Two-wavelength ghost imaging through atmospheric turbulence

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

doi:10.1364/OE.21.002050

1. Introduction

Ghost imaging [1–5] is an indirect imaging technique that gets the image of an unknown object by performing spatial correlation measurements of intensity fluctuation. In the lensless thermal light ghost imaging setup, a beam splitter is used to divide one beam into the signal and reference beams. The signal beam illuminates the object and then is measured by a single pixel bucket detector with no spatial resolution. The reference beam is measured by a spatially resolving CCD detector, without ever encountering the object. Although neither detector output alone suffices to produce the object image, the image can be recovered from spatial correlation measurements of intensity fluctuations from the two detectors. Ghost imaging becomes a novel method to mitigate atmospheric turbulence effect because its imaging resolution differs from that of classical imaging. Recently, some experiments [6–9] about ghost imaging through atmospheric turbulence have been reported and the results have shown that ghost imaging is a valuable method in the situation where the atmospheric turbulence is a serious problem, and a number of papers [10–16] have been published to investigate theoretically the effect of atmospheric turbulence imposing on ghost imaging of transmitted and reflected objects.

However, in all these studies cited above, the signal and reference beams used in ghost imaging have the same wavelength. In paper [17], the authors have proved that the two light beams in the object and reference arms can have different wavelengths for vacuum propagation. The paper [18] has reported an experiment result about two-wavelength ghost imaging using quantum light. To the best of our knowledge, the properties of two-wavelength ghost imaging through atmospheric turbulence have never been investigated so far. In this investigation, we demonstrate theoretically that two-wavelength ghost imaging through atmospheric turbulence can be carried out using the classical correlated light beams. The results show that the PSF and FOV will increase as the strength of atmospheric turbulence increases. The optimal condition in vacuum corresponding to the smallest PSF will be destroyed by atmospheric turbulence. The resolution of two-wavelength ghost imaging primarily depends on the wavelength of the light beam that illuminates the object, although more generally it also depends on the wavelength of the light beams in the reference arm. The FOV will be affected by the two wavelengths. Two-wavelength ghost imaging can produce higher resolution than that of the single-wavelength case.

More recently, a computational version of ghost imaging has been proposed [19] and demonstrated [20]. In this system, a reference wave field and high-resolution sensor CCD are not compulsory because the reference speckle wave field and the sensor output can be precomputed using the diffraction theory. But the computational reference beams used in referred papers [19,20] have the same wavelength as the signal beam. In this paper, we also give the performance of computational two-wavelength ghost imaging. According to the analysis, using shorter signal wavelength for computational case can obtain high resolution image except the strong turbulence. From two-wavelength ghost imaging and computational case for comparison, we can draw the resolution from computational case will not be worse than two-wavelength ghost imaging.

2. Theoretical analysis

For lensless thermal light ghost imaging, the light source created by a rotating ground glass is separated by a beam splitter into two correlated light beams. However, in two-wavelength ghost imaging, correlated light beams are not produced by this method. Because the phases imposed by the ground glass are different for the two wavelength beams. In paper [17], in order to get the correlated beams, the authors employed the spatial light modulating (SLM) system to impose the same amplitude modulation for the two wavelength beams. In this paper, the two wavelength beams are respectively imposed the phase modulation by two SLM systems for the sake of creating two correlated light beams. The schematic of two-wavelength lensless ghost imaging through atmospheric turbulence is shown in Fig. 1 . Two laser beams with wavelengths λ₁ and λ₂ respectively pass through two different SLM systems. The two laser beams at the output plane of the SLM systems have the same phase distribution. In this way, the two light fields are rendered spatially incoherent but in a correlated fashion and then are separated into the signal and reference beams by the dichroic mirror (DM). The signal beam with wavelength λ₁ illuminates an object after z₀ meters free space propagation through atmospheric turbulence and the reflected or transmitted light from the object travels z₂ meters through atmospheric turbulence to the single pixel bucket detector. The reference beam with wavelength λ₂ propagates z₁ meters through atmospheric turbulence and then is collected by the CCD detector. Finally, the object image can be reconstructed by computing spatial correlation fluctuations of the intensities obtained from the bucket detector and the CCD detector. In this conventional system, atmospheric turbulence exits in all three propagation paths.

Fig. 1 The schematic of two-wavelength lensless ghost imaging through atmospheric turbulence. DM represents the dichroic mirror. The u,x_r,x_t,y are the coordinators at the output plane of SLM systems, CCD detector plane, single pixel bucket detector plane and object plane, respectively.

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In the presence of atmospheric turbulence, based on the extended Huygens-Fresnel integral, the propagation function from the source plane u to the CCD detector plane x_r for the reference beam is given by

E_{r} (x_{r}) = \frac{- j \exp (\frac{2 j π z_{1}}{λ_{2}})}{λ_{2} z_{1}} \int d u E_{i r} (λ_{2}, u) \exp [\frac{j π}{λ_{2} z_{1}} {(x_{r} - u)}^{2}] \exp [ϕ_{1} (x_{r}, u)],

where the E_ir(λ₂,u) is realized by modulating the plane light field E_r’ (λ₂,u) using the SLM system 2, and ϕ₁(x_r,u) represents the random part of the complex phase due to atmospheric turbulence effects in the SLM-to-CCD-detector path. Similarly, the field at the bucket detector plane is

\begin{array}{l} E_{t} (x_{t}) = \frac{- \exp (\frac{2 j π (z_{0} + z_{2})}{λ_{1}})}{λ_{1}^{2} z_{0} z_{2}} \int d y d u E_{i t} (λ_{1}, u) \exp [\frac{j π}{λ_{1} z_{0}} {(y - u)}^{2}] \exp [ϕ_{0} (y, u)] \\ \times t (y) \exp [\frac{j π}{λ_{1} z_{2}} {(x_{t} - y)}^{2}] \exp [ϕ_{2} (x_{t}, y)], \end{array}

where the E_it(λ₁,u) is obtained by modulating the plane light field E_t’ (λ₁,u) using the SLM system 1, t(y) denotes the amplitude reflectivity coefficient of the object and ϕ₀(y,u), ϕ₂(x_t,y) characterize the atmospheric turbulence effects in the SLM-to-object path and the object-to-bucket-detector path, respectively. We assume that the fluctuations introduced by atmospheric turbulence in the three paths are statistically independent and have the same strength. The two SLM systems enforce spatial phase modulation for the two light fields to possess the same phase described by ϕ(u) in E_it(λ₁,u), and E_ir(λ₂,u). Thus the light fields at the output plane of the SLM systems can be modeled as

E_{i t} (λ_{1}, u) = E_{t'} (λ_{1}, u) \exp [j ϕ (u)], E_{i r} (λ_{2}, u) = E_{r'} (λ_{2}, u) \exp [j ϕ (u)],

where E_t’ (λ₁,u) and E_r’ (λ₂,u) are the independent light fields at the input plane of the SLM systems with wavelength λ₁ and λ₂, and the random phase ϕ(u) following Gaussian statistics is taken to possess spatial correlation property. The object information can be extracted by computing the correlation of two detector intensity fluctuations

\begin{array}{l} G (x_{t}, x_{r}) = 〈 (I_{t} (x_{t}) - 〈 I_{t} (x_{t}) 〉) (I_{r} (x_{r}) - 〈 I_{r} (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}) 〉 \\ = \frac{1}{λ_{1}^{4} λ_{2}^{2} z_{0}^{2} z_{1}^{2} z_{2}^{2}} \int d u_{1} d u_{1}^{'} d u_{2} d u_{2}^{'} d y d y^{'} Γ (u_{1}, u_{1}^{'}, u_{2}, u_{2}^{'}) < t (y) t^{*} (y^{'}) > \\ \times 〈 \exp [ϕ_{1} (u_{2}, x_{r}) + ϕ_{1}^{*} (u_{2}^{'}, x_{r})] 〉 〈 \exp [ϕ_{0} (u_{1}, y) + ϕ_{0}^{*} (u_{1}^{'}, y^{'})] 〉 \\ \times 〈 \exp [ϕ_{2} (y, x_{t}) + ϕ_{2}^{*} (y^{'}, x_{t})] 〉 \exp {\frac{j π}{λ_{2} z_{1}} [{(x_{r} - u_{2})}^{2} - {(x_{r} - u_{2}^{'})}^{2}]} \\ \times \exp {\frac{j π}{λ_{1} z_{0}} [{(y - u_{1})}^{2} - {(y^{'} - u_{1}^{'})}^{2}]} \exp {\frac{j π}{λ_{1} z_{2}} [{(x_{t} - y)}^{2} - {(x_{t} - y^{'})}^{2}]}, \end{array}

where

Γ (u_{1}, u_{1}^{'}, u_{2}, u_{2}^{'}) = 〈 E_{i t} (u_{1}) E_{i t}^{*} (u_{1}^{'}) E_{i r}^{*} (u_{2}) E_{i r} (u_{2}^{'}) 〉 - 〈 E_{i t} (u_{1}) E_{i t}^{*} (u_{1}^{'}) 〉 〈 E_{i r}^{*} (u_{2}) E_{i r} (u_{2}^{'}) 〉 .

Because the two light fields at the input plane of the SLM systems are independent, with the help of Eq. (3), Eq. (5) becomes

\begin{array}{l} Γ (u_{1}, u_{1}^{'}, u_{2}, u_{2}^{'}) = 〈 E_{t'} (λ_{1}, u_{1}) E_{t'}^{*} (λ_{1}, u_{1}^{'}) 〉 〈 E_{r'} (λ_{2}, u_{2}) E_{r'}^{*} (λ_{2}, u_{2}^{'}) 〉 \\ \times (〈 \exp {j [ϕ (u_{1}) - ϕ (u_{1}^{'}) + ϕ (u_{2}) - ϕ (u_{2}^{'})]} 〉 \\ - 〈 \exp {j [ϕ (u_{1}) - ϕ (u_{1}^{'})]} 〉 〈 \exp {j [ϕ (u_{2}) - ϕ (u_{2}^{'})]} 〉) . \end{array}

According to the central limit theorem, a large number of random variables exp(jϕ(u)) approximate Gaussian distribution, Eq. (6) becomes

\begin{array}{l} Γ (u_{1}, u_{1}^{'}, u_{2}, u_{2}^{'}) = 〈 E_{t'} (λ_{1}, u_{1}) E_{t'}^{*} (λ_{1}, u_{1}^{'}) 〉 〈 E_{r'} (λ_{2}, u_{2}) E_{r'}^{*} (λ_{2}, u_{2}^{'}) 〉 \\ \times 〈 \exp {j [ϕ (u_{1}) - ϕ (u_{2}^{'})]} 〉 〈 \exp {j [ϕ (u_{2}) - ϕ (u_{1}^{'})]} 〉 . \end{array}

We further suppose that the two light fields have the same Gaussian distribution forms, thus two beams at the output plane of the SLM systems have completely correlated, zeros-mean, Gaussian random processes, and further Eq. (7) can be rewritten as

Γ (u_{1}, u_{1}^{'}, u_{2}, u_{2}^{'}) = \exp (- \frac{u_{1}^{2} + {u^{'}}_{1}^{2} + u_{2}^{2} + {u^{'}}_{2}^{2}}{4 ω^{2}}) \exp (- \frac{{(u_{1} - u_{2}^{'})}^{2} + {(u_{2} - u_{1}^{'})}^{2}}{2 l_{c}^{2}}),

where ω is the transverse size of the laser beams and l_c defines the correlation parameter of the random phase caused by the SLM systems.

The statistical average of the phase arising from atmospheric turbulence can be described approximately by [21]

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

where ρ_i=(1.09C2(i) n(2π/λ_i)²z_i)^−3/5 is the coherence length of a spherical wave propagating through a turbulence medium and C2(i) n is the refractive index structure parameter describing the strength of atmospheric turbulence along uniform horizontal-path z_i. The standard quadratic approximation to the 5/3-power law is employed in Eq. (9) to simplify the analysis, and this approximation has been widely used for laser beam propagation through atmospheric turbulence [10–16,21].

Substituting Eqs. (8), (9) into Eq. (4) and integrating over u₁, u' 1, u₂, u' 2, we have

\begin{array}{l} G (x_{t}, x_{r}) = \frac{π^{2}}{λ_{1}^{4} λ_{2}^{2} z_{0}^{2} z_{1}^{2} z_{2}^{2} \sqrt{A B C D}} \int d y d y^{'} < t (y) t^{*} (y^{'}) > \exp [\frac{{(y - y^{'})}^{2}}{- 2 ρ_{0}^{2}} + \frac{{(y - y^{'})}^{2}}{- 2 ρ_{2}^{2}}] \\ \times exp{\frac{j π}{λ_{1} z_{0}} (y^{2} - {y^{'}}^{2}) + \frac{j π}{λ_{1} z_{2}} [(x_{t} - y)^{2} - (^{x_{t} - y^{'}) 2}]} \exp (\frac{S^{2}}{4 A} + \frac{P^{2}}{4 B} + \frac{Q^{2}}{4 C} + \frac{V^{2}}{4 D}), \end{array}

where_{$\begin{array}{l} A = \frac{1}{4 ω^{2}} + \frac{1}{2 l_{c}^{2}} + \frac{1}{2 ρ_{0}^{2}} - \frac{j π}{λ_{1} z_{0}}, B = \frac{1}{4 ω^{2}} + \frac{1}{2 l_{c}^{2}} + \frac{1}{2 ρ_{0}^{2}} + \frac{j π}{λ_{1} z_{0}} - \frac{1}{4 A ρ_{0}^{4}}, \\ C = \frac{1}{4 ω^{2}} + \frac{1}{2 l_{c}^{2}} + \frac{1}{2 ρ_{1}^{2}} - \frac{j π}{λ_{2} z_{1}} - \frac{1}{4 B l_{c}^{4}}, \\ D = \frac{1}{4 ω^{2}} + \frac{1}{2 l_{c}^{2}} + \frac{1}{2 ρ_{1}^{2}} + \frac{j π}{λ_{2} z_{1}} - \frac{1}{4 A l_{c}^{4}} - \frac{1}{16 A^{2} B l_{c}^{4} ρ_{0}^{4}} - \frac{1}{4 C ρ_{1}^{4}} - \frac{1}{64 A^{2} B^{2} C l_{c}^{8} ρ_{0}^{4}} - \frac{1}{8 A B C l_{c}^{4} ρ_{0}^{2} ρ_{1}^{2}}, \\ S = \frac{y - y^{'}}{2 ρ_{0}^{2}} + \frac{2 j π y}{λ_{1} z_{0}}, P = \frac{y - y^{'}}{2 ρ_{0}^{2}} + \frac{2 j π y^{'}}{λ_{1} z_{0}} - \frac{y - y^{'}}{4 A ρ_{0}^{4}} - \frac{j π y}{A λ_{1} z_{0} ρ_{0}^{2}}, \\ Q = \frac{P}{2 B l_{c}^{2}} - \frac{2 j π x_{r}}{λ_{2} z_{1}}, V = \frac{2 j π x_{r}}{λ_{2} z_{1}} - \frac{y - y^{'}}{4 A l_{c}^{2} ρ_{0}^{2}} - \frac{j π y}{A λ_{1} z_{0} l_{c}^{2}} + \frac{P}{4 A B l_{c}^{2} ρ_{0}^{2}} + \frac{Q}{2 C ρ_{1}^{2}} + \frac{Q}{8 A B C l_{c}^{4} ρ_{0}^{2}} . \end{array}$}

Now, considering the ghost imaging (Fig. 1), one bucket detector and one spatially resolving detector are adopted in the imaging setup. The ghost image is given by

G (x_{r}) = \int G (x_{t}, x_{r}) d (x_{t}) d x_{t},

where d(x_t) is the bucket detector function; d(x_t)=1 if x_t is inside the single pixel bucket detector, while d(x_t)=0 if x_t is outside the single pixel bucket detector. We assume the area of bucket detector is s_b. In fact, most objects encountered in real world are rough on the scale of an optical wavelength. Base on the laser radar theory [12,13], the amplitude reflectivity coefficient from rough surface can be modeled as a zeros-mean Gaussian random field with

< t (y) t^{*} (y^{'}) > = λ_{1}^{2} T (y) δ (y - y^{'}),

where T(y) is the deterministic pattern that we would like to image and δ is the delta function. Therefore, Eq. (10) can be further simplified as

G (x_{r}) = \frac{s_{b} π^{2}}{λ_{1}^{2} λ_{2}^{2} z_{0}^{2} z_{1}^{2} z_{2}^{2} \sqrt{A B C D}} \int d y T (y) \exp (\frac{{S^{'}}^{2}}{4 A} + \frac{{P^{'}}^{2}}{4 B} + \frac{{Q^{'}}^{2}}{4 C} + \frac{{V^{'}}^{2}}{4 D}),

where_{$\begin{array}{l} S^{'} = \frac{2 j π y}{λ_{1} z_{0}}, P^{'} = \frac{2 j π y}{λ_{1} z_{0}} (1 - \frac{1}{2 A ρ_{0}^{2}}), Q^{'} = \frac{j π y}{B λ_{1} z_{0} l_{c}^{2}} (1 - \frac{1}{2 A ρ_{0}^{2}}) - \frac{2 j π x_{r}}{λ_{2} z_{1}}, \\ V^{'} = \frac{2 j π x_{r}}{λ_{2} z_{1}} - \frac{j π y}{A λ_{1} z_{0} l_{c}^{2}} + \frac{P^{'}}{4 A B l_{c}^{2} ρ_{0}^{2}} + \frac{Q^{'}}{2 C ρ_{1}^{2}} + \frac{Q^{'}}{8 A B C l_{c}^{4} ρ_{0}^{2}} . \end{array}$}At last, we give Eq. (13) as the form of Gaussian function to comprehend the effect of atmospheric turbulence on two-wavelength ghost imaging

G (x_{r}) = \frac{s_{b} π^{2}}{λ_{1}^{2} λ_{2}^{2} z_{0}^{2} z_{1}^{2} z_{2}^{2} \sqrt{A B C D}} \exp (- \frac{x_{r}^{2}}{2 W_{f o v}^{2}}) \int d y T (y) \exp (- \frac{{(y - m x_{r})}^{2}}{2 W_{p s f}^{2}}),

where_{$\begin{array}{l} W_{p s f} = \sqrt{\frac{1}{2 (K_{1}^{2} + K_{2}^{2} + K_{3}^{2} + K_{5}^{2})}}, m = \frac{K_{3} K_{4} + K_{5} K_{6}}{K_{1}^{2} + K_{2}^{2} + K_{3}^{2} + K_{5}^{2}}, \\ W_{f o v} = \sqrt{\frac{1}{2 [K_{4}^{2} + K_{6}^{2} - \frac{{(K_{3} K_{4} + K_{5} K_{6})}^{2}}{K_{1}^{2} + K_{2}^{2} + K_{3}^{2} + K_{5}^{2}}]}}, \\ K_{1} = \frac{π}{\sqrt{A} λ_{1} z_{0}}, K_{2} = \frac{π}{\sqrt{B} λ_{1} z_{0}} (1 - \frac{1}{2 A ρ_{0}^{2}}), K_{3} = \frac{π}{2 B \sqrt{C} λ_{1} z_{0} l_{c}^{2}} (1 - \frac{1}{2 A ρ_{0}^{2}}), K_{4} = \frac{π}{\sqrt{C} λ_{2} z_{1}}, \\ K_{5} = \frac{π}{2 \sqrt{D} λ_{1} z_{0}} [- \frac{1}{A l_{c}^{2}} + \frac{1}{B l_{c}^{2}} (1 - \frac{1}{2 A ρ_{0}^{2}}) (\frac{1}{2 A ρ_{0}^{2}} + \frac{1}{2 C ρ_{1}^{2}} + \frac{1}{8 A B C l_{c}^{4} ρ_{0}^{2}})], \\ K_{6} = \frac{π}{\sqrt{D} λ_{2} z_{1}} (\frac{1}{2 C ρ_{1}^{2}} + \frac{1}{8 A B C l_{c}^{4} ρ_{0}^{2}} - 1) . \end{array}$}Here W_psf is the width of the point spread function (PSF) that describes the resolution of two-wavelength ghost imaging system through atmospheric turbulence as measured in object space. Also, W_fov is the field of view (FOV) as measured in image space and m is the magnification factor which is always negative in ghost imaging system. Eq. (14) gives the performance of two-wavelength ghost imaging through atmospheric turbulence. From Eq. (14), we can see that atmospheric turbulence in the path z₂ doesn’t affect the PSF. However, atmospheric turbulence in the paths z₀, z₁ disturbs the resolution of two-wavelength ghost imaging. This conclusion is consistent with the results in papers [10–13] for the single-wavelength ghost imaging through atmospheric turbulence.

Next, we will consider computational two wavelength ghost imaging system. This system removes the reference arm, and the intensity distribution in the CCD I detector can be computed by the simulation program according to the diffraction theory [19]. Thus the imaging function of computational two wavelength ghost imaging system can be obtained by setting the C2(1) n=0. Similar to the above part, the computational ghost image is proportional to

G^{'} (x_{r}) = \frac{s_{b} π^{2}}{λ_{1}^{2} λ_{2}^{2} z_{0}^{2} z_{1}^{2} z_{2}^{2} \sqrt{A^{'} B^{'} C^{'} D^{'}}} \exp (- \frac{x_{r}^{2}}{2 W_{c f o v}^{2}}) \int d y T (y) \exp (- \frac{{(y - m^{'} x_{r})}^{2}}{2 W_{c p s f}^{2}}),

where_{$\begin{array}{l} W_{c p s f} = \sqrt{\frac{1}{2 ({K^{'}}_{1}^{2} + {K^{'}}_{2}^{2} + {K^{'}}_{3}^{2} + {K^{'}}_{5}^{2})}}, m^{'} = \frac{{K^{'}}_{3} {K^{'}}_{4} + {K^{'}}_{5} {K^{'}}_{6}}{{K^{'}}_{1}^{2} + {K^{'}}_{2}^{2} + {K^{'}}_{3}^{2} + {K^{'}}_{5}^{2}}, \\ W_{c f o v} = \sqrt{\frac{1}{2 [{K^{'}}_{4}^{2} + {K^{'}}_{6}^{2} - \frac{{({K^{'}}_{3} {K^{'}}_{4} + {K^{'}}_{5} {K^{'}}_{6})}^{2}}{{K^{'}}_{1}^{2} + {K^{'}}_{2}^{2} + {K^{'}}_{3}^{2} + {K^{'}}_{5}^{2}}]}}, \\ {K^{'}}_{1} = \frac{π}{\sqrt{A^{'}} λ_{1} z_{0}}, {K^{'}}_{2} = \frac{π}{\sqrt{B^{'}} λ_{1} z_{0}} (1 - \frac{1}{2 A^{'} ρ_{0}^{2}}), {K^{'}}_{3} = \frac{π}{2 B \sqrt{C^{'}} λ_{1} z_{0} l_{c}^{2}} (1 - \frac{1}{2 A^{'} ρ_{0}^{2}}), {K^{'}}_{4} = \frac{π}{\sqrt{C^{'}} λ_{2} z_{1}}, \\ {K^{'}}_{5} = \frac{π}{2 \sqrt{D^{'}} λ_{1} z_{0}} [- \frac{1}{A^{'} l_{c}^{2}} + \frac{1}{B^{'} l_{c}^{2}} (1 - \frac{1}{2 A^{'} ρ_{0}^{2}}) (\frac{1}{2 A^{'} ρ_{0}^{2}} + \frac{1}{8 A^{'} B^{'} C^{'} l_{c}^{4} ρ_{0}^{2}})], \\ {K^{'}}_{6} = \frac{π}{\sqrt{D^{'}} λ_{2} z_{1}} (\frac{1}{8 A^{'} B^{'} C^{'} l_{c}^{4} ρ_{0}^{2}} - 1), A^{'} = \frac{1}{4 ω^{2}} + \frac{1}{2 l_{c}^{2}} + \frac{1}{2 ρ_{0}^{2}} - \frac{j π}{λ_{1} z_{0}}, \\ B^{'} = \frac{1}{4 ω^{2}} + \frac{1}{2 l_{c}^{2}} + \frac{1}{2 ρ_{0}^{2}} + \frac{j π}{λ_{1} z_{0}} - \frac{1}{4 A^{'} ρ_{0}^{4}}, C^{'} = \frac{1}{4 ω^{2}} + \frac{1}{2 l_{c}^{2}} - \frac{j π}{λ_{2} z_{1}} - \frac{1}{4 B^{'} l_{c}^{4}}, \\ D^{'} = \frac{1}{4 ω^{2}} + \frac{1}{2 l_{c}^{2}} + \frac{j π}{λ_{2} z_{1}} - \frac{1}{4 A^{'} l_{c}^{4}} - \frac{1}{16 {A^{'}}^{2} B^{'} l_{c}^{4} ρ_{0}^{4}} - \frac{1}{64 {A^{'}}^{2} {B^{'}}^{2} C^{'} l_{c}^{8} ρ_{0}^{4}} . \end{array}$}

Equation (15) presents the performance of computational two-wavelength ghost imaging through atmospheric turbulence. W_cpsf and W_cfov are the PSF and FOV of computational two-wavelength ghost imaging system. Similar to two-wavelength ghost imaging system, atmospheric turbulence in the path z₂ doesn’t affect the PSF and atmospheric turbulence in the paths z₀ disturbs the PSF and FOV. Computational two-wavelength ghost imaging system offer better or nor worse spatial resolution than above imaging system. The simulation results will approve this point.

In papers [1,17], the authors give the optimal resolution condition λ₂z₁= λ₁z₀ for two-wavelength ghost imaging in the vacuum propagation, and for single-wavelength ghost imaging through atmospheric turbulence, the results in papers [10–13] are given under the condition z₁=z₀ due to the same wavelength value. From Eqs. (14), (15), the comprehensive description of the optimal resolution condition cannot be solved. In next simulation section, we will test the optimal resolution condition λ₂z₁= λ₁z₀ for (computational) two-wavelength ghost imaging through atmospheric turbulence.

3. Simulation and results

For a realistic situation, we set the distance z₀=1km. Different wavelengths turbulence coherence lengths (m) for 1km path length are shown in Table 1 for different strength of atmospheric turbulence. In this section, we will descript the performance of two-wavelength ghost imaging and computational case, respectively.

Table 1. Different wavelengths turbulence coherence lengths (m) for 1km path length and uniform C2 n distribution.

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3.1 Two-wavelength ghost imaging system

At the beginning, we fix the wavelength λ₁ and test the resolution of two-wavelength ghost imaging under the situation that the wavelength λ₂ is changing. The results are shown in Fig. 2 , in which black dashed line represents the resolution of the single-wavelength case. We can see that the optimal resolution is obtained when λ₂z₁= λ₁z₀ for weak turbulence. However, with the increase of atmospheric turbulence strength, this phenomenon gradually disappeared. For the shorter wavelength λ₂, the condition λ₂z₁= λ₁z₀ is more susceptible to be destroyed. In the case of strong turbulence, the longer wavelength for reference beam can get better resolution. Nonetheless, based on Fig. 2(E), we can find that the resolution improvement is limited for using longer wavelength as the reference beam, and another feature in Fig. 2(A,B) is that optimal values of resolution are approximately equal for all different wavelengths of reference beam under weak turbulence situation. The reason for this phenomenon is that the wavelength of the reference beam belongs to a secondary position in the factors affecting the resolution. In all cases, the curves have first decreased and then increased trend with increasing distance z₁. The smallest W_psf values become large when C2 n is increased, which means the image quality will be worse.

Fig. 2 The W_psf as a function of distance z₁ for fixed λ₁, z₀. Parameters used are λ₁=1.2μm, ω=5cm, l_c=1mm, z₀=1km. (A-E) are the curve lines for C2 n =10⁻¹⁶, 10⁻¹⁵, 10⁻¹⁴, 10⁻¹³, 10⁻¹²m^-2/3. (F) is linear-type corresponding to the two wavelength values.

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In next experiment, we fix the wavelength λ₂ and alter the wavelength λ₁. In this situation, the capability of two-wavelength ghost imaging is shown in Fig. 3 . The smallest value of the PSF corresponding to the optimal resolution for weak turbulence is obtained when λ₂z₁= λ₁z₀, as discussed above. This phenomenon gradually disappears with increasing turbulence strength, and the smallest W_psf values also turn large as the C2 n increased. However there are several different points compared with the above experiment. Firstly, the condition λ₂z₁= λ₁z₀ is more susceptible to be destroyed for the longer wavelength λ₁. Secondly, the optimal values of resolution are very different for various wavelengths λ₁ under weak turbulence situation, which is represented in Fig. 3(A,B). Thus in order to get higher resolution than that of the single-wavelength case, the shorter wavelength λ₁ must be utilized and the distance must content the condition z₁= λ₁z₀/ λ₂in weak atmospheric turbulence. Finally, high resolution image can be obtained for the shorter wavelength λ₁ in strong turbulence situation.

Fig. 3 The W_psf as a function of distance z₁ for fixed λ₂, z₀. Parameters used are λ₂=1.2μm, ω=5cm, l_c=1mm, z₀=1km. (A-E) are the curve lines for C2 n =10⁻¹⁶, 10⁻¹⁵, 10⁻¹⁴, 10⁻¹³, 10⁻¹²m^-2/3. (F) is linear-type corresponding to the two wavelength values.

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In summary, the resolution depends on both wavelengths, but it primarily depends on the wavelength λ₁ of the signal beam. When the signal beam is fixed, the reference beam with longer wavelength could be employed to get high resolution ghost image; but when the reference beam is fixed, the shorter wavelength signal beam should be used to obtain high resolution ghost image, and the key point is that the distance z₁ must content the condition λ₂z₁= λ₁z₀ for weak and medium strength of turbulence. By changing one or both of the wavelength, the higher resolution ghost image can always be achieved than that of the single-wavelength case.

In the third experiment, we will evaluate two-wavelength ghost imaging impact on the FOV. The parameters are set the same as in the above experiment I. From Fig. 4(A,B) , we can see that the FOV for different wavelengths of reference beam have positive relationship with the distance and wavelength under weak turbulence. Figure 4(C,D) show that upward inflection points appear in the distance which meets the condition λ₂z₁= λ₁z₀ for medium strength of atmospheric turbulence. As seen from Fig. 4(E), the lines show that the inflection points shift to the small distance value z₁ compared with Fig. 4(C,D).

Fig. 4 The W_fov as a function of distance z₁ for fixed λ₁, z₀. Parameters used are λ₁=1.2μm, ω=5cm, l_c=1mm, z₀=1km. (A-E) are the curve lines for C2 n =10⁻¹⁶, 10⁻¹⁵, 10⁻¹⁴, 10⁻¹³, 10⁻¹²m^-2/3. (F) is linear-type corresponding to the two wavelength values.

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In the last experiment, we will change the wavelength λ₁ of the signal beam and fix the wavelength of the reference beam as λ₂=1.2μm. According to Fig. 5(A,B) , we can conclude that the FOV values will not change as the wavelength of the signal beam change for weak turbulence. Similar to the above experiment, the upward inflection points will appear and shift to the small distance with the increase of turbulence strength. However, as shown in Fig. 5(E), when the transmission distance z₁ is up to a certain value, the FOV values are the same for all the wavelength λ₁. From Fig. 4 and Fig. 5, we can conclude that the FOV values will be affected by the two wavelengths. The W_fov values become large with C2 n increases.

Fig. 5 The W_fov as a function of distance z₁ for fixed λ₂, z₀. Parameters used are λ₂=1.2μm, ω=5cm, l_c=1mm, z₀=1km. (A-E) are the curve lines for C2 n =10⁻¹⁶, 10⁻¹⁵, 10⁻¹⁴, 10⁻¹³, 10⁻¹²m^-2/3. (F) is linear-type corresponding to the two wavelength values.

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We can explain the phenomenon based on the ratio between turbulence coherence lengths and transverse size of the laser beams. When both turbulence coherence lengths of the two-wavelengths are much larger than transverse size of the laser beams, i.e. ρ₀ and ρ₁ >>ω, the resolution and FOV are almost not affected by turbulence, and vice versa. If we compare turbulence coherence lengths from Table 1 with transverse size of the laser beams given as 5cm, the rule will also be concluded. The condition λ₂z₁= λ₁z₀ should be applied when both turbulence coherence lengths of the two-wavelengths greater than or approximately equal to transverse size of the laser beams, i.e. ρ₀ and ρ₁ >ω, or ρ₀ and ρ₁≈ω. The reason why shorter wavelength λ₁ should be sent to the target to optimize the spatial resolution for weak turbulence can also be obtained from the above analysis. Similar to the single-wavelength case, the spatial resolution is determined by the parameter λ₁z₀/ω for weak turbulence that both turbulence coherence lengths of the two-wavelengths are much larger than transverse size of the laser beams. When the distance z₀ is fixed and the condition λ₂z₁= λ₁z₀ is employed, it is immediately obvious that two-wavelength ghost imaging using shorter wavelength λ₁ can achieve higher resolution. The FOV is determined by the average intensity pattern. It is well known that the size of the average intensity pattern is influenced by the ratio ρ₀ and ρ₁ to ω. When the ratio is larger than 1, the size of the average intensity pattern is almost not affected by turbulence, and vice versa. This principle can be used to explain the performance of the FOV in Fig. 4 and Fig. 5. Based on these numerical results and theoretical formulas, the shorter wavelength beam should be sent to illumine the object and the reference beam with the longer wavelength should be used. To do so, two-wavelength ghost imaging system can obtain high resolution image. The W_psf and W_fov values are become large as the strength of atmospheric turbulence increases.

3.2 Computational Two-wavelength ghost imaging system

In computational two-wavelength ghost imaging system, turbulence coherence lengths for wavelength λ₂ does not exist and the below mentioned turbulence coherence lengths is about wavelength λ₁. We test the resolution of computational two-wavelength ghost imaging under the situation that the wavelength λ₂ is changing and the wavelength λ₁ is fixed. The results are shown in Fig. 6 . We can see that the minimum values W_cpsf are the same for the certain strength of atmospheric turbulence in different wavelength λ₂ situation and become large with the strength of atmospheric turbulence increases. For weak turbulence, the highest resolution is almost not change that shown in Fig. 6(A,B). As seen from Fig. 6(E), the condition λ₂z₁= λ₁z₀ is destroyed in strong turbulence. Next, we will test the performance in the opposite situation. The results are given in Fig. 7 . Figure 7(A,B) show that short wavelength λ₁ corresponding to the high-resolution in weak turbulence can be concluded. As turbulence intensity become large, the resolution is get worse. For the shorter wavelength λ₁, the condition λ₂z₁= λ₁z₀ is more susceptible to be destroyed. According to Fig. 7(E) in strong turbulence, the condition λ₂z₁= λ₁z₀ is no longer valid and long-wavelength λ₁ corresponds to the high-resolution.

Fig. 6 The W_cpsf as a function of distance z₁ for fixed λ₁, z₀. Parameters used are λ₁=1.2μm, ω=5cm, l_c=1mm, z₀=1km. (A-E) are the curve lines for C2 n =10⁻¹⁶, 10⁻¹⁵, 10⁻¹⁴, 10⁻¹³, 10⁻¹²m^-2/3. (F) is linear-type corresponding to the two wavelength values.

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Fig. 7 The W_cpsf as a function of distance z₁ for fixed λ₂, z₀. Parameters used are λ₂=1.2μm, ω=5cm, l_c=1mm, z₀=1km. (A-E) are the curve lines for C2 n =10⁻¹⁶, 10⁻¹⁵, 10⁻¹⁴, 10⁻¹³, 10⁻¹²m^-2/3. (F) is linear-type corresponding to the two wavelength values.

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The performance of the FOV is reflected in Figs. 8 and 9 for the two situations. From Fig. 8(A,B), we can see that the FOV for different wavelengths of fictitious reference beam have positive relationship with the distance and wavelength under weak turbulence. Figure 8(C,D) show that upward inflection points appear in the distance which meets the condition λ₂z₁= λ₁z₀ for medium strength of atmospheric turbulence and the corresponding FOV values are equal. As seen from Fig. 8(E), the lines show that the inflection points shift to the small distance value z₁ compare with Fig. 8(C,D). According to Fig. 9(A,B), we can conclude that the FOV values will not change as the wavelength λ₁ of the signal beam change for weak turbulence. Similar to the above experiment, the upward inflection points will appear, but shorter wavelength λ₁ corresponds to larger FOV in Fig. 9(C,D). Figure 9(E) appears similar phenomenon of Fig. 8(E).

Fig. 8 The W_cfov as a function of distance z₁ for fixed λ₁, z₀. Parameters used are λ₁=1.2μm, ω=5cm, l_c=1mm, z₀=1km. (A-E) are the curve lines for C2 n =10⁻¹⁶, 10⁻¹⁵, 10⁻¹⁴, 10⁻¹³, 10⁻¹²m^-2/3. (F) is linear-type corresponding to the two wavelength values.

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Fig. 9 The W_cfov as a function of distance z₁ for fixed λ₂, z₀. Parameters used are λ₂=1.2μm, ω=5cm, l_c=1mm, z₀=1km. (A-E) are the curve lines for C2 n =10⁻¹⁶, 10⁻¹⁵, 10⁻¹⁴, 10⁻¹³, 10⁻¹²m^-2/3. (F) is linear-type corresponding to the two wavelength values.

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The phenomenon can be explained based on the ratio between turbulence coherence lengths and transverse size of the laser beams. In the case that turbulence coherence length is much larger than transverse size of the laser beams, i.e. ρ₀ >>ω, the resolution and FOV are almost not affected by turbulence, and vice versa. The rule will also be obtained by comparing turbulence coherence lengths from Table 1 with transverse size of the laser beams given as 5cm. The condition λ₂z₁= λ₁z₀ should be applied when turbulence coherence lengths greater than or approximately equal to transverse size of the laser beams, i.e. ρ₀ >ω, or ρ₀≈ω. According to the similar analysis in Section 3.1, we also can understand the reason why the shorter wavelength λ₁ is utilized to obtain higher resolution and the performance of the FOV is shown in Fig. 8 and Fig. 9. According to the above analysis, using shorter wavelength λ₁ for computational ghost imaging can obtain high resolution image. From two-wavelength ghost imaging and computational case for comparison, we can draw the resolution from computational case will not be worse than that of two-wavelength ghost imaging.

4. Conclusion

In conclusion, we have demonstrated the theoretical expressions that describe the performance of two-wavelength ghost imaging through atmospheric turbulence. With the strength of atmospheric turbulence increase, the PSF and FOV values will become large. Meanwhile, in order to get the high resolution ghost image, the analytical calculations and the numerical simulations have presented that the shorter wavelength beam should be sent to illumine the object and the reference beam with the longer wavelength should be employed. For computational case, using shorter wavelength λ₁ can obtain high resolution image. The ratio between turbulence coherence lengths and transverse size of the laser beams is the key to understand this phenomenon. We have described this in simulation part. As a unique imaging method through atmospheric turbulence, we will further study the features, such as, signal-to-noise ratio and contrast.

The reported feature of two-wavelength ghost imaging will move forward ghost imaging in atmospheric turbulence. There are still several techniques that could improve the resolution of two-wavelength ghost imaging which we have not yet to incorporate into our performance analysis. Compressed sensing could provide a high-resolution ghost image and higher convergence rate. Conventional phase compensating techniques (e.g. adaptive optics) should be utilized to further eliminate turbulence effects.

Acknowledgments

The authors are indebted to the anonymous referees for their instructive suggestions and thank Huimin Ma for helpful discussions. This work was supported by Hefei Institutes of Physical Sciences, Chinese Academy of Sciences (Grants No. 073RC11123, No. Y03RC21121 and No. XJJ-11-S106).

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C2 n (m^-2/3)	λ=0.6um	0.8um	1.0um	1.2um	1.4um	1.8um	2.4um	4.0um
10⁻¹⁶	0.2257	0.3187	0.4166	0.5185	0.6239	0.8434	1.1912	2.1989
10⁻¹⁵	0.0567	0.0801	0.1046	0.1302	0.1567	0.2119	0.2992	0.5523
10⁻¹⁴	0.0142	0.0201	0.0263	0.0327	0.0394	0.0532	0.0752	0.1387
10⁻¹³	0.0036	0.0051	0.0066	0.0082	0.0099	0.0134	0.0189	0.0348
10⁻¹²	0.0009	0.0013	0.0017	0.0021	0.0025	0.0034	0.0047	0.0088

Two-wavelength ghost imaging through atmospheric turbulence

Abstract

1. Introduction

2. Theoretical analysis

3. Simulation and results

3.1 Two-wavelength ghost imaging system

3.2 Computational Two-wavelength ghost imaging system

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (9)

Tables (1)

Equations (15)

Optics Express