Abstract

The bidirectional reflectance distribution functions of diffraction gratings were calculated by applying diffraction theory and transformed into goniospectrophotometric space curves. Gratings with parallel sinusoidal grooves having periods of 1–3.5 μm and amplitudes below 0.2 μm were analyzed. The obtained goniospectrophotometric space curves consist of lines with different slopes and possible interconnections. The slope of the lines is directly connected to the grating period and the length to the period and the amplitude. Such curves could be regarded as a simple appearance fingerprint of a diffraction grating. The ability of portable multiangle spectrophotometers to provide them for diffraction gratings is analyzed.

© 2013 Optical Society of America

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References

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  1. G. Obein, R. Bousquet, and M. E. Nadal, “New NIST reference goniospectrometer,” Proc. SPIE 5880, 58800T (2005).
    [CrossRef]
  2. L. Simonot and G. Obein, “Geometrical considerations in analyzing isotropic or anisotropic surface reflections,” Appl. Opt. 46, 2615–2623 (2007).
    [CrossRef]
  3. V. B. Podobedov, M. E. Nadal, and C. C. Miller, “Improving the performance of NIST five axis goniospectrometer for measurements of bidirectional reflectance distribution function,” Proc. SPIE 8065, 80651I (2011).
    [CrossRef]
  4. A. M. Rabal, A. Ferrero, J. Campos, J. L. Fontecha, A. Pons, A. M. Rubiño, and A. Corróns, “Automatic gonio-spectrophotometer for the absolute measurement of the spectral BRDF at in- and out-of-plane and retroreflection geometries,” Metrologia 49, 213–223 (2012).
    [CrossRef]
  5. L. Simonot, M. Hébert, and D. Dupraz, “Goniocolorimetry: from measurement to representation in the CIELAB color space,” Color Res. Appl. 36, 169–178 (2011).
    [CrossRef]
  6. E. Perales, E. Chorro, W. R. Cramer, and F. M. Martínez-Verdú, “Analysis of the colorimetric properties of goniochromatic colors using the MacAdam limits under different light sources,” Appl. Opt. 50, 5271–5278 (2011).
    [CrossRef]
  7. A. Ferrero, A. M. Rabal, J. Campos, A. Pons, and M. L. Hernanz, “Variables separation of the BRDF for better understanding color variation in special effect pigment coatings,” J. Opt. Soc. Am. A 29, 842–847 (2012).
    [CrossRef]
  8. E. Kirchner and W. Cramer, “Making sense of measurement geometries for multi-angle spectrophotometers,” Color Res. Appl. 37, 186–198 (2012).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2013

2012

A. Ferrero, A. M. Rabal, J. Campos, A. Pons, and M. L. Hernanz, “Variables separation of the BRDF for better understanding color variation in special effect pigment coatings,” J. Opt. Soc. Am. A 29, 842–847 (2012).
[CrossRef]

A. M. Rabal, A. Ferrero, J. Campos, J. L. Fontecha, A. Pons, A. M. Rubiño, and A. Corróns, “Automatic gonio-spectrophotometer for the absolute measurement of the spectral BRDF at in- and out-of-plane and retroreflection geometries,” Metrologia 49, 213–223 (2012).
[CrossRef]

E. Kirchner and W. Cramer, “Making sense of measurement geometries for multi-angle spectrophotometers,” Color Res. Appl. 37, 186–198 (2012).
[CrossRef]

2011

L. Simonot, M. Hébert, and D. Dupraz, “Goniocolorimetry: from measurement to representation in the CIELAB color space,” Color Res. Appl. 36, 169–178 (2011).
[CrossRef]

E. Perales, E. Chorro, W. R. Cramer, and F. M. Martínez-Verdú, “Analysis of the colorimetric properties of goniochromatic colors using the MacAdam limits under different light sources,” Appl. Opt. 50, 5271–5278 (2011).
[CrossRef]

V. B. Podobedov, M. E. Nadal, and C. C. Miller, “Improving the performance of NIST five axis goniospectrometer for measurements of bidirectional reflectance distribution function,” Proc. SPIE 8065, 80651I (2011).
[CrossRef]

2010

2008

2007

2006

A. Argoitia and R. Phillips, “The security enhancement of diffractive optically variable image devices,” Proc. SPIE 6075, 60750P (2006).

2005

G. Obein, R. Bousquet, and M. E. Nadal, “New NIST reference goniospectrometer,” Proc. SPIE 5880, 58800T (2005).
[CrossRef]

A. Takagi, A. Watanabe, and G. Baba, “Prediction of spectral reflectance factor distribution of automotive paint finishes,” Color Res. Appl. 30, 275–282 (2005).
[CrossRef]

L. Kotačka, T. Têthal, and V. Kolařik, “Top-quality security optical elements: from holography towards 500.000  dpi,” Proc. SPIE 5954, 59540K (2005).
[CrossRef]

2004

A. Argoitia and S. Chu, “Diffractive pigments help document security,” Eur. Coat. J. 32, 32–35 (2004).

Argoitia, A.

A. Argoitia and R. Phillips, “The security enhancement of diffractive optically variable image devices,” Proc. SPIE 6075, 60750P (2006).

A. Argoitia and S. Chu, “Diffractive pigments help document security,” Eur. Coat. J. 32, 32–35 (2004).

Baba, G.

A. Takagi, A. Watanabe, and G. Baba, “Prediction of spectral reflectance factor distribution of automotive paint finishes,” Color Res. Appl. 30, 275–282 (2005).
[CrossRef]

Bousquet, R.

G. Obein, R. Bousquet, and M. E. Nadal, “New NIST reference goniospectrometer,” Proc. SPIE 5880, 58800T (2005).
[CrossRef]

Campos, J.

A. Ferrero, A. M. Rabal, J. Campos, A. Pons, and M. L. Hernanz, “Variables separation of the BRDF for better understanding color variation in special effect pigment coatings,” J. Opt. Soc. Am. A 29, 842–847 (2012).
[CrossRef]

A. M. Rabal, A. Ferrero, J. Campos, J. L. Fontecha, A. Pons, A. M. Rubiño, and A. Corróns, “Automatic gonio-spectrophotometer for the absolute measurement of the spectral BRDF at in- and out-of-plane and retroreflection geometries,” Metrologia 49, 213–223 (2012).
[CrossRef]

Chorro, E.

Chu, S.

A. Argoitia and S. Chu, “Diffractive pigments help document security,” Eur. Coat. J. 32, 32–35 (2004).

Corróns, A.

A. M. Rabal, A. Ferrero, J. Campos, J. L. Fontecha, A. Pons, A. M. Rubiño, and A. Corróns, “Automatic gonio-spectrophotometer for the absolute measurement of the spectral BRDF at in- and out-of-plane and retroreflection geometries,” Metrologia 49, 213–223 (2012).
[CrossRef]

Cramer, W.

E. Kirchner and W. Cramer, “Making sense of measurement geometries for multi-angle spectrophotometers,” Color Res. Appl. 37, 186–198 (2012).
[CrossRef]

Cramer, W. R.

Dupraz, D.

L. Simonot, M. Hébert, and D. Dupraz, “Goniocolorimetry: from measurement to representation in the CIELAB color space,” Color Res. Appl. 36, 169–178 (2011).
[CrossRef]

Ellens, M. C.

J. K. Nisper, T. M. Richardson, M. C. Ellens, and C. Huang, “Method and system for enhanced formulation and visualization rendering,” U.S. patent application2009/0213120 A1 (27August2009).

Ferrero, A.

A. M. Rabal, A. Ferrero, J. Campos, J. L. Fontecha, A. Pons, A. M. Rubiño, and A. Corróns, “Automatic gonio-spectrophotometer for the absolute measurement of the spectral BRDF at in- and out-of-plane and retroreflection geometries,” Metrologia 49, 213–223 (2012).
[CrossRef]

A. Ferrero, A. M. Rabal, J. Campos, A. Pons, and M. L. Hernanz, “Variables separation of the BRDF for better understanding color variation in special effect pigment coatings,” J. Opt. Soc. Am. A 29, 842–847 (2012).
[CrossRef]

Fontecha, J. L.

A. M. Rabal, A. Ferrero, J. Campos, J. L. Fontecha, A. Pons, A. M. Rubiño, and A. Corróns, “Automatic gonio-spectrophotometer for the absolute measurement of the spectral BRDF at in- and out-of-plane and retroreflection geometries,” Metrologia 49, 213–223 (2012).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Hébert, M.

L. Simonot, M. Hébert, and D. Dupraz, “Goniocolorimetry: from measurement to representation in the CIELAB color space,” Color Res. Appl. 36, 169–178 (2011).
[CrossRef]

Hernanz, M. L.

Huang, C.

J. K. Nisper, T. M. Richardson, M. C. Ellens, and C. Huang, “Method and system for enhanced formulation and visualization rendering,” U.S. patent application2009/0213120 A1 (27August2009).

Intaravanne, Y.

Kirchner, E.

E. Kirchner and W. Cramer, “Making sense of measurement geometries for multi-angle spectrophotometers,” Color Res. Appl. 37, 186–198 (2012).
[CrossRef]

Klanjšek Gunde, M.

Kolarik, V.

L. Kotačka, T. Têthal, and V. Kolařik, “Top-quality security optical elements: from holography towards 500.000  dpi,” Proc. SPIE 5954, 59540K (2005).
[CrossRef]

Kotacka, L.

L. Kotačka, T. Têthal, and V. Kolařik, “Top-quality security optical elements: from holography towards 500.000  dpi,” Proc. SPIE 5954, 59540K (2005).
[CrossRef]

Martínez-Verdú, F. M.

Miller, C. C.

V. B. Podobedov, M. E. Nadal, and C. C. Miller, “Improving the performance of NIST five axis goniospectrometer for measurements of bidirectional reflectance distribution function,” Proc. SPIE 8065, 80651I (2011).
[CrossRef]

Nadal, M. E.

V. B. Podobedov, M. E. Nadal, and C. C. Miller, “Improving the performance of NIST five axis goniospectrometer for measurements of bidirectional reflectance distribution function,” Proc. SPIE 8065, 80651I (2011).
[CrossRef]

G. Obein, R. Bousquet, and M. E. Nadal, “New NIST reference goniospectrometer,” Proc. SPIE 5880, 58800T (2005).
[CrossRef]

Nisper, J.

J. Nisper, T. Richardson, and B. Teunis, “Major advances in the reliable measurement of the color and appearance of special effect paints and coatings,” presented at the American Coatings Conference, Charlotte, North Carolina, 1April, 2008.

Nisper, J. K.

J. K. Nisper, P. S. Rood, B. A. Pawlanta, T. M. Richardson, and B. D. Teunis, “Measuring an appearance property of a surface using a bidirectional reflectance distribution function,” U.S. patent application2007/0291993 A1 (20December2007).

J. K. Nisper, T. M. Richardson, M. C. Ellens, and C. Huang, “Method and system for enhanced formulation and visualization rendering,” U.S. patent application2009/0213120 A1 (27August2009).

Obein, G.

L. Simonot and G. Obein, “Geometrical considerations in analyzing isotropic or anisotropic surface reflections,” Appl. Opt. 46, 2615–2623 (2007).
[CrossRef]

G. Obein, R. Bousquet, and M. E. Nadal, “New NIST reference goniospectrometer,” Proc. SPIE 5880, 58800T (2005).
[CrossRef]

Pawlanta, B. A.

J. K. Nisper, P. S. Rood, B. A. Pawlanta, T. M. Richardson, and B. D. Teunis, “Measuring an appearance property of a surface using a bidirectional reflectance distribution function,” U.S. patent application2007/0291993 A1 (20December2007).

Perales, E.

Phillips, R.

A. Argoitia and R. Phillips, “The security enhancement of diffractive optically variable image devices,” Proc. SPIE 6075, 60750P (2006).

Podobedov, V. B.

V. B. Podobedov, M. E. Nadal, and C. C. Miller, “Improving the performance of NIST five axis goniospectrometer for measurements of bidirectional reflectance distribution function,” Proc. SPIE 8065, 80651I (2011).
[CrossRef]

Pons, A.

A. Ferrero, A. M. Rabal, J. Campos, A. Pons, and M. L. Hernanz, “Variables separation of the BRDF for better understanding color variation in special effect pigment coatings,” J. Opt. Soc. Am. A 29, 842–847 (2012).
[CrossRef]

A. M. Rabal, A. Ferrero, J. Campos, J. L. Fontecha, A. Pons, A. M. Rubiño, and A. Corróns, “Automatic gonio-spectrophotometer for the absolute measurement of the spectral BRDF at in- and out-of-plane and retroreflection geometries,” Metrologia 49, 213–223 (2012).
[CrossRef]

Rabal, A. M.

A. M. Rabal, A. Ferrero, J. Campos, J. L. Fontecha, A. Pons, A. M. Rubiño, and A. Corróns, “Automatic gonio-spectrophotometer for the absolute measurement of the spectral BRDF at in- and out-of-plane and retroreflection geometries,” Metrologia 49, 213–223 (2012).
[CrossRef]

A. Ferrero, A. M. Rabal, J. Campos, A. Pons, and M. L. Hernanz, “Variables separation of the BRDF for better understanding color variation in special effect pigment coatings,” J. Opt. Soc. Am. A 29, 842–847 (2012).
[CrossRef]

Richardson, T.

J. Nisper, T. Richardson, and B. Teunis, “Major advances in the reliable measurement of the color and appearance of special effect paints and coatings,” presented at the American Coatings Conference, Charlotte, North Carolina, 1April, 2008.

Richardson, T. M.

J. K. Nisper, P. S. Rood, B. A. Pawlanta, T. M. Richardson, and B. D. Teunis, “Measuring an appearance property of a surface using a bidirectional reflectance distribution function,” U.S. patent application2007/0291993 A1 (20December2007).

J. K. Nisper, T. M. Richardson, M. C. Ellens, and C. Huang, “Method and system for enhanced formulation and visualization rendering,” U.S. patent application2009/0213120 A1 (27August2009).

Rogelj, N.

Rood, P. S.

J. K. Nisper, P. S. Rood, B. A. Pawlanta, T. M. Richardson, and B. D. Teunis, “Measuring an appearance property of a surface using a bidirectional reflectance distribution function,” U.S. patent application2007/0291993 A1 (20December2007).

Rubiño, A. M.

A. M. Rabal, A. Ferrero, J. Campos, J. L. Fontecha, A. Pons, A. M. Rubiño, and A. Corróns, “Automatic gonio-spectrophotometer for the absolute measurement of the spectral BRDF at in- and out-of-plane and retroreflection geometries,” Metrologia 49, 213–223 (2012).
[CrossRef]

Simonot, L.

L. Simonot, M. Hébert, and D. Dupraz, “Goniocolorimetry: from measurement to representation in the CIELAB color space,” Color Res. Appl. 36, 169–178 (2011).
[CrossRef]

L. Simonot and G. Obein, “Geometrical considerations in analyzing isotropic or anisotropic surface reflections,” Appl. Opt. 46, 2615–2623 (2007).
[CrossRef]

Sumriddetchkajorn, S.

Takagi, A.

A. Takagi, A. Watanabe, and G. Baba, “Prediction of spectral reflectance factor distribution of automotive paint finishes,” Color Res. Appl. 30, 275–282 (2005).
[CrossRef]

Têthal, T.

L. Kotačka, T. Têthal, and V. Kolařik, “Top-quality security optical elements: from holography towards 500.000  dpi,” Proc. SPIE 5954, 59540K (2005).
[CrossRef]

Teunis, B.

J. Nisper, T. Richardson, and B. Teunis, “Major advances in the reliable measurement of the color and appearance of special effect paints and coatings,” presented at the American Coatings Conference, Charlotte, North Carolina, 1April, 2008.

Teunis, B. D.

J. K. Nisper, P. S. Rood, B. A. Pawlanta, T. M. Richardson, and B. D. Teunis, “Measuring an appearance property of a surface using a bidirectional reflectance distribution function,” U.S. patent application2007/0291993 A1 (20December2007).

van Renesse, R. L.

R. L. van Renesse, Optical Document Security, 3rd ed. (Artech House, 2005).

Watanabe, A.

A. Takagi, A. Watanabe, and G. Baba, “Prediction of spectral reflectance factor distribution of automotive paint finishes,” Color Res. Appl. 30, 275–282 (2005).
[CrossRef]

Appl. Opt.

Color Res. Appl.

A. Takagi, A. Watanabe, and G. Baba, “Prediction of spectral reflectance factor distribution of automotive paint finishes,” Color Res. Appl. 30, 275–282 (2005).
[CrossRef]

L. Simonot, M. Hébert, and D. Dupraz, “Goniocolorimetry: from measurement to representation in the CIELAB color space,” Color Res. Appl. 36, 169–178 (2011).
[CrossRef]

E. Kirchner and W. Cramer, “Making sense of measurement geometries for multi-angle spectrophotometers,” Color Res. Appl. 37, 186–198 (2012).
[CrossRef]

Eur. Coat. J.

A. Argoitia and S. Chu, “Diffractive pigments help document security,” Eur. Coat. J. 32, 32–35 (2004).

J. Opt. Soc. Am. A

Metrologia

A. M. Rabal, A. Ferrero, J. Campos, J. L. Fontecha, A. Pons, A. M. Rubiño, and A. Corróns, “Automatic gonio-spectrophotometer for the absolute measurement of the spectral BRDF at in- and out-of-plane and retroreflection geometries,” Metrologia 49, 213–223 (2012).
[CrossRef]

Proc. SPIE

V. B. Podobedov, M. E. Nadal, and C. C. Miller, “Improving the performance of NIST five axis goniospectrometer for measurements of bidirectional reflectance distribution function,” Proc. SPIE 8065, 80651I (2011).
[CrossRef]

L. Kotačka, T. Têthal, and V. Kolařik, “Top-quality security optical elements: from holography towards 500.000  dpi,” Proc. SPIE 5954, 59540K (2005).
[CrossRef]

A. Argoitia and R. Phillips, “The security enhancement of diffractive optically variable image devices,” Proc. SPIE 6075, 60750P (2006).

G. Obein, R. Bousquet, and M. E. Nadal, “New NIST reference goniospectrometer,” Proc. SPIE 5880, 58800T (2005).
[CrossRef]

Other

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

“Standard practice for specifying the geometry of multiangle spectrophotometers,” (American Society for Testing and Materials, 2001).

R. L. van Renesse, Optical Document Security, 3rd ed. (Artech House, 2005).

“Tolerances for automotive paint—Part 2: goniochromatic paints,” (Deutsches Institut für Normung, 1999).

J. K. Nisper, P. S. Rood, B. A. Pawlanta, T. M. Richardson, and B. D. Teunis, “Measuring an appearance property of a surface using a bidirectional reflectance distribution function,” U.S. patent application2007/0291993 A1 (20December2007).

J. K. Nisper, T. M. Richardson, M. C. Ellens, and C. Huang, “Method and system for enhanced formulation and visualization rendering,” U.S. patent application2009/0213120 A1 (27August2009).

J. Nisper, T. Richardson, and B. Teunis, “Major advances in the reliable measurement of the color and appearance of special effect paints and coatings,” presented at the American Coatings Conference, Charlotte, North Carolina, 1April, 2008.

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Figures (10)

Fig. 1.
Fig. 1.

Notation of illumination (ϑi) and diffraction (ϑd) angles and diffraction order (m).

Fig. 2.
Fig. 2.

Sample in plane x, illuminated at ϑi from plane s and measured at ϑd in plane ξ.

Fig. 3.
Fig. 3.

Normalized reflectance spectra for grating with period 3.5 μm as calculated for amplitude (a) 0.03 μm and (b) 0.2 μm. Two additional diffractions are seen for the smallest amplitude in the log scale [(a), inset]. The notations at some λ refer to Fig. 5. The numbers in square denote the diffraction order.

Fig. 4.
Fig. 4.

xDNA graphs calculated for diffraction gratings with p=3.5μm in dependence on amplitude (specified in the figure, in micrometers). All graphs emerge from (0,0) but are split vertically (on the z axis) for clarity. The individual points represent the xDNA sum for the selected λ; the points that correspond to λ=700nm are marked by stars.

Fig. 5.
Fig. 5.

xDNA graphs for gratings with p=3.5μm and a equal to (a) 0.03 μm and (b) 0.2 μm. The λ values of characteristic points are denoted on graphs (in nanometers); see also Fig. 3. The arrows indicate the direction of the corresponding graph when λ increases.

Fig. 6.
Fig. 6.

Normalized reflectance spectra for gratings with a=0.03μm and periods (p) 1, 1.5, 2, 2.5, 3, and 3.5 μm (indicated on the graph). The spectra for each grating are split vertically. The φas of the individual diffractions is labeled.

Fig. 7.
Fig. 7.

xDNA graphs in dependence on grating period p (see the legend) and a=0.03μm. All graphs emerge from (0,0) but are split vertically (on the z axis) for clarity. The lines with the same slope represent diffractions at the indicated φas. See also Fig. 6 and Table 2.

Fig. 8.
Fig. 8.

Dependence of xDNA graphs of gratings with a=0.03μm on p (indicated in the legend) calculated for G17 (open symbols) and G6 measurement sets (solid symbols). All graphs emerge from (0,0) but are split vertically (on the z axis) for clarity. The corresponding φas values are also marked.

Fig. 9.
Fig. 9.

(a) Normalized reflectance spectra and (b) the corresponding xDNA representation for a diffraction grating with p=2.5μm and a=0.03μm. The four diffractions are denoted by φas values. The characteristic λ (in nanometers) and their xDNA representations are indicated.

Fig. 10.
Fig. 10.

Dependence of the slope of lines in xDNA graphs (Figs. 7 and 8) on aspecular angle φas.

Tables (3)

Tables Icon

Table 1. Diffractions for p=3.5μm in the G6 Measurement Set: Calculated Position (λc), Theoretical Position (λt), Aspecular Angle (φas), and Diffraction Order (m)a

Tables Icon

Table 2. Diffractions Obtained for Grating Period p in the G6 Measurement Set: Calculated Position (λc), Theoretical Position (λt), Aspecular Angle (φas), and Diffraction Order (m)a

Tables Icon

Table 3. Some Properties of Diffractions in Dependence on Grating Period p in a 17-Angle Geometry: Positions in Reflectance Spectra (λc), Theoretical Position According to Eq. (1) (λt), Aspecular Angle (φas), and Diffraction Order (m)

Equations (7)

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

sinϑd+sinϑi=mλpj,
U(ξ)=eikz0eikξ2/2z0iλz0U(x)·rect(x2w)eikxξ/z0·dx,
U(x)=eikz0eikx2/2z0iλz0f(x)s1s2U0eikxs/z0ds,
d=a2(1+sin(2πx/p));
f(x)=eikna(1+sin(2πx/p)),
ϕas=ϑi+ϑd.
xDNA=,

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