## Enhancing image contrast using coherent states and photon number resolving detectors

Optics Express, Vol. 18, Issue 6, pp. 6033-6039 (2010)

http://dx.doi.org/10.1364/OE.18.006033

Acrobat PDF (350 KB)

### Abstract

We experimentally map the transverse profile of diffraction-limited beams using photon-number-resolving detectors. We observe strong compression of diffracted beam profiles for high detected photon number. This effect leads to higher contrast than a conventional irradiance profile between two Airy disk-beams separated by the Rayleigh criterion.

© 2010 Optical Society of America

## 1. Introduction

1. Seth Lloyd, “Enhanced sensitivity of photodetection via quantum illumination,” Science **321**, 1463 (2008). [CrossRef] [PubMed]

4. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: Exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. **85**, 2733–2736 (2000). [CrossRef] [PubMed]

6. J. P. Dowling, “Quantum optical metrology - the lowdown on high-NOON States,” Contemp. Phys. **49**, 125 (2008). [CrossRef]

7. Mark A. Rubin and Sumanth Kaushik , “Loss-induced limits to phase measurement precision with maximally entangled states,” Phys. Rev. A **75**, 053805 (2007), [CrossRef]

8. C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D **23**, 1693 (1981). [CrossRef]

9. Ryan T. Glasser, Hugo Cable, and Jonathan P. Dowling, Francesco De Martini, Fabio Sciarrino, and Chiara Vitelli, “Entanglement-seeded, dual, optical parametric amplification: Applications to quantum imaging and metrology,” Phys. Rev. A **78**, 012339 (2008). [CrossRef]

10. S. J. Bentley and R. W. Boyd, “Nonlinear optical lithography with ultra-high sub-Rayleigh resolution,” Opt. Express **12**, 5735 (2004). [CrossRef] [PubMed]

14. N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H.-A. Bachor, and P. K. Lam, “A Quantum Laser Pointer.” Science **301**, 940 (2003). [CrossRef] [PubMed]

*et al*. proposed that photon-number-resolving strategies could result in high-resolution images beyond the standard Rayleigh criterion [16

16. V. Giovannetti, S. Lloyd, L. Maccone, and J. Shapiro, “Sub-Rayleigh-diffraction-bound quantum imaging,” Phys. Rev. A **79**, 013827 (2009). [CrossRef]

## 2. Experimental Results

### 2.1. Setup

17. A. J. Miller, S. Nam, J. M. Martinis, and A. V. Sergienko “Demonstration of a low-noise near-infrared photon counter with multiphoton discrimination,” Appl. Phys. Lett. **83**, 791 (2003). [CrossRef]

### 2.2. Fringe Compression

*μ*m-wide slit. We scan a standard 9

*μ*m core single mode fiber coupled to the TES detector across the diffraction profile in 50

*μ*m steps, ≈23 cm from the slit, and detect a mean of ≈3.6 photons per pulse at the position of maximum irradiance. At each fiber position, we resolve photon number by integrating TES photoresponse pulses and placing them into histograms to reveal the photon number distribution [17

17. A. J. Miller, S. Nam, J. M. Martinis, and A. V. Sergienko “Demonstration of a low-noise near-infrared photon counter with multiphoton discrimination,” Appl. Phys. Lett. **83**, 791 (2003). [CrossRef]

*k*where

_{nk}*k*is the detected photon number and

*n*is the number of counts at a given

_{k}*k*[see Fig. 1(a)]. We observe the sinc

^{2}(

*d*sin

*θ*/

*λ*) dependence we expect from diffraction through a single slit, where

*d*is the slit width,

*λ*is the wavelength, and

*θ*is the angle (

*θ*≈

*x*/

_{t}*z*where

*x*is the transverse position seen in Fig. 1 and

_{t}*z*≈ 23 cm is the distance from the diffracting slit to the detection plane). We then generate a fit to this curve by minimizing the least-squares error between

*μ*· sinc

^{2}(

*d*sin

*θ*/

*λ*) and the data by varying

*μ*and

*θ*. We derive the photon-number profiles straightforwardly from the Poissonian distribution of a coherent source with a spatially varying detected mean photon number

*μ*′ =

*μ*· sinc

^{2}(

*d*sin

*θ*/

*λ*) [see Fig. 1(c)].

11. C. F. Wildfeuer, A. J. Pearlman, J. Chen, J. Fan, A. Migdall, and J. P. Dowling, “Resolution and sensitivity of a Fabry-Perot interferometer with a photon-number-resolving detector,” Phys. Rev. A , **80**, 043822 (2009). [CrossRef]

*T*[11

11. C. F. Wildfeuer, A. J. Pearlman, J. Chen, J. Fan, A. Migdall, and J. P. Dowling, “Resolution and sensitivity of a Fabry-Perot interferometer with a photon-number-resolving detector,” Phys. Rev. A , **80**, 043822 (2009). [CrossRef]

12. G. Khoury, H. S. Eisenberg, E. J. S. Fonseca, and D. Bouwmeester, “Nonlinear Interferometry via Fock-State Projection,” Phys. Rev. Lett **96**, 203601 (2006). [CrossRef] [PubMed]

*k*photons reduces to a Poisson distribution:

*T*|

^{2}is replaced by the spatial profile of the beam, which in this case is the single slit far-field diffraction profile (sinc

^{2}(

*d*sin

*θ*/

*λ*)).

### 2.3. Beam Contrast

*μ*m pinhole, obtaining a beam with an Airy disk profile. A 100 mm focal length lens, ≈155 mm after the pinhole, focuses the diffraction profile. The beam is split and recombined non-interferometrically using polarizing beam splitters in a Mach-Zehnder interferometer configuration [see Fig. 2(b)]. We approximately equalize the photon flux in the arms using a fiber polarization controller and tune the spatial separation of the two beams by moving one of the mirrors. The detection system scans across the two nearly identical images for several different beam separation values while recording all photoresponse pulses. The maximum detected mean photon number per beam for all measurements is ≈5.3, and we distinguish photon numbers up to twelve.

*J*

_{1}is a Bessel function of the first kind,

*x*and

*y*define the position in the image plane,

*f*is the focal length, and

*D*is the aperture radius of the diffracting pinhole. This separation occurs where the main lobe of one beam falls on the first minimum of the other, and the angular separation with respect to the aperture is given by 1.22

*λ f*/

*D*. The classical irradiance profile of two overlapping Airy disks is a saddle, and the contrast of this profile is defined as

*C*= (

*I*

_{max}−

*I*

_{saddle})/(

*I*

_{max}+

*I*

_{saddle}), where

*I*

_{max}and

*I*

_{saddle}are the intensities at the peak and saddle points [see Fig. 2(a)].

*k*≫

*μ*at a separation of the Rayleigh limit [see Fig. 2(c)]. At the smallest separation value studied, 90 % of the Rayleigh limit, a contrast of over 60 % is obtained for

*k*= 12. We note that the contrast value of 13 % obtained from the reconstructed classical profile just below the Rayleigh limit matches closely with the theoretical Rayleigh limit value of 15 %. As expected, the contrast of reconstructed profiles assuming the use of a conventional single photon detector is poor (< 5 %), because it does not exploit the full photon statistics. We show a sample set of measured photon-number-resolved profiles for a separation near the Rayleigh limit along with a theoretical fit derived analogously to the fit used for the single-slit configuration. Fitting parameters used account for the intensity difference of the two beams and the difference in optical path lengths. We show good agreement between the fits and data.

## 3. Discussion

*T*|

^{2}in Eq. (1) and the appropriate photon statistics. As shown in Fig. 3 for

*k*= 10, the Fock state (|n〉 = 10) shows the highest degree of narrowing of the central fringe.

16. V. Giovannetti, S. Lloyd, L. Maccone, and J. Shapiro, “Sub-Rayleigh-diffraction-bound quantum imaging,” Phys. Rev. A **79**, 013827 (2009). [CrossRef]

19. C. Sparrow, Astro Phys. **44**, 76 (1916). [CrossRef]

16. V. Giovannetti, S. Lloyd, L. Maccone, and J. Shapiro, “Sub-Rayleigh-diffraction-bound quantum imaging,” Phys. Rev. A **79**, 013827 (2009). [CrossRef]

## Acknowledgements

## References and links

1. | Seth Lloyd, “Enhanced sensitivity of photodetection via quantum illumination,” Science |

2. | T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four entangled photons,” Science |

3. | J. J. Bollinger, W. M.
Itano, D. J. Wineland, and D. J.
Heinzen, “Optimal frequency measurements with maximally correlated states,” Phys. Rev. A |

4. | N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: Exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. |

5. | M. D’Angelo, M. V. Chekhova, and Y. Shih, “Two-photon diffraction and quantum lithography,” Phys. Rev. Lett. |

6. | J. P. Dowling, “Quantum optical metrology - the lowdown on high-NOON States,” Contemp. Phys. |

7. | Mark A. Rubin and Sumanth Kaushik , “Loss-induced limits to phase measurement precision with maximally entangled states,” Phys. Rev. A |

8. | C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D |

9. | Ryan T. Glasser, Hugo Cable, and Jonathan P. Dowling, Francesco De Martini, Fabio Sciarrino, and Chiara Vitelli, “Entanglement-seeded, dual, optical parametric amplification: Applications to quantum imaging and metrology,” Phys. Rev. A |

10. | S. J. Bentley and R. W. Boyd, “Nonlinear optical lithography with ultra-high sub-Rayleigh resolution,” Opt. Express |

11. | C. F. Wildfeuer, A. J. Pearlman, J. Chen, J. Fan, A. Migdall, and J. P. Dowling, “Resolution and sensitivity of a Fabry-Perot interferometer with a photon-number-resolving detector,” Phys. Rev. A , |

12. | G. Khoury, H. S. Eisenberg, E. J. S. Fonseca, and D. Bouwmeester, “Nonlinear Interferometry via Fock-State Projection,” Phys. Rev. Lett |

13. | P. R. Hemmer, A. Muthukrishnan, M. O. Scully, and M. S. Zubairy, “Quantum Lithography with Classical Light,” Phys. Rev. Lett. |

14. | N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H.-A. Bachor, and P. K. Lam, “A Quantum Laser Pointer.” Science |

15. | Lord Rayleigh, Philos. Mag. 8, 261 (1879). |

16. | V. Giovannetti, S. Lloyd, L. Maccone, and J. Shapiro, “Sub-Rayleigh-diffraction-bound quantum imaging,” Phys. Rev. A |

17. | A. J. Miller, S. Nam, J. M. Martinis, and A. V. Sergienko “Demonstration of a low-noise near-infrared photon counter with multiphoton discrimination,” Appl. Phys. Lett. |

18. | C. C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge University Press, Cambridge, 2005). |

19. | C. Sparrow, Astro Phys. |

**OCIS Codes**

(030.5260) Coherence and statistical optics : Photon counting

(270.5290) Quantum optics : Photon statistics

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: November 23, 2009

Revised Manuscript: January 20, 2010

Manuscript Accepted: January 24, 2010

Published: March 11, 2010

**Citation**

A. J. Pearlman, A. Ling, E. A. Goldschmidt, C. F. Wildfeuer, J. Fan, and A. Migdall, "Enhancing image contrast using coherent states and photon number resolving detectors," Opt. Express **18**, 6033-6039 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-6-6033

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### References

- S. Lloyd, "Enhanced sensitivity of photodetection via quantum illumination," Science 321, 1463 (2008). [CrossRef] [PubMed]
- T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, "Beating the standard quantum limit with four entangled photons," Science 316, 726 (2007). [CrossRef] [PubMed]
- J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, "Optimal frequency measurements with maximally correlated states," Phys. Rev. A 54, R4649-R4652 (1996). [CrossRef] [PubMed]
- N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, "Quantum interferometric optical lithography: Exploiting entanglement to beat the diffraction limit," Phys. Rev. Lett. 85, 2733-2736 (2000). [CrossRef] [PubMed]
- M. D’Angelo, M. V. Chekhova, and Y. Shih, "Two-photon diffraction and quantum lithography," Phys. Rev. Lett. 87, 013602 (2001). [CrossRef] [PubMed]
- J. P. Dowling, "Quantum optical metrology - the lowdown on high-NOON States," Contemp. Phys. 49, 125 (2008). [CrossRef]
- M. A. Rubin and S. Kaushik, "Loss-induced limits to phase measurement precision with maximally entangled states," Phys. Rev. A 75, 053805 (2007), [CrossRef]
- C. M. Caves, "Quantum-mechanical noise in an interferometer," Phys. Rev. D 23, 1693 (1981). [CrossRef]
- R. T. Glasser, H. Cable, J. P. Dowling, F. De Martini, F. Sciarrino, and C. Vitelli, "Entanglement-seeded, dual, optical parametric amplification: Applications to quantum imaging and metrology," Phys. Rev. A 78, 012339 (2008). [CrossRef]
- S. J. Bentley and R. W. Boyd, "Nonlinear optical lithography with ultra-high sub-Rayleigh resolution," Opt. Express 12, 5735 (2004). [CrossRef] [PubMed]
- C. F. Wildfeuer, A. J. Pearlman, J. Chen, J. Fan, A. Migdall, and J. P. Dowling, "Resolution and sensitivity of a Fabry-Perot interferometer with a photon-number-resolving detector," Phys. Rev. A 80, 043822 (2009). [CrossRef]
- G. Khoury, H. S. Eisenberg, E. J. S. Fonseca, and D. Bouwmeester, "Nonlinear Interferometry via Fock-State Projection," Phys. Rev. Lett 96, 203601 (2006). [CrossRef] [PubMed]
- P. R. Hemmer, A. Muthukrishnan, M. O. Scully, and M. S. Zubairy, "Quantum Lithography with Classical Light," Phys. Rev. Lett. 96, 163603 (2006). [CrossRef] [PubMed]
- N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H.-A. Bachor, and P. K. Lam, "A Quantum Laser Pointer," Science 301, 940 (2003). [CrossRef] [PubMed]
- Lord Rayleigh, "Investigations in optics, with special reference to the spectroscope," Philos. Mag. 8, 261 (1879).
- V. Giovannetti, S. Lloyd, L. Maccone, and J. Shapiro, "Sub-Rayleigh-diffraction-bound quantum imaging," Phys. Rev. A 79, 013827 (2009). [CrossRef]
- A. J. Miller, S. Nam, J. M. Martinis, and A. V. Sergienko "Demonstration of a low-noise near-infrared photon counter with multiphoton discrimination," Appl. Phys. Lett. 83, 791 (2003). [CrossRef]
- C. C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge University Press, Cambridge, 2005).
- C. Sparrow, Astro Phys. 44, 76 (1916). [CrossRef]

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