Theoretical and experimental study of the color of ghost imaging

Xue-lei Yin; Yun-jie Xia; De-yang Duan

doi:10.1364/OE.26.018944

1. Introduction

Ghost imaging (GI) is an indirectly imaging technique that produces the image of an object by using the correlation between the intensity recorded at two detectors illuminated by spatially separated correlated beams. In contrast to conventional optical imaging techniques, one of the key advantages of GI is that it has the ability to form image without need for a spatial-resolving detector placed near the target. Consequently, GI is a good technique for imaging target immersed in optically harsh environment, e.g., turbid media [1], atmospheric turbulence [2,3]. Recently, the experiments of GI with X-ray source [4–7] have been reported, notarizing the power of GI in exploring and analyzing the internal of complex material, e.g. biomolecular structures and nanomaterials.

Initially demonstrated at quantum level by using entangled photon pairs as the source, GI was subsequently performed with pseudothermal light and true thermal sources [8–10]. Nevertheless, only monochromatic ghost images were obtained in the early studies in spite of the drawbacks in observation. In recent years, color GI has received extensive attention and is considered to be one of the main directions for future development. Our previous work briefly discussed the color of GI in theory [11]. To our great delight, Welsh et al. have shown the color ghost image can be obtained with multi-wavelength thermal source in experiment [12]. However, the color formation mechanism of GI has still not been tested by experiment. In the meantime, [13] theoretically predicts there is a phenomenon of color distortion in multi-wavelength GI with spatial light modulator (SLM). In order to confirm its practicability, the color of GI needs to be verified experimentally and rigorously.

In this article we theoretically and experimentally research into the color of three representative types of GI, i.e., GI with rotating ground glass plate (RGGP), GI with SLM and computational GI (CGI). The principle of the color of GI with SLM can be as a reference for that in a quantum system.

2. Theory and Experimental Result

We first consider the two-wavelength GI with RGGP. The setup is shown in Fig. 1(a). Two quasimonochromatic laser beams E₁ and E₂ with central frequencies Ω₁ and Ω₂ are get together by a dichroic mirror (DM), and then pass through a RGGP. The transmissive power generates two beams by a 50:50 beam splitter, one of which illuminates the object T(ρ⃗) and is finally collected by a bucket detector. The other beam, known as the idle one, is directly measured by a CCD camera. The ghost image is reconstructed by extracting the cross correlation between the two photocurrents arising from the bucket detector and CCD camera. Thus, the ghost image can be expressed as

C ({\vec{ρ}}_{i}, {\vec{ρ}}_{o}, t) = 〈 δ I_{im} ({\vec{ρ}}_{i}, t) δ I_{om} ({\vec{ρ}}_{o}, t) 〉,

where δI(ρ⃗, t) = I(ρ⃗, t) − 〈I(ρ⃗, t)〉, m = 1, 2. The subscripts i and o represent the idle beam and the object beam respectively.

\begin{array}{l} C ({\vec{ρ}}_{i}, {\vec{ρ}}_{o}, t) \\ = 〈 {| E_{i 1} ({\vec{ρ}}_{i}, t) |}^{2} {| E_{o 1} ({\vec{ρ}}_{o}, t) |}^{2} 〉 - 〈 {| E_{i 1} ({\vec{ρ}}_{i}, t) |}^{2} 〉 〈 {| E_{o 1} ({\vec{ρ}}_{o}, t) |}^{2} 〉 \\ + 〈 {| E_{i 2} ({\vec{ρ}}_{i}, t) |}^{2} {| E_{o 2} ({\vec{ρ}}_{o}, t) |}^{2} 〉 - 〈 {| E_{i 2} ({\vec{ρ}}_{i}, t) |}^{2} 〉 〈 {| E_{o 2} ({\vec{ρ}}_{o}, t) |}^{2} 〉 \\ = C_{1} ({\vec{ρ}}_{i}, {\vec{ρ}}_{o}, t) + C_{2} ({\vec{ρ}}_{i}, {\vec{ρ}}_{o}, t) . \end{array}

Here,

\begin{array}{l} C_{m} ({\vec{ρ}}_{i}, {\vec{ρ}}_{o}, t) \\ = \int d ω_{im} d ω_{i m}^{'} d {\vec{q}}_{i m} d {\vec{q}}_{im}^{'} d ω_{om} d ω_{om}^{'} d {\vec{q}}_{om} d {\vec{q}}_{om}^{'} \\ \times H_{im}^{*} ({\vec{ρ}}_{im}, {\vec{q}}_{im}; ω_{im}) H_{im} ({\vec{ρ}}_{im}, {\vec{q}}_{im}; ω_{im}^{'}) \\ \times H_{om}^{*} ({\vec{ρ}}_{om}, {\vec{q}}_{om}; ω_{om}) H_{om} ({\vec{ρ}}_{om}, {\vec{q}}_{rm}; ω_{om}^{'}) \\ \times e^{i (ω_{im} - ω_{im}^{'}) t} e^{i (ω_{on} - ω_{on}^{'}) t} T^{*} ({\vec{ρ}}_{o}) T ({\vec{ρ}}_{o}) \\ \times G ({\vec{q}}_{im}, {\vec{q}}_{im}^{'}, {\vec{q}}_{om}, {\vec{q}}_{om}^{'}, ω_{im}, ω_{im}^{'}, ω_{om}, ω_{om}^{'}), \end{array}

where,

\begin{array}{l} G ({\vec{q}}_{im}, {\vec{q}}_{im}^{'}, {\vec{q}}_{om}, {\vec{q}}_{om}^{'}, ω_{im}, ω_{im}^{'}, ω_{om}, ω_{om}^{'}) \\ = 〈 V^{*} ({\vec{q}}_{im}) V ({\vec{q}}_{om}^{'}) 〉 〈 V^{*} ({\vec{q}}_{om}) V ({\vec{q}}_{im}^{'}) 〉 \\ \times 〈 ε_{im}^{*} (ω_{im}) ε_{im} (ω_{im}^{'}) 〉 〈 ε_{om}^{*} (ω_{om}) ε_{om} (ω_{om}^{'}) 〉 \end{array}

is the intensity cross-correlation function of the two light fields in the spatial and temporal frequency domain evaluated at the output plane of the RGGP. Substituting Eq. (4) into Eq. (3), after some calculations, we can thus rewrite the ghost image of Eq. (3) as

\begin{array}{l} C_{m} ({\vec{ρ}}_{i}, {\vec{ρ}}_{o}, t) = \\ B {| \int d {\vec{ρ}}_{i}^{'} d {\vec{ρ}}_{o}^{'} W ({\vec{ρ}}_{i}^{'}, {\vec{ρ}}_{o}^{'}) H_{i} ({\vec{ρ}}_{i}, {\vec{ρ}}_{i}^{'}; Ω_{1}) H_{o}^{*} ({\vec{ρ}}_{o}, {\vec{ρ}}_{o}^{'}; Ω_{2}) O (\vec{ρ}) |}^{2}, \end{array}

where B = I_mI_m with I_m = 〈|∫ dω_mε_m(ω_m)e^−iω_mt|²〉 being the average intensities of the two incident light beams on the RGGP. W(ρ⃗′_i, ρ⃗′_i) is the spatial Fourier transform of 〈V(q⃗′_m)V^*(q⃗′_m)〉. The transfer functions H_i and H_o are written in position space. 〈T^*(ρ⃗₁)T(ρ⃗₂)〉 = λO(ρ⃗′₁)δ(ρ⃗′₁ − ρ⃗′₂) [14].

Fig. 1 (Color online) (a) Two-wavelength GI with RGGP. (b) CGI with two-wavelength source. (c) Two-wavelength GI with SLM. SLM: spatial light modulator; DM: dichroic mirror; BS: beam splitter; BD: bucket detector.

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Equation (2) shows that the two-wavelength GI with RGGP produces two sub-images. The term B of Eq. (5) shows that the color of ghost image arising from the single wavelength source is determined by the light source. Therefore, each light with a certain wavelength produces a specific monochromatic image. These two monochromatic sub-images are added together to form one color image. If the light source contains three wavelengths or colors (e. g., red, green, and blue one), a perfectly panchromatic ghost image appears, exactly the same as the object.

The experimental setup, shown in Fig. 1(a), is a pseudothermal reflective GI system. Two continuous wave laser beams with λ₁ = 532nm and λ₂ = 633nm are get together by a DM, and then illuminate onto a ground glass plate rotating at 3 rad/min to produce a field of randomly varying speckles. The pseudothermal light is divided by a 50:50 BS into two spatially correlated idle and object beams. The object beam illuminates a 1.2cm × 1.6cm (reflection plate) object and the reflective power is collected by a bucket detector. Meanwhile, the idle beam is directly collected by a full color CCD camera (the imaging source, DFK23U618). The distance from the BS to the bucket detector are the same as that to the CCD, L₁ = L₂ = 22cm. In the experiment, the bucket detector and camera captured 10000 frames with an exposure time of 0.01s. The reconstructed ghost image is shown in Fig. 2(b). When the single wavelength source is used independently, we can obtain the monochromatic ghost image shown in Figs. 2(c) and 2(d) correspond to laser 532nm and 633nm, respectively.

Fig. 2 (a) color object. Top row: GI with RGGP, (b)image with λ₁ = 532nm and λ₂ = 633nm, (c) image with λ₁ = 532nm, (d) image with λ₂ = 633nm. Middle row: CGI, (e) image with λ₁ = 532nm and λ₂ = 633nm, (f) image with λ₁ = 532nm, (g) image with λ₂ = 633nm. Bottom row: GI with SLM, (h) image with λ₁ = 532nm and λ₂ = 633nm, (i) object beam λ₁ = 532nm, idle beam λ₂ = 633nm, (j) object beam λ₂ = 633nm, idle beam λ₁ = 532nm.

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Then, we consider the CGI with two-wavelength source. Different from conventional GI, CGI configuration only has one object arm, as shown in Fig. 1(b). Two laser beams, modulated by an SLM, illuminates an object and is collected by a bucket detector as last. The image is reconstructed by correlating the computed patterns with the measured light field at the object arm. Consulting the calculation process of GI with RGGP, we obtain

\begin{array}{l} C ({\vec{ρ}}_{s}, {\vec{ρ}}_{d}, t) \\ = 〈 {| E_{s 1} ({\vec{ρ}}_{s}, t) |}^{2} {| E_{d 1} ({\vec{ρ}}_{d}, t) |}^{2} 〉 - 〈 {| E_{s 1} ({\vec{ρ}}_{s}, t) |}^{2} 〉 〈 {| E_{d 1} ({\vec{ρ}}_{d}, t) |}^{2} 〉 \\ + 〈 {| E_{s 2} ({\vec{ρ}}_{s}, t) |}^{2} {| E_{s 2} ({\vec{ρ}}_{d}, t) |}^{2} 〉 - 〈 {| E_{s 2} ({\vec{ρ}}_{s}, t) |}^{2} 〉 〈 {| E_{s 2} ({\vec{ρ}}_{d}, t) |}^{2} 〉 \\ + 〈 {| E_{s 1} ({\vec{ρ}}_{s}, t) |}^{2} {| E_{d 2} ({\vec{ρ}}_{d}, t) |}^{2} 〉 - 〈 {| E_{s 1} ({\vec{ρ}}_{s}, t) |}^{2} 〉 〈 {| E_{d 2} ({\vec{ρ}}_{d}, t) |}^{2} 〉 \\ + 〈 {| E_{s 2} ({\vec{ρ}}_{s}, t) |}^{2} {| E_{d 1} ({\vec{ρ}}_{d}, t) |}^{2} 〉 - 〈 {| E_{s 2} ({\vec{ρ}}_{s}, t) |}^{2} 〉 〈 {| E_{d 1} ({\vec{ρ}}_{d}, t) |}^{2} 〉 \\ = C_{1} ({\vec{ρ}}_{s}, {\vec{ρ}}_{d}, t) + C_{2} ({\vec{ρ}}_{s}, {\vec{ρ}}_{d}, t) + C_{3} ({\vec{ρ}}_{s}, {\vec{ρ}}_{d}, t) + C_{4} ({\vec{ρ}}_{s}, {\vec{ρ}}_{d}, t), \end{array}

where ρ⃗_s and ρ⃗_d represent transverse coordinates at SLM and bucket detector respectively.

Further calculations show that the image term (C_m(ρ⃗_s, ρ⃗_d, t)) has the same form as Eq. (5). However, this does not mean the color formation of CGI is consistent with that of conventional GI. In theory, the CGI has only one optical path, and the image is reconstructed by correlating the light field collected by the bucket detector and the computed patterns at the object plane. In experiment, the image is reconstructed by correlating the signal collected by the bucket detector and the grayscale map input to the SLM. Consequently, the color of the CGI is consistent with the light detected by the bucket detector. Equation (6) shows that the CGI with two-wavelength source produces four monochromatic sub-images, which is similar to the two-wavelength GI with SLM. However, the color of C₃ (C₄) obtained by nondegenerate correlations is the same as the C₂ (C₁) obtained by degenerate correlations [15]. Therefore, there is no color distortion in CGI with multiwavelength source. Consequently, if a tricolor light source is used, a full-color ghost image can be obtained.

In this experiment we adapt a CGI setup, as shown in Fig. 1(b). The two laser beams with λ₁ = 532nm and λ₂ = 633nm are get together by a DM, and then are projected onto a two-dimensional amplitude-only ferroelectric liquid crystal SLM (FLC-SLM, Meadowlark Optics A512), with bandwidth 450nm–850nm. The SLM has 512×512 pixels in the window addressable 15μm×15μm pixels. The computed light field illuminates the object placed at a distance of L₁ = 70cm and L₂ = 11cm from the SLM, and reflected power is collected by a bucket detector. The reconstructed ghost image is shown in Fig. 2(e). The monochromatic ghost images shown in Figs. 2(f) and 2(g) correspond to laser 532nm and 633nm, respectively. It is worth noting that the brightness of image Fig. 2(g) is slightly higher than that of image Fig. 2(f) because the reflectance of SLM is difference for different wavelength light.

Finally, we consider the two-wavelength GI with SLM. The setup is shown in Fig. 1(c). Different from the GI with RGGP, the RGGP is replaced by an SLM. Thus, the ghost image can be expressed as

\begin{array}{l} C ({\vec{ρ}}_{i}, {\vec{ρ}}_{o}, t) \\ = 〈 {| E_{i 1} ({\vec{ρ}}_{i}, t) |}^{2} {| E_{o 1} ({\vec{ρ}}_{o}, t) |}^{2} 〉 - 〈 {| E_{i 1} ({\vec{ρ}}_{i}, t) |}^{2} 〉 〈 {| E_{o 1} ({\vec{ρ}}_{o}, t) |}^{2} 〉 \\ + 〈 {| E_{i 2} ({\vec{ρ}}_{i}, t) |}^{2} {| E_{o 2} ({\vec{ρ}}_{o}, t) |}^{2} 〉 - 〈 {| E_{i 2} ({\vec{ρ}}_{i}, t) |}^{2} 〉 〈 {| E_{o 2} ({\vec{ρ}}_{o}, t) |}^{2} 〉 \\ + 〈 {| E_{i 1} ({\vec{ρ}}_{i}, t) |}^{2} {| E_{o 2} ({\vec{ρ}}_{o}, t) |}^{2} 〉 - 〈 {| E_{i 1} ({\vec{ρ}}_{i}, t) |}^{2} 〉 〈 {| E_{o 2} ({\vec{ρ}}_{o}, t) |}^{2} 〉 \\ + 〈 {| E_{i 2} ({\vec{ρ}}_{i}, t) |}^{2} {| E_{o 1} ({\vec{ρ}}_{o}, t) |}^{2} 〉 - 〈 {| E_{i 2} ({\vec{ρ}}_{i}, t) |}^{2} 〉 〈 {| E_{o 1} ({\vec{ρ}}_{o}, t) |}^{2} 〉 \\ = C_{1} ({\vec{ρ}}_{i}, {\vec{ρ}}_{o}, t) + C_{2} ({\vec{ρ}}_{i}, {\vec{ρ}}_{o}, t) + C_{3} ({\vec{ρ}}_{i}, {\vec{ρ}}_{o}, t) + C_{4} ({\vec{ρ}}_{i}, {\vec{ρ}}_{o}, t) . \end{array}

Equation (7) shows that two-wavelength GI with SLM produces four ghost images. Among them, C₁ and C₂ are the same as the one obtained by GI with RGGP. However, C₃ and C₄ are different because these two images arising from nondegenerate correlations [16]. The term B of Eq. (5) shows that the color of C₃ and C₄ is a superimposed color depending on the idle and object light beam. According to the principle of light color superposition [17], C₃ and C₄ show yellow. It is worth noting that C₁ and C₄, C₂ and C₃ show the same information of the object, but the color is different (i.e., C₁ shows red, C₂ shows green, C₃ and C₄ show yellow). These four monochromatic images are added together to form one color image.

In this experiment we adapt setup is shown in Fig. 1(c), by replacing the RGGP with an FLC-SLM (Meadowlark Optics A512). The computed light field is divided into two beams by a BS. Thus, the reconstructed ghost image is shown in Fig. 2(h). By replacing the BS with a DM, the idle beam and object beam has only one wavelength light, respectively. Therefore, we obtain the monochromatic ghost images in this condition, as shown in Figs. 2(i) and 2(j). Different from the GI with RGGP, the color of the output image will show distortion, i.e., the green part of the object appears verdancy and the red part appears orange. Consequently, the theoretical prediction is confirmed.

The color formation mechanism of quantum GI is the same as the GI with nondegenerate correlations (e.g., C₃, C₄). For quantum GI with multiwavelength source, there is also color distortion because there are only nondegenerate correlations [16].

3. Discussion and Conclusion

For conventional GI with RGGP, we can see that the color of ghost image is the same as the light source. Essentially, the color of GI with RGGP is a superimposed one, depending on the idle and object light beams, and following the principle of light color superposition. It is just that the light of these two paths is consistent. A full color ghost image that is the same as the color of the object can be obtained by multi-wavelength light source. For the CGI, the color of ghost image is the same as that of the light source, too. For the multi-wavelength light source, we can obtain a full color ghost image that is the same as the color of the object because the image obtained by nondegenerate correlations is the same as the light source.

In the case of GI with SLM, the color of the ghost image depends on the idle and object beam, the color of the reconstructed image follows the principle of light color superposition (Figs .2(i) and 2(j)). The color of the output image will be more complicated as the degenerate and non-degenerate correlations are obtained at the same time. The experiment shows that the degenerate correlations yield two monochromatic ghost images, i.e., red image (C₁) and green image (C₂). The nondegenerate correlations yield two monochromatic ghost images, i.e., yellow image (C₃, Fig. 2(j)) and yellow image (C₄, Fig. 2(i)). The red image and yellow image are added together to form an orange image while the green image and yellow image are added together to form a verdancy image. Finally, the output image shows orange and verdancy. The quantum GI is the same as the GI with nondegenerate correlations. There is color distortion phenomenon in both GI with SLM and quantum GI. The reason is the effect of non-degenerate correlations.

In conclusion, we have theoretically and experimentally studied the color of three representative GIs, i.e., GI with RGGP, GI with SLM and CGI. The color formation mechanism of GI, which is different from that of conventional optical imaging, has been obtained. In the meantime, the color distortion in multi-wavelength GI with SLM has been confirmed in experiment. Moreover, we found that there is also a phenomenon of color distortion in quantum GI. The color formation mechanism of GI is summarized as follows: (i) the color of GI with RGGP and CGI are the same as the light source. A full color image can be obtained with multi-wavelength source. More important, there is not color distortion in these two GIs. (ii) the color of GI with SLM and quantum GI is a superimposed one, which follows the principle of light color superposition, and depends on the idle and object light beams. A full color image can also be obtained under the condition of multiwavelength source, but with color distortion existing. As a result, the GI with RGGP and CGI has wider prospect of application, because they do not have color distortion.

Funding

This project was supported by the National Natural Science Foundation of China(Grant Nos. 11704221, 11647172, and 61675115), the Natural Science Foundation of Shandong Province, China (Grant No. ZR2016AP09).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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Theoretical and experimental study of the color of ghost imaging

Abstract

1. Introduction

2. Theory and Experimental Result

3. Discussion and Conclusion

Funding

Disclosures

References and links

Cited By

Figures (2)

Equations (7)

Optics Express