Ghost imaging via sparse structured illumination source

Han Wu; Chenglong Wang; Wenlin Gong

doi:10.1364/OE.26.004183

1. Introduction

Ghost imaging (GI), as a novel imaging mechanism, can non-locally imaging an unknown object with a single-pixel detector at the object path [1–10]. Because all the photons reflected (or transmitted) from the object illuminate the same single-pixel detector, GI has the capability of being highly sensitive in detection, and recently it has aroused wide interests in developing a new imaging method of remote sensing [6–11]. In present GI system, limited by the application mode and detection capability, pseudo-thermal light source is widely utilized instead of entangled or true thermal light source [3–10]. For GI with pseudo-thermal light source, the fluctuation light field is usually obtained by passing a pulsed laser through a rotating ground glass or a spatial light modulator [7–14]. However, the effective emitting energy or sampling speed is limited for above modulation methods in practice. For the former, the effective emitting energy is high enough but the sampling speed is restricted by the frame frequency of the reference charge-coupled device (CCD) camera [7–11]. For the latter, the sampling speed is fast enough whereas the effective emitting energy is very low because of the diffraction effect of its structure and low damage threshold [12–14]. Therefore, it is natural to ask whether GI with both high effective emitting energy and fast sampling speed can be achieved by exploring a new pseudo-thermal light source. Recently, an approach via sparse structured illumination is raised to create a pseudo-thermal light source and the principle demonstration of GI has also been implemented [15,16]. Generally speaking, this approach has a fast modulating speed (up to MHz) based on electro-optic modulators and the emitting energy can be improved by increasing the sub-source’s quantity, which may be a candidate of our expected pseudo-thermal light source. However, restricted by the center-to-center separation of two sub-sources, pseudo-thermal light source with periodic configuration illumination, like an optical grating, will generate a periodic light field, which greatly affects the effective imaging area of GI and also leads to a low energy efficiency for emitting system [15]. In this paper, in order to overcome the periodicity of pseudo-thermal light field, genetic algorithm is used to optimize the spatial configuration of sparse-array illumination source. We demonstrate by numerical simulation and experiments that the optimized sparse-array illumination can generate a better pseudo-thermal light source compared with a full-array illumination source, even if the sub-source’s quantity is reduced. In addition, GI experiments are also performed to further verify the validity of the optimized sparse structured illumination source.

2. Theory and Model

Based on the theory of statistical optics [17], pseudo-thermal light field can be achieved by a sparse illumination source, where the superposition of the light fields from more than five independent sub-sources can approach to the statistical property of thermal light field. As shown in Fig. 1, the light field emitted from a regular structured illumination source can be fully overlapped by a 2f optical system and its property can be valuated by the normalized second-order intensity correlation function, namely

g^{(2)} (x, y; x_{0}, y_{0}) = \frac{〈 I (x_{0}, y_{0}) I (x, y) 〉}{〈 I (x_{0}, y_{0}) 〉 〈 I (x, y) 〉}

where g⁽²⁾(x, y; x₀, y₀) is the normalized second-order intensity correlation function, I(x₀, y₀) is the intensity distribution at the position (x₀, y₀) of the detection plane. In addition,

〈 \cdot 〉

denotes the ensemble average of a distribution function.

Fig. 1 Optical system of pseudo-thermal light field via sparse structured illumination source.

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For the optical system depicted in Fig. 1, suppose that each sub-source’s phase is randomly modulated, which obeys to a uniform distribution in [0, 2π], then the distribution of the light field at the Fourier plane can be represented as

E (x, y) = A_{0} \sum_{m = 0}^{N - 1} \sum_{m = 0}^{N - 1} c i r c (\frac{\sqrt{{(ξ - m d)}^{2} + {(η - n d)}^{^{2}}}}{a}) \exp (j ϕ_{m, n}) \exp (\frac{2 j π}{λ f} (x ξ + y η))

where circ(ξ, η) denotes the round hole function, A₀ is the amplitude of light field, and N is the total number of sub-sources in ξ or η directions. In addition, a is the sub-source’s radius and d is the center-to-center separation between two adjacent sub-sources, ϕ_m,n is the phase distribution, and (m, n) is the ordinal of each sub-source.

According to Eq. (2), we can obtain

〈 I (x, y) 〉 = 〈 E^{*} (x, y) E (x, y) 〉 = N^{2} A_{0}^{2}

and

\begin{array}{l} 〈 I (x_{0}, y_{0}) I (x, y) 〉 = 〈 I (x_{0}, y_{0}) 〉 〈 I (x, y) 〉 + {| 〈 E^{*} (x_{0}, y_{0}) E (x, y) 〉 |}^{2} \\ = N^{4} A_{0}^{4} + A_{0}^{4} {\frac{\sin [\frac{N π (x - x_{0})}{L_{p}}] \sin [\frac{N π (y - y_{0})}{L_{p}}]}{\sin [\frac{π (x - x_{0})}{L_{p}}] \sin [\frac{π (y - y_{0})}{L_{p}}]}}^{2} \end{array}

where L_p = λf /d is the period of the light field at the detecting plane.

Substituting Eqs. (3) and (4) into Eq. (1), we have

g^{(2)} (x, y; x_{0}, y_{0}) = 1 + \frac{1}{N^{4}} {\frac{\sin [\frac{N π (x - x_{0})}{L_{p}}] \sin [\frac{N π (y - y_{0})}{L_{p}}]}{\sin [\frac{π (x - x_{0})}{L_{p}}] \sin [\frac{π (y - y_{0})}{L_{p}}]}}^{2}

from Eq. (5), it is clearly seen that the distribution of g⁽²⁾(x, y; x₀ = 0, y₀ = 0) is also periodic and its periodicity is L_p if the illumination source’s spatial configuration is periodic, which will directly influence the quality of GI [15].

In order to overcome the periodicity of g⁽²⁾(x, y; x₀ = 0, y₀ = 0), we should design the illumination source’s spatial configuration, which makes the distribution of g⁽²⁾(x, y; x₀ = 0, y₀ = 0) have only one peak as far as possible. Genetic algorithm (GA), which has an obvious advantage for the problem of combinatorial optimization [18], is utilized to optimize the spatial configuration of the illumination source. The fitness function for the optimization process is designed as

f i t n e s s (P o s) = k_{1} M_{s i d e l o b e} + k_{2} N_{s i d e l o b e}

where M_sidelobe is the peak value of the highest sidelobe and N_sidelobe is the quantity of sidelobes above the set peak value for the distribution of g⁽²⁾(x, y; x₀ = 0, y₀ = 0). k₁ and k₂ are the coefficients to adjust the weight, respectively. The lower the value of fitness(Pos) is, the better the optimization result is.

In the optimization process, the sparse structured illumination source should satisfy the following conditions: the radius of each sub-source is a and the center-to-center separation between two adjacent sub-sources is not less than d. In addition, the whole source is restricted in the area of (a + (S-1)d, a + (S-1)d) (where S is the total number of sub-sources in x/y direction for a full-array illumination source displayed in Fig. 2(a₁)), the whole number of sub-sources is M (M≤S²), and there are at least two sub-sources whose center-to-center distance is $\sqrt{2} S d$ . Therefore, the sparse structured illumination source can be expressed as a two-dimensional complex matrix in the optimization process of genetic algorithm

P o s = P o s_{1} + (O f f s e t X + j \cdot O f f s e t Y) \cdot L

where Pos is the optimized spatial configuration of the sparse structured illumination source, Pos₁ is the gene locus matrix, OffsetX and OffsetY are the offset matrices in x and y directions, respectively. In addition, L is a parameter related to the gene locus matrix Pos₁.

Fig. 2 Simulated verification of the statistical property of g⁽²⁾ (x, y; x₀ = 0, y₀ = 0) distributions in different spatial configurations. From top to bottom, the spatial configuration of spare-array sources, the corresponding distributions of g⁽²⁾ (x, y; x₀ = 0, y₀ = 0) for the source’s configurations (a₁)-(a₃), the cross-section of g⁽²⁾(x, y; x₀ = 0, y₀ = 0) along the (x, x₀ = 0) direction and the (y, y₀ = 0) direction, corresponding to (b₁)-(b₃).

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The sparse structured illumination source is obtained by decreasing the sub-source’s number of a full-array illumination source and optimizing the source’s spatial configuration. For a 7 × 7 full-array illumination source shown in Fig. 2(a₁), then S = 7. If M = 25 (namely the whole number of sub-sources is decreased from 49 to 25 for sparse structured illumination source) and L = 2d, then the gene locus matrix Pos₁ is a 5 × 5 matrix and should be expressed as follows:

P o s_{1} = [\begin{matrix} 0 & d & ... & 4 d \\ j \cdot d & d + j \cdot d & ... & 4 d + j \cdot d \\ ... & ... & ... & ... \\ j \cdot 4 d & d + j \cdot 4 d & ... & 4 d + j \cdot 4 d \end{matrix}]

For the offset matrix OffsetX, we randomly generate 5 values between 0 and 1 for each row, and then sort them as the value from minimum to maximum. However, because the center-to-center separation between two adjacent sub-sources is not less than d, the value of each element in the offset matrix OffsetY will be restricted in a certain range when the offset matrix OffsetX is fixed, which means that the spatial configuration in y direction cannot be fully optimized for the sparse structured illumination source and the statistical property of pseudo-thermal light source in y direction may be worse than that in x direction.

3. Simulated and experimental results

According to the model described by Eqs. (6)-(8), the spatial configuration of sparse structured illumination source optimized by GA method is illustrated in Fig. 2(a₃). For the schematic depicted in Fig. 1, we set a = 87.5 μm, d = 1.0 mm, λ = 660 nm and f = 150 mm. Figure 2(a₁) and Fig. 2(a₂) give the diagrams of 7 × 7 full-array illumination source and 5 × 5 spare-array illumination source where 24 sub-sources are dislodged from the spatial configuration of Fig. 2(a₁). The distributions of g⁽²⁾(x, y; x₀ = 0, y₀ = 0) for the configurations of Figs. 2(a₁)-2(a₃) are shown in Figs. 2(b₁)-2(b₃), and their corresponding cross-section of g⁽²⁾ (x, y; x₀ = 0, y₀ = 0) in both (x, x₀ = 0) and (y, y₀ = 0) directions are displayed in Figs. 2(c₁)-2(c₃) and Figs. 2(d₁)-2(d₃), respectively. It is obviously observed that the distributions of g⁽²⁾ (x, y; x₀ = 0, y₀ = 0) for the configurations of both Fig. 2(a₁) and Fig. 2(a₂) will appear a periodic spatial structure with the period of L_p in both x and y directions, and the intensity of main peak located in the position (x = 0, y = 0) is the same as that of the sidelobes, which accords with the theoretical results described by Eq. (5). However, the peak-to-sidelobe ratio (PSR) can be dramatically improved for the optimized configuration of Fig. 2(a₃). Also, we can find that PSR in the (x, x₀ = 0) direction is larger than that in the (y, y₀ = 0) direction for the optimized configuration, which is caused by the optimization strategy because the offset matrix in y direction is restricted in the optimization process.

In order to demonstrate the superiority of the optimized sparse structured illumination, Fig. 3 presents the experimental setup of performance test. The laser beam with a center-wavelength λ = 660 nm passes through a rotating ground glass disk and then goes through a diaphragm. Then the light field is collimated by the lens L₁ and illuminates the spare-array template, which is used to simulate the sparse structured source. The photons transmitted from the spare-array holes pass through the lens L₂ and then are divided by a beam splitter (BS) into a reference and an object paths. The lens L₂ is used to transform the light field located at the spare-array template plane to the far field so that all the photons transmitted from the spare-array template can be adequately interfered. In the reference path, the interfered light field at the focal plane of the lens L₂ is imaged onto a CCD camera by a lens L₅. In the object path, the interfered light field is amplified by an imaging lens L₃ and then illuminates the object. The photons transmitted from the object are collected by an imaging lens L₄ onto a bucket detector. We emphasize that the diaphragm behind the ground glass disk is used to control the speckle’s transverse size illuminating the spare-array template, which should satisfy a certain condition in relation with the center-to-center separation between two holes. The parameters of the simulated and experimental setup are set as follow: the transmission aperture of the diaphragm is D = 1.0 mm. The focal length of the lenses are f₁ = 250 mm, f₂ = 150 mm, f₃ = f₄ = 60 mm, and f₅ = 80 mm. Each hole on the spare-array template has a diameter of a = 87.5μm and the minimum center-to-center separation between two holes is d = 1.0 mm. In addition, the distances shown in Fig. 3 are l₁ = l₃ = 80 mm, l₂ = l₄ = 240 mm, l₅ = 106.7 mm, and l₆ = 320 mm, respectively. In particular, the speckle’s transverse size δx = λf₁ / D at the spare-array template plane should obey the condition of a ≤ δx ≤ d, which means that each hole can hold at most one speckle. In this way, each hole on the spare-array template is equivalent to an independent sub-source and its phase variation is realized by the rotating ground glass disk. We emphasize that although our idea is demonstrated by a ghost imaging scheme with a rotating ground glass disk in the present experiment, the sparse structured illumination source can be obtained by some fiber lasers connected with some electro-optic modulators (which is used to produce phase variation) in practical applications. On the one hand, the fiber laser has a repetition rate of up to MHz and the modulating speed of electro-optic modulator can also reach to MHz. On the other hand, the phase variation is known by controlling the electro-optic modulator. Therefore, the reference path in Fig. 3 can be removed (like computational ghost imaging) and the sampling speed can be the same as the modulating speed of electro-optic modulator (up to MHz).

Fig. 3 Experimental setup of GI via sparse structured illumination source.

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Based on the setup displayed in Fig. 3, the g⁽²⁾(x, y; x₀ = 0, y₀ = 0) distributions of the same three spare structured illumination sources listed in Figs. 2(a₁)-2(a₃) are depicted in Fig. 4. It is clearly seen that the experimental results are consistent with the simulated results shown in Fig. 2, which suggests that a high-quality pseudo-thermal light source is experimentally achieved by optimizing the configuration structure of spare-array source.

Fig. 4 Experimental verification of the statistical property of g⁽²⁾ (x, y; x₀ = 0, y₀ = 0) distributions in different spatial configurations, corresponding to Fig. 2.

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In order to further verify the validity of the optimized pseudo-thermal light source, GI experiments are performed. According to the size of L_p, as shown in Figs. 5(a₁)-5(a₃), three objects are used, whose sizes occupy the areas of L_p × L_p, 2L_p × 2L_p and 3L_p × 3L_p, respectively. The image recorded by the reference CCD camera is 168 × 168 pixels (one pixel is 5.5μm × 5.5μm) and a gradient projection for sparse reconstruction algorithm is utilized to the image reconstruction of GI [19]. Based on the sparse structured illumination sources displayed in Figs. 4(a₁)-4(a₃), Figs. 5 and 6 present the corresponding simulated and experimental demonstration results of GI. It is obviously observed that the periodicity of g⁽²⁾ (x, y; x₀ = 0, y₀ = 0) distributions will cause a periodic image (see Figs. 4-Fig. 6). When the size of the object is smaller than L_p × L_p, as illustrated in Figs. 5(b₁)-5(c₁) and Figs. 6(b₁)-6(c₁), the image of the object appears periodically but is not overlapped for the structured illumination sources of both Figs. 4(a₁) and 4(a₂). If the size of the object is larger than L_p × L_p, the reconstruction results for the sources of both Figs. 4(a₁) and 4(a₂) are overlapped so that the object’s original structure cannot be recognized (Figs. 5(b₂)-5(c₂), Figs. 5(b₃)-5(c₃), Figs. 6(b₂)-6(c₂) and Figs. 6(b₃)-6(c₃)). In addition, from Figs. 5(b₁)-5(b₃) and Figs. 5(c₁)-5(c₃), the reconstruction quality based on the structured illumination source of Fig. 4(a1) is a litter better than that of Fig. 4(a₂), which also accords with the theoretical results because the quantity of sub-sources in Fig. 4(a₁) is larger than that in Fig. 4(a₂) [17]. However, for the optimized structured illumination source, even if the object’s size is greater than 3L_p × 3L_p, as shown in Figs. 5(d₁)-5(d₃) and Figs. 6(d₁)-6(d₃), no periodic images occur and the object’s image can be still stably reconstructed, which means that we have achieved GI with a better quality by using fewer sub-sources compared with the structured illumination source of Fig. 2(a₁). Otherwise, from the results depicted in Figs. 6(d₁)-6(d₃), there are still some weak pseudo images in the vertical direction, which is related to the distribution of g⁽²⁾(x, y; x₀ = 0, y₀ = 0) in (y, y₀ = 0) direction (see Fig. 2(d₃) and Fig. 4(d₃)) because the PSR in (y, y₀ = 0) direction is smaller than that in (x, x₀ = 0) direction for the optimized structured illumination source. Therefore, further optimization of the source’s spatial configuration will be our next work.

Fig. 5 Simulated demonstration of GI via sparse structured illumination source (the measurements K = 10000). From left to right, the original objects, the reconstruction results based on the sparse structured illumination sources displayed in Fig. 4 (a₁), Fig. 4 (a₂) and Fig. 4(a₃). The objects shown in (a₁)-(a₃) occupy the areas of L_p × L_p, 2L_p × 2L_p, and 3L_p × 3L_p, respectively.

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Fig. 6 Experimental demonstration of GI via sparse structured illumination source (the measurements K = 10000), corresponding to Fig. 5.

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4. Conclusion

In conclusion, we have obtained a high-quality pseudo-thermal light field via sparse structured illumination source where the source’s spatial configuration is optimized by genetic algorithm. Compared with the full-array illumination source, we have also demonstrated experimentally that the periodicity of the light field can be efficiently overcome and ghost imaging with a better quality can be obtained even if fewer sub-sources are used. When each sub-source is fast modulated by electro-optic modulators and lots of sub-sources are utilized, a ghost imaging lidar system with both high emitting energy and high-speed sampling can be achieved, which has a potential application in remote sensing towards moving target.

Funding

This work was supported by the National Natural Science Foundation of China (61571427) and Youth Innovation Promotion Association of the Chinese Academy of Sciences.

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Ghost imaging via sparse structured illumination source

Abstract

1. Introduction

2. Theory and Model

3. Simulated and experimental results

4. Conclusion

Funding

References and links

Cited By

Figures (6)

Equations (8)

Optics Express