Abstract

Two anamorphic and achromatic Fourier processors were designed and constructed using diffractive and refractive cylindrical lenses. The diffractive lenses are holographic lenses recorded on silver halide material. In both processors the achromatic one-dimensional Fourier transform plane was obtained with two holographic lenses and one refractive cylindrical lens. The image with the same magnification in both directions at the output plane was formed with two different combinations of lenses. The differences between the two processors are analyzed, and in both cases the chromatic aberration in the Fourier plane and in the output plane is evaluated. Even though single cylindrical refractive lenses were used to image in one direction, good results were obtained.

© 2006 Optical Society of America

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  1. L. U. Almi and J. Shamir, "Pattern recognition using one-dimensional Fourier transform," Opt. Commun. 18, 304-306 (1976).
    [CrossRef]
  2. S. H. Collicott and L. Hesselink, "Analysis and design of an anamorphic optical processor for speckle metrology and velocimetry," Appl. Opt. 31, 1646-1659 (1992).
    [CrossRef] [PubMed]
  3. P. J. Marchand, P. C. Harvey, and S. C. Esener, "Motionless-head parallel-readout optical-disk system: experimental results," Appl. Opt. 34, 7604-7607 (1995).
    [CrossRef] [PubMed]
  4. K. Kubota, M. Kondo, S. Sugama, and S. Takahashi, "Hologram memory using one-dimensional Fourier-transformed image hologram," Electron. Commun. Jpn. 61-C, 108-114 (1978).
  5. D. Mendlovic, Y. Bitran, R. G. Dorsch, C. Ferreira, J. García, and H. M. Ozaktaz, "Anamorphic fractional Fourier transform: optical implementation and applications," Appl. Opt. 34, 7451-7456 (1995).
    [CrossRef] [PubMed]
  6. F. T. S. Yu, White Light Optical Signal Processing (Wiley, 1985).
  7. J. Lancis, P. Andrés, W. D. Furlan, and A. Pons, "All-diffractive achromatic Fourier-transform setup," Opt. Lett. 19, 402-404 (1994).
    [PubMed]
  8. G. M. Morris, "An ideal achromatic Fourier processor," Opt. Commun. 39, 143-147 (1981).
    [CrossRef]
  9. E. Tajahuerce, J. Lancis, V. Climent, and P. Andrés, "Hybrid (refractive-diffractive) Fourier processor: a novel optical architecture for achromatic processing with broadband point-source illumination," Opt. Commun. 151, 86-92 (1998).
    [CrossRef]
  10. P. Liang, J. Ding, Z. Jin, and G. Wenqi, "Composite binary optical elements used for multi-channel spectrum analysis," J. Mod. Opt. 50, 1411-1417 (2003).
    [CrossRef]
  11. M. V. Collados, J. Atencia, J. Tornos, and M. Quintanilla, "Construction and characterization of compound holographic lenses for multichannel one-dimensional Fourier transformation and optical parallel processing," Opt. Commun. 249, 85-94 (2005).
    [CrossRef]
  12. M. Domingo, I. Arias, and A. García, "Achromatic Fourier processor with holographic optical lenses," Appl. Opt. 40, 2267-2274 (2001).
    [CrossRef]
  13. M. V. Collados, I. Arias, A. García, J. Atencia, and M. Quintanilla, "Silver halide sensitized gelatin process effects in holographic lenses recorded on Slavich PFG-01 plates," Appl. Opt. 42, 805-810 (2003).
    [CrossRef] [PubMed]

2005 (1)

M. V. Collados, J. Atencia, J. Tornos, and M. Quintanilla, "Construction and characterization of compound holographic lenses for multichannel one-dimensional Fourier transformation and optical parallel processing," Opt. Commun. 249, 85-94 (2005).
[CrossRef]

2003 (2)

2001 (1)

1998 (1)

E. Tajahuerce, J. Lancis, V. Climent, and P. Andrés, "Hybrid (refractive-diffractive) Fourier processor: a novel optical architecture for achromatic processing with broadband point-source illumination," Opt. Commun. 151, 86-92 (1998).
[CrossRef]

1995 (2)

1994 (1)

1992 (1)

1981 (1)

G. M. Morris, "An ideal achromatic Fourier processor," Opt. Commun. 39, 143-147 (1981).
[CrossRef]

1978 (1)

K. Kubota, M. Kondo, S. Sugama, and S. Takahashi, "Hologram memory using one-dimensional Fourier-transformed image hologram," Electron. Commun. Jpn. 61-C, 108-114 (1978).

1976 (1)

L. U. Almi and J. Shamir, "Pattern recognition using one-dimensional Fourier transform," Opt. Commun. 18, 304-306 (1976).
[CrossRef]

Almi, L. U.

L. U. Almi and J. Shamir, "Pattern recognition using one-dimensional Fourier transform," Opt. Commun. 18, 304-306 (1976).
[CrossRef]

Andrés, P.

E. Tajahuerce, J. Lancis, V. Climent, and P. Andrés, "Hybrid (refractive-diffractive) Fourier processor: a novel optical architecture for achromatic processing with broadband point-source illumination," Opt. Commun. 151, 86-92 (1998).
[CrossRef]

J. Lancis, P. Andrés, W. D. Furlan, and A. Pons, "All-diffractive achromatic Fourier-transform setup," Opt. Lett. 19, 402-404 (1994).
[PubMed]

Arias, I.

Atencia, J.

M. V. Collados, J. Atencia, J. Tornos, and M. Quintanilla, "Construction and characterization of compound holographic lenses for multichannel one-dimensional Fourier transformation and optical parallel processing," Opt. Commun. 249, 85-94 (2005).
[CrossRef]

M. V. Collados, I. Arias, A. García, J. Atencia, and M. Quintanilla, "Silver halide sensitized gelatin process effects in holographic lenses recorded on Slavich PFG-01 plates," Appl. Opt. 42, 805-810 (2003).
[CrossRef] [PubMed]

Bitran, Y.

Climent, V.

E. Tajahuerce, J. Lancis, V. Climent, and P. Andrés, "Hybrid (refractive-diffractive) Fourier processor: a novel optical architecture for achromatic processing with broadband point-source illumination," Opt. Commun. 151, 86-92 (1998).
[CrossRef]

Collados, M. V.

M. V. Collados, J. Atencia, J. Tornos, and M. Quintanilla, "Construction and characterization of compound holographic lenses for multichannel one-dimensional Fourier transformation and optical parallel processing," Opt. Commun. 249, 85-94 (2005).
[CrossRef]

M. V. Collados, I. Arias, A. García, J. Atencia, and M. Quintanilla, "Silver halide sensitized gelatin process effects in holographic lenses recorded on Slavich PFG-01 plates," Appl. Opt. 42, 805-810 (2003).
[CrossRef] [PubMed]

Collicott, S. H.

Ding, J.

P. Liang, J. Ding, Z. Jin, and G. Wenqi, "Composite binary optical elements used for multi-channel spectrum analysis," J. Mod. Opt. 50, 1411-1417 (2003).
[CrossRef]

Domingo, M.

Dorsch, R. G.

Esener, S. C.

Ferreira, C.

Furlan, W. D.

García, A.

García, J.

Harvey, P. C.

Hesselink, L.

Jin, Z.

P. Liang, J. Ding, Z. Jin, and G. Wenqi, "Composite binary optical elements used for multi-channel spectrum analysis," J. Mod. Opt. 50, 1411-1417 (2003).
[CrossRef]

Kondo, M.

K. Kubota, M. Kondo, S. Sugama, and S. Takahashi, "Hologram memory using one-dimensional Fourier-transformed image hologram," Electron. Commun. Jpn. 61-C, 108-114 (1978).

Kubota, K.

K. Kubota, M. Kondo, S. Sugama, and S. Takahashi, "Hologram memory using one-dimensional Fourier-transformed image hologram," Electron. Commun. Jpn. 61-C, 108-114 (1978).

Lancis, J.

E. Tajahuerce, J. Lancis, V. Climent, and P. Andrés, "Hybrid (refractive-diffractive) Fourier processor: a novel optical architecture for achromatic processing with broadband point-source illumination," Opt. Commun. 151, 86-92 (1998).
[CrossRef]

J. Lancis, P. Andrés, W. D. Furlan, and A. Pons, "All-diffractive achromatic Fourier-transform setup," Opt. Lett. 19, 402-404 (1994).
[PubMed]

Liang, P.

P. Liang, J. Ding, Z. Jin, and G. Wenqi, "Composite binary optical elements used for multi-channel spectrum analysis," J. Mod. Opt. 50, 1411-1417 (2003).
[CrossRef]

Marchand, P. J.

Mendlovic, D.

Morris, G. M.

G. M. Morris, "An ideal achromatic Fourier processor," Opt. Commun. 39, 143-147 (1981).
[CrossRef]

Ozaktaz, H. M.

Pons, A.

Quintanilla, M.

M. V. Collados, J. Atencia, J. Tornos, and M. Quintanilla, "Construction and characterization of compound holographic lenses for multichannel one-dimensional Fourier transformation and optical parallel processing," Opt. Commun. 249, 85-94 (2005).
[CrossRef]

M. V. Collados, I. Arias, A. García, J. Atencia, and M. Quintanilla, "Silver halide sensitized gelatin process effects in holographic lenses recorded on Slavich PFG-01 plates," Appl. Opt. 42, 805-810 (2003).
[CrossRef] [PubMed]

Shamir, J.

L. U. Almi and J. Shamir, "Pattern recognition using one-dimensional Fourier transform," Opt. Commun. 18, 304-306 (1976).
[CrossRef]

Sugama, S.

K. Kubota, M. Kondo, S. Sugama, and S. Takahashi, "Hologram memory using one-dimensional Fourier-transformed image hologram," Electron. Commun. Jpn. 61-C, 108-114 (1978).

Tajahuerce, E.

E. Tajahuerce, J. Lancis, V. Climent, and P. Andrés, "Hybrid (refractive-diffractive) Fourier processor: a novel optical architecture for achromatic processing with broadband point-source illumination," Opt. Commun. 151, 86-92 (1998).
[CrossRef]

Takahashi, S.

K. Kubota, M. Kondo, S. Sugama, and S. Takahashi, "Hologram memory using one-dimensional Fourier-transformed image hologram," Electron. Commun. Jpn. 61-C, 108-114 (1978).

Tornos, J.

M. V. Collados, J. Atencia, J. Tornos, and M. Quintanilla, "Construction and characterization of compound holographic lenses for multichannel one-dimensional Fourier transformation and optical parallel processing," Opt. Commun. 249, 85-94 (2005).
[CrossRef]

Wenqi, G.

P. Liang, J. Ding, Z. Jin, and G. Wenqi, "Composite binary optical elements used for multi-channel spectrum analysis," J. Mod. Opt. 50, 1411-1417 (2003).
[CrossRef]

Yu, F. T. S.

F. T. S. Yu, White Light Optical Signal Processing (Wiley, 1985).

Appl. Opt. (5)

Electron. Commun. Jpn. (1)

K. Kubota, M. Kondo, S. Sugama, and S. Takahashi, "Hologram memory using one-dimensional Fourier-transformed image hologram," Electron. Commun. Jpn. 61-C, 108-114 (1978).

J. Mod. Opt. (1)

P. Liang, J. Ding, Z. Jin, and G. Wenqi, "Composite binary optical elements used for multi-channel spectrum analysis," J. Mod. Opt. 50, 1411-1417 (2003).
[CrossRef]

Opt. Commun. (4)

M. V. Collados, J. Atencia, J. Tornos, and M. Quintanilla, "Construction and characterization of compound holographic lenses for multichannel one-dimensional Fourier transformation and optical parallel processing," Opt. Commun. 249, 85-94 (2005).
[CrossRef]

L. U. Almi and J. Shamir, "Pattern recognition using one-dimensional Fourier transform," Opt. Commun. 18, 304-306 (1976).
[CrossRef]

G. M. Morris, "An ideal achromatic Fourier processor," Opt. Commun. 39, 143-147 (1981).
[CrossRef]

E. Tajahuerce, J. Lancis, V. Climent, and P. Andrés, "Hybrid (refractive-diffractive) Fourier processor: a novel optical architecture for achromatic processing with broadband point-source illumination," Opt. Commun. 151, 86-92 (1998).
[CrossRef]

Opt. Lett. (1)

Other (1)

F. T. S. Yu, White Light Optical Signal Processing (Wiley, 1985).

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Figures (13)

Fig. 1
Fig. 1

Schematic of the achromatic Fourier transformer. S ( λ 1 ) , S ( λ 2 ) , S ( λ 3 ) are the images of virtual source S for each value of λ, given by diffractive lens DL1. At these positions the Fourier transform planes for each λ are localized. The set of Fourier planes is imaged by DL2 in the same plane ( x F , y F ) .

Fig. 2
Fig. 2

Projections on the (a) XOZ and (b) YOZ planes of the achromatic and anamorphic Fourier transformer.

Fig. 3
Fig. 3

Construction of a holographic cylindrical divergent lens.

Fig. 4
Fig. 4

(Color online) (a) Object test with frequencies of 4.3, 5.7, 8.7, and 17   lines∕mm for each black and white band pattern. (b) Irradiance distribution at the Fourier plane in the transformer of Fig. 2 when the object is the one shown in (a).

Fig. 5
Fig. 5

(Color online) (a) Object test with frequency changing in the x direction from 14 to 7   lines∕mm . (b) Irradiance distribution across the Fourier plane in the transformer of Fig. 2 when the object is the one shown in (a).

Fig. 6
Fig. 6

(Color online) Irradiance distribution profile along a vertical line of coordinates: (a) x = 1.1   mm of Fig. 4(b) ( 4.3   lines∕mm ; band pattern); (b) x = 3.7   mm of Fig. 4(b) ( 5.7   lines∕mm band pattern); (c) x = 6.3   mm of Fig. 4(b) ( 8.7   lines∕mm band pattern); (d) x = 8.5   mm of Fig. 4(b) ( 17   lines∕mm band pattern).

Fig. 7
Fig. 7

Projection on the XOZ and YOZ planes of the achromatic and anamorphic processor in configuration I.

Fig. 8
Fig. 8

(Color online) Irradiance distribution across the output plane of the processor in configuration I when a 3 line∕mm 2D grating is used as an object.

Fig. 9
Fig. 9

(Color online) Irradiance distribution profile along a line of coordinates y = 7.1   mm for each color filter.

Fig. 10
Fig. 10

Projection on the XOZ and YOZ planes of the achromatic and anamorphic processor in configuration II.

Fig. 11
Fig. 11

(Color online) Irradiance distribution across the Fourier plane in the processor in configuration II when the object is the one shown in Fig. 4(a).

Fig. 12
Fig. 12

(Color online) Irradiance distribution profile along a vertical line of coordinates: (a) x = 2.7   mm of Fig. 11 ( 4.3   lin∕mm band pattern); (b) x = 8.6   mm of Fig. 11 ( 5.7   lin∕mm band pattern); (c) x = 16.3   mm of Fig. 11 ( 8.7   lin∕mm band pattern); (d) x = 21.9   mm of Fig. 11 ( 17   lin∕mm band pattern).

Fig. 13
Fig. 13

(Color online) Irradiance distribution across the output plane of the processor in configuration II when a 3   line∕mm 2D grating is used as an object.

Tables (2)

Tables Icon

Table 1 TCA Measured for the First- and Second-Order Frequencies of Each Band of the Object of Fig. 4(a)

Tables Icon

Table 2 TCA Measured for the First-Order Frequencies of Each Band of the Object of Fig. 11

Equations (19)

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( d 1 + ρ ) 2 = f 02 f 01 ,
0 < α < 4.
U ( x F , y F ) α exp [ i π x F 2 λ ( D y ρ ) ] exp [ i π 1 λ D y ( 1 1 D y A y ) y F 2 ]
× + + t ( x 1 , y 1 ) exp [ i2 π y F y 1 A y λ D y d 1 ( d 1 ρ ) W y ] × exp [ i2 π x F x 1 λ ( ρ D y ) ] exp [ i π λ ( ρ + d 1 d 1 ρ 1 d 1 2 W y 1 A y d 1 2 ( d 1 ρ ) 2 W y 2 ) y 1 2 ] exp [ i π D y λ ρ ( D y ρ ) x 1 2 ] d x 1 d y 1 ,
W y = 1 d 1 1 ρ + d 1 1 f 1 y ,
A y = 1 D y 1 ρ + d 1 1 W y ( ρ + d 1 ) 2 1 f 2 y ,
D y = ( 1 ρ + d 1 + λ λ 0 f 02 y λ 0 f 01 y λ ( ρ + d 1 ) 2 ) 1 ,
y F v y = λ f 1 y ρ ( ρ + d 1 ) 2 D y ,
1 d x = 1 d x + 1 f R 1 x ,
d x + d x = ρ + D y .
U ( x i , y i ) exp [ i π λ ( d i + l + l ρ ) x i 2 ] exp [ i π λ ( 1 d i ) y i 2 ] × + + d x 1 d y 1 t ( x 1 , y 1 ) exp [ i π λ f 1 y y 1 2 ] × exp [ i 2 π λ ( d i + l + l ρ ) x 1 x i ] × exp [ i π ( l + l + d i ) λ ρ ( d i + l + l ρ ) x 1 2 ] h y ( y 1 ; y i ) ,  
h y ( y 1 ; y i ) = + d y l exp [ i π λ ( 1 l 1 l 2 B y 1 f 3 y + 1 d i ) y l 2 ] × exp [ i2 π λ ( l y 1 l ρ + y i d i ) y l ] ,
B y = 1 l 1 f R 1 l 2 A y + 1 l ,
d i = [ 1 f 3 y 1 l l 2 l 2 ( 1 ρ + 1 l 1 f 2 y ) ] 1 ,
M i = d i l l ρ .
U ( x i , y i ) exp [ i π λ ( d i + l + l ρ ) x i 2 ] × exp [ i π λ ( 1 d i 1 M i 2 f 1 y ) y i 2 ] × + d x 1 t ( x 1 , y i M i ) × exp [ i 2 π λ ( d i + l + l ρ ) x 1 x i ] × exp [ i π ( l + l + d i ) λ ρ ( d i + l + l ρ ) x 1 2 ] .
f R = 9 4 f 01 y ,
f 03 y = 9 2 f 01 y .
d 1 ρ = 3 2 f 01 y .

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