Single-shot diffractive spectrometer for photonic orbital angular momentum

Since the seminal work reveals that Laguerre–Gaussian (LG) laser modes possess photonic orbital angular momentum (OAM),¹ such vortex beams with helical wavefront structures have aroused intense interest. The helical wavefront is denoted by exp(ilφ), where l is the topological charge (TC), also referring to the mode of OAM, and φ is the azimuthal coordinate. The unbounded degrees-of-freedom of OAM states provided infinite orthogonal spatial modes for various applications, including optical communication,^2,3 quantum information processing,⁴ optical measurement,^5,6 biomedical detection,^7,8 and chiral spectroscopy.⁹ Meanwhile, the generation and detection of OAM under different wavelengths have been widely studied, such as terahertz,¹⁰ extreme ultraviolet (EUV),^8,11–13 x-ray,^14,15 and electron beam,^16–18 let alone in visible and infrared light. In many of these applications, to fully utilize the infinite degree-of-freedom of the vortex beam, it is essential to decompose individual helical modes and quantitatively analyze the spectrum of OAM.^19,20 Recently, there have been some reports on OAM-induced helical dichroism analysis, such as distinguishing enantiomers and characterizing topologically complex electronic matter.^13–15 These studies opened up a new path to diagnose chiral light–matter interaction at the nanoscale. Nonetheless, such applications also led to a challenge for quantitatively diagnosing vortex beams under short-wavelength systems, such as EUV, x-ray, and electron beam.

Traditional approaches, such as geometric transformation^21–23 and interferometry,^24–26 have been used to unscramble and characterize vortex beams in visible and infrared laser systems. However, it is challenging to implement these methods in short-wavelength lasers and electron beam, because fabricating devices with the necessary resolution and alignment accuracy becomes extremely difficult or even impossible with current technologies. In contrast to conventional approaches, diffractive methods have been widely implemented in short-wavelength systems owing to their unique features, including reference-free, lens-less, and broad wavelength suitability.^27–36 The most straightforward approach is to analyze the diffracted intensity patterns of vortex beams propagating through various types of apertures^27,28 or gratings.^29,30 However, these methods are generally suitable for pure single mode instead of arbitrary multiple modes, and involving interference between aperture and beam limits the field-of-view (FoV) of the measurement system. Classic diffraction-based phase retrieval approaches, such as Gerchberg–Saxton and coherent diffraction imaging (CDI), have also been implemented to reconstruct the wavefront of vortex beams.^31–33 However, these phase retrieval methods often face difficulties in convergence, and the quality of wavefront reconstruction is highly correlated with the initial guess. Some recent studies utilize ptychography, a scanning version of the CDI method, to improve the quality of wavefront reconstruction.^12,34–36 However, the inevitable scanning and multiple measurement process in ptychography impeded its application in fast-varying scenarios, e.g., detection for ultrafast processes. Other non-interferometric methods have also been reported, such as speckle pattern analysis,^37,38 intensity distribution analysis,^39,40 and machine learning.^41,42 However, these approaches require prior knowledge of the speckle or reference intensity patterns associated with the OAM modes to be measured and are highly sensitive to slight variations in the experimental setup, such as alignment and stochastic noise. To conclude, although there have been many proposals regarding OAM mode decomposition and spectrum analysis, it is still a challenge to obtain the spectrum of OAM within a non-interferometric and single-shot manner under short-wavelength systems.

In this work, we propose a single-shot diffractive approach to analyze the spectrum of photon OAM. A coherent modulation imaging (CMI) and computational spectrum analysis scheme is implemented to reconstruct arbitrary helical wavefront and quantitatively decompose individual OAM modes. There is no need to conduct any calibration associated with the incident vortex beam to be measured, the only prior knowledge required is the transmission function of a diffusing wavefront modulator, and spectrum decomposition is realized computationally. This unique feature makes our method different from most current diffractive approaches, in which a calibration for the referenced vortex beams is usually required. Experimental results show that this method retrieves the spectrum of singular and multiple OAM modes less than −16.91 dB inter-mode crosstalk and maximum measurable TC greater than 50. The proof-of-concept visible light experiments of multiplexed OAM information transmission showed its potential application in optical communication and measurement. In addition, as our method can reconstruct the complex wavefield of arbitrary vortex beams with a high quality, the obtained wavefield can be numerically refocused, making it possible to obtain the three-dimensional wavefront structure or diagnose the beam-propagation path. Given the simplicity of system setup, feasibility of single-shot measurement, and utility for arbitrary OAM modes, we believe that this method opens up a brand-new solution for structured wavefront analysis over a broad spectral range, from visible light to EUV, x-ray, and even electron beam.

To characterize the transmission property of the modulator, ptychography is used for calibration before wavefront measurement,⁴⁵ and the transmission function of the modulator T(r_m) serves as a prior knowledge. For a modulator working under a specific wavelength, the calibration process only needs to be performed once and for all time. There is no need to conduct any calibration associated with the incident beam field that is to be measured, making our method significantly different from most existing diffractive approaches, where a calibration for the referenced vortex beam is usually required. Although a precisely calibrated T(r_m) is preferred, this strict requirement can be mitigated by using an automatic modulator refinement algorithm during the iteration process of wavefront reconstruction, which allows for rough calibration as an initial guess of T(r_m) or an even completely unknown modulator in some cases.⁴⁶ A detailed discussion of the modulator selection and calibration is provided in the supplementary material 1, Sec. S1.

Experiments are conducted to demonstrate the performances of our proposed diffractive OAM spectrometer, as shown in Fig. 2. The experiment setup is shown in Fig. 2(a). A 632.8 nm laser beam, generated by the Newport N-STP-912 He–Ne laser source, is first collimated and expanded by a set of optical lenses. Then, it propagates through a polarized beam splitter (PBS) to sort out the transverse-polarized beam and adapts to the polarization requirement of the SLM. After propagating through a non-polarized beam splitter (NPBS), the beam is modulated by a reflective liquid crystal-on-silicon (LCoS) SLM (Holoeye LETO-3-VIS-009) to carry a helical wavefront. This SLM offers a resolution of 1920 × 1080 with a pixel size of 6.4 µm and a maximum phase retardation of 2.8π under 633 nm wavelength. The OAM modes are adjusted by loading different computer-generated helical phase profiles on the SLM. The modulated vortex beam is then reflected by the NPBS and propagates to the diffractive spectrometer. Inside the spectrometer, the distances between the three planes are z₁ = 25 mm and z₂ = 59.2 mm. These distances are measured by refocusing the reconstructed transmitting function of the modulator and the incident beam field. The diameter of the pinhole at the support plane is 2 mm. The wavefront modulator is a phase mask etched on a thin glass, with randomly distributed, 15 × 15 µm² sized binary patches. The optical path difference of the patches is designed to be 316.4 nm, offering π radian phase retardation under the 632.8 nm laser. If implemented for other wavelengths, the relative phase difference should be modified to π radians under the new wavelength so that a destructive interference will reduce the strength of the zero-order beam. During calibration, the modulator is mounted on a programmable scanning stage to perform ptychography. Once the transmitting function is obtained, the scanning stage is no longer used, and the modulator can be fixed with the camera to provide a modular setup. The modulator’s calibrated modulus and phase modulation properties are shown in Fig. 2(b). The DPs are recorded using a Thorlabs CS2100M-USB complementary metal–oxide–semiconductor (CMOS) camera, with 16-bit depth, 5.04 × 5.04 µm² pixel size, and 1920 × 1080 resolution.

The results of helical wavefront reconstruction and OAM spectrum analysis are shown in Figs. 2(c)–2(e). In the singular mode test, OAM modes from l = −10 to l = 10, as well as high-order modes of l = 20, 30, 40, and 50 are included in the experiment. Here, we demonstrate the results of a singular OAM mode with low TC (l = −6), high TC (l = 50), and multiplexed OAM modes (l = −4, −2, 2, 4). The first column is the single-shot DPs collected from the camera (zoomed-in). Then, the incident OAM-carrying wavefields are reconstructed from these DPs by performing CMI phase retrieval with 500 iterations. The intensity and phase profiles of the reconstructed wavefields are shown in the second and third columns. The initial result of reconstruction is the wavefields at the support plane, and they are back-propagated to a plane near the SLM using the angular spectrum method, which led to the back-propagated phase profiles in the fourth column. The OAM spectra of the reconstructed wavefields at the support and back-propagated planes are computed, as shown in the last two columns. All results are the mean of 16 repeated experiments, and error bars in the spectrum figures are the standard deviation.

For each reconstructed vortex beam, the OAM power spectrum of the back-propagated wavefield shows a better quality than the original wavefield at the support plane, especially for the high-order helical wavefront. Regarding the singular mode in Figs. 2(c) and 2(d), the average mode purity is 93.3% and 74% at the support plane and rose to 95.2% and 93.3% after backpropagation. The standard deviation of the principal mode in Fig. 2(d) decreased from 0.0368 to 0.0017. The average intermodal crosstalk, defined by 10 log(P(l_m)), where P(l_m) is the relative power ratio of the maximum none-zero undesired OAM mode, reduced from −19.28 and −7.65 to −29.79 and −16.91 dB, respectively. This is mainly due to the phase uncertainty region at the center of the reconstructed wavefront, which originated from the missing information at the center of DP due to the dark-shaped center amplitude distribution of the vortex beams. For LG modes, the greater the TC order is, the larger the diameter of the dark center will be. This can be mitigated by using a modulator with more dramatic changes to provide stronger modulation to the wavefield and improve the speckle quality or using the perfect vortex beam with a radial intensity profile that remains independent of its TC. The diameter of the central phase uncertainty region is reduced along with backpropagation, thereby improving the mode purity in spectrum analysis. For multiplexed OAM modes in Fig. 2(e), backpropagation reduces the power ratio unevenness of individual modes. A quantitative evaluation of how the backpropagation process improves the quality of the OAM spectrum is provided in the supplementary material 1, Sec. S2. After backpropagation, the root-mean-squared errors (RMSEs) of the reconstructed wavefronts and the original phase patterns loaded on the SLM are calculated. The average RMSEs of 16 repeated experiments in Figs. 2(c) and 2(d) are 0.104 and 0.252 rad, respectively. Similarly, higher TC orders have a larger RMSE due to the phase uncertainty region in the center of the reconstructed wavefront.

In addition, the intermodal crosstalk of singular OAM modes from l = −10 to l = 10, composed of the spectra of back-propagated wavefronts, is shown in Fig. 2(f). The average mode purity is 94.52%, and the average intermodal crosstalk is −20.78 dB. Such a performance verifies that the proposed method has better spectral decomposition accuracy than existing diffraction-based methods.^37,38 In addition, in Figs. 2(c), 2(e), and 2(f), apart from the helical modes, there is inevitably a zero-order intensity in the spectrum. Apart from the phase uncertainty region, this zero-order intensity also arises from the unmodulated planar wavefront, due to the diffraction efficiency of the SLM. The average crosstalk of the zero-order modes in Fig. 2(f) is −14.47 dB. Nonetheless, it will not lead to a direct negative impact in most applications, since the zero-order state is typically not used in the orthogonal OAM multiplexing.

The maximum measurable OAM mode can be further extended by sacrificing the single-shot feature of CMI. The quality of phase retrieval can be significantly improved using the ptychography method, in which multiple scanned DPs will be recorded and accumulated to provide more information for wavefield reconstruction .⁴⁷ In the supplementary material 1, Sec. S3, 16 DPs are recorded by scanning the modulator to perform ptychography and reconstruct a wavefront. The results show that a maximum OAM mode of 150 is retrieved within a rapid convergence speed, and the phase profiles are smoother than the single-shot CMI results.

In addition, as the CMI method reconstructs the complex wavefield with a high quality, the obtained wavefield can be numerically propagated. This unique feature will be useful to reconstruct three-dimensional (3D) wavefront structures, diagnose the beam-propagation path, or analyze more complicated structured light fields, such as the spatiotemporal optical vortices.⁴⁸ As an example, Fig. 2(g) shows a 3D wavefront structure of an l = 3 vortex beam. Here, the helical wavefronts are constructed by numerically propagating an experimentally obtained wavefield for 10 mm. More details about this process are provided in the supplementary material 1, Sec. S2.

To demonstrate our proposed diffractive OAM spectrometer in practical applications, a multiplexed OAM data transmission proof-of-concept experiment was conducted, as shown in Fig. 3. We first generated a set of OAM-encoded wavefronts to denote Roman letters according to the American Standard Code for Information Interchange (ASCII) protocol. As shown in Fig. 3(a), each letter is represented by an 8-digit binary ASCII code and is then encoded using a set of LG modes with azimuthal orders ranging from −8 to 8 and takes a gap of 2, in which the LG mode −8 stands for the lowest bit and 8 stands for the highest bit. As a result, each Roman letter is encoded with a group of superposed helical wavefronts in the ROI of l = ±2, ±4, ±6, ±8. In this experiment, the name of our university, “SUSTech,” was encoded into seven multiplexed wavefronts. These multiplexed phase profiles are applied to the SLM consecutively and then recorded and analyzed by our diffractive OAM spectrometer. The experiment setup is identical to Fig. 2(a), and the data transmission process is shown in Fig. 3(b). The results show that all multiplexed wavefronts are correctly decoded into the desired Roman letters. Figure 3(c) shows a comparison of the retrieved and desired spectra of the letter “U” (ASCII code: 01010101). Here, the relative power ratios are normalized by a maximum value to demonstrate the decoding capability of the proposed OAM spectrometer for binary data. The retrieved valid modes are the same as the desired spectrum, with the maximum undesired mode (l = −2) containing a relative intensity ratio of 0.06.

Then, an image transmission experiment was conducted to further demonstrate the statistical characteristics of the proposed OAM spectrometer, as shown in Fig. 3(d). An AI-generated 8-bit grayscale image (generated by Microsoft Copilot on May 21, 2024, with the prompt “create an 8-bit grayscale image of city night scene,” https://copilot.microsoft.com) is resized into 70 × 70 pixel², shown as the “image sent.” The pixels in the image are encoded into 4900 multiplexed helical wavefronts using the aforementioned LG modes, according to their intensity levels. After transmission in sequence and analyzed by the diffractive spectrometer, the retrieved 4900 spectra are displayed in sequence, shown as the “retrieved spectra.” It is worth noting that because all wavefronts share the same modulator and optical system parameters, reconstruction for multiple wavefronts can be performed in parallel. In our experiment, reconstruction for 100 wavefronts was performed simultaneously on an NVIDIA RTX A6000 graph processing unit (GPU), taking 74.6 s for 500 iterations. The time consumption can be further reduced by using an advanced GPU with larger memory or a field programmable gate array (FPGA) to improve parallel computing. To decode the multiplexed OAM modes, binarizing all the measured spectra using a set of threshold values is critical. By setting the binary threshold values at 0.2, all 4900 wavefronts can be correctly decoded, shown as the “image received.”

If we increase the interval of threshold values to 0.18 and 0.28, there will be 6 error pixels. The rise of error rate is because some superposed OAM modes are relatively more challenging to be quantitatively decomposed in computing the angular coherence. For example, the retrieved spectrum of pixel valued 165 (binary code: 10100101) consistently matches the desired result. However, in the spectrum of pixel valued 203 (binary code: 11001011), the superposition of Hermite–Gauss (HG) modes (l = ±6, ±8) and an LG mode makes it challenging to decompose the HG modes from the LG mode-dominated wavefront by angular coherence. This is because, compared to the ring-shaped LG modes, the intensity distribution of HG modes is highly concentrated on several centrosymmetric beam spots. Consequently, for a DP generated by a superposed wavefront of LG modes and HG modes, more speckles come from the LG mode rather than the HG mode. Therefore, it is more challenging for the CMI method to reconstruct the characteristics of HG wavefronts when multiplexed with LG modes. This might be improved by modifying the modulus constraint to enhance the phase retrieval performance of CMI in the presence of missing DP data.⁴⁹ Nonetheless, the current results verified that the performance of our diffractive spectrometer is comparable to state-of-the-art counterparts,^34–38,42 laying a foundation for various applications, such as free-space optical communication, optical encryption, and OAM-spectrum-based measurement.^5,50

In this paper, we report a single-shot diffractive spectrometer for photonic OAM. A CMI phase retrieval and computational spectrum analysis scheme is implemented to reconstruct arbitrary helical wavefront and quantitatively decompose individual OAM modes, without conducting any calibration associated with the incident vortex beam field to be measured. Experiments were conducted to demonstrate the performance of the proposed method. The results show that our system can retrieve the spectrum of arbitrary OAM modes with high accuracy lower than −16.91 dB intermodal crosstalk and maximum measurable TC greater than 50. A proof-of-concept visible light demonstration of multiplexed OAM information transmission showed that the performance of our diffractive OAM spectrometer is comparable to that of the state-of-the-art approaches, laying a foundation for practical applications in optical communication and measurement.

The quantitative spatial spectrum analysis of OAM modes opens up new possibilities in fundamental and quantum physics. Although many approaches have been proposed in the visible and infrared bands, it has been a long-standing challenge under short-wavelength lasers and electron beams. As a diffractive approach, our method is universal across different wavelengths, with the system’s simplicity of being reference-free and lens-less, making it an ideal tool for various applications under EUV, x-ray, and even electron beam. For example, EUV and x-ray vortex beams have been used to study molecular chirality and its interaction with the OAM of light, such as distinguishing biochemical enantiomers and characterizing topologically complex electronic matter.^11,15 In addition, using a diffractive OAM spectrometer to diagnose resonant inelastic x-ray scattering enhanced by vibronic coupling in Laguerre–Gauss beams could reveal electronic orbital effects and vortex polarization in ferroelectrics.¹⁴ For electron beams, the quantitative diagnosis of OAM-transferring properties may lead to a new type of electron energy-loss spectroscopy (EELS), whereas the proposed diffractive method can be directly integrated into electron microscope systems.^16,17 In all these applications, our method’s single-shot feature will allow OAM spectrum analysis to be performed in a low-dose manner and also makes it possible to dynamically analyze ultrafast processes. In our future work, we will implement the proposed diffractive OAM spectrometer on a high-order harmonic generation EUV or x-ray free-electron laser and further study its application to optical metrology and OAM-induced light–matter interaction.

See the supplementary material 1 for supporting content. See Visualization 1 for a video showing how the vortex beam field, the OAM spectrum, and the 3D helical wavefront structure are reconstructed from a single-shot DP.

This work was supported by the National Natural Science Foundation of China (Grant No. 12074167) and the Shenzhen Science and Technology Innovation Program (Grant No. JCYJ20241202125334045).

The authors have no conflicts to disclose.

Y.H. and F.Z. conceived the idea. Y.H., H.Z., and T.L. conducted the experiments. Y.H., T.L., A.L., and F.Z. prepared the numerical simulations and realized the software. Y.H. wrote the original manuscript. Y.H. and H.Z. designed the figures and visualizations. All the authors discussed and contributed to the finalizing of the manuscript. F.Z. supervised the project.

Yanwei Huang: Conceptualization (lead); Data curation (lead); Investigation (lead); Methodology (equal); Software (lead); Validation (lead); Visualization (equal); Writing – original draft (lead); Writing – review & editing (lead). Hanxiao Zhang: Data curation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal). Tao Liu: Conceptualization (equal); Data curation (equal); Software (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal). Angyi Lin: Software (equal); Validation (equal); Writing – review & editing (equal). Fucai Zhang: Conceptualization (equal); Funding acquisition (lead); Methodology (equal); Project administration (lead); Software (equal); Supervision (lead); Validation (equal); Writing – review & editing (equal).

The data that support the findings of this study are openly available in the GitHub online repository “dataset_diffractive_oam_spec”: https://github.com/HuangYanwei-pixian/dataset_diffractive_oam_spec.