Abstract

Two proposals to compensate chromatic aberration of a programmable phase Fresnel lens displayed on a liquid crystal device and working under polychromatic illumination are presented. They are based on multiplexing a set of lenses, designed with a common focal length for different wavelengths, and a multicolor filter that makes each sublens work almost monochromatically. One proposal uses spatial multiplexing with mosaic aperture. The other uses a rotating scheme, a color filter against an array of lens sectors, and hybrid spatial-time integration. The central order focalization has a unique location at the focal plane. We have drastically reduced the transversal chromatic aberration of the polychromatic point spread function by properly adjusting the pupil size of each sublens. Depth of focus curves have been made coincident too for the selected wavelengths.

© 2006 Optical Society of America

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References

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2006

2005

2004

2001

1999

1998

V. Laude, "Twisted-nematic liquid-crystal pixilated active lens," Opt. Commun. 153, 134-152 (1998).
[CrossRef]

1995

1994

1992

1990

1989

1986

J. Bescós, J. H. Altamirano, J. Santamaria, and A. Plaza, "Apodizing filters in colour imaging," J. Opt. (Paris) 17, 91-96 (1986).
[CrossRef]

1970

1961

Altamirano, J. H.

J. Bescós, J. H. Altamirano, J. Santamaria, and A. Plaza, "Apodizing filters in colour imaging," J. Opt. (Paris) 17, 91-96 (1986).
[CrossRef]

Andrés, P.

Arrizón, V.

Bescós, J.

J. Bescós, J. H. Altamirano, J. Santamaria, and A. Plaza, "Apodizing filters in colour imaging," J. Opt. (Paris) 17, 91-96 (1986).
[CrossRef]

Bosch, S.

Campos, J.

Carcolé, E.

Carreón, E.

Climent, V.

Connely, S. W.

Cottrell, D. M.

Davis, J. A.

Davis, J. E.

Escalera, J. C.

Faklis, D.

D. Faklis, G. M. Morris, "Broadband imaging with holographic lenses," Opt. Eng. 28, 592-598 (1989).

Feldman, M. R.

González, L. A.

Hain, M.

Hedman, T. R.

Hirsch, P. M.

Hyde, R. A.

Iemmi, C.

Javidi, B.

Jordan, J. A.

Lancis, J.

Laude, V.

V. Laude, "Twisted-nematic liquid-crystal pixilated active lens," Opt. Commun. 153, 134-152 (1998).
[CrossRef]

Ledesma, S.

Lesem, L. B.

Lilly, R. A.

Lohmann, A. W.

Márquez, A.

Millán, M. S.

Mínguez-Vega, G.

Miyamoto, K.

Moreno, I.

Morris, G. M.

D. Faklis, G. M. Morris, "Broadband imaging with holographic lenses," Opt. Eng. 28, 592-598 (1989).

Plaza, A.

J. Bescós, J. H. Altamirano, J. Santamaria, and A. Plaza, "Apodizing filters in colour imaging," J. Opt. (Paris) 17, 91-96 (1986).
[CrossRef]

Santamaria, J.

J. Bescós, J. H. Altamirano, J. Santamaria, and A. Plaza, "Apodizing filters in colour imaging," J. Opt. (Paris) 17, 91-96 (1986).
[CrossRef]

Schmiedchen, M.

Tajahuerce, E.

Tam, E. C.

Tschudi, T.

Van Rooy, D. L.

von Spiegel, W.

Yzuel, M. J.

Zhou, S.

Appl. Opt.

J. Opt. (Paris)

J. Bescós, J. H. Altamirano, J. Santamaria, and A. Plaza, "Apodizing filters in colour imaging," J. Opt. (Paris) 17, 91-96 (1986).
[CrossRef]

J. Opt. Soc. Am.

Opt. Commun.

V. Laude, "Twisted-nematic liquid-crystal pixilated active lens," Opt. Commun. 153, 134-152 (1998).
[CrossRef]

Opt. Eng.

D. Faklis, G. M. Morris, "Broadband imaging with holographic lenses," Opt. Eng. 28, 592-598 (1989).

Opt. Express

Opt. Lett.

Other

J. W. Goodmann, Introduction to Fourier Optics, 2nd edition (McGraw-Hill, New York, 1996).

Supplementary Material (1)

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Figures (10)

Fig. 1.
Fig. 1.

(a) Mosaic color filter placed against the SLM. (b) Mosaic basic pattern consisting of N=6 cells with distances defined in the text. The i-cell, centred at the point (ai ,bi ) is characterized by its amplitude transmittance τiλ) with a very narrow bandwidth around λi .Behind the filter, the pixelated structure of the SLM with a fill factor less than unity is shown. The pixel size is Δ×Δ, but its active area is a smaller rectangle of size Δx’×Δy’.

Fig. 2.
Fig. 2.

Schematic diagram of the mosaic multichannel phase Fresnel lens building. (a) Phase Fresnel lenses Li (partial) with i=1..4 (λ 1=632nm, λ 2=543nm, λ 3=488nm, λ 4=458nm). The radius Ri marked in each lens fulfils the PSFS-condition Ri /λi =constant. (b) λi -channels lenses, obtained from Li of (a), after a double discretization of pixelation and mosaic filtering. Pixels whose distance from the center is longer than Ri , are assigned a constant phase value (CPV). (c) Integration of λi -channels lenses by spatial multiplexing according to the basic pattern (magnified). The result is the mosaic multichannel Fresnel lens.

Fig. 3.
Fig. 3.

Intensity distribution of the central order in the focal plane of the lens with mosaic aperture (Fig. 2) for λi , i=1..4 and (a) constant radius Ri =R (b) PSFS-condition, and (c) PSFI-condition.

Fig. 4.
Fig. 4.

Intensity distribution of the central order along the optical axis of the lens with mosaic aperture (Fig. 2) for λi, i=1.. 4. (a) Constant radius Ri=R, (b) PSFS-condition, and (c) PSFI-condition. The four plots coincide in (c).

Fig. 5.
Fig. 5.

Total intensity of the polychromatic PSF of the lens with mosaic aperture (Fig. 2), computed from the superposition of the intensities obtained in Figs. 3 and 4: (a) in the focal plane and (b), along the optical axis.

Fig. 6.
Fig. 6.

Scheme of the multichannel phase Fresnel lens with rotating aperture. (a) Color filter consisting of N=4 circular sectors with narrow band transmittance centered at the wavelengths λ 1=632nm, λ 2=543nm, λ 3=488nm, and λ 4=458nm. Each color filter is against the SLM that displays a part of the sublens Li with i=1..4. The radius of each lens fulfils the PSFS-condition (Ri /λi =constant). A constant phase value is assigned to pixels beyond Ri . (b) The λi -channel lenses are multiplexed using a hybrid spatial and time integration. The result is the multichannel Fresnel lens with rotating aperture. (604 KB).

Fig. 7.
Fig. 7.

(a) Intensity of the PSF of the T 1(x,y) sublens, in the top right quadrant circular sector (Fig. 6). The lack of circular symmetry is compensated when W00iL(u, v) rotates around the optical axis (b).

Fig. 8.
Fig. 8.

Intensity distribution of the central order in the focal plane of the lens with rotating aperture (Fig. 6) for λi , i=1..4 and (a) constant radius Ri =R(b) PSFS-condition, and (c) PSFI-condition.

Fig. 9.
Fig. 9.

Intensity distribution of the central order of the lens with rotating aperture (Fig. 6) along the optical axis for λi , i=1.. 4 and (a) constant Ri =R(b) PSFS-condition, and (c) PSFI-condition. The four plots coincide in (c).

Fig. 10.
Fig. 10.

Total intensity of the polychromatic PSF computed from the superposition of the intensity distributions of Figs. 8 and 9. (a) In the focal plane, (b) along the optical axis. (c) Joint representation of Figs. 5(a) and 10(a).

Equations (28)

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L i ( x , y ) = exp { j π λ i f 0 ( x 2 + y 2 ) } ,
f 0 f r ( λ i ) = M Δ 2 λ i .
M ( λ i , x , y ) = M i ( x , y ) = τ i ( Δ λ ) circ ( 1 R i ( x 2 + y 2 ) 1 2 ) n , m δ ( x [ n Δ l + a i ] , y [ m Δ s + b i ] ) ,
T i ( x , y ) = ( L i ( x , y ) M i ( x , y ) ) rect ( x Δx , y Δ y ) ,
U i ( u , v ) = exp { j 2 π λ i f 0 } j λ i f 0 T i ( u , v ) Z i ( u , v ) ,
U i ( u , v ) = exp { j 2 π λ i f 0 } j λ i f 0 Z i ( u , v ) M ˜ i ( u λ i f 0 , v λ i f 0 ) rect ( u Δ x , v Δ y ) ,
M ˜ i ( u λ i f 0 , v λ i f 0 ) = τ i ( Δ λ ) Δ l Δ s [ exp { j 2 π ( u a i λ i f 0 + v b i λ i f 0 ) } n , m δ ( u λ i f 0 n Δ l , v λ i f 0 m Δ s ) ]
π R i 2 [ 2 J 1 ( 2 π R i λ i f 0 ( u 2 + v 2 ) 1 2 ) 2 π R i λ i f 0 ( u 2 + v 2 ) 1 2 ] .
U 00 i ( u , v ) = τ i ( Δ λ ) π d i R i Δ l Δ s [ 2 J 1 ( 2 π d i ( u 2 + v 2 ) 1 2 ) 2 π d i ( u 2 + v 2 ) 1 2 ] rexct ( u Δ x , v Δ y ) ,
M L ( x , y ) = i = 1 N T i ( x , y ) = ( i = 1 N L i ( x , y ) M i ( x , y ) ) rect ( x Δ x , y Δ y ) ,
S ( λ i , r , θ ) = S i ( r , θ ) = τ i ( Δ λ ) circ ( r R i ) rect ( θ θ i A i ) ,
Q i ( x , y ) = τ i ( Δ λ ) { L i ( x , y ) circ ( r R i ) + [ circ ( r R ) circ ( r R i ) ] exp { j ϕ } }
H ( ( 1 ) i 1 x , ( 1 ) I ( i 1 2 ) y ) ,
Q i ( x , y ) = Q i L ( x , y ) + Q i B ( x , y ) ,
with
Q i L ( x , y ) = τ i ( Δ λ ) L i ( x , y ) circ ( r R i ) H ( ( 1 ) i 1 x , ( 1 ) I ( i 1 2 ) y ) ,
Q i B ( x , y ) = τ i ( Δ λ ) [ circ ( r R ) circ ( r R i ) ] exp { j ϕ } H ( ( 1 ) i 1 x , ( 1 ) I ( i 1 2 ) y ) ,
T i ( x , y ) = [ Q i ( x , y ) n , m δ ( x n Δ , y m Δ ) ] rect ( x Δ x , y Δ y ) .
U i L ( u , v ) = exp { j 2 π λ i f 0 } j λ i f 0 Z i ( u , v ) F T { τ i ( Δ λ ) circ ( r R i ) H ( ( 1 ) i 1 x , ( 1 ) I ( i 1 2 ) y )
× n , m δ ( x n Δ , y m Δ ) } rect ( u Δ x , v Δ y ) ,
U i L ( u , v ) = τ i ( Δ λ ) π R i 2 Δ 2 λ i f 0 [ 2 J 1 ( 2 π d i ( u 2 + v 2 ) 1 2 ) 2 π d i ( u 2 + v 2 ) 1 2 ] H ˜ ( ( 1 ) i 1 u λ i f 0 , ( 1 ) I ( i 1 2 ) v λ i f 0 )
n , m δ ( u λ i f 0 n Δ , v λ i f 0 m Δ ) rect ( u Δ x , v Δ y ) .
U 00 i L ( u , v ) = τ i ( Δ λ ) π R i 2 Δ 2 λ i f 0 [ 2 J 1 ( 2 π d i ( u 2 + v 2 ) 1 2 ) 2 π d i ( u 2 + v 2 ) 1 2 ]
H ˜ ( ( 1 ) i 1 u λ i f 0 , ( 1 ) I ( i 1 2 ) v λ i f 0 ) rect ( u Δ x , v Δ y ) ,
U 00 i L ( u , v ) = π Δ 2 f 0 W 00 i L ( u , v ) ⊗rect ( u Δ x , v Δ y ) ,
W 00 i L ( u , v ) = τ i ( Δ λ ) R i 2 λ i [ 2 J 1 ( 2 π d i ( u 2 + v 2 ) 1 2 ) 2 π d i ( u 2 + v 2 ) 1 2 ] H ˜ ( ( 1 ) i 1 u λ i f 0 , ( 1 ) I ( i 1 2 ) v λ i f 0 ) ,
M L ( x , y ) = i = 1 N T i ( x , y ) = i = 1 N Q i ( x , y ) n , m δ ( x n Δ , y m Δ ) rect ( x Δ x , y Δ y ) ,
M L ( x , ωt ) = i = 1 N T i ( r , ωt ) = i = 1 N Q i ( r , ωt ) n , m δ ( x n Δ , y m Δ ) rect ( x Δ x , y Δ y ) ,

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