Abstract

An array of microprisms was described by a model of multiperiod blazed gratings consisting of triangular apertures. The origins of hexagram-shaped diffraction patterns were interpreted based on multiple-beam interference and diffraction array theorem. The relation between zonal/line ghost fringes and imperfectly fabricated array structures was analyzed. Geometrical performance (e.g., the dihedral angle of the microprism) was tested by measuring the features of diffraction patterns of samples from three retroreflective sheeting manufacturers.

© 2012 Optical Society of America

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References

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    [CrossRef]
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2010

2009

K. C. Vishnubhatla, S. Venugopal Rao, R. Sai Santosh Kumar, M. Ferrari, and D. Narayana Rao, “Optical studies of two dimensional gratings in fused silica, GE 124, and Foturan TM glasses fabricated using femtosecond laser pulses,” Opt. Commun. 282, 4537–4542 (2009).
[CrossRef]

2008

2006

J. Kühn, E. Cuche, Y. Emery, T. Colomb, F. Charrière, F. Montfort, M. Botkine, N. Aspert, and C. Depeursinge, “Measurements of corner cubes microstructures by high-magnification digital holographic microscopy,” Proc. Soc. Photo-Opt. Instrum. Eng. 6188, 18804–18804, (2006).
[CrossRef]

2005

2004

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

H. Ichikawa, “Numerical analysis of microretroreflectors: transition from reflection to diffraction,” J. Opt. A: Pure Appl. Opt. 6, S121–S127 (2004).
[CrossRef]

1999

1996

1995

1960

1948

Aspert, N.

J. Kühn, E. Cuche, Y. Emery, T. Colomb, F. Charrière, F. Montfort, M. Botkine, N. Aspert, and C. Depeursinge, “Measurements of corner cubes microstructures by high-magnification digital holographic microscopy,” Proc. Soc. Photo-Opt. Instrum. Eng. 6188, 18804–18804, (2006).
[CrossRef]

Benson, G. M.

S. K. Nestegard, G. M. Benson, C. Frey, J. C. Kelliher, J. E. Lasch, K. L. Smith, and T. J. Szczech, “Dual orientation retroreflective sheeting,” U.S. patent 5,706,132 (6January1998).

Botkine, M.

J. Kühn, E. Cuche, Y. Emery, T. Colomb, F. Charrière, F. Montfort, M. Botkine, N. Aspert, and C. Depeursinge, “Measurements of corner cubes microstructures by high-magnification digital holographic microscopy,” Proc. Soc. Photo-Opt. Instrum. Eng. 6188, 18804–18804, (2006).
[CrossRef]

Chandezon, J.

Chandler, K. N.

Chang, J. C. H.

Charrière, F.

J. Kühn, E. Cuche, Y. Emery, T. Colomb, F. Charrière, F. Montfort, M. Botkine, N. Aspert, and C. Depeursinge, “Measurements of corner cubes microstructures by high-magnification digital holographic microscopy,” Proc. Soc. Photo-Opt. Instrum. Eng. 6188, 18804–18804, (2006).
[CrossRef]

Chernov, S. M.

Colomb, T.

J. Kühn, E. Cuche, Y. Emery, T. Colomb, F. Charrière, F. Montfort, M. Botkine, N. Aspert, and C. Depeursinge, “Measurements of corner cubes microstructures by high-magnification digital holographic microscopy,” Proc. Soc. Photo-Opt. Instrum. Eng. 6188, 18804–18804, (2006).
[CrossRef]

Couzin, D. I.

D. I. Couzin, “Tri-level cube corner ruling,” U.S. patent 7,334,904 (26February2008).

Cuche, E.

J. Kühn, E. Cuche, Y. Emery, T. Colomb, F. Charrière, F. Montfort, M. Botkine, N. Aspert, and C. Depeursinge, “Measurements of corner cubes microstructures by high-magnification digital holographic microscopy,” Proc. Soc. Photo-Opt. Instrum. Eng. 6188, 18804–18804, (2006).
[CrossRef]

Davies, A.

Dementev, V. A.

Denton, J. P.

Depeursinge, C.

J. Kühn, E. Cuche, Y. Emery, T. Colomb, F. Charrière, F. Montfort, M. Botkine, N. Aspert, and C. Depeursinge, “Measurements of corner cubes microstructures by high-magnification digital holographic microscopy,” Proc. Soc. Photo-Opt. Instrum. Eng. 6188, 18804–18804, (2006).
[CrossRef]

Diehl, R. D.

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

Dusséaux, R.

Edwards, D. J.

Emery, Y.

J. Kühn, E. Cuche, Y. Emery, T. Colomb, F. Charrière, F. Montfort, M. Botkine, N. Aspert, and C. Depeursinge, “Measurements of corner cubes microstructures by high-magnification digital holographic microscopy,” Proc. Soc. Photo-Opt. Instrum. Eng. 6188, 18804–18804, (2006).
[CrossRef]

Fano, U.

Farahi, F.

Faulkner, G. E.

Faure, C.

Ferralis, N.

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

Ferrari, M.

K. C. Vishnubhatla, S. Venugopal Rao, R. Sai Santosh Kumar, M. Ferrari, and D. Narayana Rao, “Optical studies of two dimensional gratings in fused silica, GE 124, and Foturan TM glasses fabricated using femtosecond laser pulses,” Opt. Commun. 282, 4537–4542 (2009).
[CrossRef]

Frey, C.

S. K. Nestegard, G. M. Benson, C. Frey, J. C. Kelliher, J. E. Lasch, K. L. Smith, and T. J. Szczech, “Dual orientation retroreflective sheeting,” U.S. patent 5,706,132 (6January1998).

Gallagher, N.

Gauer, G. J.

R. B. Nilsen, X. J. Lu, and G. J. Gauer, “Multi-orientation retroreflective structure,” U.S. patent 6,877,866 (12April2005).

Heffels, C.

Heitzmann, D.

Hirleman, E. D.

Ichikawa, H.

H. Ichikawa, “Numerical analysis of microretroreflectors: transition from reflection to diffraction,” J. Opt. A: Pure Appl. Opt. 6, S121–S127 (2004).
[CrossRef]

Kelliher, J. C.

S. K. Nestegard, G. M. Benson, C. Frey, J. C. Kelliher, J. E. Lasch, K. L. Smith, and T. J. Szczech, “Dual orientation retroreflective sheeting,” U.S. patent 5,706,132 (6January1998).

Kim, H.

Kühn, J.

J. Kühn, E. Cuche, Y. Emery, T. Colomb, F. Charrière, F. Montfort, M. Botkine, N. Aspert, and C. Depeursinge, “Measurements of corner cubes microstructures by high-magnification digital holographic microscopy,” Proc. Soc. Photo-Opt. Instrum. Eng. 6188, 18804–18804, (2006).
[CrossRef]

Kumar, R. Sai Santosh

K. C. Vishnubhatla, S. Venugopal Rao, R. Sai Santosh Kumar, M. Ferrari, and D. Narayana Rao, “Optical studies of two dimensional gratings in fused silica, GE 124, and Foturan TM glasses fabricated using femtosecond laser pulses,” Opt. Commun. 282, 4537–4542 (2009).
[CrossRef]

Lamekin, P. I.

Lasch, J. E.

S. K. Nestegard, G. M. Benson, C. Frey, J. C. Kelliher, J. E. Lasch, K. L. Smith, and T. J. Szczech, “Dual orientation retroreflective sheeting,” U.S. patent 5,706,132 (6January1998).

Lee, B.

Lu, X. J.

R. B. Nilsen, X. J. Lu, and G. J. Gauer, “Multi-orientation retroreflective structure,” U.S. patent 6,877,866 (12April2005).

Min, S.-W.

Molinet, F.

Montfort, F.

J. Kühn, E. Cuche, Y. Emery, T. Colomb, F. Charrière, F. Montfort, M. Botkine, N. Aspert, and C. Depeursinge, “Measurements of corner cubes microstructures by high-magnification digital holographic microscopy,” Proc. Soc. Photo-Opt. Instrum. Eng. 6188, 18804–18804, (2006).
[CrossRef]

Nestegard, S. K.

S. K. Nestegard, G. M. Benson, C. Frey, J. C. Kelliher, J. E. Lasch, K. L. Smith, and T. J. Szczech, “Dual orientation retroreflective sheeting,” U.S. patent 5,706,132 (6January1998).

Neudeck, G. W.

Nilsen, R. B.

W. P. Rowland and R. B. Nilsen, “Miniature micro prism retroreflector,” U.S. patent 6,206,525 (27March2001).

R. B. Nilsen, X. J. Lu, and G. J. Gauer, “Multi-orientation retroreflective structure,” U.S. patent 6,877,866 (12April2005).

O’Brien, D. C.

Ottevaere, H.

Purcell, D.

Rao, D. Narayana

K. C. Vishnubhatla, S. Venugopal Rao, R. Sai Santosh Kumar, M. Ferrari, and D. Narayana Rao, “Optical studies of two dimensional gratings in fused silica, GE 124, and Foturan TM glasses fabricated using femtosecond laser pulses,” Opt. Commun. 282, 4537–4542 (2009).
[CrossRef]

Rao, S. Venugopal

K. C. Vishnubhatla, S. Venugopal Rao, R. Sai Santosh Kumar, M. Ferrari, and D. Narayana Rao, “Optical studies of two dimensional gratings in fused silica, GE 124, and Foturan TM glasses fabricated using femtosecond laser pulses,” Opt. Commun. 282, 4537–4542 (2009).
[CrossRef]

Rowland, W. P.

W. P. Rowland and R. B. Nilsen, “Miniature micro prism retroreflector,” U.S. patent 6,206,525 (27March2001).

Scarlett, B.

Smith, K. L.

S. K. Nestegard, G. M. Benson, C. Frey, J. C. Kelliher, J. E. Lasch, K. L. Smith, and T. J. Szczech, “Dual orientation retroreflective sheeting,” U.S. patent 5,706,132 (6January1998).

Spitz, J.

Stamm, R. F.

R. F. Stamm, “Retroreflective surface,” U.S. patent 3,712,706 (23January1973).

Suratkar, A.

Szczech, T. J.

S. K. Nestegard, G. M. Benson, C. Frey, J. C. Kelliher, J. E. Lasch, K. L. Smith, and T. J. Szczech, “Dual orientation retroreflective sheeting,” U.S. patent 5,706,132 (6January1998).

Szmodis, A. W.

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

Thienpont, H.

Vishnubhatla, K. C.

K. C. Vishnubhatla, S. Venugopal Rao, R. Sai Santosh Kumar, M. Ferrari, and D. Narayana Rao, “Optical studies of two dimensional gratings in fused silica, GE 124, and Foturan TM glasses fabricated using femtosecond laser pulses,” Opt. Commun. 282, 4537–4542 (2009).
[CrossRef]

Am. J. Phys.

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

Appl. Opt.

J. Opt. A: Pure Appl. Opt.

H. Ichikawa, “Numerical analysis of microretroreflectors: transition from reflection to diffraction,” J. Opt. A: Pure Appl. Opt. 6, S121–S127 (2004).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Technol.

Opt. Commun.

K. C. Vishnubhatla, S. Venugopal Rao, R. Sai Santosh Kumar, M. Ferrari, and D. Narayana Rao, “Optical studies of two dimensional gratings in fused silica, GE 124, and Foturan TM glasses fabricated using femtosecond laser pulses,” Opt. Commun. 282, 4537–4542 (2009).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng.

J. Kühn, E. Cuche, Y. Emery, T. Colomb, F. Charrière, F. Montfort, M. Botkine, N. Aspert, and C. Depeursinge, “Measurements of corner cubes microstructures by high-magnification digital holographic microscopy,” Proc. Soc. Photo-Opt. Instrum. Eng. 6188, 18804–18804, (2006).
[CrossRef]

Other

W. P. Rowland and R. B. Nilsen, “Miniature micro prism retroreflector,” U.S. patent 6,206,525 (27March2001).

R. F. Stamm, “Retroreflective surface,” U.S. patent 3,712,706 (23January1973).

D. I. Couzin, “Tri-level cube corner ruling,” U.S. patent 7,334,904 (26February2008).

R. B. Nilsen, X. J. Lu, and G. J. Gauer, “Multi-orientation retroreflective structure,” U.S. patent 6,877,866 (12April2005).

S. K. Nestegard, G. M. Benson, C. Frey, J. C. Kelliher, J. E. Lasch, K. L. Smith, and T. J. Szczech, “Dual orientation retroreflective sheeting,” U.S. patent 5,706,132 (6January1998).

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Figures (9)

Fig. 1.
Fig. 1.

Retroreflective prisms: (a) perfect pyramid-shaped prism; (b), (c) imperfectly fabricated prisms.

Fig. 2.
Fig. 2.

Diffraction experiment in the transmission model. (a) Microprism array under collimated laser illumination, (b) photographs of diffraction patterns.

Fig. 3.
Fig. 3.

Refraction and diffraction in a prism. (a) The parallel light was separated into three refraction beams; (b) laser traveling through aperture ΔBHC, (c) diffraction pattern formed by aperture ΔBHC (theoretical image S3), (d) experimental photo S3.

Fig. 4.
Fig. 4.

Relationship between transmission apertures and their diffraction patterns. (a), (b) Triangular sub-pattern diffracted by three apertures of single prism, (c) four possible kinds of combination of adjacent prisms in c.c. array, (d) six-pointed star pattern diffracted by six apertures of biprism.

Fig. 5.
Fig. 5.

Diagonal line formed by a one-dimensional prismatic grating. Oai and Obi, adjacent apertures have opposite orientation; M1N1, one-dimensional structure constituted by periodical apertures; M1N1, diagonal line in diffraction pattern.

Fig. 6.
Fig. 6.

Diffraction patterns of retroreflective sheeting. (a) Photograph of corner-cube array, (b) schema of dual-orientation microprism structure, (c)–(e) diffraction patterns related to the variations of axis orientation of cube prism (samples 1, 2, 3; wavelength 0.63 μm), (f) deformed diffraction pattern related to the dihedral angle of sample 4 (P7, isosceles triangle; P8, equilateral triangle).

Fig. 7.
Fig. 7.

Diffraction patterns and ghost features: (a) pattern generated by c.c. array; (b), (c) enlargement of patterns in the dashed circle: (b) generated by perfect apertures, (b′) corresponding isosceles triangle aperture (schematic diagram), (c) patterns with ghost lines generated by imperfect apertures in sample 5 (wavelength 0.63 μm), (c′) corresponding obtuse triangle aperture (schematic diagram).

Fig. 8.
Fig. 8.

Ghost fringes in diffraction patterns. (a) Diagonal lines of perfect one-dimensional periodical structures; (b) single line is a ghost; (c), (d) ghost lines associated with main diagonal line in samples 6 and 7; (e) zonal ghost in sample 8 (wavelength 0.53 μm)

Fig. 9.
Fig. 9.

Refraction of light by a prism and the measuring of the dihedral angle αi.

Tables (2)

Tables Icon

Table 1. Geometrical Parameters of Retroreflective Sheetingab

Tables Icon

Table 2. Imperfections in the c.c. Array

Equations (12)

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

T⃗=Q⃗+2Q⃗×(+α·aβ·b+γ·c).
Δ42/3·nδ.
A(x,y)=Ai(x,y)Aδ,
n0·sinθ1=n1·sinθ2,
n1·sinθ3=n0·sinθ4,
n0·sinθ1=n1·(sinθ1·cosθ3+cosθ1·sinθ3),
n0·tanθ1=n1·(tanθ1·cosθ3+cosθ1·sinθ3).
αi=θ1=tan1[n1·sinθ3/(1n1·cosθ3)],
sinθ1·dθ1=n1·cnsθ2·dθ2,
n1·cosθ3·dθ3=cnsθ4·dθ4.
dθ4/dθ1=(cosθ1/cosθ2)·(cosθ3/cosθ4).
Kjkjdjtanθ3,

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