Abstract

Moiré effect is a well-known interference phenomenon occurred between repetitive structures. Conventional moiré fringes are produced by superposing gratings or dots arrays. However, when the micro-image array as base layer and the corresponding micro-focusing elements as revealing layer overlap each other, a special kind of moiré effect, so-called moiré magnifier, can be observed. Micro-image units in the base layer are enlarged and projected to moiré space. To our knowledge, there has no complete design methodology for the realization of the moiré magnifier. With the combination of the Fourier transform and spectral approach, a new algorithm based on transfer matrix is investigated, which is capable of predicting the location of any arbitrary point in the base layer mapping to the moiré space, thus it provides a simple way to explore the physical insight into the field of moiré imaging. The magnification factor and the orientation of the synthetically enlarged image are determined not only by the scaling ratio but also by the interrelation between the primitive vectors in the base and revealing layers. Experimental results are in good agreement with theoretical predictions. By using the proposed method, the moiré magnifier can extend appealing applications in esthetic security devices, highly accurate measurements and precise color printing.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2015 (1)

2014 (3)

2013 (3)

V. Saveljev and S. K. Kim, “Theoretical estimation of moiré effect using spectral trajectories,” Opt. Express 21(2), 1693–1712 (2013).
[PubMed]

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).

S. Trivedi, J. Dhanotia, and S. Prakash, “Measurement of focal length using phase shifted moiré deflectometry,” Opt. Lasers Eng. 51(6), 776–782 (2013).

2012 (2)

K. Hermann, “Periodic overlayers and moiré patterns: theoretical studies of geometric properties,” J. Phys. Condens. Matter 24(31), 314210 (2012).
[PubMed]

S. Shen, Y. Lou, J. Hu, Y. Zhou, and L. Chen, “Realization of Glass patterns by a microlens array,” Opt. Lett. 37(20), 4248–4250 (2012).
[PubMed]

2004 (1)

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

2003 (1)

1998 (2)

1997 (1)

G. L. Rogers, “A geometrical approach to moiré pattern calculations,” Opt. Acta (Lond.) 24(1), 1–13 (1997).

1994 (1)

M. C. Hutley, R. Hunt, R. F. Stevens, and P. Savander, “The moiré magnifier,” Pure Appl. Opt. 3(2), 133 (1994).

1976 (1)

K. Patorski, S. Yokozeki, and T. Suzuki, “Moiré profile prediction by using Fourier series formulism,” Jpn. J. Appl. Phys. 15(3), 443–456 (1976).

1970 (1)

Alda, J.

H. Kamal, R. Völkel, and J. Alda, “Properties of moiré magnifiers,” Opt. Eng. 37(11), 3007–3014 (1998).

Allen, J. B.

Amidror, I.

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).

Byun, S. J.

Byun, S. Y.

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).

Chen, L.

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).

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

Dhanotia, J.

S. Trivedi, J. Dhanotia, and S. Prakash, “Measurement of focal length using phase shifted moiré deflectometry,” Opt. Lasers Eng. 51(6), 776–782 (2013).

Hayashi, S.

Hermann, K.

K. Hermann, “Periodic overlayers and moiré patterns: theoretical studies of geometric properties,” J. Phys. Condens. Matter 24(31), 314210 (2012).
[PubMed]

Hersch, R. D.

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).

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

I. Amidror and R. D. Hersch, “Fourier-based analysis and synthesis of moirés in the superposition of geometrically transformed periodic structures,” J. Opt. Soc. Am. A 15(5), 1100–1113 (1998).

Hu, J.

Hunt, R.

M. C. Hutley, R. Hunt, R. F. Stevens, and P. Savander, “The moiré magnifier,” Pure Appl. Opt. 3(2), 133 (1994).

Hutley, M. C.

M. C. Hutley, R. Hunt, R. F. Stevens, and P. Savander, “The moiré magnifier,” Pure Appl. Opt. 3(2), 133 (1994).

Jianglong, W.

Johnson, W. O.

Junfei, L.

Kamal, H.

H. Kamal, R. Völkel, and J. Alda, “Properties of moiré magnifiers,” Opt. Eng. 37(11), 3007–3014 (1998).

Kim, S. K.

Kim, W. M.

Lee, J.

Lee, T. S.

Lou, Y.

Meadows, D. M.

Ogihara, S.

Patorski, K.

K. Patorski, S. Yokozeki, and T. Suzuki, “Moiré profile prediction by using Fourier series formulism,” Jpn. J. Appl. Phys. 15(3), 443–456 (1976).

Prakash, S.

S. Trivedi, J. Dhanotia, and S. Prakash, “Measurement of focal length using phase shifted moiré deflectometry,” Opt. Lasers Eng. 51(6), 776–782 (2013).

Qinwei, M.

Ri, S.

Rogers, G. L.

G. L. Rogers, “A geometrical approach to moiré pattern calculations,” Opt. Acta (Lond.) 24(1), 1–13 (1997).

Savander, P.

M. C. Hutley, R. Hunt, R. F. Stevens, and P. Savander, “The moiré magnifier,” Pure Appl. Opt. 3(2), 133 (1994).

Saveljev, V.

Shaopeng, M.

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).

Stevens, R. F.

M. C. Hutley, R. Hunt, R. F. Stevens, and P. Savander, “The moiré magnifier,” Pure Appl. Opt. 3(2), 133 (1994).

Suzuki, T.

K. Patorski, S. Yokozeki, and T. Suzuki, “Moiré profile prediction by using Fourier series formulism,” Jpn. J. Appl. Phys. 15(3), 443–456 (1976).

Trivedi, S.

S. Trivedi, J. Dhanotia, and S. Prakash, “Measurement of focal length using phase shifted moiré deflectometry,” Opt. Lasers Eng. 51(6), 776–782 (2013).

Tsuda, H.

Völkel, R.

H. Kamal, R. Völkel, and J. Alda, “Properties of moiré magnifiers,” Opt. Eng. 37(11), 3007–3014 (1998).

Wu, J. H.

Yang, X.

Yokozeki, S.

K. Patorski, S. Yokozeki, and T. Suzuki, “Moiré profile prediction by using Fourier series formulism,” Jpn. J. Appl. Phys. 15(3), 443–456 (1976).

Youqi, Z.

Zhipei, W.

Zhou, Y.

Zhu, J. C.

ACM Trans. Graph. (1)

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

Appl. Opt. (1)

J. Opt. Soc. Am. A (2)

J. Phys. Condens. Matter (1)

K. Hermann, “Periodic overlayers and moiré patterns: theoretical studies of geometric properties,” J. Phys. Condens. Matter 24(31), 314210 (2012).
[PubMed]

Jpn. J. Appl. Phys. (1)

K. Patorski, S. Yokozeki, and T. Suzuki, “Moiré profile prediction by using Fourier series formulism,” Jpn. J. Appl. Phys. 15(3), 443–456 (1976).

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).

Opt. Acta (Lond.) (1)

G. L. Rogers, “A geometrical approach to moiré pattern calculations,” Opt. Acta (Lond.) 24(1), 1–13 (1997).

Opt. Eng. (1)

H. Kamal, R. Völkel, and J. Alda, “Properties of moiré magnifiers,” Opt. Eng. 37(11), 3007–3014 (1998).

Opt. Express (4)

Opt. Lasers Eng. (1)

S. Trivedi, J. Dhanotia, and S. Prakash, “Measurement of focal length using phase shifted moiré deflectometry,” Opt. Lasers Eng. 51(6), 776–782 (2013).

Opt. Lett. (2)

Pure Appl. Opt. (1)

M. C. Hutley, R. Hunt, R. F. Stevens, and P. Savander, “The moiré magnifier,” Pure Appl. Opt. 3(2), 133 (1994).

Other (1)

I. Amidror, The Theory of the Moiré Phenomenon (Springer, 2009).

Supplementary Material (2)

NameDescription
» Visualization 1       1D reflective moire magnifier
» Visualization 2       2D transmssive moire magnifier

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Figures (8)

Fig. 1
Fig. 1

Definition of repetition vectors and angles of the base layer b1, b2, α, of the revealing layer r1, r2, β, and of the moiré image m1, m2, ϕ. The alignment angle between r1 and b1 is θ and the rotation angle between m1 and b1 is γ. The definition is the same to those in Fig. 2 in Ref [17].

Fig. 2
Fig. 2

(a) Schematic diagram of the 1D reflective moiré magnifier. (b) Surface profile of the lenticular. (c) Working principal of the 1D reflective moiré magnifier. Details of the micro-letter ‘E’ in the MPA for the magnification factors of (d) −200 and (e) + 200. The insets in (d) and (e) show the base band replication vectors ( t x , t y ).

Fig. 3
Fig. 3

Single-frame excerpt in Visualization 1 shows the 1D reflective moiré magnifier.

Fig. 4
Fig. 4

(a) Schematic diagram of the 2D transmissive moiré magnifier. The up-left inset shows the primitive vector in the revealing layer and the up-right inset shows the primitive vector in the base layer. (b) Confocal microscopic picture of the MLA. (c) The working principal of the 2D transmissive moiré magnifier. (d) The synthetically magnified image for one of the solutions and (e) microscopic picture of the corresponding micro-letter “F”. (d) The synthetically magnified image for the other solution and (g) microscopic picture of the corresponding tilt micro-letter “F”. MLA used is the same in both cases. The scale bars in (e) and (g) are 35 μm.

Fig. 5
Fig. 5

(a) Calculated magnification factors κ 1 , κ 2 , the orientation angle κ 2 , and the spanning angle ϕ as a function of the rotation angle θ .i,ii, iii, iv, and vdenote the sampling points in Visualization 2. The curve of κ 2 is almost supererimposed to that of κ 2 in the case. (b) Confocal microscopic picture of the fabricated icons ‘’ array in the base layer. (c-g) Single-frame excerpts from the video in Visualization 2 as the MLA film rotates on the top of the MPA layer, which clearly show the linear transformation of the moiré image. The red and black arrows in the circle shows the variation of the rotation angle θ.

Fig. 6
Fig. 6

Text ‘SOE’ in the MPA for the realization of the 1D reflective moiré magnifier with a magnification factor of −200X.

Fig. 7
Fig. 7

Text ‘SOE’ in the MPA for the realization of the 1D reflective moiré magnifier with a magnification factor of + 200X.

Fig. 8
Fig. 8

The array arrangement of the microtext arrays ‘SOE’ in the MPA.

Equations (25)

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[ r 1,x r 2,x r 1,y r 2,y ]= T 1 [ b 1,x b 2,x b 1,y b 2,y ].
[ R 1,x R 2,x R 1,x R 2,y ]=G[ B 1,x B 2,x B 1,x B 2,y ],
T 1 G T =I,
[ M 1,x M 2,x M 1,y M 2,y ]=[ B 1,x B 2,x B 1,y B 2,y ][ R 1,x R 2,x R 1,y R 2,y ],
[ M 1,x M 2,x M 1,y M 2,y ]=(1G)[ B 1,x B 2,x B 1,y B 2,y ].
m ¯ ¯ =[ m 1,x m 2,x m 1,y m 2,y ]=T[ b 1,x b 2,x b 1,y b 2,y ],
T (IG) T =I.
T= (I b ¯ ¯ r ¯ ¯ 1 ) 1 ,
b ¯ ¯ =[ b 1,x b 2,x b 1,y b 2,y ],
r ¯ ¯ =[ r 1,x r 2,x r 1,y r 2,y ].
[ x m y m ]=T[ x b y b ].
T= 1 1 t y T r cosθ+ t x T r sinθ [ 1 t y T r cosθ t x T r cosθ t y T r sinθ 1+ t y T r sinθ ],
m ¯ ¯ =T b ¯ ¯ = 1 1 t y T r cosθ+ t x T r sinθ [ λ(1 t y T r cosθ) t x λ t y T r sinθ t y ],
b ¯ ¯ =[ b 1 b 2 cosα 0 b 2 sinα ],
r ¯ ¯ =[ cosθ sinθ sinθ cosθ ][ r 1 r 2 0 r 2 sinβ ].
T=ΔA,
Δ= 1 b 1 b 2 sinα+ r 1 r 2 sinβ+ r 1 b 2 sin(θα) b 1 r 2 sin(θ+β) ,
A=[ r 1 r 2 sinβ r 1 b 2 sinαcosθ r 1 b 2 cosαcosθ r 2 b 1 cos(θ+β) r 1 b 2 sinαsinθ r 1 r 2 sinβ r 2 b 1 sin(θ+β)+ r 1 b 2 sinθcosβ ].
κ 1 = | m 1 | | b 1 | = r 1 Δ r 2 2 sin 2 β+ b 2 2 sin 2 α2 r 2 b 2 sinαsinβcosθ ,
κ 2 = | m 2 | | b 2 | = r 2 Δ r 1 2 sin 2 β+ b 1 2 sin 2 α2 r 1 b 1 sinαsinβcos(αβθ) ,
tanγ= b 2 sinαsinθ b 2 sinαcosθ r 2 sinβ ,
tan(γ+ϕ)= r 1 sinαsinβ b 1 sinαsin(θ+β) r 1 cosαsinβ b 1 sinαcos(θ+β) .
[ x m y m ]= 1 1 t y T r [ (1 t y T r )x+ t x T r y y ],
y x = t y T r t x .
ϕ=a tan 1 ( t y / t x ).