## Periodic diffraction correlation imaging without a beam-splitter |

Optics Express, Vol. 20, Issue 3, pp. 2956-2966 (2012)

http://dx.doi.org/10.1364/OE.20.002956

Acrobat PDF (1679 KB)

### Abstract

In this paper, we proposed and demonstrated a new correlation imaging mechanism based on the periodic diffraction effect. In this effect, a periodic intensity pattern is generated at the output surface of a periodic point source array. This novel correlation imaging mechanism can realize super-resolution imaging, Nth-order ghost imaging without a beam-splitter and correlation microscopy.

© 2012 OSA

## 1. Introduction

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). [CrossRef] [PubMed]

2. D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of two-photon ghost interference and diffraction,” Phys. Rev. Lett. **74**, 3600–3603 (1995). [CrossRef] [PubMed]

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

23. Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A **79**, 053840 (2009). [CrossRef]

24. J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A **78**, 061802(R) (2008). [CrossRef]

## 2. Periodic diffraction effect

*LiTaO*

_{3}(PPLT) crystal, with input light fields. Each point source is a sub-source. The theoretical derivation of the periodic diffraction effect is similar to that of famous Talbot effect [26–28]. Talbot effect is well-known as the self-imaging of a grating at certain distances. Essentially, Talbot effect focuses on the periodicity of a grating’s diffraction pattern along propagation direction, while the periodic diffraction effect discussed in this paper focuses on the periodicity of the diffraction pattern along directions perpendicular to propagation direction.

*U*(

*z*) of a point

*x*at the diffraction plane after a 1-D grating illuminated by a coherent light with distance

*z*is where

*ω*is angular frequency,

*k*= 2

*π*/

*λ*is the wave number of the coherent light,

*x*, and

*a*is the period of the grating. The field amplitude decreases as

*I*(

*z*) equals to 〈|

*U*(

*z*)|

^{2}〉, where 〈·〉 means statistical average, and has the form as

*z*satisfying

*n*, where

*p*is an arbitrary integer. Note that the wavelength

*λ*is normally far smaller than the period

*a*and

*z*≫

*na*for any integer

*n*, so the first-order Taylor expansion of this phase term on

*na/z*is This approximation requires that

*z*has to not overweight (

*na*)

^{2}/

*λ*, which accounts for that the traditional Talbot effect is a near-field effect observed in Fresnel region. Then, the self-imaging distance or the Talbot distance

*z*is found to satisfy where

*n*and

*p*are arbitrary integers. Here, the Talbot length

*z*is noted as

_{T}*z*=

_{T}*a*

^{2}/

*λ*. Note that the situation with a 2-D grating is similar.

*x*axis, not

*z*axis. Under Fresnel approximation, this phase term with a fixed

*z*is By further Fraunhofer approximation where

*z*≫

*x*

^{2}

*/λ*, the phase term in Eq. (5) is approximated to be

*x*axis satisfying where

*n*and

*p*are arbitrary integers. Then the period

*T*of intensity pattern along

*x*axis is found to be

*N × N*square point source array with the point-to-point distance

*a*. Here,

*N*is the point sources’ number in a side of the array. The wave-function

*U*(

*x,y*) at a point (

*x,y*) in the diffraction plane located apart from the source with a distance

*z*is where {

*N*} is a positive integer set with the maximum integer

*N*,

*A*(

*m,n*),

*ϕ*(

*m,n*) are the amplitude and phase of light emitted from the sub-source (

*m,n*) respectively, and

*m,n*) and the point (

*x,y*),

*a*is the period of the point source array. For uniform coherent light source, the lights emitted from all sub-sources have the same amplitudes and phases. While for incoherent light source, the initial amplitudes and phases of any two sub-sources are independent random values and time-varying. Cross-spectral density of the light at the output surface of the generated periodic point source array is where subscript

*k*=

*c*(coherent light) or

*i*(incoherent light), and

*ρ*is the position vector of point source (

_{m,n}*m,n*). For uniform coherent light, the cross-spectral density is where

*I*

_{0}is the average light intensity. While for uniform incoherent light, the cross-spectral density is where

*δ*is Dirac function. For

*m*

_{1}=

*m*

_{2}and

*n*

_{1}=

*n*

_{2}at the same time

*δ*(

*ρ*

_{m1,n1}−

*ρ*

_{m2,n2}) = 1, otherwise it equals to 0. The generated intensity pattern’s period

*T*only depends on the effective phase 2

*πr/λ*. Therefore, no matter with coherent light or incoherent light, the period of the generated intensity pattern is satisfying

*T*=

*λz/a*.

*N × N*point sources with a central wavelength

*λ*, the distance between adjacent point sources is

*a*. A charge-coupled device (CCD) camera used for recording the generated intensity patterns is placed after the source array with a distance

*z*. The CCD contains 200 × 200 pixel cells with each one’s size

*s*. Set

_{cell}*N*=20,

*λ*=750nm,

*a*=75um,

*s*=75um, then the grey-scale maps detected by the CCD located at

_{cell}*z*=150mm, 75mm and 37.5mm away from the source array with both coherent light and incoherent light are respectively depicted in Fig. 1. The experimental results in Fig. 1 demonstrate that the period of the generated intensity patterns at a same distance with both coherent light and incoherent light are the same. This feature is in good agreement with Eq. (7).

*T*only depends on the central wavelength of the source, the detecting distance and the period of the periodic point source array. If the point sources are randomly arranged, the generated intensity pattern will have no periodicity, which is shown in Fig. 2.

## 3. Thermal-light correlation imaging based on the periodic diffraction effect

### 3.1. Theory of second-order periodic diffraction thermal-light correlation imaging (PDCI)

*D*

_{2}is used for recording the total transmission or reflected light intensity

*I*

_{2}from the object, which is placed after the beam-splitter with a distance

*z*

_{2}in the signal path. While in the reference path, a scanning detector

*D*

_{1}is placed after the beam-splitter with a distance

*z*

_{1}to record the diffraction intensity patterns

*I*

_{1}(

*x*

_{1},

*y*

_{1}), where (

*x*

_{1},

*y*

_{1}) is the transverse position coordinate in the detecting plane of

*D*

_{1}. Set

*z*

_{1}=

*z*

_{2}, then the image of the object can be obtained by coincidence measurement because of the point-to-point correlation between the signal beam and the reference beam. In the experiments, the ghost image is retrieved by calculating 〈Δ

*I*

_{1}(

*x*

_{1},

*y*

_{1})Δ

*I*

_{2}〉.

23. Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A **79**, 053840 (2009). [CrossRef]

24. J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A **78**, 061802(R) (2008). [CrossRef]

*ρ*stands for a position vector at the source plane.

*z*≫

*ρ*

^{2}

*/λ*, the diffraction field-function collected by detector

*D*

_{1}is and the field-function collected by detector

*D*

_{2}in the object path is where

*t*(

*ρ*

_{2}) is the transmission or reflected function of the object and

*k*= 2

*π/λ*is the wave-number.

*G*

^{(2)}= 〈Δ

*I*

_{1}Δ

*I*

_{2}〉 = 〈

*I*

_{1}

*I*

_{2}〉 – 〈

*I*

_{1}〉〈

*I*

_{2}〉, because of the fact that point-to-point correlation lays in the correlation of intensity fluctuation. Substituting Eqs.(13) and (14) into Eq. (15), the function of so-called image is obtained as where

*R*is the radius of the source, the one-dimension form of

*somb*function is

*sinc*(

*x*) =

*sin*(

*x*)/

*x*and ⊗ is convolution operator. So the one-dimension function of image is given by

*x*=

*x*

_{1}, then

*t*(

*x*) =

*δ*(

*x*–

*x*

_{1}) and the retrieved image is an Airy disc described by

*N × N*mutual-independent point source array with a period

*a*. By theoretically analysis, the resolution of PDCI is where

*aN*is the side length of the source array.

*s*= 75

_{cell}*um*, and the period of the point source array

*a*=75um. The mutual-independent point source array is achieved by resetting the amplitude and phase of each point source with statistically independent random numbers for each record. The imaging distance

*z*=75mm, and the corresponding period of the generated intensity pattern

*T*=7.5mm. The intensity patterns recorded by the object CCD and the reference CCD are denoted by

*I*and

_{o}*I*, respectively. The image of a point, corresponding to the object CCD’s pixel cell (50,50), is retrieved by calculating 〈Δ

_{r}*I*(50, 50)Δ

_{o}*I*〉 with 2000 records. The retrieved images are shown in Fig. 4. The image’s size depends on CCDs’ sensing area, which is 15mm×15mm. Therefore, the retrieved image of the point (50,50) is composed of four bight specks, as shown in Fig. 4. Here, Fig. 4(a) is the retrieved image with a 10 × 10 point source array, while Fig. 4(b) is the retrieved image with a 20 × 20 point source array. The resolutions of PDCI in Fig. 4(a) and Fig. 4(b) are 0.75mm and 0.375mm, respectively, which are in good agreement with resolution expression Eq. (18).

_{r}*a*=75um, and each CCD is a 200×200 pixel cell array with

*s*=75um. The bucket detector is achieved by recording the sum of all intensity elements of matrix

_{cell}*I*for each time. The images of the mask can be retrieved by calculating 〈Δ(Σ

_{o}*I*(

_{o}*m*,

*n*))Δ

*I*〉, where

_{r}*m*,

*n*=1,...,200. Then, the retrieved images of the mask with distance

*z*=75mm and 150mm are shown in Fig. 5.(b) and Fig. 5.(c), respectively. Similar to the point-imaging experiments, four characters are obtained in Fig. 5.(b) and only one character is obtained in Fig. 5.(c). Compared with the image in Fig. 5.(c), the image in Fig. 5.(b) is more clear because it has a higher resolution according to Eq. (18).

*a*=75um, the imaging distance

*z*=75mm. Then the resolution will be 7.5um, and it can be further improved by decreasing the imaging distance

*z*. Therefore, the PDCI is capacity of realizing super-resolution imaging in theory. However, the imaging resolution of PDCI is restricted by the finite size of CCD. Because given a CCD with a fixed size, more duplicated images will be collected by using a shorter distance and each of them will be detected by the smaller amount of CCD pixels.

*T*is smaller than half of the size of CCD’s detecting area, then the PDCI can be realized with only one CCD. Part of the CCD array is used as a bucket detector, and the other part is used as a scanning detector. In addition, the PDCI has a higher resolution at a closer imaging distance in theory. Therefore, a new imaging technology similar to microscopy based on spatial correlation, named correlation microscopy, can be realized by PDCI, although the resolution reported in this paper is still far away from the useful microscopy applications. This work is being investigated.

### 3.2. High-order PDCI

*N*increases [12

12. Q. Liu, X. H. Chen, K. H. Luo, W. Wu, and L. A. Wu, “Role of multiphoton bunching in high-order ghost imaging with thermal light sources,” Phys. Rev. A **79**, 053844 (2009). [CrossRef]

*Fig.*3. Take the third-order PDCI for example, in the experiments, three CCDs are used: an object CCD (

*D*

_{1}) and two reference CCDs (

*D*

_{2}and

*D*

_{3}).

*D*

_{1}is placed after the mask, while

*D*

_{2}and

*D*

_{3}are placed in any two positions marked by dotted line square in

*Fig.*3.(

*b*). The intensity patterns synchronously recorded by

*D*

_{1},

*D*

_{2}and

*D*

_{3}are

*I*

_{1},

*I*

_{2}and

*I*

_{3}, respectively. Then the third-order correlation function of the image is given by where

*t*(

*ρ*) is the transmission function of the mask. The simulation of the third-order PDCI is similar to the second-order PDCI, except that a third intensity matrix

*I*

_{3}is introduced in the third-order PDCI. And it’s straightforward to extend to the Nth-order PDCI.

*Fig.*6. Clearly, the quality of the image obtained is improved with the increasing of the order

*N*, which is in good agreement with the theory [12

12. Q. Liu, X. H. Chen, K. H. Luo, W. Wu, and L. A. Wu, “Role of multiphoton bunching in high-order ghost imaging with thermal light sources,” Phys. Rev. A **79**, 053844 (2009). [CrossRef]

### 3.3. Shape merging effect in the PDCI

*T*in the periodic intensity pattern is named a NDP.

*T*. That’s why only one character image is obtained in Fig. 5.(c), and four character images are obtained in Fig. 5.(b). If the size of the object is larger than that of a NDP, then a shape-merged image is obtained in each NDP. In the following simulation experiment, a mask with English letters ‘sj’ at top left corner and with English letters ‘tu’ at lower right corner, shown as the Fig. 7.(a), is placed with distance

*z*= 75

*mm*. The other experimental parameters are the same to that in previous second-order PDCI experiments. Because of the relative position relation between ‘sj’ and ‘tu’, a merged ‘sjtu’ is retrieved by the PDCI, shown in Fig. 7.(b). This effect of the PDCI is named shape merging effect.

*T*in the imaging plane. The complete image can only be obtained by coincidence measurement between the sum of total intensity values detected by the senders and the intensity distribution detected by the receiver. The intensity informations detected by the two senders are actually the secret key to each other.

## 4. Conclusions

## Acknowledgments

## References and links

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 |

2. | D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of two-photon ghost interference and diffraction,” Phys. Rev. Lett. |

3. | R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Two-photon coincidence imaging with a classical source,” Phys. Rev. Lett. |

4. | A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light comparing entanglement and classical correlation,” Phys. Rev. Lett. |

5. | F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. |

6. | A. Valencia, G. Scarcelli, M. DAngelo, and Y. H. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. |

7. | M. H. Zhang, Q. Wei, X. Shen, Y. F. Liu, H. L. Liu, J. Cheng, and S. S. Han, “Lensless Fourier-transform ghost imaging with classical incoherent light,” Phys. Rev. A |

8. | Y. H. Zhai, X. H. Chen, D. Zhang, and L. A. Wu, “Two-photon interference with true thermal light,” Phys. Rev. A |

9. | X. H. Chen, Q. Liu, K. H. Luo, and L. A. Wu, “Lensless ghost imaging with true thermal light,” Opt. Lett. |

10. | Y. F. Bai and S. S. Han, “Ghost imaging with thermal light by third-order correlation,” Phys. Rev. A |

11. | J. B. Liu and Y. H. Shih, “Nth-order coherence of thermal light,” Phys. Rev. A |

12. | Q. Liu, X. H. Chen, K. H. Luo, W. Wu, and L. A. Wu, “Role of multiphoton bunching in high-order ghost imaging with thermal light sources,” Phys. Rev. A |

13. | Y. Zhou, J. Simon, J. B. Liu, and Y. H. Shih, “Third-order correlation function and ghost imaging of chaotic thermal light in the photon counting regime,” Phys. Rev. A |

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

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

16. | R. Meyers, K. S. Deacon, and Y. H. Shih, “Ghost imaging experiment by measuring reflected photons,” Phys. Rev. A |

17. | N. S. Bisht, E. K. Sharma, and H. C. Kandpal, “Experimental observation of lensless ghost imaging by measuring reflected photons,” Opt. Lasers Engineer. |

18. | L. Basano and P. Ottonello, “Diffuse-reflection ghost imaging from a double-strip illuminated by pseudo-thermal light,” Opt. Commun. |

19. | W. L. Gong, P. L. Zhang, X. Shen, and S. S. Han, “Ghost pinhole imaging in Fraunhofer region,” Appl. Phys. Lett. |

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

21. | X. B. Song, J. Xiong, X. D. Zhang, and K. G. Wang, “Second-order Talbot self-imaging with pseudothermal light,” Phys. Rev. A |

22. | K. H. Luo, X. H. Chen, Q. Liu, and L. A. Wu, “Nonlocal Talbot self-imaging with incoherent light,” Phys. Rev. A |

23. | Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A |

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

25. | S. B. He, X. Shen, H. Wang, W. L. Gong, and S. S. Han, “Ghost diffraction without a beamsplitter,” Appl. Phys. Lett. |

26. | H. F. Talbot, Fhilos. Mag. |

27. | M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. |

28. | M. V. Berry and E. Bodenschatz, “Caustics, multiply reconstructed by Talbot interference,” J. Mod. Opt. |

**OCIS Codes**

(030.1670) Coherence and statistical optics : Coherent optical effects

(110.1650) Imaging systems : Coherence imaging

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: November 2, 2011

Revised Manuscript: November 30, 2011

Manuscript Accepted: November 30, 2011

Published: January 24, 2012

**Virtual Issues**

Vol. 7, Iss. 3 *Virtual Journal for Biomedical Optics*

**Citation**

Hu Li, Zhipeng Chen, Jin Xiong, and Guihua Zeng, "Periodic diffraction correlation imaging without a beam-splitter," Opt. Express **20**, 2956-2966 (2012)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-20-3-2956

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### References

- 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). [CrossRef] [PubMed]
- D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of two-photon ghost interference and diffraction,” Phys. Rev. Lett. 74, 3600–3603 (1995). [CrossRef] [PubMed]
- R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Two-photon coincidence imaging with a classical source,” Phys. Rev. Lett. 89, 113601 (2002). [CrossRef] [PubMed]
- A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light comparing entanglement and classical correlation,” Phys. Rev. Lett. 93, 093602 (2004). [CrossRef] [PubMed]
- F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94, 183602 (2005). [CrossRef] [PubMed]
- A. Valencia, G. Scarcelli, M. DAngelo, and Y. H. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. 94, 063601 (2005). [CrossRef] [PubMed]
- M. H. Zhang, Q. Wei, X. Shen, Y. F. Liu, H. L. Liu, J. Cheng, and S. S. Han, “Lensless Fourier-transform ghost imaging with classical incoherent light,” Phys. Rev. A 75, 021803(R) (2007). [CrossRef]
- 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). [CrossRef]
- 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). [CrossRef] [PubMed]
- Y. F. Bai and S. S. Han, “Ghost imaging with thermal light by third-order correlation,” Phys. Rev. A 76, 043828 (2007). [CrossRef]
- J. B. Liu and Y. H. Shih, “Nth-order coherence of thermal light,” Phys. Rev. A 79, 023819 (2009). [CrossRef]
- Q. Liu, X. H. Chen, K. H. Luo, W. Wu, and L. A. Wu, “Role of multiphoton bunching in high-order ghost imaging with thermal light sources,” Phys. Rev. A 79, 053844 (2009). [CrossRef]
- Y. Zhou, J. Simon, J. B. Liu, and Y. H. Shih, “Third-order correlation function and ghost imaging of chaotic thermal light in the photon counting regime,” Phys. Rev. A 81, 043831 (2010). [CrossRef]
- K. W. C. Chan, M. N. O’Sullivan, and R. W. Boyd, “Two-color ghost imaging,” Phys. Rev. A 79, 033808 (2009). [CrossRef]
- S. Karmakar and Y. H. Shih, “Two-color ghost imaging with enhanced angular resolving power,” Phys. Rev. A 81, 033845 (2010). [CrossRef]
- R. Meyers, K. S. Deacon, and Y. H. Shih, “Ghost imaging experiment by measuring reflected photons,” Phys. Rev. A 77, 041801(R) (2008). [CrossRef]
- N. S. Bisht, E. K. Sharma, and H. C. Kandpal, “Experimental observation of lensless ghost imaging by measuring reflected photons,” Opt. Lasers Engineer. 48, 671–675 (2010). [CrossRef]
- L. Basano and P. Ottonello, “Diffuse-reflection ghost imaging from a double-strip illuminated by pseudo-thermal light,” Opt. Commun. 283, 2657–2661 (2010). [CrossRef]
- W. L. Gong, P. L. Zhang, X. Shen, and S. S. Han, “Ghost pinhole imaging in Fraunhofer region,” Appl. Phys. Lett. 95, 071110 (2009). [CrossRef]
- J. Cheng, “Ghost imaging through turbulent,” Opt. Express 17, 7916–7921 (2009). [CrossRef] [PubMed]
- X. B. Song, J. Xiong, X. D. Zhang, and K. G. Wang, “Second-order Talbot self-imaging with pseudothermal light,” Phys. Rev. A 82, 033823 (2010). [CrossRef]
- K. H. Luo, X. H. Chen, Q. Liu, and L. A. Wu, “Nonlocal Talbot self-imaging with incoherent light,” Phys. Rev. A 82, 033803 (2010) [CrossRef]
- Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79, 053840 (2009). [CrossRef]
- J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78, 061802(R) (2008). [CrossRef]
- S. B. He, X. Shen, H. Wang, W. L. Gong, and S. S. Han, “Ghost diffraction without a beamsplitter,” Appl. Phys. Lett. 96, 181108 (2010).
- H. F. Talbot, Fhilos. Mag. 9, 401–407 (1836).
- M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43, 2139–2164 (1996). [CrossRef]
- M. V. Berry and E. Bodenschatz, “Caustics, multiply reconstructed by Talbot interference,” J. Mod. Opt. 46, 349–365 (1999).

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