Adaptive optical ghost imaging through atmospheric turbulence

Dongfeng Shi; Chengyu Fan; Pengfei Zhang; Jinghui Zhang; Hong Shen; Chunhong Qiao; Yingjian Wang

doi:10.1364/OE.20.027992

1. Introduction

Ghost imaging [1–5] is a new kind of imaging technique that obtains the image of an object by spatial intensity correlation measurements. In the imaging setup, two correlated laser beams are employed: one beam illuminates the object and then is measured by a single pixel bucket detector, while the other beam travels some meters to a CCD detector. The image of the object can be obtained by computing spatial correlation measurements of intensity fluctuations from the two detectors. Compared with the classical area of imaging, detection and imaging are separated in ghost imaging. This feature leads ghost imaging to practical application in particularly challenging environments, such as atmospheric turbulence and scattering media. Recently, some experiments and theoretical analysis [6–15] have demonstrated that ghost imaging has a great advantage over the conventional imaging methods for imaging in atmospheric turbulence and scattering media. Nevertheless, from the theoretical studies [9–11], we know that the quality of the ghost imaging will also be degraded due to the wavefront aberrations induced by atmospheric turbulence and scattering media. At present, there are several techniques (such as, compressed sensing ghost imaging [16], two-wavelength ghost imaging [17] and high-order ghost imaging [18]) that can improve the resolution of ghost imaging.

In this paper, we consider only the problem of ghost imaging through atmospheric turbulence. Notwithstanding, if atmospheric turbulence is mitigated, ghost imaging using the methods referred above will obtain higher resolution image. A number of techniques [19–21] have been developed to compensate the effect of atmospheric turbulence. Adaptive optics (AO), which includes the adaptive mirror (AM) system, the wavefront sensing (WFS) system and a control system, refers to optical systems which adapt to compensate optical effects introduced by atmospheric turbulence between the object and its image. AO system works by determining the shape of the phase perturbation, and using AM to compensate the phase perturbation. At the present time, adaptive optics has been widely used in astronomical telescopes and laser communication systems for removing the effects of atmospheric turbulence. To date, ghost imaging via adaptive optics has not been studied. In this study, we first present adaptive optical ghost imaging (AOGI), which dramatically enhances the resolution of image for ghost imaging through atmospheric turbulence. The remainder of the paper is organized as follows. In section 2, AOGI is presented. Here, we describe our models of AOGI and then analyze theoretically the resolution of AOGI. The numerical simulation results are given in Section 3. Conclusion and a brief mention of one further development follow in section 4.

2. Adaptive optical ghost imaging

The schematic of AOGI system is shown in Fig. 1 . A thermal light is split into two beams by the beam splitter (BS). One beam travels unobstructed to the CCD Detector which records the light intensity speckle. For conventional ghost imaging, the other beam transmits through atmospheric turbulence, illuminates the object, and then is detected by the Single Pixel Bucket Detector. However, for AOGI, this beam is modulated by AM of AO before transmitting to the object. Guide star light field is used to probe the shape of the phase perturbation caused by atmospheric turbulence. This may be a natural star or an artificial star. Light from the guide star is analyzed by a WFS, and commands from control system are sent to actuators which change the surface of AM to match the phase perturbation measured by the WFS. From the references [9,11,12], atmospheric turbulence in the Object-to-Single-Pixel-Bucket-Detector path doesn’t impact on the quality of the image. Thus AO system only exists in the Light-Source-to-Object path in this study.

Fig. 1 The schematic of AOGI system. A thermal light is split into two beams by the beam splitter (BS). Z₀, Z₁ and Z₂ are the distances from the Light Source to the Object, from the Light Source to the CCD Detector and from the Object to the Single Pixel Bucket Detector, respectively. The u, y, x_r, x_t are the coordinators at the Light Source plane, object plane, CCD Detector plane and Single Pixel Bucket Detector plane, respectively. DM, WFS, AM, GS are the Dichroic Mirror, the Wave-front Sensing, the adaptive mirror and the Guide Star, respectively.

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Based on the ghost imaging theory [1–5], the object image can be obtained by computing spatial correlation measurements of intensity fluctuations from the two detectors:

\begin{array}{l} G (x_{t}, x_{r}) = 〈 I_{t} (x_{t}) I_{r} (x_{r}) 〉 - 〈 I_{t} (x_{t}) 〉 〈 I_{r} (x_{r}) 〉 \\ = 〈 E_{t}^{*} (x_{t}) E_{t} (x_{t}) E_{r}^{*} (x_{r}) E_{r} (x_{r}) 〉 - 〈 E_{t}^{*} (x_{t}) E_{t} (x_{t}) 〉 〈 E_{r}^{*} (x_{r}) E_{r} (x_{r}) 〉, \end{array}

where E_t, E_r are the light fields at the single pixel bucket plane and CCD plane, respectively. In the presence of atmospheric turbulence and AO system, the light field E_t at the single pixel bucket detector plane x_t in the quasi-monochromatic paraxial approximation is introduced by

\begin{array}{l} E_{t} (x_{t}) = \frac{- \exp (\frac{2 j π (z_{0} + z_{2})}{λ})}{λ_{}^{2} z_{0} z_{2}} \int d y d u E (u) \exp [\frac{j π}{λ z_{0}} {(y - u)}^{2}] \exp [ψ_{0} (y, u) - i ϕ_{A M} (u)] \\ \times t (y) \exp [\frac{j π}{λ z_{2}} {(x_{t} - y)}^{2}] \exp [ψ_{2} (x_{t}, y)], \end{array}

where E(u) is the light source field, t(y) is the object function, ϕ_AM is the phase distribution introduced by AM, ψ₀, ψ₂ are the complex phase perturbation caused by atmospheric turbulence in the Light-Source-to-Object and Object-to-Single-Pixel-Bucket-Detector path, respectively. Here, ψ = η + iϕ, where η and ϕ represent the turbulence-included log-amplitude and phase fluctuations, respectively. In this paper, weak amplitude fluctuation is considered and the term η is ignored. We assume that the single pixel bucket detector is large enough to be able to receive all the light field information and the phase perturbations ϕ₀ and ϕ₂ are statistically independent. Similarly, the light field E_r in the plane x_r can be expressed as

E_{r} (x_{r}) = \frac{- j \exp (\frac{2 j π z_{1}}{λ})}{λ z_{1}} \int d u E (u) \exp [\frac{j π}{λ z_{1}} {(x_{r} - u)}^{2}] .

Suppose the light source is fully spatially incoherent, and then

〈 E (u) E^{*} (u^{'}) 〉 = I_{0} \exp (- u^{2} / ρ_{s}^{2}) δ (u - u^{'}),

where I₀ is a constant, δ(u−u^’) is the Dirac delta function and ρ_s is the size of the light source.

The statistical average of the phase perturbation arising from atmospheric turbulence can be described approximately by [22]

〈 \exp [i ϕ_{i} (x, y) - i ϕ_{i}^{} (x^{'}, y^{'})] 〉 = \exp {\frac{{(x - x^{'})}^{2} + (x - x^{'}) (y - y^{'}) + {(y - y^{'})}^{2}}{- ρ_{i}^{2}}},

where

ρ_{i} = {(0.55 {(2 π / λ)}^{2} {\bar{C}}_{n}^{2 (i)} z_{i})}^{- 3 / 5}

is the atmospheric turbulence coherence length. Here,

{\bar{C}}_{n}^{2 (i)} = \int_{0}^{H} C_{n}^{2 (i)} (h) d h / H,

where

C_{n}^{2 (i)} (h)

is the refractive index structure parameter describing the strength of atmospheric turbulence at the altitude h from the ground. The standard quadratic approximation to the 5/3-power law is employed in Eq. (5) to simplify the analysis, and this approximation has been widely used for laser beam propagation through atmospheric turbulence [9–13,22].

Substituting Eqs. (2)–(5) into Eq. (1) and using the complex zero-average Gaussian random process character, we have

\begin{array}{l} G (x_{t}, x_{r}) = \frac{I_{0}^{2}}{λ^{6} z_{0}^{2} z_{1}^{2} z_{2}^{2}} \int d u_{1} d u_{2} d y d y^{'} t (y) t^{*} (y^{'}) \exp (\frac{- u_{1}^{2} - u_{2}^{2}}{ρ_{s}^{2}}) 〈 \exp [i ϕ_{2} (y, x_{t}) - i ϕ_{2}^{} (y^{'}, x_{t})] 〉 \\ \times 〈 \exp [i ϕ_{0} (u_{1}, y) - i ϕ_{0}^{} (u_{2}, y^{'}) - i ϕ_{A M} (u_{1}) + i ϕ_{A M}^{} (u_{2})] 〉 \exp {\frac{j π}{λ_{2} z_{1}} [{(x_{r} - u_{2})}^{2} - {(x_{r} - u_{1})}^{2}]} \\ \times \exp {\frac{j π}{λ_{1} z_{0}} [{(y - u_{1})}^{2} - {(y^{'} - u_{2})}^{2}]} \exp {\frac{j π}{λ_{1} z_{2}} [{(x_{t} - y)}^{2} - {(x_{t} - y^{'})}^{2}]} . \end{array}

Two special limiting cases can be easily derived from Eq. (6): (I) the AO system doesn’t work in ghost imaging through atmospheric turbulence; (II) the phase perturbation from atmospheric turbulence in the path z₀ is purely compensated by AO system. For situation I (ϕ_AM = 0), integrating over u₁, u₂, x_t, ghost image is proportional to

G (x_{r}) = {| t (y) |}^{2} \otimes \exp (- y^{2} / R_{1 Z}^{2}),

in which the constant term has been neglected. In order to get an image of the object, the condition z₀ = z₁ = z is used. In Eq. (7),

\otimes

means convolution. R_1Z defining the resolution of ghost image can be expressed as

R_{1 Z} = \frac{λ z}{\sqrt{2} π ρ_{s}^{}} \sqrt{1 + \frac{2 ρ_{s}^{2}}{ρ_{0}^{2}}} .

From Eq. (8), we can conclude that the resolution of ghost imaging is determined by the size of light source and the atmospheric turbulence coherence length in path z₀, and atmospheric turbulence in the path z₂ doesn’t affect the resolution.

For situation II, the image of AOGI has the similar expression. However, the resolution parameter R_2Z has a small value described as

R_{2 Z} = \frac{λ z}{\sqrt{2} π ρ_{s}^{}},

which is the corresponding result for ghost imaging in the absence of atmospheric turbulence although atmospheric turbulence exists in the Object-to-Single-Pixel-Bucket-Detector path.

For a realistic situation, the phase perturbation caused by atmospheric turbulence can’t be purely compensated by AO system. Thereby, the resolution of realistic AOGI is higher than that of situation I, but is lower than or equal to that of situation II. When the accurate phase ϕ₀ is obtained by the WFS in geometric optics approach, the error ε = |ϕ₀−ϕ_ΑΜ| connects with the number of adaptive mirror elements. In general, as the number of AM elements decreases, the value of the error increases. Thus the quality of the obtained image gets worse. In the next part, the simulation results accordant with the conclusion are given.

3. Simulation and results

In the simulation, the light source (wavelength 632nm) is described by a grid of $256 \times 256$ with a sample spacing 2mm. We use the HV21 model [23] to describe the refractive index structure parameter $C_{n}^{2 (i)}$ and an uplink transmission model of laser beam in atmospheric turbulence is applied. The distances are set as z₀ = z₁ = z₂ = 2km. In the first experiment, a simple double slit (slit width 2cm, slit height 8cm and separation 2cm) is employed. The size ρ_s of the light source is equal to 10cm. The ideal AO system (phase conjugation mirror (PCM) is used and time delay is ignored) is used. After statistics over 10000 samples, the images of the double slit are shown in Fig. 2 . The result shows that the ghost image is degraded by atmospheric turbulence. Comparing Figs. 2(b) with 2(c), we can see that the effect can be mitigated using AO system in ghost imaging. This confirms the analytical result in above part. To verify the computer simulation, the theory values are given in Figs. 2(d) and 2(e). The good coincidence between the simulated data (red line) and the theoretical result predicted (blue circles line) by Eqs. (8), (9) proves the validity of the simulation.

Fig. 2 The obtained images of the double slit in atmospheric turbulence. (a) corresponds to the situation II; (b) corresponds to the situation I; (c) corresponds to the realistic AOGI. The normalized horizontal sections of the images are plotted in (d,e) where red lines show the simulated data and blue lines correspond to the theoretical prediction from Eqs. (9) and (8). The blue line, green line and red line in (F) represent the normalized horizontal sections of the images (a,b,c), respectively.
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In the next simulation, we test the performance of AOGI under different strength turbulence through multiplying HV21 model by different coefficient 2.0, 3.2 and 6.8, respectively. The ideal AO system is also considered in this simulation. The letter ‘A’ is chosen as the object and other parameters are fixed the same as the above experiment. The images from convention ghost imaging through turbulence are also shown in Fig. 3 . From the results in Figs. 3(a), 3(b) and 3(c), we can see that the image quality is worse when atmospheric turbulence is stronger. This is consistent with the theoretical analysis. However, using AO system, the image quality is improved. Although convention ghost image doesn’t get the object image, AOGI can also recover the image. It is shown that AOGI is a valuable method for ghost imaging through atmospheric turbulence.

Fig. 3 The obtained images of letter ‘A’ under different strength atmospheric turbulence. (a,b,c) are the images of convention ghost imaging under different strength turbulence through multiplying HV21 model by different coefficient 2.0, 3.2 and 6.8, respectively. (d,e,f) are the images of AOGI under different strength turbulence through multiplying HV21 model by different coefficient 2.0, 3.2 and 6.8, respectively. Statistical over 10000 samples.
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At last, the influence of adaptive optics system with different numbers of adaptive mirror elements on obtained image quality will be investigated. The 127 elements AM and 61 elements AM are chosen in simulations. The results are shown in Fig. 4 . As the number of AM elements decreases, the resolution as well as the visibility of the images [Fig. 4(c), 4(d)] degrades. Nevertheless, the AOGI can produce a recognizable image. The atmospheric turbulence changes light intensity distribution in object plane. Thus intensity fluctuations from the single pixel detector also deviate, and this results in failing to get the image of the object for convention ghost imaging. For the sake of comparison, we also present the obtained images [Figs. 4(a) and 4(b)] under the situations in which atmospheric turbulence is purely compensated and AO systems of PCM is employed in ghost imaging. The highest resolution image can be obtained correlated with the intensity fluctuations which represents situation II and chosen as reference standard. Root mean square error (RMSE) is used to express the deviation between the other fluctuations and reference standard fluctuation. RMSE can be described as: $R M S E = \sqrt{〈 {(I_{t}^{'} (n) - I_{r t}^{'} (n))}^{2} 〉},$ where $I^{'} (n) = I (n) / 〈 I (n) 〉 - 1$ is the normalized intensity fluctuations from the Single Pixel Bucket Detector and I^’_rt(n) is the reference standard normalized intensity fluctuation. The values of RMSE corresponding to AO systems of PCM, 127 elements AM and 61 elements AM are equal to 0.048, 0.070, 0.077, 0.094, respectively. The smaller the RMSE value, the better the ghost image.

Fig. 4 The obtained images of letter ‘H’ for ghost imaging via different AO systems under the strength turbulence through multiplying HV21 model by coefficient 3.2. (a) is the images of ghost imaging for the situation in which atmospheric turbulence is purely compensated. (b, c, d) are the images of ghost imaging with AO systems of PCM, 127 elements AM and 61 elements AM, respectively. Statistical over 10000 samples.
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4. Conclusion

In summary, AOGI system is first presented. Meanwhile, the theoretical analysis and numerical simulations have demonstrated that AOGI can provide imaging performance superior to that of conventional ghost imaging through atmospheric turbulence. The influence of the number of the elements of the AO system’s AM on the quality of ghost image is investigated. As the number of AM elements decreases, the resolution of the images degrades. AOGI can be effectively combined with several techniques (e.g., compressed sensing, correlated imaging) to obtain further higher resolution and contrast image in atmospheric turbulence. Recently, Meyers and Deacon [6–8] took a significant step for ghost imaging to the practical applications in atmospheric turbulence. AOGI opens up significant pathway for improving the performance of ghost imaging through atmospheric turbulence. In the future study, AOGI combined with the techniques (e.g., compressed sensing) to further improve the resolution will be analyzed. Furthermore, the scintillation will be considered in further works.

Acknowledgments

The authors are indebted to the anonymous referees for their instructive comments and suggestions. This work is supported by Hefei Institutes of Physical Sciences, Chinese Academy of Sciences (Grant Nos. 073RC11123, Y03RC21121 and XJJ-11-S106).

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Adaptive optical ghost imaging through atmospheric turbulence

Abstract

1. Introduction

2. Adaptive optical ghost imaging

3. Simulation and results

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (4)

Equations (9)

Optics Express