Standoff two-color quantum ghost imaging through turbulence Yu-Lang Xue,1,2 Ren-Gang Wan,1 Fei Feng,1,2 and Tong-Yi Zhang1,* 1
State Key Laboratory of Transient Optics and Photonics, Xi’an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences, Xi’an 710119, China 2
University of Chinese Academy of Sciences, Beijing 100049, China *Corresponding author: [email protected]
Received 28 January 2014; revised 12 March 2014; accepted 3 April 2014; posted 8 April 2014 (Doc. ID 205429); published 7 May 2014
Recently, a two-color quantum ghost imaging configuration was proposed by Karmakar et al. [Phys. Rev. A 81, 033845 (2010)]. By illuminating an object located far away from the source and detector, with a signal beam of long wavelength to avoid absorption of short wavelengths in the atmosphere while a reference beam of short wavelength is detected locally, this imaging configuration can be appropriate for standoff sensing. In practice, the signal beam must propagate through atmosphere in the presence of serious turbulence. We analyzed theoretically the performance of this ghost imaging configuration through turbulence. Based on the Gaussian state source model and extended Huygens–Fresnel integral, a formula is derived to depict the ghost image formed through turbulence of a standoff reflective object. Numerical calculations are also given according to the formula. The results show that the image quality will be degraded by the turbulence, but the resolution can be improved by means of optimizing the wavelengths of the reference and signal beams even when the turbulence is very serious. © 2014 Optical Society of America OCIS codes: (280.3640) Lidar; (110.0115) Imaging through turbulent media; (100.2960) Image analysis. http://dx.doi.org/10.1364/AO.53.003035
1. Introduction
Ghost imaging is a new transverse imaging technology in which the object image is retrieved using the second-order correlation function of two spatially correlated light beams: the reference beam, which never interacts with the object and is measured by a pixelated detector, and the signal beam, which, after illuminating the object, is collected by a bucket detector with no spatial resolution capability. Since the first experiment implemented by Shih’s group using spatially entangled photon pairs [1], ghost imaging has inspired much interest [2] on account of its potential applications in quantum lithography [3–5], quantum holography [6], quantum optical coherence 1559-128X/14/143035-08$15.00/0 © 2014 Optical Society of America
tomography [7], and lensless imaging [8–12]. Recently, a new, useful application in optical security was proposed [13,14]. The feasibility of reflective ghost imaging in which the bucket detector views the target in reflection rather than in transmission has been demonstrated [15–17]. It reveals the capacity of ghost imaging for standoff sensing. In such applications, imaging through turbulent atmosphere would become unavoidable. Theoretical and experimental investigations into the effect of turbulence on the performance of ghost imaging have been reported [18–25]. However, in all the studies just cited, the signal and reference beams used in ghost imaging have the same wavelength. In Ref. [26], the authors have proved that the wavelengths of the two beams can be different (nondegenerate wavelength) for vacuum propagation. Then, two-color ghost imaging through 10 May 2014 / Vol. 53, No. 14 / APPLIED OPTICS
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atmospheric turbulence for a thermal light source was studied [27]. In Ref. [28], the authors proposed a quantum two-color ghost imaging scheme. By sending a long wavelength signal beam through the atmosphere to avoid atmospheric absorption of short wavelengths, while utilizing a reference beam with short wavelength which is local and propagates without atmosphere, this imaging scheme would be appropriate for standoff sensing. In this work, we investigate the performance of this ghost imaging configuration through turbulent atmosphere. Modeling the spontaneous parametric down conversion (SPDC) source as in a zero-mean jointly Gaussian state and using the extended Huygens–Fresnel integral, we analytically derived a formula for this ghost imaging configuration through turbulence. Based on the formula, numerical simulations are given to indicate the optimal method for ghost imaging through turbulence of different strength levels. 2. Theoretical Analysis
In order to study the properties of ghost images formed through turbulent atmosphere, we consider the schematic setup depicted in the Klyshko unfolded version [28] shown in Fig. 1. A type-II phase-matched, collinear, SPDC source emits orthogonally polarized signal and idler beams which are separated by a polarizing beam splitter to act as the signal and reference beams. The reference beam with central wavelength λr is spatially resolved by a CCD camera D2 on the image plane after direct free space propagation. The signal beam with central wavelength λs passes through a convex lens of focal length f and radius ρD, and it illuminates an object of rough surface located Lo meters away from the lens. The signal beam reflected back is then collected by a bucket detector D1 situated Lb meters away from the object. Because the optical path lengths of the two beams are different, a postdetection electronic time delay is introduced to maximize the temporal cross
Fig. 1. Klyshko unfolded version of the schematic standoff twocolor quantum ghost imaging setup. A SPDC source generates nondegenerate signal and reference beams with central wavelengths λs and λr , respectively. d1 , d2 , Lo , and Lb are distances from source to lens, source to CCD camera D2 , lens to object, and object to bucket detector D1, respectively. ρc , ρf , ρt , and ρb are coordinates on the CCD plane, lens plane, object plane, and bucket detector plane. There exists turbulence in the lens-to-object and objectto-bucket-detector paths. 3036
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correlation. Then the electric signals driven by the signal and reference fields are sent to a correlation circuit to form the ghost image of the object. According to Ref. [28], when the distance d2 from the SPDC source to the CCD is far longer than d1 to the lens, the resolution will mainly depend on λs and degrade monotonically as λs gets larger. But this is not appropriate for standoff sensing, as we should send a long-wavelength signal beam to reduce atmospheric absorption; so, to avoid this condition, we set d2 to be far shorter than d1 . Then the space between the beam splitter and SPDC crystal must be so small that we can neglect it. Lo is generally far longer than d1 and d2 (i.e., Lo ≫ d1 ≫ d2 ). The distances satisfy the modified Gaussian thin-lens equation 1∕d2 λr ∕λs d1 1∕Lo 1∕f . We assume that turbulence only exists in the lens-to-object and objectto-bucket-detector paths. We represent the optical fields of the signal and reference beams at the output of the SPDC crystal ˆ s ρs ; te−iωs t as positive-frequency field operators E p −iωr t ˆ and Er ρr ; te , which are of photon∕m2 s units [23,29,30]. Here, ρs and ρr are the transverse coordinates corresponding to the optical axis. Then, the beams incident on the CCD and object surfaces can be represented by extended Huygens–Fresnel integral [20,23,24,27] as ˆ c ρc ; t1 E
Z
ˆ r ρr ; t1 − d2 ∕cHkr ; d2 ; ρc ; ρr ; (1) dρr E
ˆ t ρt ; t2 E Z Z ˆ s ρs ; t2 − d1 Lo ∕cHks ; d1 ; ρf ; ρs dρs dρf E × Lρf Hks ; Lo ; ρt ; ρf eψ o ρt ;ρf ;
(2) 0
2
where Hk; l; ρ0 ; ρ k∕i2πleikljρ −ρj ∕2l is the freespace propagator, Lρf exp−iks ρ2f ∕2f ΘρD − jρf j is the transmission function of the lens, and Θρ is the unit step function. The turbulence is taken to be of the Kolmogorov spectrum and uniformly distributed along the lens-to-object and objectto-bucket-detector paths (i.e., o path and b path). 0 It is depicted by eψ m ρ ;ρ, where the complex phase ψ m ρ0 ; ρ represents log-amplitude and phase fluctuations imposed on a spherical wave propagating from ρ to ρ0 on path m by the turbulence. The mutual co0 herence function of eψ m ρ ;ρ taken as the square-law approximation to the rigorous 5/3-law behavior [20,23,24,27,31,32] is
heψ m ρj1 ;ρk1 eψ m ρj2 ;ρk2 i 2 ρ
e−jρj1 −ρj2 j
j1 −ρj2 ·ρk1 −ρk2 jρk1 −ρk2 j
2 ∕2ρ2 m
;
(3)
where ρm is the turbulence coherence length on path m, given by ρm 1.09k2s C2n;m Lm −3∕5 in terms of the refractive index structure parameter, C2n;m ,
describing the turbulence strength and m; j; k o; t; f or m; j; k b; b; t. The subscripts 1 and 2 represent two arbitrary coordinates on the corresponding plane. The quadratic approximation is employed here to simplify the analysis and obtain an analytical formula. The analysis procedure is also valid if we employ the mutual coherence function of other formats, but it would be difficult to obtain an analytical formula. Though the specific form of the result for other formats will be different from here, the conclusion may be similar. A real-world target which has a sufficiently rough surface can be modeled as a planar two-dimensional object with random microscopic transverse and height variations. The random field-reflection coefficient Tρt of the object, which is a zero-mean complex-valued Gaussian random process, is characterized by the autocorrelation function [23,27] hT ρ0t Tρt i λ2s Oρt δρt − ρ0t :
(4)
Here, Oρt is the average intensity-reflection coefficient we’d like to image.
ˆ p i hCρ
1 TI
Z
T I ∕2 −T I ∕2
q2 η2 Ap
dtˆip t−τd ˆib t
Z
Z dτ1
Z dτ2
Ab
dρb hc t−τd −τ1 hb t−τ2
ˆ 0† ρb ;τ2 E ˆ c ρp ;τ1 E ˆ 0T ρb ;τ2 i; ˆ †c ρp ;τ1 E ×hE T
(5)
where T I is the correlation integration time, τd is the postdetection electronic delay time, q is the electron charge, η is the quantum efficiency taken to be the same for both the bucket detector and CCD camera, Ap is the small area of the pth pixel, Ab is the active region of the bucket detector, hc t and hb t are the impulse responses of the detectors’ output circuit, ˆ 0T ρb ; t represents the field impinging on the and E bucket detector after being reflected by the object. To perform the ensemble average calculations below, we invoke the statistical independence of the target’s reflection coefficient, the signal and reference fields at the source, and the turbulence on two propagation paths. Back-propagating the object-reflected field operator to the object, we can find [23]
ˆ 0† ρb ; τ2 E ˆ c ρp ; τ1 E ˆ 0T ρb ; τ2 i ˆ †c ρp ; τ1 E hE T Z Z 0 dρ0t dρt hT ρ0t Tρt iheψ b ρb ;ρt eψ b ρb ;ρt iH ks ; Lb ; ρb ; ρ0t Hks ; Lb ; ρb ; ρt ˆ † ρ0 ; τ2 − Lb ∕cE ˆ c ρp ; τ1 E ˆ t ρt ; τ2 − Lb ∕ci ˆ †c ρp ; τ1 E × hE t t Z 1 ˆ †c ρp ; τ1 E ˆ † ρt ; τ2 − Lb ∕cE ˆ c ρp ; τ1 E ˆ t ρt ; τ2 − Lb ∕ci: 2 dρt Oρt hE t Lb
(6)
This shows that turbulence on the object-to-bucket-detector path does not affect the ghost image’s properties. The ghost imaging is only influenced by the turbulence on the lens-to-object path. ˆ p i, becomes After substituting Eqs. (1), (2), and (6) into Eq. (5), the correlation, hCρ ˆ p i q2 η2 Ap hCρ Z
Z dρf
Ab L2b
Z
Z dτ1
Z dτ2
Z
Z dρt hc t − τd − τ1 hb t − τ2 Oρt
dρr
dρ0r
Z
Z dρs
dρ0s
ˆ †r ρr ; τ1 − τr E ˆ †s ρs ; τ2 − τs E ˆ r ρ0r ; τ1 − τr E ˆ s ρ0s ; τ2 − τs i dρ0f hE
× H kr ; d2 ; ρp ; ρr Hkr ; d2 ; ρp ; ρ0r H ks ; d1 ; ρf ; ρs Hks ; d1 ; ρ0f ; ρ0s L ρf 0
× Lρ0f H ks ; Lo ; ρt ; ρf Hks ; Lo ; ρt ; ρ0f heψ o ρt ;ρf eψ o ρt ;ρf i;
The ghost image at the center coordinate ρp of the CCD’s pth pixel can be recovered by the ensemble average cross-correlation function of the photocurrents ˆip and ˆib from the pixel and the bucket detector, respectively. In practice, it can be estimated by measuring the time average of the photocurrents, which is [23,29,30]
(7)
where τr and τs are the light propagation time of the reference and signal beams, i.e., d2 ∕c and Lo Lb d1 ∕c, respectively. We assume the signal and reference beams generated by the SPDC source are in zero-mean jointly Gaussian states, which are completely characterized by their nonzero autocorrelation and cross-correlation 10 May 2014 / Vol. 53, No. 14 / APPLIED OPTICS
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functions. Hence, the fourth-order field moment can be expressed in terms of second-order moments by moment-factoring theorem [23,29]
ˆ p i hCρ
ˆ †s ρs ;τ2 −τs E ˆ r ρ0r ;τ1 −τr E ˆ s ρ0s ;τ2 −τs i ˆ †r ρr ;τ1 −τr E hE ˆ †r ρr ;τ1 −τr E ˆ †s ρs ;τ2 −τs ihE ˆ r ρ0r ;τ1 −τr E ˆ s ρ0s ;τ2 −τs i hE ˆ †r ρr ;τ1 −τr E ˆ r ρ0r ;τ1 −τr ihE ˆ †s ρs ;τ2 −τs E ˆ s ρ0s ;τ2 −τs i hE ˆ †r ρr ;τ1 −τr E ˆ s ρ0s ;τ2 −τs ihE ˆ †s ρs ;τ2 −τs E ˆ r ρ0r ;τ1 −τr i: hE (8) The autocorrelation will be assumed to have the Gaussian–Schell model form [23,29,30] ˆ K ρ2 ; t2 i ˆ † ρ1 ; t1 E hE K
2P − e πa20
jρ1 j2 jρ2 j2 a2 0
−
e
jρ1 −ρ2 j2 2ρ2 0
−
e
t1 −t2 2 2T 2 0
(9) for K s, r, where P is the photon flux, a0 is the source’s e−2 intensity radius, ρ0 is the coherence length, and T 0 is its coherence time. For a SPDC source running at high flux, the signal and reference beams are in a maximally entangled jointly Gaussian state with no phase-insensitive cross correlation, and the maximum phase-sensitive cross correlation permitted by quantum theory given their autocorrelation functions, namely [23,29,30] ˆ †s ρ1 ; t1 Er ρ2 ; t2 i 0; hE
− ˆ s ρ1 ; t1 Er ρ2 ; t2 i 2P e hE 2 πa0
jρ1 j2 jρ2 j2 a2 0
−
e
t1 −t2 2 2T 2 0
−
2μεγ 2 β2 2 A B − ; θφ α2 γ 2 Δ2PSF h i D−ε μεγ 2 β2 C−ε μεγ 2 β2 iμξγβ αiγ − B − − A α−iγ α2 γ 2 α2 γ 2 m 2μεγ 2 β2 φ A B − α2 γ2 1
1 Δ2FOV
− ·
e
APPLIED OPTICS / Vol. 53, No. 14 / 10 May 2014
m2 2θμεγ 2 β2 2 μξ C D − 2ε − Δ2PSF α2 γ 2
μξγβ2 D − εC − ε α2 γ 2
− θB
μξγβ2 μξγβ2 2 D − ε − θA C − ε2 ; α iγ2 α − iγ2
α 1∕a20 1∕2ρ20 ; 1∕2ρ2D
iks ∕2f ;
:
δ
(11)
ξ ks ∕2Lo ;
In order to perform the integral more readily, we assume that (1) the finite size of the imaging lens depicted by the step function ΘρD − jρf j can be replaced with a Gaussian function of the form exp−ρ2f ∕2ρ2D [26], (2) the detectors have the same p 2 2 Gaussian impulse response ht ΩB ∕ 8π e−ΩB t ∕8 , whose bandwidth ΩB is much smaller than that of the incident light for the SPDC source (ΩB T 0 ≪ 1), (3) the correlation integration time is sufficiently long to capture a ghost image, so T I ≫ Ω−1 B [23,30], (4) the spatial coordinates are treated as onedimensional to simplify the calculation, while maintaining the validity of the study of image quality [20–22,27]. Combining Eqs. (8) and (10), we can see that the third term in Eq. (8) equals zero. After substituting the two remaining terms into Eq. (7) and performing the integral over spatial coordinates, the correlation becomes 3038
where
(10) jρ1 −ρ2 j2 2ρ2 0
Z P2 k4s k2r q2 η2 Ap Ab dτ1 16π 5 a40 d21 L2o L2b d22 Z Z dτ2 ht − τd − τ1 ht − τ2 dρt Oρt s τ1 − τr − τ2 − τs 2 θμ × exp − 2 2 T0 α γ2 mρt − ρp 2 ρ2t × exp − exp − 2 Δ2PSF ΔFOV r υ 2α − βγ 2 2 2 ξ ρt exp −υ δ δ κ κσ 2α − βφ2 ρ2p ; (12) − σ
β 1∕2ρ20 ; ε
φ kr ∕2d2 ;
C D δ ε iξ iγ A B α −
γ ks ∕2d1 ;
1∕2ρ2o ;
γ2 ; α iγ
β2 μDγ 2 β2 iφ; α iγ α iγ2
E F δ ε iξ iγ μ−1 CD − ε2 ; υ−1 EF − ε
α − iγγ 2 ; α2 γ 2 − β 2
θ−1 AB − βγ 2 2 α γ 2 − β2
κ α2 γ 2 − β2 ;
μ2 ε2 γ 4 β4 ; α2 γ 2 2
2
;
σ α2 φ2 − β2 :
It is clear that the first term leads to a ghost image of the object and the second term leads to a featureless background. Substituting the detectors’ response function and performing the integral over temporal variables, we can get
ˆ p i hCρ
Z P2 k4s k2r q2 η2 Ap Ab dρt Oρt 16π 5 a40 d21 L2o L2b d22 s ΩB T 0 τd − τs τr 2 θμ exp − 2 × 2 2 4 T 0 16∕ΩB α γ2 mρt − ρp 2 ρ2t exp − × exp − Δ2PSF Δ2FOV r υ 2α − βγ 2 2 2 ξ ρt exp −υ δ δ κ κσ 2α − βφ2 ρ2p : (13) − σ
The temporal cross correlation is maximized when the postdetection delay time τd equals τs –τr Lo Lb d1 –d2 ∕c, and then the ghost image is Z P2 k4s k2r q2 η2 Ap Ab dρt Oρt Gρp 16π 5 a40 d21 L2o L2b d22 s ( mρt −ρp 2 ΩB T 0 θμ ρ2t exp − exp − 4 α2 γ 2 Δ2PSF Δ2FOV r υ 2α−βγ 2 2 2 ξ ρt exp −υ δ δ κ κσ ) 2α−βφ2 ρ2p : (14) − σ The image is at a magnification of m, which is negative in our system. The width of the point-spread function describing the resolution measured in image space is ΔPSF and the field of view measured in object space is ΔFOV . Equation (14) is the main result we obtain, which determines all properties of the ghost imaging through turbulence. 3. Numerical Calculations and Analyses
To make the calculation closer to real applications, we set the system parameters as: a0 1 mm, ρ0 1 μm, d1 1 m, d2 0.05 m, Lo Lb 1000 m,
and ρD 20 cm; f is determined by the modified Gaussian thin-lens equation. First, we demonstrate the effect of turbulence on the ghost imaging of a reflective double slit. The double-slit object in which the slits reflect light and the part elsewhere absorbs is shown in Fig. 2(a). The slit width is 6 cm and slit separation is 12 cm. To show the imaging property in detail, the intensity reflection coefficient of the left slit is 1 and the right slit is 0.5. The wavelengths of the reference beam λr and signal beam λs are 532 and 1064 nm, respectively. The images without the background recovered by normalized correlation for five particular turbulence strength levels are shown in Fig. 2(b). Since the magnification factor m is negative, the left slit and right slit are reversed. The imaging quality is almost not influenced when the turbulence is weak (C2n;o 10−15 m−2∕3 ). A ghost image of good quality can be obtained even through turbulence of moderate strength (C2n;o 10−14 m−2∕3 ). The image quality will be degraded seriously only when the turbulence gets very strong. Next, we investigate the effect of turbulence on the magnification factor m. The magnification as a function of the reference beam wavelength λr with λs 1064 nm and that of the signal beam wavelength λs with λr 532 nm are plotted in Figs. 3(a) and 3(b), respectively, under different conditions of turbulence. It is clear that the magnification factor is hardly influenced by the turbulence and always equals −d2 λr ∕λs d1 ∕Lo, which is the magnification when there is no turbulence [28]. The magnification approximates to −d1 ∕Lo [−0.001 in our calculation; this can also be seen from the image obtained in Fig. 2(b)] because the condition Lo ≫ d1 ≫ d2 is satisfied. Finally, we study how the turbulence affects the resolution and field of view. The point-spread function ΔPSF and field of view ΔFOV versus λr when λs is fixed at 1064 nm are shown in Figs. 4(a) and 4(b), respectively, and the ΔPSF and ΔFOV versus λs when λr is set to be 532 nm are plotted in Figs. 4(c) and 4(d). As the point-spread function depicts the resolution of the image, it is explicit that the resolution is very
Fig. 2. (a) Object of a reflective double slit to be imaged; its slit width is 6 cm and slit separation is 12 cm. The reflectivities of the left and right slits are 1 and 0.5, respectively. (b) Normalized cross section ghost image through different levels of turbulence. The image is reversed as the magnification factor is negative. 10 May 2014 / Vol. 53, No. 14 / APPLIED OPTICS
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Fig. 3. Magnification factor as a function of (a) the reference beam wavelength λr with λs 1064 nm, and (b) the signal beam wavelength λs with λr 532 nm for different turbulence strength levels.
vulnerable to the turbulence. When the turbulence gets stronger, the resolution degrades seriously. However, the ΔPSF decreases monotonically as the wavelength of the reference beam λr gets shorter, and it also decreases when a longer wavelength
signal beam is taken to propagate through the turbulence atmosphere above moderate levels. If the turbulence is very weak, the resolution gets better when a signal beam of shorter wavelength is used to illuminate the object, as shown especially in
Fig. 4. (a) Point-spread function ΔPSF describing the resolution of the ghost image, and (b) field of view ΔFOV versus λr when λs is fixed at 1064 nm for different turbulence strengths. (c) ΔPSF and (d) ΔFOV versus λs while λr is set to be 532 nm. (e) Point-spread function and (f) field of view plotted corresponding to (c) and (d), respectively, especially when the turbulence is weak. 3040
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Fig. 4(e). A remarkable fact that can be seen from Figs. 4(c) and 4(e) is that the variation of resolution is very slow with the signal wavelength when the turbulence is weak. On the other hand, the atmospheric absorption of the shorter-wavelength signal is serious, so longer signal wavelength is preferable for propagation through the atmosphere. For strong turbulence, the variation of resolution is very fast with the signal wavelength [shown in Fig. 4(c)], and hence it is always better to choose a longer signal wavelength for higher resolution. As a consequence, no matter how strong or weak the turbulence is, we should always choose a longer wavelength signal beam for propagation through the atmosphere, and a shorter wavelength reference beam to get a high resolution. Nevertheless, the field of view is unaffected by the turbulence no matter when the reference beam wavelength varies, as shown in Fig. 4(b), or when the signal beam wavelength is changed, as depicted in Figs. 4(d) and 4(f). But the field of view diminishes when λr decreases or λs increases. A trade-off should be made, therefore, between the resolution and field of view if one would like to utilize a short λr or a long λs. 4. Conclusion
In summary, the performance of a standoff two-color quantum ghost imaging scheme in the presence of turbulent atmosphere is investigated. Based on the extended Huygens–Fresnel integral, modeling the beams as in zero-mean jointly Gaussian states, and using a quadratic approximation of the turbulence structure function, we derive a formula for the twocolor quantum ghost imaging configuration through turbulence. Numerical calculations are also given based on the formula. In this imaging configuration, the magnification factor and field of view are hardly affected by the turbulence. The resolution will be degraded with strong turbulence. However, the resolution can be improved by utilizing a shorter wavelength reference beam. If the turbulence is serious (C2n;o > 10−15 m−2∕3 ), the image resolution gets better the longer the signal beam wavelength is, but while the turbulence is weak (C2n;o < 10−15 m−2∕3 ), a shorter wavelength signal beam should be chosen to obtain a good resolution. Nevertheless, a tradeoff should be made between the resolution and field of view if one would like to utilize a short wavelength reference beam or a long wavelength signal beam. Considering the serious atmospheric absorption and that the variation of resolution is very slow with the signal wavelength when the turbulence is weak, we can always send a long wavelength signal beam to illuminate the object through turbulent atmosphere with a short wavelength reference beam to get a good resolution. Hence, this two-color ghost imaging configuration is very suitable for standoff sensing. This work is supported by the NSFC (under Grant Nos. 61176084, 11204367, and 11174282), the China Postdoctoral Science Foundation (funded project
under Grant No. 2012M512040), and the “Western Light” Talent Cultivation Plan of the Chinese Academy of Sciences. References 1. T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52, R3429–R3432 (1995). 2. A. Gatti, E. Brambilla, and L. A. Lugiato, “Quantum imaging,” Prog. Opt. 51, 251 (2008). 3. M. D. Angelo, M. V. Chekhova, and Y. H. Shih, “Two-photon diffraction and quantum lithography,” Phys. Rev. Lett. 87, 013602 (2001). 4. J. Xiong, D. Z. Cao, F. Huang, H. G. Li, X. J. Sun, and K. G. Wang, “Experimental observation of classical subwavelength interference with a pseudothermal light source,” Phys. Rev. Lett. 94, 173601 (2005). 5. Y. H. Zhai, X. H. Chen, D. Zhang, and L. A. Wu, “Two-photon interference with true thermal light,” Phys. Rev. A 72, 043805 (2005). 6. A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Quantum holography,” Opt. Express 9, 498–505 (2001). 7. M. B. Nasr, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Demonstration of dispersion-canceled quantum-optical coherence tomography,” Phys. Rev. Lett. 91, 083601 (2003). 8. A. Valencia, G. Scarcelli, M. D. Angelo, and Y. H. Shih, “Twophoton imaging with thermal light,” Phys. Rev. Lett. 94, 063601 (2005). 9. D. Z. Cao, J. Xiong, and K. G. Wang, “Geometrical optics in correlated imaging systems,” Phys. Rev. A 71, 013801 (2005). 10. Y. J. Cai and F. Wang, “Lensless imaging with partially coherent light,” Opt. Lett. 32, 205–207 (2007). 11. X. H. Chen, Q. Liu, K. H. Luo, and L. A. Wu, “Lensless ghost imaging with true thermal light,” Opt. Lett. 34, 695–697 (2009). 12. Y. P. Yao, R. G. Wan, Y. L. Xue, S. W. Zhang, and T. Y. Zhang, “Positive–negative nonlocal lensless imaging based on statistical optics,” Acta Phys. Sin. 62, 154201 (2013). 13. W. Chen and X. Chen, “Ghost imaging for three-dimensional optical security,” Appl. Phys. Lett. 103, 221106 (2013). 14. W. Chen and X. Chen, “Object authentication in computational ghost imaging with the realizations less than 5% of Nyquist limit,” Opt. Lett. 38, 546–548 (2013). 15. R. Meyers, K. S. Deacon, and Y. H. Shih, “Ghost-imaging experiment by measuring reflected photons,” Phys. Rev. A 77, 041801 (2008). 16. C. Wang, D. Zhang, Y. Bai, and B. Chen, “Ghost imaging for a reflected object with a rough surface,” Phys. Rev. A 82, 063814 (2010). 17. 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, 141123 (2012). 18. R. Meyers, K. S. Deacon, and Y. H. Shih, “Turbulence-free ghost imaging,” Appl. Phys. Lett. 98, 111115 (2011). 19. R. Meyers, K. S. Deacon, and Y. H. Shih, “Positive-negative turbulence-free ghost imaging,” Appl. Phys. Lett. 100, 131114 (2012). 20. J. Cheng, “Ghost imaging through turbulent atmosphere,” Opt. Express 17, 7916–7921 (2009). 21. J. Cheng and J. Lin, “Unified theory of thermal ghost imaging and ghost diffraction through turbulent atmosphere,” Phys. Rev. A 87, 043810 (2013). 22. 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, 599–604 (2010). 23. N. D. Hardy and J. H. Shapiro, “Reflective ghost imaging through turbulence,” Phys. Rev. A 84, 063824 (2011). 24. 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, 043807 (2011). 25. Y. P. Yao, R. G. Wan, S. W. Zhang, and T. Y. Zhang, “Effect of turbulence on visibility and signal-to-noise ratio of lensless 10 May 2014 / Vol. 53, No. 14 / APPLIED OPTICS
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ghost imaging with thermal light,” Optik 124, 6973–6977 (2013). 26. K. W. C. Chan, M. N. O’Sullivan, and R. W. Boyd, “Two-color ghost imaging,” Phys. Rev. A 79, 033808 (2009). 27. D. Shi, C. Fan, P. Zhang, H. Shen, J. Zhang, C. Qiao, and Y. Wang, “Two-wavelength ghost imaging through atmospheric turbulence,” Opt. Express 21, 2050–2064 (2013). 28. S. Karmakar and Y. H. Shih, “Two-color ghost imaging with enhanced angular resolving power,” Phys. Rev. A 81, 033845 (2010).
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29. B. I. Erkmen and J. H. Shapiro, “Unified theory of ghost imaging with Gaussian-state light,” Phys. Rev. A 77, 043809 (2008). 30. B. I. Erkmen and J. H. Shapiro, “Signal-to-noise ratio of Gaussian-state ghost imaging,” Phys. Rev. A 79, 023833 (2009). 31. H. T. Yura, “Mutual coherence function of a finite cross section optical beam propagating in a turbulent medium,” Appl. Opt. 11, 1399–1406 (1972). 32. G. R. Osche, Optical Detection Theory for Laser Applications (Wiley-Interscience, 2002).
State Key Laboratory of Transient Optics and Photonics, Xi’an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences, Xi’an 710119, China 2
University of Chinese Academy of Sciences, Beijing 100049, China *Corresponding author: [email protected]
Received 28 January 2014; revised 12 March 2014; accepted 3 April 2014; posted 8 April 2014 (Doc. ID 205429); published 7 May 2014
Recently, a two-color quantum ghost imaging configuration was proposed by Karmakar et al. [Phys. Rev. A 81, 033845 (2010)]. By illuminating an object located far away from the source and detector, with a signal beam of long wavelength to avoid absorption of short wavelengths in the atmosphere while a reference beam of short wavelength is detected locally, this imaging configuration can be appropriate for standoff sensing. In practice, the signal beam must propagate through atmosphere in the presence of serious turbulence. We analyzed theoretically the performance of this ghost imaging configuration through turbulence. Based on the Gaussian state source model and extended Huygens–Fresnel integral, a formula is derived to depict the ghost image formed through turbulence of a standoff reflective object. Numerical calculations are also given according to the formula. The results show that the image quality will be degraded by the turbulence, but the resolution can be improved by means of optimizing the wavelengths of the reference and signal beams even when the turbulence is very serious. © 2014 Optical Society of America OCIS codes: (280.3640) Lidar; (110.0115) Imaging through turbulent media; (100.2960) Image analysis. http://dx.doi.org/10.1364/AO.53.003035
1. Introduction
Ghost imaging is a new transverse imaging technology in which the object image is retrieved using the second-order correlation function of two spatially correlated light beams: the reference beam, which never interacts with the object and is measured by a pixelated detector, and the signal beam, which, after illuminating the object, is collected by a bucket detector with no spatial resolution capability. Since the first experiment implemented by Shih’s group using spatially entangled photon pairs [1], ghost imaging has inspired much interest [2] on account of its potential applications in quantum lithography [3–5], quantum holography [6], quantum optical coherence 1559-128X/14/143035-08$15.00/0 © 2014 Optical Society of America
tomography [7], and lensless imaging [8–12]. Recently, a new, useful application in optical security was proposed [13,14]. The feasibility of reflective ghost imaging in which the bucket detector views the target in reflection rather than in transmission has been demonstrated [15–17]. It reveals the capacity of ghost imaging for standoff sensing. In such applications, imaging through turbulent atmosphere would become unavoidable. Theoretical and experimental investigations into the effect of turbulence on the performance of ghost imaging have been reported [18–25]. However, in all the studies just cited, the signal and reference beams used in ghost imaging have the same wavelength. In Ref. [26], the authors have proved that the wavelengths of the two beams can be different (nondegenerate wavelength) for vacuum propagation. Then, two-color ghost imaging through 10 May 2014 / Vol. 53, No. 14 / APPLIED OPTICS
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atmospheric turbulence for a thermal light source was studied [27]. In Ref. [28], the authors proposed a quantum two-color ghost imaging scheme. By sending a long wavelength signal beam through the atmosphere to avoid atmospheric absorption of short wavelengths, while utilizing a reference beam with short wavelength which is local and propagates without atmosphere, this imaging scheme would be appropriate for standoff sensing. In this work, we investigate the performance of this ghost imaging configuration through turbulent atmosphere. Modeling the spontaneous parametric down conversion (SPDC) source as in a zero-mean jointly Gaussian state and using the extended Huygens–Fresnel integral, we analytically derived a formula for this ghost imaging configuration through turbulence. Based on the formula, numerical simulations are given to indicate the optimal method for ghost imaging through turbulence of different strength levels. 2. Theoretical Analysis
In order to study the properties of ghost images formed through turbulent atmosphere, we consider the schematic setup depicted in the Klyshko unfolded version [28] shown in Fig. 1. A type-II phase-matched, collinear, SPDC source emits orthogonally polarized signal and idler beams which are separated by a polarizing beam splitter to act as the signal and reference beams. The reference beam with central wavelength λr is spatially resolved by a CCD camera D2 on the image plane after direct free space propagation. The signal beam with central wavelength λs passes through a convex lens of focal length f and radius ρD, and it illuminates an object of rough surface located Lo meters away from the lens. The signal beam reflected back is then collected by a bucket detector D1 situated Lb meters away from the object. Because the optical path lengths of the two beams are different, a postdetection electronic time delay is introduced to maximize the temporal cross
Fig. 1. Klyshko unfolded version of the schematic standoff twocolor quantum ghost imaging setup. A SPDC source generates nondegenerate signal and reference beams with central wavelengths λs and λr , respectively. d1 , d2 , Lo , and Lb are distances from source to lens, source to CCD camera D2 , lens to object, and object to bucket detector D1, respectively. ρc , ρf , ρt , and ρb are coordinates on the CCD plane, lens plane, object plane, and bucket detector plane. There exists turbulence in the lens-to-object and objectto-bucket-detector paths. 3036
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correlation. Then the electric signals driven by the signal and reference fields are sent to a correlation circuit to form the ghost image of the object. According to Ref. [28], when the distance d2 from the SPDC source to the CCD is far longer than d1 to the lens, the resolution will mainly depend on λs and degrade monotonically as λs gets larger. But this is not appropriate for standoff sensing, as we should send a long-wavelength signal beam to reduce atmospheric absorption; so, to avoid this condition, we set d2 to be far shorter than d1 . Then the space between the beam splitter and SPDC crystal must be so small that we can neglect it. Lo is generally far longer than d1 and d2 (i.e., Lo ≫ d1 ≫ d2 ). The distances satisfy the modified Gaussian thin-lens equation 1∕d2 λr ∕λs d1 1∕Lo 1∕f . We assume that turbulence only exists in the lens-to-object and objectto-bucket-detector paths. We represent the optical fields of the signal and reference beams at the output of the SPDC crystal ˆ s ρs ; te−iωs t as positive-frequency field operators E p −iωr t ˆ and Er ρr ; te , which are of photon∕m2 s units [23,29,30]. Here, ρs and ρr are the transverse coordinates corresponding to the optical axis. Then, the beams incident on the CCD and object surfaces can be represented by extended Huygens–Fresnel integral [20,23,24,27] as ˆ c ρc ; t1 E
Z
ˆ r ρr ; t1 − d2 ∕cHkr ; d2 ; ρc ; ρr ; (1) dρr E
ˆ t ρt ; t2 E Z Z ˆ s ρs ; t2 − d1 Lo ∕cHks ; d1 ; ρf ; ρs dρs dρf E × Lρf Hks ; Lo ; ρt ; ρf eψ o ρt ;ρf ;
(2) 0
2
where Hk; l; ρ0 ; ρ k∕i2πleikljρ −ρj ∕2l is the freespace propagator, Lρf exp−iks ρ2f ∕2f ΘρD − jρf j is the transmission function of the lens, and Θρ is the unit step function. The turbulence is taken to be of the Kolmogorov spectrum and uniformly distributed along the lens-to-object and objectto-bucket-detector paths (i.e., o path and b path). 0 It is depicted by eψ m ρ ;ρ, where the complex phase ψ m ρ0 ; ρ represents log-amplitude and phase fluctuations imposed on a spherical wave propagating from ρ to ρ0 on path m by the turbulence. The mutual co0 herence function of eψ m ρ ;ρ taken as the square-law approximation to the rigorous 5/3-law behavior [20,23,24,27,31,32] is
heψ m ρj1 ;ρk1 eψ m ρj2 ;ρk2 i 2 ρ
e−jρj1 −ρj2 j
j1 −ρj2 ·ρk1 −ρk2 jρk1 −ρk2 j
2 ∕2ρ2 m
;
(3)
where ρm is the turbulence coherence length on path m, given by ρm 1.09k2s C2n;m Lm −3∕5 in terms of the refractive index structure parameter, C2n;m ,
describing the turbulence strength and m; j; k o; t; f or m; j; k b; b; t. The subscripts 1 and 2 represent two arbitrary coordinates on the corresponding plane. The quadratic approximation is employed here to simplify the analysis and obtain an analytical formula. The analysis procedure is also valid if we employ the mutual coherence function of other formats, but it would be difficult to obtain an analytical formula. Though the specific form of the result for other formats will be different from here, the conclusion may be similar. A real-world target which has a sufficiently rough surface can be modeled as a planar two-dimensional object with random microscopic transverse and height variations. The random field-reflection coefficient Tρt of the object, which is a zero-mean complex-valued Gaussian random process, is characterized by the autocorrelation function [23,27] hT ρ0t Tρt i λ2s Oρt δρt − ρ0t :
(4)
Here, Oρt is the average intensity-reflection coefficient we’d like to image.
ˆ p i hCρ
1 TI
Z
T I ∕2 −T I ∕2
q2 η2 Ap
dtˆip t−τd ˆib t
Z
Z dτ1
Z dτ2
Ab
dρb hc t−τd −τ1 hb t−τ2
ˆ 0† ρb ;τ2 E ˆ c ρp ;τ1 E ˆ 0T ρb ;τ2 i; ˆ †c ρp ;τ1 E ×hE T
(5)
where T I is the correlation integration time, τd is the postdetection electronic delay time, q is the electron charge, η is the quantum efficiency taken to be the same for both the bucket detector and CCD camera, Ap is the small area of the pth pixel, Ab is the active region of the bucket detector, hc t and hb t are the impulse responses of the detectors’ output circuit, ˆ 0T ρb ; t represents the field impinging on the and E bucket detector after being reflected by the object. To perform the ensemble average calculations below, we invoke the statistical independence of the target’s reflection coefficient, the signal and reference fields at the source, and the turbulence on two propagation paths. Back-propagating the object-reflected field operator to the object, we can find [23]
ˆ 0† ρb ; τ2 E ˆ c ρp ; τ1 E ˆ 0T ρb ; τ2 i ˆ †c ρp ; τ1 E hE T Z Z 0 dρ0t dρt hT ρ0t Tρt iheψ b ρb ;ρt eψ b ρb ;ρt iH ks ; Lb ; ρb ; ρ0t Hks ; Lb ; ρb ; ρt ˆ † ρ0 ; τ2 − Lb ∕cE ˆ c ρp ; τ1 E ˆ t ρt ; τ2 − Lb ∕ci ˆ †c ρp ; τ1 E × hE t t Z 1 ˆ †c ρp ; τ1 E ˆ † ρt ; τ2 − Lb ∕cE ˆ c ρp ; τ1 E ˆ t ρt ; τ2 − Lb ∕ci: 2 dρt Oρt hE t Lb
(6)
This shows that turbulence on the object-to-bucket-detector path does not affect the ghost image’s properties. The ghost imaging is only influenced by the turbulence on the lens-to-object path. ˆ p i, becomes After substituting Eqs. (1), (2), and (6) into Eq. (5), the correlation, hCρ ˆ p i q2 η2 Ap hCρ Z
Z dρf
Ab L2b
Z
Z dτ1
Z dτ2
Z
Z dρt hc t − τd − τ1 hb t − τ2 Oρt
dρr
dρ0r
Z
Z dρs
dρ0s
ˆ †r ρr ; τ1 − τr E ˆ †s ρs ; τ2 − τs E ˆ r ρ0r ; τ1 − τr E ˆ s ρ0s ; τ2 − τs i dρ0f hE
× H kr ; d2 ; ρp ; ρr Hkr ; d2 ; ρp ; ρ0r H ks ; d1 ; ρf ; ρs Hks ; d1 ; ρ0f ; ρ0s L ρf 0
× Lρ0f H ks ; Lo ; ρt ; ρf Hks ; Lo ; ρt ; ρ0f heψ o ρt ;ρf eψ o ρt ;ρf i;
The ghost image at the center coordinate ρp of the CCD’s pth pixel can be recovered by the ensemble average cross-correlation function of the photocurrents ˆip and ˆib from the pixel and the bucket detector, respectively. In practice, it can be estimated by measuring the time average of the photocurrents, which is [23,29,30]
(7)
where τr and τs are the light propagation time of the reference and signal beams, i.e., d2 ∕c and Lo Lb d1 ∕c, respectively. We assume the signal and reference beams generated by the SPDC source are in zero-mean jointly Gaussian states, which are completely characterized by their nonzero autocorrelation and cross-correlation 10 May 2014 / Vol. 53, No. 14 / APPLIED OPTICS
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functions. Hence, the fourth-order field moment can be expressed in terms of second-order moments by moment-factoring theorem [23,29]
ˆ p i hCρ
ˆ †s ρs ;τ2 −τs E ˆ r ρ0r ;τ1 −τr E ˆ s ρ0s ;τ2 −τs i ˆ †r ρr ;τ1 −τr E hE ˆ †r ρr ;τ1 −τr E ˆ †s ρs ;τ2 −τs ihE ˆ r ρ0r ;τ1 −τr E ˆ s ρ0s ;τ2 −τs i hE ˆ †r ρr ;τ1 −τr E ˆ r ρ0r ;τ1 −τr ihE ˆ †s ρs ;τ2 −τs E ˆ s ρ0s ;τ2 −τs i hE ˆ †r ρr ;τ1 −τr E ˆ s ρ0s ;τ2 −τs ihE ˆ †s ρs ;τ2 −τs E ˆ r ρ0r ;τ1 −τr i: hE (8) The autocorrelation will be assumed to have the Gaussian–Schell model form [23,29,30] ˆ K ρ2 ; t2 i ˆ † ρ1 ; t1 E hE K
2P − e πa20
jρ1 j2 jρ2 j2 a2 0
−
e
jρ1 −ρ2 j2 2ρ2 0
−
e
t1 −t2 2 2T 2 0
(9) for K s, r, where P is the photon flux, a0 is the source’s e−2 intensity radius, ρ0 is the coherence length, and T 0 is its coherence time. For a SPDC source running at high flux, the signal and reference beams are in a maximally entangled jointly Gaussian state with no phase-insensitive cross correlation, and the maximum phase-sensitive cross correlation permitted by quantum theory given their autocorrelation functions, namely [23,29,30] ˆ †s ρ1 ; t1 Er ρ2 ; t2 i 0; hE
− ˆ s ρ1 ; t1 Er ρ2 ; t2 i 2P e hE 2 πa0
jρ1 j2 jρ2 j2 a2 0
−
e
t1 −t2 2 2T 2 0
−
2μεγ 2 β2 2 A B − ; θφ α2 γ 2 Δ2PSF h i D−ε μεγ 2 β2 C−ε μεγ 2 β2 iμξγβ αiγ − B − − A α−iγ α2 γ 2 α2 γ 2 m 2μεγ 2 β2 φ A B − α2 γ2 1
1 Δ2FOV
− ·
e
APPLIED OPTICS / Vol. 53, No. 14 / 10 May 2014
m2 2θμεγ 2 β2 2 μξ C D − 2ε − Δ2PSF α2 γ 2
μξγβ2 D − εC − ε α2 γ 2
− θB
μξγβ2 μξγβ2 2 D − ε − θA C − ε2 ; α iγ2 α − iγ2
α 1∕a20 1∕2ρ20 ; 1∕2ρ2D
iks ∕2f ;
:
δ
(11)
ξ ks ∕2Lo ;
In order to perform the integral more readily, we assume that (1) the finite size of the imaging lens depicted by the step function ΘρD − jρf j can be replaced with a Gaussian function of the form exp−ρ2f ∕2ρ2D [26], (2) the detectors have the same p 2 2 Gaussian impulse response ht ΩB ∕ 8π e−ΩB t ∕8 , whose bandwidth ΩB is much smaller than that of the incident light for the SPDC source (ΩB T 0 ≪ 1), (3) the correlation integration time is sufficiently long to capture a ghost image, so T I ≫ Ω−1 B [23,30], (4) the spatial coordinates are treated as onedimensional to simplify the calculation, while maintaining the validity of the study of image quality [20–22,27]. Combining Eqs. (8) and (10), we can see that the third term in Eq. (8) equals zero. After substituting the two remaining terms into Eq. (7) and performing the integral over spatial coordinates, the correlation becomes 3038
where
(10) jρ1 −ρ2 j2 2ρ2 0
Z P2 k4s k2r q2 η2 Ap Ab dτ1 16π 5 a40 d21 L2o L2b d22 Z Z dτ2 ht − τd − τ1 ht − τ2 dρt Oρt s τ1 − τr − τ2 − τs 2 θμ × exp − 2 2 T0 α γ2 mρt − ρp 2 ρ2t × exp − exp − 2 Δ2PSF ΔFOV r υ 2α − βγ 2 2 2 ξ ρt exp −υ δ δ κ κσ 2α − βφ2 ρ2p ; (12) − σ
β 1∕2ρ20 ; ε
φ kr ∕2d2 ;
C D δ ε iξ iγ A B α −
γ ks ∕2d1 ;
1∕2ρ2o ;
γ2 ; α iγ
β2 μDγ 2 β2 iφ; α iγ α iγ2
E F δ ε iξ iγ μ−1 CD − ε2 ; υ−1 EF − ε
α − iγγ 2 ; α2 γ 2 − β 2
θ−1 AB − βγ 2 2 α γ 2 − β2
κ α2 γ 2 − β2 ;
μ2 ε2 γ 4 β4 ; α2 γ 2 2
2
;
σ α2 φ2 − β2 :
It is clear that the first term leads to a ghost image of the object and the second term leads to a featureless background. Substituting the detectors’ response function and performing the integral over temporal variables, we can get
ˆ p i hCρ
Z P2 k4s k2r q2 η2 Ap Ab dρt Oρt 16π 5 a40 d21 L2o L2b d22 s ΩB T 0 τd − τs τr 2 θμ exp − 2 × 2 2 4 T 0 16∕ΩB α γ2 mρt − ρp 2 ρ2t exp − × exp − Δ2PSF Δ2FOV r υ 2α − βγ 2 2 2 ξ ρt exp −υ δ δ κ κσ 2α − βφ2 ρ2p : (13) − σ
The temporal cross correlation is maximized when the postdetection delay time τd equals τs –τr Lo Lb d1 –d2 ∕c, and then the ghost image is Z P2 k4s k2r q2 η2 Ap Ab dρt Oρt Gρp 16π 5 a40 d21 L2o L2b d22 s ( mρt −ρp 2 ΩB T 0 θμ ρ2t exp − exp − 4 α2 γ 2 Δ2PSF Δ2FOV r υ 2α−βγ 2 2 2 ξ ρt exp −υ δ δ κ κσ ) 2α−βφ2 ρ2p : (14) − σ The image is at a magnification of m, which is negative in our system. The width of the point-spread function describing the resolution measured in image space is ΔPSF and the field of view measured in object space is ΔFOV . Equation (14) is the main result we obtain, which determines all properties of the ghost imaging through turbulence. 3. Numerical Calculations and Analyses
To make the calculation closer to real applications, we set the system parameters as: a0 1 mm, ρ0 1 μm, d1 1 m, d2 0.05 m, Lo Lb 1000 m,
and ρD 20 cm; f is determined by the modified Gaussian thin-lens equation. First, we demonstrate the effect of turbulence on the ghost imaging of a reflective double slit. The double-slit object in which the slits reflect light and the part elsewhere absorbs is shown in Fig. 2(a). The slit width is 6 cm and slit separation is 12 cm. To show the imaging property in detail, the intensity reflection coefficient of the left slit is 1 and the right slit is 0.5. The wavelengths of the reference beam λr and signal beam λs are 532 and 1064 nm, respectively. The images without the background recovered by normalized correlation for five particular turbulence strength levels are shown in Fig. 2(b). Since the magnification factor m is negative, the left slit and right slit are reversed. The imaging quality is almost not influenced when the turbulence is weak (C2n;o 10−15 m−2∕3 ). A ghost image of good quality can be obtained even through turbulence of moderate strength (C2n;o 10−14 m−2∕3 ). The image quality will be degraded seriously only when the turbulence gets very strong. Next, we investigate the effect of turbulence on the magnification factor m. The magnification as a function of the reference beam wavelength λr with λs 1064 nm and that of the signal beam wavelength λs with λr 532 nm are plotted in Figs. 3(a) and 3(b), respectively, under different conditions of turbulence. It is clear that the magnification factor is hardly influenced by the turbulence and always equals −d2 λr ∕λs d1 ∕Lo, which is the magnification when there is no turbulence [28]. The magnification approximates to −d1 ∕Lo [−0.001 in our calculation; this can also be seen from the image obtained in Fig. 2(b)] because the condition Lo ≫ d1 ≫ d2 is satisfied. Finally, we study how the turbulence affects the resolution and field of view. The point-spread function ΔPSF and field of view ΔFOV versus λr when λs is fixed at 1064 nm are shown in Figs. 4(a) and 4(b), respectively, and the ΔPSF and ΔFOV versus λs when λr is set to be 532 nm are plotted in Figs. 4(c) and 4(d). As the point-spread function depicts the resolution of the image, it is explicit that the resolution is very
Fig. 2. (a) Object of a reflective double slit to be imaged; its slit width is 6 cm and slit separation is 12 cm. The reflectivities of the left and right slits are 1 and 0.5, respectively. (b) Normalized cross section ghost image through different levels of turbulence. The image is reversed as the magnification factor is negative. 10 May 2014 / Vol. 53, No. 14 / APPLIED OPTICS
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Fig. 3. Magnification factor as a function of (a) the reference beam wavelength λr with λs 1064 nm, and (b) the signal beam wavelength λs with λr 532 nm for different turbulence strength levels.
vulnerable to the turbulence. When the turbulence gets stronger, the resolution degrades seriously. However, the ΔPSF decreases monotonically as the wavelength of the reference beam λr gets shorter, and it also decreases when a longer wavelength
signal beam is taken to propagate through the turbulence atmosphere above moderate levels. If the turbulence is very weak, the resolution gets better when a signal beam of shorter wavelength is used to illuminate the object, as shown especially in
Fig. 4. (a) Point-spread function ΔPSF describing the resolution of the ghost image, and (b) field of view ΔFOV versus λr when λs is fixed at 1064 nm for different turbulence strengths. (c) ΔPSF and (d) ΔFOV versus λs while λr is set to be 532 nm. (e) Point-spread function and (f) field of view plotted corresponding to (c) and (d), respectively, especially when the turbulence is weak. 3040
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Fig. 4(e). A remarkable fact that can be seen from Figs. 4(c) and 4(e) is that the variation of resolution is very slow with the signal wavelength when the turbulence is weak. On the other hand, the atmospheric absorption of the shorter-wavelength signal is serious, so longer signal wavelength is preferable for propagation through the atmosphere. For strong turbulence, the variation of resolution is very fast with the signal wavelength [shown in Fig. 4(c)], and hence it is always better to choose a longer signal wavelength for higher resolution. As a consequence, no matter how strong or weak the turbulence is, we should always choose a longer wavelength signal beam for propagation through the atmosphere, and a shorter wavelength reference beam to get a high resolution. Nevertheless, the field of view is unaffected by the turbulence no matter when the reference beam wavelength varies, as shown in Fig. 4(b), or when the signal beam wavelength is changed, as depicted in Figs. 4(d) and 4(f). But the field of view diminishes when λr decreases or λs increases. A trade-off should be made, therefore, between the resolution and field of view if one would like to utilize a short λr or a long λs. 4. Conclusion
In summary, the performance of a standoff two-color quantum ghost imaging scheme in the presence of turbulent atmosphere is investigated. Based on the extended Huygens–Fresnel integral, modeling the beams as in zero-mean jointly Gaussian states, and using a quadratic approximation of the turbulence structure function, we derive a formula for the twocolor quantum ghost imaging configuration through turbulence. Numerical calculations are also given based on the formula. In this imaging configuration, the magnification factor and field of view are hardly affected by the turbulence. The resolution will be degraded with strong turbulence. However, the resolution can be improved by utilizing a shorter wavelength reference beam. If the turbulence is serious (C2n;o > 10−15 m−2∕3 ), the image resolution gets better the longer the signal beam wavelength is, but while the turbulence is weak (C2n;o < 10−15 m−2∕3 ), a shorter wavelength signal beam should be chosen to obtain a good resolution. Nevertheless, a tradeoff should be made between the resolution and field of view if one would like to utilize a short wavelength reference beam or a long wavelength signal beam. Considering the serious atmospheric absorption and that the variation of resolution is very slow with the signal wavelength when the turbulence is weak, we can always send a long wavelength signal beam to illuminate the object through turbulent atmosphere with a short wavelength reference beam to get a good resolution. Hence, this two-color ghost imaging configuration is very suitable for standoff sensing. This work is supported by the NSFC (under Grant Nos. 61176084, 11204367, and 11174282), the China Postdoctoral Science Foundation (funded project
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