Abstract

Recently, optical image coding using a circular Dammann grating (CDG) has been proposed and investigated. However, the proposed technique is intensity based and could not be used for three-dimensional (3D) image coding. In this paper, we investigate an optical image coding technique that is complex- amplitude based. The system can be used for 3D image coding. The complex-amplitude coding is provided by a circular Dammann grating through the use of a digital holographic recording technique called optical scanning holography. To decode the image, along the depth we record a series of pinhole holograms coded by the CDG. The decoded reconstruction of each depth location is extracted by the measured pinhole hologram matched to the desired depth. Computer simulations as well as experimental results are provided.

© 2011 Optical Society of America

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2010 (2)

2009 (4)

2008 (3)

2006 (1)

C.-C. Chang, J.-P. Liu, H.-Y. Lee, C.-Y. Lin, T.-C. Chang, and H.-F. Yau, “Decryption of a random-phase multiplexing recording system,” Opt. Commun. 259, 78–81 (2006).
[CrossRef]

2004 (1)

2003 (2)

1995 (1)

1985 (1)

Alfalou, A.

Brosseau, C.

Chang, C.-C.

C.-C. Chang, J.-P. Liu, H.-Y. Lee, C.-Y. Lin, T.-C. Chang, and H.-F. Yau, “Decryption of a random-phase multiplexing recording system,” Opt. Commun. 259, 78–81 (2006).
[CrossRef]

Chang, T.-C.

C.-C. Chang, J.-P. Liu, H.-Y. Lee, C.-Y. Lin, T.-C. Chang, and H.-F. Yau, “Decryption of a random-phase multiplexing recording system,” Opt. Commun. 259, 78–81 (2006).
[CrossRef]

Chen, Y. C.

Chen, Z.

Chung, P. S.

Dobson, K.

Doh, K.

Doh, K. B.

Indebetouw, G.

Javidi, B.

Jia, J.

Kim, H.

Kim, T.

Lam, E. Y

Lam, E. Y.

Lee, B.

Lee, H.-Y.

C.-C. Chang, J.-P. Liu, H.-Y. Lee, C.-Y. Lin, T.-C. Chang, and H.-F. Yau, “Decryption of a random-phase multiplexing recording system,” Opt. Commun. 259, 78–81 (2006).
[CrossRef]

Lin, C.-Y.

C.-C. Chang, J.-P. Liu, H.-Y. Lee, C.-Y. Lin, T.-C. Chang, and H.-F. Yau, “Decryption of a random-phase multiplexing recording system,” Opt. Commun. 259, 78–81 (2006).
[CrossRef]

Liu, J.-P.

C.-C. Chang, J.-P. Liu, H.-Y. Lee, C.-Y. Lin, T.-C. Chang, and H.-F. Yau, “Decryption of a random-phase multiplexing recording system,” Opt. Commun. 259, 78–81 (2006).
[CrossRef]

Liu, L.

Min, S.-W.

Ouyang, Y.

Poon, T.-C.

Refrégiér, P.

Su, W. C.

Sun, C. C.

Vo, H.

Wen, F. J.

Yau, H.-F.

C.-C. Chang, J.-P. Liu, H.-Y. Lee, C.-Y. Lin, T.-C. Chang, and H.-F. Yau, “Decryption of a random-phase multiplexing recording system,” Opt. Commun. 259, 78–81 (2006).
[CrossRef]

Zhang, X.

Zhou, C.

Adv. Opt. Photon. (1)

Appl. Opt. (7)

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

J. Opt. Soc. Korea (1)

Opt. Commun. (1)

C.-C. Chang, J.-P. Liu, H.-Y. Lee, C.-Y. Lin, T.-C. Chang, and H.-F. Yau, “Decryption of a random-phase multiplexing recording system,” Opt. Commun. 259, 78–81 (2006).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Other (1)

T.-C. Poon, Optical Scanning Holography with MATLAB (Springer, 2007).
[CrossRef]

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Figures (18)

Fig. 1
Fig. 1

Schematic setup of an OSH system: ⊗, electronic multipliers; and LPF, low-pass filter.

Fig. 2
Fig. 2

Real part of the optical field just behind the Dammann grating.

Fig. 3
Fig. 3

(a) Real part, (b) imaginary part, and (c) absolute value of the diffraction pattern at the back focal plane of L1 ( z 0 = 0 ).

Fig. 4
Fig. 4

(a) Real part, (b) imaginary part, and (c) absolute value of the diffraction pattern at the plane z 0 = 4.5 cm .

Fig. 5
Fig. 5

Object to be coded.

Fig. 6
Fig. 6

(a) Real part and (b) imaginary part of the coded complex hologram for the object located at the back focal plane of L1 ( z 0 = 0 ).

Fig. 7
Fig. 7

Images reconstructed at (a)  z 0 = 0 and (b)  z 0 = 4.5 cm while the object is located at z 0 = 0 .

Fig. 8
Fig. 8

(a) Real part and (b) imaginary part of the hologram for the object located at z 0 = 4.5 cm .

Fig. 9
Fig. 9

Images reconstructed at (a)  z 0 = 0 and (b)  z 0 = 4.5 cm while the object is located at z 0 = 4.5 cm .

Fig. 10
Fig. 10

(a) Using Fresnel diffraction and (b) applying a decoding Dammann grating of period 100 μm to reconstruct the coded complex hologram shown in Fig. 8. The reconstruction distance is at z 0 = 4.5 cm .

Fig. 11
Fig. 11

Optical system for coding using a CDG as the coding pupil: BE1, BE2, beam expanders; AOM1, AOM2, acousto-optic frequency shifter; BS, beam splitter.

Fig. 12
Fig. 12

(a) Real part and (b) imaginary part of a coded complex hologram, where the object “VT” is located at z 0 = 4.5 cm .

Fig. 13
Fig. 13

Using the Fresnel diffraction to reconstruct the coded complex hologram shown in Fig. 12. The object “VT” is located at z 0 = 4.5 cm . The standard holographic reconstruction of the coded complex hologram is not recognizable as the hologram has been coded.

Fig. 14
Fig. 14

Pinhole hologram measured at z 0 = 4.5 cm : (a) real part of the hologram and (b) imaginary part of the hologram.

Fig. 15
Fig. 15

Reconstruction of the complex hologram shown in Fig. 12 using the pinhole hologram shown in Fig. 14. The hologram is clearly reconstructed at the correct location of z 0 = 4.5 cm .

Fig. 16
Fig. 16

Complex hologram of a 3D object. “V” is located at z 0 = 0 , and “T” is located at z 0 = 4.5 cm : (a) real part of the coded complex hologram and (b) imaginary part of the coded complex hologram.

Fig. 17
Fig. 17

Pinhole hologram measured at z 0 = 0 : (a) real part of the hologram and (b) imaginary part of the hologram.

Fig. 18
Fig. 18

Reconstructions using the pinhole holograms shown in Figs. 14, 17. (a) “V” is reconstructed correctly at z 0 = 0 and (b) “T” at z 0 = 4.5 cm .

Equations (18)

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i c = Re { i Ω p ( x , y ) } ,
i s = Im { i Ω p ( x , y ) } ,
i Ω p ( x , y ) = i c + j i s = F 1 { F { T ( x , y ; z 0 ) } × OTF ( k x , k y ; z 0 ) } = T ( x , y ; z 0 ) F 1 { OTF ( k x , k y ; z 0 ) } ,
OTF ( k x , k y ; z ) = exp [ j z 2 k 0 ( k x 2 + k y 2 ) ] × p 1 * ( x , y ) p 2 ( x + f k 0 k x , y + f k 0 k y ) exp [ j z f ( x k x + y k y ) ] d x d y ,
OTF ( k x , k y ; z ) = exp [ j z 2 k 0 ( k x 2 + k y 2 ) ] = OTF OSH ( k x , k y ; z ) .
i Ω p ( x , y ) F 1 { OTF OSH * ( f x , f y ; z 0 ) } = F 1 { F { T ( x , y ; z 0 ) } × OTF OSH ( k x , k y ; z 0 ) × OTF OSH * ( k x , k y ; z 0 ) } = T ( x , y ; z 0 ) ,
OTF OSH ( k x , k y ; z ) × OTF OSH * ( k x , k y ; z ) } = 1 ,
i Ω p ( x , y ) = m = 1 M T m ( x , y ; z m ) F 1 { OTF ( k x , k y ; z m ) } ,
i Ω p ( x , y ) F 1 { OTF * ( k x , k y ; z 0 ) } = F 1 { F { T ( x , y ; z 0 ) } × OTF ( k x , k y ; z 0 ) × OTF * ( k x , k y ; z 0 ) } .
OTF coding ( k x , k y ; z 0 ) = OTF ( k x , k y ; z 0 ) ,
OTF decoding ( k x , k y ; z 0 ) = OTF * ( k x , k y ; z 0 ) ,
OTF coding × OTF decoding = OTF ( k x , k y ; z 0 ) × OTF * ( k x , k y ; z 0 ) } = 1 ,
F 1 { OTF ( k x , k y ; z 0 ) } F 1 { OTF * ( k x , k y ; z 0 ) } = δ ( x , y ) ,
OTF coding ( k x , k y ; z ) = exp [ j z 2 k 0 ( k x 2 + k y 2 ) ] × p 1 * ( f k 0 k x , f k 0 k y ) = OTF OSH ( k x , k y ; z ) × p 1 * ( f k 0 k x , f k 0 k y ) .
OTF coding ( k x , k y ; z ) × OTF decoding ( k x , k y ; z ) = ( OTF OSH × p 1 * ) × ( OTF OSH * × p 1 ) = p 1 * × p 1 ,
OTF decoding ( k x , k y ; z ) = 1 OTF coding ( k x , k y ; z ) + ε ,
OTF encoding ( k x , k y ; z ) × OTF decoding ( k x , k y ; z ) 1.
OTF coding ( k x , k y ; z ) × OTF decoding ( k x , k y ; z ) = p 1 * × p 1 = 1 .

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