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Unbalanced lensless ghost imaging with thermal light Lu Gao,1,3 Xiao-long Liu,1 Zhiyuan Zheng,1 and Kaige Wang2,* 2

1 School of Science, China University of Geosciences, Beijing 100083, China Department of Physics, Applied Optics Beijing Area Major Laboratory, Beijing Normal University, Beijing 100875, China 3 e-mail: [email protected] *Corresponding author: [email protected]

Received December 19, 2013; accepted February 18, 2014; posted March 3, 2014 (Doc. ID 203190); published March 31, 2014 Lensless ghost imaging (LGI) with thermal light requires an equal length between the test and reference arms. When this condition is not met, the image becomes blurred. Here we propose an experimental scheme of LGI where the lengths of the two arms are not equal. Our experiment shows that when a glass rod is inserted into a longer arm, the clear image can be formed in the intensity correlation measurement of the two arms. The theoretical analysis can well explain the experimental results. The unbalanced LGI may provide an alternative scheme in practical application. © 2014 Optical Society of America OCIS codes: (110.1650) Coherence imaging; (270.0270) Quantum optics. http://dx.doi.org/10.1364/JOSAA.31.000886

1. INTRODUCTION Thermal light can mimic a two-photon entangled state in quantum imaging, such as ghost imaging, ghost interference, and subwavelength interference etc. [1–21]. Due to the different features in the second-order field spatial correlation between the thermal light and entangled light [6], these quantum imaging phenomena are not exactly same. For example, thermal light can perform ghost imaging without using a lens while entangled light cannot. A lensless ghost imaging (LGI) scheme was first proposed by Cao et al. [8], who pointed out that “for the classical correlation source, a real correlated image can be formed through the beam splitter (BS) without using a lens … and object and real imaging have an equal distance from the source.” They further explained the effect by “… the classical thermal source behaves as a phase conjugate mirror which reflects an object to itself.” The phase conjugate mirror is naturally related to a balanced system. In order to obtain a perfect in-focus image, the test arm and the reference arm must be in balance; that is, the distance from thermal light source to scanning detector in the reference arm is equal to that from source to object in the test arm. LGI with thermal light was soon demonstrated and further discussed in experiments [17,18,20]. These experimental studies concerned a debate on the nature of spatial correlation in thermal light. However, the fact is that a ghost image is infocus when two arms are in balance and blurred when they are unbalanced. The authors in [20] and [21] introduced the longitudinal correlation length [22] in LGI, which can describe the out-of-focus case. Furthermore, optical coherence tomography (OCT) based on the intensity correlation of thermal light has been proposed, where the concept of the imaging longitudinal coherence length is introduced to evaluate the longitudinal resolution of the OCT system [23]. In this paper, we report an experimental observation of unbalanced LGI with thermal light, in which the in-focus image can be formed when the two arms have a certain length difference. In the unbalanced scheme of LGI, where a blurred 1084-7529/14/040886-05$15.00/0

image appears, a glass rod is inserted into a long arm and an in-focus ghost image can be recovered immediately, although the optical path difference between two arms is increased. Our experimental scenario can be used in some special situation where the two arms cannot have the same length. It also provides a method of recovering the blurred imaging without moving the detector in the reference arm; thus it can be applied, for example, to a recent proposal of thermal light OCT [23].

2. EXPERIMENT AND RESULTS The experimental setups are sketched in Fig. 1, where (a) is the balanced LGI scheme; (b) and (c) are the unbalanced LGI schemes with the same geometry, but in (c) a K95 glass rod is inserted in the longer arm. The glass rod is machined and well polished with the tolerances in alignment and length less than 0.1 mm. The pseudothermal light source is obtained by passing a He-Ne laser beam of wavelength 632.8 nm through a slowly rotating ground glass disk, G. P 1 and P 2 are two polarizers for modulating light intensity. N is a lens that expands the laser beam illuminating on G. P 3 is a diaphragm whose dimension is adjustable to control the transverse size of the thermal light source. P 3 has a distance of 4.5 cm from G. By setting the diameter 1.5 mm of P 3 , the average size of the speckles in the object plane is about 166 μm. T is the mask object of the acronym CUGB (China University of Geosciences, Beijing), and the line width of the characters is about 250 μm, which is larger than the speckle size for the imaging resolution. The quasi-thermal light beam is separated into two daughter beams by a 50∕50 nonpolarizing BS at a distance of 6 cm from P 3 . Here the pseudothermal light beams in the two arms are both completely polarized, since it can reach the best visibility in ghost imaging [24]. The transmitted beam propagates freely in the reference arm, while the reflected beam impinges on the object in the test arm. D1 and D2 are charge-coupled devices (CCD), where D1 is close to object in the test arm and acts as a bucket detector. © 2014 Optical Society of America

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(a)

d1

d 2=d 1 (b)

d1

(c)

d2

d1

d2

L

Fig. 1. Experimental setups of LGI schemes. (a) Lengths of the two arms are equal. (b) Lengths of the two arms are not equal. (c) Lengths of the two arms are not equal, and a glass rod is inserted into the longer arm. M, reflective mirror; P 1 and P 2 , polarizers; P 3 , diaphragm; G, ground glass; N, lens; BS, 50∕50 nonpolarizing beam splitter; D1 and D2 , CCDs; T, transmissive object; L, K95 cylindrical glass rod; C, correlation computation.

In Fig. 1(a), the distances from the source G to both D1 and D2 are d1  35.5 cm. In Fig. 1(b), D2 is moved to a distance of d2  42.3 cm from G while D1 remains unchanged. However, in Fig. 1(c) the glass rod with the refractive index 1.51 has the length of l  20 cm and the diameter of 3 cm in the reference arm. Both refractive index and length of the transmission medium must be properly chosen to keep the same effective diffraction distance for the two arms. The theoretical explanations will be given in Section 3. The normalized intensity correlation function between the two detectors is defined as g2 x1 ; x2   hI 1 x1 I 2 x2 i∕ hI 1 x1 ihI 2 x2 i, where I 1 x1  and I 2 x2  are the intensities detected by D1 and D2 at positions x1 and x2 across the beams, respectively. Fixing x1  0 and scanning x2 , we obtain the normalized second-order field correlation function, i.e., Hanbury–Brown–Twiss (HBT) curve, and the corresponding 2D contour. The HBT curves and contours of the three schemes in Figs. 1(a), 1(b), and 1(c) are shown in Figs. 2(a), 2(b), and 2(c), respectively. Comparing Figs. 2(b) with 2(a), we see clearly that the sharp HBT curve becomes blunt when the balance of the two arms is broken. However, when the glass rod is inserted into the longer arm in the scheme of Fig. 1(b), the sharp HBT curve is recovered in Fig. 2(c). We now observe the speckle patterns of the two arms. For the scheme of Fig. 1(a), Figs. 3(a) and 3(b) present the speckle patterns detected by D1 in the object arm and D2 in the reference arm, respectively. The two patterns are symmetrically correlated. When the reference arm prolongs in Fig. 1(b),

Fig. 2. Experimental results of HBT curves and contours of the normalized second-order field correlation functions by scanning detector D2 and fixing detector D1 . The circles are the experimental results, and the solid lines are the theoretical simulation. (a), (b), and (c) are obtained in the experimental setups of Figs. 1(a), 1(b), and 1(c), respectively.

the corresponding speckle pattern is recorded in Fig. 3(c), and the correlation with that in Fig. 3(a) becomes unclear. Finally, the symmetric correlation can be recovered in Fig. 3(d) when the glass rod is set. LGI is obtained by the intensity correlation measurement of the bucket intensity I 1 in the test arm and the scanning intensity I 2 x2  in the reference arm. The experimental results are shown in Figs. 4(a)–4(c), corresponding to the schemes of Figs. 1(a)–1(c), respectively. The image of the acronym CUGB is in-focus for the balanced LGI of Fig. 1(a) and blurred when the reference arm is extended by 6.8 cm in Fig. 1(b). The infocus image is recovered after the glass rod has been inserted in the reference arm in Fig. 1(c).

Fig. 3. Speckle patterns observed in the experiments. (a) Speckle pattern recorded by D1 in the object arm; (b), (c), and (d) are speckle patterns recorded by D2 in the reference arm for the experimental setups of Figs. 1(a), 1(b), and 1(c), respectively.

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~ 0 − x0   S 0 δx0 − x0 , where S 0 describes the source Sx 0 0 intensity. Let Tx be the transmissive function of the object. The impulse response functions of the object and reference arms are, respectively, given by [7,8] s   k iπ x − x0 2 Tx1  exp −  ikd1  ik 1 h1 x1 ; x0   2πd1 4 2d1 (3) and h2 x2 ; x00 

s   x − x00 2 k iπ :  exp −  ikd2  ik 2 2πd2 4 2d2

(4)

Substituting Eqs. (3) and (4) into Eq. (2), we obtain hI 1 x1 ihI 2 x2 i 

S 20 k2 jTx1 j2 4π 2 d1 d2

(5)

and hE1 x1 E2 x2 i 

S k p0 expikd2 − d1 T  x1  2π d1 d2   Z ikx2 − x0 2 ikx1 − x0 2 × exp dx0 : (6) − 2d2 2d1

Equation (6) can be further calculated as s   k iπ  T x1  exp ikd2 − d1   2πd1 − d2  4   ikx1 − x2 2 × exp ; (7) 2d2 − d1 

hE 1 x1 E 2 x2 i  S 0

Fig. 4. Experimental results of ghost imaging for the object of acronym CUGB: (a), (b), and (c) are the images in the experimental setups of Figs. 1(a), 1(b), and 1(c), respectively. Statistical averages are taken over 10,000 frames.

3. THEORETICAL ANALYSIS The intensity spatial correlation of thermal light satisfies [5,6] hI 1 x1 I 2 x2 i  hE1 x1 E 2 x2 E 2 x2 E1 x1 i  hI 1 x1 ihI 2 x2 i  jhE 1 x1 E2 x2 ij2 ;

(1)

where E j xj  is the field amplitude for beam j and I j  E j E j . Here, the proportionality factor to relate the intensity and electric field is set as 1. The average intensity and the firstorder field correlation function can be written as hEi xi E j xj i  i; j  1; 2;

Z

~ 0 − x0 dx0 dx0 ; hi xi ; x00 hj xj ; x0 Sx 0 0

when d2 ≠ d1 and hE 1 x1 E2 x2 i  S 0 T  x1 δx1 − x2  when d2  d1 :

For LGI,R the bucket detection is applied to the object arm; Rthat is, hI 1 x1 I 2 x2 idx1 . Hence the integration of Eq. (5), hI 1 x1 ihI 2 x2 idx1 , shows a homogeneous pattern. For the unbalanced scheme using Eq. (7), the integration of the second term of Eq. (1) reads Z

jhE1 x1 E 2 x2 ij2 dx1 

where hj xj ; x0  j  1; 2 is the impulse response function of ~ 0 − x0  is the first-order field the field propagation, and Sx 0 correlation function of the source. In the broadband limit,

S 20 k 2πjd1 − d2 j

Z

jTx1 j2 dx1  Const: (9)

There is no image formed in the unbalanced case. For the standard LGI, d1  d2 , with Eq. (8) the bucket detection shows an in-focus image: Z

(2)

(8)

jhE 1 x1 E2 x2 ij2 dx1  S 20 jTx2 j2 :

(10)

We now consider the unbalanced scheme of Fig. 1(c), where the reference arm consists of two propagation media. The impulse response function of the reference arm can be written as

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h2 x2 ; x00  

Z

1 0 h2 2c x2 ; xh2c x; x0 dx;

(11)

h2 2c x2 ; x

s k  2πd2 − l   x − x00 2 iπ ; × exp −  ikd2 − l  ik 4 2d2 − l

r   nk iπ x − x2 exp −  inkl  ink 2  ; 2 πl 4 2l

(12)

(13)

where n and l are the refractive index and length of the medium, respectively. With Eqs. (11)–(13), we arrive at h2 x2 ;x00  

ACKNOWLEDGMENTS This work was supported by the Beijing Natural Science Foundation (4133086). K. Wang acknowledges financial support by the National Natural Science Foundation of China, project no. 11174038, and the National High Technology Research and Development Program of China, project grant no. 2013AA122902.

where 0 h1 2c x; x0 

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s   ikx2 − x00 2 k iπ ; exp −  ikd2  n − 1l  4 2π d¯ 2 2d¯ 2 (14)

where d¯ 2  d2 − n − 1∕nl is defined as the effective diffraction length. The effective diffraction length is the equivalent Fresnel diffraction length when the optical field travels in a vacuum [25]. Obviously, Eq. (14) has the form comparable to Eq. (4). As a result, if d1  d¯ 2 , the first-order field correlation function is the same as Eq. (8). The in-focus ghost image is realized, although the geometric lengths of the two arms are not equal, i.e., d1 ≠ d2 . In our experiment of Fig. 1(c), the effective diffraction length of the reference arm is d¯ 2  42.3 − 1.51–1.0∕1.51 × 20  35.5 cm, equal to the length of the object arm d1  35.5 cm. So the in-focus ghost image of Fig. 4(c) can be recovered as Fig. 4(a). The position of the glass rod will not affect the image recovery, and the glass rod can be set at any position along the reference arm.

4. DISCUSSION AND CONCLUSION When monochromatic light propagates in more than one medium, the effective diffraction length of light may be different from the real geometric length. In balanced and unbalanced geometries of LGI, the in-focus ghost image exists only when the effective diffraction lengths of the object and reference arms are equal. In general, the refractive index of the medium is larger than 1, so the effective diffraction length is less than the real geometric length; therefore the medium should be placed in the longer arm. It should be noted that, in the unbalanced LGI, an optical path difference between the two arms has been introduced, although their effective diffraction lengths are equal. When the path difference between the two arms exceeds the longitudinal coherence length of the beam, the spatial correlation disappears and ghost imaging will never exist [22]. Although ghost imaging with thermal light in the unbalanced geometry can be implemented with the help of a lens, unbalanced LGI may provide an alternative scheme in practical applications. Keeping “lensless” strategy in ghost imaging, an adjustable medium can play a role similar to that of a focus lens, avoiding any lens defects, such as aberrations.

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Unbalanced lensless ghost imaging with thermal light.

Lensless ghost imaging (LGI) with thermal light requires an equal length between the test and reference arms. When this condition is not met, the imag...
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