Abstract

We propose a new function of the two-dimensional lens array that is composed of many gradient-index lenses. The array forms three-dimensional (3D) images. The characteristics of the 3D images depend on the length of the gradient-index lens. Especially, if the length of the lens is an odd-integer multiple of the half period of the optical path, 3D images are pseudoscopic with a reversed depth. The two lens arrays are positioned at a suitable distance, so that orthoscopic 3D images with the correct depth are formed in front of the lens array.

© 2002 Optical Society of America

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References

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  13. A. P. Sokolov, “Autostereoscopy and integral photography by Professor Lippmann’s method,” Izd.MGU, Moscow State University Press (1911).
  14. J. Arai, F. Okano, H. Hoshino, I. Yuyama, “Gradient-index lens array method based on real-time integral photography for three-dimensional images,” Appl. Opt. 37, 2034–2045 (1998).
    [CrossRef]
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    [CrossRef]
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  17. D. C. Leiner, “Correction of chromatic aberrations in GRIN endoscopes,” Appl. Opt. 22, 383 (1998).
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1999

F. Okano, J. Arai, H. Hoshino, I. Yuyama, “Three-dimensional video system based on integral photography,” Opt. Eng. 38, 1072–1077 (1999).
[CrossRef]

1998

1986

1975

H. Ochi, “An Analysis for Multiline of Light-Focusing Fiber Optics,” J. Inst. Electron. Eng. Jpn. 4, 13–21 (1975).

1970

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matumura, “Optical characteristics of a light-focusing fiber guide and its applications,” IEEE J. Quantum Electron. 6, 606–612 (1970).
[CrossRef]

1965

1964

D. Marcuse, S. E. Miller, “Analysis of a tubular gas lens,” Bell Syst. Tech. J. 43, 1759–1782 (1964).
[CrossRef]

1931

Arai, J.

F. Okano, J. Arai, H. Hoshino, I. Yuyama, “Three-dimensional video system based on integral photography,” Opt. Eng. 38, 1072–1077 (1999).
[CrossRef]

J. Arai, F. Okano, H. Hoshino, I. Yuyama, “Gradient-index lens array method based on real-time integral photography for three-dimensional images,” Appl. Opt. 37, 2034–2045 (1998).
[CrossRef]

Caldwell, J. B.

DeMontebello, R. L.

R. L. DeMontebello, “Wide-angle integral photography-The integral system,” Proc., 1977 SPIE Annu. Tech. Conf., San Diego, Seminar 10, No. 120-08, Tech. Digest pp. 73–91 (1977)

Furukawa, M.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matumura, “Optical characteristics of a light-focusing fiber guide and its applications,” IEEE J. Quantum Electron. 6, 606–612 (1970).
[CrossRef]

Hoshino, H.

Ichikawa, H.

Isono, H.

Ives, H. E.

Kitano, I.

H. Nishi, H. Ichikawa, M. Toyama, I. Kitano, “Gradient-index lens for the compact disk system,” Appl. Opt. 25, 3340–3344 (1986).
[CrossRef]

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matumura, “Optical characteristics of a light-focusing fiber guide and its applications,” IEEE J. Quantum Electron. 6, 606–612 (1970).
[CrossRef]

Kitano, L.

Kogelnik, H.

Koike, Y.

Koizumi, K.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matumura, “Optical characteristics of a light-focusing fiber guide and its applications,” IEEE J. Quantum Electron. 6, 606–612 (1970).
[CrossRef]

Leiner, D. C.

Lippmann, G.

G. Lippmann, Comptes-Rendus, Vol. 146 (Elsevier, New York, 1908) pp. 446–451.

Marchand, E. W.

Marcuse, D.

D. Marcuse, S. E. Miller, “Analysis of a tubular gas lens,” Bell Syst. Tech. J. 43, 1759–1782 (1964).
[CrossRef]

Matumura, H.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matumura, “Optical characteristics of a light-focusing fiber guide and its applications,” IEEE J. Quantum Electron. 6, 606–612 (1970).
[CrossRef]

Miller, S. E.

D. Marcuse, S. E. Miller, “Analysis of a tubular gas lens,” Bell Syst. Tech. J. 43, 1759–1782 (1964).
[CrossRef]

Moore, D. T.

Nishi, H.

Ochi, H.

H. Ochi, “An Analysis for Multiline of Light-Focusing Fiber Optics,” J. Inst. Electron. Eng. Jpn. 4, 13–21 (1975).

Ohtsuka, Y.

Okano, F.

Sokolov, A. P.

A. P. Sokolov, “Autostereoscopy and integral photography by Professor Lippmann’s method,” Izd.MGU, Moscow State University Press (1911).

Sumi, Y.

Toyama, M.

Uchida, T.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matumura, “Optical characteristics of a light-focusing fiber guide and its applications,” IEEE J. Quantum Electron. 6, 606–612 (1970).
[CrossRef]

Ueno, H.

Yuyama, I.

Appl. Opt.

Bell Syst. Tech. J.

D. Marcuse, S. E. Miller, “Analysis of a tubular gas lens,” Bell Syst. Tech. J. 43, 1759–1782 (1964).
[CrossRef]

IEEE J. Quantum Electron.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matumura, “Optical characteristics of a light-focusing fiber guide and its applications,” IEEE J. Quantum Electron. 6, 606–612 (1970).
[CrossRef]

J. Inst. Electron. Eng. Jpn.

H. Ochi, “An Analysis for Multiline of Light-Focusing Fiber Optics,” J. Inst. Electron. Eng. Jpn. 4, 13–21 (1975).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Eng.

F. Okano, J. Arai, H. Hoshino, I. Yuyama, “Three-dimensional video system based on integral photography,” Opt. Eng. 38, 1072–1077 (1999).
[CrossRef]

Other

G. Lippmann, Comptes-Rendus, Vol. 146 (Elsevier, New York, 1908) pp. 446–451.

R. L. DeMontebello, “Wide-angle integral photography-The integral system,” Proc., 1977 SPIE Annu. Tech. Conf., San Diego, Seminar 10, No. 120-08, Tech. Digest pp. 73–91 (1977)

A. P. Sokolov, “Autostereoscopy and integral photography by Professor Lippmann’s method,” Izd.MGU, Moscow State University Press (1911).

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Figures (11)

Fig. 1
Fig. 1

Refractive-index distribution of the GRIN lens.

Fig. 2
Fig. 2

Elemental lens with length of θ = 2kπ. This figure shows the optical path in the case of k = 1. The object Q s (x s , z s ) and its image Q c (x c , z c ) are shown.

Fig. 3
Fig. 3

Part of the GRIN lens array. θ = 2kπ. The object Q s (x s , z s ) and its image Q c (x c , z c ) are shown.

Fig. 4
Fig. 4

Elemental lens with length of θ = (2k - 1)π, otherwise the same as Fig. 2.

Fig. 5
Fig. 5

Part of the GRIN lens array. θ = (2k - 1)π. The object Q s (x s , z s ) and its image Q f (x f , z f ) are shown. The Q c (x c , z c ) is the image formed by the elemental lens (m = 1). The position of the Q c (x c , z c ) is different from that of Q f (x f , z f ).

Fig. 6
Fig. 6

Combined two lens arrays. The intermediate image Q f1 is formed by the first array. The final image Q f2 is formed by the second array. The length of the first GRIN lens array: L G1 = (2k 1 - 1)π/AG. The length of the second GRIN lens array: L G2 = (2k 2- 1)π/AG.

Fig. 7
Fig. 7

Two-dimensional arrangement of the GRIN lens array.

Fig. 8
Fig. 8

Overview of the object for the combined two lens arrays.

Fig. 9
Fig. 9

Intermediate image formed by the first array of the combined two lens arrays. When these photographs were taken by a camera, the second array was removed. (a) Left viewpoint, (b) right viewpoint, (c) upper viewpoint, (d) lower viewpoint.

Fig. 10
Fig. 10

Final image formed by the combined two lens arrays. (a), (b), (c), (d) as in Fig. 9.

Fig. 11
Fig. 11

Spread of the rays from the GRIN lens array. The object Q s (x s ,z s ) and its image Q f (x f ,z f ) are shown. The output rays of each elemental lens are equivalently emitted from Q c (x c ,z c ), which is formed by the elemental lens (m = 1). The position of the Q f (x f ,z f ) is different from that of Q c (x c ,z c ).

Tables (1)

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Table 1 Gradient-Index Lens Array Used in the Experiment

Equations (47)

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nr=n01-12 AGr2,
rcrc=cos θ1n0AGsin θ-n0AG sin θcos θ rsrs,
zc-LG=-n0zsAG cos θ+sin θn0AG-n0zsAG sin θ-cos θ,
Mz=dzcdzs=1n0zsAG sin θ+cos θ2,
Mr=-1-n0zsAG sin θ-cos θ=1n0AG sin θzs+cot θn0AG,
θ=AGLG,
rcrc=1001 rsrs.
rs=drsdz=rs-xs-zs,
rc=drcdz=rc-xcLG-zc.
xc=rs1-zc-LGzs + zc-LGzs xs,
zc=zs+LG=zs+2kπAG,
xc=xS.
rcmrcm=1001 rsmrsm,
rsm=drsmdz=rsm+mP-xs-zs,
rcm=drcmdz=rcm+mP-xcLG-zc,
xc=rsm+mP1-zc-LGzs+zc-LGzs xs.
zc=zs+LG,
xc=xs.
Mz=ΔZcΔZs=ΔZsΔZs=1,
Mr=ΔxcΔxs=ΔxsΔxs=1,
rcrc=-100-1 rsrs.
xc=rszc-LGzs-1-zc-LGzs xs,
zc=zs+LG,
xc=-xs.
rcmrcm=-100-1 rsmrsm.
rsm=drsmdz=rsm+mP-xs-zs,
rcm=drcmdz=rcm+mP-xcLG-zc.
xc=rsmzc-LGzs-1+mPzc-LGzs+1-zc-LGzs xs.
xc=mPzc-LGzs+1-zc-LGzs xs.
zc=-zs+LG=zf,
xc=xs=xf,
Mz=ΔzfΔzs=-ΔzsΔzs=-1,
Mr=ΔxfΔxs=ΔxsΔxs=1,
D1=-zs,
D2=-Ld-D1=-Ld+D1=-Ld-zs,
D3=-D2=Ld-D1=Ld+zs.
zf2=LG1+Ld+LG2+D3=LG1+LG2+2Ld+zs,
xf2=xf1=xs.
Mz=Δzf2Δzs=ΔzsΔzs=1,
Mr=Δxf2Δxs=ΔxsΔxs=1,
Ldzs,
-LdD20,
LdD30.
Ldzs,
D20,
D30.
xc=-2rsm+xs=xf.

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