Abstract

Nyquist sampling theorem in an image calculation with angular spectrum method restricts a propagation distance and a focal length of a lens. In order to avoid these restrictions, we studied suitable expressions for the image computations depending on their conditions. Additionally, a lateral scale in an observation plane can be magnified freely by using a scaled convolution in each expression.

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References

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2009

2007

2006

1999

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164(4-6), 233–245 (1999).
[CrossRef]

1989

1988

Bernardo, L. M.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164(4-6), 233–245 (1999).
[CrossRef]

Ferreira, C.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164(4-6), 233–245 (1999).
[CrossRef]

Garcia, J.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164(4-6), 233–245 (1999).
[CrossRef]

Mansuripur, M.

Marinho, F.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164(4-6), 233–245 (1999).
[CrossRef]

Mas, D.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithms for free-space diffraction patterns calculation,” Opt. Commun. 164(4-6), 233–245 (1999).
[CrossRef]

Matsushima, K.

Muffoletto, R. P.

Nishiwaki, S.

Shen, F.

Shimobaba, T.

Tohline, J. E.

Tyler, J. M.

Wang, A.

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Figures (4)

Fig. 1
Fig. 1

A schematic model of an aberration-free lens. ξ denotes vector (ξ,η). Details are shown in the text.

Fig. 2
Fig. 2

Airy disc images using the scaled convolutions (λ = 0.6 um, F = 28 um, M = 212-1). (a) Type-II: NA = 0.2, magnification 1x. (b) Type-II: NA = 0.2, magnification 20x. (c) Type-III: NA = 0.9, magnification 1x. (d) Type-III: NA = 0.9, magnification 625x.

Fig. 3
Fig. 3

The intensity distributions in the observation planes far from the focal planes in distance −10 mm using (a) Type-I. (b) Type-II. (c) Type-III. (d) Type-IV, when aberration-free lens having f = 11.456 mm, NA = 0.4, L = 10 mm and D = 9.999 mm in λ = 0.6 um with M = 214.

Fig. 4
Fig. 4

The intensity distributions in the observation planes far from the focal planes in distance 0.1 mm using (a) Type-I. (b) Type-II. (c) Type-III. (d) Type-IV, when aberration-free lens having f = 0.128 mm, NA = 0.9, L = 10 mm and D = 0.779 mm in λ = 0.6 um with M = 214.

Tables (1)

Tables Icon

Table 1 Summary of the suitable formulas and the conditions

Equations (29)

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NA = d / 2 f .
a ( u ) = f 2 ξ ξ + f 2 T ( ξ ) A ( ξ ) exp ( 2 π i u ξ ) d ξ 2 ,
A ( x : z ) = a ( u ) exp ( 2 π i z λ 2 u u ) exp ( 2 π i u x ) d u 2 ,
T ( ξ ) = exp [ 2 π i λ ( ξ ξ + f 2 f ) ] .
z M ( Δ ξ ) 2 λ , f M ( Δ ξ ) 2 λ ,
A ( x : z ) = A ( ξ ) H ( x ξ ) d ξ 2 ,
H ( ξ ) = z exp ( 2 π i f ξ ξ + z 2 ) ξ ξ + z 2 [ 1 2 π ξ ξ + z 2 + i 1 λ ] .
a ( u ) = g ( p ) t ( u p : f ) d p 2 ,
t ( p : f ) = exp [ 2 π i f ( λ 2 p p 1 λ ) ] λ 2 p p [ 1 2 π λ λ 2 p p + i f λ ] ,
g ( p ) = f 2 ξ ξ + f 2 A ( ξ ) exp ( 2 i π p ξ ) d ξ 2 .
A ( x : z ) = a ( u ) [ H ( ξ ) exp ( 2 π i u ξ ) d ξ 2 ] exp ( 2 π i u x ) d u 2 ,
f ( x x 0 ) = F ( u ) exp [ 2 π i u ( x x 0 ) ] d u 2 .
f [ ( m M ) Δ x x 0 ] = p = 0 2 M 1 F p M exp [ 2 π i ( p M ) ( ( m M ) Δ x x 0 ) Δ u ] ,
f [ ( m M ) Δ x / α x 0 ] = exp [ π i γ M ( m m 2 M m + 4 M 2 ) ] p = 0 2 M 1 F p M × exp ( π i γ M ( p p 2 M p ) 2 π i L ( p M ) x 0 ) exp [ π i γ M ( m p ) 2 ] ,
Δ ξ Δ x = γ .
f [ ( m M ) Δ x / γ x 0 ] = exp [ π i γ M ( m m 2 M m + 4 M 2 ) ] × FFT ( IFFT [ F p M exp [ π i γ M ( p p 2 M p ) 2 π i L ( p M ) x 0 ] ] × IFFT [ exp ( π i γ M ( p M ) 2 ) ] ) .
d Ω = sin θ d θ d φ ,
d Ω = 1 cos θ d α d β = cos 3 θ d ξ d η ,
ξ = f tan θ cos φ η = f tan θ sin φ
α = f sin θ cos φ β = f sin θ sin φ .
d α d β = cos 4 θ d ξ d η .
d α d β = cos 2 θ d ξ d η .
d α d β = f 2 ξ ξ + f 2 d ξ d η .
t ( p : f ) = exp [ i k ( ξ ξ + f 2 f ) ] exp ( 2 i π p ξ ) d ξ 2 = exp ( i k f ) k i ξ ξ + f 2 exp [ i k ξ ξ + f 2 ] exp ( 2 i π p ξ ) d ξ 2 .
t ( p : f ) = exp ( i k f ) k d s 2 d ξ 2 1 λ 2 s s × exp [ 2 π i f λ 2 s s ] exp [ 2 i π ( p + s ) ξ ] ,
exp ( i k ξ ξ + f 2 ) ξ ξ + f 2 = i λ 2 s s exp [ 2 π i ( s ξ + f λ 2 s s ) ] d s 2 .
t ( p : f ) = exp [ 2 π i f ( λ 2 p p 1 λ ) ] λ 2 p p [ 1 2 π λ λ 2 p p + i f λ ] .
a ( u ) = exp [ i k ( ξ ξ + f 2 f ) ] A ( ξ ) exp ( 2 π i u ξ ) d ξ 2 = g ( p ) t ( u p : f ) d p 2 ,
g ( p ) = A ( ξ ) exp ( 2 i π p ξ ) d ξ 2 .

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