Laser blindness can reduce or disable the information acquisition ability of photoelectric imaging systems. In this paper, numerical simulation and experimental verification are both performed to systematically study the laser protection performance of the vortex phase mask. First, the imaging model and laser transmission model of the vortex wavefront coding imaging system are introduced in detail. Then, the experimental setup of the imaging system is built, and the imaging result of the imaging system is obtained. Finally, the influence of propagation distance on the maximum single-pixel receiving power and suppression ratio of the imaging system is measured experimentally. The simulation and experimental results both show that the energy suppression ratio of this method can reach more than two orders of magnitude compared with the conventional imaging system, and the probability of laser blindness can be effectively reduced.

## I. INTRODUCTION

A photoelectric imaging system is one of the most important tools to obtain information, which is widely used in entertainment, medical, and security fields. The rapid development and widespread application of laser technology^{1,2} and the high gain characteristics of conventional imaging (CI) systems have resulted in a sudden increase in the probability of blindness for the photoelectric imaging system, that is, the laser will cause permanent damage to the detector in the photoelectric system, making it lose the sensing or imaging capability.^{3,4} Thus, the study of laser blinding protection in the photoelectric imaging system is particularly important. There are two feasible technical ways to prevent laser blindness in the photoelectric imaging system: one is to use laser protective materials and the other is to develop new photoelectric imaging systems. Laser protective material technology utilizes linear materials,^{5,6} nonlinear materials,^{7–10} and phase change materials^{11–14} to make light filters for the protection of photoelectric imaging devices, which is a relatively common laser protection method; however, so far, there are still disadvantages, such as narrow protective bandwidth, slow reaction speed, and low protective strength.^{15} At the same time, the increasing power and wavelength broadening of lasers^{16–18} make laser protection of photoelectric imaging systems more difficult. Researchers also study new technologies, such as computational imaging, including light field imaging, wavefront coding imaging (WFCI), etc., which is expected to make up for the lack of laser protection materials and improve the laser protection ability of photoelectric imaging systems.

Typical techniques of computational imaging,^{19} light-field imaging,^{20,21} and wavefront coding^{22,23} have been innovatively applied to the field of laser blinding protection, and the results show that it can redistribute the spot energy on the imaging plane. Simulations show that light-field imaging can reduce the maximum single-pixel receiving power on the imaging plane by more than 78% and, thus, effectively improve the laser blinding threshold by nearly one order of magnitude.^{24} Further experimental studies show that the energy suppression ratio of light-field imaging under optimized parameters can reach two orders of magnitude.^{25} However, light-field imaging has the disadvantage of reducing the spatial resolution of the imaging system. Wavefront coding can be a good solution to this problem when applied to laser blinding protection. Simulation studies prove that wavefront coding imaging based on a vortex phase mask can effectively reduce the peak light intensity on the imaging plane by two orders of magnitude^{26} and, at the same time, restore the incoherent scene to achieve high-quality imaging,^{27} which is an ideal laser blinding protection method. This method shows the advantages of compact system structure, mature optical technology, and no need for the brightness, wavelength, or polarization of the laser. Further research experimentally verified the blinding protection effect and imaging capacity of the vortex, axicon, and cubic phase mask by using a spatial light modulator.^{28} Experimental studies show that the cubic phase mask can effectively improve the laser damage threshold of the imaging system.^{29} Phase mask is the key component of the wavefront coding imaging system, which determines the focal depth extension, imaging quality, and laser protection performance of the imaging system. At present, the research on the laser blinding protection capability of the vortex phase mask has the problems of restricted scenarios and insufficiently scientific evaluation indices and is still limited to plane waves, with no relevant research carried out for Gaussian beams, and consideration of more comprehensive scenarios and reasonable evaluation indices can more thoroughly help to understand the laser protection performance of the vortex phase mask.

Therefore, in this paper, the imaging model and laser transmission model of the vortex wavefront coding imaging system are established, and the imaging and laser protection capabilities of the imaging system are simulated and investigated. A more reasonable evaluation index for blinding protection is proposed—the maximum single-pixel receiving power, i.e., the maximum power that can be received by a single pixel on the imaging plane. Furthermore, the imaging and laser protection capabilities of the imaging system are experimentally verified, and the influence of the array structure of the image sensor on the protection performance is analyzed.

## II. THEORETICAL MODEL

### A. Imaging model

Figure 1 shows the imaging model of the vortex wavefront coding imaging system. After being modulated by the vortex phase mask and focused by the imaging lens, the object reaches the image sensor and is converted into a digital signal. The imaging lens is an ideal thin lens, and the principal plane and pupil plane of the imaging system coincide with the imaging lens.

*I*

_{i}is the intensity image of the imaging plane, $xi,yi$ is the spatial coordinates of the imaging plane,

*h*represents the point spread function, that is, the complex amplitude distribution on the imaging plane after the unit pulse on the object passes through the diffractive limited system,

*I*

_{g}represents the intensity image of the geometric optical ideal image, and $hxi,yi2$ is the intensity point spread function. Equation (1) indicates that the imaging plane intensity distribution of the linear spatially invariant imaging system under incoherent illumination is the convolution of the intensity distribution of the ideal image and the intensity point spread function. The frequency-domain form corresponding to Eq. (1) can be obtained by Fourier transform,

*P*is the pupil function of the imaging system,

*λ*is the wavelength, and

*d*

_{i}is the image distance. The circular pupil function of the conventional imaging system is

*D*is the diameter of the imaging lens. In essence, the wavefront coding imaging system is a kind of aberration system. When calculating the generalized pupil function of the wavefront coding system, the optical path difference $Wx,y$ introduced by the phase mask should be considered, and the corresponding phase difference is expressed as $kWx,y$. The phase difference introduced by the vortex phase mask can be expressed as

*l*is the topological charge. Accordingly, the generalized pupil function of wavefront coding imaging system can be expressed as

*K*is a tentative parameter, and $Gifx,fy$ is the normalized spectrum of the coded image $Iix,y$. The decoded image can be obtained using Wiener filtering, written as

### B. Laser transmission model

To study the blinding protection performance of the vortex wavefront coding imaging system, the laser transmission model of the wavefront coding imaging system is established, as shown in Fig. 2, where VPM represents the vortex phase mask and L represents the lens. The model considers that the phase mask and the imaging lens are closely combined, and the two are regarded as an equivalent phase plane. The model consists of three main planes, namely, the Gaussian beam waist plane, the equivalent phase plane of the vortex phase mask and the imaging lens, and the imaging plane. The three planes are defined as plane 0, plane 1, and plane 2 in turn, and the complex amplitudes at the planes are marked with subscripts 0, 1, and 2, respectively. The Gaussian beam with a distance of *z*_{g} between the beam waist plane and the equivalent phase plane is incident on the equivalent phase plane, and after being modulated by the phase mask and focused by the lens, it diffracts to the imaging plane with a distance of *d*_{i} and forms a focused spot. The ideal image distance *d*_{i} follows the Gaussian imaging formula 1/*z*_{g} + 1/*d*_{i} = 1/*f*, where *z*_{g} is the object distance, *d*_{i} is the ideal image distance, and *f* is the focal length of the imaging lens.

^{30}the complex amplitude distribution of the Gaussian beam in front of the equivalent phase plane is written as follows:

*A*

_{0}is the power related constant,

*ω*

_{0}is the beam waist size,

*k*is the wave number,

*λ*is the wavelength, $\omega zg$ is the spot radius of the Gaussian beam on the equivalent phase plane, and $Rzg$ is the curvature radius of the Gaussian beam on the equivalent phase plane,

*D*is the lens diameter,

*f*is the lens focal length, and

*l*is the topological charge. The complex amplitude transmittance function of the equivalent phase plane can be obtained by multiplying Eqs. (13) and (14),

^{31,32}the propagation of the light field from the back surface of the equivalent phase plane to the imaging plane satisfies Fresnel diffraction, and the complex amplitude distribution on the imaging plane is calculated as follows:

*d*

_{i}is the image distance. By multiplying the complex amplitude by its conjugate, the profile and intensity distributions of the spot on the imaging plane are calculated as

The intensity distribution is then used to calculate the quantitative index of protection performance: the maximum single-pixel receiving power. The specific calculation method of the maximum single-pixel receiving power is to find the position of peak light intensity on the imaging plane, take this position as the center, and integrate the focused spot power within the equivalent area of a single pixel. The evaluation criteria are as follows: for the same interference laser parameters, the smaller the maximum single-pixel receiving power, the better the blinding protection performance.

## III. EXPERIMENTAL SETUP

The experimental device of the vortex wavefront coding imaging system mainly includes vortex phase mask VPM, lens L, and CMOS black-and-white camera C. The experimental device is shown in Fig. 3. The camera (CS165CU/M, Thorlabs Inc.) and the imaging lens (No. 59-873, Edmound Optics) are fixed together through the C mount, the vortex phase mask is fixed on the five-axis optical adjustment frame (AMM5-1AC, LBTEK), and the frame is connected to the camera through the 30 mm coaxial system. Two pinhole apertures are placed in the optical path to realize the collimation between the Gaussian beam and the imaging system by observing the reflected spot. The CMOS sensor is 1/2.9 in., 1440 × 1080 pixels (1.6 megapixels), and each pixel is a 3.45 *µ*m square. The Gaussian beam is output by a laser with power-stabilized rms < 0.5%, attenuated by an attenuator, modulated by a vortex phase mask, and focused by an imaging lens before reaching a CMOS black-and-white sensor.

## IV. RESULTS AND DISCUSSION

### A. Imaging performance

In this section, the imaging effect of conventional and vortex wavefront coding imaging systems is compared by numerical simulation and experimental verification. The specific parameters of the imaging system in numerical simulation are listed in Table I.

Parameter . | Value . |
---|---|

Topological charge | 16 |

Imaging lens diameter | 20 mm |

Imaging lens focal length | 50 mm |

Object distance | 2000 m |

Image distance | 50.0 mm |

Imaging plane length | 6.65 mm |

Tentative parameter | 9 × 10^{−3} |

Parameter . | Value . |
---|---|

Topological charge | 16 |

Imaging lens diameter | 20 mm |

Imaging lens focal length | 50 mm |

Object distance | 2000 m |

Image distance | 50.0 mm |

Imaging plane length | 6.65 mm |

Tentative parameter | 9 × 10^{−3} |

Figure 4 shows the simulation imaging results of conventional and vortex wavefront coding imaging systems, wherein the leftmost panel shows the imaging result of the conventional imaging system, and the middle and rightmost panels show the encoded image and decoded image of the wavefront coding imaging system, respectively. The results show that the image of the conventional system is clearest. In contrast, the encoded image is blurred due to the modulation of the vortex phase mask. By selecting the tentative parameter *K* = 9 × 10^{−3}, the decoded image is close to the imaging result of the conventional system in the focusing state. Therefore, the vortex wavefront coding imaging system obtains excellent imaging effects through the optimization of hardware and software. There is a slight ringing effect in the decoded image, resulting in a slight degradation of the decoded image.

Figure 5 shows the experimental imaging results of conventional and vortex wavefront coding imaging systems, wherein the leftmost panel shows the imaging results of the conventional imaging system, and the middle and rightmost panels show the encoded and decoded images of the wavefront coding imaging system, respectively. It can be observed that the experimental results are consistent with the simulation results, verifying that the vortex wavefront coding imaging system can obtain a clear image by decoding operation.

### B. Laser protection performance

In this section, the laser protection performance of the imaging system at different propagation distances is calculated by numerical simulation and verified experimentally. The parameters of the laser, optical system, and image sensor are listed in Table II.

Parameter . | Value . |
---|---|

Incident laser power | 11.65 mW |

Incident laser wavelength | 532 nm |

Gaussian beam waist size | 0.46 mm |

Topological charge | 16 |

Imaging lens focal length | 50 mm |

Imaging lens size | Φ20 mm |

Imaging plane detector pixel size | 3.45 × 3.45 µm^{2} |

Parameter . | Value . |
---|---|

Incident laser power | 11.65 mW |

Incident laser wavelength | 532 nm |

Gaussian beam waist size | 0.46 mm |

Topological charge | 16 |

Imaging lens focal length | 50 mm |

Imaging lens size | Φ20 mm |

Imaging plane detector pixel size | 3.45 × 3.45 µm^{2} |

Figure 6 shows the intensity distribution of the focused spot on the imaging plane of the conventional imaging system when the propagation distances are 5, 10, 15, and 20 m. The spot size on the imaging plane decreases with the increase in the propagation distance, resulting in an increase in the maximum single-pixel receiving power.

Figure 7 shows the intensity distribution of the focused spot on the imaging plane of the vortex wavefront coding imaging system when the propagation distances are 5, 10, 15, and 20 m. The phase modulation of the phase mask causes the light spot on the imaging plane to be distributed in a symmetrical ring, and the redistribution of the light spot energy can effectively reduce the peak light intensity reaching the detector, thus reducing the maximum single-pixel receiving power.

As given in Table III, when the propagation distances are 5, 10, 15, and 20 m, the experimental maximum single-pixel receiving power of the conventional imaging system is 6.79 × 10^{−7}, 18.68 × 10^{−7}, 13.15 × 10^{−7}, and 21.45 × 10^{−7} *µ*W, respectively. The experimental maximum single-pixel receiving power of the vortex wavefront coding imaging system is 2.84 × 10^{−9}, 10.16 × 10^{−9}, 25.18 × 10^{−9}, and 46.77 × 10^{−9} *µ*W, respectively. The experimental results show that the vortex wavefront coding imaging can significantly reduce the maximum single-pixel receiving power and is capable of achieving laser protection. There is a big difference between the experimental and simulated maximum single-pixel receiving power. The main reasons are the difference between the actual and nominal transmittance of the attenuator, the tolerance of the attenuator, and the relative position of the focused spot and the image sensor.

Propagation distance (m) | 5 | 10 | 15 | 20 |

Simulated MSPRP of CI (×10^{−6} µW) | 3.28 | 8.51 | 10.98 | 11.15 |

Experimental MSPRP of CI (×10^{−7} µW) | 6.79 | 18.68 | 13.15 | 21.45 |

Confidence interval (×10^{−7} µW) | [4.20, 51.52] | [5.81, 130.92] | [5.99, 166.44] | [5.91, 168.11] |

Simulated MSPRP of WFCI (×10^{−9} µW) | 9.34 | 33.88 | 73.46 | 136.29 |

Experimental MSPRP of WFCI (×10^{−9} µW) | 2.84 | 10.16 | 25.18 | 46.77 |

Confidence interval (×10^{−9} µW) | [1.65, 13.35] | [6.23, 49.76] | [13.7, 109.85] | [24.8, 205.67] |

Propagation distance (m) | 5 | 10 | 15 | 20 |

Simulated MSPRP of CI (×10^{−6} µW) | 3.28 | 8.51 | 10.98 | 11.15 |

Experimental MSPRP of CI (×10^{−7} µW) | 6.79 | 18.68 | 13.15 | 21.45 |

Confidence interval (×10^{−7} µW) | [4.20, 51.52] | [5.81, 130.92] | [5.99, 166.44] | [5.91, 168.11] |

Simulated MSPRP of WFCI (×10^{−9} µW) | 9.34 | 33.88 | 73.46 | 136.29 |

Experimental MSPRP of WFCI (×10^{−9} µW) | 2.84 | 10.16 | 25.18 | 46.77 |

Confidence interval (×10^{−9} µW) | [1.65, 13.35] | [6.23, 49.76] | [13.7, 109.85] | [24.8, 205.67] |

^{a}

MSPRP stands for the maximum single-pixel receiving power.

We first discuss the attenuator tolerance. Laser power is attenuated using an absorbing neutral density filter kit, model NEK01 (Thorlabs Inc.) in the experiment. NE20A, NE30A, and NE40A filters with optical density (OD) of 2, 3, and 4 were selected to form a combined attenuation filter with OD 9. The optical density, nominal transmittance, actual transmittance, and optical density tolerance at 532 nm of NE20A, NE30A, and NE40A filters are given in Table IV. The nominal transmittance of the filter is used in the simulation, and the transmittance of the filter in the experiment is the actual transmittance at 532 nm. The actual transmittance should consider both the fact that transmittance is wavelength-dependent and optical-density-tolerant.

Filter . | NE20A . | NE30A . | NE40A . |
---|---|---|---|

Optical density | 2 | 3 | 4 |

Nominal transmittance | 1 × 10^{−2} | 1 × 10^{−3} | 1 × 10^{−4} |

Actual transmittance @532 nm | 9.621 × 10^{−3} | 7.651 × 10^{−4} | 7.251 × 10^{−5} |

Optical density tolerance | ±0.10 | ±0.15 | ±0.20 |

Filter . | NE20A . | NE30A . | NE40A . |
---|---|---|---|

Optical density | 2 | 3 | 4 |

Nominal transmittance | 1 × 10^{−2} | 1 × 10^{−3} | 1 × 10^{−4} |

Actual transmittance @532 nm | 9.621 × 10^{−3} | 7.651 × 10^{−4} | 7.251 × 10^{−5} |

Optical density tolerance | ±0.10 | ±0.15 | ±0.20 |

We next analyze the influence of the relative position of the focused spot and the image sensor. We calculate the radius, inner diameter, outer diameter, and width of the simulated spots and set the boundary value as 1/e^{2} of the peak light intensity. The results are given in Table V.

Propagation distance (m) | 5 | 10 | 15 | 20 |

Spot radius (μm) | 4.7 | 2.4 | 1.7 | 1.5 |

Spot inner diameter (μm) | 20.5 | 10.4 | 7.6 | 7.2 |

Spot outer diameter (μm) | 83.7 | 45.8 | 30.4 | 21.7 |

Spot width (μm) | 63.2 | 35.4 | 22.8 | 14.5 |

Propagation distance (m) | 5 | 10 | 15 | 20 |

Spot radius (μm) | 4.7 | 2.4 | 1.7 | 1.5 |

Spot inner diameter (μm) | 20.5 | 10.4 | 7.6 | 7.2 |

Spot outer diameter (μm) | 83.7 | 45.8 | 30.4 | 21.7 |

Spot width (μm) | 63.2 | 35.4 | 22.8 | 14.5 |

The pixel size of the image sensor in the imaging system is 3.45 × 3.45 *µ*m^{2}. The results show that when the propagation distances are 5, 10, 15, and 20 m, and the radius of the focused spot of the conventional imaging system is less than two times the pixel width, and even less than the pixel width. Therefore, the relative position of the focused spot and image sensor will affect the maximum single-pixel receiving power measured experimentally. Figure 8 takes the conventional imaging circular spot as an example to show the relative position relationship between the focused spot and the image sensor under three typical circumstances, in which the redder area in the spot center indicates stronger light intensity. The results show that the change in the relative position relationship will lead to a change in the maximum single-pixel receiving power. The confidence intervals of maximum single-pixel receiving power for different relative positions are calculated through simulation, as shown in Fig. 9. It can be observed that the experiential results are all located inside the confidence intervals of the simulation.

The ratio of the maximum single-pixel receiving power between conventional and vortex wavefront coding imaging systems is defined as the suppression ratio, which is used to characterize the laser protection performance of the imaging system. In different application scenarios, the propagation distance is different, and the properties of the Gaussian beam are different resulting in the variation of the suppression ratio. As given in Table VI, when the propagation distances are 5, 10, 15, and 20 m, the experimental suppression ratios of the imaging system are 239.1, 183.9, 52.5, and 45.9 respectively, while the simulation suppression ratios are 351.8, 251.1, 149.4, and 81.8, respectively. Considering the change of the relative position relationship between the focused spot and the image sensor, the confidence intervals of the suppression ratio were obtained as [235.3, 370.6], [87.4, 249.4], [40.8, 144.4], and [21.5, 80.4], respectively, and the experimental suppression ratios are all inside the confidence interval.

Propagation distance (m) | 5 | 10 | 15 | 20 |

Simulated suppression ratio | 351.8 | 251.1 | 149.4 | 81.8 |

Experimental suppression ratio | 239.1 | 183.9 | 52.2 | 45.9 |

Confidence interval | [235.3, 370.6] | [87.4, 249.4] | [40.8, 144.4] | [21.5, 80.4] |

Propagation distance (m) | 5 | 10 | 15 | 20 |

Simulated suppression ratio | 351.8 | 251.1 | 149.4 | 81.8 |

Experimental suppression ratio | 239.1 | 183.9 | 52.2 | 45.9 |

Confidence interval | [235.3, 370.6] | [87.4, 249.4] | [40.8, 144.4] | [21.5, 80.4] |

Finally, we simulate the maximum single-pixel receiving power within a much larger range of propagation distance for both the conventional and vortex wavefront coding imaging systems and calculate the suppression ratio, as shown in Fig. 10. It can be observed that the maximum single-pixel receiving power of the vortex wavefront coding imaging system (blue dotted line) is always lower than that of the conventional system (blue solid line), indicating that the vortex wavefront coding imaging system possesses laser protection capability regardless of the laser propagation distance. Then, for quantitative analysis, Fig. 10 plots the suppression ratio vs propagation distance (red solid line). The laser protection performance of the vortex wavefront coding imaging system is relatively stronger when the laser source is close and approaches maximum at the propagation distance of 4 m. Specifically, the suppression ratio first grows and forms a peak with a propagation distance increasing to 4 m and then drastically decreases as the propagation distance increases to 100 m. With further increasing the propagation distance to 10 000 m, the suppression ratio reduces slowly and finally maintains stability in the vicinity of 25. Therefore, although the vortex wavefront coding imaging system has laser protection capability in a wide range of the propagation distance, it has an optimum working distance of around 4 m.

## V. CONCLUSION

This paper presents a method of laser blinding protection for the photoelectric imaging system based on a vortex phase mask. First, the imaging model and laser transmission model of the imaging system are established, and then, the imaging results of the conventional and vortex wavefront coding imaging systems are obtained by building an experimental device of the imaging system. Then, the effects of different propagation distances on the maximum single-pixel receiving power and energy suppression ratio of the vortex wavefront coding imaging system are measured experimentally. When the propagation distances are 5, 10, 15, and 20 m, the energy suppression ratio of the imaging system reaches 239.1, 183.9, 52.2, and 45.9, respectively. The numerical difference between the experimental and simulation energy suppression ratios is mainly caused by the relative position of the focused spot and image sensor. The confidence intervals of the suppression ratios are calculated to be [235.3, 370.6], [87.4, 249.4], [40.8, 144.4], and [21.5, 80.4] for propagation distances of 5, 10, 15, and 20 m, respectively, and the experimentally measured suppression ratios are all within the confidence intervals. The results demonstrate that vortex wavefront coding can achieve the function of laser blinding protection while ensuring clear imaging, which shows important potential for applications in security and other fields.

## ACKNOWLEDGMENTS

Yangliang Li thanked the Technology Domain Fund of the Basic Strengthening Plan for supporting this work.

This work was supported by the Technology Domain Fund of Basic Strengthening Plan (Grant Nos. 2021-JCJQ-JJ-0284 and 2022-JCJQ-JJ-0237), the CAST Creative Foundation (Grant No. 1020J20210117), the Research Project of National University of Defense Technology (Grant No. ZK2041), the Advanced Laser Technology Laboratory Foundation of Anhui Province (Grant Nos. AHL2021QN03 and AHL2022ZR03), and the Postdoctoral Fellowship Program of China Postdoctoral Science Foundation (Grant No. GZC20233531).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

Y.L. and H.L. contributed to this work equally.

**Yangliang Li**: Investigation (lead); Methodology (equal); Software (lead); Validation (equal); Writing – review & editing (equal). **Haoqi Luo**: Methodology (equal); Validation (equal); Writing – original draft (equal). **Qing Ye**: Conceptualization (equal); Data curation (equal); Funding acquisition (equal); Supervision (equal); Visualization (equal). **Yunlong Wu**: Data curation (lead); Funding acquisition (equal); Visualization (equal). **Junyu Zhang**: Formal analysis (equal); Software (equal); Validation (equal). **Dake Chen**: Resources (equal). **Xiaoquan Sun**: Supervision (lead).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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