## Nanometrology optical ruler imaging system using diffraction from a quasiperiodic structure |

Optics Express, Vol. 18, Issue 20, pp. 20827-20838 (2010)

http://dx.doi.org/10.1364/OE.18.020827

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### Abstract

This work demonstrates wafer-scale, path-independent, atomically-based long term-stable, position nanometrology. This nanometrology optical ruler imaging system uses the diffraction pattern of an atomically stabilized laser from a microfabricated quasiperiodic aperture array as a two-dimensional optical ruler. Nanometrology is accomplished by cross correlations of image samples of this optical ruler. The quasiperiodic structure generates spatially dense, sharp optical features. This work demonstrates new results showing positioning errors down to 17.2 nm over wafer scales and long term stability below 20 nm over six hours. This work also numerically demonstrates robustness of the optical ruler to variations in the microfabricated aperture array and discretization noise in imagers.

© 2010 OSA

## 1. Introduction

^{−4}precision over a 75 mm wafer.

2. N. Yoshimizu, A. Lal, and C. R. Pollock, “Nanometrology Using a Quasiperiodic Pattern Diffraction Optical Ruler,” J. Microelectromech. Syst. **19**(4), 865–870 (2010). [CrossRef]

## 2. Quasiperiodic diffraction

*x*,

*y*,

*z*) due to the diffraction of amplitude U at (

*ξ*,

*η*) by an optical field of wave vector

*k*and wavelength

*λ*[3]. The first factor decreases the optical power density as the optical field diffracts away from its origin. The second factor corresponds to a decrease in the optical field away from its center. The diffraction pattern will have more optical power towards the center of the image, and the optical power will generally decay towards the outer parts of the diffraction pattern. The double integral contains the information for the features in the diffraction pattern, a critical component in achieving high precision in NORIS.

*ξ*,

*η*) at the diffraction plane. If the amplitude is constant across the aperture array, the double integral is a Fourier transform of the geometric shape of the aperture array where the openings are a constant intensity and the nontransmitting regions are zero. In order to maximize the precision of the NORIS system, a diffraction pattern, or optical ruler, is needed whereby image cross correlation techniques can be used to yield the highest precision in the estimates of displacement. The precision can be estimated based on the mean square error of the image registration at some offset

**r**[4

4. D. Robinson and P. Milanfar, “Fundamental performance limits in image registration,” IEEE Trans. Image Process. **13**(9), 1185–1199 (2004). [CrossRef] [PubMed]

*J*is, which is the negative expectation value of the partial derivatives of the log of the likelihood function. Here, the likelihood function refers to the image registration, or the calculation of the translational offset of two images.

*U*(

*ξ*,

*η*) is a periodic array of rect functions. As expected from the readily available analytical solution, the diffraction pattern or Fourier transform of the screen aperture is a sum of shifted sinc functions. The diffraction pattern is a number of peaks with a decaying, periodic envelope of the intensity which decays away from the center of the diffraction pattern. In Fig. 2 , we show the one-dimensional diffraction pattern from a periodic, seven aperture screen. A Fourier transform is used to estimate the resulting diffraction pattern.

5. M. Tanibayashi, “Diffraction of light by quasi-periodic gratings,” J. Phys. Soc. Jpn. **61**(9), 3139–3145 (1992). [CrossRef]

7. N. Ferralis, A. W. Szmodis, and R. D. Diehl, “Diffraction from one- and two-dimensional quasicrystalline gratings,” Am. J. Phys. **72**(9), 1241–1246 (2004). [CrossRef]

8. R. K. P. Zia and W. J. Dallas, “A Simple Derivation of Quasi-Crystalline Spectra,” J. Phys. Math. Gen. **18**(7), L341–L345 (1985). [CrossRef]

11. D. Levine and P. J. Steinhardt, “Quasicrystals. I. Definition and structure,” Phys. Rev. B Condens. Matter **34**(2), 596–616 (1986). [CrossRef] [PubMed]

## 3. Experimental results

^{−7}.

## 4. Imager bit resolution

_{x}Δx-k

_{y}Δy phase shift between the Fourier transforms of the two images. Calculating the displacement is a matter of calculating this phase gradient across the images, such as by using a linear regression fit. As the number of pixels increase, the wave vectors k

_{x}and k

_{y}at which this gradient is calculated is increased, resulting in a better approximation of the displacement between the two images. At higher pixel numbers the gradient is sampled at smaller and smaller wave vectors which approximate the gradient more poorly, which may explain the reason why the marginal improvement in precision get smaller for higher numbers of pixels of the same size.

## 5. Aperture array variations

## 5. Conclusion

^{−8}is the base error of the system, and each component or source of error is added until the resulting 2 × 10

^{−7}that we observe. The optical processing also affects the precision, such as if the individual pixel gain or analog to digital conversion are unstable, which in turn causes errors in the cross correlation image registration. The optical ruler has a spatial bandwidth of approximately 10 μm, for which an error of 20 nm would be a 0.1% error in the pixel intensity, or a substantial error of half a least significant bit on the 8-bit gain control of the imager. It is most likely that the current 17 nm limit we observe is due to the mechanical stability of the optical components in the system. For example, a four inch object which changes temperature by 0.1 K will increase its total length by 100 nm at 10 ppm/K, which is a shift of 50 nm within a 50 mm region. Further improvements will require additional work to stabilize the environmental sources of noise.

## Acknowledgements

## References and links

1. | N. Yoshimizu, A. Lal, and C. R. Pollock, “MEMS Diffractive Optical Nanoruler Technology for Tip-Based Nanofabrication and Metrology,” in |

2. | N. Yoshimizu, A. Lal, and C. R. Pollock, “Nanometrology Using a Quasiperiodic Pattern Diffraction Optical Ruler,” J. Microelectromech. Syst. |

3. | J. W. Goodman, |

4. | D. Robinson and P. Milanfar, “Fundamental performance limits in image registration,” IEEE Trans. Image Process. |

5. | M. Tanibayashi, “Diffraction of light by quasi-periodic gratings,” J. Phys. Soc. Jpn. |

6. | R. D. Diehl, J. Ledieu, N. Ferralis, A. W. Szmodis, and R. McGrath, “Low-energy electron diffraction from quasicrystal surfaces,” J. Phys. Condens. Matter |

7. | N. Ferralis, A. W. Szmodis, and R. D. Diehl, “Diffraction from one- and two-dimensional quasicrystalline gratings,” Am. J. Phys. |

8. | R. K. P. Zia and W. J. Dallas, “A Simple Derivation of Quasi-Crystalline Spectra,” J. Phys. Math. Gen. |

9. | V. Elser, “The Diffraction of Projected Structures,” Acta Crystallogr. A |

10. | F. Gähler and J. Rhyner, “Equivalence of the Generalized Grid and Projection Methods for the Construction of Quasiperiodic Tilings,” J. Phys. Math. Gen. |

11. | D. Levine and P. J. Steinhardt, “Quasicrystals. I. Definition and structure,” Phys. Rev. B Condens. Matter |

12. | J. E. S. Socolar and P. J. Steinhardt, “Quasicrystals. II. Unit-cell configurations,” Phys. Rev. B Condens. Matter |

13. | C. Janot, |

14. | M. Senechal, |

15. | N. G. de Bruijn, “Algebraic theory of Penrose’s non-periodic tilings of the plane I & II,” Ned. Akad. Wetensch. Proc. Ser. A |

**OCIS Codes**

(050.1940) Diffraction and gratings : Diffraction

(120.3940) Instrumentation, measurement, and metrology : Metrology

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: August 10, 2010

Revised Manuscript: September 8, 2010

Manuscript Accepted: September 9, 2010

Published: September 16, 2010

**Citation**

Norimasa Yoshimizu, Amit Lal, and Clifford R. Pollock, "Nanometrology optical ruler imaging system using diffraction from a quasiperiodic structure," Opt. Express **18**, 20827-20838 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-20-20827

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### References

- N. Yoshimizu, A. Lal, and C. R. Pollock, “MEMS Diffractive Optical Nanoruler Technology for Tip-Based Nanofabrication and Metrology,” in Proceedings of the IEEE MEMS (Institute of Electrical and Electronics Engineers, New York, 2009), pp. 547–550.
- N. Yoshimizu, A. Lal, and C. R. Pollock, “Nanometrology Using a Quasiperiodic Pattern Diffraction Optical Ruler,” J. Microelectromech. Syst. 19(4), 865–870 (2010). [CrossRef]
- J. W. Goodman, Introduction to Fourier Optics (Roberts & Company Publishers, Greenwood Village, CO, 2005).
- D. Robinson and P. Milanfar, “Fundamental performance limits in image registration,” IEEE Trans. Image Process. 13(9), 1185–1199 (2004). [CrossRef] [PubMed]
- M. Tanibayashi, “Diffraction of light by quasi-periodic gratings,” J. Phys. Soc. Jpn. 61(9), 3139–3145 (1992). [CrossRef]
- R. D. Diehl, J. Ledieu, N. Ferralis, A. W. Szmodis, and R. McGrath, “Low-energy electron diffraction from quasicrystal surfaces,” J. Phys. Condens. Matter 15(3), R63–R81 (2003). [CrossRef]
- N. Ferralis, A. W. Szmodis, and R. D. Diehl, “Diffraction from one- and two-dimensional quasicrystalline gratings,” Am. J. Phys. 72(9), 1241–1246 (2004). [CrossRef]
- R. K. P. Zia and W. J. Dallas, “A Simple Derivation of Quasi-Crystalline Spectra,” J. Phys. Math. Gen. 18(7), L341–L345 (1985). [CrossRef]
- V. Elser, “The Diffraction of Projected Structures,” Acta Crystallogr. A 42(1), 36–43 (1986). [CrossRef]
- F. Gähler and J. Rhyner, “Equivalence of the Generalized Grid and Projection Methods for the Construction of Quasiperiodic Tilings,” J. Phys. Math. Gen. 19(2), 267–277 (1986). [CrossRef]
- D. Levine and P. J. Steinhardt, “Quasicrystals. I. Definition and structure,” Phys. Rev. B Condens. Matter 34(2), 596–616 (1986). [CrossRef] [PubMed]
- J. E. S. Socolar and P. J. Steinhardt, “Quasicrystals. II. Unit-cell configurations,” Phys. Rev. B Condens. Matter 34(2), 617–647 (1986). [CrossRef] [PubMed]
- C. Janot, Quasicrystals: A Primer (Cambridge University Press, Oxford, 1997).
- M. Senechal, Quasicrystals and Geometry (Cambridge University Press, Oxford, 1995).
- N. G. de Bruijn, “Algebraic theory of Penrose’s non-periodic tilings of the plane I & II,” Ned. Akad. Wetensch. Proc. Ser. A 43, 39–66 (1981).

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