Defective states of Hermite-Gaussian modes for long-distance image transmission and high-capacity encoding

Abstract

Structured light brings a breakthrough in information capacity carried by the laser field, finding an ideal utility in optical information transmission. Advancements in optical intensity-based imaging have facilitated the use of structured light for simple information enconding and decoding. Here, we propose a method for extremely high-capacity information encoding, as well as image direct transmission, by modulating the structured light to defective states. Using well-designed two-dimensional binary hologram gratings to generate distinct defects within a single Hermite-Gaussian mode, we achieve over 10ⁿ (n > 10) of laser states for encoding, corresponding to information capacity being tens of bits. These defective states are recognized by image processing method for quick decoding. In addition, various image patterns can also be generated and are possible to achieve long-distance transmission with high fidelity. It means that images can be directly transmitted for long distance without digital encoding process, which paves a simple way for information transmission.

The authors propose a method for high-capacity information encoding and long-distance image transmission by utilising Hermite-Gaussian eigenmodes in defective states. The enables the generation over 10ⁿ varying laser states for encoding, or an information capacity of tens of bits.

Introduction

Structured light, extensively studied for both scientific research^1–3 and various applications^4–8, holds significant promise in addressing the bottleneck problem of the capacity crunch of optical communications^9–17. One of the most significant applications of structured light in optical communications is to utilize its rich spatial characteristics as recognizable encoding parameters, and so to achieve ultra-high-capacity information transmission by generating a large amount of uniquely structured light modes^11,18. Similar to the shift-keying techniques in temporal optical communication, this approach allows the transmission of multibit data within a single time interval. For instance, quadrature amplitude modulation (e.g., 256QAM) has been successfully used to encode 8-bit data, and by further increasing the value of amplitude and phase levels, 12-bit data encoding has been feasible. However, QAM-based techniques face limitations when attempting to scale to higher bit depths. The breakthrough of recent advances in structured light has opened up the possibility of high-capacity shift-keying data transmission in the spatial domain.

Mode-encoding using structured light can neglect the orthogonal requirement of the channels and only needs a large number of patterns to be different from each other, with proper decoding methods. To achieve a maximum number of beam patterns within a certain order, leveraging the superpositions of laser eigenmodes both in degeneracy¹⁹ or nondegeneracy²⁰ becomes inevitable, albeit resulting in increasingly complex beam structures. Optical methods with gratings^21–23, hologram gratings^24,25 and other optical elements^26–28 for such complex superposed modes decoding are not suitable, then self-organizing map^29–31 based shallow neural networks and convolutional neural networks (CNN)^20,32–36 based deep learning methods have been widely investigated. However, with the quantity of beam patterns increasing, recognition of complex patterns becomes more challenging, requiring increasingly complex CNN architectures. Despite advancements, the total pattern quantity is still at a relatively low level, with the highest value reported to be 1680³⁷, which asks for the highest mode order of 16 and a quite time-consuming network design. Therefore, there is an urgent need for a much more efficient encoding method to provide considerable beam patterns without increasing decoding burdens.

From a broader perspective, current optical information transmission works are mainly based on well-developed radio communication schemes, which transmit digital bits in the time domain. However, the most direct way to achieve information transfer using optics is by leveraging the spatial domain. Information of images and texts can be transmitted from A to B by optical waves directly. Imaging technology is the most common method to obtain spatial information on targets, and it belongs to the detection area, but not the communication area. If the image information of both entities and texts can be transferred directly in free space without digital encoding and decoding processes like imaging technology, optical communication may embrace a revolution, while the problem is that the optical waves carrying images are not easy for long-distance non-diffraction propagation to deliver the original images with high fidelity. Thanks to the spatial expansion characteristics of laser-beam-based structured light bringing great advances, one can achieve the generation of both non-diffraction beams^38–40 and laser patterns carrying complex images⁴¹. Nevertheless, the beam lacks image information, or the laser mode is not possible to invariantly propagate too far. Additionally, projectors and holography^42,43 may realize the image’s direct transmission, but also with a relatively short distance due to the relatively large divergence angle of the light field. So far, direct image transmission for long distances is still impossible.

In this work, we define an information encoding method by using a defective state of structured light and reveal that the defective state can propagate through long distances in a uniform medium with high fidelity. By manipulating the structure defects’ distribution, arbitrary image patterns can be generated for spatial information direct transmission and captured by machine vision for fast recognition. Based on this principle, we also propose an effective information transmission link. By manipulating the existence of each bright spot in an Hermite-Gaussian (HG) eigenmode, an extremely large number of defective patterns are obtained for high-capacity encoding and fast decoding. It greatly breaks through previous limitations on recognizable structured light, opening useful avenues for the application of structured light in information transmission.

Results

Concepts and principle

The eigenmodes of lasers are solutions to the Helmholtz equation under the paraxial approximation, which means that their intensity distribution remains invariant during propagation, with only a scaling effect due to beam divergence as the propagation distance changes. This characteristic of the eigenmodes also implies that the spatial intensity information carried by the laser can be transmitted over long distances in a vacuum or a homogeneous medium without significant distortion, providing a solid foundation for direct image transmission. If the two-dimensional intensity information of an image to be transmitted is loaded onto a two-dimensional higher-order eigenmode, the intensity distribution of the eigenmode will be modulated by the corresponding spatial information of the image. Specifically, if one wants to obtain an HG_m,n mode with some local defects at a certain distance d, the targeted light field should be given as E_tar, and inverse propagation calculation should be made for its near-field E_near, according to the angular spectrum theory. E_near can be calculated as below,

$$E_{HG}\left(x,y,d\right)={\mathbb{F}}^{-1}\left\{\mathbb{F}\left[E_{HG}\left(x,y,0\right)\right]\cdot H\left(d\right)\right\}$$ 1

$$E_{tar}\left(x,y,d\right)=E_{HG}\left(x,y,d\right)\cdot T\left(x,y\right)$$ 2

$$E_{near}\left(x,y,0\right)={\mathbb{F}}^{-1}\left\{\mathbb{F}\left[E_{tar}\left(x,y,d\right)\right]/H\left(d\right)\right\}={\mathbb{F}}^{-1}\left\{\mathbb{F}\left[E_{tar}\left(x,y,d\right)\right]\cdot H\left(-d\right)\right\}$$ 3

where, $\mathbb{F}$ and ${\mathbb{F}}^{-1}$ are the Fourier and inverse Fourier transformations, T is the transmittance function with specially designed local defects, and $H\left(d\right)=\exp \left(ikd\right)\exp \left[-i\pi \lambda d\left(f_x^2+f_y^2\right)\right]$ is the transfer function to the distance d. After getting the function of E_near, the binary hologram grating to be loaded on the spatial light modulators (SLM) can be obtained to generate the required light field. Please refer to the “Methods” section for details.

By further studying the complete propagation features of the generated light field, the one at z = 0 can be propagated forward over an arbitrary distance D, yielding the complex amplitude of the light wave at z = D, denoted as E_out,

$$E_{out}\left(x,y\right)={\mathbb{F}}^{-1}\left\{\mathbb{F}\left[E_{near}\left(x,y\right)\right]\cdot H\left(D\right)\right\}$$ 4

Subsequently, the complex amplitude of the light wave at z = 0 can be expressed in the convolutional form of Fresnel diffraction, substituted into the wave at z = D, i.e. the complex amplitude of the emitting light wave expanded in the same convolutional form,

$$E_{out}\left(x,y,D\right)=e^{jkL}{E}_{tar}\left(x,y,d\right)\otimes \left[\frac{1}{{\lambda }^{2}Dd}{e}^{j\frac{k}{2\left(D-d\right)}\left(x^2+y^2\right)}\right]$$ 5

The function of the emitting light field can be expressed as the convolution of the complex amplitude of the target field with transformation function A(x,y), followed by multiplication with the phase factor φ,

$$E_{out}\left(x,y,D\right)=\phi \left[E_{HG}\left(x,y,d\right)T\left(x,y\right)\right]\otimes A\left(x,y\right)$$ 6

where,$A\left(x,y\right)=\frac{1}{{\lambda }^{2}Dd}{e}^{j\frac{k}{2\left(D-d\right)}\left(x^2+y^2\right)}$.

Specifically, when studying the transmission distance, assuming the target field is placed at a large distance d, the approximation of the Fourier transform of the transformation function A(x, y) to a Dirac delta function (when L = D–d ≈ 0) ensures that the transmission characteristics of the target field are minimally affected. This allows the designed intensity information to be transmitted over long distances to the target area nearby z = d. However, when the Fourier transform of A(x, y) does not approximate to delta function (L > 0), the image encoded in the light field can only be transmitted within a limited range, of which size is related to the value of d, maintaining approximate invariance. However, as HG mode is an eigenmode of laser cavity, its intensity distribution has the property of transmission-invariance (nothing variant but spatial scaling), which significantly increases the similarity-maintaining range L of the target light field (for detailed analysis in Supplementary Information S1). And it is important to notice that, the similarity between E_out (x, y) and E_tar (x, y) is almost always satisfied when d has a large value (located at the far field, compared with the Rayleigh length), but unrelated to the exact distribution of T(x, y).

The loaded image function only modulates the intensity of the eigenmode without altering its phase. When the intensity of the eigenmode is modulated by a slowly varying function since the Fourier transform of the modulation function approximates a Dirac delta function, the transmission characteristics of the eigenmode are minimally affected, allowing the intensity information it carries to be transmitted over long distances. However, when the Fourier transform of the modulation function no longer approximates a Dirac delta function, the image loaded onto the optical field will only maintain near-invariant transmission over a limited region. Despite these variations, the common advantage in all these cases is the ability to achieve long-distance image transmission with the laser beam’s small divergence angle. Figure 1 presents a concept diagram of the image transmission method proposed in this paper.

Fig. 1. Principle of the generation and transmission of defect beams.

The position z = d corresponds to the location of the target light field, and L represents the transmission range within which the field maintains a high fidelity. I: the design of the target field, with a standard HG eigenmode light field, modulated by a gray-image-based transmittance function; II: the near field of the target field is derived from the inverse propagation (IP) of the target field, and then the two-dimensional grating can be achieved by the computer-generated holography (CGH) method.

Image pattern generation and propagation

Following the concept above, the HG eigenmode is possible to be spatially modulated by different image patterns. As shown in Fig. 2a, various image patterns are generated successfully, including the digits, letters, Chinese characters and geometries. Here, the HG_8,8 eigenmode is used as the carrier beam to be encoded to various relatively simple symbols. Comparing the target characters with the simulated and experimentally measured patterns, a high degree of consistency is observed, demonstrating that this method is effective in obtaining images of simple symbols.

Fig. 2. The generation and propagation of defective HG modes for different images.

a The generation of digits, letters, Chinese characters and cubic geometries. From the first to the fourth row are the target defect mode patterns, the corresponding holograms, the simulated far field of the +1^st diffraction order beam and the experimentally measured patterns. b1 Institution emblem images generated using defective states within 10 m inside the laboratory. b2 Character patterns generated within 50 m outside the laboratory. c Variation of the similarities between the random defective target field and the light field with different propagation distances. d Variation of similarity-maintaining range L (SSIM > 0.9) of HG modes (taking HG_3,3 as an example) with random defects over long-distance propagation, with ω₀ = 1, 2, and 3 mm, respectively.

By increasing the indexes m and n of the HG mode to enhance the array resolution, more complex images are possible to be generated. The on-hand digital micromirror device (DMD) (TI, V-7001) system is possible to generate an HG mode with the highest indexes of m = n = 24 by the super pixel method with a unit possessing 4×4 pixels (see details in Supplementary Information S2). As the intensity modulation function, the institution emblem was used to load onto the eigenmode of HG_24,24, and a discrete emblem pattern was obtained, whose transmission properties were subsequently studied. Fig. 2b1 first presents the intensity distributions of discretized institution emblem pattern under different similarities SSIM = 1, 0.95, and 0.9. Under the laboratory condition, the target positions of the emblem pattern were set at d = 2 and 6 m. The figure shows the intensity distributions of the emblem pattern at and around the target positions. It can be seen that the emblem patterns obtained at the target position exhibit a well-preserved structure, with the original sub-spots of the HG mode maintaining their characteristic profile. By comparing the images transmitted before and after the target positions, a variation in similarity (with calculated SSIM to be around 0.8) with respect to the target image can be observed. To further verify the transmission performance over long distances, the generated defective-state light field was propagated outside the laboratory, achieving a 50 m transmission along the building hallway (see details in Supplementary Information S3.3). Figure 2b2 presents the character transmission results at various target distances d. It is evident that when d is larger, a broader spatial interval around the target position yields higher similarity, which is consistent with the previous theoretical analyses.

After confirming the feasibility and validity of the proposed method, a simulation-based statistical analysis was conducted to investigate the transmission properties of defective-state beams under more general conditions. Figure 2c presents simulations of the transmission behavior of defective HG modes with varying sparsity levels of randomly introduced defects, whose corresponding fundamental Gaussian mode has a beam waist of 3 mm, resulting in a Rayleigh length z_R of 53 m. The diagram also shows the evolution of image similarity for HG_3,3, HG_5,5, HG_7,7, and HG_9,9 modes under random defect sparsity levels of 1/4, 1/2, and 3/4, with two target positions, d₁ = 200 m and d₂ = 800 m, selected for evaluation, where SSIM = 1.0. In the SSIM calculation, each curve is obtained by taking the averaged value of 10 different random defective states. The purple arrow in the figure indicates that the curve for d (200 <d < 800 m) has a variation tendency from the one with d₁ = 200 m to d₂ = 800 m. Through comparative analysis, several key observations can be made. First, for all defective-mode orders, the image similarity region (defined as the interval where SSIM > 0.9) at d₂ = 800 m is significantly larger than that at d₁ = 200 m. Second, as the order of the HG mode increases, corresponding to a reduction in the size of individual defect spots, the similarity region tends to decrease accordingly. For instance, at d₁ = 200 m, the average similarity region narrows from 130 m for the HG_3,3 mode to 47 m for the HG_9,9 mode. Additionally, the similarity curves for various defect sparsity levels are found to be closely aligned across all HG modes, especially in the high-fidelity regions, indicating that defect sparsity has a relatively minor impact on the image similarity after transmission. To more intuitively illustrate how the similarity region changes with different target distances d, a numerical statistical analysis is performed, using the HG_3,3 mode with random defects at a sparsity of 1/2, as shown in Fig. 2d, where the evolution of the similarity region L with respect to d for Gaussian beam waist radii ω₀ = 1, 2, and 3 mm. It is evident that for all ω₀ values, the similarity region L increases more rapidly with larger d. A smaller beam waist radius corresponds to a shorter Rayleigh length, allowing the beam to reach the far field earlier. It is worth noting that although the similarity range L of defective modes varies significantly with different selected target positions d, it has little impact on practical information transmission, since the value of d can be accurately controlled according to the known distance between the transmitter and receiver. What’s more, the range L provides a margin of distance tolerance approximately equal to the value of d. Naturally, the larger the value of L, the greater the dynamic range for information transmission, or the more flexibility in supporting multiple receivers located at different distances can be achieved.

One application strategy

Spatial images have been used as identification codes, such as the widely used Quick Respond (QR) code. The investigation in this study makes the image of the QR code possible to transmit directly for identification in the area far away. Considering the large number of QR codes and the high information capacity contained within a single pattern, the HG mode, which has a quite similar structure to the QR code, is quite adapted for sub-structure modulations. According to the physical characteristics of HG_m,n mode, intensity segmented by phase singularity, it has (m + 1)(n + 1) bright spots on its light intensity distribution. As the HG_m,n mode is composed of the array of bright spots, and if we specially design the transmission function T (x, y), it’s able to modulate each single spot in the array. Local defects of the spots at particular positions can be generated by eliminating some bright spots in the mode, and then a large number of different patterns can be generated for coding.

By encoding an HG mode with its sub-spots in different defective states, it is theoretically possible to directly generate all the arrangement conditions in the mode pattern. Taking HG_m,n as an example, the mode has (m + 1)(n + 1) bright spots, if each bright spot is treated as a binary switch and converted into a digital 0 and 1, each bright spot can be regarded as a carrier of 1-bit of information, and the whole mode can carry data with (m + 1)(n + 1)-bit, corresponding to 2^(m+1)(n+1) status in total. It can be derived that as the order of the HG mode increases, the information carried by a mode increases exponentially.

Similar to the QR code, the distributions of the sub-spots need to be accurately located so that the patterns can be quickly and accurately recognized by the decoding process. For example, when there is only one single sub-spot left in the mode, it is easy to produce large errors when confirming its position. Therefore, to facilitate its positioning, combined with the principle of the coordinate axis, one column and one row of sub-spots need to be all reserved for auxiliary positioning. In the practical strategy, the number of available codes from HG_m,m mode is the same as the total number of defective states of HG_m-1,m-1 mode. Considering that HG_4,4 is upstream based on relatively low order and high information capacity, it is selected to take as an example. When the positioning row and column are anchored, exactly four light spots remain in each row, and two rows of these four sub-spots can form a byte, which is shown in Fig. 3a. Then, each of these defective mode states can carry two bytes of information, shown as in Fig. 3b. By the presented principle, one can easily obtain the capacity for coding with the HG_m,m mode, as shown in Fig. 3c. For example, the defective states of HG_7,7 mode can obtain the coding capacity of 49-bit, corresponding to a total defect mode number of 0.56 × 10¹⁵.

Fig. 3. The encoding principle, generation method and propagation property of the defective states of HG modes.

a The encoding principle by generating local defects of bright spots in an HG mode. b Examples of encoding with defective states of HG_4,4 mode. c Coding capacity with different HG_m,m modes. d The calculation process of the diffraction grating on the DMD for the generation of defective states. e Simulated and experimentally generated defective HG_4,4 mode.

In the previous section, a method for loading specific patterns onto structured modes by superposing varying distribution information was discussed. This approach enhances information transmission by embedding pattern data into the beams while preserving higher fidelity during transmission. However, it requires higher-order structured light, which demands higher system design and technical implementation in terms of generation and manipulation. To simplify the system design, the utilization of lower-order light beams with defect structures was investigated. Despite their lower orders, these structured light beams can still carry a large amount of information, thus offering the promise to enhance transmission efficiency through more simplified structures while maintaining information transmission functions. As discussed in the aforementioned QR-code-related section, the exponential increase in information-carrying capacity provides significant potential for using defective HG modes in information transmission.

To achieve the generation of defective HG beams, the simplified method shown in Fig. 3d was employed. Specifically, the HG_m,n mode was first occluded with a mask T to obtain the target light field E_tar, then performed an inverse Fourier transform calculation (matching the preset distance) to obtain the corresponding near-field of E_tar. The calculated holographic pattern by CGH was loaded onto a DMD, and the desired target field E_tar was produced in the +1^st-order diffraction of a vertically incident Gaussian beam. Figure 3e shows the simulated and experimental results of defective HG_3,3 and HG_4,4 modes, validating that HG modes generated by the method in Fig. 3d can generate defects at any local position.

More specifically, when using an HG_4,4 mode beam with a waist radius of ω₀ = 3 mm for transmission, selecting a location 20 times the Rayleigh range (~1 km) for information loading and occlusion results in a beam size of ω_z = 20 mm at that distance. At this range, the transverse distribution of the HG_4,4 beam still preserves the target defect mode structure, albeit with an enlarged beam size, which not only improves the recognizability of the pattern, but also enhances the stability of signal propagation. At this distance, occlusion and information loading operations ensure that the HG_4,4 beam retains its defect structure during subsequent long-distance transmission. When specific patterns are loaded onto higher-order HG modes, their distinctive defective modes remain stable over considerable distances, despite diffusion in free space. This stability offers significant advantages for transmitting complex information over long distances. Furthermore, utilizing lower-order HG beams reduces the complexity of generating and controlling high-order modes, enhancing the practicality and feasibility of defective structure-based beam transmission in real-world applications.

In the experiment, the diffraction gratings calculated by CGH and loaded on DMD can generate the desired structured beams. As illustrated in Fig. 4a, the experimental setup enables the direct generation of defective structured beams by modulating the holograms on the DMD. A Gaussian beam produced by a 532 nm solid-state laser, after collimated through a beam expander, is directly incident on the DMD surface and covers the hologram completely. The emerging light, carrying encoded information, is emitted at 24° relative to the DMD normal direction and enters the 4f system composed of lens 1, aperture and lens 2 in sequence. Within this system, lens 1 and the aperture filter the beam, selecting the +1^st diffraction order light field, and lens 2 collimates the output. After propagating for a certain distance, a focusing lens 3 is used to simulate the far-field propagation effect, and finally, the beam is concentrated onto the charge-coupled device (CCD) detector to capture the beam intensity image.

Fig. 4. Schematic of the experimental setup and real-time decoding results.

a The experimental setups and the whole procedure for the encoding and decoding strategies. b The image processing procedure of the measured beam pattern. c Real-time pattern recognition accuracy with 50 fps rate. d Measured defective HG_4,4 modes. e Measured defective HG_7,7 modes.

The digital image signals acquired by the CCD detector are transmitted to the host computer using the USB 3.0 interface. The real-time digital image signals are acquired by using the OpenCV-Python library in the host computer, and the parameters of the CCD are controlled by using the Software Development Kit of the corresponding CCD with Python. The main parameters of the CCD are the pixel format, acquisition frame rate and exposure degree: the pixel format is Mono8; the acquisition frame rate matches the frame rate of the transmitter; the appropriate exposure degree helps to reduce the effects of the rapid transformation of the light spot residual effects and improve the acquisition clarity. Compared to the machine learning method for complex structured light decoding, the recognition of HG mode with defects requires much simpler algorithms and is much faster. Decoding patterns with local defects can be achieved by an image processing method comprising several key steps, including pre-processing, Suzuki’s algorithm⁴⁴, circumscribed circles, and matrix scanning, as depicted in Fig. 4b. Before contour detection, images should first undergo some basic pre-processing to increase the accuracy of contour detection. This includes resizing the source images to a uniform scale, converting them to grayscale format, and applying binary thresholding. Morphological transformations, such as closing small holes within foreground objects and noise removal, are used to further refine the images. Suzuki’s algorithm is then employed for contour tracing, resulting in the detection of outer contours, as depicted in the third experimental image of Fig. 4b.

The real-time information transmission based on defective HG_4,4 and HG_7,7 modes was accomplished. In the experiment, the entire defective states of HG_4,4 mode (with a total number of 2¹⁶ = 65,536) and partial that of HG_7,7 mode (the amount is 2962, which were randomly generated) were used to encode the information, and the defective patterns collected by the CCD camera were decoded directly by the algorithm presented above in real-time. For the defective states of HG_4,4 mode, all the 65,536 states were divided into 60 groups based on the encoding sequence for decoding experiments, with each of the first 59 groups containing 1100 different states, and the last group containing the remaining 636 states. Compared to the defective states of HG_4,4 mode, HG_7,7 mode has a finer structure and higher recognition requirements, so it uses a conversion rate of 20 fps, while defective HG_4,4 mode has a faster recognition speed with a conversion rate of 50 fps. The recognition speed is now limited by the performance of our personal computer and the ordinary camera device, while it’s possible to improve the data transfer rate by improving the hardware. By recognizing the defective patterns with the method in the “Methods” section, each defective state is identified as a data string with 16-bit to HG_4,4 mode and 49-bit to HG_7,7 mode. The experimental results for defective states of HG_4,4 mode show an average recognition accuracy of 98.04% with a variance of 0.04%, as illustrated in Fig. 4c, indicating no significant difference in recognition accuracy among the defective states with different sparsities. The accuracy of recognition on defective states of HG_7,7 mode with the capacity of 2962 is 91.24%. Furthermore, the experimentally obtained defective patterns from HG_4,4 and HG_7,7 modes are shown in Fig. 4d and e, which correspond very well with the targeted patterns.

Discussion

By utilizing the single HG eigenmode to make spatial modulations, we obtain as many as 10¹⁵ different defective patterns for encoding (corresponding to 49 bits), which is tens of orders higher than the current method in structured-light-based shift keying FSO applications. What’s more, this still does not include all regimes in which the defective HG modes could be used. Compared to temporal QAM techniques, this method also exhibits a clear advantage in terms of mode capacity. Admittedly, current SLM and imaging devices still lag behind the temporal devices in modulation and response speed. However, with the continuous development in performance of these spatial optical components, the proposed technique will promisingly become more competitive. Moreover, to further enhance its communication capacity, two additional advanced strategies can be applied to this method. First, temporal modulations can be added to the defective HG modes to achieve the time division multiplexing along with shift keying. So that, the information transmission capacity can be greatly improved again. Second, one can also use the complementarity of the defective distribution of the HG modes to achieve two (or multi) beam space division multiplexing for higher capacity information transmission. To further explore the potential of this strategy, the recombination could be realized at the transmitter end using polarization or spectral combining techniques, while correspondingly, the separation of the two could also be realized in accordance with the chosen combining method. Additionally, it is possible to change the two (or multi) beam multiplexing states freely, which is very difficult to decipher as there are so many complementary combination conditions.

In this work, the defective state of structured light is proposed for high-capacity information transmission. The generation method and propagation property of high-order defective HG eigenmode are investigated to show its feasibility for spatial information direct transmission. Image information spatially modulated to the HG mode can be well maintained within long-distance propagation. Based on this principle, a method to greatly increase the encoding capacity of structured light for image-based decoding is also presented. Single HG eigenmode in different locally defective states is generated by computer-generated holograms for information encoding. With an HG_m,m eigenmode, we obtained defect patterns encoded with an information capacity of m²-bit. A simple image-processing-based decoding method achieved an up to 50 fps image recognition rate, with an accuracy of above 98%. What’s more, the defective HG beams can also have good performance under moderate turbulences (see Supplementary Information S4). This work may pave a useful way for structured-light-based optical information transmission, due to its good image transmission ability and extremely high information encoding capacity.

Methods

Mode generation

Structured light can be well generated by an SLM. DMD is one of the SLMs, and it can generate the perfect structure of a mode with very fast speed and spectrum robustness. DMD provides a two-dimensional binary grating for the transformations of a Gaussian mode to a target structured mode on the +1^st diffraction order. The two-dimensional binary grating is calculated according to the holography principle of the input and output optical fields. For a near field structured mode E_out from a nearly plane wave, the binary grating can be calculated as below^45–48,

$$T\left(x,y\right)=\frac{1}{2}+\frac{1}{2}\mathrm{sgn}\left(\cos \left[2\pi \frac{x}{x_0}+\pi p\left(x,y\right)\right]\right)-\cos \left[\pi \omega \left(x,y\right)\right]$$ 7

Here, it is easy to check that as $\omega \left(x,y\right)$ and $p\left(x,y\right)$ are slowly varying, this formula reproduces the pulse train described above. We can find the corresponding $\omega \left(x,y\right)$ and $p\left(x,y\right)$ functions for a general complex scalar field $A\left(x,y\right)e^{i\varphi \left(x,y\right)}$ according to the relations below,

$$\omega \left(x,y\right)=\frac{1}{\pi }\arcsin \left[A\left(\mathrm{x},\mathrm{y}\right)\right]$$ 8

$$p\left(x,y\right)=\frac{1}{\pi }\varphi \left(x,y\right)$$ 9

Local spots coordinate extraction methods

In the spot coordinate extraction, we applied two methods. First, the directional projection method can be used to identify the sub-spots in HG_m,n mode. The original grayscale image is converted into a binary image, and the pixel points are added and summed in the vertical and horizontal directions on the binary image to obtain two projection histograms. According to the vertical and horizontal projection histogram, the vertical peaks on both sides [Y₁, Y_m+1] and horizontal peaks on both sides [X₁, X_n+1] are obtained. The accuracy of the direction projection method is relatively high. However, due to the randomness in the shape and size of the actual laser spot, the intensity curve’s smoothness is poor, resulting in a large computational load and long computation time when searching for peak points in the intensity curve. After extracting the spot coordinates, the next step is defect spot localization. The simplest location method is the three-point localization. This method uses positioning rows and columns to transform the spots’ coordinates from the digital image coordinate system to the positioning coordinate system, determining whether a laser spot exists at the coordinates (x_i, y_j). Through spot coordinate extraction, we obtain the coordinates used in three-point localization (X₁, Y₁), (X₁, Y_m+1), (X_n+1, Y_m+1), the coordinate mark of other possible laser spots in the HG_m,n mode as (x_i, y_j), which is shown in Fig. 5a, and the coordinates are calculated as followed,

$$\left\{\begin{array}{c}\hfill x_i=X_1+\left(\frac{X_{n+1}-X_1}{n}\right)\times \left(i-1\right),1\le i\le n+1\hfill \\ \hfill y_j=Y_1+\left(\frac{Y_{\mathrm{m}+1}-Y_1}{m}\right)\times \left(j-1\right),1\le j\le m+1\hfill \end{array}\right)$$ 10

Fig. 5. Local spot coordinate extraction methods.

a Directional projection method to the spot coordinate extraction. The coordinates of blue circles are determined by the intensity peaks, while the coordinates of white circles are calculated from the blue ones. b Coordinate comparative method to the spot coordinate extraction. The coordinates of red circles are given from the standard HG mode’s spots center location (marked with light blue grid), while whether there is a circle is determined by the extraction of circles in (a).

Given the slow extraction of coordinates by the directional projection method, the spot recognition method is used to replace it, as shown in Fig. 5b. Following pre-processing preparations, Suzuki’s algorithm is employed to trace and detect the outer contours. After identifying the contour of each spot in defective HG_m,n mode, the center point of each contour is determined by using the outer circle obtained by contour fitting. With the vertical coordinate Y=[Y₁, Y₂, Y₃,…, Y_m+1] with the smallest x value as positioning column, and horizontal coordinate X=[X₁, X₂, X₃,…, X_n+1] with the smallest y value as positioning row, the spots of HG_m,n can be marked as K=[(x₁, y₁), (x₂, y₂),…, (x_k, y_k)](1≤k≤mn).

Also, in order to speed up the defect spot localization process, a coordinate comparative method is introduced to distinguish different spots. To calibrate the coordinate points in array K. We select the coordinate value in array X that is the closest to coordinate $x_k$ as the value of $x_i$. Similarly, select the coordinate value in array Y that is the closest to coordinate $y_k$ as the value of $y_j$. In the coordinate comparative method, as shown in Fig. 5b, the index mapping formula for coordinates (x_i, y_j) is as follows,

$$\left\{\begin{array}{c}i=index\left(min\left(\left[X_1-x_k,X_2-x_k,X_3-x_k,\cdots ,X_{n+1}-x_k\right]\right)\right)\hfill \\ j=index\left(min\left(\left[Y_1-y_k,Y_2-y_k,Y_3-y_k,\cdots ,Y_{m+1}-y_k\right]\right)\right)\hfill \end{array}.\right)$$ 11

Therefore, several scattered points in K are all calibrated. The comparison method, relative to the triangulation method, requires less computational effort. It makes more efficient use of the available spot position information, effectively enhancing recognition speed and accuracy. This approach performs well in specific spot recognition tasks. Once the existing spots and the coordinate axes are successfully recognized, the missing spots can be easily localized. By reading the locations of the missing spots, the information on the defective mode can be decoded.

Author contributions

Z.Z. conceived the idea and basic theory. Y.W. performed the theoretical calculations and primary simulations. L.Z. performed the decoding process. Z.Z., Y.W., and L.Z. carried out the experiments and analyzed the results. H.Y., S.Z., and W.H. helped with the CGH process. X.P. and Y.M. helped with the equations’ derivation. Z.Z. and Y.W. wrote the primary manuscript, assisted by L.Z., X.P., and L.K., L.X., and C.Z. revised the manuscript. All authors contributed to the final version of the manuscript. Z.Z. supervised this project.

Peer review

Peer review information

Nature Communications thanks the anonymous reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Data availability

All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Information. Source data are provided with this paper. The source data underlying Figs. 2c,d and 4c and Supplementary Fig. S10f, g are provided as Source data file. Source data are provided with this paper.

Code availability

The codes for the current study are available from the corresponding author on request.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-025-63100-2.