Photoelectric imaging systems, composed of an optical imaging lens, photodetector (sensor), and signal processing part, serve as the fundamental basis for the advancement of modern optics and image technology. The primary lens focuses point-source incident light onto the image plane sensor and forms a tiny Airy spot with a huge optical gain when satisfying the Gauss image formula, thus obtaining high-quality images [1].

However, excessive local power can lead to point damage, line damage, and surface damage of a photodetector with permanent incapacity [2, 3]. On the other hand, with the rapid development of laser technology, compact lasers with high power density and directivity have been widely used in scientific research and even daily life [4–6], and the probability of sensor damage for photoelectric imaging systems has significantly increased.

Therefore, the demand for laser protection methods for imaging systems is urgent. Conventional laser protection employs amplitude limitation in the frequency and time domain by using frequency-dependent optical filters, which are typically fabricated using linear materials [7], nonlinear materials [8–10] and phase change materials [11–14] possessing self-tunable transmittance accompanied by the interference laser power. With the improvement of laser power and the wide application of supercontinuum lasers and femtosecond lasers, the amplitude limitation method has disadvantages including low suppression ratio, narrow bandwidth, and slow response.

Computational imaging adds functionalities to the imaging system by modifying the point spread function (PSF) and cooperating with algorithms to achieve intriguing imaging effects such as increasing depth of field [15], improving dynamic range [16, 17], seeing through obstructions [18], and recording spatial depth information [19]. Recently, computational imaging has also been innovatively extended to the field of laser protection [20–25], showcasing great tolerance to the intensity, wavelengths, and polarization of interference lasers. Light field imaging can raise the laser damage threshold of a sensor by two orders of magnitude, and effectively reduce the risk in a complex laser environment [20, 21]. However, it sacrifices spatial resolution and significantly deteriorates image quality. Wavefront coding (WFC) imaging is another computational imaging technology for which phase-mask engineering can build special PSFs to unlock new functionalities for imaging systems [15–18]. Cubic, spiral and axicon phase masks (CPM, SPM, and APM) can achieve laser protection at a similar level with light field imaging, while restoring higher-quality incoherent images by using the knowledge of the PSF.

These WFC imaging systems are considered ideal platforms to realize anti-laser imaging systems [22–24]. The current research, however, is mostly limited to scenarios of infinite interference, where the laser beam is regarded as a plane wave. Actually, the laser source can be positioned at a specific finite distance from the imaging system and may even be focused. Therefore, it is imperative to explore the influence of laser propagation distance on both laser damage risk and the laser protection performance of the WFC imaging system. Furthermore, our previous experiment and simulation results have demonstrated that the laser protection performance of a CPM is weak for a close-range laser source [25]. Therefore, it is crucial to develop a series of phase masks with full-range laser protection capability.

In this paper, we propose a spiral axicon phase mask (SAPM) WFC imaging system to achieve high-performance laser protection at almost arbitrary laser interference distances. Firstly, a laser transmission model is established for the WFC system, and the field distributions of the interference laser on the image plane are calculated for both the conventional and WFC systems. Then, the influence of laser propagation distance on suppression ratios of maximum intensity and single-pixel power is revealed for WFC imaging systems. The SAPM modulates the minuscule laser spot into a rotational symmetric ring featuring a tunable radius and full-width half-maximum (FHWM). This design shows robustness and remarkable laser protection performance against interference laser irradiation at almost arbitrary laser propagation distances. In addition, we investigate the sensor damage risk versus interference distance and observe the existence of a critical sensor hazard distance where the maximum intensity or single-pixel power on sensor approach peaks, regardless of the conventional or WFC systems. Finally, imaging simulation is performed for the WFC systems to verify the imaging capacity.

Figure 1 shows a schematic diagram of a WFC imaging system and laser transmission model. The WFC system comprises a phase mask, primary lens, and detector, wherein the phase mask and primary lens with focal length f and diameter d are combined and regarded as an equivalent pupil plane. We exclusively consider the phase modulation effect of the phase mask and ignore the structural details. The experimental realization of the phase mask can be achieved with various methods, including a phase-only liquid crystal spatial light modulator, phase plate, diffractive optical element and metasurface. The laser transmission model includes three planes, namely the waist plane of a Gaussian beam, the equivalent pupil plane, and the image plane on the sensor, which are defined as plane 0, plane 1, and plane 2 in turn, and the complex amplitude distributions at the planes are marked with subscripts 0, 1 and 2, respectively. Specifically, a Gaussian beam with the propagation distance z_g between the beam waist plane and the equivalent pupil plane is generated from the laser source, modulated by the phase mask and focused by the primary lens, then diffracted to the image plane sensor at distance d_i behind them to form a facula.

Figure 1. Schematic diagram of wavefront coding (WFC) imaging system (upper) and laser transmission model (lower).

A Gaussian beam with waist ω₀ and propagation distance z_g propagates to the equivalent pupil plane, and the complex amplitude distribution in front of plane 1 can be written by a Gaussian beam propagation equation as:

where (x, y) are the spatial coordinates of the equivalent pupil plane, A₀ is the constant related to laser power, k is the wave vector, $\omega \left(z_g\right)={\omega }_0\sqrt{1+{\left(2z_g/k{\omega }_0^2\right)}^2}$ is the spot radius of the Gaussian beam, and $R\left(z_g\right)=z_g\left[1+{\left(k{\text{ω}}_0^2/2z_g\right)}^2\right]$ is the curvature radius of the equiphasic surface. The light field distribution behind the pupil can be given as

where T_M is the transmittance of the phase mask and $T_L=\text{circ}\left(\frac{\sqrt{x_1^2+y_1^2}}{D/2}\right)\exp \left[-i\frac{k}{2f}\left(x_1^2+y_1^2\right)\right]$ for the primary lens. According to the Fresnel diffraction approximation, the complex amplitude distribution of diffracted light at the image plane sensor is obtained as

where (x₂, y₂) is the spatial coordinate of the image plane. The intensity profile of the modulated laser facula is thus expressed as $I_2\left(x_2,y_2\right)=U_2\left(x_2,y_2\right)U_2^{\ast }\left(x_2,y_2\right)$. In the following, the laser protection performance of the WFC imaging system is quantitatively analyzed by calculating and comparing intensity profiles of the laser facula with and without the modulation of the phase mask.

3.1. Distributions of the Interference Laser Facula

The phase mask is the core optical element of the WFC imaging system and determines the spatial distribution of the interference laser facula on image plane sensor. Different facula shapes result in different degrees of energy diffusion, which thus affects the laser protection performance. The focus of this paper is primarily on SAPM, with an emphasis on its laser protection characteristics in relation to laser propagation distance through a comparison with CPM and SPM (SPM is a special case of the SAPM, and the other case of APM is given in Section 3.4 to simplify the discussion). The phase modulation functions φ of the phase mask are listed in Table 1, where $x_{\text{norm}}=2x_1/D$ and $y_{\text{norm}}=2y_1/D$ represent the normalized spatial coordinates of the equivalent pupil plane, with values ranging from −1 to 1. Parameter α is the phase modulation coefficient and l is for the topological charge. The transmittance of the phase mask is obtained as $T_M=\exp \left(i\varphi \right)$.

TABLE 1. Phase modulation functions of phase mask.

Category	Phase Function
CPM	αxnorm3+ynorm3
SPM	larctanynormxnorm, xnorm≠0
SAPM	αxnorm2+ynorm2+larctanynormxnorm, xnorm≠0

The parameters of the imaging system used in the numerical simulation are shown in Table 2.

TABLE 2. Parameters of imaging system.

Parameter	Value
Laser Power (W)	10
Laser Wavelength (nm)	532
Gaussian Beam Waist Size (mm)	2.5
Phase Modulation Coefficient (rad)	200
Topological Charge	10
Imaging Lens Focal Length (mm)	100
Imaging Lens Size (mm)	50
Image Plane Detector Pixel Size (μm)	4.5 × 4.5

Figure 2(a) depicts the light intensity distribution on image plane sensors of conventional, CPM, SPM, and SAPM imaging systems when the propagation distance of the interference laser is 1 m, and Figs. 2(b) and 2(c) for propagation distances of 100 m and 10,000 m, respectively. It can be observed in panel (i) in Figs. 2(a)–2(c) that the facula of the interference laser focused by the conventional imaging system is a rotationally symmetric Airy disk, and the size gradually decreases with laser propagation distance. In contrast, it can be observed in panels (ii)–(iv) that all three kinds of WFC imaging systems show different facula shapes due to light field manipulation of the phase mask, which redistributes facula energy to effectively reduce maximum light intensity on the sensor, thus achieving the purpose of laser protection.

Figure 2. Facula shapes of interference laser: The light field distribution (in units of W/m²) on the image plane of a conventional imaging system [panel (i)], CPM [panel (ii)], SPM [panel (iii)], and SAPM [panel (iv)] WFC imaging system when the laser propagation distance is (a) 1 m, (b) 100 m, and (c) 10,000 m. CPM, cubic phase masks; SPM, spiral phase mask; SAPM, spiral axicon phase mask; WFC, wavefront coding.

The diffraction facula shapes vary among the three phase masks, resulting in the difference in their laser protection performance. The diffraction facula by CPM resembles an Airy beam whose nodes exhibit L-shaped non-rotationally symmetric distribution. The light field generated by SPM and SAPM are both vortex beams that display a rotationally symmetric ring distribution. Notably, the faculae produced by different phase masks are affected by laser propagation distance in distinct ways. When the laser source is close to imaging system, the light field modulation of the CPM is slight, and the facula shape is like the Airy disk in the conventional imaging system. With the increase of laser propagation distance, the diffraction facula gradually evolves into the L-shaped distribution, and thus the laser energy gradually disperses. The evolution process of the SPM facula with laser propagation distance is the inverse of the CPM. The radius and FWHM of the ring both decrease rapidly with laser propagation distance, and the energy dispersion effect accordingly weakens. Despite both featuring a ring-shaped distribution with angular momentum, the diffraction facula of the SAPM exhibits different properties from the SPM versus laser propagation distance. The radius of the ring facula remains almost the same and only FWHM decreases with the laser propagation distance. It is noteworthy that the energy dispersion effect of the SAPM shows robustness to laser propagation distance, giving it a tremendous application foreground in laser protection.

3.2. Suppression Ratio of Maximum Light Intensity and Single-pixel Power

In this section, the laser protection capability of the WFC imaging system with different phase masks is quantitatively analyzed. Figure 3(a) plots the sensor maximum intensity of the interference laser facula versus laser propagation distance for CPM, SPM, and SAPM WFC imaging systems, and compared with conventional imaging systems. The maximum light intensity at the image plane is determined by two factors, the total energy incident into the image system, and the facula morphology. The spot radius of the interference laser gradually increases with propagation distance according to the nature of the Gaussian beam, resulting in a reduction of total energy entering the imaging system. On the other hand, the facula size on the image plane may diminish within a specific range of propagation distance, leading to an increase in energy density. Therefore, it can be observed in Fig. 3(a) that the maximum intensity decreases nonmonotonically with the propagation distance, and there exists a peak at a specific propagation distance, which we refer to as the most hazardous distance (TMHD). For conventional imaging systems, the maximum light intensity occurs at the facula center, which can be written quantitatively by Fourier-Bessel transform:

Figure 3. Maximum light intensity and its suppression ratio: (a) Sensor maximum light intensity versus laser propagation distance for different imaging systems. (b) Intensity suppression ratio versus laser propagation distance for different wavefront coding (WFC) imaging systems.

The derivation of Eq. (4) shows that it reaches the maximum when relation $2\omega {\left(z_g\right)}^2/2\omega {\left(z_g\right)}^2+D^2\exp \left[D^2/4\omega {\left(z_g\right)}^2\right]=1$ is satisfied, and the corresponding TMHD is 10^2.5, consistent with the numerical simulation result. For the rotationally symmetric SPM and SAPM, TMHDs are close to that of conventional imaging systems, which are 10^2.6 and 10^2.4, respectively. For the CPM, TMHD is relatively smaller as 10^1.8 m.

Figure 3(b) shows the intensity suppression ratio of different WFC imaging systems compared with the conventional system, which is defined as $S_{\text{intensity}}=I_{\text{CON}}/I_{\text{WFC}}$, where I_CON and I_WFC are the sensor maximum light intensity for the conventional and WFC imaging systems, respectively. It can be observed that the laser protection capability of the CPM is inadequate (S_intensity < 10) when the interference laser propagation distance is short (less than 100 m). Meanwhile, both SPM and SAPM show excellent laser protection performance, for which the maximum light intensity can be suppressed below 1%. With the increase in laser propagation distance, the L-shaped facula generated by CPM modulation gradually expands, and its intensity suppression ratio increases accordingly, and exceeds 100 after the laser propagates a distance of 1,000 m. Because the ring radius of the hollow facula on the SPM image plane decreases, the intensity suppression ratio falls below 100 when the laser propagation distance is longer than 1,000 m. In contrast, SAPM can disperse the energy of the interference laser facula on the sensor effectively regardless of the laser propagation distance, and its laser protection capability is therefore more robust against the laser propagation distance. SAPM enhances the intensity suppression ratio two orders of magnitude higher than the CPM when the laser source is near, and one order of magnitude higher than the SPM when the laser source is far away, giving it considerable laser protection capability.

Next, we explore the impact of the nonlocal property of the sensor on laser protection performance because the photoelectric sensor generates and transfers charges in pixel units. Therefore, a new physical parameter called maximum single-pixel power P_pixel is defined, which is the laser power within the equivalent area of the unit pixel centered on the position of the intensity peak. Figure 4(a) shows the maximum single-pixel power versus laser propagation distance for both conventional and WFC imaging systems. Although the curve shape changes compared with the maximum light intensity displayed in Fig. 3(a), it still possesses identical characteristics that the maximum single-pixel power reaches the peak at a specific propagation distance. TMHD is slightly reduced to 10^2.2 for the conventional imaging system, whereas it is almost unchanged for the WFC imaging systems. Figure 4(b) presents the suppression ratio of maximum single-pixel power $S_{\text{pixel}}=P_{\text{pixel,CON}}/P_{\text{pixel,WFC}}$ between conventional (P_pixel,CON) and WFC (P_pixel,WFC) imaging systems. The suppression ratio of maximum single-pixel power is almost the same as that of maximum intensity depicted in Fig. 3(a) for a laser propagation distance range of less than 10 m. The suppression ratio undergoes changes as the propagation distance increases and eventually restabilizes until the laser propagation distance exceeds 1,000 m. In this case, the laser spot of conventional imaging systems is reduced to the level of single-pixel size, and the energy accumulation effect is much smaller than that of WFC systems. Therefore, the suppression ratio of maximum single-pixel power is lower than that of light intensity. It should also be pointed that the laser protection performance of the SAPM remains robust to a laser propagation distance under the pixel metric, reducing the maximum single-pixel power to below 1.2% at an almost arbitrary laser propagation distance, thus achieving full-range laser protection for photodetectors.

Figure 4. Maximum single-pixel power and its suppression ratio: (a) Sensor maximum single-pixel power versus laser propagation distance for different imaging systems. (b) Maximum single-pixel power suppression ratio versus laser propagation distance for different wavefront coding (WFC) imaging systems.

3.3. Laser Protection Performance for Defocus Scenario

For actual working conditions of an imaging system, the detection target may be separate from the interference laser source. The imaging system focuses on the detection target to obtain a clear image, and the interference laser may simultaneously be out of focus. Thus, it is necessary to study the laser protection performance of the WFC imaging system when it is out of focus. With the aim of finding a general application scenario for an imaging system in detecting scenes at infinity, this section studies the protection capability of the WFC imaging system against laser interference with varying propagation distances. To quantitatively assess the defocus condition of the imaging system, Fig. 5(a) illustrates the defocus phase difference of the imaging system for interference laser with different propagation distances, which is defined as the maximum optical path difference at the edge of the circular pupil. The greater value of phase difference indicates the higher defocus amount. Because Gaussian beam can be regarded as a plane wave with and infinitely large curvature radius at both the beam waist (i.e., zero propagation distance) and infinity, the defocus phase difference initially increases and then decreases with laser propagation distance, as shown in Fig. 5(a). Figure 5(b) depicts the maximum single-pixel power versus laser propagation distance for both the conventional and diverse WFC imaging systems in a defocus scenario, wherein the system parameters are consistent with Fig. 4. By comparing Figs. 5(b) and 4(a), it can be observed that the conventional imaging systems exhibit the highest sensitivity to defocus, and the curve of maximum single-pixel power changes in propagation distance range corresponding to a large defocus phase difference. Defocus enlarges the laser spot on the sensor of conventional imaging systems, which reduces maximum single-pixel power and eliminates the peak at TMHD, leading to power stabilization within a propagation distance of less than 1,000 m. In contrast, the maximum single-pixel power of the WFC imaging system is relatively less affected by defocus, and the evolution characteristics with distance are maintained the same as the focus situation.

Figure 5. Laser protection performance for defocus scenario: (a) Defocus phase difference versus laser propagation distance. (b) Maximum single-pixel power versus laser propagation distance for both conventional and various wavefront coding (WFC) imaging systems in defocus scenario. (c) Maximum single-pixel power versus defocus phase difference at a laser propagation distance of 100 m.

It should be pointed out that the maximum single-pixel power of the CPM WFC imaging system is higher than that of the conventional system in a specific range of laser propagation (around 100 m) due to defocus, which means that the additional CPM may even improve the sensor hazard risk. In contrast, the maximum single-pixel power of the SPM and SAPM WFC imaging systems is always lower than that of the conventional system, indicating stable laser protection capability in the defocus scenario. To show the influence of defocus on the laser protection performance of WFC imaging systems more intuitively, Fig. 5(c) plots the maximum single-pixel power versus defocus phase difference at a laser propagation distance of 100 m. The maximum single-pixel power of the conventional imaging system decreases with the defocus phase difference, while the CPM stabilizes the maximum single-pixel power due to the quasi-non-diffraction nature of the PSF. When defocus phase difference is larger than 5λ, the maximum single-pixel power of the CPM WFC imaging system surpasses that of conventional imaging systems, and thus laser protection capability vanishes. Because both the SAM and SAPM possess a high basic suppression ratio, their laser protection capability is more robust to defocus.

3.4. Influence of the SAPM Parameters on Laser Protection Performance

In pervious sections we investigated the dependence of laser protection performance on propagation distance and defocusing amount of interference laser for the WFC imaging systems and demonstrated the superiority of the SAPM on laser protection. In this section, the influence of the SAPM phase modulation coefficient on laser protection performance is studied to provide guidance for SAPM design. Figure 6(a) presents the suppression ratio of maximum single-pixel power versus laser propagation distance for SAPMs with α = 0 (SPM), 100, 200, 300, and a fixed topological charge of 10. The radial phase coefficient α of the SAPM primarily affects the laser protection performance in the case of long propagation distances, but slightly in the case of short propagation distances, and the overall suppression ratio of maximum single-pixel power increases with α. Figure 6(b) depicts facula profiles corresponding to different radial phase coefficients when the laser propagation distance equals 0.1 m and Fig. 6(c) for 10,000 m, where the panels (i)–(iv) correspond to the four parameters used in Fig. 6(a). When the laser propagation distance is 0.1 m [Fig. 6(b)], the ring radiuses are 34.5 μm, 49.6 μm, 78.5 μm, and 110.1 μm, and the FHWMs of the ring facula are 30.7 μm, 24.8 μm, 20.1 μm, and 17.8 μm with the increase of α. The ring radius increases while FHWM decreases with α, and thus the change of the suppression ratio of maximum single-pixel power is slight. When the laser propagation distance equals 10,000 m [Fig. 6(c)], the ring radiuses are 4.6 μm, 34.6 μm, 68.3 μm and 101.9 μm, and the FHWMs of the ring facula are 1.4 μm, 2.4 μm, 2.3 μm, and 2.2 μm with the increase of α. The ring radius increases with α while the FHWM change is comparatively negligible, and thus the suppression ratio of maximum single-pixel power is gradually improved.

Figure 6. Influence of SAPM parameters on laser protection performance: (a) Sensor maximum single-pixel power suppression ratio versus laser propagation distance for different SAPM radial phase coefficients. (b, c) Laser facula profiles on the sensor in SAPM WFC imaging system when laser propagation distances are (b) 0.1 m and (c) 10,000 m, panels (i)–(iv) correspond to the radial phase coefficients of 0, 100, 200, 300. (d) Sensor maximum single-pixel power suppression ratio versus laser propagation distance for different SAPM topological charges. (e, f) Laser facula profiles on sensor in SAPM WFC imaging system when laser propagation distances are (e) 0.1 m and (f) 10,000 m, panels (i)–(iv) correspond to the topological charges of 0, 5, 10, and 15. SAPM, spiral axicon phase mask; WFC, wavefront coding.

Figure 6(d) shows the suppression ratio of maximum single-pixel power versus laser propagation distance for the SAPMs with l = 0 (APM), 5, 10, 15 and a fixed radial phase coefficient of 200. The topological charge l of the SAPM mainly contributes to laser protection performance in the case of short propagation distances, but is almost unrelated for long propagation distances. The suppression ratio of maximum single-pixel power increases with l in the context of short-range laser interference. Figure 6(e) presents the facula profiles corresponding to the different topological charges when the laser propagation distance is 0.1 m and Fig. 6(f) for 10,000 m, where panels (i)–(iv) correspond to the four parameters shown in Fig. 6(d). When the laser propagation distance equals 0.1 m [Fig. 6(e)], the ring radiuses are 67.3 μm, 72.5 μm, 79.1 μm, and 87.4 μm, and the FHWMs of the ring facula are 12.6 μm, 14.1 μm, 20.3 μm, and 30.2 μm with the increase of l. Both the ring radius and FHWM increase with l, and thus the suppression ratio of maximum single-pixel power is enhanced. When the laser propagation distance is 10,000 m [Fig. 6(f)], the ring radiuses are 67.8 μm, 67.9 μm, 68.3 μm, and 68.5 μm, and the FHWMs of the ring facula are 2.1 μm, 2.1 μm, 2.3 μm, and 2.5 μm with the increase of l. Both the ring radius and FHWM remain stable, and therefore the change of the suppression ratio of maximum single-pixel power is negligible.

To sum up, increasing both the radial phase coefficient and topological charge can improve the laser protection performance of the SAPM, but they are different in emphasis. However, considering the practical fabrication difficulty of phase masks, the parameters of the phase mask should be reasonably optimized to obtain better fabrication compatibility while meeting the requirements of laser protection capability. If the imaging system is primarily aimed at long-distance laser protection, it is necessary to increase the radial phase coefficient instead of the topological charge. If the imaging system pays more attention to the scenario involving short-range laser interference, increasing the topological charge can yield a more pronounced effect than the radial phase coefficient.

In this section, the imaging model of the WFC imaging system is established, and imaging simulation is performed to demonstrate that the WFC imaging system can effectively detect the target scene while possessing laser protection capability. Assuming that the scene is illuminated by incoherent light, the coded image captured by sensor I_coded can be obtained by convolution of the geometric image of the target I_geom with the incoherent point spread function |h(x_i, y_i)|²:

where (x_i, y_i) are spatial coordinates of the image plane. By performing the Fourier transform on Eq. (5), the frequency domain expression of the object-image relationship can be obtained:

where ${\mathcal{G}}_{\text{coded}}\left(f_x,f_y\right)=\mathcal{F}\left[I_{\text{coded}}\left(x_i,y_i\right)\right]$ and ${\mathcal{G}}_{\text{geom}}\left(f_x,f_y\right)=\mathcal{F}\left[I_{\text{geom}}\left(x_i,y_i\right)\right]$ are normalized spatial frequency spectra of I_coded and I_geom, respectively, and $\mathcal{H}\left(f_x,f_y\right)$ is the optical transfer function (OTF). By performing inverse Fourier transform on Eq. (6), the incoherent captured image can be obtained:

Therefore, deducing the OTF of the imaging system is the key for imaging simulation. The OTF equals the normalized autocorrelation of coherent transfer function H( f_x, f_y):

and the coherent transfer function of a diffraction-limited imaging system can be derived from the generalized pupil function:

The WFC imaging system is essentially a defocusing system, and the generalized pupil function includes both amplitude and phase transmittances, which are written as:

where (x, y) are spatial coordinates at the pupil plane and φ is the modulation function of the phase mask, as shown in Table 1. By substituting Eqs. (8)–(10) into Eq. (7), the incoherent captured image of the system can be obtained.

The image captured directly by the WFC imaging system is a fuzzy image coded by a phase mask, which can be restored to a clear image by Wiener filter:

where ${\mathcal{G}}_{decoded}\left(f_x,f_y\right)$ is the normalized frequency spectrum of the decoded image and K is the tentative parameter. By performing inverse Fourier transform on the Eq. (11), the decoded image can be obtained as $I_{\text{decoded}}\left(x_i,y_i\right)={\mathcal{F}}^{-1}\left[{\mathcal{G}}_{\text{decoded}}\left(f_x,f_y\right)\right]$.

Figure 7 shows the imaging simulation results of the CPM, SPM, and SAPM WFC imaging systems for infinite scenes of artificial and natural targets, and the parameters of the phase mask and imaging system are consistent with those used in Section 3.1. It can be observed in Figs. 7(b) and 7(e) that the SAPM-coded image is the fuzziest, followed by CPM, and the SPM-coded image is the clearest. Since the laser protection capability of the WFC imaging system is stronger, the corresponding PSF is more dispersed, and the coded image is fuzzier. The imaging results indicate that the laser protection performance of the SAPM is most excellent, which coincides with the conclusion obtained by using the laser transmission model in Section 3.2. Figures 7(c) and 7(f) depict the restored images by decoding using a Wiener filter for artificial and natural targets, respectively. The three kinds of WFC imaging systems feature comparable decoded image quality, and all the intricate details containing high-frequency information are restored effectively, whether artificial or natural targets. Overall, except for an extremely slight ringing effect, the imaging quality of the WFC imaging systems is close to the level of conventional imaging systems without defocus [Figs. 7(a) and 7(d)], and possesses imaging capability.

Figure 7. Imaging simulation of WFC imaging system: (a), (d) Image of conventional imaging systems for (a) artificial target and (d) natural scene. (b), (e) Coded images and (c), (f) decoded images of the CPM, SPM, and SAPM WFC imaging systems for (b), (c) artificial target and (e), (f) natural scene. The tentative parameter of the Wiener filter is set to 10⁻⁴ for CPM and SPM and 10⁻⁵ for SAPM. The detector possesses Gaussian noise with a standard deviation of 0.001. CPM, cubic phase masks; SPM, spiral phase mask; SAPM, spiral axicon phase mask; WFC, wavefront coding.

In this paper, an SAPM WFC imaging system is proposed for laser protection against photoelectric detector damage, which has superior laser protection performance compared with CPM or SPM WFC imaging systems. Distinct from CPM and SPM with laser protection capability sensitive to the interference laser propagation distance, SAPM disperses the interference laser spot on a sensor into a hollow ring, which can reduce the maximum intensity and single-pixel power to levels of 1% and 1.2%, respectively, compared with conventional imaging systems for full-range laser protection. We also investigate the impact of the SAPM phase modulation coefficients on laser protection performance and provide design recommendations. Specifically, the radial phase coefficient primarily modulates long-range laser protection capability, while the topological charge is for short-range scenarios. Most importantly, we find that both the WFC and conventional imaging systems feature the most hazardous distance from the interference laser source, where the maximum single-pixel power of sensor reaches a peak. Therefore, it is imperative to ensure that imaging systems are not placed within this dangerous range when exposed to intense laser irradiation. Our findings can guide the laser protection design of photoelectric imaging systems and may have potential applications in security monitoring, autonomous driving, and intense laser experiments.

Postdoctoral Fellowship Program of China Postdoctoral Science Foundation (GZC20233531); Technology Domain Fund of Basic Strengthening Plan (2021-JCJQ-JJ-0284 and 2022-JCJQ-JJ-0237); Research Project of National University of Defense Technology (ZK2041 and ZK23-49); Advanced Laser Technology Laboratory Foundation of Anhui Province (AHL2021QN03 and AHL2022ZR03); State Key Laboratory Foundation of Pulsed Power Laser Technology (SKL2022ZR09); Young Doctor Foundation of Electronic Engineering College of the National University of Defense Technology (KY21C218).

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data underlying the results presented in this paper are not publicly available at the time of publication, but may be obtained from the authors upon reasonable request.