.
The phase difference between the two polarization states introduced by the crystal is obtained as follows:
Fig. 6. Measurement results of crystal uniformity. (a) Phase difference Δϕ; (b) refractive index inhomogeneity δn2; (c) interferometric measurement result.
4. Vector beam measurement
The proposed method can also be used to measure vector beams, which can simultaneously provide information on the intensity, phase, and polarization states. To test this capability, we construct a vector light with a spatial light modulator (SLM). The front part of the setup was similar to that shown in Fig. 3, with the crystal replaced by an SLM (HOLOEYE LC2012), and the approach shown in Fig. 1(b) was implemented in the rear part; that is, the diffractive object was placed in front of the Wollaston prism.
The polarization direction of the polarizer was along the reference x-axis, and the image in Fig. 7(a) was loaded onto the SLM. The exit light from the SLM is a vector beam P with two orthogonal components, P1 and P2, which have different complex amplitude distributions. Figures 7(b) and (c) show the intensity distributions of the two orthogonal components directly recorded by the CCD on the SLM image plane. The diffractive object was still a resolution test target (rotated by π/6) and was placed near the SLM image plane.
Fig. 7. (a) Image loaded onto the SLM; intensity distributions of orthogonal components; (b) P1 and (c) P2 on the SLM image plane.
Therefore, by recording a set of diffraction patterns [ Fig. 8(a)] containing two orthogonal components during the scanning process of the object, P1 and P2 can be reconstructed using the proposed mixed-state algorithm, as shown in Fig. 8. The resolution achieved in Fig. 8(b) is also 13.92 µm, indicating that there is no difference in resolution between the two optical paths in Fig. 1. The reconstructed amplitudes of P1 and P2 are consistent with those in Figs. 7(b) and (c), both in terms of detail and intensity variation. As shown in Figs. 8(d) and (f), the phases of P1 and P2 are different.
Fig. 8. Reconstructed results. (a) One of recorded diffraction patterns; (b) amplitude of object; (c) amplitude and (d) phase of P1; (e) amplitude and (f) phase of P2.
The vector beam P can be expressed as:
Fig. 9. Calculation result. (a) Amplitude ratio $\tan \gamma $; (b) phase difference δ; (c) state of polarization, close-up of polarization state on the small square region in (b).
To verify the reconstruction ability of this method in phase, an experiment using standard phase-type object is performed for quantitative evaluation. The results are shown in Fig. 10. The resolution achieved is approximately 12.40 µm, and the average phase difference between the etched area and the unetched area is 3.972. The measured etching depth is 877.199 nm according to $\varphi = 2\pi h({n - 1} )/\lambda $ where n = 1.456. Compared to the etching depth 879.573 nm provided in the product report, the deviation is 0.27%.
Fig. 10. Reconstructed results with phase-type object. (a) Phase of object; (b) etching depth h at location indicated by black line in (a).
5. Conclusions
In summary, we presented polarization-sensitive ptychography that enables full optical property measurement of vector light. The proposed method integrates a Wollaston prism into mixed-state ptychography and solves the reconstruction ambiguities caused by polarization aliasing. It avoids multiple image acquisitions with various polarizer configurations and significantly improves the measurement accuracy by correlating the intensity and position of different polarization components with single object reconstruction. It also maintains the inherent characteristics of ptychography, with a simple optical path, high resolution, and good robustness. Two schemes were employed to successfully reconstruct the probe polarization properties. Moreover, we propose a new computational reconstruction algorithm to address the issues of fuzziness and nonconvergence caused by inaccurate propagation directions. The method first exploits two object guesses to calibrate the propagation direction (that is, the oblique wavefront) in a feedback manner, and then performs faithful retrieval of the probe based on a single object guess and a mixed-state ptychography algorithm. Measurements performed on a crystal confirmed the power of the method for quantifying anisotropic optical parameters. High-precision measurements of the refractive-index inhomogeneity of the crystals were achieved by compensating for the length errors introduced by crystal processing. Measurements performed on a vector light confirmed its ability to obtain the full properties contained within the Jones matrix, high-resolution complex amplitude information of each polarization state, and a polarization state map of the vector beam. Both measurements achieved a resolution of 13.92 µm. Our experimental method is simple and enables a single measurement combined with a single-shot ptychography. As an important extension of ptychography, the proposed method can find several uses in monitoring vector beams and birefringent properties of optical materials, and can also be applied to anisotropic biological samples.
Funding
National Natural Science Foundation of China (NSFC) (61827816); project of the Ministry of Industry and Information Technology (TC220H05L); National Natural Science Foundation of China Youth Fund (12102394).
Disclosures
The authors declare no conflicts of interest.
Data availability
The data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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