Abstract

Two superposed layers of transparent cylindrical lenslet gratings create classical moiré fringes, when illuminated from behind. We rely on this observation to conceive special devices made of superposed lenslet gratings that produce animated moirés when they are tilted against the light. One-dimensional moirés can show a message moving back and forth along a given direction or radially expanding towards the exterior of a disk. These 1D moirés are conceived by fabricating two layers of micro-lenses on both sides of a transparent substrate. The top layer is a rectilinear grating of cylindrical lenslets and the bottom layer is an arrangement of smaller lenslets of different sizes and orientations that create a high contrast. Moirés created by superpositions of lenslets are visually striking and can be challenging to fabricate. Therefore they have a high potential for art, decoration, and document security.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]

2017 (1)

2014 (1)

S. M. Chosson and R. D. Hersch, “Beating shapes relying on moiré level lines,” ACM Trans. Graph. 34(1), 1–11 (2014).
[Crossref]

2013 (1)

V. J. Cadarso, S. Chosson, K. Sidler, R. D. Hersch, and J. Brugger, “High-resolution 1d moirés as counterfeit security features,” Light: Sci. Appl. 2(7), e86 (2013).
[Crossref]

2004 (1)

R. D. Hersch and S. Chosson, “Band moiré images,” ACM Trans. Graph. (Proc. Siggraph) 23(3), 239–247 (2004).
[Crossref]

2001 (1)

1998 (1)

J. A. Hala Kamal and Reinhard Voelkel, “Properties of moire magnifiers,” Opt. Eng. 37(11), 3007–3014 (1998).
[Crossref]

1997 (1)

P. Nussbaum, R. Völkel, H. P. Herzig, M. Eisner, and S. Haselbeck, “Design, fabrication and testing of microlens arrays for sensors and microsystems,” Pure Appl. Opt. 6(6), 617–636 (1997).
[Crossref]

1994 (2)

I. Amidror, “A generalized fourier-based method for the analysis of 2d moiré envelope-forms in screen superpositions,” J. Mod. Opt. 41(9), 1837–1862 (1994).
[Crossref]

M. C. Hutley, R. Hunt, R. F. Stevens, and P. Savander, “The moire magnifier,” Pure Appl. Opt. 3(2), 133–142 (1994).
[Crossref]

1990 (1)

D. Daly, R. F. Stevens, M. C. Hutley, and N. Davies, “The manufacture of microlenses by melting photoresist,” Meas. Sci. Technol. 1(8), 759–766 (1990).
[Crossref]

1988 (1)

1975 (1)

O. Mikami, “New image-rotation using moiré lenses,” Jpn. J. Appl. Phys. 14(7), 1065–1066 (1975).
[Crossref]

1974 (1)

1965 (1)

1964 (1)

1887 (1)

A. Righi, “Sui fenomeni che si producono colla sovrapposizione di due peticoli e sopra alcune loro applicazioni,” Il Nuovo Cimento (1877-1894) 21(1), 203–228 (1887).
[Crossref]

Amidror, I.

I. Amidror, “A generalized fourier-based method for the analysis of 2d moiré envelope-forms in screen superpositions,” J. Mod. Opt. 41(9), 1837–1862 (1994).
[Crossref]

I. Amidror, The Theory of the Moiré Phenomenon - Volume I: Periodic Layers (Springer-Verlag, 2009), 2nd ed.

Besson, T.

T. Walger, T. Besson, V. Flauraud, R. D. Hersch, and J. Brugger, “Level-line moirés by superposition of cylindrical micro-lens gratings; submitted to josa-a,” JOSA-A (2019).

Bruckstein, A. M.

Brugger, J.

V. J. Cadarso, S. Chosson, K. Sidler, R. D. Hersch, and J. Brugger, “High-resolution 1d moirés as counterfeit security features,” Light: Sci. Appl. 2(7), e86 (2013).
[Crossref]

T. Walger, T. Besson, V. Flauraud, R. D. Hersch, and J. Brugger, “Level-line moirés by superposition of cylindrical micro-lens gratings; submitted to josa-a,” JOSA-A (2019).

Bryngdahl, O.

Cadarso, V. J.

V. J. Cadarso, S. Chosson, K. Sidler, R. D. Hersch, and J. Brugger, “High-resolution 1d moirés as counterfeit security features,” Light: Sci. Appl. 2(7), e86 (2013).
[Crossref]

Chosson, S.

V. J. Cadarso, S. Chosson, K. Sidler, R. D. Hersch, and J. Brugger, “High-resolution 1d moirés as counterfeit security features,” Light: Sci. Appl. 2(7), e86 (2013).
[Crossref]

R. D. Hersch and S. Chosson, “Band moiré images,” ACM Trans. Graph. (Proc. Siggraph) 23(3), 239–247 (2004).
[Crossref]

Chosson, S. M.

S. M. Chosson and R. D. Hersch, “Beating shapes relying on moiré level lines,” ACM Trans. Graph. 34(1), 1–11 (2014).
[Crossref]

Connell, G. A. N.

Daly, D.

D. Daly, R. F. Stevens, M. C. Hutley, and N. Davies, “The manufacture of microlenses by melting photoresist,” Meas. Sci. Technol. 1(8), 759–766 (1990).
[Crossref]

Davies, N.

D. Daly, R. F. Stevens, M. C. Hutley, and N. Davies, “The manufacture of microlenses by melting photoresist,” Meas. Sci. Technol. 1(8), 759–766 (1990).
[Crossref]

Eisner, M.

P. Nussbaum, R. Völkel, H. P. Herzig, M. Eisner, and S. Haselbeck, “Design, fabrication and testing of microlens arrays for sensors and microsystems,” Pure Appl. Opt. 6(6), 617–636 (1997).
[Crossref]

Flauraud, V.

T. Walger, T. Besson, V. Flauraud, R. D. Hersch, and J. Brugger, “Level-line moirés by superposition of cylindrical micro-lens gratings; submitted to josa-a,” JOSA-A (2019).

Gao, Y.

Hala Kamal, J. A.

J. A. Hala Kamal and Reinhard Voelkel, “Properties of moire magnifiers,” Opt. Eng. 37(11), 3007–3014 (1998).
[Crossref]

Haselbeck, S.

P. Nussbaum, R. Völkel, H. P. Herzig, M. Eisner, and S. Haselbeck, “Design, fabrication and testing of microlens arrays for sensors and microsystems,” Pure Appl. Opt. 6(6), 617–636 (1997).
[Crossref]

Hecht, E.

E. Hecht, Optics, Chapter 5 (Pearson, 2017), 5th ed.

Hersch, R. D.

S. M. Chosson and R. D. Hersch, “Beating shapes relying on moiré level lines,” ACM Trans. Graph. 34(1), 1–11 (2014).
[Crossref]

V. J. Cadarso, S. Chosson, K. Sidler, R. D. Hersch, and J. Brugger, “High-resolution 1d moirés as counterfeit security features,” Light: Sci. Appl. 2(7), e86 (2013).
[Crossref]

R. D. Hersch and S. Chosson, “Band moiré images,” ACM Trans. Graph. (Proc. Siggraph) 23(3), 239–247 (2004).
[Crossref]

T. Walger, T. Besson, V. Flauraud, R. D. Hersch, and J. Brugger, “Level-line moirés by superposition of cylindrical micro-lens gratings; submitted to josa-a,” JOSA-A (2019).

Herzig, H. P.

P. Nussbaum, R. Völkel, H. P. Herzig, M. Eisner, and S. Haselbeck, “Design, fabrication and testing of microlens arrays for sensors and microsystems,” Pure Appl. Opt. 6(6), 617–636 (1997).
[Crossref]

Hunt, R.

M. C. Hutley, R. Hunt, R. F. Stevens, and P. Savander, “The moire magnifier,” Pure Appl. Opt. 3(2), 133–142 (1994).
[Crossref]

Hurt, M.

R. Steenblik, M. Hurt, and G. Jordan, “Micro-optic security and image presentation system,” (2008). US Patent 7,333,268.

Hutley, M. C.

M. C. Hutley, R. Hunt, R. F. Stevens, and P. Savander, “The moire magnifier,” Pure Appl. Opt. 3(2), 133–142 (1994).
[Crossref]

D. Daly, R. F. Stevens, M. C. Hutley, and N. Davies, “The manufacture of microlenses by melting photoresist,” Meas. Sci. Technol. 1(8), 759–766 (1990).
[Crossref]

Jordan, G.

R. Steenblik, M. Hurt, and G. Jordan, “Micro-optic security and image presentation system,” (2008). US Patent 7,333,268.

Lebanon, G.

Liu, N.

Liu, Y.

Mikami, O.

O. Mikami, “New image-rotation using moiré lenses,” Jpn. J. Appl. Phys. 14(7), 1065–1066 (1975).
[Crossref]

Nishijima, Y.

Nussbaum, P.

P. Nussbaum, R. Völkel, H. P. Herzig, M. Eisner, and S. Haselbeck, “Design, fabrication and testing of microlens arrays for sensors and microsystems,” Pure Appl. Opt. 6(6), 617–636 (1997).
[Crossref]

Oster, G.

Popovic, Z. D.

Righi, A.

A. Righi, “Sui fenomeni che si producono colla sovrapposizione di due peticoli e sopra alcune loro applicazioni,” Il Nuovo Cimento (1877-1894) 21(1), 203–228 (1887).
[Crossref]

Savander, P.

M. C. Hutley, R. Hunt, R. F. Stevens, and P. Savander, “The moire magnifier,” Pure Appl. Opt. 3(2), 133–142 (1994).
[Crossref]

Shen, S.

Sidler, K.

V. J. Cadarso, S. Chosson, K. Sidler, R. D. Hersch, and J. Brugger, “High-resolution 1d moirés as counterfeit security features,” Light: Sci. Appl. 2(7), e86 (2013).
[Crossref]

Sprague, R. A.

Steenblik, R.

R. Steenblik, M. Hurt, and G. Jordan, “Micro-optic security and image presentation system,” (2008). US Patent 7,333,268.

Stevens, R. F.

M. C. Hutley, R. Hunt, R. F. Stevens, and P. Savander, “The moire magnifier,” Pure Appl. Opt. 3(2), 133–142 (1994).
[Crossref]

D. Daly, R. F. Stevens, M. C. Hutley, and N. Davies, “The manufacture of microlenses by melting photoresist,” Meas. Sci. Technol. 1(8), 759–766 (1990).
[Crossref]

Tollenaar, D.

D. Tollenaar, Moiré: interferentieverschijnselen bij rasterdruk, Moiré interference phenomena in halftone printing English translation published in 1964, reprinted in Indebetouw G. Czarnek R. (Eds.). Selected Papers on Optical Moiré and Applications, SPIE Milestone Series, Vol. MS64, 1992, pp. 618-633 (Instituut voor Grafische Techniek, 1945).

Voelkel, Reinhard

J. A. Hala Kamal and Reinhard Voelkel, “Properties of moire magnifiers,” Opt. Eng. 37(11), 3007–3014 (1998).
[Crossref]

Völkel, R.

P. Nussbaum, R. Völkel, H. P. Herzig, M. Eisner, and S. Haselbeck, “Design, fabrication and testing of microlens arrays for sensors and microsystems,” Pure Appl. Opt. 6(6), 617–636 (1997).
[Crossref]

Walger, T.

T. Walger, T. Besson, V. Flauraud, R. D. Hersch, and J. Brugger, “Level-line moirés by superposition of cylindrical micro-lens gratings; submitted to josa-a,” JOSA-A (2019).

Zheng, W.

ACM Trans. Graph. (1)

S. M. Chosson and R. D. Hersch, “Beating shapes relying on moiré level lines,” ACM Trans. Graph. 34(1), 1–11 (2014).
[Crossref]

ACM Trans. Graph. (Proc. Siggraph) (1)

R. D. Hersch and S. Chosson, “Band moiré images,” ACM Trans. Graph. (Proc. Siggraph) 23(3), 239–247 (2004).
[Crossref]

Appl. Opt. (2)

Il Nuovo Cimento (1877-1894) (1)

A. Righi, “Sui fenomeni che si producono colla sovrapposizione di due peticoli e sopra alcune loro applicazioni,” Il Nuovo Cimento (1877-1894) 21(1), 203–228 (1887).
[Crossref]

J. Mod. Opt. (1)

I. Amidror, “A generalized fourier-based method for the analysis of 2d moiré envelope-forms in screen superpositions,” J. Mod. Opt. 41(9), 1837–1862 (1994).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

Jpn. J. Appl. Phys. (1)

O. Mikami, “New image-rotation using moiré lenses,” Jpn. J. Appl. Phys. 14(7), 1065–1066 (1975).
[Crossref]

Light: Sci. Appl. (1)

V. J. Cadarso, S. Chosson, K. Sidler, R. D. Hersch, and J. Brugger, “High-resolution 1d moirés as counterfeit security features,” Light: Sci. Appl. 2(7), e86 (2013).
[Crossref]

Meas. Sci. Technol. (1)

D. Daly, R. F. Stevens, M. C. Hutley, and N. Davies, “The manufacture of microlenses by melting photoresist,” Meas. Sci. Technol. 1(8), 759–766 (1990).
[Crossref]

Opt. Eng. (1)

J. A. Hala Kamal and Reinhard Voelkel, “Properties of moire magnifiers,” Opt. Eng. 37(11), 3007–3014 (1998).
[Crossref]

Opt. Express (1)

Pure Appl. Opt. (2)

P. Nussbaum, R. Völkel, H. P. Herzig, M. Eisner, and S. Haselbeck, “Design, fabrication and testing of microlens arrays for sensors and microsystems,” Pure Appl. Opt. 6(6), 617–636 (1997).
[Crossref]

M. C. Hutley, R. Hunt, R. F. Stevens, and P. Savander, “The moire magnifier,” Pure Appl. Opt. 3(2), 133–142 (1994).
[Crossref]

Other (5)

R. Steenblik, M. Hurt, and G. Jordan, “Micro-optic security and image presentation system,” (2008). US Patent 7,333,268.

D. Tollenaar, Moiré: interferentieverschijnselen bij rasterdruk, Moiré interference phenomena in halftone printing English translation published in 1964, reprinted in Indebetouw G. Czarnek R. (Eds.). Selected Papers on Optical Moiré and Applications, SPIE Milestone Series, Vol. MS64, 1992, pp. 618-633 (Instituut voor Grafische Techniek, 1945).

I. Amidror, The Theory of the Moiré Phenomenon - Volume I: Periodic Layers (Springer-Verlag, 2009), 2nd ed.

T. Walger, T. Besson, V. Flauraud, R. D. Hersch, and J. Brugger, “Level-line moirés by superposition of cylindrical micro-lens gratings; submitted to josa-a,” JOSA-A (2019).

E. Hecht, Optics, Chapter 5 (Pearson, 2017), 5th ed.

Supplementary Material (5)

NameDescription
» Visualization 1       Video of a fabricated sample depicting a "Valid OK" 1D moire´, illuminated from an office window with plants and sky in the background.
» Visualization 2       Diffracting 1D moirés formed by a horizontal revealer lenslet grating of repetition period 400 µm and bases with different lens parameters.
» Visualization 3       Video of the "VALID" and "OK" 1D lenslet grating moiré moving in opposite directions.
» Visualization 4       Video where the circular moiré "valid official document" is evolving radially.
» Visualization 5       Video of moiré displaying petals rotating circularly when tilting the lenslet device.

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Figures (14)

Fig. 1.
Fig. 1. (a) Schematic cross-section of the new 1D moiré samples. (b) Photograph of a fabricated sample depicting a "Valid OK" 1D moiré, illuminated from an office window with plants and sky in the background (see Visualization 1). The revealer is made of a rectilinear vertical cylindrical lenslet grating of period 400 µm. The inside of the characters is filled with a grating of horizontal cylindrical lenslets of period 16 µm. The outside of the characters is filled with a grating of vertical cylindrical lenslets of period 6 µm.
Fig. 2.
Fig. 2. (a) Photograph of moiré fringes produced by the superposition of two layers of lenticular gratings. (b) Example of a 1D moiré formed by repetitive base bands superposed with a sampling revealing line grating [3].
Fig. 3.
Fig. 3. Top (a) and (b): raised cosinusoidal gratings and (c) their multiplicative superposition in the image domain. Bottom (d) and (e): impulse pairs obtained by the Fourier transform of the corresponding individual layers and (f) the new additive and subtractive impulse pairs obtained by convolution of the frequency pairs $(\vec {f_1}, -\vec {f_1})$ and $(\vec {f_2}, -\vec {f_2})$.
Fig. 4.
Fig. 4. Representation of the base bands (gray) replicated by vector $\vec {t}$, of the revealing layer sampling line grating (red) having period $T_r$ as well as the corresponding vertically-orientated (vector $\vec {s}$) moiré letters (blue) laid out horizontally (vector $\vec {u}$) replicated along vector $\vec {p}$.
Fig. 5.
Fig. 5. (a) Bitmap showing the interior and exterior of the compressed letter shape "P" within successive base bands with black segments forming the interior of the letter shapes, to be filled with cylindrical lenslets. The letters are mirrored in the base to obtain the correctly-oriented text when looking through the revealer. (b) Magnified area of the bitmap image. (c) Optical micrograph of the fabricated lenslet base layer. The depicted area corresponds to the design area shown in (a). (d) Magnified area of (c) that corresponds to the design area shown in (b). The rectilinear cylindrical lenslets have a repetition period of 16 µm and are laid out at 15$^\circ$ within the interior regions of the letter shapes. Randomly-placed spherical lenslets of diameters varying between 4 µm and 7 µm are laid out in the exterior regions.
Fig. 6.
Fig. 6. Section through the transparent element comprising on one side a cylindrical lens that is part of the revealing sampling layer and on the other side a base band with both small cylindrical lenses forming the interior of the base band shapes and smaller randomly-placed spherical lenses of various sizes forming the exterior of the base band shapes.
Fig. 7.
Fig. 7. (a) Part of a cylindrical lenslet on top of a substrate. (b) Rectangular section before reflow (top) and corresponding lens section after reflow (bottom) with the lenslet parameters: $S_{circ}$: surface of lenslet section above the substrate, $R$: radius of curvature, $w$: width of lenslet, $h$: sag-height, $f$: focal length, $d$: mono-lens substrate thickness.
Fig. 8.
Fig. 8. Diffracting 1D moirés formed by a horizontal revealer lenslet grating of repetition period 400 µm, of base layer shape foreground (inside) lenslets of period (a) 16 µm horizontal, (b) 27 µm at 15$^\circ$, (c) 27 µm horizontal and of base layer shape background (outside) lenslets of period (a) 8 µm vertical, (b) 12 µm at 105$^\circ$, and (c) 12 µm vertical. (d) Optical micrograph of the letter "P" within the base of (b), where the text in the base is mirrored to obtain a correctly-oriented text when looking through the revealer. (e) Magnified area of (d).
Fig. 9.
Fig. 9. Process flow for the micro-fabrication of the lenslet gratings.
Fig. 10.
Fig. 10. Effect of the rotation of the revealer on a horizontal moiré vector ($\vec {u}$ in Fig. 4). The considered moirés are the "ok" and the "valid" moirés shown in Fig. 11. For each curve, the base layer size has been calculated to yield the same moiré size. Even small rotations of the revealer have a large impact on the moiré shapes. We can also observe that the smaller the period $T_r$ of the revealer, the larger the impact that misalignments have on the moiré.
Fig. 11.
Fig. 11. Capture of the "VALID" and "OK" 1D lenslet grating moiré moving in opposite directions, see the video (Visualization 3).
Fig. 12.
Fig. 12. (a) Base layer design in which the white areas correspond to the interior of the shapes and the black areas correspond to the outside. From this design, the base can be filled with lenslets. (b) Superposition of the base and the revealer to visualize the message formed by the moiré. This specific moiré design was used to fabricate the sample shown in Fig. 13(a).
Fig. 13.
Fig. 13. (a) Part of frame extracted from the video showing the tilted lenslet device (Visualization 4). (b) Optical micrograph of an area of the base. The interior of the shapes is filled with rectilinear cylindrical lenses of period 16 µm at 15$^\circ$. The exterior of the shapes is filled with randomly-placed spherical lenslets.
Fig. 14.
Fig. 14. (a) Part of frame extracted from the video showing the tilted lenslet device (Visualization 5). (b) Bitmap showing the lens layout in the center of the design. The black lines indicate the place of the cylindrical lenslets. (c) Optical micrograph of the fabricated lenslet base layer. The interior of the shapes is filled with rectilinear cylindrical lenses of period 16 µm at 15$^\circ$. The exterior of the shapes is filled with randomly-placed spherical lenslets.

Equations (18)

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

[xy]=[1txTrty0TrTrty][xy]
x=mx(xt,yt)=hx(xt,yt)+(hy(xt,yt)gy(xt,yt))txTrtyy=my(xt,yt)=hy(xt,yt)TrTrtygy(xt,yt)tyTrty
(Rh)2=R2(w/2)2
R=w28h+h2
f=nSnSnairR
h=fd
R24m(m2)+R8d(1m)+w2+4d2=0
R=bb24ac2awitha=4m28mb=8d(1m)c=w2+4d2
S=2(R2θ2w2(Rh)2)
S=R2θR2sinθcosθ
S=R22(2θsin(2θ))
[xy]=Mt[xtyt]
Mt=1Trty[Trty+txsinθtx(1cosθ)tysinθTrtycosθ]
[xtyt]=Mt1[xy]
Mt1=Trty(Trty+txsinθ)(Trtycosθ)tx(1cosθ)tysinθ[Trtycosθtx(1cosθ)tysinθTrty+txsinθ]
hx(xt,yt)=(gy(xt,yt)my(xt,yt))txTr+mx(xt,yt)hy(xt,yt)=gy(xt,yt)tyTr+my(xt,yt)TrtyTr
x=mx(xt,yt)=πarctan(ytcy,xtcx)2πwxy=my(xt,yt)=cm(xtcx)2+(ytcy)2
hx(xt,yt)=(ytcm(xtcx)2+(ytcy)2)txTr+πarctan(ytcy,xtcx)2πwxhy(xt,yt)=cm(xtcx)2+(ytcy)2TrtyTr+yttyTr

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