## Entangled-photon coincidence fluorescence imaging

Optics Express, Vol. 16, Issue 20, pp. 16189-16194 (2008)

http://dx.doi.org/10.1364/OE.16.016189

Acrobat PDF (395 KB)

### Abstract

We describe fluorescence imaging using the second-order correlation of entangled photon pairs. The proposed method is based on the principle that one photon of the pair carries information on where the other photon has been absorbed and has produced fluorescence in a sample. Because fluorescent molecules serve as “detectors” breaking the entanglement, multiply-scattered fluorescence photons within the sample do not cause image blur. We discuss experimental implementations.

© 2008 Optical Society of America

## 1. Introduction

1. J. P. Dowling and G. J. Milburn, “Quantum technology: the second quantum revolution,” Phil. Trans. R. Soc. London A **361**, 1655–1674 (2003); [CrossRef]

2. N. Gisin and R. Thew, “Quantum communication,” Nat. Photonics **1**, 165–171 (2007). [CrossRef]

3. 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]

5. 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]

7. A. Valencia, G. Scarcelli, M. D’Angelo, and Y. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. **94**, 063601 (2005). [CrossRef] [PubMed]

8. A. F. Abouraddy, P. R. Stone, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Entangled-photon imaging of a pure phase object,” Phys. Rev. Lett. **93**, 213903 (2004). [CrossRef] [PubMed]

## 2. Working principle

*τ*

_{1}-

*τ*

_{2}, is somewhat uncertain because of random optical and electrical time delays in the system. For example, fluorescence emission is a stochastic process with a lifetime, and the photon-counting module has a finite electronic timing jitter. These timing uncertainties, however, can be accommodated by having a correlation time window,

*T*, longer than these time delays [Fig. 1(b) and (c)]. In each correlation window, no more than one entangled photon pair should be dealt with, ideally, in order to avoid any erroneous correlation count.

## 3. Theory

9. R. J. Glauber, “Quantum theory of optical coherence,” Phys. Rev. **130**, 2529 (1963). [CrossRef]

*G*

^{(2)}is the intrinsic second-order correlation function computed in the case of ideal detectors, whereas

*G*

^{(2)}

*is the correlation function taking into account the temporal convolution imposed by the finite response time of fluorescence generation and photoelectric conversion;*

_{F}*F*(

*r*⃗

_{1},

*t*) denotes the probability function of fluorescence generation,

*F*(

*r*⃗

_{1},

*t*)=

*f*(

*r*⃗

_{1})exp(-

*t*/

*τ*

_{F})

*H*(

*t*)/

*τ*

*where F(*

_{F}*r*⃗

_{1}) describes the spatial distribution and concentration of fluorescence molecules,

*τ*

_{F}is the fluorescence lifetime, typically 0.2–5 ns, and

*H*(

*t*) is the Heaviside step function.

*γ*

_{1},

_{2}(

*t*) denotes the response function of the photo-electric detectors, that can be approximated to a Gaussian function with a width of 0.2–1 ns. D1 integrates all the fluorescence photons emitted in a sample volume

*V*. Therefore, the final measured quantity is the marginal coincidence counting rate

*R*

*:*

_{C}*f*(

*r*⃗

_{1}) from the measurement of

*R*

*(*

_{c}*r*⃗

_{2}).

*r*⃗

*in the SPDC crystal and have perfectly phase-matched wave vectors,*

_{c}*k*⃑

_{1}and

*k*⃑

_{2}, i.e.

*k*⃑

_{1}+

*k*⃑

_{2}=

*k*⃑

*where*

_{p}*k*⃑

*is the wave vector of the pump photon. From these properties, the beam propagation paths linking two points*

_{p}*r*⃗

_{1}and

*r*⃗

_{2}can be identified, as represented by a red shaded region in Fig. 2. From the quantum optical viewpoint, the second-order correlation is established as coherent linear superposition of all of the correlation beam paths. Nevertheless, the classical ray picture is adequate to understand the imaging relationship as follows. The phase matching condition leads to a Snell’s-law-like relation at the crystal interface: λ2 sinθ1=λ1sinθ2 where λ1 and λ2 denote the wavelengths of probe and reference photons, respectively. Simple ray tracing leads to an imaging equation [10

10. A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Entangled-photon Fourier optics,” J. Opt. Soc. Am. B **19**, 1174–1184 (2002). [CrossRef]

*f*

*denotes the focal length of the objective lens. This relation asserts that the imaging*

_{obj}*s*

_{1}, can be varied in the sample remotely by changing the distance,

*b*, of the

*r*⃗

_{1}, the conjugate location to

*r*⃗

_{2}. In this sense, the imaging system considered here is similar to classical wide-field fluorescence imaging.

10. A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Entangled-photon Fourier optics,” J. Opt. Soc. Am. B **19**, 1174–1184 (2002). [CrossRef]

*h*

_{1},

_{2}are the 3D classical impulse response function of the probe and reference arms, respectively. When the imaging relationship given by Eq. (4) is established with 1:1 magnification, we get

*I*

^{(2)}(

*r*⃗

_{1};

*r*⃗

_{2})≈

*f*(

*r*⃗

_{1})δ(

*r*⃗

_{1}-

*r*⃗

_{2}) and

*R*

*(*

_{C}*r*⃗

_{2})∝

*f*(

*r*⃗

_{2}) from Eq. (3). Imaging resolution can be analyzed by considering a point object

*f*(

*r*⃗

_{1})=

*δ*(

*r*⃗

_{1}). From standard diffraction calculation, it can be shown that the system’s resolution is identical to that of a classical standard microscope using the wavelength of

*λ*

_{1}.

## 4. Practical considerations

*R*

*≈*

_{C}*N*

_{0}(1-

*e*-

*)*

^{mεL}*η*

_{F}*η*

_{c}*η*

_{1}

*η*

_{2}, where

*N*

_{0}is the number of entangled photon pairs generated per second,

*m*the molar concentration,

*ε*the extinction coefficient,

*L*the sample thickness,

*η*

*the quantum efficiency of the fluorophore,*

_{F}*η*

*the geometrical collection efficiency of the bucket detector D1, and*

_{c}*η*

_{1},

_{2}are the quantum efficiency of the detectors. As an example, we consider a near-infrared fluorescence dye, Alexa-Fluor 700 (Invitrogen), which is widely used in biological imaging. Its peak absorption wavelength 700 nm is suited for a SPDC source pumped by an Argon laser at 351 nm [3

3. 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]

*ε*=~20 M

^{-1}µm

^{-1},

*η*

*=0.25, and*

_{F}*τ*

*=1 ns. Let us consider an experiment where D2 is a large scale array [10*

_{F}10. A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Entangled-photon Fourier optics,” J. Opt. Soc. Am. B **19**, 1174–1184 (2002). [CrossRef]

*N*

_{0}=4×10

^{6}pairs per second,

*T*=10 ns, and a pixel dead-time of 50 ns. Based on the Poisson distribution, the error probability of having more than one photon pair in 10 ns is only 0.08 %. For a sample with

*m*=100 µM and

*L*=20 µm, we estimate that about 4% of probe photons are absorbed in the sample. Using

*η*

*=0.5, and*

_{c}*η*

_{1},

_{2}=0.7, we get

*R*

*=10,000 counts per second. At this rate, the pixel acquisition time for collecting average 100 counts per pixel would be ~10 ms. Therefore, it will take about 100 sec to acquire an 8-bit 100×100 image. Alternatively, a single- or few-element detector can be scanned in 2D at the expense of acquisition time.*

_{C}## 5. Spectrally-encoded entangled fluorescence spectroscopy and imaging

*ω*

_{1}+

*ω*

_{2}=

*ω*

_{P}where

*ω*

_{1},

_{2},

*are the frequencies of probe, reference, and pump photons, respectively. This indicates that one can determine*

_{P}*ω*

_{1}, without directly analyzing the spectrum of the probe photon, by measuring

*ω*

_{2}of the reference photon [12

12. G. Scarcelli, A. Valencia, S. Gompers, and Y. Shih, “Remote spectral measurement using entangled photons,” Appl. Phys. Lett. **83**, 5560–5562 (2003). [CrossRef]

13. J. T. Motz, D. Yelin, B. J. Vakoc, B. E. Bouma, and G. J. Tearney, “Spectral- and frequency-encoded fluorescence imaging,” Opt. Lett. **30**, 2760–2762 (2005). [CrossRef] [PubMed]

## 6. Discussion and conclusion

*π*microscopy) have been used for fluorescence imaging. Second, multiple scattering of fluorescence photons in a sample does not blur images. Figure 4(a) illustrates a situation in which a fluorescence photon undergoes elastic scattering before reaching the detector D1. Scattering of the fluorescence photon may affect the detection time

*τ*

_{1}, but does not perturb the image retrieval because fluorescence is an incoherent process so that no position correlation exists between the fluorescence and reference photons. On the other hand, any elastic scattering events of the probe photon before it reaches the fluorophore [Fig. 4(b)] does affect the correlation property between the probe and reference photons, resulting in speckle-type image degradation. Third, only a single-element detector is required in the probe arm, which may collect all the fluorescence photons. This may be advantageous where a 2D detector array is not readily available at the fluorescence wavelength or when it is difficult to place it near the sample; for instance, in endoscopy. Two-photon microscopy also uses a bucket detector to collect fluorescence photons generated by nonlinear absorption, but it requires a high numerical aperture (NA) objective lens, mode-locked femtosecond laser, and beam scanning device [14

14. W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science **248**, 73–76 (1990). [CrossRef] [PubMed]

## References and links

1. | J. P. Dowling and G. J. Milburn, “Quantum technology: the second quantum revolution,” Phil. Trans. R. Soc. London A |

2. | N. Gisin and R. Thew, “Quantum communication,” Nat. Photonics |

3. | 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 |

4. | A. V. Belinskii and D. N. Klyshko, “Two-Photon Optics - Diffraction, Holography and Transformation of 2-Dimensional Signals,” JETP |

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

6. | A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Correlated imaging, quantum and classical,” Phys. Rev. A |

7. | A. Valencia, G. Scarcelli, M. D’Angelo, and Y. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. |

8. | A. F. Abouraddy, P. R. Stone, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Entangled-photon imaging of a pure phase object,” Phys. Rev. Lett. |

9. | R. J. Glauber, “Quantum theory of optical coherence,” Phys. Rev. |

10. | A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Entangled-photon Fourier optics,” J. Opt. Soc. Am. B |

11. | S. Bellis, R. Wilcock, and C. Jackson, “Photon counting imaging: the digital APD,” Proc. SPIE |

12. | G. Scarcelli, A. Valencia, S. Gompers, and Y. Shih, “Remote spectral measurement using entangled photons,” Appl. Phys. Lett. |

13. | J. T. Motz, D. Yelin, B. J. Vakoc, B. E. Bouma, and G. J. Tearney, “Spectral- and frequency-encoded fluorescence imaging,” Opt. Lett. |

14. | W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science |

**OCIS Codes**

(170.2520) Medical optics and biotechnology : Fluorescence microscopy

(270.0270) Quantum optics : Quantum optics

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: July 8, 2008

Revised Manuscript: August 22, 2008

Manuscript Accepted: August 29, 2008

Published: September 26, 2008

**Virtual Issues**

Vol. 3, Iss. 11 *Virtual Journal for Biomedical Optics*

**Citation**

Giuliano Scarcelli and Seok H. Yun, "Entangled-photon coincidence fluorescence imaging," Opt. Express **16**, 16189-16194 (2008)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-16-20-16189

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### References

- J. P. Dowling and G. J. Milburn, "Quantum technology: the second quantum revolution," Phil. Trans. R. Soc. London A 361, 1655-1674 (2003); [CrossRef]
- N. Gisin and R. Thew, "Quantum communication," Nat. Photonics 1, 165-171 (2007). [CrossRef]
- 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]
- A. V. Belinskii and D. N. Klyshko, "Two-Photon Optics - Diffraction, Holography and Transformation of 2-Dimensional Signals," JETP 105, 487-493 (1994).
- 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, "Correlated imaging, quantum and classical," Phys. Rev. A 70, 013802 (2004).Q4 [CrossRef]
- A. Valencia, G. Scarcelli, M. D'Angelo, and Y. Shih, "Two-photon imaging with thermal light," Phys. Rev. Lett. 94, 063601 (2005). [CrossRef] [PubMed]
- A. F. Abouraddy, P. R. Stone, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, "Entangled-photon imaging of a pure phase object," Phys. Rev. Lett. 93, 213903 (2004). [CrossRef] [PubMed]
- R. J. Glauber, "Quantum theory of optical coherence," Phys. Rev. 130, 2529 (1963). [CrossRef]
- A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, "Entangled-photon Fourier optics," J. Opt. Soc. Am. B 19, 1174-1184 (2002). [CrossRef]
- S. Bellis, R. Wilcock, and C. Jackson, "Photon counting imaging: the digital APD," Proc. SPIE 6068, 6068D (2006).
- G. Scarcelli, A. Valencia, S. Gompers, and Y. Shih, "Remote spectral measurement using entangled photons," Appl. Phys. Lett. 83, 5560-5562 (2003). [CrossRef]
- J. T. Motz, D. Yelin, B. J. Vakoc, B. E. Bouma, and G. J. Tearney, "Spectral- and frequency-encoded fluorescence imaging," Opt. Lett. 30, 2760-2762 (2005). [CrossRef] [PubMed]
- W. Denk, J. H. Strickler, and W. W. Webb, "Two-photon laser scanning fluorescence microscopy," Science 248, 73-76 (1990). [CrossRef] [PubMed]

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