1. Introduction

Ghost imaging (GI) is a special imaging scheme, which has been developed for decades since the first GI demonstration using entangled photon sources [1]. Subsequently, various light sources such as thermal light [2,3], pseudothermal light [4], sunlight [5], x-ray [6,7] and LED arrays [8] were employed for GI. In traditional GI, signals of the reference arm and the test arm need to be detected experimentally at the same time. Shapiro proposed the computational ghost imaging (CGI) [9], where the spatial light modulator (SLM) was introduced and the ghost images could be reconstructed only by detecting the test arm while the reference arm was replaced by computer simulation [10]. In this way, CGI provides a broad space for one to manipulate the image quality and the imaging performance.

It is well known that the spatial resolution of ghost images is mainly determined by the average grain size of speckles, which is in general affected by the roughness of the scattering media and the light source area [11]. On the other hand, the visibility of ghost images is decided by the intensity fluctuation of the light speckles, which could be customized through the probability density function (PDF) of the speckle intensities [12,13]. Various structured light sources [14–22] and reconstruction algorithms [23–30] were proposed to improve the image quality and the image reconstruction efficiency. In addition, the position of the ghost images, both in the transverse and longitudinal dimensions, can be controlled through the speckle fields [31–33]. For instance, long distance non-diffracting GI [33], periodically reproducible GI [34] and GI based on the Talbot effect [35] have been reported.

In this paper, we developed designable speckle light fields through a modified Gerchberg-Saxton (GS) algorithm [36] based on the Fresnel diffraction. Both the visibility and the grain size of the designed speckle fields are controllable independently. Based on the designed speckle fields, ghost images with controllable visibility and spatial resolution were demonstrated experimentally with much better performance than that with a pseudothermal light. Surprisingly, it was found that the spatial resolution of ghost image was independent of the area of the light source, which is very different from that of the traditional ghost images. In addition, we also demonstrated the capability to reconstruct simultaneously multiple ghost images at different positions in the longitudinal dimension, which may have potential applications such as optical encryption and optical tomography.

2. Design of controllable light sources

The controllable light source is implemented by loading a customized phase distribution to a phase-only SLM, which is illuminated by a plane wave light. The schematic flow diagram to produce the customized phase distribution is depicted in Fig. 1(a), and the phase and intensity distribution of the light fields on the corresponding planes are shown in Fig. 1(b).

 

Fig. 1. (a) The schematic diagram to generate the phase distribution of the target light field. Lens $L_1$ with a focal length $f_1$ and lens $L_2$ with a focal length $f_2$ form a confocal system. Here, (a1) an axicon at the front focal plane of lens $L_1$; (a2) the confocal plane of lenses $L_1$ and $L_2$; (a3) the random phase matrix at the confocal plane; (a4) the rear focal plane of lens $L_2$; (a5) the target field plane with a distance $z$ away from the SLM plane (a6). (b) the phase or intensity distributions of the light fields at different planes. Here, (b1) the phase distribution induced by the axicon; (b2) the annular ring at the rear focal plane of lens $L_1$; (b3) the phase distribution of the random phase matrix; (b4) the speckle pattern at the rear focal plane of lens $L_2$; (b5) the target speckle pattern generated by applying a power term $n$ onto the complex field amplitude in the rear focal plane of lens $L_2$; (b6) the designed phase distribution retrieved through the GS iteration algorithm.

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Following the phase generation flow chart, an axicon placed at the front focal plane of lens $L_1$ (with a focal length $f_1$) is illuminated by a plane wave light. The phase distribution induced by the axicon, as shown in Fig. 1(b1), is $\varphi _a=2\pi \rho _a/\rho _0$ with $\rho _a=\sqrt {x_a^2+y_a^2}$ and $\rho _0=\lambda /\left (n_a-1\right )\gamma$, where $\left (x_a,y_a\right )$ is the coordinates on the front focal plane of lens $L_1$, $\lambda$ is the wavelength of light, $n_a$ is the refractive index and $\gamma$ is the conical angle of the axicon [37,38], respectively. The axicon-induced phase distribution shown in Fig. 1(b1) can also be generated digitally through a SLM with a pixel size $\delta$, in which $\rho _0$ is expressed as $\rho _0=P\delta$ with the parameter $P=\lambda /(n_a-1)\gamma \delta$ being a constant associated with the structure parameters $n_a$ and $\gamma$ of the axicon. An annular ring as seen in Fig. 1(b2) is generated on the back focal plane of lens $L_1$. Then a random phase matrix as shown in Fig. 1(b3) is superimposed with the annular ring field, and the transmitted field is transformed by lens $L_2$ with a focal length $f_2$. Here $L_1$ and $L_2$ form a confocal system, and the random phase matrix is placed at the confocal plane of lenses $L_1$ and $L_2$. The speckle pattern, with a complex amplitude $\tilde {E}_o=E_oe^{i\varphi _o}$, is recorded at the rear focal plane of lens $L_2$, as shown in Fig. 1(b4).

Theoretically, the desired light field with its intensity distribution as shown in Fig. 1(b5) is designed by applying a power term $n$ (a positive integer) onto the complex field amplitude in the rear focal plane of lens $L_2$, and therefore its field amplitude becomes $\tilde {E}_z= E_o^n e^{i\varphi _z}$. This target field is expected to be generated at a distance $z$ (the target plane of Fig. 1(a5)) away from a SLM (at the position of Fig. 1(a6)) under the illumination of a plane wave light, on which an appropriate phase distribution is loaded, as seen in Figs. 1(a5) and 1(a6). So the key to produce the target field is to solve the inverse Fresnel diffraction problem to get the appropriate phase distribution loaded on the SLM. Here, we will employ the GS algorithm [36] to get the phase distribution, which is an effective algorithm for the phase retrieval problem when only the amplitude information of the target field is known, and has been widely applied in various disciplines of optical subjects where the phase retrieval is required [39–41].

According to the GS iteration algorithm based on the Fresnel diffraction between two planes (the target plane at the position of Fig. 1(a5) and the SLM plane at the position of Fig. 1(a6)), the iteration begins with the light field at the SLM plane with a complex amplitude $\tilde {E}_s=E_se^{i\varphi _s}$, where $E_s=1$ is the amplitude of the plane wave light and $\varphi _s$ is initially set to be random. After a Fresnel diffraction of a distance $z$ in free space to the target plane of Fig. 1(a5), the light field develops to be $\tilde {E}_z= E_ze^{i\varphi _z}$. Next, one replaces the amplitude $E_z$ with the target field amplitude $E_o^n$, while maintaining the phase $\varphi _z$ unchanged. The constructed field transmits back to the SLM plane of Fig. 1(a6) through the inverse Fresnel diffraction, resulting in an updated field $\tilde {E}^{1}_s= E^{1}_se^{i\varphi ^{1}_s}$. Then one substitutes the amplitude $E_s=1$ for $E^{1}_s$, that is $E^{1}_s=1$, and keeps the phase $\varphi ^{1}_s$ unchanged. So far, one iteration loop of the GS algorithm is accomplished. The iteration process continues for $k$ times until the algorithm converges, that is, the mean square error (MSE) between the calculated and the desired speckle intensity distributions converges [42]. In this way, the expected phase distribution $\varphi ^k_s$ is retrieved (see Fig. 1(b6)), which will be loaded onto the SLM to get the target field with the desired intensity fluctuation and speckle grain size.

3. Correlation properties of the designed speckle fields

It is known that the spatial resolution and the visibility of ghost image are associated with the average grain size and the intensity fluctuation of the speckle fields, respectively, which are characterized by the normalized second-order intensity correlation function. In this section, we will give a detailed characterization on the correlation properties of the designed speckle fields in Sec. 2.

The normalized second-order correlation function of the designed speckle fields in Sec. 2 is calculated as (see Appendix A1 for the details)

The detailed calculations about the visibility and the average grain size of the speckles can be found in Appendix A2.

The experimental setup to generate the target speckle fields is shown in Fig. 2(a). A single-mode continuous-wave laser beam operating at 532 nm was expanded and collimated by a beam expander BE, and the power of the laser beam was controlled by a combination of a half-wave plate and a polarizer. The beam was then transmitted through an aperture with a diameter of 6 mm and was reflected by a beam splitter BS to a reflection-type phase-only SLM (HEO 1080P from HOLOEYE Photonics AG, Germany) with $1920\times 1080$ addressable pixels (pixel size: $8\ \mu \mathrm {m}\times 8\ \mu \mathrm {m}$), on which the target phase distribution retrieved through the GS iteration algorithm (see Fig. 1(b6)) was loaded to generate the target speckle field. In addition, a blazed grating was superposed on the SLM to separate different diffraction orders and eliminate the influence of stray light. The target speckle field was designed to appear at the position $z=30$ cm away from the SLM, and the speckle patterns were detected by a charge coupled device (CCD) camera with an exposure time of 1 ms for each frame.

 

Fig. 2. (a) The experimental setup to generate the target speckle fields. (b) Part of the experimental setup for GI on a single plane. (c) Part of the experimental setup for GI on multiple planes, where the yellow, the green and the purple planes represent the positions with different distances $z_1$, $z_2$ and $z_3$ away from the SLM, on which the designed speckles were generated. Here, $\lambda /2$: half-wave plate; BE: beam expander; P: polarizer; A: aperture; BS: $50:50$ beam splitter; Object: amplitude transmitting object.

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Figures 3(a1-a3) and 3(b1-b3) show the simulated and the experimentally measured speckle patterns with $n=1,\, 2$ and 3, respectively, where the parameter $P$ was set to be 20. The normalized second-order correlation functions are displayed in the corresponding bottom-right insets, which are of the profile of Bessel function of zero order (see Fig. 3(c)), just as predicted by Eq. (1). It is seen from the speckle patterns that the intensity fluctuation of the speckles becomes more intense with the increase of $n$, which will also modify the visibility (Fig. 3(c)) and the probability density function (PDF) (Fig. 3(d)) of speckles. In the case with $n=1$, the PDF obeys a negative exponential distribution and the visibility of speckles is $V_{n=1}=0.33$, which is the same as that of thermal light. While as the power term $n$ increases, the light energy is redistributed among the speckles, resulting in a dramatic intensity fluctuation among the speckles and a significant deviation of the PDF from the negative exponential distribution, and therefore a higher visibility of the speckles, for example, $V_{n=2}=0.60$ and $V_{n=3}=0.82$, respectively. One sees that the larger the power term $n$, the stronger the light intensity fluctuation, and the higher the speckle visibility. This is in excellent consistent with the theoretical prediction of Eq. (2). Note that, although the designed target fields with different power term $n$ are related with each other because theoretically all target intensity speckle patterns are designed by applying a power term $n$ onto the same complex field amplitude $\tilde {E}_o=E_oe^{i\varphi _o}$ on the rear focal plane of lens $L_2$ in Fig. 1, the statistics properties of these target speckle fields with different power term $n$ are quite different. It can be verified that the target speckle pattern with $n=1$ is the Rayleigh speckles, and those with $n>1$ are the super-Rayleigh speckles. The details about the statistics properties of the designed target speckle fields with different power term can be found in Appendix A3. It is the different statistics properties of the designed target speckle fields with different power term $n$ that result in their different correlation properties and ghost imaging performance.

 

Fig. 3. (a1-a3) Simulated and (b1-b3) experimental speckle patterns with $f_1/f_2=1$, $\delta =8\ \mu$m and $P=20$ for the cases $n=1,\ 2$ and 3 from top to bottom, respectively. The inset at the bottom-right corner of each figure is the second-order correlation function of the speckles. (c) The normalized second-order correlation curves of the speckles, where the solid color curves and the dashed or dash-dotted color curves are the simulated and the experimental results, respectively. (d) The PDF curves of the speckle intensities, where the solid color curves are the simulated results and the filled symbols are the experimental results, respectively. The solid black curve is a fit to the negative exponential function $P_I\left (I_1\right )$. (e) The dependence of the average speckle size on the parameter $P$ and the ratio $f_1/f_2$. The solid curves are the theoretical ones plotted based on Eq. (3). The filled and the hollow symbols are the simulated and the experimental speckle sizes, respectively.

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According to the normalized second-order correlation curves in Fig. 3(c), the simulated speckle grain size was $D_{n=1,2,3}=64\ \mu$m, and the experimental one was measured to be $D_{n=1,2,3}=64.5\ \mu$m, respectively, which is in good agreement with that predicted by Eq. (3), that is $D_{n=1,2,3}=61.3\ \mu$m, with the parameters $f_1/f_2=1$, $\delta =8\ \mu$m and $P=20$. Note that the speckle grain size is independent of the power term $n$. In addition, we plotted the dependence of the average speckle grain size on the parameter $P$ and the ratio $f_1/f_2$ in Fig. 3(e), where the pixel size $\delta$ of the SLM was $8\ \mu$m. It is seen from the curves that a smaller $P$ and a larger $f_1/f_2$ will lead to smaller speckle grains, verifying the tendence theoretically predicted by Eq. (3).

4. Ghost images based on the designed speckle fields

The experimental setup for GI is described in Fig. 2(b), which is similar to the scheme in Fig. 2(a) except that an object was placed at the position $z=30$ cm away from the SLM. The CCD was placed behind the object and served as a bucket detector by integrating over all pixel signals carrying the object information. Here, we used a CCD for signal acquisition because the CCD can not only serve as a bucket detector but also record directly the intensity speckle pattern on the detection plane for further analyzing the statistics properties of the designed speckle fields, therefore, simplifying the experimental setup. The use of CCD for signal acquisition can also be found in Refs. [23,24,27,31,32]. Following the procedure for CGI, the numerically calculated reference signal $\mathcal {R}_i$ and the experimentally measured bucket signal $\mathcal {B}_i$ were correlated, and the ghost image of the object $I_{obj}$ was reconstructed as

4.1 Visibility of the ghost images

Following the procedure in Ref. [43], the visibility of the ghost images based on the designed speckle fields can be expressed as

In the experiment, a double-slit with a slit width $a=300\ \mu$m, a slit separation distance $d=600\ \mu$m and a slit height $h=1500\ \mu$m was used as the object. The ghost images of the double-slit were reconstructed experimentally based on Eq. (4) with different parameters $P$ and $n$ while keeping the ratio $f_1/f_2=1$ the same, and the results are shown in Fig. 4(a). Here, an ensemble of 10,000 frames were taken and used to calculate the ghost image for each picture in Fig. 4(a). The visibility of the ghost images for both the experimental and the simulated cases are shown in Fig. 4(b), and are marked with black and red numbers on the pillar of the histogram, respectively. One sees that, as expected, the visibility of the ghost image increases with the increase of the power term $n$, for instance, with the same parameter $P=20$, the visibility of the ghost image with $n=3$ is 2.42%, about 10 times higher than that of the case with $n=1$. In addition, the visibility of the ghost image can also be improved by using a large parameter $P$, as shown in Fig. 4(b), which is in good agreement with the theoretical prediction by Eq. (5). It is also noted that, under the same parameters of the designed speckle fields, the visibility of the experimental ghost images is smaller than that of the simulated ones. This may be due to the imperfect modulation of the SLM on the light field and the stray light in the process of experiments. On the other hand, it is possible to improve further on the quality of ghost images based on the designed speckle fields by employing schemes such as the conditional positive-negative ghost imaging [27,44] and the fractional-order ghost imaging [45].

 

Fig. 4. (a) Experimental ghost images with different parameters $P$ and $n$. The color bar is normalized by the maximum and minimum values of the nine images. Ghost images are obtained by taking the ensemble average of 10 000 pictures. (b) The simulated and experimental visibilities of the double-slit ghost images. The red and the black numbers marked on the pillar of the histogram are the visibility of the ghost images for the simulated and the experimental cases, respectively. The spatial resolution was calculated to be 30.64 ${\rm \mu} m$, 45.96 ${\rm \mu} m$ and 61.28 ${\rm \mu} m$ with the parameter $P = 10$, 15 and 20, respectively.

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4.2 Spatial resolution of the ghost images

It is well known that the spatial resolution of the ghost images is determined by the average speckle grain size of the speckle light field, therefore, according to Eq. (3), it is possible to improve the spatial resolution of the ghost images with the designed speckle fields by modifying the parameters such as $P$.

In the experiment, the USAF 1951 resolution target was used as the object. Ghost images of a resolution target 4-3 (spatial resolution: 49.6 $\mu$m) were reconstructed under the illumination of the designed speckle fields with different parameters $P=10,\, 15$ and 20, as shown in Fig. 5. Here, the power term $n$ was set to be 1 in all cases. One sees that the ghost image with a parameter $P=10$ is clearly resolved while that with $P=20$ is blurred, verifying that the spatial resolution of ghost images is higher in the case with a smaller parameter $P$, in good agreement with that predicted by Eq. (3). What’s more, according to Eq. (3), the average speckle grain size $D$ is independent of the area of the light source. This means the spatial resolution of the ghost images with the designed speckle field is also independent of the light source area, which is very different from the case with a thermal light in which the spatial resolution of the ghost images is higher with a larger area of thermal sources. This was also confirmed experimentally. Figure 6 shows the ghost images of a resolution target 4-5 (spatial resolution: 39.38 $\mu$m) with the pseudothermal light sources (Figs. 6(a1-a3)) and the designed speckle light fields (Figs. 6(b1-b3)) of different source areas. Here, the designed speckle fields were generated with $n=1$ and $P=10$, and the diameter of the light sources were controlled by the aperture in Fig. 2(a). The pseudothermal sources were generated by directly loading random phase (equally distributed within [0, 2$\pi$)) onto the SLM in Fig. 2. One sees that, in the case with pseudothermal source, the ghost image is clearly resolved with a source diameter of 6 mm in Fig. 6(a3), while the ghost image is blurry in Fig. 6(a1) with a small source diameter of 4 mm. On the other hand, in the case with the designed speckle fields, all ghost images are clearly resolved regardless of the source diameter, as shown in Figs. 6(b1-b3). Therefore, the spatial resolution of the ghost imaging with the designed speckle field could be designed to be better than that of the ghost imaging with thermal light.

 

Fig. 5. (a), (b) and (c) Ghost images of a USAF 1951 resolution target 4-3 with the designed speckle fields of different parameters $P=10,\ 15$ and 20, respectively. Here the power term $n$ was set to be 1 and the source diameter was 6 mm in all cases.

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Fig. 6. Ghost images of a USAF 1951 resolution target 4-5 with the pseudothermal sources ((a1-a3)) and the designed speckle fields ((b1-b3)) of different source areas. The diameter of the pseudothermal sources and the designed speckle fields was set to be 4 mm ((a1) and (b1)), 5 mm ((a2) and (b2)) and 6 mm ((a3) and (b3)), respectively. The power term $n=1$ and the parameter $P=10$ were used to generate the designed speckle fields, and the diameter of the sources was controlled by the aperture A in Fig. 2(a).

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4.3 Ghost images on multiple planes

In previous sections, ghost images on a specific plane were demonstrated and characterized. In this section, we will demonstrate the capability to reconstruct ghost images on multiple planes with the same phase distribution loaded on the SLM. For this purpose, we employ the parallel GS algorithm [46,47] to design the desired phase distribution loaded on the SLM. Here, multiple target field amplitudes $E_{o_j}^n$ ($j\in \{1,2,\ldots,M\}$), which are expected to generate the desired speckle fields at $M$ different imaging planes, are generated with different sets of random phase matrixes of Fig. 1(b3). In the program, in each iteration step of the parallel GS algorithm, the Fresnel diffraction field $\tilde {E}_{z_j}$ at position $z_j$ away from the SLM is calculated. Then the amplitude $E_{z_j}$ is respectively replaced by the target amplitude $E_{o_j}^n$, and the constructed field successively undergoes the inverse Fresnel diffraction, leading to a new phase distribution $\varphi _{s_j}$ at the SLM plane. Then these $M$ phase distributions $\varphi _{s_j}$ are averaged out, resulting in a new phase matrix $\varphi _s=\frac {1}{M} \sum _{j=1}^M\varphi _{s_j}$, which serves as the initial phase for the next iteration of GS algorithm. The iteration continues until the program converges, and the target phase distribution is obtained. The target speckle fields on $M$ different planes can be generated by loading the obtained target phase distribution onto the SLM under the illumination of a plane wave. The ghost image at the corresponding plane $z_j$ is reconstructed through

The schematic experimental setup for GI on multiple planes is shown in Fig. 2(c). As a proof of principle while at the same time without loss of the generality, we will demonstrate ghost images on three different planes at the positions $z_1=30$ cm, $z_2=50$ cm and $z_3=70$ cm away from the SLM, respectively. Firstly, we considered the relatively simple case that the target speckle fields at three different planes were the same, that is, $E_{o_1}^n=E_{o_2}^n= E_{o_3}^n$. The same double-slit as that used in Fig. 4 was used as the object. In the experiments, the double slit was placed in sequence at the positions $z_1=30$ cm, $z_2=50$ cm and $z_3=70$ cm away from the SLM, and the ghost images on the corresponding planes were reconstructed according to Eq. (6). Figures 7(a1-a3) and Figs. 7(b1-b3) show the simulated and the experimental ghost images at three different planes, respectively, with a power term $n=3$. It is seen that all ghost images on different planes are clearly displayed, and the visibilities of the experimental ghost images in Figs. 7(b1-b3) were measured to be $0.67\%$, $1.12\%$ and $1.01\%$, respectively, verifying the capability to reconstruct the ghost images on multiple different planes. In addition, similar to the case of GI on a specific plane, the visibility of the ghost images on multiple planes also increases with the increase of the power term $n$, for example, the visibility of the ghost images is only about $0.25\%$ for the case with a power term $n=1$, much lower than those with a power term $n=3$. More interestingly, when the object is placed only on one of the three planes, one can reconstruct ghost images on all pre-set multiple planes. Figures 7(c1-c3) show the simulated ghost images on three different planes when one placed an object "NK" at the position $z_2=50$ cm away from the SLM with a power term $n=3$, where the ghost images were reconstructed through the correlation between the bucket signal $\mathcal {B}_{z_2}$ at the position $z_2=50$ cm and the corresponding reference signals $\mathcal {R}_{z_j}$ at the positions $z_1=30$ cm, $z_2=50$ cm and $z_3=70$ cm away from the SLM, respectively. This is capable because the target speckle fields on these three different planes are designed to be identical.

 

Fig. 7. (a1-a3) The simulated and (b1-b3) the experimental ghost images of a double-slit at three different positions. (c1-c3) The simulated ghost images of the character "NK" at three different positions. Here, the target speckle fields at three different positions are customized to be the same by setting $E_{o_1}^n=E_{o_2}^n=E_{o_3}^n$. (d1-d3) The simulated ghost images of three different characters "N", "K" and "U" at three different positions with three different speckle fields by setting $E_{o_1}^n\neq E_{o_2}^n\neq E_{o_3}^n$. In all cases, the parameters used to design the target fields were $P=20$ and $n=3$, and the three imaging planes were at the positions $z_1=30$ cm, $z_2=50$ cm and $z_3=70$ cm away from the SLM, respectively.

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The target speckle fields on different imaging planes can also be customized to be different, which can be achieved by setting $E_{o_1}^n\neq E_{o_2}^n\neq E_{o_3}^n$. In this case, the ghost image at the $j$th position is reconstructed with the respective bucket signal $\mathcal {B}_{z_j}$ and reference signal $\mathcal {R}_{z_j}$ based on Eq. (6). Figures 7(d1-d3) show the simulated ghost images at three different positions $z_1=30$ cm, $z_2=50$ cm and $z_3=70$ cm away from the SLM, respectively, where three different characters "N", "K" and "U" were used in sequence as the objects, and the power term $n$ was set to be 3 again. One sees that the ghost images are clearly displayed on the respective imaging planes. Surely, speckle fields can be customized at much more different positions, which may provide the potential for optical image encryption and optical tomography. In addition, these results indicate that, just as the case for thermal light [48,49], complex natural scenes can also be reconstructed based on our designed speckle fields because the imaging principle is the same for both the thermal light and our designed speckle fields.

5. Conclusion

In summary, based on the modified GS algorithm with Fresnel diffraction, speckle light fields with independently controllable visibility and speckle grain size were customized, with which high performance ghost images with controllable visibility and spatial resolution were demonstrated experimentally. The visibility and the spatial resolution of the ghost images based on the designed speckle fields could be improved significantly as compared to those of ghost images with a pseudothermal light. It was verified that the spatial resolution of the ghost images was independent of the area of the light source, which is very different from that of the traditional ghost images. Moreover, speckle fields capable of multiple ghost images on different imaging planes were customized, and multiple ghost images on different imaging planes were demonstrated simultaneously with the same customized phase distribution loaded on the SLM. These results may provide potential application prospects on 3D imaging, optical encryption, optical tomography and so on.

Appendix A

In the scheme shown in Fig. 1, the complex amplitude $\tilde {E}_o\left (x,y\right )$ of the speckle field on the rear focal plane of the lens $L_2$ is given by

(A1)$$\tilde{E}_o\left(x,y\right)=\frac{1}{i\lambda f_2}\iint \tilde{E}_r\left(x_r,y_r\right)t\left(x_r,y_r\right)\mathrm{exp}\left(\frac{ik}{2f_2} \left(\left(x_r^2+y_r^2\right)-2\left(xx_r+yy_r\right)\right)\right)\mathrm{d}x_r \mathrm{d}y_r.$$

Here, $\lambda$ is the wavelength of the light, $k$ is the wave number of the light, $f_2$ is the focal length of the lens $L_2$. $\tilde {E}_r\left (x_r,y_r\right )$ is the complex amplitude of the annular field on the confocal plane of lenses $L_1$ and $L_2$ (see Fig. 1) with a radius $R'=\lambda f_1/P\delta$[50], where $f_1$ is the focal length of the lens $L_1$, $P$ is the parameter associated with the axicon phase and $\delta$ is the pixel size of the SLM, $t\left (x_r,y_r\right )$ represents the transmittance function of the random phase matrix on the confocal plane of lenses $L_1$ and $L_2$ in Fig. 1.

A1. Normalized second-order intensity correlation function of the designed speckle fields

The normalized second-order intensity correlation function of the designed speckle field is expressed as

(A2)$$g_n^{\left(2\right)}\left(x_1,x_2;y_1,y_2\right)=\frac{\left\langle I_n\left(x_1,y_1\right) I_n\left(x_2,y_2\right)\right\rangle}{\left\langle I_n\left(x_1,y_1\right)\right\rangle\left\langle I_n\left(x_2,y_2\right)\right\rangle},$$

where $\left\langle\cdots \right\rangle$ represents the ensemble average, $I_n\left (x_q,y_q\right )$ is the speckle intensity on the observation plane $\left (x_q,y_q\right )$ $\left (q=1,2\right )$ and $\left\langle I_n\left (x_q,y_q\right )\right\rangle$ is the average intensity.

The complex amplitude of the designed speckle field with the power term $n$ is

(A3)$$\tilde{E}_n\left(x,y\right)=E_o^n\left(x,y\right)e^{i\phi_n\left(x,y\right)},$$

where $E_o(x,y)$ is the field magnitude on the rear focal plane of lens $L_2$ and $\phi _n\left (x,y\right )$ is the phase of the designed speckle field. The intensity distribution of the designed speckle field with the power term $n$ is

(A4)$$I_n\left(x,y\right)=\tilde{E}_n\left(x,y\right)\tilde{E}_n^*\left(x,y\right)=|E_o(x,y)|^{2n}=I_o^{n}\left(x,y\right).$$

Therefore, the normalized second-order correlation function can be expressed as

(A5)$$g_n^{\left(2\right)}\left(x_1,x_2;y_1,y_2\right)=\frac{\left\langle I_o^n\left(x_1,y_1\right) I_o^n\left(x_2,y_2\right)\right\rangle}{\left\langle I_o^n\left(x_1,y_1\right)\right\rangle\left\langle I_o^n\left(x_2,y_2\right)\right\rangle}=\frac{\left\langle I_o^n\left(x_1,y_1\right) I_o^n\left(x_2,y_2\right)\right\rangle}{\left(n!\right)^2\left\langle I_o\left(x_1,y_1\right)\right\rangle^n\left\langle I_o\left(x_2,y_2\right)\right\langle^n},$$

where the relationship $\left\langle I_o^n\right\rangle=n!\left\langle I_o\right\rangle^n$ is used. According to Ref. [51], one can get

(A6)$$g_n^{\left(2\right)}\left(x_1,x_2;y_1,y_2\right)=\sum_{m=0}^n\left(C_n^m\right)^2 \left|\mu\left(x_1,x_2;y_1,y_2\right)\right|^{2m}\, ,$$

where $C_n^m=n!/m!\left (n-m\right )!$ and $\mu \left (x_1,x_2;y_1,y_2\right )$ being the normalized first-order field correlation function

(A7)$$\mu\left(x_1,x_2;y_1,y_2\right)=\frac{\left\langle \tilde{E}_o\left(x_1,y_1\right) \tilde{E}_o^*\left(x_2,y_2\right)\right\rangle}{\sqrt{\left\langle I_o\left(x_1,y_1\right)\right\rangle \left\langle I_o\left(x_2,y_2\right)\right\rangle}}.$$

By substituting Eq. (A1) into Eq. (A7), the first-order field correlation of the speckle field is calculated as

(A8)$$\begin{aligned}&\left\langle\tilde{E}_o\left(x_1,y_1\right) \tilde{E}_o^*\left(x_2,y_2\right)\right\rangle\\ &=\frac{1}{\lambda^2 f_2^2}\iint |\tilde{E}_r\left(x_r,y_r\right)|^2\mathrm{exp}\left[-\frac{1}{2f_2^2} \left(x_r^2+y_r^2\right)\right]\mathrm{exp}\left[-\frac{ik}{f_2}\left(\Delta xx_r+\Delta yy_r\right)\right]\mathrm{d}x_r \mathrm{d}y_r, \end{aligned}$$

where the notations $\Delta x=x_1-x_2$, $\Delta y=y_1-y_2$ and the relationship $\left\langle t\left (x_r,y_r\right )t^*\left (x_r',y_r'\right )\right\rangle=\delta \left (x_r-x_r',y_r-y_r'\right )$ are employed [52,53]. For convenience, we will use the cylindrical coordinate in the following. The coordinates in the confocal plane of lenses $L_1$ and $L_2$ in Fig. 1 are related with the formulas $x_r=r\mathrm {cos}\theta$ and $y_r=r\mathrm {sin}\theta$, where $r$ and $\theta$ are the polar radius and the polar angle in the confocal plane of lenses $L_1$ and $L_2$. The position shifts along $x$ and $y$ directions in the speckle plane (i.e., the rear focal plane of lens $L_2$) are taken as $\Delta x=\Delta \rho \mathrm {cos}\Delta \phi$ and $\Delta y=\Delta \rho \mathrm {sin}\Delta \phi$, where $\Delta \rho =\sqrt {\Delta x^2+\Delta y^2}$ is the shift in the polar radius and $\Delta \phi$ is the difference in the polar angle. Thus, one can deduce

(A9)$$\left\langle\tilde{E}_o\left(x_1,y_1\right)\tilde{E}_o^*\left(x_2,y_2\right)\right\rangle{=}\frac{2\pi R'}{\lambda^2 f_2^2}J_0\left(\frac{kR'\Delta\rho}{f_2}\right),$$

where $J_0\left (\cdot \right )$ is the Bessel function of zero order of the first kind. In the calculation, the function $J_0(x)=\frac {1}{2\pi }\int _{-\pi }^\pi \mathrm {exp}\left (ix\mathrm {cos}\psi \right )\mathrm {d}\psi$ and the condition $E\left (r\right )=\delta \left (r-R'\right )$ [54] are used for an extremely narrow annular ring with a radius of $R'$ .

The average speckle intensity is calculated as

(A10)$$\left\langle I_o\left(x_1,y_1\right)\right\rangle{=}\left\langle I_o\left(x_2,y_2\right)\right\rangle{=}\frac{2\pi R'}{\lambda^2 f_2^2}.$$

Eventually, by substituting Eqs. (A7), (A9) and (A10) into Eq. (A6), one can get the expression of the normalized second-order intensity correlation function

(A11)$$g_n^{\left(2\right)}\left(\Delta\rho\right)=\sum_{m=0}^n\left(C_n^m\right)^2\left| J_0\left(\frac{2\pi f_1}{P\delta f_2}\Delta\rho\right)\right|^{2m}.$$

A2. Visibility and the grain size of the designed speckle fields

The visibility and the average grain size of the designed speckle fields can be deduced from the normalized second-order intensity correlation function. According to Eq. (A11), the visibility of the designed speckle fields is obtained as

(A12)$$V=\frac{\left(g_n^{\left(2\right)}\right)_{max}-\left(g_n^{\left(2\right)}\right)_{min}}{\left(g_n^{\left(2\right)}\right)_{max}+\left(g_n^{\left(2\right)}\right)_{min}}= \frac{C_{2n}^n-1}{C_{2n}^n+1}=1-\frac{2}{\frac{\left(2n\right)!}{\left(n!\right)^2}+1}.$$

The average grain size $D$ of the speckles is defined as the distance from the central maximum to the first minimum of the intensity correlation function of Eq. (A11), which yields

(A13)$$D=0.383\frac{P\delta f_2}{f_1}.$$

A3. Speckle contrast and probability density function of the designed speckle fields

The statistics properties of speckle fields can be characterized by the speckle contrast $C$ [12] and the probability density function (PDF). For the designed target speckle fields with a power term $n$, the speckle contrast $C_n$ is defined as

(A14)$$C_n=\sqrt{\frac{\left\langle I_n^2\right\rangle}{\left\langle I_n\right\rangle^2}-1}\, .$$

The speckles obey Rayleigh statistics for $C_n = 1$, super-Rayleigh statistics for $C_n > 1$ and sub-Rayleigh statistics for $C_n < 1$. As an example, we calculated the speckle contrast $C_n$ of the experimentally generated speckle fields in Figs. 3(b1-b3) in the main text, and the results are shown in Fig. 8. One sees that the speckle contrasts are $C_1=0.99$, $C_2=1.58$ and $C_3=2.50$ for the designed target speckle fields with $n = 1,\, 2$ and 3, respectively, indicating that the speckle field with $n = 1$ is a Rayleigh speckle field while those with $n = 2,\, 3$ are super-Rayleigh speckle fields, respectively. It is confirmed that the speckle contrast is larger for a speckle field with a larger power term $n$.

 

Fig. 8. The speckle intensity distributions and the speckle contrasts of the designed target speckle fields with different power term $n = 1,\, 2$ and 3, respectively. Other parameters are the same as those in Figs. 3(b1-b3) in the main text.

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The PDF of the designed target speckle fields with a power term $n$ is calculated as

(A15)$$P_I\left(I_n\right)=\frac{I_n^{\left(1-n\right)/n}}{n\left\langle I_o\right\rangle}\mathrm{exp}{\left(-\frac{I_n^{1/n}}{\left\langle I_o\right\rangle}\right)}.$$

It can be seen from Eq. (A15) that the PDF of the designed speckle fields with $n=1$ is in the form of negative exponential function, just as that of the standard Rayleigh speckle field, while those with $n>1$ deviate from the negative exponential function because the speckle fields with $n > 1$ are the super-Rayleigh speckle field.

Funding

National Natural Science Foundation of China (91750204); 111 Project (B07013); Fundamental Research Funds for the Central Universities.

Disclosures

The authors declare no conflict of interest.

Data availability

Data may be obtained from the authors upon reasonable request.

References

1. T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52(5), R3429–R3432 (1995). [CrossRef]  

2. D. Zhang, Y.-H. Zhai, L.-A. Wu, and X.-H. Chen, “Correlated two-photon imaging with true thermal light,” Opt. Lett. 30(18), 2354–2356 (2005). [CrossRef]  

3. A. Valencia, G. Scarcelli, M. D’Angelo, and Y. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. 94(6), 063601 (2005). [CrossRef]  

4. G. Scarcelli, V. Berardi, and Y. Shih, “Phase-conjugate mirror via two-photon thermal light imaging,” Appl. Phys. Lett. 88(6), 061106 (2006). [CrossRef]  

5. X.-F. Liu, X.-H. Chen, X.-R. Yao, W.-K. Yu, G.-J. Zhai, and L.-A. Wu, “Lensless ghost imaging with sunlight,” Opt. Lett. 39(8), 2314–2317 (2014). [CrossRef]  

6. H. Yu, R. Lu, S. Han, H. Xie, G. Du, T. Xiao, and D. Zhu, “Fourier-transform ghost imaging with hard x rays,” Phys. Rev. Lett. 117(11), 113901 (2016). [CrossRef]  

7. D. Pelliccia, A. Rack, M. Scheel, V. Cantelli, and D. M. Paganin, “Experimental x-ray ghost imaging,” Phys. Rev. Lett. 117(11), 113902 (2016). [CrossRef]  

8. Z.-H. Xu, W. Chen, J. Penuelas, M. Padgett, and M.-J. Sun, “1000 fps computational ghost imaging using LED-based structured illumination,” Opt. Express 26(3), 2427–2434 (2018). [CrossRef]  

9. J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78(6), 061802 (2008). [CrossRef]  

10. Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79(5), 053840 (2009). [CrossRef]  

11. Y. Cai and F. Wang, “Lensless imaging with partially coherent light,” Opt. Lett. 32(3), 205–207 (2007). [CrossRef]  

12. Y. Bromberg and H. Cao, “Generating non-Rayleigh speckles with tailored intensity statistics,” Phys. Rev. Lett. 112(21), 213904 (2014). [CrossRef]  

13. N. Bender, H. Yilmaz, Y. Bromberg, and H. Cao, “Customizing speckle intensity statistics,” Optica 5(5), 595–600 (2018). [CrossRef]  

14. C. Luo and J. Cheng, “Ghost imaging with shaped incoherent sources,” Opt. Lett. 38(24), 5381–5384 (2013). [CrossRef]  

15. S. M. M. Khamoushi, Y. Nosrati, and S. H. Tavassoli, “Sinusoidal ghost imaging,” Opt. Lett. 40(15), 3452–3455 (2015). [CrossRef]  

16. L. Zhang, Y. Lu, D. Zhou, H. Zhang, L. Li, and G. Zhang, “Superbunching effect of classical light with a digitally designed spatially phase-correlated wave front,” Phys. Rev. A 99(6), 063827 (2019). [CrossRef]  

17. K. Kuplicki and K. W. C. Chan, “High-order ghost imaging using non-Rayleigh speckle sources,” Opt. Express 24(23), 26766–26776 (2016). [CrossRef]  

18. L. Wang and S. Zhao, “Fast reconstructed and high-quality ghost imaging with fast Walsh-Hadamard transform,” Photonics Res. 4(6), 240–244 (2016). [CrossRef]  

19. K. M. Czajkowski, A. Pastuszczak, and R. Kotyński, “Single-pixel imaging with Morlet wavelet correlated random patterns,” Sci. Rep. 8(1), 466 (2018). [CrossRef]  

20. W. Gong, “Sub-Nyquist ghost imaging by optimizing point spread function,” Opt. Express 29(11), 17591–17601 (2021). [CrossRef]  

21. X. Nie, F. Yang, X. Liu, X. Zhao, R. Nessler, T. Peng, M. S. Zubairy, and M. O. Scully, “Noise-robust computational ghost imaging with pink noise speckle patterns,” Phys. Rev. A 104(1), 013513 (2021). [CrossRef]  

22. B. Jack, J. Leach, J. Romero, S. Franke-Arnold, M. Ritsch-Marte, S. M. Barnett, and M. J. Padgett, “Holographic ghost imaging and the violation of a Bell inequality,” Phys. Rev. Lett. 103(8), 083602 (2009). [CrossRef]  

23. K. W. C. Chan, M. N. O’Sullivan, and R. W. Boyd, “High-order thermal ghost imaging,” Opt. Lett. 34(21), 3343–3345 (2009). [CrossRef]  

24. X.-H. Chen, I. N. Agafonov, K.-H. Luo, Q. Liu, R. Xian, M. V. Chekhova, and L.-A. Wu, “High-visibility, high-order lensless ghost imaging with thermal light,” Opt. Lett. 35(8), 1166–1168 (2010). [CrossRef]  

25. F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett. 104(25), 253603 (2010). [CrossRef]  

26. B. Sun, S. S. Welsh, M. P. Edgar, J. H. Shapiro, and M. J. Padgett, “Normalized ghost imaging,” Opt. Express 20(15), 16892–16901 (2012). [CrossRef]  

27. R. E. Meyers, K. S. Deacon, and Y. Shih, “Positive-negative turbulence-free ghost imaging,” Appl. Phys. Lett. 100(13), 131114 (2012). [CrossRef]  

28. W. Wang, X. Hu, J. Liu, S. Zhang, J. Suo, and G. Situ, “Gerchberg-Saxton-like ghost imaging,” Opt. Express 23(22), 28416–28422 (2015). [CrossRef]  

29. W. Wang, Y.-P. Wang, J. Li, X. Yang, and Y. Wu, “Iterative ghost imaging,” Opt. Lett. 39(17), 5150–5153 (2014). [CrossRef]  

30. O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. 95(13), 131110 (2009). [CrossRef]  

31. H. Li, Z. Chen, J. Xiong, and G. Zeng, “Periodic diffraction correlation imaging without a beam-splitter,” Opt. Express 20(3), 2956–2966 (2012). [CrossRef]  

32. H. Li, Y. Zhang, J. Shi, and G. Zeng, “Experimental realization of reflection-type periodic diffraction correlation imaging,” Appl. Phys. Lett. 102(20), 201901 (2013). [CrossRef]  

33. D. B. Phillips, R. He, Q. Chen, G. M. Gibson, and M. J. Padgett, “Non-diffractive computational ghost imaging,” Opt. Express 24(13), 14172–14182 (2016). [CrossRef]  

34. L. Li, P. Hong, and G. Zhang, “Transverse revival and fractional revival of the Hanbury Brown and Twiss bunching effect with discrete chaotic light,” Phys. Rev. A 99(2), 023848 (2019). [CrossRef]  

35. X. Wang, Z. Li, F. Wen, H. Wang, W. Guo, N. Gao, Z. Li, and P. Mai, “Controllable visibility and resolution of Nth-order Talbot imaging with pseudo-thermal light,” Opt. Express 25(12), 13455–13464 (2017). [CrossRef]  

36. R. W. Gerchberg, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35(2), 237–246 (1972).

37. D. M. Cottrell, J. M. Craven, and J. A. Davis, “Nondiffracting random intensity patterns,” Opt. Lett. 32(3), 298–300 (2007). [CrossRef]  

38. G. Scott and N. McArdle, “Efficient generation of nearly diffraction-free beams using an axicon,” Opt. Eng. 31(12), 2640–2643 (1992). [CrossRef]  

39. G. Zheng, H. Mühlenbernd, M. Kenney, G. Li, T. Zentgraf, and S. Zhang, “Metasurface holograms reaching 80% efficiency,” Nat. Nanotechnol. 10(4), 308–312 (2015). [CrossRef]  

40. J. S. Liu and M. R. Taghizadeh, “Iterative algorithm for the design of diffractive phase elements for laser beam shaping,” Opt. Lett. 27(16), 1463–1465 (2002). [CrossRef]  

41. H.-E. Hwang, H. T. Chang, and W.-N. Lie, “Multiple-image encryption and multiplexing using a modified Gerchberg-Saxton algorithm and phase modulation in Fresnel-transform domain,” Opt. Lett. 34(24), 3917–3919 (2009). [CrossRef]  

42. C.-F. Ying, H. Pang, C. J. Fan, and W. D. Zhou, “New method for the design of a phase-only computer hologram for multiplane reconstruction,” Opt. Eng. 50(5), 055802 (2011). [CrossRef]  

43. X.-F. Liu, M.-F. Li, X.-R. Yao, W.-K. Yu, G.-J. Zhai, and L.-A. Wu, “High-visibility ghost imaging from artificially generated non-Gaussian intensity fluctuations,” AIP Adv. 3(5), 052121 (2013). [CrossRef]  

44. K. Luo, B. Huang, W. Zheng, and L. Wu, “Nonlocal imaging by conditional averaging of random reference measurements,” Chin. Phys. Lett. 29(7), 074216 (2012). [CrossRef]  

45. D. Cao, Q. Li, X. Zhuang, C. Ren, S. Zhang, and X. Song, “Ghost images reconstructed from fractional-order moments with thermal light,” Chin. Phys. B 27(12), 123401 (2018). [CrossRef]  

46. G. Sinclair, J. Leach, P. Jordan, G. Gibson, E. Yao, Z. J. Laczik, M. J. Padgett, and J. Courtial, “Interactive application in holographic optical tweezers of a multi-plane Gerchberg-Saxton algorithm for three-dimensional light shaping,” Opt. Express 12(8), 1665–1670 (2004). [CrossRef]  

47. P. Zhou, Y. Bi, M. Sun, H. Wang, F. Li, and Y. Qi, “Image quality enhancement and computation acceleration of 3D holographic display using a symmetrical 3D GS algorithm,” Appl. Opt. 53(27), G209–G213 (2014). [CrossRef]  

48. B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D computational imaging with single-pixel detectors,” Science 340(6134), 844–847 (2013). [CrossRef]  

49. C. Zhao, W. Gong, M. Chen, E. Li, H. Wang, W. Xu, and S. Han, “Ghost imaging lidar via sparsity constraints,” Appl. Phys. Lett. 101(14), 141123 (2012). [CrossRef]  

50. M.-D. Wei, W.-L. Shiao, and Y.-T. Lin, “Adjustable generation of bottle and hollow beams using an axicon,” Opt. Commun. 248(1-3), 7–14 (2005). [CrossRef]  

51. D.-Z. Cao, J. Xiong, S.-H. Zhang, L.-F. Lin, L. Gao, and K. Wang, “Enhancing visibility and resolution in Nth-order intensity correlation of thermal light,” Appl. Phys. Lett. 92(20), 201102 (2008). [CrossRef]  

52. K. Uno, J. Uozumi, and T. Asakura, “Speckle clustering in diffraction patterns of random objects under ring-slit illumination,” Opt. Commun. 114(3-4), 203–210 (1995). [CrossRef]  

53. C. R. Alves, A. J. Jesus-Silva, and E. J. S. Fonseca, “Effect of the spatial coherence length on the self-reconfiguration of a speckle field,” Phys. Rev. A 94(1), 013835 (2016). [CrossRef]  

54. M. Ibrahim, J. Uozumi, and T. Asakura, “Longitudinal correlation properties of speckles produced by ring-slit illumination,” Opt. Rev. 5(3), 129–137 (1998). [CrossRef]