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Ghost imaging based on Fermat spiral laser array designed for remote sensing

Wenchang Lai, Guozhong Lei, Qi Meng, Yanxing Ma, Wenda Cui, Dongfeng Shi, Hao Liu, Yan Wang, and Kai Han

Open Access Open Access

Abstract

We propose a Fermat spiral laser array as illumination source in ghost imaging. Due to the aperiodic structure, the Fermat spiral laser array generates illuminating light field without spatial periodicity on the normalized second-order intensity correlation function. A single-pixel detector is used to receive the signal light from object for image reconstruction. The effects of laser array parameters on the quality of ghost imaging are analyzed comprehensively. Through experimental demonstration, the Fermat spiral laser array successfully achieves ghost imaging with high quality by combining with the compressive sensing reconstruction algorithm. This method is expected to be applied in remote sensing by combining with phased and collimated fiber laser array equipped with the high emitting power and high-speed modulation frequency.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Ghost imaging (GI) is a nonlocal imaging technique with a single-pixel detector receiving the total signal light intensity at the object path [1,2]. The object image is reconstructed by simultaneously measuring the illumination light field and the corresponding reflected (transmitted) light intensity from object for correlated computation. Thanks to the high detection sensitivity of single-pixel detector, GI has many applications such as remote sensing [3,4], 3D imaging [5,6] and so on. The first GI experiment was conducted by Pittman and co-workers using two entangled photons as light source in 1995 [7]. In fact, quantum entanglement was not necessary in GI and the classical thermal light sources were also successfully applied in GI experiment [8,9]. Owing to the detection difficulties for true thermal light, the pseudo-thermal light sources were usually utilized in present GI experiments [10–12].

Nowadays, in order to obtain fluctuating pseudo-thermal light field, different types of modulated light sources were proposed for GI, including dynamic diffuser (rotating ground glass plate) [11–13], liquid-crystal spatial light modulator (SLM) [14–16], digital micro-mirror device (DMD) [17–19], LED-based array [20–22] and Silicon-based optical phased array (OPA) [23,24]. However, the above light sources cannot promise both high emitting energy for remote sensing and high imaging speed at the same time. The imaging speed of GI experiment mainly depends on the refreshing speed of illumination light field and the reconstruction algorithm. As for the rotating ground glass plate and the SLM, the modulating frequency does not exceed hundreds of hertz. The refreshing frequency of DMD is no more than 40 kHz. Recently, a 2000000 fps 2D and 3D imaging of periodic or reproducible scenes has been achieved by using DMD and a single-pixel detector [19]. The LED-based array has high refreshing speed which can be up to MHz level. The GI scheme realized by LED-based array has achieved a frame rate of 1000 fps with 32${\times} $32 pixels resolution [20]. However, the emitted light from LED-based array is not collimated and has a large divergence angle, which limits the application in remote sensing. In order to further improve the refreshing speed of illumination light field, a chip-scaled integrated Silicon OPA has been used as a compact wavefront modulation device for GI experiments [23,24]. The electro-optic phase-shifters in OPA enable a high-speed switching frequency beyond 100 MHz. A multimode fiber (MMF) is used to project the illumination speckle patterns on object. Due to the limitation of integrated chip and MMF, the OPA cannot emit high power which limits the application in remote sensing.

In addition to the above light sources, laser array can also be used as the pseudo-thermal light source for GI experiments [25,26], which is expected to achieve high emitting energy and high modulating frequency. Take the phased fiber laser array as an example, the illumination light filed is formed when the sub-sources are coherently interfered in the far field. The emitting power can be improved by increasing the sub-laser’s output power, especially the single-frequency all-fiber laser has achieved hundreds of Watts output power [27]. Besides, each sub-fiber laser can be equipped with electro-optic phase modulator with up to tens of gigahertz (GHz) modulation frequency. Therefore, the phased fiber laser array can generate illumination light filed equipped with both high emitting power and high-speed refreshing frequency. As for GI experiment, the imaging quality is closely related to the normalized second-order intensity correlation function (g(2)(x, y; x0, y0)) of the illuminating light filed. The latter mainly depends on the geometrical configuration of the laser array. The periodic distribution of laser array will lead to periodic g(2)(x, y; x0, y0) of the illuminating light filed, which impairs the imaging quality of GI. In 2018, Gong, et al has proposed an optimized spatial configuration of sparse structured illumination source based on genetic algorithm to suppress the periodicity of g(2)(x, y; x0, y0) [25]. In the same year, another GI scheme with optical fiber phased array and a low-pixel APD array is reported [26]. The laser array has square distribution and the low-pixel detector array is applied to avoiding the image aliasing induced by spatial periodicity of the speckles. However, the researchers carried out a detailed theoretical analysis without any experimental demonstrations.

In this manuscript, we introduced a Fermat spiral laser array acted as illumination source for GI system. The aperiodic configuration of Fermat spiral laser array can generate illumination light filed without periodicity. Different from the optimized spatial configuration of sparse structured illumination source in Ref. [25], the sub-laser source of Fermat spiral laser array has determined coordinates satisfying the Fermat spiral function. Moreover, the Fermat spiral laser array can have a compact configuration to enable small optical system with more convenience. The effects of Fermat spiral laser array parameters on ghost imaging quality are analyzed in detail. We also established a demonstration experiment system based on a single-pixel detector to present the GI performance of Fermat spiral laser array.

2. Model and theory

2.1. Far-field distribution of coherent laser array

To facilitate the analysis for the GI performance of coherent laser array, the mathematical properties of the far-field, as well as the illumination light field, is necessary to be illustrated. Here, a N-element coherent fiber laser array is taken as an example for analyzation. The concept design of laser array system is shown in Fig. 1. The laser source is divided by a splitter and coupled into N polarization-maintaining fibers. Each single-road fiber laser is modulated by an independent high-speed phase modulator driven by random voltage signal. For each sub-laser source, the laser beams are tilted uniformly. The modulated fiber lasers are collimated through collimator array and enter into the free space. To obtain the far-field of coherent fiber laser array, a focusing lens is placed at the emitting plane after collimator array. Then the light field at the focal plane of lens can be regarded as the far-field of laser array.

 figure: Fig. 1.

Fig. 1. The schematic system of coherent fiber laser array.

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At the emitting plane of laser array, each laser beam output from single-mode fiber can be regarded as Gaussian beam with size of w0. Here we assume that each laser beam is truncated by a circular aperture with a diameter of D. Then the light field of the N laser beams at emitting plane can be expressed by

$$\begin{array}{l} {E_e}(x,y,z = 0) = \sum\limits_{n = 1}^N {{A_n}} \exp ( - \frac{{{{(x - {x_n})}^2} + {{(y - {y_n})}^2}}}{{w_0^2}})\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \times circ(\frac{{\sqrt {{{(x - {x_n})}^2} + {{(y - {y_n})}^2}} }}{{(D/2)}})\exp (j{\phi _n}) \end{array}$$
where An is the amplitude of the nth laser beam. (xn, yn) is the center coordinate of the nth laser beam. ϕn is the random phase of the nth laser beam. circ(r0) denotes the function of the circular aperture, which is defined as
$$circ({r_0}) = \left\{ \begin{array}{l} 1{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {r_0} \le 1{\kern 1pt} \\ 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {r_0} > 1 \end{array} \right.$$

As we know, the focusing lens with a focal length of f will bring a phase change of -ik(x2 + y2)/2f to the light field at emitting plane. k denotes the wavenumber of laser beam. According to the Fraunhofer diffraction theory, the light field at the focal plane can be expressed as [28]

$${E_r}(u,v,z = f) = \frac{{{e^{ikf}}}}{{i\lambda f}}\int_{ - \infty }^\infty {\int_{ - \infty }^\infty {{E_e}(x,y,z = 0) \cdot {e^{ - \frac{{ik}}{{2f}}({x^2} + {y^2})}} \cdot {e^{\frac{{ik}}{{2f}}[{{(u - x)}^2} + {{(v - y)}^2}]}}} } dxdy$$
where λ is the wavelength of laser beam and (u, v) is the coordinate parameter at focal plane. Accordingly, the intensity distribution of light field at focal plane can be calculated as
$${I_r}(u,v,z = f) = {E_r}(u,v,z = f) \cdot E_r^ \ast (u,v,z = f)$$

In the next sections, the theoretical calculation of far-field of laser array will be employed to analyze the spatial correlation properties and GI performance.

2.2. Spatial correlation function of laser arrays with different geometric configurations

As we know, the spatial correlation property of illumination light field can directly influence the quality of GI [25,26]. A typical parameter to evaluate the spatial correlation property is the normalized second-order intensity correlation function, which is defined as

$${g^{(2)}}(x,y;{x_0},{y_0}) = \frac{{\left\langle {I({x_0},{y_0})I(x,y)} \right\rangle }}{{\left\langle {I({x_0},{y_0})} \right\rangle \left\langle {I(x,y)} \right\rangle }}$$
where I(x0, y0) is the light intensity at the position (x0, y0) of the observed plane. means the ensemble average value of intensity I over time. As we can see, g(2)(x, y; x0, y0) describes the spatial cross-correlation coefficient of given two points, (x, y) and (x0, y0) on the observed plane.

According to the above analysis, the g(2)(x, y; x0, y0) distribution at far-field mainly depends on the geometric configuration of laser array. The most laser array system are usually designed to be placed along square distribution [26], hexagon distribution [29] and their derivative distributions. Especially, hexagon laser array has been proved to be most effective in the coherent beam combining (CBC) system owing to the high compactness [29]. However, in GI system, the regular distribution of laser array will lead to the periodicity of g(2)(x, y; x0, y0) [25,26], which can impair the quality of GI seriously. To overcome the periodicity and achieve only one peak of g(2)(x, y; x0, y0) at the observed plane, here we introduce an aperiodic laser array distribution called Fermat spiral. The Fermat spiral distribution has been proven to be useful for suppressing the sidelobe in CBC system [30]. It has also been applied in radars and acoustics for designing the synthetic aperture devices [31,32]. However, to the best of our knowledge, the Fermat spiral has never been used for designing laser array in GI system.

In the Fermat spiral laser array, each laser beam center is arranged along the Fermat spiral, which can be described as

$$\left\{ \begin{array}{l} {r_n} = s\sqrt {\frac{n}{\pi }} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} n = 1,2,3\ldots N{\kern 1pt} \\ {\theta_n} = 2\pi \beta n{\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} n = 1,2,3\ldots N \end{array} \right.$$
where (rn, θn) is the polar coordinate of the nth laser beam. s and β are the radial and angular parameters that control the compactness of Fermat spiral. In this manuscript, without loss of generality, 2πβ is set to be 137.5° to provide a relatively uniform distribution [30]. Then the compactness of Fermat spiral mainly depends on the value of s. In addition, the cartesian coordinate of nth laser beam can be expressed by
$$\left\{ \begin{array}{l} {x_n} = {r_n}cos({\theta_n}),{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} n = 1,2,3\ldots N{\kern 1pt} \\ {y_n} = {r_n}\sin ({\theta_n}){\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} n = 1,2,3\ldots N \end{array} \right.$$

Based on the above theoretical analysis, the far-field g(2)(x, y; x0, y0) distributions of different laser arrays are calculated and the results are shown in Fig. 2. There are three types of laser arrays being analyzed, which are square array, hexagon array and Fermat spiral array shown in Fig. 2(a1-a3), respectively. The laser wavelength is equal to be 1064 nm here. And the total number of laser beams is set to be N = 36 in square array, N = 37 in hexagon array and Fermat spiral array. Without loss of generality, the size of single Gaussian laser beam w0 and the diameter of circular aperture D are set to be w0 = D/2 = 3 mm. As for the square and hexagon array, the center-to-center distance between two adjacent laser beams L is equal to be 7 mm. The fill factor of the square and hexagon array can be calculated to be D/L = 0.86. In the Fermat spiral array, the radial parameter is set to be s = 0.7D to promise similar compactness compared with the other two arrays. To obtain the accurate g(2)(x, y; x0, y0) distribution, the sampling number of light field is set to be 3000. Figure 2(b1-b3) shows the 2D distribution of g(2)(x, y; x0 = 0, y0 = 0) on the observed plane for the corresponding laser arrays in Fig. 2(a1-a3). Figure 2(c1-c3) and Fig. 2(d1-d3) show the cross-sections of g(2)(x, y; x0 = 0, y0 = 0) along the (x, y = 0) and the (y, x = 0) directions, respectively. As we can see, the distributions of g(2)(x, y; x0 = 0, y0 = 0) for the square and hexagon array have obvious periodicity at the observed plane. The sidelobes of g(2)(x, y; x0 = 0, y0 = 0) are equal to the main peak in the position (x = 0, y = 0). As for the Fermat spiral array, there is only one peak in the distribution of g(2)(x, y; x0 = 0, y0 = 0) and no periodic characteristics are found. The sidelobes are suppressed well and present good uniformity. The far-field of Fermat spiral laser array shows great superiority at the peak-to-sidelobe ratio (PSR) property.

 figure: Fig. 2.

Fig. 2. Simulated results of the g(2)(x, y; x0 = 0, y0 = 0) distributions in different laser array configurations. From top to bottom, the spatial configuration of laser arrays (a1-a3), the corresponding g(2)(x, y; x0 = 0, y0 = 0) distributions (b1-b3), the cross-section of g(2)(x, y; x0 = 0, y0 = 0) along the (x, y = 0) direction (c1-c3) and the (y, x = 0) direction (d1-d3).

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In order to further verify the feasibility and superiority of Fermat spiral laser array for GI, we perform computational GI experiments to compare the GI results of the above three types of laser arrays. The typical triple slit is used as the object, which is placed at the focal plane of lens with focal length of 3 m. And the size of object image is the same as that of illumination light field, which is about 1.3 mm calculated by the diameter of single laser beam and the focal length of lens. From the results in Fig. 2, we can see that the light field (or the object) occupies about two periodicities of g(2)(x, y; x0 = 0, y0 = 0). The size of the triple slit is one-third that of the illumination light field (or the object image). The total object image has a resolution of 128${\times} $128 pixels. The sampling patterns is set to be 3000, which corresponds to a sampling ratio of 0.18. And two typical reconstruction algorithms, including differential ghost imaging (DGI) [33] and sparse representation prior compressive sensing (CS) [8,34,35], are utilized to the image reconstruction. The GI results of different laser arrays are shown in Fig. 3. Figure 3(a1-a3) show the reconstructed images by DGI algorithm and Fig. 3(b1-b3) show the reconstructed results of CS algorithm. It is obvious that the square and hexagon arrays have periodic GI results owing to the periodicity in the corresponding g(2)(x, y; x0 = 0, y0 = 0) distributions. The periodicity does not appear in the reconstructed images conducted by Fermat spiral laser array. Besides, the CS algorithm has better reconstructing image quality compared with the DGI algorithm. Especially, as shown in Fig. 3(b3), the Fermat spiral laser array combined with the CS algorithm can achieve a clear reconstructed image with no periodicity and little background noise.

 figure: Fig. 3.

Fig. 3. Simulated demonstration of GI via different laser arrays.

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2.3. Ghost imaging analysis of Fermat spiral laser array

The above section has proved that the Fermat spiral laser array can achieve excellent GI results by combining CS reconstruction algorithm. This section will present how to design the Fermat spiral laser array to obtain better GI quality. The main influence factors are the element number and the compactness of Fermat spiral laser array. Firstly, we calculate the GI results when the number of laser array elements N is equal to 7, 19, 37, 61 and 91, which are shown in Fig. 4. As N increases from 7 to 91, the compactness parameter of Fermat spiral laser array s is maintained to be unchanged, which is set to be s = 0.7D. In addition to the binary image of triple slit, the typical gray image Cameraman is also taken as an object for analysis. Both object images have a resolution of 128${\times} $128 pixels and the sampling ratio is about 0.18. It is obviously observed that the laser array with more laser beams can achieve clearer GI results for both binary and gray images.

 figure: Fig. 4.

Fig. 4. Simulated GI results of Fermat spiral laser arrays with different sub-laser source numbers. From top to bottom, the laser array distributions of different numbers, the reconstructed images of binary image triple slit, the reconstructed images of gray image Cameraman.

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In order to perform precise analysis, we calculate the root mean squared error (RMSE) of the reconstructed images. The RMSE is expressed by

$$RMSE = \sqrt {\frac{{\sum\limits_{x = 1}^M {\sum\limits_{y = 1}^N {{{[{I_R}(x,y) - I(x,y)]}^2}} } }}{{M \times N}}} $$
where IR(x, y) and I(x, y) are the intensity values of reconstructed and original images at the specific coordinate of (x, y). M and N are the pixel numbers of image along x and y directions. M and N are equal to 128 in this section. It is obvious that the smaller RMSE means the better reconstruction image quality. The calculated RMSEs of Fermat spiral laser array with different element numbers are shown in Fig. 5(a). The RMSE values of triple slit and Cameraman image fall down as the element number N increases. When the element number N exceeds up to 61, the falling speed of RMSE becomes slow and the image quality is maintained basically unchanged. Besides, as shown in Fig. 5(b), we also calculate the g(2)(x, y; x0 = 0, y0 = 0) distribution along the (x, y = 0) direction of different Fermat spiral laser arrays. As we can see, the linewidth of the g(2)(x, y; x0 = 0, y0 = 0) becomes narrower when N increases. It means that the resolution ability of illumination light field is improved. Therefore, the GI quality is improved obviously when the element number of laser array increases.

 figure: Fig. 5.

Fig. 5. (a) The RMSEs of Fermat spiral laser array with different element numbers, (b) the corresponding g(2)(x, y; x0 = 0, y0 = 0) distribution along the (x, y = 0) direction

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Analogous to the number of sub-sources, the compactness is another important consideration factor while designing the Fermat spiral laser array. The compactness of Fermat spiral laser array mainly depends on the radial and angular parameters. From the previous analyzation, without loss of generality, the angular parameter 2πβ is set to be 137.5° to provide a relatively uniform distribution [30]. Here we mainly analyze the influence of radial parameter s on the GI effects. Similarly, both binary (triple slit) and gray (Cameraman) images are acted as the objects. The element number of laser array N is equal to 37 for convenience. The radial parameter s is set to be 0.7D, 0.8D, 0.9D, D and 1.1D, respectively. The GI results are shown in Fig. 6. And the calculated RMSEs and g(2)(x, y; x0 = 0, y0 = 0) distributions are shown in Fig. 7. When the radial parameter s varies from 0.7D to 1.1D, all the Fermat spiral laser arrays can achieve clear ghost imaging. The background noise is much less when s is smaller. As for the RMSEs results in Fig. 7(a), the RMSE increases along with the radial parameter s. The growth trend is more significant for gray (Cameraman) image. The main lobe of g(2)(x, y; x0 = 0, y0 = 0) distribution in Fig. 7(b) does not show obvious difference in different compactness laser arrays. However, when the s grows, there was also some growth on the sidelobe of g(2)(x, y; x0 = 0, y0 = 0). Therefore, the more compact Fermat spiral laser array can achieve better GI results. Besides, the more compact laser array can bring significant convenience for the whole optical system.

 figure: Fig. 6.

Fig. 6. Simulated GI results of Fermat spiral laser arrays with different compactness. From top to bottom, the laser array distributions of different compactness, the reconstructed images of binary image triple slit, the reconstructed images of gray image Cameraman.

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 figure: Fig. 7.

Fig. 7. (a) The RMSEs of Fermat spiral laser array with different compactness, (b) the corresponding g(2)(x, y; x0 = 0, y0 = 0) distribution along the (x, y = 0) direction

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3. Experimental results

We also build a concept demonstration experiment system to verify the superiority of Fermat spiral laser array. The experimental setup is shown in Fig. 8. A single-frequency laser source with a central wavelength of 1064 nm is coupled with a delivery fiber and then passes through a collimator (CO). The collimated output laser is expanded by ten times through two lenses (Lens 1 and Lens 2). Then the expanded laser goes through a beam splitter (BS) and the transmission light illuminates on a phase type spatial light modulator (SLM). The modulated laser is reflected by the BS and passes through a mask. The mask can provide determined sub-apertures forming different arrays. To be precise, there are 36 sub-apertures in square array, 37 sub-apertures in hexagon and Fermat spiral arrays on the mask. In order to be similar with the above theoretical analysis, the square and hexagon arrays on the mask are designed to have a fill factor of 0.86. And the radial parameter of Fermat spiral laser array on the mask is designed to be s = 0.7D. The phase type SLM only provides random piston phase modulation for the sub-apertures corresponding to the mask. Then the phase modulated laser array is focused by a lens with 1 m focus length. The focusing light is divided by another BS. A CCD camera located on the focal plane receives the reflected light to capture the illumination light field distribution. And the transmission light illuminates on the target which is also located on the focal plane. The transmission light from target is collected by a single-pixel (S-P) detector. A computer is utilized to controlling the phase modulation and the synchronous data acquisitions of the CCD camera and S-P detector.

 figure: Fig. 8.

Fig. 8. Experiment setup of GI via different laser arrays. CO, collimator; BS, beam splitter; SLM, spatial light modulator; S-P detector, single-pixel detector.

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Based on the experimental setup shown in Fig. 8, the g(2)(x, y; x0 = 0, y0 = 0) distributions of the three types of laser arrays are measured by the CCD camera and calculated according to Eq. (5), which is shown in Fig. 9. To precisely present the g(2)(x, y; x0 = 0, y0 = 0) distributions, the CCD camera captures 1000 illumination light fields for each laser array. It is obviously observed that the experimental results are consistent with the simulated results shown in Fig. 2. The g(2)(x, y; x0 = 0, y0 = 0) distributions of square and hexagon arrays are obviously periodic. The g(2)(x, y; x0 = 0, y0 = 0) distribution of Fermat spiral laser array, as shown in Fig. 9(b3), has no periodicity on experiment, which can provide a high-quality illumination light field for GI experiment. Furtherly, the corresponding experimental GI results of the three laser arrays are shown in Fig. 10. The typical triple slit is used as the target which is designed to be consistent with that in above theoretical analysis. The sparse representation prior compressive sensing (CS) algorithm is utilized for image reconstruction with sampling number of 200. The reconstructed images have a resolution of 64${\times} $64 pixels. Obviously, the periodicity appears in the reconstructed images of square and hexagon arrays shown in Fig. 10 (b) and (c). And for the Fermat spiral laser array, as shown in Fig. 10(d), there is no periodicity in the reconstructed image and the object-triple slit can be recognized clearly. It means that the Fermat spiral laser array can achieve better GI quality by combining CS reconstruction algorithm. However, there are some weak background noises on the reconstructed image in Fig. 10(d) impairing the image quality. According to our analysis, the image noises mainly come from the intensity distribution heterogeneity of illuminating light field. As shown in the experimental setup, the illuminating light field is generated by an expanded fiber laser source which has a Gaussian intensity distribution. It means that the most energy concentrates on the central area. Therefore, the sub-apertures on the arrays have heterogeneous intensity distribution where the sub-sources at central area have large emitting energy. The next work will focus on establishing a collimated fiber laser array for GI system with uniform and high emitting energy.

 figure: Fig. 9.

Fig. 9. Experimental results of the g(2)(x, y; x0 = 0, y0 = 0) distributions in different laser array configurations. From top to bottom, the spatial configuration of laser arrays (a1-a3), the corresponding g(2)(x, y; x0 = 0, y0 = 0) distributions (b1-b3), the cross-section of g(2)(x, y; x0 = 0, y0 = 0) along the (x, y = 0) direction (c1-c3) and the (y, x = 0) direction (d1-d3).

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 figure: Fig. 10.

Fig. 10. The experimental GI results of different laser arrays. From left to right, (a) the original image, (b) the reconstructed image by square laser array, (c) the reconstructed image by hexagon laser array, (d) the reconstructed image by Fermat spiral laser array.

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4. Conclusion

In summary, we have successfully applied Fermat spiral laser array in GI experiments as pseudo-thermal light source. The illumination light field has no spatial periodicity on the normalized second-order intensity correlation function (g(2)(x, y; x0, y0)) due to the aperiodic structure of Fermat spiral laser array. Therefore, the reconstructed images based on Fermat spiral laser array has better quality compared with those reconstructed from regular arrays such as square and hexagon arrays. Besides, the structure parameters of Fermat spiral laser arrays have also been analyzed for obtaining better GI quality, which provides guidance on the design of laser array. When the laser array is formed by collimated fiber laser sources equipped with electro-optic modulators, the illumination light field with high emitting power and high refreshing speed can be achieved in GI experiments which is expected to be applied in remote sensing. Besides, to achieve fast imaging with high quality, the next work is mainly focused on the reconstructing algorithm such as deep learning or machine learning.

Funding

Youth Innovation Promotion Association of the Chinese Academy of Science (2020438).

Disclosures

The authors declare no conflicts of interest.

Data availability

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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Data availability

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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Figures (10)
Fig. 1.
Fig. 1. The schematic system of coherent fiber laser array.

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Fig. 2.
Fig. 2. Simulated results of the g(2)(x, y; x0 = 0, y0 = 0) distributions in different laser array configurations. From top to bottom, the spatial configuration of laser arrays (a1-a3), the corresponding g(2)(x, y; x0 = 0, y0 = 0) distributions (b1-b3), the cross-section of g(2)(x, y; x0 = 0, y0 = 0) along the (x, y = 0) direction (c1-c3) and the (y, x = 0) direction (d1-d3).

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Fig. 3.
Fig. 3. Simulated demonstration of GI via different laser arrays.

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Fig. 4.
Fig. 4. Simulated GI results of Fermat spiral laser arrays with different sub-laser source numbers. From top to bottom, the laser array distributions of different numbers, the reconstructed images of binary image triple slit, the reconstructed images of gray image Cameraman.

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Fig. 5.
Fig. 5. (a) The RMSEs of Fermat spiral laser array with different element numbers, (b) the corresponding g(2)(x, y; x0 = 0, y0 = 0) distribution along the (x, y = 0) direction

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Fig. 6.
Fig. 6. Simulated GI results of Fermat spiral laser arrays with different compactness. From top to bottom, the laser array distributions of different compactness, the reconstructed images of binary image triple slit, the reconstructed images of gray image Cameraman.

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Fig. 7.
Fig. 7. (a) The RMSEs of Fermat spiral laser array with different compactness, (b) the corresponding g(2)(x, y; x0 = 0, y0 = 0) distribution along the (x, y = 0) direction

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Fig. 8.
Fig. 8. Experiment setup of GI via different laser arrays. CO, collimator; BS, beam splitter; SLM, spatial light modulator; S-P detector, single-pixel detector.

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Fig. 9.
Fig. 9. Experimental results of the g(2)(x, y; x0 = 0, y0 = 0) distributions in different laser array configurations. From top to bottom, the spatial configuration of laser arrays (a1-a3), the corresponding g(2)(x, y; x0 = 0, y0 = 0) distributions (b1-b3), the cross-section of g(2)(x, y; x0 = 0, y0 = 0) along the (x, y = 0) direction (c1-c3) and the (y, x = 0) direction (d1-d3).

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Fig. 10.
Fig. 10. The experimental GI results of different laser arrays. From left to right, (a) the original image, (b) the reconstructed image by square laser array, (c) the reconstructed image by hexagon laser array, (d) the reconstructed image by Fermat spiral laser array.

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Equations (8)

Equations on this page are rendered with MathJax. Learn more.

E e ( x , y , z = 0 ) = ∑ n = 1 N A n exp ⁡ ( − ( x − x n ) 2 + ( y − y n ) 2 w 0 2 ) × c i r c ( ( x − x n ) 2 + ( y − y n ) 2 ( D / 2 ) ) exp ⁡ ( j ϕ n )
c i r c ( r 0 ) = { 1 r 0 ≤ 1 0 r 0 > 1
E r ( u , v , z = f ) = e i k f i λ f ∫ − ∞ ∞ ∫ − ∞ ∞ E e ( x , y , z = 0 ) ⋅ e − i k 2 f ( x 2 + y 2 ) ⋅ e i k 2 f [ ( u − x ) 2 + ( v − y ) 2 ] d x d y
I r ( u , v , z = f ) = E r ( u , v , z = f ) ⋅ E r ∗ ( u , v , z = f )
g ( 2 ) ( x , y ; x 0 , y 0 ) = ⟨ I ( x 0 , y 0 ) I ( x , y ) ⟩ ⟨ I ( x 0 , y 0 ) ⟩ ⟨ I ( x , y ) ⟩
{ r n = s n π , n = 1 , 2 , 3 … N θ n = 2 π β n , n = 1 , 2 , 3 … N
{ x n = r n c o s ( θ n ) , n = 1 , 2 , 3 … N y n = r n sin ⁡ ( θ n ) , n = 1 , 2 , 3 … N
R M S E = ∑ x = 1 M ∑ y = 1 N [ I R ( x , y ) − I ( x , y ) ] 2 M × N
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