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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 2 — Jan. 28, 2013
  • pp: 2050–2064
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Two-wavelength ghost imaging through atmospheric turbulence

Dongfeng Shi, Chengyu Fan, Pengfei Zhang, Hong Shen, Jinghui Zhang, Chunhong Qiao, and Yingjian Wang  »View Author Affiliations


Optics Express, Vol. 21, Issue 2, pp. 2050-2064 (2013)
http://dx.doi.org/10.1364/OE.21.002050


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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

Ghost imaging [1

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

5

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]

] 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

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

9

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]

] 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

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

16

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

] 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

17. K. W. C. Chan, M. N. O'Sullivan, and R. W. Boyd, “Two-color ghost imaging,” Phys. Rev. A 79(3), 033808 (2009). [CrossRef]

], 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

18. S. Karmakar and Y. H. Shih, “Two-color ghost imaging with enhanced angular resolving power,” Phys. Rev. A 81(3), 033845 (2010). [CrossRef]

] 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

19. J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78, 061802(R) (2008).

] and demonstrated [20

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

]. 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

19. J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78, 061802(R) (2008).

,20

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

] 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

17. K. W. C. Chan, M. N. O'Sullivan, and R. W. Boyd, “Two-color ghost imaging,” Phys. Rev. A 79(3), 033808 (2009). [CrossRef]

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

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 xr for the reference beam is given by
Er(xr)=jexp(2jπz1λ2)λ2z1duEir(λ2,u)exp[jπλ2z1(xru)2]exp[ϕ1(xr,u)],
(1)
where the Eir2,u) is realized by modulating the plane light field Er’2,u) using the SLM system 2, and ϕ1(xr,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
Et(xt)=exp(2jπ(z0+z2)λ1)λ12z0z2dyduEit(λ1,u)exp[jπλ1z0(yu)2]exp[ϕ0(y,u)]          ×t(y)exp[jπλ1z2(xty)2]exp[ϕ2(xt,y)],
(2)
where the Eit1,u) is obtained by modulating the plane light field Et’1,u) using the SLM system 1, t(y) denotes the amplitude reflectivity coefficient of the object and ϕ0(y,u), ϕ2(xt,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 Eit1,u), and Eir2,u). Thus the light fields at the output plane of the SLM systems can be modeled as
Eit(λ1,u)=Et'(λ1,u)exp[jϕ(u)],Eir(λ2,u)=Er'(λ2,u)exp[jϕ(u)],
(3)
where Et’1,u) and Er’2,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
G(xt,xr)=(It(xt)It(xt))(Ir(xr)Ir(xr))=It(xt)Ir(xr)It(xt)Ir(xr)             =Et*(xt)Et(xt)Er*(xr)Er(xr)Et*(xt)Et(xt)Er*(xr)Er(xr)              =1λ14λ22z02z12z22du1du1du2du2dydyΓ(u1,u1,u2,u2)<t(y)t*(y)>             ×exp[ϕ1(u2,xr)+ϕ1*(u2,xr)]exp[ϕ0(u1,y)+ϕ0*(u1,y)]             ×exp[ϕ2(y,xt)+ϕ2*(y,xt)]exp{jπλ2z1[(xru2)2(xru2)2]}             ×exp{jπλ1z0[(yu1)2(yu1)2]}exp{jπλ1z2[(xty)2(xty)2]},
(4)
where
Γ(u1,u1,u2,u2)=Eit(u1)Eit*(u1)Eir*(u2)Eir(u2)Eit(u1)Eit*(u1)Eir*(u2)Eir(u2).
(5)
Because the two light fields at the input plane of the SLM systems are independent, with the help of Eq. (3), Eq. (5) becomes
Γ(u1,u1,u2,u2)=Et'(λ1,u1)Et'*(λ1,u1)Er'(λ2,u2)Er'*(λ2,u2)                         ×(exp{j[ϕ(u1)ϕ(u1)+ϕ(u2)ϕ(u2)]}                         exp{j[ϕ(u1)ϕ(u1)]}exp{j[ϕ(u2)ϕ(u2)]}).
(6)
According to the central limit theorem, a large number of random variables exp((u)) approximate Gaussian distribution, Eq. (6) becomes

Γ(u1,u1,u2,u2)=Et'(λ1,u1)Et'*(λ1,u1)Er'(λ2,u2)Er'*(λ2,u2)                       ×exp{j[ϕ(u1)ϕ(u2)]}exp{j[ϕ(u2)ϕ(u1)]}.
(7)

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(u1,u1,u2,u2)=exp(u12+u12+u22+u224ω2)exp((u1u2)2+(u2u1)22lc2),
(8)
where ω is the transverse size of the laser beams and lc 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

21. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 2005).

]
exp[ϕi(x,y)+ϕi*(x,y)]=exp{(xx)2+(xx)(yy)+(yy)22ρi2},
(9)
where ρi=(1.09C2(i) n(2π/λi)2zi)−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 zi. 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

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

16

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

,21

21. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 2005).

].

Substituting Eqs. (8), (9) into Eq. (4) and integrating over u1, u' 1, u2, u' 2, we have
G(xt,xr)=π2λ14λ22z02z12z22ABCDdydy<t(y)t*(y)>exp[(yy)22ρ02+(yy)22ρ22]              ×exp{jπλ1z0(y2y2)+jπλ1z2[(xty)2(xty)2]}exp(S24A+P24B+Q24C+V24D),
(10)
whereA=14ω2+12lc2+12ρ02jπλ1z0,  B=14ω2+12lc2+12ρ02+jπλ1z014Aρ04,C=14ω2+12lc2+12ρ12jπλ2z114Blc4,D=14ω2+12lc2+12ρ12+jπλ2z114Alc4116A2Blc4ρ0414Cρ14164A2B2Clc8ρ0418ABClc4ρ02ρ12,S=yy2ρ02+2jπyλ1z0,  P=yy2ρ02+2jπyλ1z0yy4Aρ04jπyAλ1z0ρ02,Q=P2Blc22jπxrλ2z1,  V=2jπxrλ2z1yy4Alc2ρ02jπyAλ1z0lc2+P4ABlc2ρ02+Q2Cρ12+Q8ABClc4ρ02.

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(xr)=G(xt,xr)d(xt)dxt,
(11)
where d(xt) is the bucket detector function; d(xt)=1 if xt is inside the single pixel bucket detector, while d(xt)=0 if xt is outside the single pixel bucket detector. We assume the area of bucket detector is sb. In fact, most objects encountered in real world are rough on the scale of an optical wavelength. Base on the laser radar theory [12

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

,13

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]

], the amplitude reflectivity coefficient from rough surface can be modeled as a zeros-mean Gaussian random field with
<t(y)t*(y)>=λ12T(y)δ(yy),
(12)
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(xr)=sbπ2λ12λ22z02z12z22ABCDdyT(y)exp(S24A+P24B+Q24C+V24D),
(13)
whereS=2jπyλ1z0,  P=2jπyλ1z0(112Aρ02),  Q=jπyBλ1z0lc2(112Aρ02)2jπxrλ2z1,V=2jπxrλ2z1jπyAλ1z0lc2+P4ABlc2ρ02+Q2Cρ12+Q8ABClc4ρ02.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(xr)=sbπ2λ12λ22z02z12z22ABCDexp(xr22Wfov2)dyT(y)exp((ymxr)22Wpsf2),
(14)
whereWpsf=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(112Aρ02),  K3=π2BCλ1z0lc2(112Aρ02),  K4=πCλ2z1,K5=π2Dλ1z0[1Alc2+1Blc2(112Aρ02)(12Aρ02+12Cρ12+18ABClc4ρ02)],K6=πDλ2z1(12Cρ12+18ABClc4ρ021).Here Wpsf 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, Wfov 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

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

13

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]

] 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

19. J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78, 061802(R) (2008).

]. 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(xr)=sbπ2λ12λ22z02z12z22ABCDexp(xr22Wcfov2)dyT(y)exp((ymxr)22Wcpsf2),
(15)
whereWcpsf=12(K12+K22+K32+K52),m=K3K4+K5K6K12+K22+K32+K52, Wcfov=12[K42+K62(K3K4+K5K6)2K12+K22+K32+K52],K1=πAλ1z0,  K2=πBλ1z0(112Aρ02),  K3=π2BCλ1z0lc2(112Aρ02),  K4=πCλ2z1,K5=π2Dλ1z0[1Alc2+1Blc2(112Aρ02)(12Aρ02+18ABClc4ρ02)],K6=πDλ2z1(18ABClc4ρ021),A=14ω2+12lc2+12ρ02jπλ1z0,  B=14ω2+12lc2+12ρ02+jπλ1z014Aρ04,C=14ω2+12lc2jπλ2z114Blc4,D=14ω2+12lc2+jπλ2z114Alc4116A2Blc4ρ04164A2B2Clc8ρ04.

Equation (15) presents the performance of computational two-wavelength ghost imaging through atmospheric turbulence. Wcpsf and Wcfov 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.

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

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

table-icon
View This Table
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

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
Fig. 2 The Wpsf as a function of distance z1 for fixed λ1, z0. Parameters used are λ1=1.2μm, ω=5cm, lc=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.
, 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 Wpsf values become large when C2 n is increased, which means the image quality will be worse.

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)
Fig. 4 The Wfov as a function of distance z1 for fixed λ1, z0. Parameters used are λ1=1.2μm, ω=5cm, lc=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.
, 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).

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)
Fig. 5 The Wfov as a function of distance z1 for fixed λ2, z0. Parameters used are λ2=1.2μm, ω=5cm, lc=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.
, 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 Wfov values become large with C2 n increases.

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 Wpsf and Wfov 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
Fig. 6 The Wcpsf as a function of distance z1 for fixed λ1, z0. Parameters used are λ1=1.2μm, ω=5cm, lc=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.
. We can see that the minimum values Wcpsf 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
Fig. 7 The Wcpsf as a function of distance z1 for fixed λ2, z0. Parameters used are λ2=1.2μm, ω=5cm, lc=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.
. 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.

The performance of the FOV is reflected in Figs. 8
Fig. 8 The Wcfov as a function of distance z1 for fixed λ1, z0. Parameters used are λ1=1.2μm, ω=5cm, lc=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.
and 9
Fig. 9 The Wcfov as a function of distance z1 for fixed λ2, z0. Parameters used are λ2=1.2μm, ω=5cm, lc=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.
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).

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).

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. 1110, 417–422 (2009).

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. 96(6), 063602 (2006). [CrossRef] [PubMed]

4.

B. I. Erkmen and J. H. Shapiro, “Unified theory of ghost imaging with Gaussian-state light,” Phys. Rev. A 77(4), 043809 (2008). [CrossRef]

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]

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. 100(6), 061126 (2012). [CrossRef]

8.

R. E. Meyers, K. S. Deacon, and Y. Shih, “Positive-negative turbulence-free ghost imaging,” Appl. Phys. Lett. 100(13), 131114 (2012). [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]

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 99(3), 599–604 (2010). [CrossRef]

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]

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 84(4), 043807 (2011). [CrossRef]

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 83(5), 051803 (2011). [CrossRef]

16.

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

17.

K. W. C. Chan, M. N. O'Sullivan, and R. W. Boyd, “Two-color ghost imaging,” Phys. Rev. A 79(3), 033808 (2009). [CrossRef]

18.

S. Karmakar and Y. H. Shih, “Two-color ghost imaging with enhanced angular resolving power,” Phys. Rev. A 81(3), 033845 (2010). [CrossRef]

19.

J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78, 061802(R) (2008).

20.

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

21.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 2005).

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/oe/abstract.cfm?URI=oe-21-2-2050


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References

  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.1110, 417–422 (2009).
  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.96(6), 063602 (2006). [CrossRef] [PubMed]
  4. B. I. Erkmen and J. H. Shapiro, “Unified theory of ghost imaging with Gaussian-state light,” Phys. Rev. A77(4), 043809 (2008). [CrossRef]
  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]
  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.100(6), 061126 (2012). [CrossRef]
  8. R. E. Meyers, K. S. Deacon, and Y. Shih, “Positive-negative turbulence-free ghost imaging,” Appl. Phys. Lett.100(13), 131114 (2012). [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. Express17(10), 7916–7921 (2009). [CrossRef] [PubMed]
  11. 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]
  12. N. D. Hardy and J. H. Shapiro, “Reflective ghost imaging through turbulence,” Phys. Rev. A84(6), 063824 (2011). [CrossRef]
  13. N. D. Hardy and J. H. Shapiro, “Ghost imaging in reflection: resolution, contrast, and signal-to-noise ratio,” Proc. SPIE7815, 78150L (2010). [CrossRef]
  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. A84(4), 043807 (2011). [CrossRef]
  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. A83(5), 051803 (2011). [CrossRef]
  16. P. Zhang, W. Gong, X. Shen, and S. Han, “Correlated imaging through atmospheric turbulence,” Phys. Rev. A82(3), 033817 (2010). [CrossRef]
  17. K. W. C. Chan, M. N. O'Sullivan, and R. W. Boyd, “Two-color ghost imaging,” Phys. Rev. A79(3), 033808 (2009). [CrossRef]
  18. S. Karmakar and Y. H. Shih, “Two-color ghost imaging with enhanced angular resolving power,” Phys. Rev. A81(3), 033845 (2010). [CrossRef]
  19. J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A78, 061802(R) (2008).
  20. O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett.95(13), 131110 (2009). [CrossRef]
  21. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 2005).

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