## Abstract

Ghost imaging through turbulent atmospheres are theoretically studied. Based on the extended Huygens-Fresnel integral, we obtain an analytical imaging formula. The ghost image can be viewed as the convolution of the original object and a point-spread function (PSF). The imaging quality is determined by the size of the PSF. Increasing the turbulence strength and propagation distance, or decreasing the source size, will increase the size of the PSF, and lead to the degradation of the imaging quality.

©2009 Optical Society of America

## 1. Introduction

In recent years, ghost imaging (GI) has attracted a lot of attentions in the field of quantum optics [1]. As a kind of nonlocal imaging method, intensity correlation measurements are performed in GI experiments to get the image of an unknown object. Both quantum entangled and classically correlated incoherent light sources can be used to realize GI [2–17]. A very interesting application of the GI with classically incoherent sources is that it can be used to generate images without the use of lenses. Lensless imaging is very useful in many applications. We have studied the realization of lensless ghost diffraction and its applicability in x-ray diffraction [4, 13]. Some other groups have shown that true images also can be produced from lensless ghost imaging (LGI) systems [14, 15]. Very recently, researchers have proposed the idea of ghost camera and realized reflected ghost imaging experiment [16].

However, in all these studies cited above, the light fields are propagated in free space. To the best of our knowledge, the properties of GI through turbulent atmosphere have never been investigated so far. Beam propagation in a turbulent atmosphere is a topic of a long history and still attracts many researcher’s interests [18–24]. In this paper, we try to investigate the possibility to realize ghost imaging through turbulent atmosphere. Using the extended Huygens-Fresnel integral, we analytically derive the ghost imaging formula in atmospheric turbulence. The ghost image is a convolution of the object and a Gaussian function. Strong turbulence, large propagation distance, and small source size lead to bad imaging quality. Our results may find applications in spatial measurement and observation.

## 2. Theory

Let’s consider the lensless ghost imaging (LGI) scheme [14, 15] shown in Fig. 1. An incoherent thermal or pseduo-thermal source is split into two beams by the beam splitter BS and travel through two different imaging systems, one is a test imaging system which contains an unknown object and a test detector *D _{t}* without resolution, the other is a reference imaging system which is irrelevant with the object and has a high resolution reference detector

*D*. The intensity distribution recorded in

_{r}*D*and

_{t}*D*are correlated by a correlator to obtain the correlation function of the intensity fluctuations. To get an image of the object,

_{r}*z*

_{0}=

*z*

_{1}is needed to be satisfied. Here we assume there exist atmospheric turbulence in all optical paths. A more realistic situation, in which only the test arm experiences atmospheric turbulence and the reference arm is local, can be viewed as a special case of the general theory by assuming the turbulence strength in the reference arm to be zero. When there is no atmosphere, the imaging properties of this LGI system are well known now.

In the reference path, based on the extended Huygens-Fresnel integral [19], the field at *D _{r}* is

where *f*(*u*) is the source distribution, *ϕ*
_{1}(*u*,*x*
_{1}) represents the random part of the complex phase due to the turbulent atmosphere. In the test path, similarly at the plane of *D _{t}*, we have

where *t*(*y*) is the object and *ϕ _{i}* characterize the turbulent effects.

Then the intensity correlation function is given by

$$\phantom{\rule[-0ex]{.2em}{0ex}}=\genfrac{}{}{0.1ex}{}{1}{{\lambda}^{3}{z}_{0}{z}_{1}{z}_{2}}\u3008\int d{u}_{1}d{u\prime}_{1}d{u}_{2}d{u\prime}_{2}\mathrm{dydy}\prime f\left({u}_{1}\right){f}^{*}\left({u\prime}_{1}\right)f\left({u}_{2}\right)f\left({u\prime}_{2}\right){f}^{*}\left({u\prime}_{2}\right)t\left(y\right){t}^{*}\left(y\prime \right)$$

$${e}^{\genfrac{}{}{0.1ex}{}{\mathrm{j\pi}}{\lambda {z}_{1}}\left[{\left({x}_{1}-{u}_{1}\right)}^{2}-{\left({x}_{1}-{u\prime}_{1}\right)}^{2}\right]}{e}^{\genfrac{}{}{0.1ex}{}{\mathrm{j\pi}}{\lambda {z}_{0}}\left[{\left(y-{u}_{2}\right)}^{2}-{\left(y\prime -{u\prime}_{2}\right)}^{2}\right]}{e}^{\genfrac{}{}{0.1ex}{}{\mathrm{j\pi}}{\lambda {z}_{2}}\left[{\left({x}_{2}-y\right)}^{2}-{\left({x}_{2}-y\prime \right)}^{2}\right]}$$

$${e}^{{\varphi}_{1}({u}_{1},{x}_{1})+{\varphi}_{1}^{*}({u\prime}_{1},{x}_{1})}{e}^{{\varphi}_{0}({u}_{2},y)+{\varphi}_{0}^{*}({u\prime}_{2},y\prime )}{e}^{{\varphi}_{2}({x}_{2},y)+{\varphi}_{2}^{*}({x}_{2},y\prime )}\u3009$$

$$=\genfrac{}{}{0.1ex}{}{1}{{\lambda}^{3}{z}_{0}{z}_{1}{z}_{2}}\int d{u}_{1}d{u\prime}_{1}d{u}_{2}d{u\prime}_{2}\mathrm{dydy}\prime \u3008f\left({u}_{1}\right){f}^{*}\left({u\prime}_{1}\right)f\left({u}_{2}\right)f\left({u\prime}_{2}\right)\u3009\u3008{e}^{{\varphi}_{1}\left({u}_{1}{x}_{1}\right)+{\varphi}_{1}^{*}({u\prime}_{1},{x}_{1})}\u3009$$

$$\u3008{e}^{{\varphi}_{0}({u}_{2},y)+{\varphi}_{0}^{*}({u\prime}_{2},y\prime )}\u3009\u3008{e}^{{\varphi}_{2}({x}_{2},y)+{\varphi}_{2}^{*}({x}_{2},y\prime )}\u3009$$

$$t\left(y\right){t}^{*}{e}^{\genfrac{}{}{0.1ex}{}{\mathrm{j\pi}}{\lambda {z}_{1}}\left[{\left({x}_{1}-{u}_{1}\right)}^{2}-{\left({x}_{1}-{u\prime}_{1}\right)}^{2}\right]}{e}^{\genfrac{}{}{0.1ex}{}{\mathrm{j\pi}}{\lambda {z}_{0}}\left[{\left(y-{u}_{2}\right)}^{2}-{\left(y\prime -{u\prime}_{2}\right)}^{2}\right]}{e}^{\genfrac{}{}{0.1ex}{}{\mathrm{j\pi}}{\lambda {z}_{2}}\left[{\left({x}_{2}-y\right)}^{2}-{\left({x}_{2}-y\prime \right)}^{2}\right]}$$

where we have assumed the statistics of the source, and the three propagation regimes are independent.

It is now well known that the statistical averages caused by the turbulent atmosphere can be described approximately by

where *ρ _{i}*, = (0.55

*C*

^{2(i)}

_{n}

*k*

^{2}

*z*)

_{i}^{-3/5}is the coherence length of a spherical wave propagating through a turbulent medium and

*C*

^{2(i)}

*n*is the refractive-index structure parameter describing the strength of the atmospheric turbulence in the path

*z*. It is important to point out that a quadratic approximation of the Rytov’s phase structure function has been used in Eq. (4) in order to simplify the analysis and obtain an analytical formula [19, 20, 21, 22]. This quadratic approximation has been widely used in literatures and is usually accepted to be valid in the weak fluctuation regime. Even for strong fluctuations cases, the approximation also seems useful.

_{i}The sources used in GI generally can be described as a zero-average Gaussian random process, so

Suppose the source is fully incoherent, we have 〈*f*(*u*)*f*
^{*}(*u*′)〉 = *I*(*u*)*δ*(*u* - *u*′).

Introducing several new parameters, *γ _{i}* =

*π*/

*λz*,

_{i}*β*=

_{i}*ρ*

^{-2}

*i*= (0.55

*C*

^{2(i)}

_{n}

*k*

^{2}

*z*)

_{i}^{6/5}, then the correlation of intensity fluctuations is

$$\phantom{\rule[-0ex]{3.5em}{0ex}}=\genfrac{}{}{0.1ex}{}{{\gamma}_{0}{\gamma}_{1}{\gamma}_{2}}{{\pi}^{3}}\int d{u}_{1}d{u}_{2}\mathrm{dydy}\prime t\left(y\right){t}^{*}\left(y\prime \right)I\left({u}_{1}\right)I\left({u}_{2}\right)$$

$$\phantom{\rule[-0ex]{3.7em}{0ex}}{e}^{j{\gamma}_{1}\left[{\left({x}_{1}-{u}_{1}\right)}^{2}-{\left({x}_{1}-{u}_{2}\right)}^{2}\right]+j{\gamma}_{0}\left[{\left(y-{u}_{2}\right)}^{2}-{\left(y\prime -{u}_{1}\right)}^{2}\right]+j{\gamma}_{2}\left[{\left({x}_{2}-y\right)}^{2}-{\left({x}_{2}-y\prime \right)}^{2}\right]}$$

$${e}^{-{\beta}_{1}{\left({u}_{1}-{u}_{2}\right)}^{2}-{\beta}_{0}\left[{\left({u}_{2}-{u}_{1}\right)}^{2}+\left({u}_{2}-{u}_{1}\right)(y-y\prime )+{\left(y-y\prime \right)}^{2}\right]-{\beta}_{2}{\left(y-y\prime \right)}^{2}},$$

The image information is contained in this function.

Suppose the source intensity has a Gaussian distribution, *I*(*u*) = *e*
^{-u2}/*ρ*
^{2}
_{s}, defining *α* = *ρ*
^{-2}
_{s} we can integrate Eq. (5) to get,

where the kernel function

with *A* = *α* + *β*
_{0} + *β*
_{1} + *jγ _{0}* -

*jγ*,

_{1}*B*= -2

*β*

_{0}- 2

*β*

_{1},

*C*=

*α*+

*β*

_{0}+

*β*

_{1}-

*jγ*

_{0}+

*jγ*

_{1},

*D*=

*β*

_{0}(

*y*′ -

*y*) + 2

*jγ*

_{1}

*x*

_{1}- 2

*jγ*

_{0}

*γ*

_{1},

*E*= -

*β*

_{0}(

*y*′ -

*y*) - 2

*jγ*

_{1}

*x*

_{1}+ 2

*jγ*

_{0}

*y*.

Now, when the imaging condition of a LGI system is satisfied (*z*
_{0} = *z*
_{1} = *z*), *γ*
_{0} = *γ*
_{1} = *γ* = *n*/*λz*, the kernel function is simplified to be

Combined this Eq. and Eq. (6), we obtain a general formula to describe the lensless ghost imaging through turbulent atmosphere. If the test detector *D _{t}* is a bucket detector, then the ghost image is proportional to

$$\phantom{\rule[-0ex]{.2em}{0ex}}=\genfrac{}{}{0.1ex}{}{{\gamma}^{2}}{{\pi}^{2}}\int \mathrm{dydy}\prime \delta \left(y-y\prime \right)t\left(y\right){t}^{*}\left(y\prime \right)h\left(y,y\prime ,{x}_{1}\right){e}^{j\left({\gamma}_{0}+{\gamma}_{2}\right)\left({y}^{2}-{y\prime}^{2}\right)-\left({\beta}_{0}+{\beta}_{2}\right){\left(y-y\prime \right)}^{2}}$$

$$\phantom{\rule[-0ex]{.2em}{0ex}}=\genfrac{}{}{0.1ex}{}{{\gamma}^{2}}{{\pi}^{2}}\int \mathrm{dy}{\mid t\left(y\right)\mid}^{2}h\left(y,y,{x}_{1}\right)$$

Eq. (9) can be represented as:

in which a z-independent factor $\sqrt{\genfrac{}{}{0.1ex}{}{\pi}{2\alpha}}$ has been neglected. In Eq. (10), ⊗ means convolution *h _{z}*(

*y*) can be considered as the point-spread function (PSF) at the distance

*z*,

and the z-dependent radius

Eq. (10) is the main result in this paper, it clearly determines how the ghost image is degraded in the atmospheric turbulence.

## 3. Numerical calculations and analysis

For a realistic situation, in which only the test arm experiences atmospheric turbulence, by setting *C*
^{2(1)}
_{n} = 0 and *C*
^{2(0)}
_{n} = *C*
^{2}
_{n}, we can use numerical calculations to demonstrate atmospheric effects on ghost imaging. In the first example, we set *λ* = 0.785 *μ*m, *ρ _{s}* = 2.5 cm,

*C*

^{2}

_{n}= 10

^{-14}m

^{-2/3}. The object is a double-slit, with slit width 2 cm and separation 6 cm. From Fig. 2, we can directly see the deformation of the images when the propagation length is increased. Compared with the free space, at large distance, the atmospheric turbulence may significantly decrease the imaging quality. But when the propagation distance is small, even in turbulent atmosphere, high quality ghost images still can be formed. The test arm and reference arm have a common path before the BS, but this distance is very small compared with

*z*

_{0}, and can not affect our results.

Since the ghost images are determined by the PSF, changing *C*
^{2}
_{n} from the weak turbulence (10^{-16}m^{-2/3}) to strong turbulence (10^{-13}m^{-2/3}), we plot *R _{z}* as the function of

*z*in Fig. 3(a). We can see

*R*become larger and larger when

_{z}*C*

^{2}

_{n}is increased, which means the imaging quality will be worse and worse. Eq. (12) also relies on the source size

*ρ*, so we have plotted

_{s}*R*(

_{z}*z*) with 5 different

*ρ*in Fig. 3(b). We find increasing

_{s}*z*can decrease

*R*, leading to the improvement of imaging quality. But the improvement will be saturated when

_{z}*ρ*is enough large.

_{s}Based on these numerical results and theoretical formulas, we find the main features of GI through turbulent atmosphere are:

- It is very clear, when
*R*→ 0, the PSF*h*_{0}(*y*) →*δ*(*y*), so the ghost image will takes the form $M\left({x}_{1}\right)\to \genfrac{}{}{0.1ex}{}{1}{{\lambda}_{z}}{\mid t\left({x}_{1}\right)\mid}^{2}$, which is exactly the image of the object, i.e. , one can realize perfectly imaging. - The ghost image is the convolution of the original object and a PSF, so the imaging quality is totally determined by the PSF. Since the PSF is a Gaussian function, the image will be greatly degraded when
*R*is increased to a large value._{z} - Since
*R*is a monotone increasing function of_{z}*z*, the imaging quality will be worse when the propagating distance is longer. - Strong fluctuation (large
*C*^{2}_{n})) will significantly degraded the imaging quality because it leads to a large*R*._{z} - Increasing the source size may improve the imaging quality, but the improvement will be saturated when ps is large enough.

## 4. Conclusion

In conclusion, we have investigated the ghost imaging through turbulent atmosphere. Based on the extended Huygens-Fresnel integral, using a quadratic approximation of the structure function, analytical formulas for the ghost images have been derived. We find the ghost image can be viewed as the convolution of the original object and a point-spread function. The imaging quality is determined from the size of the PSF. The images will be significantly degraded with strong turbulence and large propagation distance. Increasing the source size can improve imaging quality, but the improvement will be saturated for enough large sources.

## Acknowledgments

Support from the National Natural Science Foundation of China (10774047), and South China University of Technology is acknowledged.

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