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

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  1. 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.
  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]
  3. J. W. Goodman, Introduction to Fourier Optics (Roberts & Company Publishers, Greenwood Village, CO, 2005).
  4. D. Robinson and P. Milanfar, “Fundamental performance limits in image registration,” IEEE Trans. Image Process. 13(9), 1185–1199 (2004).
    [CrossRef] [PubMed]
  5. M. Tanibayashi, “Diffraction of light by quasi-periodic gratings,” J. Phys. Soc. Jpn. 61(9), 3139–3145 (1992).
    [CrossRef]
  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 15(3), R63–R81 (2003).
    [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]
  9. V. Elser, “The Diffraction of Projected Structures,” Acta Crystallogr. A 42(1), 36–43 (1986).
    [CrossRef]
  10. 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]
  11. D. Levine and P. J. Steinhardt, “Quasicrystals. I. Definition and structure,” Phys. Rev. B Condens. Matter 34(2), 596–616 (1986).
    [CrossRef] [PubMed]
  12. 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]
  13. C. Janot, Quasicrystals: A Primer (Cambridge University Press, Oxford, 1997).
  14. M. Senechal, Quasicrystals and Geometry (Cambridge University Press, Oxford, 1995).
  15. 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).

2010 (1)

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]

2004 (2)

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

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]

2003 (1)

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]

1992 (1)

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

1986 (4)

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]

1985 (1)

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]

1981 (1)

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).

Dallas, W. J.

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]

de Bruijn, N. G.

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).

Diehl, R. D.

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. 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]

Elser, V.

V. Elser, “The Diffraction of Projected Structures,” Acta Crystallogr. A 42(1), 36–43 (1986).
[CrossRef]

Ferralis, N.

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. 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]

Gähler, F.

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]

Lal, A.

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]

Ledieu, J.

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]

Levine, D.

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

McGrath, R.

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]

Milanfar, P.

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

Pollock, C. R.

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]

Rhyner, J.

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]

Robinson, D.

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

Socolar, J. E. S.

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]

Steinhardt, P. J.

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]

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

Szmodis, A. W.

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. 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]

Tanibayashi, M.

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

Yoshimizu, N.

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]

Zia, R. K. P.

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]

Acta Crystallogr. A (1)

V. Elser, “The Diffraction of Projected Structures,” Acta Crystallogr. A 42(1), 36–43 (1986).
[CrossRef]

Am. J. Phys. (1)

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]

IEEE Trans. Image Process. (1)

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

J. Microelectromech. Syst. (1)

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. Phys. Condens. Matter (1)

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]

J. Phys. Math. Gen. (2)

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]

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]

J. Phys. Soc. Jpn. (1)

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

Ned. Akad. Wetensch. Proc. Ser. A (1)

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).

Phys. Rev. B Condens. Matter (2)

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]

Other (4)

C. Janot, Quasicrystals: A Primer (Cambridge University Press, Oxford, 1997).

M. Senechal, Quasicrystals and Geometry (Cambridge University Press, Oxford, 1995).

J. W. Goodman, Introduction to Fourier Optics (Roberts & Company Publishers, Greenwood Village, CO, 2005).

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.

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Figures (13)

Fig. 1
Fig. 1

Nanometrology Optical Ruler Imaging System schematic: An external cavity laser is frequency stabilized within 6 MHz, or a relative accuracy of 1.5 × 10−8, to a saturated resonance (F = 2 to 1) of the D2-line of 85Rb. A 22.77 ± 0.03 °C temperature stabilized microfabricated Penrose vertices grating diffracts the laser beam (fabricated using ebeam lithography on SOI device layer; device layer Si etch; through carrier wafer backside KOH etch; buffered HF release; Ti/Au evaporation). A wafer-scale optical ruler is shown. Tip/CMOS imager is mounted on a commercial stage. Upsampled Fourier transform cross correlation calculates the CMOS imager position within the optical ruler.

Fig. 2
Fig. 2

One dimensional diffraction pattern from a seven element, periodic aperture array, estimated by a Fourier transform. The resulting pattern, a sum of sinc functions, results in evenly spaced peaks with an intensity envelope.

Fig. 3
Fig. 3

One dimensional diffraction pattern from a seven element, aperiodic aperture array in an attempt to increase features in the diffraction pattern. The aperture positions are nearly periodic, located at slight displacements from their periodic locations in Fig. 2. Some low intensity structure appears to emerge, but some of the larger structures seen in the periodic case have disappeared.

Fig. 4
Fig. 4

One dimensional diffraction pattern from a seven element, quasiperiodic structure based on the Fibonacci sequence. Comparing this diffraction pattern to that in Fig. 2, we see that already several new peaks are introduced, some marked by arrows. The increase in sharp features increases the precision of NORIS.

Fig. 5
Fig. 5

a: Electron micrograph of a metal thin film quasiperiodic aperture array generated by using the vertices of a Penrose tiling. b: CMOS imager sample of the resulting diffraction pattern. Note the high density of sharp peaks, due to the diffraction from a quasiperiodic structure.

Fig. 6
Fig. 6

NORIS positioning errors across 3 μm displacement, compared to short range, high precision capacitive sensor in a piezoelectric flexural stage.

Fig. 7
Fig. 7

NORIS positioning errors across over six hours.

Fig. 8
Fig. 8

Precision in NORIS affected by 640 × 480 imager bit resolution, using same data set as in Fig. 6. Resolution from 14.6 to 6 bits exhibit nearly identical performance. At 4 bits, a linear drift is induced and a large deviation is observed at 3 bit imager resolution. NORIS demonstrates great robustness against imager bit resolution.

Fig. 9
Fig. 9

Precision in NORIS affected by imager bit resolution as in Fig. 8, but with a reduced imager size of 576 x 432 pixels. Again, imager bit resolutions down to 6 exhibit the same precision, 4 bit resolution results in a linear drift, and finally at 3 bit resolution results in poor results.

Fig. 10
Fig. 10

Linear drift error in NORIS as the bit resolution is changed. Data shows robustness against orders of magnitude reduction in bit resolution, but shows significant linear drift error in NORIS as the number of pixels in the imager is decreased slightly.

Fig. 11
Fig. 11

Root mean error of NORIS about the linear drift described in Fig. 10. Again, the root mean error is constant through orders of magnitude reduction in bit resolution, but is sensitive to the number of pixels in the imager.

Fig. 12
Fig. 12

Histogram of aperture diameter and radial positioning errors, for simulated errors of 50 nm in microfabrication.

Fig. 13
Fig. 13

NORIS positioning errors due to microfabrication errors. Positions are calculated by positioning NORIS based on image registration to error aperture array to zero error aperture array. Errors of up to 200 nm appear to have no impact on positioning error; a 500 nm fabrication error shows an effect on NORIS positioning error. As an example, image shows calculated diffraction pattern from 20 nm errors.

Equations (3)

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U ( x , y , z ) = e i k z / ( i λ z ) × e i k ( x 2 + y 2 ) / 2 z × ± U ( ξ , η ) exp [ i 2 π ( x ξ + y η ) / ( λ z ) ] d ξ d η ,
M S E ( r ) J 1 ( r ) ,
[ J ( r ) ] i j = E [ 2 ( log f ) / ( r i r j ) ] ,

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