• Photonics Research
  • Vol. 13, Issue 6, 1620 (2025)
Mingli Sun1,2,†, Yingming Xu2,4,†,*, Yuanyuan Liu3..., Chiye Li2, Bijun Xu1, Xiaogang Wang1, Junhui Shi2,5,* and Qiwen Zhan3,6,*|Show fewer author(s)
Author Affiliations
  • 1School of Science, Zhejiang University of Science and Technology, Hangzhou 310023, China
  • 2Research Center for Novel Computational Sensing and Intelligent Processing, Zhejiang Lab, Hangzhou 311100, China
  • 3School of Optical-Electrical and Computer Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China
  • 4e-mail: yingmingxu@163.com
  • 5e-mail: junhuishi@outlook.com
  • 6e-mail: qwzhan@usst.edu.cn
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    DOI: 10.1364/PRJ.554521 Cite this Article Set citation alerts
    Mingli Sun, Yingming Xu, Yuanyuan Liu, Chiye Li, Bijun Xu, Xiaogang Wang, Junhui Shi, Qiwen Zhan, "Single-shot common-path encoded coherent diffraction imaging with OAM multiplexing," Photonics Res. 13, 1620 (2025) Copy Citation Text show less

    Abstract

    Single-shot multi-frame phase imaging plays an important role in detecting continuous extreme physical phenomena, particularly suitable for measuring the density of media with non-repeatable changes and uncertainties. However, traditional single-pattern multiplexed imaging faces challenges in retrieving amplitude and phase information of multiple frames without sacrificing spatial resolution and phase accuracy. In this study, we demonstrate single-shot common-path encoded coherent diffraction imaging with orbital angular momentum (OAM) multiplexing. It employs a sequence of vortex illumination fields, combined with encoding wavefront modulation and a vortex multiplexing phase retrieval algorithm, to achieve the retrieval of complex amplitudes from dynamic samples in single shots. Our experimental validation demonstrated the capability to achieve 9-frame high-resolution phase imaging of the dynamic sample in a single diffraction pattern. The spatial resolution and phase accuracy improve to 9.84 μm and 4.7% with this lensless multiplexed imaging system, which is comparable to single-mode imaging. This technology provides a multiplexed dimension with orbital angular momentum and holds potential in the study of transient continuous phenomena.

    1. INTRODUCTION

    Single-shot multi-frame imaging technology is crucial for detecting non-repeatable phenomena, and the sequence depth of frames is an important parameter [1]. The measurement of transient phenomena traditionally relies on the pump-probe method [24]. However, extreme physical phenomena initiated by high-energy pulses are non-repeatable and probabilistic in occurrence [57]. Traditional high-sequence depth imaging techniques, such as framing cameras [8], high-speed video cameras [9], or streak cameras [10], have achieved the performance of multi-frame imaging. However, these methods are limited by the hardware devices and require computational retrieval algorithms to further improve the sequence depth in a single diffraction pattern. For example, compressed ultrafast photography (CUP) [11] can effectively improve the sequence depth of streak cameras and has demonstrated the capability to achieve 103 frames [12]. By constructing a pulse sequence, single-shot multi-frame imaging can be realized by distinguishing different time information in the spatial division [1315], spatial frequency division [16], or wavelength division [17,18]. However, these methods are incapable of retrieving phase information, which limits the applications in phase-sensitive imaging.

    Phase information plays an important role in determining the change of medium density or tracing the direction of energy flow. The Hartmann sensor [19,20] and holographic interference [21,22] are common approaches for single-shot phase imaging. The Hartmann sensor allows for single-shot wavefront reconstruction and multi-frame phase imaging from different detector regions [23], but it can produce significant error for wavefronts with large phase distortions, limiting spatial resolution and phase accuracy. Holographic interference methods enable multi-frame phase measurement through the interference of several ultra-short pulses, and complex amplitude information at different times can be resolved from the intensity pattern of interference fringes [24,25]. Although this method can retrieve temporal information in single-shot, it requires precise time synchronization of multiple ultra-short pulses and often suffers from a reduction in spatial resolution as the number of frames increases due to the impact of frequency domain diffraction orders. Coherent diffraction imaging (CDI) is another technique employed in multi-frame phase imaging. For example, the manner of dispersing multiple diffraction patterns to different spatial regions on the detector is adopted to reconstruct multiple wavefronts based on the Gerchberg-Saxton (GS) [26] or hybrid input-output (HIO) [27,28] iterative algorithms [29]. This method offers the advantages of simple and fast calculation, but this algorithm suffers from low convergence due to the small amount of recorded information, and the demand for the detector target size will increase dramatically as the sequence depth of the frames increases. Multiplexed CDI [30,31] provides an opportunity to maximize the use of detector target size and the high-frequency information of a multi-frame light field. Multi-wavelength coherent modulation imaging (CMI) can reconstruct complex amplitude information of two different wavelengths in a single diffraction pattern [32]. Although the use of phase modulation in CMI greatly improves the convergence performance [33], the spatial resolution of the multi-wavelength CMI method decreases significantly as the sequence depth increases [34]. For multiplexed single-shot ptychography [3538], the single diffraction pattern at multiple angles is recorded, and the diffraction patterns at different moments overlap with each other. Based on the multiplexed iterative algorithm, multi-frame information is successfully retrieved, and this method can effectively improve the convergence. However, crosstalk exists in high-frequency information from different angles, which limits the improvement of spatial resolution. Furthermore, single-shot ultrafast multiplexed CDI (SUM-CDI) is used to analyze the complex amplitude information at different moments from the intensity of a single diffraction pattern [39]. The advantages of SUM-CDI lie in preserving the diffraction information recorded by the detector as much as possible, and reconstructing the complex amplitude of different field modes by means of high dynamic range capabilities. However, an increase in the number of frames leads to a decline in convergence performance due to insufficient information constraints, which causes a significant increase in the noise of reconstruction results. Fortunately, the orbital angular momentum (OAM) light field has become irreplaceable in optical phase imaging [4042]. As a wavefront modulation tool [43,44], it can provide important convergence conditions when combined with multiplexed retrieval algorithms [45].

    In this paper, we propose a technique of single-shot common-path encoded coherent diffraction imaging with OAM multiplexing (OAMM-eCDI). The convergence performance of the vortex multiplexing phase retrieval algorithm was effectively improved by the constraint of different OAM modes. Moreover, the spatial resolution and phase accuracy of single-shot multi-frame imaging can be enhanced. In the experiment, a spatial light modulator (SLM) is used to generate dynamic vortex beams with 9 different topological charge (TC) combinations, which serves as the illumination light for the moving sample. The detector records a single diffraction pattern from a 9-frame intensity superposition, and the complex amplitude information of the moving sample corresponding to different OAM modes is reconstructed by OAMM-eCDI. This technique can effectively distinguish different mode information by the phase modulation difference of vortex beams without the need for multi-wavelength or multi-angle illumination, and enables the reconstruction of richer dynamic information with a limited number of detector pixels. The reconstruction results maintain high spatial resolution and phase accuracy, comparable to single-mode imaging. Based on the above characteristics, this method can be used for ultrafast phase imaging at a high sequence depth of frames, which holds significant applications in dynamic cell detection [46], laser damage detection [47], and plasma transient measurement [48].

    2. METHODS

    OAMM-eCDI employs diffraction propagation characteristics of vortex beams with different TC combinations to distinguish the moving sample. The multi-vortex beams [49] possess distinct OAM distributions and energy distributions, which facilitate the reduction of information crosstalk between different modes. This characteristic allows the iterative results to rapidly converge to the true value of light field. Thus, this method maintains high spatial resolution and phase measurement accuracy while retrieving information at a high sequence depth of frames, with the advantages of spatial multiplexing and lensless imaging.

    The schematic diagram of the optical path is shown in Fig. 1(a). The dynamic sample is illuminated by a series of multi-position low-topological-charge (MPLTC) vortex fields carrying different TCs combinations. Nine different MPLTC vortex fields illuminate the moving biological sample from time t1 to t9. The transmitted light field includes a sequence of a 9-frame vortex pulse, and each frame successively carries complex amplitude information of the dynamic sample at different time points. After modulation by the encoding plate, the incoherent superposition of diffraction intensities from multi-vortex fields is recorded by the detector. Based on the OAM multiplexed encoding phase retrieval algorithm, the complex amplitude information of the dynamic sample at different times is resolved from a single superimposed diffraction pattern. The dynamic sample can move randomly in the xy plane, and the speed of movement needs to match 9 frames of the vortex illumination beam at different moments. In Fig. 1(a), the dynamic sample moves at a constant speed v in the y-direction. The modulated 9 multi-vortex beams sequentially pass through the sample and an encoding plate, after which the detector records the intensity superposition of these beams. Although the multiple diffraction patterns completely overlap at different moments, the association with the phases of different OAM modes enables the retrieval results to rapidly converge to the true amplitude and phase of the sample.

    Schematic diagram of OAMM-eCDI. (a) Optical path of MPLTC vortex illumination imaging. (b) Iterative process of the vortex multiplexing phase retrieval algorithm.

    Figure 1.Schematic diagram of OAMM-eCDI. (a) Optical path of MPLTC vortex illumination imaging. (b) Iterative process of the vortex multiplexing phase retrieval algorithm.

    Figure 1(b) illustrates the iterative principle diagram of the vortex multiplexing phase retrieval algorithm, and the iteration is adopted among the constraint position, sample position, encoding plate position, and detector position. For each of the 9 MPLTC vortex fields, the input light field is expressed as Ein(x,y,z=0,tn)=exp(x2+y2w02)×m=1M0exp(ilmθm),where n=1,,9, w0 is the beam width of the light field, M0 is the total number of vortices, and lm is TC of the mth vortex. θm represents the phase angle of (xdxm)+i(ydym), where (dxm,dym) denotes the position vector of the mth vortex core, expressed by the following formula: {dxm=d0cos[(m1)2π/M0]dym=d0sin[(m1)2π/M0].

    In this case, the off-axis vortices of m=1 to m=M0 are located evenly with respect to the optical axis, and M0 takes 3 in this paper. Each vortex is equal in distance from the optical axis, with an off-axis distance of d0. It should be noted that the generated times tn of the pulse train for the multi-vortex fields correspond to t1t9, respectively. The transmittance function of the dynamic sample is expressed as O(x,y,tn)=A(x,y,tn)eiφ(x,y,tn), with A(x,y,tn) and φ(x,y,tn) being the amplitude and phase distribution of the sample, respectively. The transmittance function of the encoding plate is denoted by T(x,y), which can be calibrated based on the scanning CDI method, e.g., extended ptychography iterative engine (ePIE) [50]. The phase modulation of the encoding plate is a binary random distribution (0, π), and the minimum unit size is 9  μm×9  μm. The vortex fields at the constraint position are represented as F(x,y,tn). The wavefront intensities are concentrated in different shapes, and the convergence is enhanced according to this characteristic. The amplitudes of the diffraction patterns at the detector position corresponding to different times are represented by γ(x,y,tn), and the superposition of the diffraction intensities is expressed as η(x,y). The intensity of the diffraction pattern recorded by the detector is I(x,y). In Fig. 1(b), () and 1() denote the forward and backward propagation of the light field in free space, respectively. The specific iteration process of the vortex multiplexing phase retrieval algorithm is shown in Appendix A.

    3. EXPERIMENTAL SECTION

    A. Experimental Setup

    Figure 2 shows the schematic of OAMM-eCDI experimental setup based on multiple vortex beams generated by the spatial light modulator (SLM). Figure 2(a) depicts the module for generating the MPLTC vortex beams. In the experiment, a continuous laser beam with a wavelength of 632.8 nm passes through the lens L1, pinhole P1, and lens L2 for expanding the beam size enough to cover the functional area of the spatial light modulator (SLM, pixel pitch of 8  μm×8  μm, 1280×1024  pixels). Since the SLM can only modulate a vertically polarized beam, two half-wave plates (HWPs) and a polarization beam splitter (PBS) are used to generate the vertically polarized beam and control the power incident on the SLM. The encoded computer-generated hologram (CGH) including the complex information of the targeted vortex beam is loaded onto the SLM, and the modulated beam is reflected toward a 4f imaging system consisting of a couple of identical lenses with a focal length of 250 mm. To separate the targeted parts from the central bright spot, a linear phase is introduced in the CGH, and a low-pass spatial filter (P2) at the Fourier plane of the lens L3 selects the modulated zeroth-order diffraction wave. The targeted vortex beam is generated at the back focal plane of lens L4, and the intensity profiles can be checked by an industrial camera (pixel pitch of 4.5  μm  ×4.5  μm). It should be noted that 9 CGHs are loaded on the SLM, and 9 frames of three-vortex beams with different TCs are generated successively by controlling the switching frequency of CGHs. The CGHs loaded on the SLM are encoded by the single-pixel checkerboard method [51]. First, two phase elements extracted from a targeted vortex field E0(x,y) are expressed as ϕ1(x,y)=φ(x,y)+arccos[M(x,y)],ϕ2(x,y)=φ(x,y)arccos[M(x,y)],where M(x,y) and φ(x,y)=arg[E0(x,y)] refer to the normalized amplitude and the phase of the light field, respectively. Then, we considered two complementary checkerboard amplitude functions Q1(x,y) and Q2(x,y), which represent two-dimensional transmission complementary binary gratings and can be expanded by the Fourier series as Q1,2(x,y)=12p=q=Λ1,2(p,q)exp(i2πpxd)exp(i2πqyd),Λ1,2(p,q)=sinc(p2)sinc(q2)cos[π2(p±q)],where d is the period of the two-dimensional binary grating; p and q denote binary diffraction orders. The two-dimensional phase information loaded on the SLM is expressed by Φ(x,y)=Q1(x,y)ϕ1(x,y)+Q2(x,y)ϕ2(x,y)+αx.

    Schematic of the OAMM-eCDI experimental setup. (a) The module for generating MPLTC vortex beams. (b) The module of multiplexed complex amplitude imaging. (c) The recorded diffraction pattern and illumination sequence. P, pinhole; L, lens; HWP, half-wave plate; PBS, polarization beam splitter; M, mirror; SLM, spatial light modulator; EP, encoding plate.

    Figure 2.Schematic of the OAMM-eCDI experimental setup. (a) The module for generating MPLTC vortex beams. (b) The module of multiplexed complex amplitude imaging. (c) The recorded diffraction pattern and illumination sequence. P, pinhole; L, lens; HWP, half-wave plate; PBS, polarization beam splitter; M, mirror; SLM, spatial light modulator; EP, encoding plate.

    In Eq. (5), αx represents a linear shift along the x direction, and α is a constant.

    Subsequently, the 9 generated vortex beams pass through the moving sample and the encoding plate before being captured by the detector. Note that the moving sample is positioned at the imaging plane of the 9 vortex beams, where the 9 vortex beams propagate with the common-path. However, after propagation and modulation by the encoding plate, different vortex beams produce different diffraction angles on the detector plane. Based on the differences in diffraction patterns, the complex amplitude information of the moving sample carried by different vortex beams can be retrieved through the iterative calculation. Figure 2(b) is the multiplexed complex amplitude imaging module. After passing through the sample, the MPLTC illumination beam is modulated by the encoding plate, and the detector records the diffraction pattern of multiple modes (illumination beams 1–9). The sample moves in one direction at a speed of v, and during the time period that each sub-illumination beam passes through it, the sample can be approximated as being in a stationary state. Figure 2(c) shows the diffraction pattern and illumination sequence recorded by the detector. The illumination sequence is composed of 9 different vortex beams, with a time interval of 1/20 s between each pair of adjacent sub-beams. This time interval represents the maximum refresh speed of the SLM model and is not the highest imaging speed achievable by this method. The exposure time of the detector is longer than the total duration of 9 illumination beams, and the detector records the incoherent superposition of the diffraction patterns from each sub-illumination beam. Figure 2(c) displays the retrieved diffraction patterns of I1I9 and the recorded diffraction pattern of detector, denoted as Is=n=1N=9In. OAMM-eCDI is employed to retrieve the complex amplitude information of samples with different OAM modes based on this single diffraction pattern Is.

    B. Encoding Plate and MPLTC Vortex Information

    In the experiment, the encoding plate employed binary phase modulation, with a binary modulation distribution of (0, π) for the 632.8 nm wavelength. Figure 3(a) shows the amplitude and phase distribution of the encoding plate, and the insets are the enlarged view of the amplitude and phase information within the area marked by the red box. Figure 3(b) shows the intensity distribution of the diffraction pattern of the 9 vortex beams after passing through the phase encoding plate. Figure 3(c) is a one-dimensional phase diagram of the area corresponding to the red dashed line in Fig. 3(a), from which the minimum modulation unit can be obtained as 9 μm. The amplitude and phase distribution of 9 different vortex beams are shown in Fig. 3(d), with a beam diameter of 1.5 mm. A total of three vortex singularities are distributed at different positions, each with different TCs, represented as (l1, l2, l3). The 9 sets of vortex beams are used in the experiment with the following TC combinations: (1,1,1), (1,2,3), (1,3,2) (2,1,3) (2,2,2) (2,3,1) (3,1,2), (3,2,1), and (3,3,3). Due to the phase difference of the 9 vortex beams, there is recognizability of diffraction patterns in different OAM modes. This phase difference can be accurately calibrated using ePIE as prior information in the vortex multiplexing phase retrieval algorithm. The intensity distribution of the vortex beam at the phase singularity position is characterized by a hollow dark core. As the TC increases, the diameter of the dark core increases. As the illumination beam for the sample, the increase in the diameter of the dark core of the vortex beam can affect the reconstruction of the complex amplitude of the sample at the dark core position. In the experiment, a vortex pulse with 1–3 TC distribution was used to minimize this effect.

    Encoding plate and MPLTC vortex beams. (a) The amplitude and phase distribution of the encoding plate. (b) The recorded diffraction pattern of the 9 vortex beams after passing through the phase encoding plate. (c) The phase distribution of the area corresponding to the red dashed line in (a). (d) The amplitude and phase distribution of 9 vortex beams with different TC combinations.

    Figure 3.Encoding plate and MPLTC vortex beams. (a) The amplitude and phase distribution of the encoding plate. (b) The recorded diffraction pattern of the 9 vortex beams after passing through the phase encoding plate. (c) The phase distribution of the area corresponding to the red dashed line in (a). (d) The amplitude and phase distribution of 9 vortex beams with different TC combinations.

    C. Experimental Results

    In the experiment, we measured the moving pumpkin stem slice biological sample, spatial amplitude resolution plate, and phase step plate. The measurement results of the dynamic pumpkin stem slice are shown in Fig. 4. Figure 4(a) is the schematic diagram of the principle, in which 9 illumination beams arrive at the pumpkin stem sample at different times, and then the detector records the superposition of 9 diffraction patterns modulated by the encoding plate. Figure 4(b) shows the amplitude and phase information of the sample retrieved by OAMM-eCDI at the time of arrival of 9 illumination beams. Below Fig. 4(b) is a magnified view of the area indicated by the red dashed line. The enlarged image demonstrates that this method effectively reconstructs the details of the dynamic biological sample, which moves at a speed v along the direction of the red arrow. By calculation, the moving distance between adjacent frames is 250 μm, with a time interval of 1/20 s, resulting in a moving speed of 5 mm/s. In the first frame, the phase protrusion in the S region is caused by the phase singularity of the vortex beam, but this defect point does not affect the reconstruction of other regions.

    Measurement results of the dynamic sample. (a) Principle of the optical path in the experiment of dynamic sample imaging. (b) The amplitude and phase distributions of the moving sample retrieved by OAMM-eCDI, corresponding to 9 illumination light fields.

    Figure 4.Measurement results of the dynamic sample. (a) Principle of the optical path in the experiment of dynamic sample imaging. (b) The amplitude and phase distributions of the moving sample retrieved by OAMM-eCDI, corresponding to 9 illumination light fields.

    For the calibration of spatial resolution and phase measurement accuracy, the amplitude resolutions of the dynamic USAF1951 and the designed phase step plate were measured in the experiment, and the retrieval results of the vortex multiplexing phase retrieval algorithm are shown in Fig. 5. Figures 5(a) and 5(b) show the reconstruction results of the dynamic USAF1951 resolution plate and phase step plate, respectively. There is no imaging device in the detection optical path of this system. Figure 5(c) presents an enlarged view of the red box from the fifth frame, and the fifth element of the fifth group in USAF 1951 can be clearly reconstructed, which corresponds to a linewidth of 9.84 μm. Figure 5(d) displays the reconstruction result in the single-mode imaging. By comparing the enlarged areas in the red and blue boxes in Figs. 5(c) and 5(d), it can be seen that the vortex multiplexing phase retrieval algorithm can maintain a spatial resolution almost the same as that of the single mode. Figure 5(e) displays the one-dimensional intensity curve of the fifth element of the fifth group in the single-mode and multi-mode cases. The modulation transfer functions (MTFs) of single mode and multi-mode calculated from the intensity curve are 0.4 and 0.364, respectively. In the multi-mode case, the decrease of MTF is less than 10% compared to the single-mode case, while the image still maintains a high signal-to-noise ratio. The average phase difference between region P and region Q in Fig. 5(b) is 2.19 rad, and the error is 4.7% compared with the design value 2.3 rad. By analyzing the results of the amplitude resolution plate and the phase resolution plate, it is proved that this method offers a high signal-to-noise ratio and phase measurement accuracy and ensures high spatial resolution. Therefore, OAMM-eCDI can be used for single-shot ultrafast multiplexed phase imaging and has application prospects in multiplexed imaging of coaxial multi-pulse beams.

    Spatial resolution and phase measurement accuracy verification. (a) and (b) The retrieval results of the dynamic USAF 1951 resolution plate and phase step plate, respectively. (c) The enlarged pattern of the red box corresponding to the fifth frame in (a). (d) The retrieval result of the single-mode calculation corresponding to the fifth frame in (a). (e) The one-dimensional intensity curve for the fifth element of the fifth group in the single-mode and multi-mode cases, respectively.

    Figure 5.Spatial resolution and phase measurement accuracy verification. (a) and (b) The retrieval results of the dynamic USAF 1951 resolution plate and phase step plate, respectively. (c) The enlarged pattern of the red box corresponding to the fifth frame in (a). (d) The retrieval result of the single-mode calculation corresponding to the fifth frame in (a). (e) The one-dimensional intensity curve for the fifth element of the fifth group in the single-mode and multi-mode cases, respectively.

    4. DISCUSSION

    Now we will discuss the influence of the topological number of the vortex light field on multiplexed imaging. This work proposes a vortex multiplexing phase imaging method based on multi-vortex beam illumination, which can improve the single-shot phase imaging sequence depth of frames and spatial resolution. However, the singularity at the vortex core can induce small-scale amplitude defects and phase mutations in the imaging results, and this effect is related to the TC. Therefore, it is necessary to analyze the impact of increasing TC on the imaging results, and the multiplexing imaging results are simulated under 9 single-position high-topological-charge (SPHTC) vortex illumination conditions with a TC of l=19. The iteration results of the vortex multiplexing phase retrieval algorithm are shown in Fig. 6. Figure 6(a) shows the amplitude and phase ground truth of the sample. The sample is illuminated at different times by vortex beams of different OAM modes, and the sample is rotated by 20 degrees to achieve single-shot imaging of the dynamic scene. Figure 6(b) shows the phase information of the encoding plate, which features a binary phase step distribution of 0π at the wavelength of 632.8 nm, with an amplitude set to a constant of 1. Figure 6(c) illustrates the diffraction pattern intensity recorded by the detector, which is the incoherent superposition of the diffraction pattern intensities of 9 beams with different OAM modes. Based on this single diffraction pattern, the complex amplitude information of the sample in different OAM modes can be reconstructed in single-shot. However, the phase singularity at the core of the vortex beam leads to amplitude defects and phase distortion in the reconstruction result. Figure 6(d) shows the reconstructed amplitude information of the sample illuminated by different OAM modes, where the insets are the phase information of the vortex beams with 1–9 TCs. Figure 6(e) shows the phase information of the sample corresponding to different frames. As the TC increases, the amplitude defect at the phase singularity position becomes larger. The one-dimensional curves are drawn along this position as shown in Fig. 6(f), which shows the change in the amplitude defect range for TCs of l=59. The Ll is the defect depression diameter, and the maximum diameter is Ll=9=45  μm. Figure 6(g) displays the one-dimensional phase curves at the horizontal dotted line in Fig. 6(e), where the maximum diameter of the depression is Vl=9=18  μm, and the maximum phase mutation value is Pl=9=3.1  rad. This phase change is caused by the vortex phase singularity. From the changing pattern of the curves, it can be obtained that as the TC increases, the amplitude and phase distortions become more serious. Therefore, a method to utilize MPLTC vortex fields is proposed in the above experiment. In addition, Appendix B explores the comparison of the reconstructed images when using MPLTC vortex beams and SPHTC vortex beams as illumination. The comparison results show that the method used in this experiment (MPLTC vortex beams) weakens the disadvantages of phase singularity in the vortex while retaining the advantages of phase modulation. In the experiment, the selection of three vortex singularities is determined by the number of reconstructed modes and the measured aperture of the object. This approach ensures high constraint performance and spatial resolution while balancing the challenges associated with localized distortions, calibration errors, and hardware requirements. Specifically, if the number of reconstructed modes or the measured aperture of the object increases, the number of vortex singularities needs to be correspondingly increased to maintain system performance. Moreover, employing a vortex beam with distinct TCs for each mode enhances phase sensitivity, reduces crosstalk, and improves spatial resolution. These factors collectively ensure that the system can accurately distinguish between different modes and achieve high-quality reconstruction results.

    The impact of high-topological-charge vortices on retrieval results. (a) The amplitude and phase ground truth of the sample. (b) The phase information of the encoding plate. (c) The diffraction pattern recorded by the detector. (d) The retrieval amplitude of the sample illuminated by different OAM modes. (e) The retrieval phase of sample corresponding to different frames in (d). (f) One-dimensional intensity curves corresponding to the dotted line in (d). (g) The one-dimensional phase curves corresponding to the dotted line in (e).

    Figure 6.The impact of high-topological-charge vortices on retrieval results. (a) The amplitude and phase ground truth of the sample. (b) The phase information of the encoding plate. (c) The diffraction pattern recorded by the detector. (d) The retrieval amplitude of the sample illuminated by different OAM modes. (e) The retrieval phase of sample corresponding to different frames in (d). (f) One-dimensional intensity curves corresponding to the dotted line in (d). (g) The one-dimensional phase curves corresponding to the dotted line in (e).

    5. CONCLUSION

    In summary, we proposed a technique of common-path coherent diffraction imaging with OAM multiplexing (OAMM-eCDI). It employs the illumination of multiple OAM combinations [52] to provide precise phase modulation for dynamic phenomena, and effectively improves the convergence speed of the single-shot multi-frame imaging system. Simulations and experiments demonstrated that this OAM multiplexing retrieval method achieves a spatial resolution of 9.84 μm and a phase error of 4.7%, which is comparable to single-mode imaging. Without the need for elements of varying wavelengths or illumination angles, it provides a new multiplexed dimension in single-shot multi-frame imaging. In future research, by combining an ultrashort pulse sequence, ultrafast phase imaging of more than 9 frames in a single shot can be achieved, while ensuring both considerable spatial resolution and phase accuracy. Furthermore, such a lensless imaging technology is suitable for phase detection imaging in the extreme ultraviolet or X-ray band to achieve precise and high-speed dynamic phase imaging of semiconductor defects or extreme physical phenomena.

    APPENDIX A: THE ITERATION PROCESS OF THE VORTEX MULTIPLEXING PHASE RETRIEVAL ALGORITHM

    The iteration process of the vortex multiplexing phase retrieval algorithm is shown in Fig. 7. It begins with k=1 as the iteration number, and as an initial estimate, the complex amplitude of the dynamic sample is denoted as On=rand(x,y,tn)ei·rand(x,y,tn). After passing through the sample, the vortex pulse sequence is represented as ϕ(x,y,tn)=En(x,y,tn)On. At this point, the field propagating to the encoding plate position at different times tn is expressed as ψ(x,y,tn)=(ϕ(x,y,tn),LME), where LME is the distance between the dynamic sample and the encoding plate. After modulation by the encoding plate, the wavefront ξ(x,y,tn)=ψ(x,y,tn)T(x,y) is diffracted to the sensing target plane of the detector in free space. The detector records the intensity superposition of the diffraction patterns, which is expressed as η(x,y)=n=1N=9|γ(x,y,tn)|2, where γ(x,y,tn)=(ξ(x,y,tn),LED), and LED is the distance between the encoding plate and the detector. The iterative estimated amplitude is constrained by the following multi-mode update formula: γ(x,y,tn)=I(x,y)η(x,y)γ(x,y,tn). The light field γ(x,y,tn) is propagated backward to the encoding plate position, denoted as ξ(x,y,tn)=1(γ(x,y,tn),LED), and then the light field at the front of the encoding plate can be obtained based on the following formula: ψ(x,y,tn)=ψnχ1conj(T)(ξnξn)max(T*T), where χ1(0,1), and the value used in the experiment is 0.8. The wavefront can be gradually converged to the true value of the complex amplitude by the phase modulation of the encoding plate. The updated wavefront propagates to the constraint position in free space, which is expressed as F(x,y,tn)=1(ψ(x,y,tn),LEF), where LEF is the distance between the encoding plate and the constraint position, and then different aperture constraints are applied to the wavefronts at different times (corresponding to different OAM modes): F(x,y,tn)=F(x,y,tn)H(x,y,tn), where H(x,y,tn) is the aperture constraint function. This process can effectively constrain the wavefront phase and improve the phase accuracy as well as the spatial resolution. Then, F(x,y,tn) is propagated forward to the dynamic sample position ϕ(x,y,tn)=(F(x,y,tn),LMF), where LMF is the distance between the dynamic sample and the constraint position, where the formula Onk+1=Onkχ2conj(En)(ϕnϕn)max(En*En) is used, and the value χ2(0,1) used in the experiment is 0.8. It should be noted that the parameters χ1 and χ2 are regularization terms that determine the update step size in the iterative reconstruction process. Their values are influenced by the size of the diffraction pattern matrix and the complexity of the reconstruction task. For simpler, less data-intensive reconstructions, a larger update step size is typically used. In contrast, for more complex, data-intensive reconstructions, a smaller update step size is preferred to enhance the accuracy of the reconstruction. This is because larger matrices introduce greater discrepancies between the recorded and retrieval diffraction patterns. While a smaller update step size improves the reconstruction signal-to-noise ratio, it also increases the number of iterations and computational time. The next iteration is carried out after the updated complex amplitude of dynamic sample is obtained, and the iterative process ends when the iterative error Error=|η(x,y)I(x,y)|I(x,y) is less than the set value, which is determined according to the size of the matrix and the intensity. Alternatively, the iterative process can also be ended by setting a fixed number of iterations. It is worth mentioning that in the above iterative process, En(x,y,tn) is the vortex field with different OAM modes. The modulated wavefront enables rapid convergence of the dynamic sample to the true complex amplitude and effectively improves the sequence depth of frames, spatial resolution, and phase measurement accuracy. The algorithm has been implemented on a system equipped with an Intel® Core™ i9-9900K CPU at 3.60 GHz, 64 GB RAM, and an NVIDIA A5000 GPU with 24 GB of memory. The computation time for each iteration ranges from 5 to 7 s.

    Flow chart of the iteration calculation process of the vortex multiplexing phase retrieval algorithm.

    Figure 7.Flow chart of the iteration calculation process of the vortex multiplexing phase retrieval algorithm.

    APPENDIX B: COMPARISON OF ILLUMINATION CONDITIONS OF THE MULTI-POSITION LOW-TOPOLOGICAL-CHARGE VORTEX AND THE SINGLE-POSITION HIGH-TOPOLOGICAL CHARGE VORTEX

    In single-shot multi-frame phase imaging with multi-vortex beam illumination, the topological charge (TC) of the vortex will affect the amplitude and phase information of the reconstructed sample. As analyzed in the discussion, an increase in the TC number leads to distortions of the retrieval sample in amplitude and phase. In the discussion for the case of a single-position high-topological-charge (SPHTC) vortex, when the TC is 9, the amplitude defect diameter can reach Ll=45  μm, the phase defect diameter is Vl=18  μm, and the phase defect value is Pl=3.1  rad. Therefore, in order to avoid such defects, we employ a multi-position low-topological-charge (MPLTC) vortex beam as the illumination light. The reconstruction results and error comparison data are shown in Fig. 8. Figures 8(a) and 8(b) are the amplitude and phase information of the sample corresponding to vortex beams with different TC combinations, respectively. The sample is rotated by 20 degrees between adjacent frames, and the phase information of the illumination light is shown in the inset of Fig. 8(a). The illumination light is the vortex beam with different orbital angular momentum (OAM) modes, and the TC of each three-vortex light is a combination of 1–3. The TCs of the vortex beam in frames 1–9 are the same as those in the experiment. In this vortex combination, a representation of the maximum defect position of the reconstructed sample is shown in Fig. 8(c), and the insets are the enlarged areas of the red box in Figs. 8(a) and 8(b). Figure 8(c) shows the one-dimensional curves of intensity and phase drawn along the red dotted line. It can be obtained that the maximum defect diameter of the amplitude is Ll=22.5  μm, the maximum mutation diameter of the phase is Vl=9  μm, and the maximum mutation value of the phase is Pl=0.33  rad, which are reduced by 50%, 50%, and 89%, respectively, compared with the SPHTC vortex case. In the case of the MPLTC vortex, the influence of vortex phase singularity on the reconstructed sample is reduced. In addition, Fig. 8(d) displays the error curves of the sample amplitude for the two cases calculated using the following error formula: Errorsample=sum(||Eestimate(x,y)||Etruth(x,y)||)sum(|Etruth(x,y)|),where Eestimate(x,y) and Etruth(x,y) are the complex amplitude matrices of the iterative estimation sample and true sample, respectively, and sum() is the summation operation of matrix elements. From the variation of error with the iteration number in Fig. 8(d), it can be calculated that the errors of the two cases are almost consistent, with the difference of error within 1.22%. The illustrations show the amplitude information of the reconstructed sample in two cases. In the case of MPLTC, the information can also be reconstructed with high convergence while the effect of the phase distortion in the vortex core is reduced. Therefore, the MPLTC vortex beam is used in the experiment in this work.

    Simulation results and error analysis of MPLTC vortex conditions. (a) The retrieval amplitude of the sample illuminated by different OAM modes. (b) The retrieval phase of the sample corresponding to different frames in (a). (c) One-dimensional intensity and phase curves corresponding to the dotted line in the illustration, which corresponds to the enlarged view of the red box in (a) and (b). (d) The error curves of MPLTC and SPHTC vortex conditions.

    Figure 8.Simulation results and error analysis of MPLTC vortex conditions. (a) The retrieval amplitude of the sample illuminated by different OAM modes. (b) The retrieval phase of the sample corresponding to different frames in (a). (c) One-dimensional intensity and phase curves corresponding to the dotted line in the illustration, which corresponds to the enlarged view of the red box in (a) and (b). (d) The error curves of MPLTC and SPHTC vortex conditions.

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    Mingli Sun, Yingming Xu, Yuanyuan Liu, Chiye Li, Bijun Xu, Xiaogang Wang, Junhui Shi, Qiwen Zhan, "Single-shot common-path encoded coherent diffraction imaging with OAM multiplexing," Photonics Res. 13, 1620 (2025)
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