1. Introduction

Underwater exploration is vital in marine and underwater rescue, and obtaining underwater images is crucial to understanding oceans. Currently, there are two main types of underwater detection techniques: optical imaging [1,2] and acoustic imaging [3,4], with underwater optical imaging having the advantage of high resolution, unlike acoustic imaging. Further, there are two types of underwater optical imaging: scanning and nonscanning. Scanning optical imaging techniques include conventional laser point scanning imaging [5,6], streak tube laser imaging [7,8], and vortex optical imaging [9,10], while non-scanning optical imaging methods include underwater range-gated intensified charge coupled device (ICCD) imaging [11,12], submerged single-photon array detector imaging [13] and optical polarization imaging [14–18]. Underwater range-gated ICCD imaging and optical polarization imaging require high system emission energy and relatively short imaging distances. Single-photon array detector imaging has a longer detection distance, but the cost of the sensor is higher. Scanning optical imaging has a low resolution that is insufficient for capturing small and moving underwater targets. Streak tube laser imaging exhibits a large amount of noise, which is not conducive to target identification, and vortex light imaging can filter out scattered light but requires a complex modulation system. Underwater ghost imaging is a non-scanning gaze imaging method and an important computational imaging method. Ghost imaging uses a bucket detector to sample the target multiple times and then reconstructs the target image with the patterns of the reference light path. Because ghost imaging only requires detection of the sum of the light intensities reflected from the target, it has higher detection sensitivity and longer imaging distances than conventional array imaging, which reduces the requirement for transmit power. Since ghost imaging can eliminate the effect of optical transmission process on resolution by determining the point spread function in the optical path, it has higher resolution than scanning imaging, and can better image and identify small underwater targets [19–22]. Meanwhile, for small moving targets, scanning imaging cannot realize the high frequency tracking and imaging like ghost imaging and array imaging. As a result, ghost imaging techniques have been increasingly emphasized in underwater detection.

Underwater ghost imaging has been investigated regarding a number of different research fields. Several studies have validated the unique advantages of ghost imaging in the imaging of scattering media from different perspectives, including Gong’s early application of ghost imaging to the detection of targets in turbid liquids and the validation of ghost imaging as being resistant to interference [23]; Le et al. investigated the effects of turbidity and field of view on computational ghost imaging [24]. To address the effect of water body backscatter on ghost imaging, Chen et al. analyzed the system and water body parameters affecting the imaging performance in pulsed laser ghost imaging according to the underwater LiDAR equation to control the effect of backscatter [25]. Some other scholars have combined polarization imaging with ghost imaging to achieve anti-scattering effects [26–28]. To analyze the role of underwater turbulence in ghost imaging, Luo and Zhang developed physical models for underwater turbulence ghost imaging [29,30]. In a study on underwater illumination pattern optimization, Yang verified that the Hadama pattern could effectively improve the quality of underwater ghost imaging [31]. In terms of algorithms, differential ghost imaging [32], binary method ghost imaging [33], the rectified temporally corrected correlation algorithm [34], and neural networks [35,36] are some of the schemes that can effectively improve the imaging quality of underwater ghost imaging. There were also a number of studies related to ghost imaging and scattering-assisted techniques in different frequency ranges [37,38].

The imaging capability limitations of underwater ghost imaging are important metrics for underwater imaging research. Previous studies have investigated the limitations of the imaging capability of underwater ghost imaging systems, mainly in terms of energy and signal-to-noise ratio. Water is a typical light absorbing and scattering medium, and for ghost imaging, in addition to the energy attenuation caused by the absorption and scattering effects of the water body during light transmission underwater, the “smoothing” effect of the scattering of the water body on the intensity fluctuations of the light field can also limit the imaging distance and image quality of the ghost imaging system. In this research, we used the Wells model [39] and the approximate Sahu-Shanmugam (S-S) scattering phase function [40,41] related to the intrinsic optical parameters of the water body to construct an underwater optical transmission model. We then used the second-order Glauber function of the optical field [42] to investigate the degradation of the scattering field during the underwater transmission process as well as to analyze the effect of the water body on ghost imaging. The performance of underwater ghost imaging was verified experimentally, and the consistency between the simulation and experimental results was verified in terms of the degrading effect of underwater scattering on ghost imaging results. Gated photomultiplier tube (PMT) was employed in the system to minimize the effects of backward scattering so that a larger gain can be set for longer range detection of targets. In the experiments, the diffuse-reflection target imaging distance reached 6.4 ALs and the high-reflective target imaging distance reached 9.3 ALs.

2. Theoretical analysis of underwater ghost imaging

In this study, the effects of the inherent optical parameters (e.g., scattering coefficient b and absorption coefficient a) of a homogeneous stationary water body on ghost imaging were investigated, and the effect of turbulence was temporarily excluded from the scope of this study. The absorption coefficient is mainly caused by the absorption of light at specific wavelengths by water molecules and colored dissolved organic matter (e.g., chlorophyll), which reduces the transmission energy of light [26] and shortens the detection distance. Scattering is mainly caused by tiny particles in water, dominated by Mie scattering [26,27], which deflects light from its original propagation direction and causes degradation of the speckle field.

Ghost imaging only requires the collection of fluctuations in the relative intensity of light reflected from a target in the form of a bucket detector. With sufficient detection energy, the static scattering medium in the return optical path does not affect the detection of relative intensity fluctuations by ghost imaging. The main focus of this study was the effect of speckle field transmission from the emitting side to the object surface in a water medium on ghost imaging.

A principle diagram of underwater ghost imaging is presented in Fig. 1. Pseudothermal light is generated by a laser beam passing through a rotating hair glass, which is split into detection and reference light paths by a beam splitter. In the detection light path, the pattern was directed through the water body onto the object, and the light reflected from the object was received by bucket detector $I_{Bh}^{(i)}$ through the water medium. In the reference light path, the pattern propagates directly through air to the charge coupled device (CCD) camera $I_r^{(i)}({x,y} )$ (where $i = 1 \cdots M$ is the number of samples).

 

Fig. 1. Principle diagram of underwater pseudo-thermal light ghost imaging.

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The optical field at the target surface $E_t^{(i )}({x,y} )$ is related to that at the emission surface of the ground glass $E_0^{(i )}({x,y} )$, the coherent transfer function of the detection optical path within system ${h_{t1}}({x,y} )$, and the coherent transport process in the water body between the target and the system. In the simulation performed in this study, the intensity distribution of pseudo-thermal light coherently transmitted to the target surface is convolved with the incoherent point spread function (PSF) to obtain the target surface light field. The scattered light phase is disrupted when coherent light passes through the dynamic scattering medium, and the target surface cannot form a pattern. Only unscattered channel light could be coherently transmitted to form a pattern field. Channel light propagation in water is equivalent to coherent transmission in a medium without scattering substances and with the same refractive index as that of pure water. While the phase of the scattered light becomes random in the water body, its scattering transmission process is similar to the incoherent light propagation in the water body. The speckle pattern intensity distribution of the target surface in water is expressed as:

The optical field of the CCD imaging surface in the reference optical path $E_r^{(i )}({x,y} )$ is the convolution of the transmitted optical field of the ground glass surface $E_0^{(i )}({x,y} )$ with the coherent transfer function of the reference optical path within the system ${h_r}({x,y} )$. The pattern intensity distribution recorded by the reference optical path CCD is expressed as:

To evaluate the effect of pattern degradation caused by aqueous media on ghost imaging, a second-order mutual correlation (i.e., the normalized second-order Glauber function) was adopted in this study to reveal the irregularity of thermal light intensity fluctuations on the object surface [35]. The normalized second-order Glauber function for two points $({x_1},{y_1},{t_1})$ and $({x_2},{y_2},{t_2})$ in space–time is defined as:

The scattering and absorption effects of water diminish beam spread and intensity. Many scholars have described the transmission of light in stable homogeneous water as a PSF or modulation transfer function (MTF) related to inherent optical properties (IOPs) (i.e., scattering phase function $s(\theta )$, absorption coefficient, and scattering coefficient) and transmission distance $z$ [43]. Duntley, Wells, and Dolin obtained respective PSF and MTF models through experimental and theoretical derivations [39,44,45]. Wells derived an MTF model in the frequency domain from the firstness principle [39], which can accurately describe the transmission process of an underwater scattered beam in a small-angle approximation. The correctness of the model was verified in [46] and other studies. In this study, the Sahu-Shanmugam (S-S) scattering phase function proposed in the literature was linearly approximated in a small angular range (${0.1^ \circ } \sim {5^ \circ }$) in a logarithmic coordinate system [41]. The scattering phase function was then Hankel-transformed and substituted into the Wells’ MTF model to obtain the MTF of the pattern propagation in water. The MTF is the frequency-domain expression of the PSF that can describe the degrading effect of seawater on the beam.

As shown in Appendix 1, the fitting accuracies of the proposed approximate linear logarithmic scattering phase function, S-S scattering phase function, and F-F scattering phase function were much better than those of the OTHG and TTHG scattering phase functions in the range of ${0.1^ \circ } \sim {5^ \circ }$. The scattering phase function proposed in this study can not only approximate the S-S scattering phase function well to achieve a good fitting effect, but also has a simpler mathematical form than the S-S and F-F scattering phase functions, which is more conducive to practical derivation and engineering applications. An approximate derivation of the scattering phase function is presented in Appendix 1. The normalized scattering phase function is Hankel-transformed to obtain:

Substituting Eq. (7) into the Wells model:

The scattering phase-function parameters in the simulation were considered as the conditions for ${B_p} = 0.0183$. As shown in Figs. 2(b) and 2(d), for different distances (z = 22.8, 31, 37.6, 45.8, 54 m) and scattering albedo $\omega = b/c = 0.4$, $g_{MAX}^{(2 )}$ and $g_{FWHM}^{(2 )}$ decrease with increasing b; Figs. 2(a) and 2(c) show the decrease in $g_{MAX}^{(2 )}$ and $g_{FWHM}^{(2 )}$ with increasing distance z for albedos $\omega = 0.4$ and different scattering coefficients ($b$= 0.03, 0.06, 0.09, 0.12, 0.15 /m). Thus, the simulation results reflect the effect of water scattering on pseudo-thermal optical ghost imaging.

 

Fig. 2. Comparison of trends of $g_{MAX}^{(2 )}$ and $g_{FWHM}^{(2 )}$ at different b or z. (a) FWHM of ${g^{(2 )}}({x,y} )$ with distance z; (b) FWHM of ${g^{(2 )}}({x,y} )$ with b; (c) Maximum value of ${g^{(2 )}}({x,y} )$ with distance z; (d) Maximum value of ${g^{(2 )}}({x,y} )$ with scattering coefficient b.

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3. Experimental research

The experimental device comprised a transmitting modulation system, receiving system, control system, and sealed housing, using laser irradiation-rotating ground glass to produce intensity fluctuations of the patterns. All experiments in this research were performed in the same underwater sealed box. The sealed box ensures that the optoelectronic system operates in a relatively stable environment. On the one hand, the sealing box pass-through aperture is planar glass, which only makes the emission angle smaller due to the refraction of light and does not affect the intensity distribution of the patterns. On the other hand, the relative fluctuation in the intensity of the received reflected light from the target is not influenced. Therefore the watertight box does not affect underwater ghost imaging. A bucket detector gating technique was employed to obtain the reflected light intensity of the target. To reduce the impact of backward scattering as much as possible, the receiving and transmitting systems were placed separately in a double-layer sealed box. A pulsed laser with a central wavelength of 532 nm is incorporated to irradiate the rotating ground glass and produce pseudo-thermal light. The pseudo-thermal light source generated by the rotating ground glass has a higher peak damage power compared to DMD and SLM, which is suitable for long-range high-power ghost imaging. The laser has an operating frequency of 2KHz, a single pulse energy of 2 mJ, a pulse width of 10 ns, an average power of 4W and a peak power of $2 \times {10^5}$W. The single pulsed light energy of the system output is 37 µJ. The receiving bucket detector constructed by a PMT. The PMT uses a HAMAMATSU photoelectric sensor module with a gating function. The sampling frequency of PMT was set to 1 GHz. The highest sampling frequency of the PMT can achieve is 2 GHz, so the system can reach a range accuracy of 15 cm. A diagram of the transmitting and receiving systems is shown in Fig. 3. The CCD in the transmitting system collects the reference arm pattern intensity distribution while the PMT in the receiving system collects the object arm bucket detection values reflected from the target. The underwater emission angle of the system was 0.43°. Both the reference arm pattern and resulting image were 256 × 256 pixels.

 

Fig. 3. Transmitting and receiving device diagram. (a) Optical system package; (b) Underwater experimental seal; (c) Receiving optical system; (d) Transmitting optical system. The receiving and transmitting systems were placed separately in a double-layer sealed box. The receiving system uses the pulsed laser of 532 nm wavelength to irradiate the rotating rough glass and produce pseudo-thermal light. The reference arm CCD records patterns of the reference arm. The PMT in the receiving system collects the object arm bucket detection values reflected from the target. The monitor CCD in the receiving system is used to adjust the position of the objects in the field of view.

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Pulsed laser ghost imaging can obtain the echo energy near the target location. The device used a gated PMT to sample the target echoes. The gated PMT differs from the conventional PMT in that the PMT opening time can be manually adjusted when sampling synchronously with the reference arm CCD. Thus, the backward scattering peak that was much larger than the PMT threshold was placed outside the gating time. It ensures that the system can set large PMT gain values without over-saturation and damage. For reconstruction calculation, the intensity values in the range of 10 ns before and after the peak ${\tau _0}$ of the target echo recorded by the PMT are summed as the one sample value of the detection arm. As shown in Fig. 4, the gated PMT can effectively skip the backscatter peak, thus protecting the PMT from damage when the gain is large.

 

Fig. 4. Comparison of gated PMT and non-gated PMT echo curves. The echo intensities of diffuse-reflection targets “Turtle” and backscattering were detected using two PMTs. The backscatter peaks were removed by controlling the gate opening time of each PMT sample. The peak position of the target is the moment marked by ${\tau _0}$, which represents the same location of the target. The moment of gated PMT opening in each sample was set to 150 ns position. (a) Non-gated PMT echo curve; (b) Gated PMT echo curve.

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The underwater environment for the experiments was a large ship-towing pond rich in algae and suspended microscopic particles that played a very important role in the absorption and scattering of light. The scattering and absorption coefficients of the water bodies were measured using an AC-S hyperspectral meter (WET Labs, USA). This instrument measures the attenuation coefficients c and absorption coefficients a for each wavelength of light in water and calculates the scattering coefficient $b = c - a$. According to the Beer-Lambert Law, the attenuation effect of light in water can be defined as $I = {I_0}{e^{ - cz}}$. I and ${I_0}$ are the light intensity after propagation distance z and that at the light source, respectively. Meanwhile, $cz ={-} \ln ({{I / {{I_0}}}} )= 1$ is defined as one attenuation length, which represents the propagation distance at which the light intensity is attenuated 1/e = 0.3679 times.

To more specifically characterize the effect of the aqueous medium on imaging quality, underwater pseudo-thermal optical ghost imaging simulations and experiments with highly reflective underwater targets of the corresponding size were performed for the three-slit image displayed in Fig. 5. To verify the theory presented in this study, experiments and simulations were performed under the same conditions: the emission angle of the system was 0.43°; the pattern size at the emission end was 3 cm × 3 cm; and the reference arm pattern and resulting image were both 256 pixels × 256 pixels. To exclude the effect of sampling rate on imaging, the number of experimental and simulation samples were set to 10,000 in Fig. 5.

 

Fig. 5. Comparison of experimental and simulation results. (a1)–(d1) Simulation result; (a2)–(d2) Experimental result; (a3)–(d3) Echo curve obtained from the PMT acquired in the experiment. Figure (e) is the original diagram “three slits” used for the simulation. Figure (f) shows the resolution curves at the (a1)-(d1) frames marked for the simulation results. Figure (g) shows the resolution curves at the (a2)-(d2) frames marked for the experimental results. The simulation results correspond to the experimental results one by one. The parameters in the simulation were the same as those in the experiment. (a1)–(d1) and (a2)–(d2) are the imaging results for targets of different sizes at different distances to guarantee the same angular resolution. The target slit spacings of the “three slits” used in (a1)–(d1) and (a2)–(d2) are 3.5, 4.5, 5.6, and 6.8 mm; the underwater distances from the target to the system are 22.8, 30.1, 37.6, and 45.8 m; and the gains of the PMT are 0.36, 0.48, 0.65 and 0.75, respectively. Scattering coefficients b of the water bodies measured in simulations (a1)–(d1) and in experiments (a2)–(d2) were 0.081, 0.066, 0.066, and 0.061, while absorption coefficients a were 0.076, 0.079, 0.079, and 0.061, respectively.

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Because the emission angle was fixed, we ensured that the imaging results of the objects had the same scale by changing the target distance and size. As shown in Fig. 4, the original GI algorithm was adopted for both the simulation and the experiment. The difference is that the simulation did not consider the detection signal-to-noise ratio. The results of the simulation and experimental reconstruction were compared, as shown in Figs. 2 and 4, demonstrating similarity between the decreasing trends of resolution and speckle degradation of the detection arm under water, and the experimental and MTF theoretical simulation results are consistent.

Figures 6(a) and 6(b) present the experimental results. It is clear that the echo energy of the target at 54 m was weak and the signal-to-noise ratio was poor. To further clarify the effect of water on ghost imaging, simulations were performed for imaging at longer distances (Figs. 6(c–g) with $b = \textrm{0}\textrm{.06}{\textrm{m}^{ - 1}}$ and $\omega = b/c = 0.4$. The target slit spacings of the “three slits” used in Figs. 6(c)-(f) are 7.5, 10.5, 13.5, and 16.5 mm. The simulation and experimental reconstruction results were obtained at 10,000 samples. The resolution curves in Figs. 6(g) are normalized by intercepting the three slits marked in the frame diagram. The results show that the resolution of the target gradually decreases with increasing distance at the same angular resolution. The target was resolved exactly at three slits, z = 70 m from the system (refer to the Rayleigh criterion; the intensity value at the depression was 81% of the intensity value at the peak). The simulation results indicate that a system with the underwater angular resolution of 0.15 mrad can remain resolvable for 10.5 ALs if the energy emitted is sufficiently large.

 

Fig. 6. (1) Experimental results: (a) Second-order correlation calculation results for a three-slit target with slit width of 8 mm at distance of 54 m underwater and PMT gain of 0.9; (b) PMT echo curve of the experiment. (2) Simulation results for underwater distances of (c) 50 m, (d) 70 m, (e) 90 m, and (f) 110 m; (g) Resolution curves obtained for varying z.

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Ghost imaging is a kind of computational imaging, and the image reconstruction algorithm has a great influence on the imaging results. Currently, in ghost imaging research, the ghost imaging (GI) algorithm gives an unbiased estimate of the target and is generally used as a benchmark algorithm. Differential ghost imaging (DGI) is a well-established linear image restoration algorithm, because DGI has been used in other underwater ghost imaging studies, the reconstruction results with DGI algorithm is also presented in this study to facilitate comparison with other studies [32]. The total variation augmented lagrangian alternating direction (TVAL3) algorithm is a typical compressed sensing algorithm that has been commonly used in ghost imaging studies, and it can achieve much better image restoration than linear algorithms, making it a more practical algorithm than the GI algorithms that are usually used only as a test benchmark for image restoration [47].

In order to characterize the effect of different material targets on imaging in this study, experiments were conducted using two different types of different targets. And two diffuse-reflection targets (i.e., “Turtle” and “White propeller”) and two high-reflective targets (i.e., “frogman” and “highly reflective propeller”) were used to verify the longest distance that can be imaged by this system. The target is depicted in Fig. 7. The size of the turtle was 12 cm × 13 cm; the diameter of the white propeller was 20 cm; the length of the frogman was 43 cm; and the diameter of the highly reflective propeller was 31 cm. Diffuse-reflection target “Turtle” is a toy made of plastic. Diffuse-reflection target “White propeller” is a white metal propeller used on model ships. The high-reflective target is made of 3M’s Diamond Grade white reflective film (Type 4090 T, conforming to ASTM4956-2001, Class XI reflective film, mainly used for traffic signs).

 

Fig. 7. Diffuse-reflection targets and high-reflective targets. (1) Diffuse-reflection targets: (a) “Turtle” and (b) “White propeller.” (2) High-reflective targets: (c) “Frogman” and (d) “Highly reflective propeller.”

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As shown in Fig. 8, as the distance increased, the PMT detector gating time was adjusted to move toward the target position such that the backscatter was outside the gating range as much as possible. However, as the PMT gain increased, the sensor noise also increased, and the echo signal-to-noise ratio at the target position deteriorated. In the imaging experiment of the diffuse-reflection target, scattering coefficient b = 0.081 m^-1 and attenuation coefficient c = 0.1569 m^-1 were measured. The sampling rate of the diffuse-reflection targets was 10,000 times. The images show that the turtle was imaged at a distance of approximately 34.9 m. The white propeller had a greater reflectivity and was imaged at a distance of approximately 41.2 m. Meanwhile, the imaging distances for the diffuse-reflection targets of “turtle” and “white propeller” were calculated to be 5.46 ALs and 6.4 ALs, respectively.

 

Fig. 8. “Turtle” and “white propeller” imaging results at different distances. Reconstruction results of (1) and (5) GI algorithm, (2) and (6) DGI algorithm, and (3) and (7) TVAL3 algorithm. (4) and (8) Curve of echo signals. (1)–(4) Imaging results of “turtle” target, the imaging distances from (a) to (e) are 22.8, 25.8, 28.8, 31.8, and 34.8 m, respectively; the corresponding PMT gains are 0.55, 0.62, 0.75, 0.82, and 0.90. (5)–(8) Imaging results of “white propeller” target, the imaging distances from (a) to (e) are 29.1, 32.2, 35.2, 38.0, and 41.2 m, respectively; the corresponding PMT gains are 0.60, 0.65, 0.72, 0.75, and 0.90.

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The imaging results of the high-reflective target are shown in Fig. 9. In the imaging experiment of the high-reflective target, scattering coefficient b = 0.052 m^-1 and attenuation coefficient c = 0.1426 m^-1 were measured. In order to avoid the effect on the longest imaging distance of the system due to the small number of samples, the reconstruction results were calculated with the maximum 50,000 samples that the system can accommodate. The image quality decreased as the received signal-to-noise ratio decreased. The experimental high-reflective target was imaged underwater at the distance of 65.2 m. Combining the optical parameters of the water column, the maximum ALs was calculated to be 9.3 ALs. As can be seen from the PMT amplification gain variation and echo curves, the signal detected at the detection distance limit of the system contained a large amount of noise, resulting in eventual failure of the image.

 

Fig. 9. “Frogman” and “highly reflective propeller” imaging results at different distances. Reconstruction results of (1) and (5) GI algorithm, (2) and (6) DGI algorithm, (3) and (7) TVAL3 algorithm. (4) and (8) Curves of echo signals. (1)–(4) Imaging results of “frogman” target, the imaging distances from (a) to (e) are 55.1, 57.0, 59.1, 61.1, 63.1, and 65.2 m, respectively; the corresponding PMT gains are 0.67, 0.72, 0.78, 0.83, 0.88, and 0.95. (5)–(8) Imaging results of “highly reflective propeller” target, the imaging distances from (a) to (e) are 55.0, 57.0, 59.0, 61.1, 63.1, and 65.1 m, respectively; the corresponding PMT gains are 0.67, 0.72, 0.78, 0.83, 0.86, and 0.92.

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4. Discussion

1. Pseudo-thermal light ghost imaging has a small field of view and high resolution. This requires measurement and fitting of very small angular scattering phase functions (<0.1°). In this study, the field of view was 0.43°. The instrumental accuracy of the measured scattering function was only 0.1°. In this study, the volume scattering function (or scattering phase function) for less than 0.1° was fitted using a log-linear approximation in Cartesian coordinates, and the accuracy can be further improved. A more accurate PSF requires a more accurate VSF, and the limitations of VSF measurement instruments do not allow for more accurate measurements of the VSF. Instruments that measure the volume scattering function at very small angles will allow for a more accurate PSF and facilitate the development of underwater high-resolution optical imaging.

2. In our experiments, combining a pulsed laser with distance-selective slice imaging was applied to filter out most of the backscattering noise.The small amount of remaining noise has been further removed with reasonable optical design. Backscatter might be better filtered by combining other methods. For example, combining ghost imaging with coherent detection can utilize the optical coherence property to filter out the scattered light in the echo signal.

3. Because of the different properties of water and air, the steep attenuation of energy cannot be ignored in underwater optical detection. For practical systems, the maximum effective imaging (required resolution) distance is related not only to the degradation of the spot that shines on the target, as analyzed in this study, but also to the reflected light energy and signal-to-noise ratio of the target obtained by the detector. The energy and detection S/N ratios can be derived from the underwater lidar equations and are related to the actual transmitting power of the system, transmitting angle, receiving aperture, angle between the transmitting and receiving axes, reflective properties of the target, and inherent optical parameters of the water body (scattering coefficient, absorption coefficient, attenuation coefficient, body scattering function, backward scattering rate, etc.).

5. Conclusions

In this study, the Wells model and an approximate S-S scattering phase function were adopted to reveal the degrading effect of water on the spot transmission of the detection optical path. A normalized second-order Glauber function was considered to investigate the effects of water on ghost imaging. In contrast to conventional ghost imaging, gated PMT was used to filter out backscatter peaks, allowing greater gain to be set for longer-range detection of the target. The diffuse-reflection target imaging distance reached 6.4 ALs and the high-reflective target imaging distance reached 9.3 ALs. If the target is enlarged in equal proportion according to the system emission angle (i.e., the target has the same duty cycle in the field of view), according to the underwater lidar equation and the Beer-Lambert Law under the premise that the laser energy is large enough, the longest detection distance of 9.3 ALs obtained from the original experiment can be can be equivalent to 193.7 m distance under Jerlov-I water body. This study provides a valid reference for long-range underwater pseudothermal optical ghost imaging. Further, long range can be achieved for underwater ghost imaging through optimization of the emission energy and detection sensitivity of the system.

Appendix 1 Deduction and simulation of scattering phase function

Because of the small field-of-view of ghost imaging (imaging field-of-view of the experimental device was 0.43°), this research was only concerned with the scattering phase-function matching accuracy in the small-angle range. The S-S scattering phase function [41] is based on the Mie scattering theory to approximate the polydisperse particle system (typical marine environment) to a monodisperse particle system to obtain the scattering phase function $s(\theta )$ fitted in the angle range of 0.1–5°:

(11)$$\lg [{s(\theta )} ]= {P_1} \cdot {[{\ln (\theta )} ]^2} + {P_2} \cdot \ln (\theta ) + {P_3}$$

where $\theta $ is the scattering angle and ${P_i}$ (i = 1,2,3) is a function of the slope of the particle size distribution ($\xi $) and the real index of refraction of the particles ($n$). $\xi $, n, and ${P_i}$ can be found using backward scattering rate ${B_P}$ and optical wavelength $\lambda $, and the specific parameters are calculated referring to literature [41]. Moreover, referring to the method of Petzold [48] for the logarithmic linear extension of the phase function for very small angular scattering, the logarithmic function linear approximation of the S-S scattering phase function in Eq. (11) in the range of 0.1–5° is presented as follows:

(12)$$\lg [{s(\theta )} ]= p \cdot \lg (\theta )+ q$$

where p is the slope of the linearly fitted log-log curve and q is the intercept. The range of $\theta $ extends from 0 to 5°. Equations (11) and (12) are equal on both sides of the equal sign; i.e., ${P_1} \cdot {[{\ln (\theta )} ]^2} + {P_2} \cdot \ln (\theta ) + {P_3} = p \cdot \lg (\theta )+ q$. The approximations in Eqs. (11) and (12) (using $\ln (10)\ln (\theta ) \approx{-} 5$and $\ln (10) \approx 2.3$) are obtained by solving for the minimum root-mean-square error (RMSE) of the two functions in the range of 0.1–5°. The scattering phase function was obtained as follows:

(13)$$s(\theta ) = {10^{p \cdot \lg (\theta ) + q}} = {\theta ^p} \cdot {10^q} = {\theta ^{\ln (10) \cdot [{{P_1} \cdot \ln (\theta ) + {P_2}} ]}} \cdot {10^{{P_3}}} \approx {\theta ^{ - 5{P_1} + 2.3{P_2}}} \cdot {10^{{P_3}}}$$

Equation (13) is an approximation of Eq. (11). The scattering phase function is a normalized bulk scattering function whose integral over the spherical solid angle is 1: We normalized the obtained scattering phase function by dividing its integral over the spherical solid angle from 0 to π:

(14)$$\tilde{s}(\theta )= \frac{{s(\theta )}}{B} = {10^{{P_3}}}{B^{ - 1}}{\theta ^{ - 5{P_1} + 2.3{P_2}}}, $$

(15)$$B = 2\pi \int_0^\pi {s(\theta )} \sin (\theta )d\theta = 2\pi \int_0^\pi {{\theta ^{ - 4{P_1} + 2.3{P_2}}} \cdot {{10}^{{P_3}}}} \sin (\theta )d\theta . $$

When the optical property parameters of the water body are constant, B is the normalization constant. The normalized scattering phase function is Hankel transformed to obtain:

(16)$$\tilde{S}(\psi ) = {10^q}{[{\Gamma ( - {p / 2})} ]^{ - 2}}{B^{ - 1}}{\pi ^{ - p}} \cdot {[{\sin ({ - {{p\pi } / 2}} )} ]^{ - 1}}{(\psi )^{ - p - 2}}$$

where $\Gamma (u) = \int_0^\infty {{e^{ - t}}{t^{u - 1}}dt} $ is the second type of Euler integral function.

All phase functions were calculated for the corresponding backscattering ratio of B_p = 0.183, which is the ${B_p}$ value for Petzold’s average particle phase function, taken as the reference phase function in the present study. The curves for each phase function from 0.1° to 180° are shown in Fig. 10; however, this study was only concerned with approximating the small-angle range. As shown in Fig. 10, the scattering phase function of the experimental water body is measured by LISST-VSF (which is a volume scattering function measurement instrument developed by SEQUOIA). The error of the scattering phase function values for large angles is large due to the limitation of the instrument accuracy. It is observed from the figure that the experimental water scattering phase function is consistent with the Petzold phase function. Petzold's average phase function is used in many literatures as an evaluation criterion for the studies of scattering phase function in natural water bodies. Defining the RMSE rate, the approximate linear logarithmic phase function, the Sahu-Shanmugam (S-S) scattering phase function [41,49], the Fournier-Forand (F-F) phase function [50], the One-Term Henyey-Greenstein (OTHG) phase function [51], Haltrin's Two-Term Henyey-Greenstein (TTHG) phase function [52] and Petzold’s average phase function [48,53] were compared in the range of 0.1 to 5°. The parameters of the approximately linear logarithmic scattering phase function (proposed in this research), S-S scattering phase function, and F-F scattering phase function are the slope of the particle scale distribution $\xi = 3.4586$ and the refractive index of the main body of the particle $n = 1.16$. The RMSE rate between the scattering phase functions ${s_1}(\theta )$ and ${s_2}(\theta )$ is defined as:

(17)$$RMSE = {\left\{ {\frac{1}{K}\sum\nolimits_{i = 1}^K {{{\left[ {\frac{{{s_1}(\theta ) - {s_2}(\theta )}}{{0.5 \cdot [{{s_1}(\theta ) + {s_2}(\theta )} ]}}} \right]}^2}} } \right\}^{0.5}}$$

where K is the number of samples of the phase function measured by Petzold in the range of 0.1–5°. The RMSE rate between each scattering phase function and Petzold’s average particle phase function is listed in Table 1. The RMSE of the S-S scattering phase function and the linear approximate logarithmic phase function in this study are nearly equal, approximately 0.26. The fitting accuracies of the proposed approximate linear logarithmic scattering phase function, S-S scattering phase function, and F-F scattering phase function were much better than those of the OTHG and TTHG scattering phase functions in the range of 0.1–5°. The scattering phase function proposed in this study can not only approximate the S-S scattering phase function well to achieve a good fitting effect, but also has a simpler mathematical form than the S–S and F-F scattering phase functions, which is more conducive to practical derivation and engineering applications.

 

Fig. 10. Comparison of phase functions obtained through different models for scattering angles of 0.1–180° to emphasize the large deviation of different phase functions.

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Table 1. RMSE of different phase functions

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Funding

National Natural Science Foundation of China (61991454); CAS Interdisciplinary Innovation Team Project Grant.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data of Appendix 1 underlying the results presented in this paper are available in Ref. [48] and Ref. [53]. Other data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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