Abstract

We looked for design methodologies that cope with optical specifications described in terms of trajectories in the CIE (Commission Internationale de l’Eclairage) 1976 chromaticity diagram in the context of low-cost mass-reproduction processes that inevitably introduce changes in the design of a diffractive device for security applications. The mathematics of the design process can be strongly simplified if the theory of planar waveguides (in integrated optics) is used to estimate, with sufficient accuracy, the position of Wood singularities, responsible for the more-interesting visual features of a grating. We show how to use such a model to assess color dynamics variations that are due to production and to estimate domains within the space of grating parameters that enable both first- and second-level security features to be implemented simultaneously. All the results are compared with the values obtained by rigorous coupled-wave analysis.

© 1999 Optical Society of America

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References

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  1. M. T. Gale, “Diffraction, beauty, and commerce,” Phys. World 2(10), 24–28 (1990).
  2. M. T. Gale, K. Knop, R. Morf, “Zero-order diffractive microstructures for security applications,” in Optical Security and Anticounterfeiting Systems, W. F. Fagan, ed., Proc. SPIE1210, 83–89 (1990).
    [CrossRef]
  3. M. T. Gale, “Replication,” in Micro-Optics: Elements, Systems, and Applications, H. P. Herzig, ed. (Taylor & Francis, Washington, D.C., 1997), Chap. 6.
  4. G. L. Erwin, E. Popov, “Replication of gratings,” in Diffraction Gratings and Applications, B. J. Thompson, ed. (Marcel Dekker, New York, 1997), Chap. 7.
  5. M. R. Descour, W. C. Sweatt, A. K. Ray-Chaudhuri, K. D. Krenz, D. A. Tichenor, R. H. Stulen, “Mass-producible, microtags for security applications: tolerance analysis by rigorous coupled-wave analysis,” in Micromachining and Imaging, T. A. Michalske, M. A. Wendman, eds., Proc. SPIE3009, 15–24 (1997).
    [CrossRef]
  6. R. W. G. Hunt, The Reproduction of Color (Fountain Press, Surrey, UK, 1995), Chap. 7.
  7. T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
    [CrossRef]
  8. M. G. Moharam, E. B. Grann, D. A. Pommet, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1073 (1995).
    [CrossRef]
  9. D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics series (Springer-Verlag, Berlin, 1980), Chap. 3.
    [CrossRef]
  10. D. Dobson, “A variational method for electromagnetic diffraction in biperiodic structures,” Model. Math. Anal. 28, 419–439 (1994).
  11. GSOLVER V2.0c, Grating Solver Development Company, P.O. Box 353, Allen, Tex. 75013.
  12. J. A. Cox, “Inverse and optimal design problems for imaging and diffractive optical systems,” in Inverse Problems and Optimal Design in Industry, H. Engl, J. McLaughlin, eds. (B. G. Teubner, Stuttgart, Germany, 1994), pp. 29–36.
  13. G. Bao, D. C. Dobson, J. A. Cox, “Mathematical studies in rigorous grating theory,” J. Opt. Soc. Am. A 12, 1029–1042 (1995).
    [CrossRef]
  14. P. Lalanne, G. M. Morris, “Highly improved convergence of coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
    [CrossRef]
  15. L. Li, J. Chandezon, G. Granet, J.-P. Plumey, “Rigorous and efficient grating-analysis method made easy for optical engineers,” Appl. Opt. 38, 304–313 (1999).
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  16. J. M. Santos, L. Bernardo, “Antireflection structures with use of multilevel subwavelength zero-order gratings,” Appl. Opt. 36, 8935–8938 (1997).
    [CrossRef]
  17. R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–400 (1902).
    [CrossRef]
  18. J. W. S. Rayleigh, “Note on the remarkable case of diffraction spectra described by Prof. Wood,” Philos. Mag. 14, 60–65 (1907).
    [CrossRef]
  19. S. S. Wang, R. Magnusson, J. S. Bagby, M. G. Moharam, “Guided-mode resonances in planar dielectric-layer diffraction gratings,” J. Opt. Soc. Am. A 8, 1470–1475 (1990).
    [CrossRef]
  20. S. S. Wang, R. Magnusson, “Theory and applications of guided-mode resonance filters,” Appl. Opt. 32, 2606–2613 (1993).
    [CrossRef] [PubMed]
  21. R. Marz, Integrated Optics Design and Modeling (Artech House, Norwood, Mass., 1994), Chap. 3.
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    [CrossRef]
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    [CrossRef]
  25. J. G. Fleming, C. C. Barron, B. Stallard, S. Kaushik, “Fabrication of large area gratings with sub-micron pitch using mold micromachining,” in Micromachining and Imaging, T. A. Michalske, M. A. Wendman, eds., Proc. SPIE3009, 7–14 (1997).
    [CrossRef]
  26. B. Schnabel, E. B. Kley, “Fabrication and application of sub-wavelength gratings,” in Miniaturized Systems with Micro-Optics and Micromechanics II, M. Motamedi, L. J. Hornbeck, K. S. Pister, eds., Proc. SPIE3008, 233–241 (1997).
    [CrossRef]

1999 (1)

1997 (1)

1996 (1)

1995 (2)

1994 (1)

D. Dobson, “A variational method for electromagnetic diffraction in biperiodic structures,” Model. Math. Anal. 28, 419–439 (1994).

1993 (1)

1990 (2)

S. S. Wang, R. Magnusson, J. S. Bagby, M. G. Moharam, “Guided-mode resonances in planar dielectric-layer diffraction gratings,” J. Opt. Soc. Am. A 8, 1470–1475 (1990).
[CrossRef]

M. T. Gale, “Diffraction, beauty, and commerce,” Phys. World 2(10), 24–28 (1990).

1985 (1)

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

1977 (1)

1907 (1)

J. W. S. Rayleigh, “Note on the remarkable case of diffraction spectra described by Prof. Wood,” Philos. Mag. 14, 60–65 (1907).
[CrossRef]

1902 (1)

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–400 (1902).
[CrossRef]

Bagby, J. S.

S. S. Wang, R. Magnusson, J. S. Bagby, M. G. Moharam, “Guided-mode resonances in planar dielectric-layer diffraction gratings,” J. Opt. Soc. Am. A 8, 1470–1475 (1990).
[CrossRef]

Bao, G.

Barron, C. C.

J. G. Fleming, C. C. Barron, B. Stallard, S. Kaushik, “Fabrication of large area gratings with sub-micron pitch using mold micromachining,” in Micromachining and Imaging, T. A. Michalske, M. A. Wendman, eds., Proc. SPIE3009, 7–14 (1997).
[CrossRef]

Bernardo, L.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959).

Chandezon, J.

Cox, J. A.

G. Bao, D. C. Dobson, J. A. Cox, “Mathematical studies in rigorous grating theory,” J. Opt. Soc. Am. A 12, 1029–1042 (1995).
[CrossRef]

J. A. Cox, “Inverse and optimal design problems for imaging and diffractive optical systems,” in Inverse Problems and Optimal Design in Industry, H. Engl, J. McLaughlin, eds. (B. G. Teubner, Stuttgart, Germany, 1994), pp. 29–36.

Descour, M. R.

M. R. Descour, W. C. Sweatt, A. K. Ray-Chaudhuri, K. D. Krenz, D. A. Tichenor, R. H. Stulen, “Mass-producible, microtags for security applications: tolerance analysis by rigorous coupled-wave analysis,” in Micromachining and Imaging, T. A. Michalske, M. A. Wendman, eds., Proc. SPIE3009, 15–24 (1997).
[CrossRef]

Dobson, D.

D. Dobson, “A variational method for electromagnetic diffraction in biperiodic structures,” Model. Math. Anal. 28, 419–439 (1994).

Dobson, D. C.

Erwin, G. L.

G. L. Erwin, E. Popov, “Replication of gratings,” in Diffraction Gratings and Applications, B. J. Thompson, ed. (Marcel Dekker, New York, 1997), Chap. 7.

Fleming, J. G.

J. G. Fleming, C. C. Barron, B. Stallard, S. Kaushik, “Fabrication of large area gratings with sub-micron pitch using mold micromachining,” in Micromachining and Imaging, T. A. Michalske, M. A. Wendman, eds., Proc. SPIE3009, 7–14 (1997).
[CrossRef]

Gale, M. T.

M. T. Gale, “Diffraction, beauty, and commerce,” Phys. World 2(10), 24–28 (1990).

M. T. Gale, “Replication,” in Micro-Optics: Elements, Systems, and Applications, H. P. Herzig, ed. (Taylor & Francis, Washington, D.C., 1997), Chap. 6.

M. T. Gale, K. Knop, R. Morf, “Zero-order diffractive microstructures for security applications,” in Optical Security and Anticounterfeiting Systems, W. F. Fagan, ed., Proc. SPIE1210, 83–89 (1990).
[CrossRef]

Gaylord, T. K.

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Granet, G.

Grann, E. B.

Hunt, R. W. G.

R. W. G. Hunt, The Reproduction of Color (Fountain Press, Surrey, UK, 1995), Chap. 7.

Kaushik, S.

J. G. Fleming, C. C. Barron, B. Stallard, S. Kaushik, “Fabrication of large area gratings with sub-micron pitch using mold micromachining,” in Micromachining and Imaging, T. A. Michalske, M. A. Wendman, eds., Proc. SPIE3009, 7–14 (1997).
[CrossRef]

Kley, E. B.

B. Schnabel, E. B. Kley, “Fabrication and application of sub-wavelength gratings,” in Miniaturized Systems with Micro-Optics and Micromechanics II, M. Motamedi, L. J. Hornbeck, K. S. Pister, eds., Proc. SPIE3008, 233–241 (1997).
[CrossRef]

Knop, K.

M. T. Gale, K. Knop, R. Morf, “Zero-order diffractive microstructures for security applications,” in Optical Security and Anticounterfeiting Systems, W. F. Fagan, ed., Proc. SPIE1210, 83–89 (1990).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Theory of optical waveguides,” in Guided-Wave Optoelectronics, T. Tamir, ed. (Springer-Verlag, Berlin, 1990), Chap. 2.
[CrossRef]

Krenz, K. D.

M. R. Descour, W. C. Sweatt, A. K. Ray-Chaudhuri, K. D. Krenz, D. A. Tichenor, R. H. Stulen, “Mass-producible, microtags for security applications: tolerance analysis by rigorous coupled-wave analysis,” in Micromachining and Imaging, T. A. Michalske, M. A. Wendman, eds., Proc. SPIE3009, 15–24 (1997).
[CrossRef]

Lalanne, P.

Li, L.

Magnusson, R.

S. S. Wang, R. Magnusson, “Theory and applications of guided-mode resonance filters,” Appl. Opt. 32, 2606–2613 (1993).
[CrossRef] [PubMed]

S. S. Wang, R. Magnusson, J. S. Bagby, M. G. Moharam, “Guided-mode resonances in planar dielectric-layer diffraction gratings,” J. Opt. Soc. Am. A 8, 1470–1475 (1990).
[CrossRef]

Marz, R.

R. Marz, Integrated Optics Design and Modeling (Artech House, Norwood, Mass., 1994), Chap. 3.

Maystre, D.

D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics series (Springer-Verlag, Berlin, 1980), Chap. 3.
[CrossRef]

Moharam, M. G.

M. G. Moharam, E. B. Grann, D. A. Pommet, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1073 (1995).
[CrossRef]

S. S. Wang, R. Magnusson, J. S. Bagby, M. G. Moharam, “Guided-mode resonances in planar dielectric-layer diffraction gratings,” J. Opt. Soc. Am. A 8, 1470–1475 (1990).
[CrossRef]

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Morf, R.

M. T. Gale, K. Knop, R. Morf, “Zero-order diffractive microstructures for security applications,” in Optical Security and Anticounterfeiting Systems, W. F. Fagan, ed., Proc. SPIE1210, 83–89 (1990).
[CrossRef]

Morris, G. M.

Plumey, J.-P.

Pommet, D. A.

Popov, E.

G. L. Erwin, E. Popov, “Replication of gratings,” in Diffraction Gratings and Applications, B. J. Thompson, ed. (Marcel Dekker, New York, 1997), Chap. 7.

Ray-Chaudhuri, A. K.

M. R. Descour, W. C. Sweatt, A. K. Ray-Chaudhuri, K. D. Krenz, D. A. Tichenor, R. H. Stulen, “Mass-producible, microtags for security applications: tolerance analysis by rigorous coupled-wave analysis,” in Micromachining and Imaging, T. A. Michalske, M. A. Wendman, eds., Proc. SPIE3009, 15–24 (1997).
[CrossRef]

Rayleigh, J. W. S.

J. W. S. Rayleigh, “Note on the remarkable case of diffraction spectra described by Prof. Wood,” Philos. Mag. 14, 60–65 (1907).
[CrossRef]

Santos, J. M.

Schnabel, B.

B. Schnabel, E. B. Kley, “Fabrication and application of sub-wavelength gratings,” in Miniaturized Systems with Micro-Optics and Micromechanics II, M. Motamedi, L. J. Hornbeck, K. S. Pister, eds., Proc. SPIE3008, 233–241 (1997).
[CrossRef]

Stallard, B.

J. G. Fleming, C. C. Barron, B. Stallard, S. Kaushik, “Fabrication of large area gratings with sub-micron pitch using mold micromachining,” in Micromachining and Imaging, T. A. Michalske, M. A. Wendman, eds., Proc. SPIE3009, 7–14 (1997).
[CrossRef]

Stulen, R. H.

M. R. Descour, W. C. Sweatt, A. K. Ray-Chaudhuri, K. D. Krenz, D. A. Tichenor, R. H. Stulen, “Mass-producible, microtags for security applications: tolerance analysis by rigorous coupled-wave analysis,” in Micromachining and Imaging, T. A. Michalske, M. A. Wendman, eds., Proc. SPIE3009, 15–24 (1997).
[CrossRef]

Sweatt, W. C.

M. R. Descour, W. C. Sweatt, A. K. Ray-Chaudhuri, K. D. Krenz, D. A. Tichenor, R. H. Stulen, “Mass-producible, microtags for security applications: tolerance analysis by rigorous coupled-wave analysis,” in Micromachining and Imaging, T. A. Michalske, M. A. Wendman, eds., Proc. SPIE3009, 15–24 (1997).
[CrossRef]

Tichenor, D. A.

M. R. Descour, W. C. Sweatt, A. K. Ray-Chaudhuri, K. D. Krenz, D. A. Tichenor, R. H. Stulen, “Mass-producible, microtags for security applications: tolerance analysis by rigorous coupled-wave analysis,” in Micromachining and Imaging, T. A. Michalske, M. A. Wendman, eds., Proc. SPIE3009, 15–24 (1997).
[CrossRef]

Wang, S. S.

S. S. Wang, R. Magnusson, “Theory and applications of guided-mode resonance filters,” Appl. Opt. 32, 2606–2613 (1993).
[CrossRef] [PubMed]

S. S. Wang, R. Magnusson, J. S. Bagby, M. G. Moharam, “Guided-mode resonances in planar dielectric-layer diffraction gratings,” J. Opt. Soc. Am. A 8, 1470–1475 (1990).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959).

Wood, R. W.

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–400 (1902).
[CrossRef]

Yariv, A.

Yeh, P.

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Model. Math. Anal. (1)

D. Dobson, “A variational method for electromagnetic diffraction in biperiodic structures,” Model. Math. Anal. 28, 419–439 (1994).

Philos. Mag. (2)

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–400 (1902).
[CrossRef]

J. W. S. Rayleigh, “Note on the remarkable case of diffraction spectra described by Prof. Wood,” Philos. Mag. 14, 60–65 (1907).
[CrossRef]

Phys. World (1)

M. T. Gale, “Diffraction, beauty, and commerce,” Phys. World 2(10), 24–28 (1990).

Proc. IEEE (1)

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Other (13)

D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics series (Springer-Verlag, Berlin, 1980), Chap. 3.
[CrossRef]

GSOLVER V2.0c, Grating Solver Development Company, P.O. Box 353, Allen, Tex. 75013.

J. A. Cox, “Inverse and optimal design problems for imaging and diffractive optical systems,” in Inverse Problems and Optimal Design in Industry, H. Engl, J. McLaughlin, eds. (B. G. Teubner, Stuttgart, Germany, 1994), pp. 29–36.

M. T. Gale, K. Knop, R. Morf, “Zero-order diffractive microstructures for security applications,” in Optical Security and Anticounterfeiting Systems, W. F. Fagan, ed., Proc. SPIE1210, 83–89 (1990).
[CrossRef]

M. T. Gale, “Replication,” in Micro-Optics: Elements, Systems, and Applications, H. P. Herzig, ed. (Taylor & Francis, Washington, D.C., 1997), Chap. 6.

G. L. Erwin, E. Popov, “Replication of gratings,” in Diffraction Gratings and Applications, B. J. Thompson, ed. (Marcel Dekker, New York, 1997), Chap. 7.

M. R. Descour, W. C. Sweatt, A. K. Ray-Chaudhuri, K. D. Krenz, D. A. Tichenor, R. H. Stulen, “Mass-producible, microtags for security applications: tolerance analysis by rigorous coupled-wave analysis,” in Micromachining and Imaging, T. A. Michalske, M. A. Wendman, eds., Proc. SPIE3009, 15–24 (1997).
[CrossRef]

R. W. G. Hunt, The Reproduction of Color (Fountain Press, Surrey, UK, 1995), Chap. 7.

R. Marz, Integrated Optics Design and Modeling (Artech House, Norwood, Mass., 1994), Chap. 3.

H. Kogelnik, “Theory of optical waveguides,” in Guided-Wave Optoelectronics, T. Tamir, ed. (Springer-Verlag, Berlin, 1990), Chap. 2.
[CrossRef]

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959).

J. G. Fleming, C. C. Barron, B. Stallard, S. Kaushik, “Fabrication of large area gratings with sub-micron pitch using mold micromachining,” in Micromachining and Imaging, T. A. Michalske, M. A. Wendman, eds., Proc. SPIE3009, 7–14 (1997).
[CrossRef]

B. Schnabel, E. B. Kley, “Fabrication and application of sub-wavelength gratings,” in Miniaturized Systems with Micro-Optics and Micromechanics II, M. Motamedi, L. J. Hornbeck, K. S. Pister, eds., Proc. SPIE3008, 233–241 (1997).
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

CIE 1976 UCS chromaticity diagram.

Fig. 2
Fig. 2

Geometric model of a grating with the necessary parameters to describe it according to the coupled-wave analysis. E and H are the electric and the magnetic fields, respectively; k = 2π/λ is the wave number; K is the grating vector; λ is the wavelength; Λ is the grating period; w is the distance between adjacent planes of different permittivities; θ′ is the incident angle; and ϕ and Ψ are the angles between the grating vector, the incident direction, and incident plane, respectively.

Fig. 3
Fig. 3

Laminar structure of a planar waveguide. ∊(x) is the permittivity variation.

Fig. 4
Fig. 4

Parametric regions of a waveguide grating for ±1 and ±2 orders able to diffract. ∊I = 2.074, ∊II = 3.386, ∊III = 2.250, Λ = 280 nm, and TE polarization.

Fig. 5
Fig. 5

Solutions of the transcendental eigenvalue equation normalized to the grating period; ∊I = 2.074, ∊II = 3.386, ∊III = 2.250, θ′ = 10°.

Fig. 6
Fig. 6

Diffraction efficiency spectra for a rectangular grove grating; ∊I = 2.074, ∊II = 3.386, ∊III = 2.250, Λ = 280 nm, d = 400 nm, θ = 10°, and TE polarization.

Fig. 7
Fig. 7

Color behavior of a grating when the incident angle changes; ∊I = 2.074, ∊II = 3.386, ∊III = 2.250, Λ = 280 nm, d = 400 nm, θ = 0°–50°. The arrow represents the angle increase.

Fig. 8
Fig. 8

Color behavior of a grating when the incident angle changes; ∊I = 2.074, ∊II = 3.386, ∊III = 2.250, Λ = 280 nm, θ = 0°–50°; (a) d = 360 nm, (b) d = 440 nm. The arrow represents the angle increase.

Fig. 9
Fig. 9

(a) Predicted position difference of the anomalies between a grating with d = 400 nm and d = 360 nm (thin curve) and with d = 440 nm and d = 400 nm (thick curve). (b) Calculated position difference of the anomalies (RCWA) between a grating with d = 400 nm and d = 360 nm (thin curve) and with d = 440 nm and d = 400 nm (thick curve).

Fig. 10
Fig. 10

Difference for each grating thickness between the predicted and the calculated anomalies position; d = 360 nm (dotted–dashed curve), d = 400 nm (dashed curve), and d = 440 nm (solid cuve).

Fig. 11
Fig. 11

Color behavior of a grating when the incident angle changes; ∊I = 2.074, ∊II = 3.386, ∊III = 2.250, d = 400 nm, θ = 0°–50°; (a) Λ = 266 nm, (b) Λ = 294 nm. The arrow represents the angle increase.

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

12π2d2Sˆizdz2-201/2 sin θλ-iΛ2-0λ2Sˆiz+Δλ2Sˆi-1z+Sˆi+1z=0.
Emx, y, z=emx, yexpiqmz.
max1, 3<m<2.
d2eymdx2+k2x-meym=0.
tanδm=2Fm1-Fm2,
Fm=V2-δm2δm2
Fm=213V2-δm2δm21/2,
d2Sˆizdz2+k20-k01/2 sin θ-iλΛSˆiz=0.
qi=k0sin θ-iλΛ=km.
tankid=kiγi+ηiki2-γiηi.
tankid=2ki3γi+1ηi13ki2-γiηi,
ki=2k2-qi21/2,γi=qi2-1k21/2,δi=qi2-3k21/2.

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