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High visibility temporal ghost imaging with classical light
Optics Communications 410 (2018) 824–829
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
High visibility temporal ghost imaging with classical light Jianbin Liu a , Jingjing Wang a , Hui Chen a , Huaibin Zheng a , Yanyan Liu c , Yu Zhou b, *, Fu-li Li b , Zhuo Xu a a
Electronic Materials Research Laboratory, Key Laboratory of the Ministry of Education, and International Center for Dielectric Research, Xi’an Jiaotong University, Xi’an 710049, China b MOE Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter, Department of Applied Physics, Xi’an Jiaotong University, Xi’an 710049, China c Science and Technology on Electro-Optical Information Security Control Laboratory, China
a r t i c l e
i n f o
Keywords: Ghost imaging Two-photon bunching Superbunching pseudothermal light Photon statistics Optical coherence theory
a b s t r a c t High visibility temporal ghost imaging with classical light is possible when superbunching pseudothermal light is employed. In the numerical simulation, the visibility of temporal ghost imaging with pseudothermal light, equaling (4.7 ± 0.2)%, can be increased to (75 ± 8)% in the same scheme with superbunching pseudothermal light. The reasons for that the retrieved images are different for superbunching pseudothermal light with different values of degree of second-order coherence are discussed in detail. It is concluded that high visibility and high quality temporal ghost image can be obtained by collecting sufficient number of data points. The results are helpful to understand the difference between ghost imaging with classical light and entangled photon pairs. The superbunching pseudothermal light can be employed to improve the image quality in ghost imaging applications. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Ghost imaging is a new imaging technique that is based on the second- or higher-order correlation of light [1]. In a typical ghost imaging scheme, there are two light beams, which are usually emitted by the same light source. One is signal beam and is incident onto an object. The transmitted or reflected light from the object is collected by a bucket detector without position information. The other beam is reference beam, which does not interact with the object and is detected by a scannable detector with position information. The image of the object cannot be obtained by either signal from these two detectors. Because the image information is collected by the bucket detector without position resolution and the position resolution of the reference detector is able to resolve the object, while the object does not interact with the reference light beam. However, the image of the object can be retrieved by correlating the signals from these two detectors. This peculiar property is the reason why the imaging technique is named after ghost imaging [1]. The developing history of ghost imaging is a typical example of how scientific research advances itself by the efforts of people with different opinions [2–20]. Ghost imaging was first performed with entangled photon pairs [2,3]. It was then thought that entangled photon
pairs was necessary for ghost imaging [4]. However, Bennink et al. employed classical correlated laser light beams to realize coincidence imaging, which is similar to the scheme of ghost imaging [5]. It was argued that the coincidence imaging performed by Bennink et al. was different from ghost imaging [6,7]. Shortly after the work of Bennink et al., several groups proved that ghost imaging can be performed with thermal light [8–11]. Since then, the discussion about the physics of ghost imaging with thermal light never stops [12–20]. Scarcelli et al. suggested that ghost imaging with thermal light might have to be described quantum mechanically [12]. Gatti et al. disagreed with the conclusion and claimed that classical theory could also be employed to interpret thermal light ghost imaging [13,14]. Shapiro proposed a new type of ghost imaging called computational ghost imaging and pointed out that the nature of ghost imaging is classical [15]. On the other hand, Rogy et al. employed quantum discord to study ghost imaging and concluded that there is quantum correlation in thermal light ghost imaging [16]. Although the debate about the physics of thermal light ghost imaging still goes on [1,17–20], it is well accepted that ghost imaging with thermal light can mimic all the behaviors of ghost imaging with entangled photon pairs except the visibility of the former is much lower than the latter [8]. The reason is that the degree of second-order coherence, 𝑔 (2) (0), equals 2 for thermal light; while it can be much
* Corresponding author.
E-mail address: [email protected] (Y. Zhou). https://doi.org/10.1016/j.optcom.2017.11.054 Received 18 August 2017; Received in revised form 16 November 2017; Accepted 18 November 2017 0030-4018/© 2017 Elsevier B.V. All rights reserved.
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Optics Communications 410 (2018) 824–829
larger than 2 for entangled photon pairs [21]. If there is one type of classical light with large value of 𝑔 (2) (0), high visibility ghost imaging with classical light should also be possible. Different methods, such as differential ghost imaging [22], normalized ghost imaging [23], correspondence ghost imaging [24,25], highorder ghost imaging [26–28], etc., have been employed to improve the visibility of ghost imaging with classical light. However, all the existing methods employ mathematical algorithms to improve the visibility, which require extra calculations and is different from the one in ghost imaging with entangled photon pairs [22–27]. Recently, we have proposed a new type of classical light called superbunching pseudothermal light, in which 𝑔 (2) (0) can be much larger than 2 [29,30]. With the help of superbunching pseudothermal light, high visibility temporal ghost imaging can be implemented with normal correlation algorithm similar to that in ghost imaging with entangled photon pairs [3]. Our results show that high visibility ghost imaging can also be realized with classical light, which should be one important step in the development of ghost imaging and be helpful to understand the physics of ghost imaging. Most ghost imaging experiments were in the spatial domain [2,3,8– 16,22–28]. Based on the space–time duality [31–33], temporal ghost imaging should also be possible. Shirai et al. first proved that temporal ghost imaging with classical pulsed light can be realized by analogy of spatial ghost imaging with thermal light [34]. It was soon realized that temporal ghost imaging can also be performed with entangled photon pairs [35,36], chaotic laser light [37,38], and chaotic light [39–41]. Recently, computational temporal ghost imaging was also experimentally implemented [42]. There are also two light beams in a typical temporal ghost imaging scheme [34–41]. The signal beam is modulated by a temporal object and then is detected by a slow detector, which cannot follow the variation of the signal. The role of this detector is the same as that of bucket detector in spatial ghost imaging. The reference beam is directly detected by a fast detector to record the variation of light intensity. The role of the detector is the same as the one of scannable detector in spatial ghost imaging. The image of the temporal object cannot be obtained by either signal from these two detectors. However, the image can be retrieved by correlating the signals from these two detectors [38], just like ghost imaging in spatial domain [1]. The rest of the paper is organized as follows. Theoretical and numerical study of high visibility temporal ghost imaging with superbunching pseudothermal light is in Section 2. The discussions about the reasons why the visibility of temporal ghost imaging can be increased by changing the light source and the difference in the retrieved images when there are different number of RGs in superbunching pseudothermal light source are in Section 3. Our conclusions are in Section 4.
Fig. 1. Temporal ghost imaging scheme. IM: intensity modulator. BS: non-polarizing 50 ∶ 50 beam splitter. O: temporal object. D1 : temporal bucket detector, which integrates the intensity within one period. D2 : reference detector with high temporal resolution. CC: second-order correlation measurement system.
Fig. 2. Process of temporal ghost imaging within one period. The horizontal and vertical axes are time (𝑡) and intensity of light (𝐼), respectively. S: signal beam. O: temporal object. R: reference beam. 𝑇 : period length of temporal object. 𝑆 ⋅ 𝑂: the product of signal light beam and temporal object, which corresponds to modulating the temporal object with the signal light beam. Sum(𝑆 ⋅𝑂) is the output of the bucket detector within one period, which is a non-negative real number. Sum(𝑆 ⋅𝑂)×𝑅 represents the intensity of the reference beam multiplied by the output of the bucket detector.
temporal ghost imaging, D2 is scanned longitudinally to measure the second-order temporal correlation between the reference and signal light beams. D1 and D2 are in transversely symmetrical positions in temporal ghost imaging so that the light intensity fluctuations received by these two detectors are identical. The process of temporal ghost imaging within one period is described in Fig. 2. S and R represent signal and reference light beams, respectively. The intensity fluctuations of these two light beams are identical. O is a temporal object, which is a periodic double-slit in our scheme. The period of the temporal object is 𝑇 , which is shown in the left panel of Fig. 2. 𝑆 ⋅ 𝑂 means modulating the temporal object with the signal light beam. In the modulating process, the variation of the signal beam should be faster than the temporal object. Otherwise, the information of the object may be lost [34]. The modulating process is shown in the middle panel of Fig. 2. Sum(𝑆 ⋅ 𝑂) is the output of the bucket detector, D1 , which integrates the modulated signals within a period. The second-order correlation between signal and reference light beams is implemented via Sum(𝑆 ⋅ 𝑂) × 𝑅. The image of the temporal object can be retrieved by correlating the outputs of D1 and D2 and repeating the process in Fig. 2 many times [34–40]. The temporal ghost imaging with classical light in Fig. 2 can be mathematically expressed as
2. Temporal ghost imaging with superbunching pseudothermal light 2.1. Theoretical study The scheme of temporal ghost imaging with superbunching pseudothermal light is shown in Fig. 1, which is similar to that of spatial ghost imaging with thermal light [12]. IM is an intensity modulator, which is employed to modulate the intensity of the incident light. It is a combination of rotating ground glasses (RGs), pinholes, lenses, etc., in superbunching pseudothermal light source [29,30]. BS is a nonpolarizing 50 ∶ 50 beam splitter. O is a temporal object, which can be simulated by an electro-optic modulator [38]. D1 is a slow detector, which cannot resolve the temporal object by itself. D2 is a fast detector, which can follow the modulation of light intensity after IM. CC is a second-order correlation measurement system. The distance between the light source and the temporal object planes is equal to the one between the light source and D2 planes. The difference between spatial and temporal ghost imaging schemes is as follows. In spatial ghost imaging, D2 is scanned transversely to measure the second-order spatial correlation between the reference and signal light beams. While in
𝐺𝑇(2) (𝑡2 ) = ⟨𝐼(𝑡2 )
𝑇
∫0
𝐼(𝑡1 )𝑂(𝑡1 )𝑑𝑡1 ⟩,
(1)
where 𝐼(𝑡2 ) is the intensity of the reference light beam recorded by D2 , 𝐼(𝑡1 ) is the intensity of the signal light beam, 𝑂(𝑡1 ) is the temporal 𝑇 object, and ∫0 𝐼(𝑡1 )𝑂(𝑡1 )𝑑𝑡1 is the output of D1 within one period. ⟨⋯ ⟩ is ensemble average, which is equivalent to a long time average for an 825
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ergodic and stationary process [43]. The temporal object remains the same in different periods. Hence Eq. (1) can be re-arranged as 𝐺𝑇(2) (𝑡2 ) =
𝑇
⟨𝐼(𝑡2 )𝐼(𝑡1 )⟩𝑂(𝑡1 )𝑑𝑡1 ,
∫0
(2)
where ⟨𝐼(𝑡2 )𝐼(𝑡1 )⟩ is the second-order correlation function [43]. For a stationary process, the second-order correlation function is only dependent on the time difference between these two intensities [43]. Eq. (2) can be simplified as 𝐺𝑇(2) (𝑡2 ) =
𝑇
𝐺(2) (𝑡2 − 𝑡1 )𝑂(𝑡1 )𝑑𝑡1 ,
∫0
(3)
where 𝐺(2) (𝑡2 − 𝑡1 ) is the second-order correlation function [44,45]. Eq. (3) is similar to that of lensless spatial ghost imaging with thermal light [1,12] and is consistent with the results of temporal ghost imaging with classical light [34,37]. If the superbunching pseudothermal light proposed in [29,30] is employed in temporal ghost imaging, Eq. (3) can be expressed as 𝐺𝑇(2) (𝑡2 ) ∝
𝑁 𝑇 ∏
∫0
[1 + sinc2
𝛥𝜔𝑗 (𝑡2 − 𝑡1 )
𝑗=1
2
]𝑂(𝑡1 )𝑑𝑡1 ,
(4)
Fig. 3. Simulated temporal ghost imaging with superbunching pseudothermal light. 𝑔𝑇(2) (𝑡) is the normalized temporal ghost imaging, which is normalized according to the constant background in Eq. (4). 𝑂(𝑡) is the temporal object. 𝑡 is time and 𝑎𝑟𝑏.𝑢. is short for arbitrary unit of time. The black squares are simulated results and the red lines are theoretical results. 𝑉 is visibility and 𝑁 is the number of RGs in superbunching pseudothermal light source. (a)–(f) are the retrieved images of the temporal object in (g) for different number of RGs in superbunching pseudothermal light source. (h) is the visibility of temporal ghost images in (a)–(f), in which the hollow black squares and red product signs are simulated and theoretical results, respectively.
where 𝑁 is the number of RGs in superbunching pseudothermal light source, 𝛥𝜔𝑗 is the frequency bandwidth of pseudothermal light scattered by the 𝑗th RG, and sinc(𝑥) equals sin(𝑥)∕𝑥 [30]. When the time difference between two intensities is larger than the coherence time of the light field, 1∕𝛥𝜔𝑗 , sinc2 [𝛥𝜔𝑗 (𝑡2 − 𝑡1 )∕2] approximately equals 0, which indicates that there is no correlation between these two light beams. When 𝑡1 equals 𝑡2 , sinc2 [𝛥𝜔𝑗 (𝑡2 − 𝑡1 )∕2] gets its maximum value, 1. There exists extra correlation comparing to that the time difference is larger than the coherence time. Eq. (4) becomes temporal ghost imaging with thermal light when 𝑁 equals 1.
A third RG can be added after the second RG and the intensity after the third RG is a conditional negative exponential distribution based on the intensity distribution given by Eq. (6). Treating 𝑃𝐼 (𝐼) in Eq. (6) as 𝑃𝑥 (𝑥) and repeating the process in Eq. (6) will obtain the probability distribution of the intensity after the third RG. The process can be repeated for 𝑁 RGs. The intensity distribution for any number of RGs in the superbunching pseudothermal light source can be obtained in the same way [29]. The numerical simulation of temporal ghost imaging follows the process in Fig. 2. 100 random numbers following certain statistics are generated by the method introduced before to simulate the intensity fluctuations of superbunching pseudothermal light. Two copies of the random numbers are employed to represent the intensities of signal and reference light beams, respectively. The intensity of the signal beam is modulated by the temporal object. The output of the bucket detector, D2 , is obtained by summing the modulated signals within one period. Then the intensity of the reference light beam is multiplied by the output of D2 . The image of the temporal object is retrieved by repeating the process 105 times. Fig. 3 shows the numerical results of temporal ghost imaging with superbunching pseudothermal light. The temporal object is shown in Fig. 3(g). 𝑂(𝑡) represents the amplitude of the temporal object. 𝑡 is time and 𝑎𝑟𝑏.𝑢. is short for arbitrary unit of time. The width and height of the first temporal slit are 10 and 0.5, respectively. The width and height of the second temporal slit are 5 and 1, respectively. The heights of all other points are zero. Fig. 3(a)–(f) are the simulated temporal ghost images of the object with superbunching pseudothermal light when 𝑁 equals 1, 2, 3, 4, 5, and 6, respectively. 𝑔𝑇(2) (𝑡) is the normalized temporal ghost image, which is normalized according to the constant background in Eq. (4). The constant background in Eq. (4) corresponds to accidental two-photon coincidence counts [1]. The photon detection events are independent in this condition. Hence the normalized temporal ghost image, 𝑔𝑇(2) (𝑡), is calculated by setting the constant background in Eq. (4) equaling 1. The black squares are numerical results. The red lines in Fig. 3(a)–(f) are theoretical results. The meaning of the horizontal axis in Fig. 3(a)–(f) is the same as the one in Fig. 3(g). When 𝑁 equals 1, the
2.2. Numerical simulations The statistics of the light intensity of superbunching pseudothermal light is dependent on the number of RGs in the superbunching pseudothermal light source [29]. If single-mode continuous-wave laser light is employed as the input of superbunching pseudothermal light source, the intensity of the incident light before the first RG is constant. The intensity, 𝐼, after the first RG follows negative exponential distribution [46] 𝐼 1 exp(− ), (5) 𝑥 𝑥 where 𝑥 is the average intensity of the scattered light and is proportional to the intensity of the incident light. If the generated pseudothermal light within one coherence area is filtered by a pinhole and set as the incident light of the second RG, the intensity after the second RG follows a conditional negative exponential distribution, which is conditioned on the average intensity of the scattered light after the second RG. If there is no photon loss in the scattering process, the average intensity of the scattered light after the second RG is proportional to the intensity of the incident light before the second RG. The process can be expressed as 𝑦 = 𝑥∕𝛼, in which 𝑦 is the average intensity of the scattered light after the second RG, 𝑥 is the intensity of the incident light before the second RG, and 𝛼 is a constant. The intensity distribution of superbunching pseudothermal light after two RGs is [46] 𝑃𝐼|𝑥 (𝐼|𝑥) =
∞
𝑃𝐼 (𝐼) =
𝑃𝐼|𝑦 (𝐼|𝑦)𝑃𝑥 (𝑥)𝑑𝑥
∫0 ∞
=
∫0
1 𝐼 1 𝑥 exp(− ) ⋅ exp(− )𝑑𝑥, 𝑦 𝑦 ⟨𝐼⟩ ⟨𝐼⟩
(6)
where 𝑃𝑥 (𝑥) is the intensity distribution after the first RG and ⟨𝐼⟩ is the average intensity of the scattered light after the first RG. Eq. (6) can be calculated by employing the relation 𝑦 = 𝑥∕𝛼 and we have assumed 𝛼 = 1 to simplify the calculations. 826
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usually measured in the Hanbury Brown and Twiss (HBT) interferometer [49,50]. Imaging is based on the point-to-point correlation in both classical imaging with lens and ghost imaging [1]. In classical imaging with lens, the point-to-point correlation between the object and image planes is due to the constructive first-order interference of light via the imaging lens. From the same point of view, the point-to-point correlation between the object and image planes in ghost imaging is due to the second-order interference of light. The point-to-point correlations in ghost imaging can be described by 𝑔 (2) (0). Different from there is no constant background in classical imaging with lens, there is a constant background in thermal light ghost imaging, which is caused by the accidental two-photon coincidence counts [8–12]. When 𝑔 (2) (0) equals 1, there is no extra correlation comparing to the background. It is impossible to form ghost imaging with this type of light, which is the reason why single-mode laser light cannot be employed to realize ghost imaging. When 𝑔 (2) (0) is larger than 1, there is extra correlation, based on which ghost imaging can be implemented. In fact, the experimental setup measuring 𝑔 (2) (0) can be treated as a special type of ghost imaging. 𝑔 (2) (0) is measured in the HBT interferometer by fixing the position of one detector and scanning the other detector in either temporal or spatial regimes [49,50]. Let us take the spatial case for example. If the object in ghost imaging is a small pinhole, the light passing through the pinhole is collected by the bucket detector, which is equivalent to the fixed detector in the HBT interferometer. Scanning the other detector to obtain the ghost image of the pinhole is equivalent to measuring 𝑔 (2) (0) in the HBT interferometer. Hence the experimental setup to measure 𝑔 (2) (0) is equivalent to the ghost imaging scheme when the object is a small pinhole. It is easy to prove that similar conclusion holds for temporal ghost imaging. Hence the larger value 𝑔 (2) (0) is, the higher visibility ghost imaging will be. Theoretically, 𝑔 (2) (0) of the light is fixed if the light source is specified. However, only limited number of intensities are employed to calculate 𝑔 (2) (0) in both experiments and simulations. The calculated values of 𝑔 (2) (0) are inevitably different in different runs for the same light source. The variance of the calculated 𝑔 (2) (0) will cause image distortion in ghost imaging. If the variance of 𝑔 (2) (0) is small, the retrieved image will be consistent with the object due to the calculated correlations are similar for different points on the image plane. On the other hand, the calculated correlations may be very different for different points on the imaging plane if the variance of 𝑔 (2) (0) is large, in which there is image distortion. One can eliminate or decrease the image distortion in ghost imaging by decreasing the variance of 𝑔 (2) (0) in the measurement or simulation. Fig. 4 shows that 𝑔 (2) (0) increases with the number of RGs in the superbunching pseudothermal light source. 𝑔 (2) (0) is the degree of second-order coherence and 𝑁 is the number of RGs. The black squares are simulated results, which are calculated by generating 2×104 random numbers following the statistics for superbunching pseudothermal light with different values of 𝑁. The error bar is the standard deviation (SD) and is calculated with 50 independent runs for each value of 𝑁. The red line is the theoretical value of 2𝑁 . The results in Fig. 4 can be employed to explain why the quality of the retrieved images are different in Fig. 3(a)–(f). The lengths of the error bars for 𝑁 equaling 1 and 2 are less than the size of the black square, which indicates that the fluctuations of the calculated second-order correlations are small in different runs. As the number of RGs in superbunching pseudothermal light source increases, the SD of the calculated 𝑔 (2) (0) increases, in which the calculated correlations vary in different runs. This is the reason why the retrieved images are consistent with the temporal object in Fig. 3(a)– (d) and are distorted in Fig. 3(e)–(f). The reason why the SD of the calculated 𝑔 (2) (0) increases with 𝑁 can be understood as follows. Fig. 5 shows the histograms of the generated random intensities for different number of RGs in superbunching pseudothermal light source. All the results in Fig. 5 are drawn based on 5 × 104 generated random intensities following the statistics when there are 1, 2, 3, 4, 5, and 6 RGs in the superbunching pseudothermal light
scattered light is pseudothermal light [47]. The visibility of temporal ghost image with pseudothermal light in Fig. 3(a) is (4.7 ± 0.2)%, which is calculated by employing 𝑉 =
max[𝑔𝑇(2) (𝑡)] − min[𝑔𝑇(2) (𝑡)] max[𝑔𝑇(2) (𝑡)] + min[𝑔𝑇(2) (𝑡)]
(7)
.
Max[𝑔𝑇(2) (𝑡)] is the maximal value of the normalized second-order correlation function and is calculated by averaging the data points at the peak of the second slit in the retrieved image. Min[𝑔𝑇(2) (𝑡)] is the minimal value and is calculated by averaging the data points in the background. The visibility of temporal ghost image increases as 𝑁 increases, which can be seen from the results in Fig. 3(h). 𝑉 is the visibility of temporal ghost image and 𝑁 is the number of RGs in superbunching pseudothermal light source. The hollow black squares are numerical results. The red product signs are theoretical results calculated by employing Eq. (9). The visibility of temporal ghost imaging with superbunching pseudothermal light can be much higher than the maximal visibility of ghost imaging with thermal light, 33% [1]. For instance, the visibility in Fig. 3(f) is (75 ± 8)%. In the simulation of Fig. 3, the coherence time of superbunching pseudothermal light is assumed to be infinity short due to the correlation between these two copies of random intensities disappears when they are displaced by one number. The sinc function in this case can be approximately treated as 𝛿(𝑡1 − 𝑡2 ). 𝛿(𝑡1 − 𝑡2 ) equals 1 when 𝑡1 equals 𝑡2 . Otherwise, 𝛿(𝑡1 − 𝑡2 ) equals 0. In this condition, Eq. (4) can be simplified as 𝐺𝑇(2) (𝑡2 ) ∝
𝑁 𝑇 ∏
∫0
[1 + 𝛿 2 (𝑡2 − 𝑡1 )]𝑂(𝑡1 )𝑑𝑡1
𝑗=1 𝑁
(8)
= 10 + (2 − 1)𝑂(𝑡2 ), 𝑇 ∫0
in which the constant 10 is a result of the integral, 𝑂(𝑡1 )𝑑𝑡1 , by taking the employed temporal double slits in Fig. 3(g) into account. The integral of all the other 2𝑁 − 1 terms including 𝛿 2𝑚 (𝑡2 − 𝑡1 ) (𝑚 is positive integer) will give the same results, 𝑂(𝑡2 ). The red lines in Fig. 3(a)–(f) are drawn by setting 𝑁 equals 1, 2, 3, 4, 5, and 6 in Eq. (8), respectively. The visibility of temporal ghost imaging when there are 𝑁 RGs in the superbunching pseudothermal light source can be calculated by substituting the results in Eq. (8) into Eq. (7), 20 , (9) 2𝑁 + 19 where 20 and 19 are from the constant 10 in Eq. (8). Eq. (9) indicates that the visibility may vary if the temporal object changes when all the other conditions are fixed, which is consistent with the conclusion in spatial ghost imaging [48]. The numerical results are consistent with the theoretical results in Fig. 3. It is predicted from Eq. (9) that the visibility of temporal ghost imaging with superbunching pseudothermal light will approach 1 when 𝑁 further increases. 𝑉𝑁 = 1 −
3. Discussions In last section, we have theoretically and numerically proved that high visibility temporal ghost imaging can be realized with superbunching pseudothermal light. Comparing the results in Fig. 3(a)–(f) with the temporal object in Fig. 3(g), it is easy to see that the retrieved images in Fig. 3(a)–(d) are consistent with the temporal object while the retrieved images in Fig. 3(e)–(f) are somewhat distorted. In this section, we will discuss why the retrieved images are different for superbunching pseudothermal light with different number of RGs and how to eliminate or decrease the image distortions. As shown in Eq. (3), temporal ghost imaging is closely related to the second-order coherence function of the employed light. The ability to implement ghost imaging and the visibility of ghost image can be analyzed via the degree of second-order coherence, 𝑔 (2) (0), which is 827
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Fig. 4. The degree of second-order coherence vs. the number of RGs in superbunching pseudothermal light source. 𝑔 (2) (0) is the degree of second-order coherence and 𝑁 is the number of RGs. The black squares are the simulated results and the red line is the theoretical value. The scale of y axis is logarithmic.
Fig. 6. Standard deviation vs. the number of random intensities in a single run. 𝑔 (2) (0) is the degree of second-order coherence and 𝑛 is the number of random intensities in a single run. 1000 groups of data are generated to calculate the SD for each data point. The black lines connecting the data points are served as the guidance for the eye. (a)–(f) are the results for 1, 2, 3, 4, 5, and 6 RGs in the superbunching pseudothermal light source, respectively.
calculated with 1000 different runs for each data point. 𝑁 is the number of RGs in superbunching pseudothermal light source and 𝑛 is the number of random intensities in a single run. The SD in Fig. 6(a)–(b) decreases as 𝑛 increases. When 𝑁 is larger than 2, the calculated SD fluctuates when 𝑛 increases. For instance, the calculated SD in Fig. 6(c)–(d) first increases when 𝑛 increases from 100 to 200 and then decreases as 𝑛 increases. While the SD in Fig. 6(e)–(f) increases when 𝑛 increases from 100 to 5000. The results in Figs. 4–6 can be understood via the definition of the degree of second-order coherence [43], 𝑔 (2) (0) = Fig. 5. Histograms of numerically generated intensities for different number of RGs in superbunching pseudothermal light source. 5 × 104 generated intensities are employed to draw the histogram in each sub-figure. 𝑃 (𝐼) is the probability for the intensity being 𝐼 and 𝐼 is the intensity of light. The average of the generated intensities is 5k and 𝑎𝑟𝑏.𝑢. is short for arbitrary unit of light intensity.
⟨𝐼 2 ⟩ , ⟨𝐼⟩2
(10)
where the intensities detected by two detectors are assumed to be identical and ⟨⋯ ⟩ is ensemble average. Ensemble average means taking all the possible realizations into account [43]. When there are sufficient number of random intensities to calculate 𝑔 (2) (0), the calculated value will be close to the theoretical one. When there are not sufficient number of random intensities, the calculated value is a partial ensemble average and may be very different from the theoretical one. Different distributions require different number of random intensities to obtain the results close to the theoretical ones. For instance, 20 000 × 1000 intensities are sufficient calculate 𝑔 (2) (0) for 1 RG in superbunching pseudothermal light source, which can be seen from the last data point in Fig. 6(a). While the same number of random intensities are not sufficient to calculate 𝑔 (2) (0) for 6 RGs, which can be seen from the last data point in Fig. 6(f). However, no matter what distribution the intensity follows, it is always possible to obtain 𝑔 (2) (0) close to the theoretical value by employing sufficient number of random intensities. Hence the image distortion in Fig. 3(e)–(f) can be eliminated or decreased by including more number of random intensities.
source, respectively. When 𝑁 equals 1, the intensity follows negative exponential distribution, which is shown in Fig. 5(a). The probability, 𝑃 (𝐼), decreases as the intensity, 𝐼, increases. It is possible to obtain sufficient number of intensities to calculate the degree of second-order coherence with 2×104 random numbers following the same distribution. Hence the calculated value with limited number of random intensities will be close to the theoretical one when 𝑁 equals 1, as shown in Fig. 4. When the number of RGs increases, 𝑃 (𝐼) concentrates more on the far left column in the histograms. For instance, the probability for 𝐼 being in the far left column equals 0.98 in Fig. 5(f). The calculated 𝑔 (2) (0) with the same number of random intensities will fluctuate more than the ones with less RGs in Fig. 5(a)–(e). Hence the SD of the calculated 𝑔 (2) (0) will increase as the number of RGs increases, too. The results in Fig. 5 are consistent with the ones in Fig. 4. The SD of the calculated 𝑔 (2) (0) can be decreased by increasing the number of random intensities. Fig. 6 shows how the SD changes with the number of random intensities for different number of RGs in superbunching pseudothermal light source. The degrees of secondorder coherence in Fig. 6 are calculated for 100, 200, 500, 1000, 2000, 5000, 10 000, and 20 000 random intensities, respectively. The SD is
4. Conclusions In conclusion, we have proved that high visibility temporal ghost imaging with classical light is possible. The superbunching pseudothermal light is helpful to improve the visibility of temporal ghost image. The visibility of temporal ghost image with superbunching pseudothermal light can approach 1 when more rotating ground glasses are 828
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employed in superbunching pseudothermal light source. The visibility of ghost imaging with classical light can be as large as the one of ghost imaging with entangled photon pairs. The results are helpful to understand the physics of ghost imaging and increase the image quality in future temporal ghost imaging applications.
[20] M.J. Padgett, R.W. Boyd, Phil. Trans. R. Soc. A 375 (2017) 20160233. [21] R. Loudon, The Quantum Theory of Light, third ed., Oxford University Press, New York, 2000, p. 203, Eq. (5514). Squeezed vacuum state can be treated as two-photon entangled state generated by spontaneous parametric down conversion, p. 268. [22] F. Ferri, D. Magatti, L.A. Lugiato, A. Gatti, Phys. Rev. Lett. 104 (2010) 253603. [23] B.Q. Sun, S.S. Welsh, M.P. Edgar, J.H. Shapiro, M.J. Padgett, Opt. Express 20 (2012) 16892. [24] K.H. Luo, B.Q. Huang, W.M. Zheng, L.A. Wu, Chin. Phys. Lett. 29 (2012) 074216. [25] H. Chen, T. Peng, Y.H. Shih, Phys. Rev. A 88 (2013) 023808. [26] D.Z. Cao, J. Xiong, S.H. Zhang, L.F. Lin, L. Gao, K.G. Wang, Appl. Phys. Lett. 92 (2008) 201102. [27] X.H. Chen, I.N. Agafonov, K.H. Luo, Q. Liu, R. Xian, M.V. Chekhova, L.A. Wu, Opt. Lett. 35 (2010) 1166. [28] X.F. Liu, M.F. Li, X.R. Yao, W.K. Yu, G.J. Zhai, L.A. Wu, AIP Adv. 3 (2013) 052121. [29] Y. Zhou, B. Bai, H. Chen, J.B. Liu, F.L. Li, Z. Xu, H.B. Zheng, Phys. Rev. A 95 (2017) 053809. [30] B. Bai, J.B. Liu, Y. Zhou, H.B. Zheng, H. Chen, S.L. Zhang, Y.C. He, F.L. Li, Z. Xu, J. Opt. Soc. Amer. B 34 (2017) 2081. [31] B.H. Kolner, IEEE. J. Quantum Electron. 30 (19944) 1951. [32] V. Torres-Company, J. Lancis, P. Andres, Prog. Opt. 56 (2011) 1. [33] R. Salem, M.A. Foster, A.L. Gaeta, Adv. Opt. Photonics 5 (2013) 274. [34] T. Shirai, T. Setälä, Ari T. Friberg, J. Opt. Soc. Amer. B 27 (2010) 2549. [35] K. Cho, J. Noh, Opt. Commun. 285 (2012). [36] S. Denis, F. Devaux, E. Lantz, P.A. Moreau, 2016. arXiv:quant-ph/1612.05723v1. [37] Z.P. Chen, H. Li, Y. Li, J.H. Shi, G.H. Zeng, Opt. Eng. 52 (2013) 076103. [38] P. Ryczkowski, M. Barbier, Ari T. Friberg, J.M. Dudley, G. Genty, Nat. Photon. 10 (2016) 167. [39] F. Devaux, K.P. Huy, S. Denis, E. Lantz, P.A. Moreau, J. Opt. 19 (2017) 024001. [40] Y. O-oka, S. Fukatsu, Appl. Phys. Lett. 111 (2017) 061106. [41] P. Ryczkowski, M. Barbier, Ari T. Friberg, J.M. Dudley, G. Genty, APL Photonics 2 (2017) 046102. [42] F. Devaux, P.A. Moreau, S. Denis, E. Lantz, Optica 3 (2016) 698. [43] L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press, New York, 1995. [44] R.J. Glauber, Phys. Rev. 130 (1963) 2529. [45] R.J. Glauber, Phys. Rev. 131 (1963) 2766. [46] J.W. Goodman, Speckle Phenomena in Optics: Theory and Applications, Ben Roberts & Company, Greenwood Village, CO, 2007. [47] W. Martienssen, E. Spiller, Amer. J. Phys. 32 (1965) 919. [48] A. Gatti, M. Bache, D. Magatti, E. Brambilla, F. Ferri, L.A. Lugiato, J. Mod. Phys. 53 (2006) 739. [49] R. Hanbury Brown, R.Q. Twiss, Nature 177 (1956) 27. [50] R. Hanbury Brown, R.Q. Twiss, Nature 178 (1956) 1046.
Acknowledgments This project is supported by National Natural Science Foundation of China (NSFC) (Grant No. 11404255), National Basic Research Program of China (973 Program) (Grant No. 2015CB654602), 111 Project of China (Grant No. B14040), and Fundamental Research Funds for the Central Universities. References [1] Y.H. Shih, An Introduction to Quantum Optics, Taylor and Francis Group, LLC, FL, 2011. [2] D.N. Klyshko, Zh. Eksp. Teor. Fiz. 94 (1998) 82. [3] T.B. Pittman, Y.H. Shih, D.V. Strekalov, A.V. Sergienko, Phys. Rev. A 52 (1995) 3429(R). [4] A.F. Abouraddy, B.E.A. Saleh, A.V. Sergienko, M.C. Teich, Phys. Rev. Lett. 87 (2001) 123602. [5] R.S. Bennink, S.J. Bentley, R.W. Boyd, Phys. Rev. Lett. 89 (2002) 113601. [6] M. D’Angelo, Y.H. Shih, 2013. arXiv:quant-ph/0302146v2. [7] M.H. Rubin, 2003. arXiv:quant-ph/0303188v2. [8] A. Gatti, E. Brambilla, M. Bache, L.A. Lugiato, Phys. Rev. Lett. 93 (2004) 093602. [9] J. Chen, S.S. Han, Phys. Rev. Lett. 92 (2004) 093903. [10] A. Valencia, G. Scarcelli, M. D’Angelo, Y.H. Shih, Phys. Rev. Lett. 94 (2005) 063601. [11] Y. Cai, S.Y. Zhu, Phys. Rev. E 71 (2005) 056607. [12] G. Scarcelli, V. Berardi, Y.H. Shih, Phys. Rev. Lett. 96 (2006) 063602. [13] A. Gatti, M. Bondani, L.A. Lugiato, M.G.A. Paris, C. Fabre, Phys. Rev. Lett. 98 (2007) 039301. [14] G. Scarcelli, V. Berardi, Y.H. Shih, Phys. Rev. Lett. 98 (2007) 039302. [15] J.H. Shapiro, Phys. Rev. A 78 (2008) 061802(R). [16] S. Ragy, G. Adesso, Sci. Rep. (2012) 0–651. [17] J.H. Shapiro, R.W. Boyd, Quantum Inf. Process. (2012). http://dx.doi.org/10.1007/ s11128-011-03656-5. [18] Y.H. Shih, Quantum Inf. Process. (2012). http://dx.doi.org/10.1007/s11128-0120396-5. [19] J.H. Shapiro, R.W. Boyd, Quantum Inf. Process. (2012). http://dx.doi.org/10.1007/ s11128-012-0399-2.
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
High visibility temporal ghost imaging with classical light Jianbin Liu a , Jingjing Wang a , Hui Chen a , Huaibin Zheng a , Yanyan Liu c , Yu Zhou b, *, Fu-li Li b , Zhuo Xu a a
Electronic Materials Research Laboratory, Key Laboratory of the Ministry of Education, and International Center for Dielectric Research, Xi’an Jiaotong University, Xi’an 710049, China b MOE Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter, Department of Applied Physics, Xi’an Jiaotong University, Xi’an 710049, China c Science and Technology on Electro-Optical Information Security Control Laboratory, China
a r t i c l e
i n f o
Keywords: Ghost imaging Two-photon bunching Superbunching pseudothermal light Photon statistics Optical coherence theory
a b s t r a c t High visibility temporal ghost imaging with classical light is possible when superbunching pseudothermal light is employed. In the numerical simulation, the visibility of temporal ghost imaging with pseudothermal light, equaling (4.7 ± 0.2)%, can be increased to (75 ± 8)% in the same scheme with superbunching pseudothermal light. The reasons for that the retrieved images are different for superbunching pseudothermal light with different values of degree of second-order coherence are discussed in detail. It is concluded that high visibility and high quality temporal ghost image can be obtained by collecting sufficient number of data points. The results are helpful to understand the difference between ghost imaging with classical light and entangled photon pairs. The superbunching pseudothermal light can be employed to improve the image quality in ghost imaging applications. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Ghost imaging is a new imaging technique that is based on the second- or higher-order correlation of light [1]. In a typical ghost imaging scheme, there are two light beams, which are usually emitted by the same light source. One is signal beam and is incident onto an object. The transmitted or reflected light from the object is collected by a bucket detector without position information. The other beam is reference beam, which does not interact with the object and is detected by a scannable detector with position information. The image of the object cannot be obtained by either signal from these two detectors. Because the image information is collected by the bucket detector without position resolution and the position resolution of the reference detector is able to resolve the object, while the object does not interact with the reference light beam. However, the image of the object can be retrieved by correlating the signals from these two detectors. This peculiar property is the reason why the imaging technique is named after ghost imaging [1]. The developing history of ghost imaging is a typical example of how scientific research advances itself by the efforts of people with different opinions [2–20]. Ghost imaging was first performed with entangled photon pairs [2,3]. It was then thought that entangled photon
pairs was necessary for ghost imaging [4]. However, Bennink et al. employed classical correlated laser light beams to realize coincidence imaging, which is similar to the scheme of ghost imaging [5]. It was argued that the coincidence imaging performed by Bennink et al. was different from ghost imaging [6,7]. Shortly after the work of Bennink et al., several groups proved that ghost imaging can be performed with thermal light [8–11]. Since then, the discussion about the physics of ghost imaging with thermal light never stops [12–20]. Scarcelli et al. suggested that ghost imaging with thermal light might have to be described quantum mechanically [12]. Gatti et al. disagreed with the conclusion and claimed that classical theory could also be employed to interpret thermal light ghost imaging [13,14]. Shapiro proposed a new type of ghost imaging called computational ghost imaging and pointed out that the nature of ghost imaging is classical [15]. On the other hand, Rogy et al. employed quantum discord to study ghost imaging and concluded that there is quantum correlation in thermal light ghost imaging [16]. Although the debate about the physics of thermal light ghost imaging still goes on [1,17–20], it is well accepted that ghost imaging with thermal light can mimic all the behaviors of ghost imaging with entangled photon pairs except the visibility of the former is much lower than the latter [8]. The reason is that the degree of second-order coherence, 𝑔 (2) (0), equals 2 for thermal light; while it can be much
* Corresponding author.
E-mail address: [email protected] (Y. Zhou). https://doi.org/10.1016/j.optcom.2017.11.054 Received 18 August 2017; Received in revised form 16 November 2017; Accepted 18 November 2017 0030-4018/© 2017 Elsevier B.V. All rights reserved.
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larger than 2 for entangled photon pairs [21]. If there is one type of classical light with large value of 𝑔 (2) (0), high visibility ghost imaging with classical light should also be possible. Different methods, such as differential ghost imaging [22], normalized ghost imaging [23], correspondence ghost imaging [24,25], highorder ghost imaging [26–28], etc., have been employed to improve the visibility of ghost imaging with classical light. However, all the existing methods employ mathematical algorithms to improve the visibility, which require extra calculations and is different from the one in ghost imaging with entangled photon pairs [22–27]. Recently, we have proposed a new type of classical light called superbunching pseudothermal light, in which 𝑔 (2) (0) can be much larger than 2 [29,30]. With the help of superbunching pseudothermal light, high visibility temporal ghost imaging can be implemented with normal correlation algorithm similar to that in ghost imaging with entangled photon pairs [3]. Our results show that high visibility ghost imaging can also be realized with classical light, which should be one important step in the development of ghost imaging and be helpful to understand the physics of ghost imaging. Most ghost imaging experiments were in the spatial domain [2,3,8– 16,22–28]. Based on the space–time duality [31–33], temporal ghost imaging should also be possible. Shirai et al. first proved that temporal ghost imaging with classical pulsed light can be realized by analogy of spatial ghost imaging with thermal light [34]. It was soon realized that temporal ghost imaging can also be performed with entangled photon pairs [35,36], chaotic laser light [37,38], and chaotic light [39–41]. Recently, computational temporal ghost imaging was also experimentally implemented [42]. There are also two light beams in a typical temporal ghost imaging scheme [34–41]. The signal beam is modulated by a temporal object and then is detected by a slow detector, which cannot follow the variation of the signal. The role of this detector is the same as that of bucket detector in spatial ghost imaging. The reference beam is directly detected by a fast detector to record the variation of light intensity. The role of the detector is the same as the one of scannable detector in spatial ghost imaging. The image of the temporal object cannot be obtained by either signal from these two detectors. However, the image can be retrieved by correlating the signals from these two detectors [38], just like ghost imaging in spatial domain [1]. The rest of the paper is organized as follows. Theoretical and numerical study of high visibility temporal ghost imaging with superbunching pseudothermal light is in Section 2. The discussions about the reasons why the visibility of temporal ghost imaging can be increased by changing the light source and the difference in the retrieved images when there are different number of RGs in superbunching pseudothermal light source are in Section 3. Our conclusions are in Section 4.
Fig. 1. Temporal ghost imaging scheme. IM: intensity modulator. BS: non-polarizing 50 ∶ 50 beam splitter. O: temporal object. D1 : temporal bucket detector, which integrates the intensity within one period. D2 : reference detector with high temporal resolution. CC: second-order correlation measurement system.
Fig. 2. Process of temporal ghost imaging within one period. The horizontal and vertical axes are time (𝑡) and intensity of light (𝐼), respectively. S: signal beam. O: temporal object. R: reference beam. 𝑇 : period length of temporal object. 𝑆 ⋅ 𝑂: the product of signal light beam and temporal object, which corresponds to modulating the temporal object with the signal light beam. Sum(𝑆 ⋅𝑂) is the output of the bucket detector within one period, which is a non-negative real number. Sum(𝑆 ⋅𝑂)×𝑅 represents the intensity of the reference beam multiplied by the output of the bucket detector.
temporal ghost imaging, D2 is scanned longitudinally to measure the second-order temporal correlation between the reference and signal light beams. D1 and D2 are in transversely symmetrical positions in temporal ghost imaging so that the light intensity fluctuations received by these two detectors are identical. The process of temporal ghost imaging within one period is described in Fig. 2. S and R represent signal and reference light beams, respectively. The intensity fluctuations of these two light beams are identical. O is a temporal object, which is a periodic double-slit in our scheme. The period of the temporal object is 𝑇 , which is shown in the left panel of Fig. 2. 𝑆 ⋅ 𝑂 means modulating the temporal object with the signal light beam. In the modulating process, the variation of the signal beam should be faster than the temporal object. Otherwise, the information of the object may be lost [34]. The modulating process is shown in the middle panel of Fig. 2. Sum(𝑆 ⋅ 𝑂) is the output of the bucket detector, D1 , which integrates the modulated signals within a period. The second-order correlation between signal and reference light beams is implemented via Sum(𝑆 ⋅ 𝑂) × 𝑅. The image of the temporal object can be retrieved by correlating the outputs of D1 and D2 and repeating the process in Fig. 2 many times [34–40]. The temporal ghost imaging with classical light in Fig. 2 can be mathematically expressed as
2. Temporal ghost imaging with superbunching pseudothermal light 2.1. Theoretical study The scheme of temporal ghost imaging with superbunching pseudothermal light is shown in Fig. 1, which is similar to that of spatial ghost imaging with thermal light [12]. IM is an intensity modulator, which is employed to modulate the intensity of the incident light. It is a combination of rotating ground glasses (RGs), pinholes, lenses, etc., in superbunching pseudothermal light source [29,30]. BS is a nonpolarizing 50 ∶ 50 beam splitter. O is a temporal object, which can be simulated by an electro-optic modulator [38]. D1 is a slow detector, which cannot resolve the temporal object by itself. D2 is a fast detector, which can follow the modulation of light intensity after IM. CC is a second-order correlation measurement system. The distance between the light source and the temporal object planes is equal to the one between the light source and D2 planes. The difference between spatial and temporal ghost imaging schemes is as follows. In spatial ghost imaging, D2 is scanned transversely to measure the second-order spatial correlation between the reference and signal light beams. While in
𝐺𝑇(2) (𝑡2 ) = ⟨𝐼(𝑡2 )
𝑇
∫0
𝐼(𝑡1 )𝑂(𝑡1 )𝑑𝑡1 ⟩,
(1)
where 𝐼(𝑡2 ) is the intensity of the reference light beam recorded by D2 , 𝐼(𝑡1 ) is the intensity of the signal light beam, 𝑂(𝑡1 ) is the temporal 𝑇 object, and ∫0 𝐼(𝑡1 )𝑂(𝑡1 )𝑑𝑡1 is the output of D1 within one period. ⟨⋯ ⟩ is ensemble average, which is equivalent to a long time average for an 825
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ergodic and stationary process [43]. The temporal object remains the same in different periods. Hence Eq. (1) can be re-arranged as 𝐺𝑇(2) (𝑡2 ) =
𝑇
⟨𝐼(𝑡2 )𝐼(𝑡1 )⟩𝑂(𝑡1 )𝑑𝑡1 ,
∫0
(2)
where ⟨𝐼(𝑡2 )𝐼(𝑡1 )⟩ is the second-order correlation function [43]. For a stationary process, the second-order correlation function is only dependent on the time difference between these two intensities [43]. Eq. (2) can be simplified as 𝐺𝑇(2) (𝑡2 ) =
𝑇
𝐺(2) (𝑡2 − 𝑡1 )𝑂(𝑡1 )𝑑𝑡1 ,
∫0
(3)
where 𝐺(2) (𝑡2 − 𝑡1 ) is the second-order correlation function [44,45]. Eq. (3) is similar to that of lensless spatial ghost imaging with thermal light [1,12] and is consistent with the results of temporal ghost imaging with classical light [34,37]. If the superbunching pseudothermal light proposed in [29,30] is employed in temporal ghost imaging, Eq. (3) can be expressed as 𝐺𝑇(2) (𝑡2 ) ∝
𝑁 𝑇 ∏
∫0
[1 + sinc2
𝛥𝜔𝑗 (𝑡2 − 𝑡1 )
𝑗=1
2
]𝑂(𝑡1 )𝑑𝑡1 ,
(4)
Fig. 3. Simulated temporal ghost imaging with superbunching pseudothermal light. 𝑔𝑇(2) (𝑡) is the normalized temporal ghost imaging, which is normalized according to the constant background in Eq. (4). 𝑂(𝑡) is the temporal object. 𝑡 is time and 𝑎𝑟𝑏.𝑢. is short for arbitrary unit of time. The black squares are simulated results and the red lines are theoretical results. 𝑉 is visibility and 𝑁 is the number of RGs in superbunching pseudothermal light source. (a)–(f) are the retrieved images of the temporal object in (g) for different number of RGs in superbunching pseudothermal light source. (h) is the visibility of temporal ghost images in (a)–(f), in which the hollow black squares and red product signs are simulated and theoretical results, respectively.
where 𝑁 is the number of RGs in superbunching pseudothermal light source, 𝛥𝜔𝑗 is the frequency bandwidth of pseudothermal light scattered by the 𝑗th RG, and sinc(𝑥) equals sin(𝑥)∕𝑥 [30]. When the time difference between two intensities is larger than the coherence time of the light field, 1∕𝛥𝜔𝑗 , sinc2 [𝛥𝜔𝑗 (𝑡2 − 𝑡1 )∕2] approximately equals 0, which indicates that there is no correlation between these two light beams. When 𝑡1 equals 𝑡2 , sinc2 [𝛥𝜔𝑗 (𝑡2 − 𝑡1 )∕2] gets its maximum value, 1. There exists extra correlation comparing to that the time difference is larger than the coherence time. Eq. (4) becomes temporal ghost imaging with thermal light when 𝑁 equals 1.
A third RG can be added after the second RG and the intensity after the third RG is a conditional negative exponential distribution based on the intensity distribution given by Eq. (6). Treating 𝑃𝐼 (𝐼) in Eq. (6) as 𝑃𝑥 (𝑥) and repeating the process in Eq. (6) will obtain the probability distribution of the intensity after the third RG. The process can be repeated for 𝑁 RGs. The intensity distribution for any number of RGs in the superbunching pseudothermal light source can be obtained in the same way [29]. The numerical simulation of temporal ghost imaging follows the process in Fig. 2. 100 random numbers following certain statistics are generated by the method introduced before to simulate the intensity fluctuations of superbunching pseudothermal light. Two copies of the random numbers are employed to represent the intensities of signal and reference light beams, respectively. The intensity of the signal beam is modulated by the temporal object. The output of the bucket detector, D2 , is obtained by summing the modulated signals within one period. Then the intensity of the reference light beam is multiplied by the output of D2 . The image of the temporal object is retrieved by repeating the process 105 times. Fig. 3 shows the numerical results of temporal ghost imaging with superbunching pseudothermal light. The temporal object is shown in Fig. 3(g). 𝑂(𝑡) represents the amplitude of the temporal object. 𝑡 is time and 𝑎𝑟𝑏.𝑢. is short for arbitrary unit of time. The width and height of the first temporal slit are 10 and 0.5, respectively. The width and height of the second temporal slit are 5 and 1, respectively. The heights of all other points are zero. Fig. 3(a)–(f) are the simulated temporal ghost images of the object with superbunching pseudothermal light when 𝑁 equals 1, 2, 3, 4, 5, and 6, respectively. 𝑔𝑇(2) (𝑡) is the normalized temporal ghost image, which is normalized according to the constant background in Eq. (4). The constant background in Eq. (4) corresponds to accidental two-photon coincidence counts [1]. The photon detection events are independent in this condition. Hence the normalized temporal ghost image, 𝑔𝑇(2) (𝑡), is calculated by setting the constant background in Eq. (4) equaling 1. The black squares are numerical results. The red lines in Fig. 3(a)–(f) are theoretical results. The meaning of the horizontal axis in Fig. 3(a)–(f) is the same as the one in Fig. 3(g). When 𝑁 equals 1, the
2.2. Numerical simulations The statistics of the light intensity of superbunching pseudothermal light is dependent on the number of RGs in the superbunching pseudothermal light source [29]. If single-mode continuous-wave laser light is employed as the input of superbunching pseudothermal light source, the intensity of the incident light before the first RG is constant. The intensity, 𝐼, after the first RG follows negative exponential distribution [46] 𝐼 1 exp(− ), (5) 𝑥 𝑥 where 𝑥 is the average intensity of the scattered light and is proportional to the intensity of the incident light. If the generated pseudothermal light within one coherence area is filtered by a pinhole and set as the incident light of the second RG, the intensity after the second RG follows a conditional negative exponential distribution, which is conditioned on the average intensity of the scattered light after the second RG. If there is no photon loss in the scattering process, the average intensity of the scattered light after the second RG is proportional to the intensity of the incident light before the second RG. The process can be expressed as 𝑦 = 𝑥∕𝛼, in which 𝑦 is the average intensity of the scattered light after the second RG, 𝑥 is the intensity of the incident light before the second RG, and 𝛼 is a constant. The intensity distribution of superbunching pseudothermal light after two RGs is [46] 𝑃𝐼|𝑥 (𝐼|𝑥) =
∞
𝑃𝐼 (𝐼) =
𝑃𝐼|𝑦 (𝐼|𝑦)𝑃𝑥 (𝑥)𝑑𝑥
∫0 ∞
=
∫0
1 𝐼 1 𝑥 exp(− ) ⋅ exp(− )𝑑𝑥, 𝑦 𝑦 ⟨𝐼⟩ ⟨𝐼⟩
(6)
where 𝑃𝑥 (𝑥) is the intensity distribution after the first RG and ⟨𝐼⟩ is the average intensity of the scattered light after the first RG. Eq. (6) can be calculated by employing the relation 𝑦 = 𝑥∕𝛼 and we have assumed 𝛼 = 1 to simplify the calculations. 826
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usually measured in the Hanbury Brown and Twiss (HBT) interferometer [49,50]. Imaging is based on the point-to-point correlation in both classical imaging with lens and ghost imaging [1]. In classical imaging with lens, the point-to-point correlation between the object and image planes is due to the constructive first-order interference of light via the imaging lens. From the same point of view, the point-to-point correlation between the object and image planes in ghost imaging is due to the second-order interference of light. The point-to-point correlations in ghost imaging can be described by 𝑔 (2) (0). Different from there is no constant background in classical imaging with lens, there is a constant background in thermal light ghost imaging, which is caused by the accidental two-photon coincidence counts [8–12]. When 𝑔 (2) (0) equals 1, there is no extra correlation comparing to the background. It is impossible to form ghost imaging with this type of light, which is the reason why single-mode laser light cannot be employed to realize ghost imaging. When 𝑔 (2) (0) is larger than 1, there is extra correlation, based on which ghost imaging can be implemented. In fact, the experimental setup measuring 𝑔 (2) (0) can be treated as a special type of ghost imaging. 𝑔 (2) (0) is measured in the HBT interferometer by fixing the position of one detector and scanning the other detector in either temporal or spatial regimes [49,50]. Let us take the spatial case for example. If the object in ghost imaging is a small pinhole, the light passing through the pinhole is collected by the bucket detector, which is equivalent to the fixed detector in the HBT interferometer. Scanning the other detector to obtain the ghost image of the pinhole is equivalent to measuring 𝑔 (2) (0) in the HBT interferometer. Hence the experimental setup to measure 𝑔 (2) (0) is equivalent to the ghost imaging scheme when the object is a small pinhole. It is easy to prove that similar conclusion holds for temporal ghost imaging. Hence the larger value 𝑔 (2) (0) is, the higher visibility ghost imaging will be. Theoretically, 𝑔 (2) (0) of the light is fixed if the light source is specified. However, only limited number of intensities are employed to calculate 𝑔 (2) (0) in both experiments and simulations. The calculated values of 𝑔 (2) (0) are inevitably different in different runs for the same light source. The variance of the calculated 𝑔 (2) (0) will cause image distortion in ghost imaging. If the variance of 𝑔 (2) (0) is small, the retrieved image will be consistent with the object due to the calculated correlations are similar for different points on the image plane. On the other hand, the calculated correlations may be very different for different points on the imaging plane if the variance of 𝑔 (2) (0) is large, in which there is image distortion. One can eliminate or decrease the image distortion in ghost imaging by decreasing the variance of 𝑔 (2) (0) in the measurement or simulation. Fig. 4 shows that 𝑔 (2) (0) increases with the number of RGs in the superbunching pseudothermal light source. 𝑔 (2) (0) is the degree of second-order coherence and 𝑁 is the number of RGs. The black squares are simulated results, which are calculated by generating 2×104 random numbers following the statistics for superbunching pseudothermal light with different values of 𝑁. The error bar is the standard deviation (SD) and is calculated with 50 independent runs for each value of 𝑁. The red line is the theoretical value of 2𝑁 . The results in Fig. 4 can be employed to explain why the quality of the retrieved images are different in Fig. 3(a)–(f). The lengths of the error bars for 𝑁 equaling 1 and 2 are less than the size of the black square, which indicates that the fluctuations of the calculated second-order correlations are small in different runs. As the number of RGs in superbunching pseudothermal light source increases, the SD of the calculated 𝑔 (2) (0) increases, in which the calculated correlations vary in different runs. This is the reason why the retrieved images are consistent with the temporal object in Fig. 3(a)– (d) and are distorted in Fig. 3(e)–(f). The reason why the SD of the calculated 𝑔 (2) (0) increases with 𝑁 can be understood as follows. Fig. 5 shows the histograms of the generated random intensities for different number of RGs in superbunching pseudothermal light source. All the results in Fig. 5 are drawn based on 5 × 104 generated random intensities following the statistics when there are 1, 2, 3, 4, 5, and 6 RGs in the superbunching pseudothermal light
scattered light is pseudothermal light [47]. The visibility of temporal ghost image with pseudothermal light in Fig. 3(a) is (4.7 ± 0.2)%, which is calculated by employing 𝑉 =
max[𝑔𝑇(2) (𝑡)] − min[𝑔𝑇(2) (𝑡)] max[𝑔𝑇(2) (𝑡)] + min[𝑔𝑇(2) (𝑡)]
(7)
.
Max[𝑔𝑇(2) (𝑡)] is the maximal value of the normalized second-order correlation function and is calculated by averaging the data points at the peak of the second slit in the retrieved image. Min[𝑔𝑇(2) (𝑡)] is the minimal value and is calculated by averaging the data points in the background. The visibility of temporal ghost image increases as 𝑁 increases, which can be seen from the results in Fig. 3(h). 𝑉 is the visibility of temporal ghost image and 𝑁 is the number of RGs in superbunching pseudothermal light source. The hollow black squares are numerical results. The red product signs are theoretical results calculated by employing Eq. (9). The visibility of temporal ghost imaging with superbunching pseudothermal light can be much higher than the maximal visibility of ghost imaging with thermal light, 33% [1]. For instance, the visibility in Fig. 3(f) is (75 ± 8)%. In the simulation of Fig. 3, the coherence time of superbunching pseudothermal light is assumed to be infinity short due to the correlation between these two copies of random intensities disappears when they are displaced by one number. The sinc function in this case can be approximately treated as 𝛿(𝑡1 − 𝑡2 ). 𝛿(𝑡1 − 𝑡2 ) equals 1 when 𝑡1 equals 𝑡2 . Otherwise, 𝛿(𝑡1 − 𝑡2 ) equals 0. In this condition, Eq. (4) can be simplified as 𝐺𝑇(2) (𝑡2 ) ∝
𝑁 𝑇 ∏
∫0
[1 + 𝛿 2 (𝑡2 − 𝑡1 )]𝑂(𝑡1 )𝑑𝑡1
𝑗=1 𝑁
(8)
= 10 + (2 − 1)𝑂(𝑡2 ), 𝑇 ∫0
in which the constant 10 is a result of the integral, 𝑂(𝑡1 )𝑑𝑡1 , by taking the employed temporal double slits in Fig. 3(g) into account. The integral of all the other 2𝑁 − 1 terms including 𝛿 2𝑚 (𝑡2 − 𝑡1 ) (𝑚 is positive integer) will give the same results, 𝑂(𝑡2 ). The red lines in Fig. 3(a)–(f) are drawn by setting 𝑁 equals 1, 2, 3, 4, 5, and 6 in Eq. (8), respectively. The visibility of temporal ghost imaging when there are 𝑁 RGs in the superbunching pseudothermal light source can be calculated by substituting the results in Eq. (8) into Eq. (7), 20 , (9) 2𝑁 + 19 where 20 and 19 are from the constant 10 in Eq. (8). Eq. (9) indicates that the visibility may vary if the temporal object changes when all the other conditions are fixed, which is consistent with the conclusion in spatial ghost imaging [48]. The numerical results are consistent with the theoretical results in Fig. 3. It is predicted from Eq. (9) that the visibility of temporal ghost imaging with superbunching pseudothermal light will approach 1 when 𝑁 further increases. 𝑉𝑁 = 1 −
3. Discussions In last section, we have theoretically and numerically proved that high visibility temporal ghost imaging can be realized with superbunching pseudothermal light. Comparing the results in Fig. 3(a)–(f) with the temporal object in Fig. 3(g), it is easy to see that the retrieved images in Fig. 3(a)–(d) are consistent with the temporal object while the retrieved images in Fig. 3(e)–(f) are somewhat distorted. In this section, we will discuss why the retrieved images are different for superbunching pseudothermal light with different number of RGs and how to eliminate or decrease the image distortions. As shown in Eq. (3), temporal ghost imaging is closely related to the second-order coherence function of the employed light. The ability to implement ghost imaging and the visibility of ghost image can be analyzed via the degree of second-order coherence, 𝑔 (2) (0), which is 827
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Fig. 4. The degree of second-order coherence vs. the number of RGs in superbunching pseudothermal light source. 𝑔 (2) (0) is the degree of second-order coherence and 𝑁 is the number of RGs. The black squares are the simulated results and the red line is the theoretical value. The scale of y axis is logarithmic.
Fig. 6. Standard deviation vs. the number of random intensities in a single run. 𝑔 (2) (0) is the degree of second-order coherence and 𝑛 is the number of random intensities in a single run. 1000 groups of data are generated to calculate the SD for each data point. The black lines connecting the data points are served as the guidance for the eye. (a)–(f) are the results for 1, 2, 3, 4, 5, and 6 RGs in the superbunching pseudothermal light source, respectively.
calculated with 1000 different runs for each data point. 𝑁 is the number of RGs in superbunching pseudothermal light source and 𝑛 is the number of random intensities in a single run. The SD in Fig. 6(a)–(b) decreases as 𝑛 increases. When 𝑁 is larger than 2, the calculated SD fluctuates when 𝑛 increases. For instance, the calculated SD in Fig. 6(c)–(d) first increases when 𝑛 increases from 100 to 200 and then decreases as 𝑛 increases. While the SD in Fig. 6(e)–(f) increases when 𝑛 increases from 100 to 5000. The results in Figs. 4–6 can be understood via the definition of the degree of second-order coherence [43], 𝑔 (2) (0) = Fig. 5. Histograms of numerically generated intensities for different number of RGs in superbunching pseudothermal light source. 5 × 104 generated intensities are employed to draw the histogram in each sub-figure. 𝑃 (𝐼) is the probability for the intensity being 𝐼 and 𝐼 is the intensity of light. The average of the generated intensities is 5k and 𝑎𝑟𝑏.𝑢. is short for arbitrary unit of light intensity.
⟨𝐼 2 ⟩ , ⟨𝐼⟩2
(10)
where the intensities detected by two detectors are assumed to be identical and ⟨⋯ ⟩ is ensemble average. Ensemble average means taking all the possible realizations into account [43]. When there are sufficient number of random intensities to calculate 𝑔 (2) (0), the calculated value will be close to the theoretical one. When there are not sufficient number of random intensities, the calculated value is a partial ensemble average and may be very different from the theoretical one. Different distributions require different number of random intensities to obtain the results close to the theoretical ones. For instance, 20 000 × 1000 intensities are sufficient calculate 𝑔 (2) (0) for 1 RG in superbunching pseudothermal light source, which can be seen from the last data point in Fig. 6(a). While the same number of random intensities are not sufficient to calculate 𝑔 (2) (0) for 6 RGs, which can be seen from the last data point in Fig. 6(f). However, no matter what distribution the intensity follows, it is always possible to obtain 𝑔 (2) (0) close to the theoretical value by employing sufficient number of random intensities. Hence the image distortion in Fig. 3(e)–(f) can be eliminated or decreased by including more number of random intensities.
source, respectively. When 𝑁 equals 1, the intensity follows negative exponential distribution, which is shown in Fig. 5(a). The probability, 𝑃 (𝐼), decreases as the intensity, 𝐼, increases. It is possible to obtain sufficient number of intensities to calculate the degree of second-order coherence with 2×104 random numbers following the same distribution. Hence the calculated value with limited number of random intensities will be close to the theoretical one when 𝑁 equals 1, as shown in Fig. 4. When the number of RGs increases, 𝑃 (𝐼) concentrates more on the far left column in the histograms. For instance, the probability for 𝐼 being in the far left column equals 0.98 in Fig. 5(f). The calculated 𝑔 (2) (0) with the same number of random intensities will fluctuate more than the ones with less RGs in Fig. 5(a)–(e). Hence the SD of the calculated 𝑔 (2) (0) will increase as the number of RGs increases, too. The results in Fig. 5 are consistent with the ones in Fig. 4. The SD of the calculated 𝑔 (2) (0) can be decreased by increasing the number of random intensities. Fig. 6 shows how the SD changes with the number of random intensities for different number of RGs in superbunching pseudothermal light source. The degrees of secondorder coherence in Fig. 6 are calculated for 100, 200, 500, 1000, 2000, 5000, 10 000, and 20 000 random intensities, respectively. The SD is
4. Conclusions In conclusion, we have proved that high visibility temporal ghost imaging with classical light is possible. The superbunching pseudothermal light is helpful to improve the visibility of temporal ghost image. The visibility of temporal ghost image with superbunching pseudothermal light can approach 1 when more rotating ground glasses are 828
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employed in superbunching pseudothermal light source. The visibility of ghost imaging with classical light can be as large as the one of ghost imaging with entangled photon pairs. The results are helpful to understand the physics of ghost imaging and increase the image quality in future temporal ghost imaging applications.
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