## Two-wavelength ghost imaging through atmospheric turbulence |

Optics Express, Vol. 21, Issue 2, pp. 2050-2064 (2013)

http://dx.doi.org/10.1364/OE.21.002050

Acrobat PDF (2373 KB)

### Abstract

Recent work has indicated that ghost imaging might find useful application in standoff sensing where atmospheric turbulence is a serious problem. There has been theoretical study of ghost imaging in the presence of turbulence. However, most work has addressed signal-wavelength ghost imaging. Two-wavelength ghost imaging through atmospheric turbulence is theoretically studied in this paper. Based on the extended Huygens-Fresnel integral, the analytical expressions describing atmospheric turbulence effects on the point spread function (PSF) and field of view (FOV) are derived. The computational case is also reported.

© 2013 OSA

## 1. Introduction

5. Y. Cai and S.-Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **71**(5), 056607 (2005). [CrossRef] [PubMed]

6. R. E. Meyers, K. S. Deacon, and Y. Shih, “Turbulence-free ghost imaging,” Appl. Phys. Lett. **98**(11), 111115 (2011). [CrossRef]

9. C. Zhao, W. Gong, M. Chen, E. Li, H. Wang, W. Xu, and S. Han, “Ghost imaging lidar via sparsity constraints,” Appl. Phys. Lett. **101**(14), 141123 (2012). [CrossRef]

10. J. Cheng, “Ghost imaging through turbulent atmosphere,” Opt. Express **17**(10), 7916–7921 (2009). [CrossRef] [PubMed]

16. P. Zhang, W. Gong, X. Shen, and S. Han, “Correlated imaging through atmospheric turbulence,” Phys. Rev. A **82**(3), 033817 (2010). [CrossRef]

20. O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. **95**(13), 131110 (2009). [CrossRef]

20. O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. **95**(13), 131110 (2009). [CrossRef]

## 2. Theoretical analysis

*u*to the CCD detector plane

*x*for the reference beam is given bywhere the

_{r}*E*(λ

_{ir}_{2},

*u*) is realized by modulating the plane light field

*E*(λ

_{r’}_{2},

*u*) using the SLM system 2, and ϕ

_{1}(

*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

*E*(λ

_{it}_{1},

*u*) is obtained by modulating the plane light field

*E*(λ

_{t’}_{1},

*u*) using the SLM system 1,

*t*(

*y*) denotes the amplitude reflectivity coefficient of the object and ϕ

_{0}(

*y*,

*u*), ϕ

_{2}(

*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*(λ

_{it}_{1},

*u*), and

*E*(λ

_{ir}_{2},

*u*). Thus the light fields at the output plane of the SLM systems can be modeled aswhere

*E*(λ

_{t’}_{1},

*u*) and

*E*(λ

_{r’}_{2},

*u*) are the independent light fields at the input plane of the SLM systems with wavelength λ

_{1}and λ

_{2}, 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

*jϕ*(

*u*)) approximate Gaussian distribution, Eq. (6) becomes

*ρ*(1.09

_{i}=*C*2(

*i*)

*n*(2π/λ

*)*

_{i}^{2}

*z*)

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

*C*2(

*i*)

*n*is the refractive index structure parameter describing the strength of atmospheric turbulence along uniform horizontal-path

*z*. 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

_{i}10. J. Cheng, “Ghost imaging through turbulent atmosphere,” Opt. Express **17**(10), 7916–7921 (2009). [CrossRef] [PubMed]

16. P. Zhang, W. Gong, X. Shen, and S. Han, “Correlated imaging through atmospheric turbulence,” Phys. Rev. A **82**(3), 033817 (2010). [CrossRef]

*u*

_{1},

*u*' 1,

*u*

_{2},

*u' 2*, we have

_{A=14ω2+12lc2+12ρ02−jπλ1z0, B=14ω2+12lc2+12ρ02+jπλ1z0−14Aρ04,C=14ω2+12lc2+12ρ12−jπλ2z1−14Blc4,D=14ω2+12lc2+12ρ12+jπλ2z1−14Alc4−116A2Blc4ρ04−14Cρ14−164A2B2Clc8ρ04−18ABClc4ρ02ρ12,S=y−y′2ρ02+2jπyλ1z0, P=y−y′2ρ02+2jπy′λ1z0−y−y′4Aρ04−jπyAλ1z0ρ02,Q=P2Blc2−2jπxrλ2z1, V=2jπxrλ2z1−y−y′4Alc2ρ02−jπyAλ1z0lc2+P4ABlc2ρ02+Q2Cρ12+Q8ABClc4ρ02.}

*d*(

*x*) is the bucket detector function;

_{t}*d*(

*x*)=1 if

_{t}*x*is inside the single pixel bucket detector, while

_{t}*d*(

*x*)=0 if

_{t}*x*is outside the single pixel bucket detector. We assume the area of bucket detector is

_{t}*s*. In fact, most objects encountered in real world are rough on the scale of an optical wavelength. Base on the laser radar theory [12

_{b}12. N. D. Hardy and J. H. Shapiro, “Reflective ghost imaging through turbulence,” Phys. Rev. A **84**(6), 063824 (2011). [CrossRef]

13. N. D. Hardy and J. H. Shapiro, “Ghost imaging in reflection: resolution, contrast, and signal-to-noise ratio,” Proc. SPIE **7815**, 78150L (2010). [CrossRef]

*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 aswhere

_{S′=2jπyλ1z0, P′=2jπyλ1z0(1−12Aρ02), Q′=jπyBλ1z0lc2(1−12Aρ02)−2jπxrλ2z1,V′=2jπxrλ2z1−jπyAλ1z0lc2+P′4ABlc2ρ02+Q′2Cρ12+Q′8ABClc4ρ02.}At last, we give Eq. (13) as the form of Gaussian function to comprehend the effect of atmospheric turbulence on two-wavelength ghost imagingwhere

_{Wpsf=12(K12+K22+K32+K52),m=K3K4+K5K6K12+K22+K32+K52, Wfov=12[K42+K62−(K3K4+K5K6)2K12+K22+K32+K52],K1=πAλ1z0, K2=πBλ1z0(1−12Aρ02), K3=π2BCλ1z0lc2(1−12Aρ02), K4=πCλ2z1,K5=π2Dλ1z0[−1Alc2+1Blc2(1−12Aρ02)(12Aρ02+12Cρ12+18ABClc4ρ02)],K6=πDλ2z1(12Cρ12+18ABClc4ρ02−1).}Here

*W*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,

_{psf}*W*is the field of view (FOV) as measured in image space and

_{fov}*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 z

_{2}doesn’t affect the PSF. However, atmospheric turbulence in the paths z

_{0}, z

_{1}disturbs the resolution of two-wavelength ghost imaging. This conclusion is consistent with the results in papers [10

10. J. Cheng, “Ghost imaging through turbulent atmosphere,” Opt. Express **17**(10), 7916–7921 (2009). [CrossRef] [PubMed]

13. N. D. Hardy and J. H. Shapiro, “Ghost imaging in reflection: resolution, contrast, and signal-to-noise ratio,” Proc. SPIE **7815**, 78150L (2010). [CrossRef]

*C*2(1)

*n*=0. Similar to the above part, the computational ghost image is proportional towhere

_{Wcpsf=12(K′12+K′22+K′32+K′52),m′=K′3K′4+K′5K′6K′12+K′22+K′32+K′52, Wcfov=12[K′42+K′62−(K′3K′4+K′5K′6)2K′12+K′22+K′32+K′52],K′1=πA′λ1z0, K′2=πB′λ1z0(1−12A′ρ02), K′3=π2BC′λ1z0lc2(1−12A′ρ02), K′4=πC′λ2z1,K′5=π2D′λ1z0[−1A′lc2+1B′lc2(1−12A′ρ02)(12A′ρ02+18A′B′C′lc4ρ02)],K′6=πD′λ2z1(18A′B′C′lc4ρ02−1),A′=14ω2+12lc2+12ρ02−jπλ1z0, B′=14ω2+12lc2+12ρ02+jπλ1z0−14A′ρ04,C′=14ω2+12lc2−jπλ2z1−14B′lc4,D′=14ω2+12lc2+jπλ2z1−14A′lc4−116A′2B′lc4ρ04−164A′2B′2C′lc8ρ04.}

*W*and

_{cpsf}*W*are the PSF and FOV of computational two-wavelength ghost imaging system. Similar to two-wavelength ghost imaging system, atmospheric turbulence in the path z

_{cfov}_{2}doesn’t affect the PSF and atmospheric turbulence in the paths z

_{0}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.

## 3. Simulation and results

_{0}=1

*km*. Different wavelengths turbulence coherence lengths (

*m*) for 1

*km*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.

### 3.1 Two-wavelength ghost imaging system

_{1}and test the resolution of two-wavelength ghost imaging under the situation that the wavelength λ

_{2}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 λ

_{2}z

_{1}= λ

_{1}z

_{0}for weak turbulence. However, with the increase of atmospheric turbulence strength, this phenomenon gradually disappeared. For the shorter wavelength λ

_{2}, the condition λ

_{2}z

_{1}= λ

_{1}z

_{0}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 z

_{1}. The smallest

*W*values become large when

_{psf}*C*2

*n*is increased, which means the image quality will be worse.

_{2}and alter the wavelength λ

_{1}. 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 λ

_{2}z

_{1}= λ

_{1}z

_{0}, as discussed above. This phenomenon gradually disappears with increasing turbulence strength, and the smallest

*W*values also turn large as the

_{psf}*C*2

*n*increased. However there are several different points compared with the above experiment. Firstly, the condition λ

_{2}z

_{1}= λ

_{1}z

_{0}is more susceptible to be destroyed for the longer wavelength λ

_{1}. Secondly, the optimal values of resolution are very different for various wavelengths λ

_{1}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 λ

_{1}must be utilized and the distance must content the condition z

_{1}= λ

_{1}z

_{0}/ λ

_{2}in weak atmospheric turbulence. Finally, high resolution image can be obtained for the shorter wavelength λ

_{1}in strong turbulence situation.

_{1}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 z

_{1}must content the condition λ

_{2}z

_{1}= λ

_{1}z

_{0}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.

_{2}z

_{1}= λ

_{1}z

_{0}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 z

_{1}compared with Fig. 4(C,D).

_{1}of the signal beam and fix the wavelength of the reference beam as λ

_{2}=1.2μ

*m*. 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 z

_{1}is up to a certain value, the FOV values are the same for all the wavelength λ

_{1}. From Fig. 4 and Fig. 5, we can conclude that the FOV values will be affected by the two wavelengths. The

*W*values become large with

_{fov}*C*2

*n*increases.

*ρ*and

_{0}*ρ*>>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 5

_{1}*cm*, the rule will also be concluded. The condition λ

_{2}z

_{1}= λ

_{1}z

_{0}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.

*ρ*and

_{0}*ρ*>ω, or

_{1}*ρ*and

_{0}*ρ*≈ω. The reason why shorter wavelength λ

_{1}_{1}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 λ

_{1}z

_{0}/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 z

_{0}is fixed and the condition λ

_{2}z

_{1}= λ

_{1}z

_{0}is employed, it is immediately obvious that two-wavelength ghost imaging using shorter wavelength λ

_{1}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

*ρ*and

_{0}*ρ*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

_{1}*W*and

_{psf}*W*values are become large as the strength of atmospheric turbulence increases.

_{fov}### 3.2 Computational Two-wavelength ghost imaging system

_{2}does not exist and the below mentioned turbulence coherence lengths is about wavelength λ

_{1}. We test the resolution of computational two-wavelength ghost imaging under the situation that the wavelength λ

_{2}is changing and the wavelength λ

_{1}is fixed. The results are shown in Fig. 6 . We can see that the minimum values

*W*are the same for the certain strength of atmospheric turbulence in different wavelength λ

_{cpsf}_{2}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 λ

_{2}z

_{1}= λ

_{1}z

_{0}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 λ

_{1}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 λ

_{1}, the condition λ

_{2}z

_{1}= λ

_{1}z

_{0}is more susceptible to be destroyed. According to Fig. 7(E) in strong turbulence, the condition λ

_{2}z

_{1}= λ

_{1}z

_{0}is no longer valid and long-wavelength λ

_{1}corresponds to the high-resolution.

_{2}z

_{1}= λ

_{1}z

_{0}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 z

_{1}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 λ

_{1}of the signal beam change for weak turbulence. Similar to the above experiment, the upward inflection points will appear, but shorter wavelength λ

_{1}corresponds to larger FOV in Fig. 9(C,D). Figure 9(E) appears similar phenomenon of Fig. 8(E).

*ρ*>>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 5

_{0}*cm*. The condition λ

_{2}z

_{1}= λ

_{1}z

_{0}should be applied when turbulence coherence lengths greater than or approximately equal to transverse size of the laser beams, i.e.

*ρ*>ω, or

_{0}*ρ*≈ω. According to the similar analysis in Section 3.1, we also can understand the reason why the shorter wavelength λ

_{0}_{1}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 λ

_{1}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

_{1}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.

## Acknowledgments

## References and links

1. | J. Shapiro and R. Boyd, “The physics of ghost imaging,” Quantum Inf. Process. DOI 10.1007 (2012). |

2. | B. I. Erkmen and J. H. Shapiro, “Ghost imaging: from quantum to classical to computational,” AIP Conf. Proc. |

3. | G. Scarcelli, V. Berardi, and Y. Shih, “Can two-photon correlation of chaotic light be considered as correlation of intensity fluctuations?” Phys. Rev. Lett. |

4. | B. I. Erkmen and J. H. Shapiro, “Unified theory of ghost imaging with Gaussian-state light,” Phys. Rev. A |

5. | Y. Cai and S.-Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

6. | R. E. Meyers, K. S. Deacon, and Y. Shih, “Turbulence-free ghost imaging,” Appl. Phys. Lett. |

7. | R. E. Meyers, K. S. Deacon, A. D. Tunick, and Y. Shih, “Virtual ghost imaging through turbulence and obscurants using Bessel beam illumination,” Appl. Phys. Lett. |

8. | R. E. Meyers, K. S. Deacon, and Y. Shih, “Positive-negative turbulence-free ghost imaging,” Appl. Phys. Lett. |

9. | C. Zhao, W. Gong, M. Chen, E. Li, H. Wang, W. Xu, and S. Han, “Ghost imaging lidar via sparsity constraints,” Appl. Phys. Lett. |

10. | J. Cheng, “Ghost imaging through turbulent atmosphere,” Opt. Express |

11. | C. Li, T. Wang, J. Pu, W. Zhu, and R. Rao, “Ghost imaging with partially coherent light radiation through turbulent atmosphere,” Appl. Phys. B |

12. | N. D. Hardy and J. H. Shapiro, “Reflective ghost imaging through turbulence,” Phys. Rev. A |

13. | N. D. Hardy and J. H. Shapiro, “Ghost imaging in reflection: resolution, contrast, and signal-to-noise ratio,” Proc. SPIE |

14. | K. W. C. Chan, D. S. Simon, A. V. Sergienko, N. D. Hardy, J. H. Shapiro, P. B. Dixon, G. A. Howland, J. C. Howell, J. H. Eberly, M. N. O’Sullivan, B. Rodenburg, and R. W. Boyd, “Theoretical analysis of quantum ghost imaging through turbulence,” Phys. Rev. A |

15. | P. Dixon, G. A. Howland, K. W. C. Chan, C. O'Sullivan-Hale, B. Rodenburg, N. D. Hardy, J. H. Shapiro, D. S. Simon, A. V. Sergienko, R. W. Boyd, and J. C. Howell, “Quantum ghost imaging through turbulence,” Phys. Rev. A |

16. | P. Zhang, W. Gong, X. Shen, and S. Han, “Correlated imaging through atmospheric turbulence,” Phys. Rev. A |

17. | K. W. C. Chan, M. N. O'Sullivan, and R. W. Boyd, “Two-color ghost imaging,” Phys. Rev. A |

18. | S. Karmakar and Y. H. Shih, “Two-color ghost imaging with enhanced angular resolving power,” Phys. Rev. A |

19. | J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A |

20. | O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. |

21. | L. C. Andrews and R. L. Phillips, |

**OCIS Codes**

(030.6600) Coherence and statistical optics : Statistical optics

(110.2990) Imaging systems : Image formation theory

(110.0115) Imaging systems : Imaging through turbulent media

(070.6120) Fourier optics and signal processing : Spatial light modulators

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: July 6, 2012

Revised Manuscript: November 5, 2012

Manuscript Accepted: December 18, 2012

Published: January 18, 2013

**Virtual Issues**

Vol. 8, Iss. 2 *Virtual Journal for Biomedical Optics*

**Citation**

Dongfeng Shi, Chengyu Fan, Pengfei Zhang, Hong Shen, Jinghui Zhang, Chunhong Qiao, and Yingjian Wang, "Two-wavelength ghost imaging through atmospheric turbulence," Opt. Express **21**, 2050-2064 (2013)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-21-2-2050

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### References

- J. Shapiro and R. Boyd, “The physics of ghost imaging,” Quantum Inf. Process. DOI 10.1007 (2012).
- B. I. Erkmen and J. H. Shapiro, “Ghost imaging: from quantum to classical to computational,” AIP Conf. Proc.1110, 417–422 (2009).
- G. Scarcelli, V. Berardi, and Y. Shih, “Can two-photon correlation of chaotic light be considered as correlation of intensity fluctuations?” Phys. Rev. Lett.96(6), 063602 (2006). [CrossRef] [PubMed]
- B. I. Erkmen and J. H. Shapiro, “Unified theory of ghost imaging with Gaussian-state light,” Phys. Rev. A77(4), 043809 (2008). [CrossRef]
- Y. Cai and S.-Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.71(5), 056607 (2005). [CrossRef] [PubMed]
- R. E. Meyers, K. S. Deacon, and Y. Shih, “Turbulence-free ghost imaging,” Appl. Phys. Lett.98(11), 111115 (2011). [CrossRef]
- R. E. Meyers, K. S. Deacon, A. D. Tunick, and Y. Shih, “Virtual ghost imaging through turbulence and obscurants using Bessel beam illumination,” Appl. Phys. Lett.100(6), 061126 (2012). [CrossRef]
- R. E. Meyers, K. S. Deacon, and Y. Shih, “Positive-negative turbulence-free ghost imaging,” Appl. Phys. Lett.100(13), 131114 (2012). [CrossRef]
- C. Zhao, W. Gong, M. Chen, E. Li, H. Wang, W. Xu, and S. Han, “Ghost imaging lidar via sparsity constraints,” Appl. Phys. Lett.101(14), 141123 (2012). [CrossRef]
- J. Cheng, “Ghost imaging through turbulent atmosphere,” Opt. Express17(10), 7916–7921 (2009). [CrossRef] [PubMed]
- C. Li, T. Wang, J. Pu, W. Zhu, and R. Rao, “Ghost imaging with partially coherent light radiation through turbulent atmosphere,” Appl. Phys. B99(3), 599–604 (2010). [CrossRef]
- N. D. Hardy and J. H. Shapiro, “Reflective ghost imaging through turbulence,” Phys. Rev. A84(6), 063824 (2011). [CrossRef]
- N. D. Hardy and J. H. Shapiro, “Ghost imaging in reflection: resolution, contrast, and signal-to-noise ratio,” Proc. SPIE7815, 78150L (2010). [CrossRef]
- K. W. C. Chan, D. S. Simon, A. V. Sergienko, N. D. Hardy, J. H. Shapiro, P. B. Dixon, G. A. Howland, J. C. Howell, J. H. Eberly, M. N. O’Sullivan, B. Rodenburg, and R. W. Boyd, “Theoretical analysis of quantum ghost imaging through turbulence,” Phys. Rev. A84(4), 043807 (2011). [CrossRef]
- P. Dixon, G. A. Howland, K. W. C. Chan, C. O'Sullivan-Hale, B. Rodenburg, N. D. Hardy, J. H. Shapiro, D. S. Simon, A. V. Sergienko, R. W. Boyd, and J. C. Howell, “Quantum ghost imaging through turbulence,” Phys. Rev. A83(5), 051803 (2011). [CrossRef]
- P. Zhang, W. Gong, X. Shen, and S. Han, “Correlated imaging through atmospheric turbulence,” Phys. Rev. A82(3), 033817 (2010). [CrossRef]
- K. W. C. Chan, M. N. O'Sullivan, and R. W. Boyd, “Two-color ghost imaging,” Phys. Rev. A79(3), 033808 (2009). [CrossRef]
- S. Karmakar and Y. H. Shih, “Two-color ghost imaging with enhanced angular resolving power,” Phys. Rev. A81(3), 033845 (2010). [CrossRef]
- J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A78, 061802(R) (2008).
- O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett.95(13), 131110 (2009). [CrossRef]
- L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 2005).

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