## Quantum interference fringes beating the diffraction limit

Optics Express, Vol. 15, Issue 21, pp. 14244-14250 (2007)

http://dx.doi.org/10.1364/OE.15.014244

Acrobat PDF (560 KB)

### Abstract

Spatially formed two-photon interference fringes with fringe periods smaller than the diffraction limit are demonstrated. In the experiment, a fringe formed by two-photon NOON states with wavelength λ=702.2 nm is observed using a specially developed near-field scanning optical microscope probe and two-photon detection setup. The observed fringe period of 328.2 nm is well below the diffraction limit (351 nm = λ/2). Another experiment with a path-length difference larger than the coherent length of photons confirms that the observed fringe is due to two-photon interference.

© 2007 Optical Society of America

## 1. Introduction

2. 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]

7. T. B. Pittman, Y. H. Shih, A. V. Sergienko, and M. H. Rubin, “Experimental tests of Bell’s inequalities based on space-time and spin variables,” Phys. Rev. A **51**, 3495 – 3498 (1995). [CrossRef] [PubMed]

5. E. J. S. Fonseca, C. H. Monken, and S. Páuda, “Measurement of the de Broglie wavelength of a multiphoton wave packet,” Phys. Rev. Lett. **82**, 2868–2871 (1999). [CrossRef]

6. M. D–Angelo, M. V. Chekhova, and Y. Shih
, “Two-photon diffraction and quantum lithography,” Phys. Rev. Lett. **87**, 013602 (2001). [CrossRef]

10. Y. H. Kim, S. P. Kulik, and Y. Shih, “High-intensity pulsed source of space-time and polarization double-entangled photon pairs,” Phys. Rev. A **62**, 011802 (2000). [CrossRef]

11. J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-NOT gate,” Nature **426**, 264–267 (2003). [CrossRef] [PubMed]

12. E. Betzig, J. K. Trautman, T. D. Harris, J. S. Weiner, and R. L. Kostelak, “Breaking the diffraction barrier: optical microscopy of a nanometric scale,” Science **251**, 1468–1470 (1991). [CrossRef] [PubMed]

*λ*intersect at an angle of 2

*θ*(incidence angle

*θ*), the intensity profile of the laser shows fringes due to interference:

*I*=

*I*

_{0}(1 +

*cos*(2

*πr*/

*p*)), with a fringe period

*p*=

*λ*/2

*sinθ*and where

*r*is the position along the intersecting plane (Fig. 1). The period can never be smaller than

*p*=

_{min}*λ*/2, which is realized when

*θ*=

*π*/2. This is called the Rayleigh diffraction limit and is the best fringe resolution that can be achieved classically. However, quantum lithography [2

2. 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]

*N*entangled photons in spatially different optical modes (optical paths): |

*ψ*〉= (|

*N*〉

_{α}|0〉

_{β}〉 +|0〉

_{α}|

*N*〉

_{β})/√2 as an input state. In this case, the probability to find

*N*photons at a position

*r*is given by

*N*. By using a multi-photon absorbing photo-resist material, it is possible in principle to use this phenomenon to improve the resolution of the lithography.

## 2. Experimental setup

12. E. Betzig, J. K. Trautman, T. D. Harris, J. S. Weiner, and R. L. Kostelak, “Breaking the diffraction barrier: optical microscopy of a nanometric scale,” Science **251**, 1468–1470 (1991). [CrossRef] [PubMed]

^{-5}) of the conventional NSOM probe is insufficient for coincidence counting, for which the event detection probability is proportional to the square of the collection efficiency. We overcame this problem by using a specially developed NSOM probe with an elliptical opening (Fig. 2 inset, minor axis: 0.2 μm, major axis: 1.8 μm) based on a polarization NSOM probe (JASCO International Co., Ltd.). The collection efficiency was improved to 3.3×10

^{-3}maintaining the sub-wavelength resolution along the axis perpendicular to the fringes, giving a 2.0×10

^{3}= (3.3×10

^{-3}/ 7.5×10

^{-5})

^{2}improvement in the coincidence rate.

*VV*〉

_{a}+|

*HH*〉

_{a})/√2, via spontaneous parametric down conversion (SPDC) [10

10. Y. H. Kim, S. P. Kulik, and Y. Shih, “High-intensity pulsed source of space-time and polarization double-entangled photon pairs,” Phys. Rev. A **62**, 011802 (2000). [CrossRef]

*b*and

*c*, according to the polarization [11

11. J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-NOT gate,” Nature **426**, 264–267 (2003). [CrossRef] [PubMed]

*VV*〉

_{b}+|

*HH*〉

_{c})/√2. Then, the state is changed into a two-photon NOON state [15

15. C. K. Hong, Z. Y. Ou, and L. Mandel
, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. **59**, 2044~2046 (1987). [CrossRef] [PubMed]

16. C. C. Gerry and R. A. Campos, “Generation of maximally entangled photonic states with a quantum-optical Fredkin gate,” Phys. Rev. A **64**, 063814 (2001). [CrossRef]

_{d}|0〉

_{e}+|0〉

*|2〉*

_{d}_{e})/√2 by rotating the polarization in path b by 90 degrees. A coupling lens with a large numerical aperture (L; clear aperture = 3.6 mm, NA = 0.65) is used to superpose the two modes with a large intersecting angle 2

*θ*to give an interference fringe of small period (Eq. (1)). Since the two photons in a pair pass through the same optical components, the fluctuation of the path-length difference is minimized and results in super-stable interference fringes, which remain unchanged for about ten hours. Then, an NSOM probe (inset) is scanned with a piezo-actuator along the focal plane. The single-mode fiber output of the probe is divided by a 50:50 fiber coupler and detected by single-photon counting modules (SPCM-AQR14FC, PerkinElmer). The coincidence events between the two SPCMs are recorded by a gated photon counter (SR-400, Stanford Research Systems).

## 3. Results and discussion

*I*-

_{max}*I*)/(

_{min}*I*+

_{max}*I*)) of 70.7 ± 7.8%. It should be mentioned that for such a high-visibility interference fringe, considerable work was required to improve the quality of the entangled photons incident on the calcite crystal via a single-mode fiber (shown in Fig. 2) using polarization compensators (not shown in Fig. 2). The data shown in Figs. 3(a) and 3(b) were measured by scanning the probe position from 0 μm to 1.5 μm (single scan), changing the setup for single-count data (Fig. 3(a)) and coincidence data (Fig. 3(b)) using computer-controlled motorized stages. For reference, single count rates of one of the two detectors measured at the same time with the coincidence count rate are shown as blue dots. A slight modulation (about 10% on average) with a period of about 656 nm is observed, which may be caused by single photon interference of the two-photon source, which has still imperfection compared to an ideal source.

_{min}9. K. Edamatsu, R. Shimizu, and T. Itoh, “Measurement of the photonic de Broglie wavelength of entangled photon pairs generated by spontaneous parametric down-conversion,” Phys. Rev. Lett. **89**, 213601 (2002). [CrossRef] [PubMed]

9. K. Edamatsu, R. Shimizu, and T. Itoh, “Measurement of the photonic de Broglie wavelength of entangled photon pairs generated by spontaneous parametric down-conversion,” Phys. Rev. Lett. **89**, 213601 (2002). [CrossRef] [PubMed]

## 4. Conclusions

17. K. Edamatsu, G. Oohata, R. Shimizu, and T. Itoh, “Generation of ultraviolet entangled photons in a semiconductor,” Nature **431**, 167–170 (2004). [CrossRef] [PubMed]

18. R. M. Stevenson, R. J. Young, P. Atkinson, K. Cooper, D. A. Ritchie, and A. J. Shields, “A semiconductor source of triggered entangled photon pairs,” Nature **439**, 179–182 (2006). [CrossRef] [PubMed]

19. K.-S. Lee, D.-Y. Yang, S. H. Park, and R. H. Kim, “Recent developments in the use of two-photon polymerization in precise 2D and 3D microfabrications,” Polym. Adv. Technol. **17**, 72–82 (2006). [CrossRef]

*N*-photon interference is not

*λ*/2 but

*λ*/2

*N*[2

2. 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]

8. T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four entangled photons,” Science **316**, 726–729 (2007). [CrossRef] [PubMed]

9. K. Edamatsu, R. Shimizu, and T. Itoh, “Measurement of the photonic de Broglie wavelength of entangled photon pairs generated by spontaneous parametric down-conversion,” Phys. Rev. Lett. **89**, 213601 (2002). [CrossRef] [PubMed]

20. P. Walther, J.-W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, “De Broglie wavelength of a non-local four-photon state,” Nature **429**, 158–161 (2004). [CrossRef] [PubMed]

8. T. Nagata, R. Okamoto, J. L. O’Brien, K. Sasaki, and S. Takeuchi, “Beating the standard quantum limit with four entangled photons,” Science **316**, 726–729 (2007). [CrossRef] [PubMed]

## Acknowledgements

^{st}Century COE Program, and Special Coordination Funds for Promoting Science and Technology.

## References and links

1. | L. Rayleigh, “Investigations in optics, with special reference to the spectroscope,” Phil. Mag. |

2. | 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. |

3. | P. Kok, A. N. Boto, D. S. Abrams, C. P. Williams, S. L. Braunstein, and J. P. Dowling, “Quantum-interferometric optical lithography: towards arbitrary two-dimensional patterns,” Phys Rev. A |

4. | G. Bjork and L. L. Sánchez-Soto, “Entangled-state Lithography: Tailoring any pattern with a single state,” Phys. Rev. Lett. |

5. | E. J. S. Fonseca, C. H. Monken, and S. Páuda, “Measurement of the de Broglie wavelength of a multiphoton wave packet,” Phys. Rev. Lett. |

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

7. | T. B. Pittman, Y. H. Shih, A. V. Sergienko, and M. H. Rubin, “Experimental tests of Bell’s inequalities based on space-time and spin variables,” Phys. Rev. A |

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

9. | K. Edamatsu, R. Shimizu, and T. Itoh, “Measurement of the photonic de Broglie wavelength of entangled photon pairs generated by spontaneous parametric down-conversion,” Phys. Rev. Lett. |

10. | Y. H. Kim, S. P. Kulik, and Y. Shih, “High-intensity pulsed source of space-time and polarization double-entangled photon pairs,” Phys. Rev. A |

11. | J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-NOT gate,” Nature |

12. | E. Betzig, J. K. Trautman, T. D. Harris, J. S. Weiner, and R. L. Kostelak, “Breaking the diffraction barrier: optical microscopy of a nanometric scale,” Science |

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

14. | J. Xiong, D. Z. Cao, F. Huang, H. G. Li, X. J. Sun, and K. Wang, “Experimental observation of classical subwavelength interference with a pseudothermal light source,” Phys. Rev. Lett. |

15. | C. K. Hong, Z. Y. Ou, and L. Mandel
, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. |

16. | C. C. Gerry and R. A. Campos, “Generation of maximally entangled photonic states with a quantum-optical Fredkin gate,” Phys. Rev. A |

17. | K. Edamatsu, G. Oohata, R. Shimizu, and T. Itoh, “Generation of ultraviolet entangled photons in a semiconductor,” Nature |

18. | R. M. Stevenson, R. J. Young, P. Atkinson, K. Cooper, D. A. Ritchie, and A. J. Shields, “A semiconductor source of triggered entangled photon pairs,” Nature |

19. | K.-S. Lee, D.-Y. Yang, S. H. Park, and R. H. Kim, “Recent developments in the use of two-photon polymerization in precise 2D and 3D microfabrications,” Polym. Adv. Technol. |

20. | P. Walther, J.-W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, “De Broglie wavelength of a non-local four-photon state,” Nature |

**OCIS Codes**

(270.5290) Quantum optics : Photon statistics

(270.5585) Quantum optics : Quantum information and processing

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: August 23, 2007

Revised Manuscript: October 11, 2007

Manuscript Accepted: October 11, 2007

Published: October 12, 2007

**Virtual Issues**

Vol. 2, Iss. 11 *Virtual Journal for Biomedical Optics*

**Citation**

Yoshio Kawabe, Hideki Fujiwara, Ryo Okamoto, Keiji Sasaki, and Shigeki Takeuchi, "Quantum interference fringes beating the diffraction limit," Opt. Express **15**, 14244-14250 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-21-14244

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### References

- L. Rayleigh, "Investigations in optics, with special reference to the spectroscope," Phil. Mag. 8, 261-274 (1879).
- 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]
- P. Kok, A. N. Boto, D. S. Abrams, C. P. Williams, S. L. Braunstein, and J. P. Dowling, "Quantum-interferometric optical lithography: towards arbitrary two-dimensional patterns," Phys Rev. A 63, 063407 (2001). [CrossRef]
- G. Bjork and L. L. Sánchez-Soto, "Entangled-state Lithography: Tailoring any pattern with a single state," Phys. Rev. Lett. 86, 4516-4519 (2001). [CrossRef] [PubMed]
- E. J. S. Fonseca, C. H. Monken, and S. Páuda, "Measurement of the de Broglie wavelength of a multiphoton wave packet," Phys. Rev. Lett. 82, 2868-2871 (1999). [CrossRef]
- M. D’Angelo, M. V. Chekhova, and Y. Shih, "Two-photon diffraction and quantum lithography," Phys. Rev. Lett. 87, 013602 (2001). [CrossRef]
- T. B. Pittman, Y. H. Shih, A. V. Sergienko, and M. H. Rubin, "Experimental tests of Bell’s inequalities based on space-time and spin variables," Phys. Rev. A 51, 3495 - 3498 (1995). [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, 726-729 (2007). [CrossRef] [PubMed]
- K. Edamatsu, R. Shimizu, and T. Itoh, "Measurement of the photonic de Broglie wavelength of entangled photon pairs generated by spontaneous parametric down-conversion," Phys. Rev. Lett. 89, 213601 (2002). [CrossRef] [PubMed]
- Y. H. Kim, S. P. Kulik, and Y. Shih, "High-intensity pulsed source of space-time and polarization double-entangled photon pairs," Phys. Rev. A 62, 011802 (2000). [CrossRef]
- J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, "Demonstration of an all-optical quantum controlled-NOT gate," Nature 426, 264-267 (2003). [CrossRef] [PubMed]
- E. Betzig, J. K. Trautman, T. D. Harris, J. S. Weiner, and R. L. Kostelak, "Breaking the diffraction barrier: optical microscopy of a nanometric scale," Science 251, 1468-1470 (1991). [CrossRef] [PubMed]
- Y. H. Zhai, X.-H. Chen, D. Zhang, L.-A. Wu, "Two-photon interference with true thermal light," Phys. Rev. A 72, 043805 (2005). [CrossRef]
- J. Xiong, D. Z. Cao, F. Huang, H. G. Li, X. J. Sun, and K. Wang, "Experimental observation of classical subwavelength interference with a pseudothermal light source," Phys. Rev. Lett. 94, 173601 (2005). [CrossRef] [PubMed]
- C. K. Hong, Z. Y. Ou, and L. Mandel, "Measurement of subpicosecond time intervals between two photons by interference," Phys. Rev. Lett. 59, 2044~2046 (1987). [CrossRef] [PubMed]
- C. C. Gerry and R. A. Campos, "Generation of maximally entangled photonic states with a quantum-optical Fredkin gate," Phys. Rev. A 64, 063814 (2001). [CrossRef]
- K. Edamatsu, G. Oohata, R. Shimizu, and T. Itoh, "Generation of ultraviolet entangled photons in a semiconductor," Nature 431, 167-170 (2004). [CrossRef] [PubMed]
- R. M. Stevenson, R. J. Young, P. Atkinson, K. Cooper, D. A. Ritchie, and A. J. Shields, "A semiconductor source of triggered entangled photon pairs," Nature 439, 179-182 (2006). [CrossRef] [PubMed]
- K.-S. Lee, D.-Y. Yang, S. H. Park, and R. H. Kim, "Recent developments in the use of two-photon polymerization in precise 2D and 3D microfabrications," Polym. Adv. Technol. 17, 72-82 (2006). [CrossRef]
- P. Walther, J.-W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, "De Broglie wavelength of a non-local four-photon state," Nature 429, 158-161 (2004). [CrossRef] [PubMed]

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