Abstract

We consider diffraction by pixelated lenses when the lens size is significantly smaller than the diffraction pattern of single pixels. In that case, the diffraction orders show shapes that have not been identified in earlier studies and that are quite sensitive to the pixel filling ratio as well as to decentering.

© 2010 Optical Society of America

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References

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    [CrossRef]
  6. G. J. Swanson, “The theory and design of multi-level diffractive optical elements,” Binary Optics Technology, MIT Tech. Rep. 854 (MIT, 1989)
  7. G. J. Swanson“Theoretical limits on the diffraction efficiency of multilevel diffractive optical elements,” in Binary Optics Technology, MIT Tech. Rep. 914 (MIT, 1991).
  8. This is compatible with a centered pupil of size N×N pixels only if N is an odd integer. But the extension to even values of N would be straightforward. However, the parity of N is an inessential factor in the present study and we shall not develop it in any detail.
    [CrossRef]
  9. S. Wang, E. Barnabeu, and J. Alda, “Unified and generalized Fresnel numbers,” Opt. Quantum Electron. 24, 1351-1358(1992).

1999

1995

1994

1992

S. Wang, E. Barnabeu, and J. Alda, “Unified and generalized Fresnel numbers,” Opt. Quantum Electron. 24, 1351-1358(1992).

1990

Alda, J.

S. Wang, E. Barnabeu, and J. Alda, “Unified and generalized Fresnel numbers,” Opt. Quantum Electron. 24, 1351-1358(1992).

Arrizon, V.

Barnabeu, E.

S. Wang, E. Barnabeu, and J. Alda, “Unified and generalized Fresnel numbers,” Opt. Quantum Electron. 24, 1351-1358(1992).

Bosch, S.

Campos, J.

Carcolé, E.

Carreon, E.

Cottrell, D. M.

Davis, J. A.

Gonzalez, L. A.

Hedman, T. R.

Juvells, I.

Lilly, R. A.

Moneo, J. R. de F.

Swanson, G. J.

G. J. Swanson, “The theory and design of multi-level diffractive optical elements,” Binary Optics Technology, MIT Tech. Rep. 854 (MIT, 1989)

G. J. Swanson“Theoretical limits on the diffraction efficiency of multilevel diffractive optical elements,” in Binary Optics Technology, MIT Tech. Rep. 914 (MIT, 1991).

Wang, S.

S. Wang, E. Barnabeu, and J. Alda, “Unified and generalized Fresnel numbers,” Opt. Quantum Electron. 24, 1351-1358(1992).

Appl. Opt.

Opt. Quantum Electron.

S. Wang, E. Barnabeu, and J. Alda, “Unified and generalized Fresnel numbers,” Opt. Quantum Electron. 24, 1351-1358(1992).

Other

G. J. Swanson, “The theory and design of multi-level diffractive optical elements,” Binary Optics Technology, MIT Tech. Rep. 854 (MIT, 1989)

G. J. Swanson“Theoretical limits on the diffraction efficiency of multilevel diffractive optical elements,” in Binary Optics Technology, MIT Tech. Rep. 914 (MIT, 1991).

This is compatible with a centered pupil of size N×N pixels only if N is an odd integer. But the extension to even values of N would be straightforward. However, the parity of N is an inessential factor in the present study and we shall not develop it in any detail.
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

(a) Continuous lens. (b) Diffractive lens. (c) Pixelated lens. (d) Diffractive pixelated lens.

Fig. 2
Fig. 2

Scheme illustrating the notation used in the basic mathematical formulation.

Fig. 3
Fig. 3

Shape of the sinc function and shape of the derivative of the sinc function.

Fig. 4
Fig. 4

Central order of a centered pixelated lens: (a)  a = p = 10 μm , (b)  a = p = 200     μm .

Fig. 5
Fig. 5

First order of a centered pixelated lens: (a)  a = p = 10 μm , (b)  a = p = 200 μm .

Fig. 6
Fig. 6

First order of a centered pixelated lens: a = p / 2 = 5 μm .

Fig. 7
Fig. 7

Observation of the repartition of light in the focal plane of a pixelated lens. Dashed frame, the central order; solid frame, order (0,1).

Fig. 8
Fig. 8

Observation of the shape of the central order.

Fig. 9
Fig. 9

Observation of the shape of order (0,1).

Fig. 10
Fig. 10

Vertical cross section of the image in Fig. 9. The pixel pitch distance is 45 µm .

Fig. 11
Fig. 11

Order 1 in the focal plane of a decentered pixelated lens: a = p , X = λ f / ( 3 p ) .

Fig. 12
Fig. 12

Normalized intensity in the focal plane of a decentered pixelated lens: a = p / 2 , X = 2 λ f / 5 p . (a) Curve associated with direct calculation; (b) curve associated with sinc ( a ξ / λ f ) .

Tables (2)

Tables Icon

Table 1 Normalized Intensity of Orders 0, 1, and 2

Tables Icon

Table 2 Normalized Intensity of Orders 1 , 0 and 1 in the Focal Plane of a Decentered Pixelated Lens a

Equations (44)

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t c ( r ) = P ( r ) exp ( i π r 2 λ f )
t Δ ( r ) = t c ( r ) Ш p , p ( r ) ,
( x , y ) 2 , Ш p , p ( x , y ) = n x = + δ ( x n x p ) × n y = + δ ( y n y p ) .
t ( r ) = t Δ ( r ) * Π 2 , a ( r ) .
FGN = r 0 2 λ f ,
Hypothesis ( H 1 ) : FGN = a 2 2 λ f 1.
u 0 ( ρ ) = i U 0 λ f t c ( 0 ) exp ( 2 π i f λ + i π ρ 2 λ f ) Π ˜ 2 , a ( ρ λ f ) .
u j ( ρ ) = i U 0 λ f t c ( j p ) exp ( 2 π i f λ + i π ρ j p 2 λ f ) Π ˜ 2 , a ( ρ j p λ f ) .
f j ( ρ ) P ( j p ) exp ( 2 i π j p · ρ λ f ) Π ˜ 2 , a ( ρ j p λ f )
u j ( ρ ) = i U 0 λ f exp [ 2 π i λ ( f + ρ 2 2 f ) ] f j ( ρ ) .
S ( ρ ) = j 2 f j ( ρ ) = j 2 P ( j p ) exp ( 2 π i j p · ρ λ f ) Π ˜ 2 , a ( ρ j p λ f ) .
Hypothesis ( H 2 ) : A p λ f 1 ,
Π ˜ 1 , a ( ξ j p λ f ) Π ˜ 1 , a ( ξ λ f ) j p λ f Π 1 , a ( ξ λ f ) .
x , Ш p ( x ) = n x = + δ ( x n x p ) .
S ( ξ ) = FT [ P ( x ) Ш p ( x ) ] ξ / λ f Π ˜ 1 , a ( ξ λ f ) 1 λ f FT [ x P ( x ) Ш p ( x ) ] ξ / λ f Π ˜ 1 , a ( ξ λ f ) ,
S ( ξ ) = ( P ˜ * Ш 1 / p ) ξ / λ f Π ˜ 1 , a ( ξ λ f ) + 1 2 π i λ f ( P ˜ * Ш 1 / p ) ξ / λ f Π ˜ 1 , a ( ξ λ f ) .
S ( ξ ) = S A ( ξ ) + S B ( ξ )
S A ( ξ ) = n S A , n ( ξ λ f ) = Π ˜ 1 , a ( ξ λ f ) n P ˜ ( ξ λ f n p ) , S B ( ξ ) = n S B , n ( ξ λ f ) = Π ˜ 1 , a ( ξ λ f ) 2 π i λ f n P ˜ ( ξ λ f n p ) .
S ( ξ ) = n S n ( ξ λ f )
S A ( ξ ) = a sinc ( ξ λ f a ) n A sinc [ ( ξ λ f n p ) A ] , S B ( ξ ) = a 2 2 π i λ f sinc ( ξ λ f a ) n A 2 sinc [ ( ξ λ f n p ) A ] .
S A ( ξ ) = p sinc ( ξ λ f p ) n A sinc [ ( ξ λ f n p ) A ] , S B ( ξ ) = p 2 2 π i λ f sinc ( ξ λ f p ) n A 2 sinc [ ( ξ λ f n p ) A ] .
S 0 ( ξ ) = S A , 0 ( ξ ) + S B , 0 ( ξ ) S A , 0 ( ξ ) = p A sinc ( p ξ / λ f ) sinc ( A ξ / λ f ) n 0 , S n ( ξ ) S B , n ( ξ ) = ( p 2 sinc ( p ξ / λ f ) / 2 π i λ f ) A 2 sinc [ ( ξ / λ f n / p ) A ] .
n , S n ( ξ ) S A , n ( ξ ) = a sinc ( ξ λ f a ) A sinc [ ( ξ λ f n p ) A ] .
u ( ξ , f ) = i U 0 λ f exp ( 2 π i f λ ) t ( x ) exp ( + i π ( x ξ ) 2 λ f ) d x ,
Δ = max ξ D | f ( ξ ) g ( ξ ) | ,
S A ( ξ ) = a sinc ( a ξ λ f ) n A sinc [ ( ξ λ f X λ f n p ) A ] , S B ( ξ ) = a 2 2 π i λ f sinc ( ξ λ f a ) n A 2 sinc [ ( ξ λ f X λ f n p ) A ] .
S n X ( ξ ) S A , n X ( ξ ) = p A sinc ( p ξ / λ f ) sinc ( A ξ / λ f ) , n n X , S n ( ξ ) S B , n ( ξ ) = ( p 2 sinc ( p ξ / λ f ) / 2 π i λ f ) A 2 sinc [ ( ξ / λ f X / λ f n / p ) A ] ,
n , S n ( ξ ) S A , n ( ξ ) = a sinc ( ξ λ f a ) A sinc [ ( ξ λ f X λ f n p ) A ] .
n , S n ( ξ ) S A , n ( ξ ) = a sinc ( ξ λ f a ) A sinc [ ( ξ λ f X λ f n p ) A ] .
x , floor ( x ) = max { n , n x } , x , ceil ( x ) = min { n , n x } ,
S = j ( , ) f j ( ρ ) = j ( , ) P ( j p ) exp ( 2 i π j p · ρ λ f ) Π ˜ 2 , a ( ρ j p λ f ) ,
Π ˜ a ( ρ j p λ f ) Π ˜ a ( ξ λ f , η λ f ) j x p λ f Π ˜ 2 , a ξ ( ξ λ f , η λ f ) j y p λ f Π ˜ 2 , a η ( ξ λ f , η λ f ) .
S ( ξ ) = FT [ P ( x , y ) Ш p , p ( x , y ) ] ( ξ / λ f , η / λ f ) Π ˜ 2 , a ( ξ λ f , η λ f ) 1 λ f FT [ x P ( x , y ) Ш p , p ( x , y ) ] ( ξ / λ f , η / λ f ) Π ˜ 2 , a ξ ( ξ λ f , η λ f ) 1 λ f FT [ y P ( x , y ) Ш p , p ( x , y ) ] ( ξ / λ f , η / λ f ) Π ˜ 2 , a η ( ξ λ f , η λ f ) .
S ( ξ ) = ( P ˜ * Ш ( 1 / p , 1 / p ) ) ρ / λ f Π ˜ 2 , a ( ρ λ f ) + 1 2 π i λ f ( P ˜ ξ * Ш ( 1 / p , 1 / p ) ) ρ / λ f Π ˜ 2 , a , ξ ( ρ λ f ) + 1 2 π i λ f ( P ˜ η * Ш ( 1 / p , 1 / p ) ) ρ / λ f Π ˜ 2 , a , η ( ρ λ f ) .
S ( ξ ) = S A ( ξ ) + S B , ξ ( ξ ) + S B , η ( ξ ) = Π ˜ 2 , a ( ρ λ f ) n ( , ) P ˜ ( ρ λ f n p ) + Π ˜ 2 , a , ξ ( ρ λ f ) 2 π i λ f n ( , ) P ˜ ξ ( ρ λ f n p ) + Π ˜ 2 , a , η ( ρ λ f ) 2 π i λ f n ( , ) P ˜ η ( ρ λ f n p ) .
t c ( x ) = P ( x ) exp ( i π ( x X ) 2 / λ f ) .
u 0 ( ξ ) = i U 0 λ f t c ( 0 ) exp ( 2 π i f λ + i π ξ 2 λ f ) Π ˜ 1 , a ( ξ λ f ) ,
u j x ( ξ ) = i U 0 λ f t c ( j x p ) Π ˜ 1 , a ( ξ j x p λ f ) exp ( 2 π i f λ + i π ( ξ j x p ) 2 λ f ) .
f j x ( ξ ) P ( j x p ) exp ( 2 i π j x p ξ λ f ) Π ˜ 1 , a ( ξ j x p λ f ) exp ( 2 i π X j x p λ f ) ,
u j x ( ξ ) = i U 0 λ f exp [ 2 π i λ ( f + ξ 2 2 f ) ] exp ( i π X 2 λ f ) f j x ( ξ ) .
S ( ξ ) = j x f j x ( ξ ) = j x P ( j x p ) exp ( 2 π i ξ j x p λ f ) Π ˜ 1 , a ( ξ j x p λ f ) exp ( 2 π i X j x p λ f ) .
S ( ξ ) = FT [ P ( x ) Ш p ( x ) ] ( ξ X ) / λ f Π ˜ 1 , a ( ξ λ f ) 1 λ f FT [ x P ( x ) Ш p ( x ) ] ( ξ X ) / λ f Π ˜ 1 , a ( ξ λ f ) .
S ( ξ ) = ( P ˜ * Ш 1 / p ) ( ξ X ) / λ f Π ˜ 1 , a ( ξ λ f ) + 1 2 π i λ f ( P ˜ * Ш 1 / p ) ( ξ X ) / λ f Π ˜ 1 , a ( ξ λ f ) .
S ( ξ ) = S A ( ξ ) + S B ( ξ ) = Π ˜ 1 , a ( ξ λ f ) n P ˜ ( ξ λ f X λ f n p ) + Π ˜ 1 , a ( ξ / λ f ) 2 π i λ f n P ˜ ( ξ λ f X λ f n p ) .

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