A conventional photoelectric imaging system is mainly composed of optical elements, photoelectric detectors, and signal-processing parts. The autofocus mechanism improves the imaging quality of the system, and at the same time the photodetector is illuminated by the incident laser and becomes a secondary light-emitting point. A considerable part of the reflected light returns along the original incident light path, resulting in a strong cat-eye effect, making the imaging device easily discovered and positioned by a laser active-detection system [1, 2]. The photoelectric detector is also easily damaged, due to the focusing effect of the optical system and the high gain of the sensor. Therefore, laser protection of the photoelectric imaging system includes two aspects: Anti-laser detection and anti-laser damage [3]. To reduce the probability of the photoelectric imaging system being detected and damaged, reasonable and feasible technical means are used to reduce the cat-eye effect of the imaging system and the light intensity reaching the photodetector, thereby improving the safety of photoelectric imaging equipment.

The typical technical solutions for anti-laser detection are to add different devices for filtering, blocking, or optical isolation into the optical path of the imaging system, to prevent the generation of cat-eye echoes. Alternatively, the photodetector of the imaging system is placed out of focus, to reduce the intensity of cat-eye echoes [4^–6]. Typical technical solutions to prevent laser damage in photoelectric imaging systems include adding filters and using nonlinear window materials [7]. However, those typical technical solutions have problems, such as limited working band and deterioration of imaging quality. Focusing on solving the contradiction between laser-protection performance and the imaging quality of the photoelectric imaging system, researchers have used the sizeable focal-depth characteristics of the wavefront-coding technology [8, 9] to improve the laser-protection capability by defocusing the imaging plane [10, 11]. The phase mask is a vital part of a wavefront-encoding imaging system that determines the focal depth of the imaging system, the signal-to-noise ratio of the decoded image, and the laser-protection performance. Previous studies have not compared the laser-protection performance of different phase masks, and the influence of propagation distance on laser-protection performance has not been studied. This paper focuses on studying and comparing the anti-laser damage and detection performance of different asymmetric phase masks at different defocus distances and propagation distances, aiming to explore the variation rule of the laser-protection performance of different asymmetric phase masks. The conclusions can guide the selection of appropriate asymmetric phase masks in different application scenarios, and improve the laser-protection capability of imaging systems.

The theoretical model of this paper is consistent with that of reference [10]; both expand the wavefront-coding imaging system into an equivalent 4F system to calculate the spot profiles and intensity distributions reaching the imaging plane and the observation plane. The laser-propagation model of the imaging system is established to clarify the physical process. Figure 1 illustrates the equivalent 4F system for studying the anti-laser damage and detection performance of an imaging system. The process can be expressed as follows: A Gaussian beam with a waist-to-imaging-lens distance equal to z_Gauss is incident upon the optical system, which is modulated by an imaging lens L and a phase-mask plate (PMP) and focuses on the imaging plane at a distance d_i behind the phase mask. If considering the defocus of the imaging plane, the distance between the defocused imaging plane and the ideal imaging plane is ∆d. Then the reflection component of the focused spot on the imaging plane in turn propagates through the phase mask plate PMP and the imaging lens L again, and a cat-eye echo is formed on the observation plane at a distance z_obser from the optical system.

Figure 1. Laser-propagation model of a wavefront-coding imaging system. PMP, phase-mask plate.

Specify that the phase lag is positive. The Gaussian beam of waist width ω₀ propagates to the front surface of imaging lens L, and the complex amplitude distribution can be written as the Gaussian-beam propagation equation [12]:

(1)$$U_1^{-}\left(x_1,y_1\right)=A_0\frac{{\omega }_0}{\omega \left(z\right)}\text{exp}\left(-\frac{x_1^2+y_1^2}{{\omega }^2\left(z\right)}\right)\times \text{exp}\left\{i\left[k\left(z+\frac{x_1^2+y_1^2}{2R\left(z\right)}\right)-\text{arctan}\left(\frac{\lambda z}{\pi {\omega }_0^2}\right)\right]\right\},$$

where z = z_Gauss, which is defined as the distance between the Gaussian beam’s waist and the imaging lens’s front surface. A₀ is a constant related to power. k and λ are respectively the wave number and wavelength of the incident laser. ω(z) and R(z) are respectively the spot size and equiphase surface curvature radius of the Gaussian beam’s wavefront on the imaging lens’s front surface:

(2)$$\omega \left(z\right)={\omega }_0\sqrt{1+{\left(\frac{\lambda z}{\pi {\omega }_0^2}\right)}^2}$$

(3)$$R\left(z\right)=z\left[1+{\left(\frac{\pi {\omega }_0^2}{\lambda z}\right)}^2\right]$$

Assuming that the phase-mask plate is in very close contact with the rear surface of imaging lens L, the two components can be equivalent to a single phase plane, and its generalized transmittance function could be expressed as

(4)$$T\left(x_1,y_1\right)=\text{circ}\left(\frac{\sqrt{x_1^2+y_1^2}}{d/2}\right)\times \text{exp}\left[-i\left(\frac{k}{2f}\left(x_1^2+y_1^2\right)-ph\left(x_1,y_1\right)\right)\right],$$

where d is the diameter of the imaging lens and f is the focal length of the imaging lens. ph(x, y) is the phase function of the PMP. The complex amplitude distribution on the rear surface of the PMP is calculated as:

(5)$$U_1^{+}\left(x_1,y_1\right)=U_1^{-}\left(x_1,y_1\right)T\left(x_1,y_1\right)$$

With the assumption of the Fresnel approximation [13, 14], the complex amplitude distribution on the imaging plane is

(6)$$U_2\left(x_2,y_2\right)=\frac{\text{exp}\left(ik{d}_i\right)}{i\lambda d_i}\text{exp}\left[i\frac{k}{2d_i}\left(x_2^2+y_2^2\right)\right]\times {\displaystyle \iint U_1^{+}\left(\xi ,\eta \right)\text{exp}\left[i\frac{k}{2d_i}\left({\xi }^2+{\eta }^2\right)\right]}\times \text{exp}\left[-i\frac{k}{d_i}\left(\xi x_2+\eta y_2\right)\right]d\xi d\eta .$$

To simplify the analytical model, we mainly consider the reflection of the detector’s silicon substrate. Assuming that its reflectivity is ρ, the complex amplitude distribution on the rear surface of the PMP is

(7)$$U_3^{-}\left(x_3,y_3\right)=\frac{\text{exp}\left(ik{d}_i\right)}{i\lambda d_i}\text{exp}\left[i\frac{k}{2d_i}\left(x_3^2+y_3^2\right)\right]\times {\displaystyle \iint \rho U_2\left(\xi ,\eta \right)\text{exp}\left[i\frac{k}{2d_i}\left({\xi }^2+{\eta }^2\right)\right]}\times \exp \left[-i\frac{k}{d_i}\left(\xi x_3+\eta y_3\right)\right]d\xi d\eta .$$

Through the secondary modulation of the phase mask plate PMP and the imaging lens L, the complex amplitude distribution on the rear surface of L is calculated as

(8)$$U_3^{+}\left(x_3,y_3\right)=U_3^{-}\left(x_3,y_3\right)T\left(x_3,y_3\right)$$

Based on the Fresnel diffraction principle, the complex amplitude distribution of the cat-eye echo formed on the observation plane can be expressed as follows:

(9)$$U_4\left(x_4,y_4\right)=\frac{\exp \left(ik{z}_{\text{obser}}\right)}{i\lambda z_{\text{obser}}}\text{exp}\left[i\frac{k}{2z_{\text{obser}}}\left(x_4^2+y_4^2\right)\right]\times {\displaystyle \iint U_3^{+}\left(\xi ,\eta \right)}\cdot \text{exp}\left[i\frac{k}{2z_{\text{obser}}}\left({\xi }^2+{\eta }^2\right)\right]\times \text{exp}\left[-i\frac{k}{z_{\text{obser}}}\left(\xi x_4+\eta y_4\right)\right]d\xi d\eta .$$

Finally, the spot profile and intensity distribution on the imaging plane and on the observation plane can be calculated by multiplying the complex amplitude by its conjugate.

The difference between this paper and [10] is that to characterize the degree of defocus, we use the defocus distance instead of the defocus wave aberration, because the defocus distance is more intuitive, while the defocus wave aberration is relatively abstract. The defocus wave aberration W₂₀ can be expressed as

(10)$$W_{20}=\frac{D^2}{8}\left(\frac{1}{f}-\frac{1}{d_0}-\frac{1}{d_i}\right)$$

where D is the pupil diameter, f is the focal length of the imaging lens, d_o is the object distance, and d_i is the image distance. The defocusing image distance can be expressed as

(11)$$d_i={\left[\frac{1}{f}-\frac{1}{d_0}-\frac{8W_{20}}{D^2}\right]}^{-1}$$

Based on the Gaussian imaging formula 1/d_o + 1/d_i0 = 1/f, the calculated defocus distance is

(12)$$\Delta d=d_i-d_{i0}={\left[\frac{1}{f}-\frac{1}{d_0}-\frac{8W_{20}}{D^2}\right]}^{-1}-{\left[\frac{1}{f}-\frac{1}{d_0}\right]}^{-1}$$

where d_i0 represents the ideal imaging distance. The above is the relationship between the defocus wave aberration and the defocus distance.

The phase masks are the core optical elements of the wavefront-encoding imaging system that determine the imaging system’s focal depth and imaging quality. In addition, the theoretical model clarifies that the phase mask directly affects the anti-laser damage and detection performance. The modulation transfer functions (MTFs) of asymmetric phase masks tend to be consistent in a larger defocus range, which is more advantageous than for the case of symmetric phase masks [15]. Typical asymmetric phase masks include the cubic phase mask (CPM) [8], generalized cubic phase mask (GCPM) [16], logarithmic phase mask (LPM) [17], exponential phase mask (EPM) [18], free-form phase mask (FPM) [19], rational phase mask (RPM) [20], improved logarithmic phase mask (improved-LPM) [21, 22], sinusoidal phase mask (SPM) [23], improved sinusoidal phase mask (improved-SPM) [24], tangent phase mask (TPM) [25], arcsine phase mask (ASPM) [26], etc. The CPM is the most original phase mask. The improved-LPM can attain the optimized phase profile. The SPM is superior in extending the focal depth. The TPM can get a relatively stable defocus MTF and excellent defocus stability, while the ASPM achieves optimal defocus stability and image-reconstruction performance. Therefore, studying and comparing the laser-protection performance of the CPM, improved-1 LPM, improved-2 LPM, SPM, TPM, and ASPM is representative. The phase functions of the asymmetric phase masks are listed in Table 1. The symbols x and y in the table represent the normalized spatial coordinates of the aperture plane respectively, and the value range is [−1, 1]. α and β are the phase-mask parameters (in radians) that control the magnitude of phase deviation, and sgn(∙) represents the sign function.

Table 1 . Phase functions of asymmetric phase masks.

Category	Phase Function
CPM	α (x3 + y3)
Improved-1 LPM	α⋅sgnx⋅x2⋅logx+β+α⋅sgny⋅y2⋅logy+β
Improved-2 LPM	α⋅sgnx⋅x4⋅logx+β+α⋅sgny⋅y4⋅logy+β
SPM	α⋅x4⋅sinβx+α⋅y4⋅sinβy
TPM	α⋅x2⋅tanβx+α⋅y2⋅tanβy
ASPM	α⋅x2⋅arcsinx+α⋅y2⋅arcsiny

CPM, cubic phase mask; Improved-LPM, improved logarithmic phase mask; SPM, sinusoidal phase mask; TPM, tangent phase mask; ASPM, arcsine phase mask..

3.1. Evaluation Index

To explore the influence of asymmetric phase masks on the laser-protection performance of the wavefront-coding imaging system, we need to define reasonable evaluation criteria to compare the laser-protection performance of different phase masks. Excessive light intensity can damage the photodetector. Since the charge-coupled device (CCD) array detector performs the charge generation and transfer in units of pixels, the maximum single-pixel receiving power is superior to the maximum light intensity in characterizing the anti-laser damage performance. The maximum single-pixel receiving power is used as the evaluation index when comparing anti-laser damage performance figures. The maximum single-pixel receiving power is the laser power within the equivalent area of a single pixel centered on the position of peak light intensity. Under the same parameters for the incident laser and optical system, the smaller the maximum single-pixel receiving power, the better the anti-laser damage performance of the phase mask.

The laser-active detection equipment includes two main parts: the transmitting system, and the receiving system. As their names imply, the transmitting and receiving systems realize the functions of emitting the laser and receiving echoes. The specific design, optimization, and signal-processing methods of different receiving systems (echo-detection systems) are diverse, and the particular working modes of various echo-detection systems are not considered. Therefore, the echo-detection system can be simplified to an echo-detection area coaxial with the transmitting system, and the echo power received in this area is calculated to characterize the anti-laser detection performance of the imaging system. The echo-detection receiving power is used as the evaluation index when comparing anti-laser detection performance figures. The echo-detection receiving power is the echo power within the echo-detection area, simplified from the echo-detection system centered on the optical axis. Under the same parameters for the incident laser and optical system, the smaller the echo-detection receiving power, the better the anti-laser detection performance of the phase mask.

3.2. Simulation Parameters

The parameters of the asymmetric phase masks affect the defocusing range of the imaging system and the signal-to-noise ratio of the decoded image, which further limits the laser protection performance. Therefore, to compare the anti-laser damage and detection performance of different phase masks, we need to optimize the phase-mask parameters under the same conditions; that is, we need to obtain the optimal parameters of the phase masks under the same constraints. The principle of optimization is this: Within the designed defocus range, strive to achieve imaging characteristics as similar as possible to obtain a better defocus range, and at the same time use constraints to prevent overoptimization, ensure the signal-to-noise ratio of the decoded image, and prevent the encoded image from being appropriately decoded. Optimization based on Fisher information (FI) is a commonly used method that can be used to evaluate the similarity of the point-spread function (PSF) or MTF. FI can measure the shifting degree of PSF or MTF in the defocusing process. FI equal to zero indicates that the defocusing PSF or MTF is consistent within the designed defocusing range, which means that the PSF or MTF is absolutely insensitive to defocusing error. At the same time, to prevent overoptimization, the integral area of the MTF is used as a constraint to control the strength of the MTF. Optimizing the phase-modulation coefficient amounts to finding the minimum value of the FI under the constraints. To simplify the mathematical expression, we adopt a one-dimensional form of the phase function in the optimization process, and the optimization of the phase-mask parameters can be expressed as:

(13)$$\left\{\begin{array}{c}\text{min}\text{FI}={\displaystyle {\int }_{-\psi }^{\psi }\left(\int {\left|\frac{\partial }{\partial W_{20}}H\right|}^{2}du\right)d{W}_{20}}\\ s.t.\int H\left(u,\alpha ,\beta ,W_{20}=0\right)du\ge Th\end{array}\right.$$

where H represents the modulation transfer function, W₂₀ is the wave aberration, ψ is the defocus parameter, α and β represent the phase mask parameters, and Th is the chosen threshold. When the threshold of the MTF area is set to 0.33 in normalized pupil coordinates, the defocus parameter ψ is set to 0, 2π, 4π, 6π, 8π, 10π, and the calculated phase mask parameters are listed in Table 2.

Table 2 . Optimized parameters of asymmetric phase masks.

Category	Parameter α	Parameter β
CPM	74.73	-
Improved-1 LPM	−268.96	−1.52
Improved-2 LPM	275.38	0.97
SPM	148.76	1.83
TPM	37.59	1.27
ASPM	42.11	-

On the other hand, the numerical simulation involves the parameters of the incident laser, optical system, and detection system. The specific simulation parameters are listed in Table 3.

Table 3 . Numerical simulation parameters.

Parameter	Value
Incident Laser Power (W)	25
Incident Laser Wavelength (nm)	532
Gaussian Beam Waist Size (mm)	5
Imaging Lens Focal Length (mm)	100
Imaging Lens Size (mm)	Ф50
Imaging Plane Detector Pixel Size (μm)	2.4 × 2.4
Imaging Plane Detector Reflectivity	0.2
Echo Detector Size (mm)	Ф50

Through numerical simulation, we calculate the laser protection performance of the imaging system at different defocus distances. The Gaussian beam with a waist 1,000 m away from the imaging lens passes through the imaging system, and finally forms a cat-eye echo at the observation plane 1,000 m away from the imaging lens. The spot profile and intensity distribution are detected by detectors placed on the imaging and observation planes.

4.1. Simulation Results for Anti-laser Damage

Figure 2 shows the variation of the maximum single-pixel receiving power P_pixel with defocus distance on the imaging plane of the conventional and asymmetric-phase-mask wavefront-coding imaging systems. With the increase of defocus distance, the maximum single-pixel receiving power of the conventional imaging system decreases sharply. The reason is that the focused spot size on the imaging plane increases rapidly with increasing defocus distance, resulting in a rapid decrease of the light intensity. With increasing defocus distance, the CPM’s maximum single-pixel receiving power basically remains unchanged, and the maximum single-pixel receiving power of improved-1 LPM, improved-2 LPM, SPM, TPM and ASPM decreases slowly, among which improved-1 LPM declines the fastest.

Figure 2. Variation of maximum single-pixel receiving power with defocus distance.

On the premise of ensuring the imaging quality, the Rayleigh criterion [27] indicates that for clear imaging the maximum wave aberration allowed by a conventional imaging system is W₂₀ = λ/4. Using Eq. (12), the calculated maximum negative defocus distance is ∆d = −4.26 μm, and the maximum positive defocus distance is ∆d = 4.26 μm.

The designed maximum wave aberration is W₂₀ = 5λ when optimizing the phase-mask parameters. Therefore, on the premise of ensuring the imaging quality, the maximum wave aberration allowed by the wavefront-coding imaging system for clear imaging is W₂₀ = 5λ. Using Eq. (12), the calculated maximum negative defocus distance is ∆d = −85.1 μm, and the maximum positive defocus distance is ∆d = 85.2 μm.

Figures 3 and 4 show the focused spot profiles, intensity distributions, and corresponding maximum single-pixel receiving powers P_pixel on the imaging plane of the conventional and asymmetric-phase-mask wavefront-encoding imaging systems at different defocus distances. In conventional imaging systems, the spot size on the imaging plane increases rapidly with increasing defocus distance, resulting in a rapid decrease in the maximum single-pixel receiving power. In wavefront-coding imaging systems, the phase-modulation effect of the phase masks makes the spot on the imaging plane present an L-shaped distribution that is not rotationally symmetrical. The redistribution of the spot energy can reduce the maximum light intensity reaching the imaging plane. At the same time, the L-shaped nondiffracting Airy beam can ensure that the spot profile and light-intensity distribution on the imaging plane remain stable in a larger defocus range.

Figure 3. Spot profiles and corresponding maximum single-pixel receiving powers at different defocus distances.

Figure 4. Spot profiles and corresponding maximum single-pixel receiving powers at different defocus distances.

Figure 3 illustrates that when the defocus distance is 0 μm, the maximum single-pixel receiving power of the conventional imaging system is 20,352.94 mW; When the defocus distances are −34.0 μm, −85.1 μm, 34.1 μm, and 85.2 μm, the maximum single-pixel receiving power is 924.19 mW, 149.98 mW, 911.45 mW, and 149.32 mW respectively.

When the defocus distance is 0 μm, the maximum single-pixel receiving power of CPM, improved-1 LPM, and improved-2 LPM is 1,018.12 mW, 2,348.88 mW, and 2,199.24 mW respectively. Compared to the conventional imaging system without defocusing, the decreasing amplitude of the maximum single-pixel receiving power is 95.00%, 88.46%, and 89.19% respectively. When the defocus distance is −85.1 μm, the maximum single-pixel receiving power of CPM, improved-1 LPM, and improved-2 LPM is reduced to 1,007.25 mW, 695.64 mW, and 766.55 mW respectively. When the defocus distance is 85.2 μm, the maximum single-pixel receiving power of CPM, improved-1 LPM, and improved-2 LPM is reduced to 1,003.16 mW, 694.12 mW, and 727.98 mW respectively.

Figure 4 shows that when the defocus distance is 0 μm, the maximum single-pixel receiving power of SPM, TPM, and ASPM is 1,935.07 mW, 1,538.27 mW, and 1,866.42 mW respectively. Compared to the conventional imaging system without defocusing, the maximum single-pixel receiving power decreases by 90.49%, 92.44%, and 90.83% respectively. When the defocus distance is −85.1 μm, the maximum single-pixel receiving power of SPM, TPM, and ASPM is reduced to 751.77 mW, 1,196.89 mW, and 1,551.16 mW respectively. When the defocus distance is 85.2 μm, the maximum single-pixel receiving power of SPM, TPM, and ASPM is reduced to 733.51 mW, 1,192.17 mW, and 1,548.50 mW respectively.

On the premise of ensuring image quality, the maximum negative defocus distance allowed by the conventional imaging system for clear imaging is ∆d = −4.26 μm, and the corresponding maximum single-pixel receiving power is 18,317.45 mW, which is 10% lower than that without defocusing. The maximum positive defocus distance is ∆d = 4.26 μm, and the corresponding maximum single-pixel receiving power is 17,891.92 mW, 12.09% lower than without defocusing.

The maximum negative defocus distance of the wavefront-coding imaging system is ∆d = −85.1 μm. The corresponding maximum single-pixel receiving power reduction of CPM, improved-1 LPM, improved-2 LPM, SPM, TPM, and ASPM respectively reaches 95.05%, 96.58%, 96.23%, 96.31%, 94.12% and 92.38%, compared to the conventional imaging system without defocusing. The maximum positive defocus distance of the wavefront-coding imaging system is ∆d = 85.2 μm. The corresponding maximum single-pixel receiving power reduction of CPM, improved-1 LPM, improved-2 LPM, SPM, TPM, and ASPM respectively reaches 95.07%, 96.59%, 96.42%, 96.40%, 94.14%, and 92.39%, compared to the conventional imaging system without defocusing.

In general, the anti-laser damage performance for different asymmetric phase masks is slightly different. The maximum single-pixel receiving power decreases by 92.38%-96.59%, about one order of magnitude. Improved-1 LPM achieves the best anti-laser damage effect, with a decrease of 96.59%. Therefore, the wavefront-coding imaging system can reduce the maximum single-pixel receiving power on the imaging plane through defocusing a larger distance on the premise of ensuring the imaging quality, which theoretically proves the potential of the wavefront-coding imaging system to achieve anti-laser damage.

4.2. Simulation Results for Anti-laser Detection

Figure 5 shows the variation of the echo-detection receiving power P_dec with defocus distance on the observation plane of the conventional and asymmetric-phase-mask wavefront-coding imaging systems. The results show that the echo-detection receiving power decreases rapidly near the imaging plane and tends to remain the same as the defocus distance continues to increase. For the same defocus distance, the cat-eye echo spot size of the wavefront-coding imaging system is close to that of the conventional imaging system. The echo-detection receiving power fluctuates slightly, compared to the conventional imaging system. Still, the difference gradually decreases and eventually converges with increasing defocus distance. The main reason for this phenomenon is analyzed: The cat-eye echo spot size increases significantly with increasing defocus distance, resulting in the gradual decrease of the proportion between the echo-detection area size and the cat-eye echo size, and the power difference between the imaging systems decreases accordingly.

Figure 5. Variation of echo-detection receiving power with defocus distance.

The analysis in Section 4.1 shows that the maximum negative defocus distance allowed by a conventional imaging system for clear imaging is ∆d = −4.26 μm, and the maximum positive defocus distance is ∆d = 4.26 μm. The maximum negative defocus distance allowed by a wavefront-coding imaging system is ∆d = −85.1 μm, and the maximum positive defocus distance is ∆d = 85.2 μm.

Figures 6 and 7 show the cat-eye echo spot profiles, intensity distributions, and corresponding echo-detection receiving powers on the observation plane for the conventional and asymmetric-phase-mask wavefront-coding imaging systems at different defocus distances. The results show that the echo spot size gradually increases with increasing defocus distance. The proportion of spot size received by echo-detection area to overall cat-eye echo size decreases, resulting in a decline of echo-detection receiving power. In addition, the reduction of the proportion causes the difference in echo-detection receiving power for different imaging systems to decrease accordingly.

Figure 6. Echo spot profiles and corresponding echo-detection receiving powers at different defocus distances.

Figure 7. Echo spot profiles and corresponding echo-detection receiving powers at different defocus distances.

Figure 6 shows that when the defocus distance is 0 mm, the echo-detection receiving power of a conventional imaging system is 849.01 mW; when the defocus distance is −34.0 μm, −85.1 μm, 34.1 μm, and 85.2 μm, the echo-detection receiving power is 31.33 mW, 5.00 mW, 31.26 mW, and 5.02 mW respectively.

When the defocus distance is 0 μm, the echo-detection receiving power of CPM, improved-1 LPM, and improved-2 LPM are 845.67 mW, 655.32 mW, and 652.19 mW respectively. Compared to a conventional imaging system without defocusing, the echo-detection receiving power is reduced by 3.34 mW, 193.69 mW, and 196.82 mW respectively. When the defocus distance is −85.1 μm, the echo-detection receiving power of CPM, improved-1 LPM, and improved-2 LPM is reduced to 5.03 mW, 5.13 mW, and 5.19 mW respectively. When the defocus distance is 85.2 μm, the echo-detection receiving power of CPM, improved-1 LPM, and improved-2 LPM is reduced to 5.00 mW, 6.28 mW, and 6.31 mW respectively.

Figure 7 shows that when the defocus distance is 0 μm, the echo-detection receiving power of SPM, TPM, and ASPM is 753.21 mW, 775.27 mW, and 841.73 mW respectively. Compared to a conventional imaging system without defocusing, the echo-detection receiving power is reduced by 95.80 mW, 73.74 mW, and 7.28 mW respectively. When the defocus distance is −85.1 μm, the echo-detection receiving power of SPM, TPM, and ASPM is reduced to 5.26 mW, 5.09 mW, and 5.03 mW respectively. When the defocus distance is 85.2 μm, the echo-detection receiving power of SPM, TPM, and ASPM is reduced to 5.20 mW, 5.19 mW, and 5.09 mW respectively.

The maximum negative defocus distance allowed by the conventional imaging system for clear imaging is ∆d = −4.26 μm, and the corresponding echo-detection receiving power is 789.98 mW, which is 6.95% lower than that without defocusing. The maximum positive defocus distance is ∆d = 4.26 μm, and the corresponding echo-detection receiving power is 783.58 mW, 7.71% lower than without defocusing.

The maximum negative defocus distance of the wavefront-coding imaging system is ∆d = −85.1 μm. The corresponding echo-detection receiving power reduction of CPM, improved-1 LPM, improved-2 LPM, SPM, TPM, and ASPM respectively reaches 99.41%, 99.40%, 99.39%, 99.38%, 99.40%, and 99.41%, compared to a conventional imaging system without defocusing. The maximum positive defocus distance of the wavefront-coding imaging system is ∆d = 85.2 μm. The corresponding echo-detection receiving power reduction of CPM, improved-1 LPM, improved-2 LPM, SPM, TPM, and ASPM respectively reaches 99.41%, 99.26%, 99.26%, 99.39%, 99.39%, and 99.40%, compared to a conventional imaging system without defocusing.

In general, the anti-laser detection performance for different asymmetric phase masks tends to be consistent, and the echo-detection receiving power decreases by 99.26%^–99.41%, about two orders of magnitude. Therefore, the wavefront-coding imaging system can reduce the echo-detection receiving power through defocusing a larger distance on the premise of ensuring the imaging quality, which theoretically proves the potential of the wavefront-coding imaging system to achieve anti-laser detection.

The laser-propagation distance is different in different application scenarios. Exploring the influence of the propagation distance on the anti-laser damage and detection performance helps us to select the appropriate imaging system for specific application scenarios. We simulate the anti-laser damage and detection performance of an imaging system without defocusing, when the propagation distance varies from 100 m to 100 km.

5.1. Simulation Results for Anti-laser Damage

Figure 8 shows the variation of the maximum single-pixel receiving power on the imaging plane with propagation distance. With the continuous increase of propagation distance, on the one hand, the interval between the beam waist of the transmitted beam and the imaging plane gradually decreases, and the beam waist size gradually decreases until it approaches zero. The position of the imaging plane is determined by the image distance calculated by the Gaussian imaging formula.

Figure 8. Variation of the maximum single-pixel receiving power with propagation distance.

The position of the beam waist needs to be explained in combination with the propagation properties of the Gaussian beam. The focal length of the imaging lens is f, the beam waist size of the incident Gaussian beam is ω0, and the propagation distance between the beam waist and the imaging lens is l. After the action of the imaging lens, the beam waist size ω′₀ of the transmitted beam and the distance l′ from the beam waist to the imaging lens can be expressed as

(14)$${{\omega }^{\prime }}^2{}_0=\frac{f^2{\omega }_0^2}{{\left(f-l\right)}^2+\left(\frac{\pi {\omega }_0^2}{\lambda }\right)}$$

(15)$$l^{\prime }=f+\frac{\left(l-f\right)f^2}{{\left(l-f\right)}^2+\left(\frac{\pi {\omega }_0^2}{\lambda }\right)}$$

When $\left(\frac{\pi {\omega }_0^2}{\lambda }\right)\ll {\left(l-f\right)}^2$, l and l′ satisfy the imaging formula, namely 1/l + 1/l′ = 1/f. When l > f, as the propagation distance gradually increases, the interval ∆d′ = d_i0 − l′ between the beam waist and the imaging plane gradually decreases, and the beam waist size of the transmitted beam also gradually decreases.

On the other hand, the spot size reaching the imaging lens gradually increases with increasing propagation distance. The modulation effect of the phase masks gradually increases when the spot size gradually increases to become equal to the size of the imaging lens, because the modulation effect in the central area of the phase masks is relatively weak, while at the edge it is strong. Further increase in spot size causes the proportion of the spot size received by the imaging lens to the overall spot size to decrease gradually; that is, the proportion of the received energy decreases progressively.

When the propagation distance varies from 100 m to 200 m, the maximum single-pixel receiving power of the conventional and wavefront-coding imaging systems rises rapidly. At the same time, the maximum single-pixel receiving power of the conventional imaging system is close to or even lower than that of the wavefront-coding imaging system. As the propagation distance increases, the interval ∆d′ between the beam waist and the imaging plane decreases, and the waist size of the transmitted beam gradually decreases, which together lead to the increase of light intensity on the imaging plane. The larger focal depth of the wavefront-coding imaging system is equivalent to a smaller interval ∆d′ between the beam waist and the imaging plane, resulting in a higher maximum single-pixel receiving power. This viewpoint can be verified by comparing the variation of the maximum single-pixel receiving power at the beam waist position with the propagation distance (100 m to 1,000 m), shown in the inset of Figure 8.

When the propagation distance varies from 200 m to 525 m, the maximum single-pixel receiving power of the conventional imaging system gradually increases. In this process, the interval ∆d′ between the beam waist and the imaging plane continues to decrease, and the waist size of the transmitted beam continues to decrease, which together lead to a further increase of the light intensity on the imaging plane. However, the maximum single-pixel receiving power of wavefront-coding imaging system gradually decreases. Because the spot size reaching the imaging lens gradually increases until it is equal to the size of the imaging lens, the modulation effect of the phase masks becomes more and more prominent. A part of the energy is dispersed to the side lobes of the L-shaped beam, resulting in a decrease in the maximum single-pixel receiving power.

When the propagation distance varies from 525 m to 2,000 m, the maximum single-pixel receiving power of the conventional imaging system begins to decrease. The maximum single-pixel receiving power of the wavefront-coding imaging system continues to decrease. Because the spot size reaching the imaging lens begins to exceed the size of the imaging lens and continues to increase, the proportion of the spot size received by the imaging system to the overall spot size gradually decreases, resulting in a decrease of the maximum single-pixel receiving power.

When the propagation distance varies from 2,000 m to 100 km, the maximum single-pixel receiving power of both conventional and wavefront-coding imaging systems remains stable, because the laser beam has been transformed from a spherical wave to a plane wave. The proportion of the spot size received by the imaging system to the overall spot size remains stable, and the maximum single-pixel receiving power also remains steady. The maximum single-pixel receiving power of the conventional imaging system is the largest, about 16.497 W, and that of the CPM is smallest, about 0.603 W, with a decrease of 96.34%.

Figure 9 shows that when the propagation distance varies from 200 m to 2,000 m, the decreasing amplitude of the maximum single-pixel receiving power increases rapidly, and the decreasing amplitude of CPM is the largest. When the propagation distance is 200 m, the decreasing amplitude of the maximum single-pixel receiving power of CPM, improved-1 LPM, improved-2 LPM, SPM, TPM, and ASPM is 47.49%, 31.28%, 21.74%, 24.96%, 33.76%, and 27.27% respectively. When the propagation distance is 2,000 m, the decreasing amplitude increases to 96.00%, 91.75%, 91.52%, 92.27%, 93.88%, and 92.55% respectively. When the propagation distance varies from 2,000 m to 100 km, the decreasing amplitude increases extremely slowly and remains basically stable. When the propagation distance is 100 km, the decreasing amplitude of the maximum single-pixel receiving power of CPM, improved-1 LPM, improved-2 LPM, SPM, TPM, and ASPM reaches 96.35%, 92.57%, 92.22%, 92.91%, 94.41%, and 93.17% respectively.

Figure 9. Variation of the decreasing amplitude of the maximum single-pixel receiving power with propagation distance.

5.2. Simulation Results for Anti-laser Detection

Figure 10 shows the variation of the echo-detection receiving power with propagation distance on the observation plane of the conventional and asymmetric-phase-mask wavefront-coding imaging systems. When the propagation distance varies from 100 m to 300 m, the echo-detection receiving power of both conventional and wavefront-coding imaging systems is stable at about 1,000 mW, because the echo spot size is smaller than the echo-detection area size, and the echo-detection area completely receives the echo spot. When the propagation distance varies from 300 m to 100 km, the echo-detection receiving power of the conventional and wavefront-coding imaging systems decreases rapidly until it is close to zero. Because the imaging system’s cat-eye echo size increases rapidly with increasing propagation distance, the proportion of the echo size received by the echo detection area to the overall echo size gradually decreases, and the echo-detection receiving power decreases accordingly. It is noteworthy that the curves for improved-1 LPM and improved-2 LPM overlap and the echo-detection receiving power is basically the same, with a difference of less than 1.17 mW. The echo-detection receiving power of improved-1 LPM and improved-2 LPM is the smallest among the six types of asymmetric phase masks between 100 m and 100 km.

Figure 10. Variation of the echo-detection receiving power with propagation distance.

This paper studies and compares the variation of the anti-laser damage and detection performance of asymmetric phase masks with defocus distance and propagation distance, when the optimal phase mask parameters are used. When exploring the influence of defocus distance, the anti-laser damage performance of different asymmetric phase masks is slightly different. Compared to conventional imaging, the maximum single-pixel receiving power is reduced by about one order of magnitude. The anti-laser detection performance of various asymmetric phase masks tends to be consistent. Compared to conventional imaging, the echo-detection receiving power is reduced by about two orders of magnitude.

When exploring the influence of propagation distance, the results illustrate that the maximum single-pixel receiving power of a conventional imaging system rises rapidly between 100 m and 525 m, and falls off rapidly between 525 m and 2,000 m. The maximum single-pixel receiving power of a wavefront-coding imaging system rises rapidly between 100 m and 200 m, and falls off rapidly between about 200 m and 2,000 m. The maximum single-pixel receiving power of both conventional and wavefront-coding imaging systems is basically stable between 2,000 m and 100 km. Comparing the anti-laser damage performance of asymmetric phase masks, the results show that CPM is the best between 175 m and 100 km. The echo-detection receiving power of both conventional and wavefront-coding imaging systems is stable between 100 m and 300 m, and drops rapidly between 300 m and 100 km until it approaches zero. Comparing the anti-laser detection performance of various asymmetric phase masks, the results show that improved-1 LPM and improved-2 LPM are the best between 100 m and 100 km.

In conclusion, asymmetric phase masks have the potential to improve the anti-laser damage and detection performance of imaging systems by one and two orders of magnitude respectively, through defocusing. The simulation results can guide the selection of appropriate asymmetric phase masks according to the laser protection performance requirements. For example, CPM can be selected for an imaging system needing anti-laser damage performance, and improved-1 LPM or improved-2 LPM can be selected for a system needing anti-laser detection performance. A suitable defocus distance can be further selected to meet the performance requirements of the imaging system.

The authors declare no conflicts of interest.

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.

Technology Domain Fund of 173 Project (2021-JCJQ-JJ-0284); Anhui Provincial Natural Science Foundation (1908085QF275); Natural Science Foundation of Anhui Province (1908085MF199); Research Project of the National University of Defense Technology (ZK20-41).