In 2008, Shapiro et al. proposed the concept of computational ghost imaging, laying a robust groundwork for the practical application of ghost imaging techniques^[1]. In the same year, in the field of computer vision, Duarte et al. proposed the concept of a single-pixel camera combined with the theory of compressed sensing (CS)^[2]. Since then, computational ghost imaging and the single-pixel camera as two kinds of implementations of single-pixel imaging technology have significantly advanced. Compared with the traditional imaging method, which employs planar array detectors, the characteristic of single-pixel imaging is that only a single point detector without spatial resolution is used to record the image information. Specifically, the object is sequentially modulated by a series of mask patterns, and the aggregate light intensity is recorded by a single-pixel detector. Finally, the object image can be reconstructed by correlating a sequence of total light intensity readings with their corresponding mask patterns. Single-pixel detectors boast several advantages: lower dark noise levels, higher sensitivity, faster response time, and lower cost. In addition, they demonstrated excellent performance over almost the entire spectrum range. Due to its unique imaging mechanism, single-pixel imaging has been applied to many fields, such as terahertz imaging^[3–5], remote sensing imaging^[6], three-dimensional imaging^[7–9], multispectral imaging^[10], and optical information encryption^[11,12].

In single-pixel imaging based on structured light modulation, the choice of modulated mask patterns is critically tied to the efficiency and quality of the reconstructed images. Inspired by quantum ghost imaging, the initial single-pixel imaging uses random speckle patterns for illumination. However, random pattern illumination necessitates oversampling, which consequently prolongs the data acquisition time. To address this issue, researchers have suggested employing orthogonal illumination patterns, such as the Hadamard basis^[13], the Fourier basis^[14], and the cosine basis^[15]. Due to the orthogonality and completeness of these bases, it is possible to attain images with high signal-to-noise ratios by employing a number of samples equal to the number of pixels in the reconstructed image. Furthermore, a new technique named single-pixel holography is derived by combining single-pixel imaging and digital holographic technology^[16].

Single-pixel holography is an effective method for simultaneous phase and amplitude imaging with a single point detector, which records phase information by introducing a reference light to interfere with the signal light. According to different interference methods, it can be categorized into two types: one is based on double-path interference and the other is based on common-path interference^[17]. As early as 2012, Clemente et al. utilized a spatial light modulator (SLM) to generate random phase patterns and integrated it with double-path phase-shifting interferometry, successfully realizing phase object imaging with a single-pixel detector^[18]. Furthermore, the CS algorithm is employed to reduce the sampling times^[19]. In order to improve the speed of imaging, a digital micromirror device (DMD) with a faster modulation speed is used^[20,21]. Hu et al. used the Fourier basis combined with four-step phase-shifting to realize phase imaging of objects^[22]. Wu et al. used the Hadamard basis to develop a high-throughput single-pixel holography system that quadrupled the imaging speed by introducing a frequency difference between the signal light and the reference light^[23]. Because double-path interferometers are prone to environmental disturbances, researchers have also proposed common-path interference schemes^[24–26]. By encoding the signal light and reference light into a single hologram, the complex amplitude imaging of an object can be realized with a single beam, which is more robust against external interference. However, all of the methods mentioned above make use of real-value orthogonal bases such as Fourier and Hadamard bases. These bases have some drawbacks and restrictions. On the one hand, these masks cannot propagate stably in free space and are difficult to adapt to the requirements of long-range imaging. On the other hand, amplitude modulation alone is not sufficient to distinguish the fuzziness caused by the complex conjugations of the phase objects^[27].

Laguerre–Gaussian (LG) modes, consisting of a complete orthogonal basis, represent a set of eigenmodes of the paraxial wave equation, which is able to maintain stable propagation in free space^[28]. Since Allen et al. revealed in 1992 that LG beams with a helical phase wavefront carry well-defined orbital angular momentum (OAM)^[29], LG beams have been widely adopted in many fields^[30–32], including single-pixel imaging. Zhao et al. used LG modes as mask patterns for single-pixel imaging to measure complex OAM spectrum and carried out a simple complex amplitude imaging experiment^[33]. Gao et al. further combined it with the CS algorithm to reduce the sampling rate^[34]. However, the mechanism of the LG mode interaction with complex objects is not fully discussed and the advantages of single-pixel imaging based on LG modes still need to be explored further.

In this Letter, we propose and experimentally demonstrate a computational ghost holographic imaging scheme based on LG modes. Unlike conventional approaches that demand exact correspondence between the number of imaging pixels and modulation modes, our method uses 4128 distinct LG modes for illumination and reconstructs the image of complex objects with an adjustable pixel number. Using the second-order correlation (SOC) and TVAL3 CS algorithms, the image of 512pixel×512pixel is reconstructed, and the actual spatial resolution obtained by the experiment is 70.13 µm. When imaging objects with rotational symmetry, the LG spectrum distribution is sparse, which means that the symmetric features of objects can be easily identified and exploited for data compression. We also compare two different signal reception methods: bucket detection and zero-frequency detection. The bucket detection method can only obtain the amplitude information of the object, while the zero-frequency detection method can obtain both amplitude and phase information. Nonetheless, the bucket detection method is less sensitive to external disturbances. During the experiments, we significantly improved the image quality by properly calibrating the orthogonality of the LG modes destroyed by holographic coding. In addition, by preprocessing the LG modes, we facilitate seamless integration of image processing functionalities including scaling and edge detection.

In cylindrical coordinates, the light field distribution of an LG mode at the beam waist position (z=0) is $$\begin{gathered} LGlp(r,φ)=2p!π(p+|l|)!1ω0(r2ω0)|l| \\ ×exp(−r2ω02)Lp|l|(2r2ω02)exp(ilφ), \end{gathered}$$(1)where ω0 is the waist radius, l and p are the azimuthal index and radial index, respectively, and Lp|l| is the generalized Laguerre polynomial. l is also known as the topological charge. The LG modes are the eigen-solutions of the paraxial wave equation and constitute a complete orthogonal basis.

For any complex amplitude object, we can use the LG modes to decompose it as $$\begin{gathered} O(r,φ)=A(r,φ)exp[iϕ(r,φ)]=∑lBl(r)exp(ilφ) \\ =∑l,pAl,pLGlp(r)exp(ilφ), \end{gathered}$$(2)where A(r,φ) and ϕ(r,φ) represent the amplitude and phase distributions of the object in cylindrical coordinates, respectively. Bl(r) represents the decomposition coefficient of the azimuthal index for the object, i.e., the spiral spectrum. Al,p represents the decomposition coefficient of the object into the entire set of LG modes, i.e., the two-dimensional LG complex spectrum. Al,p can be represented as the overlap integral between the complex amplitude of the object and the individual complex conjugated LG mode. Although calculating the LG complex spectrum of the object alone is sufficient, analyzing parameter Bl(r) provides insights into how the azimuthal and radial dimensions of the LG modes separately contribute to the reconstruction of the object. Bl(r) and Al,p can be expressed as Bl(r)=∫02πO(r,φ)[exp(ilφ)]*dφ,(3)Al,p=∫02πdφ∫0+∞O(r,φ)[LGlp(r)exp(ilφ)]*rdr.(4)

As the single-pixel detector can only respond to the intensity of light and lose the phase information, a four-step phase-shifting method is used to obtain Al,p optically. Specifically, we create the interference light field composed of the LG mode with a desired phase shift and a reference light to illuminate the object. By obtaining the complex coefficient corresponding to each LG mode in the sequence, the two-dimensional LG complex spectrum is obtained, and the complex amplitude of the object can be reconstructed by Eq. (2). Due to the use of the four-step phase-shifting method, the required number of samples is four times the total number of the LG modes employed. According to different receiver strategies, there exist two methods for single-pixel detection, zero-frequency detection, and bucket detection^[35,36]. Zero-frequency detection requires filtering the zero-frequency component through a pinhole at the focal point of the collecting lens (i.e., the Fourier plane), while bucket detection directly receives the total light intensity passing through the collection lens without spatial filtering.

For zero-frequency detection, the light intensity passing through the pinhole can be expressed as $$\begin{gathered} In=∬|[O(r,φ)LGlp(r,φ)*·exp(inπ/2)+O(r,φ)·R(r,φ)]k→=0|2rdrdφ, \\ n=0,1,2,3, \end{gathered}$$(5)where exp(inπ/2) represents the additional phase shift to the LG modes, R(r,φ) represents the complex amplitude of the reference light, and k⇀=0 represents the spatial frequency of zero, i.e., the central point of the Fourier plane. For a specified object, O(r,φ)·R(r,φ)≡αe−iϕα is a global complex factor. Therefore, Eq. (5) can be expressed as $$\begin{gathered} In=∬|O(r,φ)LGlp(r,φ)*|2rdrdφ+α2 \\ +2∬rdrdφ|O(r,φ)LGlp(r,φ)*|α·cos[ϕ(r,φ)+ϕα+n·π2],n=0,1,2,3, \end{gathered}$$(6)where ϕ(r,φ)=angle|O(r,φ)LGlp(r,φ)*|. The four-step phase-shifting corresponds to the illumination beam with four different additional phase shifts, and after interacting with the object, the LG complex spectrum can be obtained through the four detected intensity values, which can be expressed as $$\begin{gathered} Al,p∝(I0−I2)+i(I1−I3) \\ =4αeiϕα∬O(r,φ)LGlp(r,φ)*rdrdφ. \end{gathered}$$(7)

The final calculated Al,p will have a phase shift of ϕα. However, the results of the phase imaging are not affected because the phase shift ϕα is a fixed value common to all LG modes.

For bucket detection, the total light intensity received by the detector after the collection lens can be expressed as $$\begin{gathered} In′=∬|O(r,φ)|2|LGlp(r,φ)*·exp(inπ/2)+R(r,φ)|2rdrdφ, \\ n=0,1,2,3. \end{gathered}$$(8)

At this point, the LG spectrum can be represented as $$\begin{gathered} Al,p′∝(I0−I2)+i(I1−I3) \\ =4∬|O(r,φ)|2LGlp(r,φ)*R(r,φ)*rdrdφ. \end{gathered}$$(9)

From the measured LG spectrum Al,p′, the amplitude of the object can be retrieved. It is noted that zero-frequency detection is essentially a coherent detection, which can obtain the amplitude and phase of an object at the same time. However, a pinhole for spatial filtering is required, which will reduce the measured light intensity and increase the complexity of the experiment. Bucket detection is a form of incoherent detection which only retrieves the amplitude information of an object. However, it requires merely measuring the total light intensity, making experimental realization relatively straightforward.

The quantity of the LG modes is crucial to the quality of imaging. With the increase in the azimuthal and radial indices, the size of the LG modes also enlarges. During simulation, we strive to align high-order LG modes (l=64, p=0; l=0, p=31) tangentially with the square field of view to ensure coverage of the entire field. It should be noted that since LG modes are circularly symmetrical, the imaging field is circular rather than square. In simulation settings, we utilize 4128 LG modes to sample the object, with the azimuthal index l ranging from −64 to 64 and the radial index p spanning from 0 to 31. Each mask pattern is a superposition of an LG mode and the reference light field, with 512pixel×512pixel. The mesh size set in the simulation is 0.0157 mm, the wavelength is 633 nm, and the waist radius ω0 is 0.5 mm. It can be seen from Eq. (2) that the spiral spectrum of the object is closely related to the LG modes that need to be projected. In other words, if the object to be measured is sparse in the spiral spectrum, that is, for most of the azimuthal indices Bl=0, then the object can be reconstructed with partial l components.

First, we choose two binary amplitude objects with rotational symmetry (a clover and a pentagram) for simulation, as shown in Figs. 1(a2) and 1(b2). The signal detection method is bucket detection. The spiral spectra of the objects are shown in Figs. 1(a1) and 1(b1). As can be seen, the spiral spectrum mainly concentrates in the low-frequency region, and it only possesses components with the azimuthal index equal to integer multiples of the object’s symmetry order. Upon further radial decomposition of Bl(r), the LG complex spectrum can be obtained. The LG amplitude spectra for the cloverleaf and pentagram are illustrated in Figs. 1(a3) and 1(b3), respectively. It is evident that the amplitude spectra primarily concentrate in the low-frequency region both azimuthally and radially. Figures 1(a4) and 1(b4) and 1(a5) and 1(b5) show the imaging results using the SOC and TVAL3 CS^[37] algorithms, respectively. Table 1 shows the quantitative analysis of structural similarity (SSIM) between the two algorithms. The SSIM(O,O¯) is calculated using the following equation: SSIM(O,O¯)=(2μOμO¯+c1)(σOO¯+c2)(μO2+μO¯2+c1)(σO2+σO¯2+c2),(10)where μO and μO¯ are the mean of the original image O(r,φ) and reconstructed image O¯(r,φ), respectively, σO2 and σO¯2 are the corresponding variance, σOO¯ is the covariance of two images, and c1, c2 are calculated as c1=(K1L)2,c2=(K2L)2,(11)where K1 and K2 are two small constants, and L is the dynamic range of the pixel intensity. The value of the SSIM is [0, 1], and the larger the value, the smaller the image distortion. Both algorithms can reconstruct the original object well. In practice, the symmetry of the object can be directly identified based on the LG amplitude spectrum. For symmetric objects, it is feasible to employ solely LG modes whose azimuthal indices are multiples of the object’s symmetry order for accurate reconstruction, as any non-multiple components exhibit null amplitudes.

Then, we perform a simulation with a more general complex amplitude object mimicking the Chinese character “hang.” Different phases 2π/3 and 4π/3 are assigned to its left and right sides, respectively. The amplitude and phase are depicted in Figs. 6(c1) and 6(d1), accordingly. The results from the zero-frequency detection and the bucket detection, respectively, correspond to the first row and the second row in Fig. 2. Figure 2(a) shows the LG amplitude spectrum of the Chinese character object. Figure 2(b) shows the LG phase spectrum. Both LG amplitude spectra primarily concentrate in the low-frequency region. Consequently, we can effectively reconstruct the object with good fidelity by employing a limited number of LG modes. The LG amplitude spectrum is symmetric about l=0 under bucket detection, as shown in Fig. 2(a2). The reconstructed results for the two detection methods are displayed in Figs. 2(c) and 2(d), respectively. Both detection methods effectively capture the amplitude information of the original object. However, only zero-frequency detection can retrieve the phase information while bucket detection cannot. This is consistent with our theoretical analysis. In each detection scheme, SOC and TVAL3 CS algorithms are used, respectively. The last three rows of Table 1 show the SSIM of different detection methods and different algorithms. Both algorithms can reconstruct the original object well, and the image quality is similar. To better showcase the performance of the two algorithms, the data in Table 1 specifically compares the regions with non-zero amplitude in the images.

To verify the proposed scheme, we designed the experimental setup as shown in Fig. 3. A He–Ne laser with a wavelength of 633 nm is used as a light source. The laser beam passes through a polarizer and is modulated as horizontally polarized light, allowing the liquid-crystal phase-only SLMs to operate at their maximum efficiency. Subsequently, the light passes through a spatial filtering system composed of an objective lens, a pinhole, and lens L1, which serves to generate a high-quality Gaussian beam. Simultaneously, the light beam is expanded to an appropriate size so that it can fully cover the working area of the SLM. The SLM can only modulate the phase of light, so here we adopt the method from Ref. [38] to achieve complex amplitude modulation. SLM1 successively loads holograms that encode the superposition of different LG modes and reference light. The hologram on SLM1 uses a checkerboard encoding approach^[26], where half the pixels are used to modulate the LG modes, and the other half are used to modulate the reference light. The illumination light field generated by SLM1 is projected onto SLM2 through a 4f system consisting of lenses L2 and L3. The same blazed grating is pre-loaded on SLM1 and SLM2 with the aim of filtering all the diffraction spectra except for the first-order diffracted light to separate the effective modulation from the reflected light. In the 4f system, an iris aperture is used to select the first-order diffracted light from SLM1. SLM2 is used to simulate the complex amplitude objects to be measured. Both SLM1 and SLM2 are configured with pixels of 1024×1024. The number of LG modes modulated by SLM1 is 4128, where p has a range of 0 to 31 and l has a range of −64 to 64.

In addition to emulating complex amplitude objects using SLM2, we also perform imaging experiments with actual transmissive objects. A Fourier transform lens L4 is placed behind the object, which transforms the wavefront at the object plane into its spatial frequency spectrum. For zero-frequency detection, an appropriately sized pinhole is placed at the focal plane to allow only the zero-frequency component of the modulated light field to pass through. A single-pixel detector collects the zero-frequency light transmitted through the pinhole. The size of the pinhole is approximately λf/d, where λ is the wavelength, f is the focal length of L4, and d is the maximum possible diameter of the beam in front of the lens. For bucket detection, all modulated lights are collected without spatial filtering. The output signal of the photodetector is collected by an acquisition card to extract the intensity values, which are correlated with the corresponding LG modes for object reconstruction.

In theory, the LG mode is an eigen-solution of the paraxial wave equation and can form a complete orthogonal basis. In the experiment, the LG modes are prepared using the SLM. In order to utilize the full bit-depth of the SLM, the amplitudes of the LG modes need to be normalized. However, when modulated light fields illuminate the objects for single-pixel imaging, this unintentionally breaks the orthogonality of the LG modes^[39]. Figure 4 shows the correlation matrix of the LG modes, where Figs. 4(a1) and 4(a2) present the scenario with a fixed radial index p=5 while the azimuthal index l varies from −16 to 16. Figures 4(b1) and 4(b2) depict the case with a constant azimuthal index l=5 as the radial index p changes from 0 to 31.

In the matrix, diagonal elements correspond to the self-correlation of LG modes, and non-diagonal elements represent the cross-correlation of different LG modes. In Figs. 4(a1) and 4(b1), LG modes are perfectly orthogonal in both the azimuthal and radial directions, where the weights along the diagonal are uniformly consistent. However, after normalization of the amplitudes, the orthogonality in both the azimuthal and radial dimensions is no longer maintained, as shown in Figs. 4(a2) and 4(b2). Meanwhile, the weights on the diagonal become uneven. The broken orthogonality of the LG modes will affect the quality of the image reconstruction, so the detection value should be corrected during image reconstruction. To rectify the orthogonality of the LG modes, the detected intensity value of each mode should be multiplied by the maximum value from its corresponding mode matrix.

First, we perform experiments on two symmetric amplitude objects (a cloverleaf and a pentagram). The pixel number of the reconstructed image is 512×512, and the size of the illumination spot is 8.5mm×8.5mm. The detection method is bucket detection. The experimental results are shown in Fig. 5. The LG amplitude spectra of the two objects are respectively shown in Figs. 5(a1) and 5(a2), which are basically consistent with the simulation results. The middle two columns and the last two columns in Figs. 5(b)–5(e) represent the object reconstruction results before and after the correction of the LG modes’ orthogonality, respectively. The first and third rows represent the reconstruction results of the SOC algorithm, and the second and fourth rows represent the reconstruction results of the TVAL3 CS algorithm. Both the SOC algorithm and TVAL3 CS algorithm perform comparably in this regard.

The reconstructed images after correction exhibit fewer noises and higher contrast than those before correction. Before correction, the reconstructed pentagram has a darker central region and brighter edges. This is because the edge details of the image are mainly determined by the high-frequency components of the LG modes, while the central part is determined by its low-frequency components. In the uncorrected case, the low-frequency component (i.e., when l and p are relatively small) of the LG modes has a lower weight, which results in a darker central region, consistent with the analysis presented in Figs. 4(a2) and 4(b2). Column c of Fig. 5 represents the reconstructions based on the full sampling of all modes, while Column d depicts the reconstructions obtained by selectively partial LG modes with azimuthal indices that are multiples of 3 or 5. The image quality of the latter case is even better than that of the former one because, for a symmetric object, the measured component with the azimuthal index not equal to multiples of the object’s symmetry order is actually equivalent to the noise. Therefore, the number of samples can be greatly reduced when imaging an object with known symmetry. Inversely, when imaging unknown objects, the symmetry of the object can be quickly determined according to the measured LG spectrum.

Then, to demonstrate the complex amplitude imaging capability of the scheme, two complex amplitude objects are imaged using zero-frequency detection and bucket detection, respectively. The complex amplitude of the image object is encoded by SLM2. In Fig. 6, the first column is the amplitude and phase of the object to be imaged. The amplitude of the first object is limited to a circle, with the phase of a Chinese character “hang” and the surrounding background set to 4π/3 and 2π/3, respectively. The amplitude of the second object takes the form of the Chinese character ‘hang’, where the phases on the left and right sides of the character are, respectively, assigned as 2π/3 and 4π/3. The middle two columns show the reconstruction results of the object amplitude and phase under zero-frequency detection, while the last two columns show the reconstruction effects under bucket detection. The bucket detection method can only provide the amplitude of the object, resulting in a constant phase across the object, thus losing the original phase details. In contrast, under zero-frequency detection, the amplitude and phase of the object can be effectively extracted simultaneously. It is noted that the abrupt discontinuities on the right side of the phase images in Figs. 6(b2) and 6(b3) are due to the phase wrapping during reconstruction.

By comparing Figs. 6(c2) and 6(c3) with 6(c4) and 6(c5), it is observed that the amplitude images obtained through bucket detection yield better signal-to-noise ratios than those acquired via zero-frequency detection. This is because zero-frequency detection requires the use of a pinhole to filter the zero-frequency component, which is susceptible to environmental disturbances, thereby introducing noise into the measurements. On the contrary, bucket detection is easy to implement and is not susceptible to environmental disturbance. To sum up, both bucket detection and zero-frequency detection present unique advantages and disadvantages and can be chosen according to practical situations. Additionally, the imaging performance of the CS and the SOC algorithms are essentially equivalent.

To test the resolution of the imaging scheme, we image a standard USAF1951 resolution target. The imaging results are shown in Fig. 7, where Figs. 7(a) and 7(b) are the reconstruction results using the CS and SOC algorithms under the pixel number of 512×512, respectively. The performance of these two algorithms is comparable. The smallest units that can be distinguished are both element 6 in group 2, as shown in the red box in Figs. 7(a) and 7(b). Thus, the practically achievable spatial resolution is approximately 7.13 lp/mm, which equates to 70.13 µm. When reconstructing images using the SOC algorithm, we can increase the pixel number of the LG modes to produce more refined images with the same LG spectrum. As shown in Figs. 7(c) and 7(d), the reconstructed images have pixels of 1024×1024 and 2048×2048, respectively. By comparing Figs. 7(b)–7(d), it can be found that the imaging resolution does not change significantly as the number of pixels increases, but the image with a higher pixel count remains clear when magnified by the same factor. The hot dot in the upper right corner of Figs. 7(b)–7(d) is the enlarged image of the yellow dotted box area in Fig. 7(b), which becomes clearer as the number of pixels increases.

In addition, in the reconstruction process, the LG modes can be preprocessed to directly achieve specific image processing functionalities. As the first example, by varying the waist radius of the LG modes during reconstruction, the image size can be arbitrarily enlarged or reduced. Figures 8(a)–8(c) represent the reconstruction results with a waist radius of 0.3, 0.5, and 0.6 mm, respectively. It shows that the image is magnified proportionally with the increase of the waist radius. As another useful application, the edge detection of the original object can also be realized by simply differentiating the LG modes used for reconstruction^[40]. Specifically, the differential expression for the LG modes in the polar coordinate is given as $$\begin{gathered} ∂LGlp(r,φ)∂r+∂LGlp(r,φ)r∂φ \\ =[|l|r−2rω02−4rω02Lp−1|l|+1(2r2ω02)Lp|l|(2r2ω02)+ilr]LGlp(r,φ). \end{gathered}$$(12)

The two-dimensional edge of the image can be obtained directly using the differential transformation of the LG modes for image reconstruction. As shown in Figs. 8(d)–8(f), the edges of the cloverleaf, pentagram, and Chinese character “hang” are all clearly detected.

When LG modes are used for single-pixel imaging, there is no strict correspondence between the number of pixels in the reconstructed image and the number of projected masks. Objects can be reconstructed with high fidelity as long as their LG spectra are sparse. LG modes have three variable parameters, the azimuth index l, the radial index p, and the waist radius ω0. During the experiment, the range of l determines the angular resolution of the imaging object^[41], and the range of p determines the size of the imaging field of view. In addition, properly adjusting ω0 can make a trade-off between the imaging field of view and the resolution^[33]. Because of the symmetry of the LG modes, the rotational symmetry of the objects can be directly identified from the obtained LG spectrum, which may also facilitate the imaging of the rotating objects^[42]. In terms of image reconstruction quality, the SOC algorithm and the TVAL3 CS algorithm perform comparably. This may be attributed to the relatively high imaging pixels and relatively simple objects, which make the advantages of the CS algorithm not obvious. Furthermore, compared with the traditional modulation patterns in single-pixel imaging, such as random speckle, Hadamard basis, and Fourier basis, the LG modes can propagate stably in free space and may be advantageous in long-range imaging.

As shown in Table 2, we compare other works on single-pixel holography using Fourier and Hadamard bases, focusing primarily on three parameters: imaging field of view, resolution, and required number of modes. Since these parameters vary across different studies, we have defined a metric called basis efficiency, which is calculated as Field of view/Resolution/Numberofmodes. Theoretically, the basis efficiency is 1 when fully sampled, and a higher base efficiency indicates better imaging performance under undersampled conditions. The last column of Table 2 displays the basis efficiencies of various studies, showing notably better results using the LG modes. Here, the field of view is uniformly defined as the number of pixels of the used modulator multiplied by the pixel size. It should be noted that because the LG modes are radially symmetric, the imaging field of view is circular and tangential to the edges of the image. In experiments, the actual size of the light spot is approximately 8.5mm×8.5mm. It is undeniable that the modulation rate of the DMD is significantly faster than that of the SLM. Here, we have not considered the imaging time. Our experiments are based on a SLM with a refresh rate of 60 Hz and thus the imaging speed is limited. Fortunately, with the development of high-speed SLMs^[44], the generation of LG modes at kHz or even higher rates can be accessible.

In conclusion, we have proposed and experimentally demonstrated an efficient method for computational ghost holography based on LG modes. We utilized an SLM to prepare a sequence of LG modes of different azimuthal and radial indices to illuminate the object. As for the receiver design, we compared two distinct reception methods: zero-frequency detection and bucket detection. Zero-frequency detection allows for simultaneous acquisition of both the amplitude and the phase of an object, whereas bucket detection only retrieves amplitude but proves to be more robust to disturbance. The objects with rotational symmetry can be quickly identified and reconstructed by analyzing the obtained LG spectrum. In addition, we have also discussed the resolution of the imaging scheme and realized the function of edge detection directly through the preprocessing of the LG modes. Our work may find widespread applications in target recognition and microscopic imaging.