Abstract

A Fourier hologram with the distribution of a radial harmonic function creates a nondiverging beam in the far field. The properties of this beam are analyzed and the beam is demonstrated. We also give a set of theorems that describe the relations between the hologram and the longitudinal distribution of the beam.

© 1995 Optical Society of America

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References

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  11. J. Rosen, A. Yariv, “Snake beam: a paraxial arbitrary focal line,” submitted to Opt. Lett.
  12. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), Chap. 7, pp. 222–254.
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1995 (2)

1994 (3)

1992 (2)

1991 (1)

1987 (2)

J. Durnin, “Exact solutions for diffraction-free beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[CrossRef]

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

1964 (1)

1960 (1)

1954 (1)

Bará, S.

Davidson, N.

Durnin, J.

J. Durnin, “Exact solutions for diffraction-free beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[CrossRef]

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Friesem, A. A.

Hasman, E.

Herman, R. M.

Jaroszewicz, Z.

Kolodziejczyk, A.

Liu, H.-K.

McCutchen, C. W.

McLeod, J. H.

Miceli, J. J.

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), Chap. 7, pp. 222–254.

Piestun, R.

Rosen, J.

Salik, B.

Shamir, J.

Sochacki, J.

Welford, W. T.

Wiggins, T. A.

Yariv, A.

Appl. Opt. (2)

J. Opt. Soc. Am. (3)

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

Opt. Lett. (5)

Phys. Rev. Lett. (1)

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Other (2)

J. Rosen, A. Yariv, “Snake beam: a paraxial arbitrary focal line,” submitted to Opt. Lett.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), Chap. 7, pp. 222–254.

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Figures (8)

Fig. 1
Fig. 1

Schematic system used to obtain the PNDB.

Fig. 2
Fig. 2

Computer-simulated axial intensity distribution of the PNDB (solid curve) and of the ordinary focused beam (dashed curve).

Fig. 3
Fig. 3

Transverse cross section of the PNDB at (a) z = 0, (b) z = 500, (c) z = 1000, (d) z = 1500, (e) z = 2000, and (f) z = 2500.

Fig. 4
Fig. 4

(a) Output intensity distribution in the xz plane obtained by illumination of the hologram shown in (g) with a plane wave. Transverse cross sections are shown at distances (b) 33.5 cm (z = −z2), (c) 38.5 cm (z = −z1), (d) 40 cm (z = 0), (e) 41.5 cm (z = z1), and (f) 46.5 cm (z = z2) from the lens (g) The Fourier hologram that generates this beam. The hologram’s distribution is given in Eq. (25), where dx = 5.13 mm−1, b = 3.72 mm, and R0 = 6.2 mm.

Fig. 5
Fig. 5

As Fig. 4 with α = −0.31 mm−2, da = 20 mm, b = 3.6 mm, dx = 5.13 mm−1, and R0 = 6.2 mm. Transverse cross sections are shown at distances (b) 34 cm, (c) 35.5 cm, (d) 40 cm, (e) 44.5 cm, and (f) 46 cm from the lens. (g) The Fourier hologram that generates this beam.

Fig. 6
Fig. 6

As Fig. 4 with α =0.31 mm−2, b = 4 mm, da = dx = 0, and R0 = 6.2 mm. Transverse cross sections are shown at distances (b) 28 cm, (c) 29.5 cm, (d) 33 cm, (e) 37.5 cm, (f) 40 cm, (g) 44.5 cm, (h) 47 cm, (i) 49.5 cm, and (j) 52 cm from the lens. (k) The Fourier hologram that generates this beam.

Fig. 7
Fig. 7

Schematic imaging system with extended depth of focus.

Fig. 8
Fig. 8

Imaging results of three kinds of imaging system: with pupil filtering at (a) z = 0, (d) z = 64 pixels, and (g) z = 128 pixels; with a clear pupil of diameter 2b at (b) z = 0, (e) z = 64 pixels; and (h) z = 128 pixels; and with a clear pupil of diameter 2R0 at (c) z = 0, (f) z = 64 pixels, and (i) z = 128 pixels.

Equations (55)

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u ( r , z ) = exp [ j k ( z + 2 f ) ] j λ f 0 g ( r i ) J 0 ( 2 π r r i λ f ) × exp ( j π z r i 2 λ f 2 ) r i d r i ,
u ( 0 , z ) = exp [ j k ( z + 2 f ) ] j 2 λ f 0 g ( ρ i ) exp ( j k z ρ i 2 f 2 ) d ρ i ,
u ( x , y , z ) = FST { g ( x i , y i ) } = exp [ j k ( z + 2 f ) ] j λ f g ( x i , y i ) × exp ( j 2 π ( x x i + y y i ) λ f j π z ( x i 2 + y i 2 ) λ f 2 ) × d x i d y i .
FST { g * ( x i , y i ) } = exp ( j 4 k f j π ) u * ( x , y , z ) .
FST { g ( x i , y i ) exp [ j 2 π ( d x x i + d y y i ) ] } = u ( x λ f d x , y λ f d y , z ) .
FST { g ( x i , y i ) exp [ j 2 π α r i 2 ) } = exp ( j 4 π f 2 α ) × u ( x , y , z 2 λ f 2 α ) .
FST { g ( x i d a , y i d b ) } = exp ( j k χ ) × u ( x ¯ sec θ , y ¯ sec φ , z ¯ cos θ cos φ ) ,
χ = x tan θ + y tan φ z 2 ( tan 2 θ + tan 2 φ ) ,
FST { g ( s r i ) } = exp [ jkz ( 1 s 2 ) ] s 2 ( r s , z s 2 ) .
ū ( x , y , z ) = A ( x , y , z ) u ( x , , z ) ,
( x , , z ) = f 2 z ( f d ) + f 2 ( x , y , z ) , A ( x , y , z ) = f 2 exp [ j k ( z + d f + ( f d ) r 2 2 f 2 z 2 [ z ( f d ) + f 2 ] ) ] z ( f d ) + f 2 .
| u ( 0 , z ) | 2 = const . , z Δ z ,
max r { | u ( r , z ) | } = | u ( 0 , z ) | , z Δ z ,
lim r | u ( r , z ) | = 0 , z Δ z ,
even if 0 R | u ( r , z ) | 2 r d r R .
g p ( r i ) = ( r i / a ) exp [ j 2 π ( r i / b ) p ] , p 4 ,
u ( 0 , z ) = exp [ j k ( z + 2 f ) ] j λ f a ( 4 p ) / 2 0 r i ( p 2 ) / 2 exp ( j 2 π r i p b p j π z r i 2 λ f 2 ) d r i .
u ( 0 , z ) exp [ j k ( z + 2 f ) + j π / 4 ] j λ f b p p ( p 2 ) a p 4 × exp ( j π γ z p / ( p 2 ) ) , z > z 1 ,
γ = ( p 2 ) ( b 2 p λ f 2 ) p / ( p 2 ) = p = 4 b 4 8 λ 2 f 4 ,
z 1 = p λ f 2 b 2 ( 2 p 4 ) ( p 2 ) / p = p = 4 2 λ f 2 b 2 .
k μ ( r s ) = 2 π r s p b p π z 1 r s 2 λ f 2 = π 2 ,
r s = ( b p z p λ f 2 ) 1 / ( p 2 )
z 2 p λ f 2 R 0 p 3 b p [ R 0 ( p 2 ) b p 2 p R 0 p 2 ] = p = 4 2 λ f 2 ( 2 R 0 2 b 2 ) b 4 .
u ( 0 , z ) exp [ j k ( z + 2 f ) ] j λ f a p 4 { b p p ( p 2 ) exp ( j γ z p / ( p 2 ) + j π / 4 ] λ f 2 ( 2 R 0 ) ( p 4 ) / 2 2 π ( λ f 2 p R 0 p 2 b p z ) × exp ( j k R 0 2 z 2 f 2 + j 2 π R 0 p b p ) } , z 1 < z < z 2 .
u ( 0 , z ) p = 4 exp [ j k ( z + 2 f ) ] j λ f [ b 2 2 2 exp ( j γ z 2 + j π / 4 ) λ f 2 / 2 π ( 4 λ f 2 R 0 2 / b 4 ) z exp ( j k R 0 2 z 2 f 2 + j 2 π R 0 4 b 4 ) λ f 2 2 π z ] , z 1 < z < z 2 .
u ( r , z ) exp [ j k ( z + 2 f ) + j π / 4 ] j λ f J 0 [ 2 π r λ f ( b p z p λ f 2 ) 1 / ( p 2 ) ] × b p p ( p 2 ) a p 4 exp [ j γ z p / ( p 2 ) ] , z Δ z .
W P ( z ) 2.26 2 π ( p λ p 1 f p b p z ) 1 / ( p 2 ) = p = 4 0.72 f 2 b 2 λ 3 z .
W P ( z 1 ) 2.26 λ f 2 π b ( 2 p 4 ) 1 / p = p = 4 0.51 λ f b .
W P ( z 2 ) 0.36 λ f R 0 ,
Δ z p Δ z G = p λ 2 f 2 R 0 p 2 4 W 2 b p .
Δ z P Δ z G W = W P ( z 1 ) = 1.93 p ( 2 p 4 ) 2 / p ( R 0 b ) p 2 p = 4 4 ( R 0 b ) 2 .
Δ z P Δ z G W = W P ( z 2 ) = 1.93 p ( R 0 b ) p p = 4 8 ( R 0 b ) 4 .
Δ z p 16 [ W P ( z 1 ) ] 4 NA 2 λ 3 8.3 [ W P ( z 1 ) ] 4 λ [ W P ( z 2 ) ] 2 ,
u ( r , z ) r λ f R 0 p 1 / b p exp [ j k ( z + 2 f ) ] j λ f r 0 r i ( p 3 ) / 2 × exp ( j 2 π r p b p j π z r i 2 λ f 2 ) × cos ( 2 π r r i λ f ) d r i .
I ( r , z ) r r p / ( p 1 ) for z = 0.
I ( r , z ) r r 1 r ¯ s p 3 [ ( p 1 ) p λ b p r ¯ s p 2 2 z f 2 ] 1 , z Δ z ,
r ¯ s = r [ ( p λ f b p r p 2 ) 1 / ( p 1 ) 2 z f ( p 1 ) ] 1 .
d μ d r i = p r i p 1 b p z r i λ f 2 + r λ f = 0 .
r m = p 2 p 1 ( z p 1 b p λ f P p ( p 1 ) ) 1 / ( P 2 ) = p = 4 b 2 z 3 / 2 3 f 2 3 λ
2 r m ( z 1 ) = 4 2 λ f 3 3 b , 2 r m ( z 2 ) = 16 λ f R 0 3 3 3 b 4
| u ( r , z ) | < | u ( 0 , z ) | , z Δ z .
h ( x i , y i ) = 1 2 + 1 4 g 4 ( r i ) exp ( j 2 π d x x i ) + 1 4 g 4 * ( r i ) exp ( j 2 π d x x i ) ,
h ( x i , y i ) = 1 2 + 1 4 g 4 ( x i d a , y i ) exp [ j 2 π ( d x x i + α r i 2 ) ] + 1 4 g 4 * ( x i d a , y i ) exp ( j 2 π ( d x x i + α r i 2 ) ] ,
h ( x i , y i ) = 1 2 + 1 4 g 4 ( r i ) exp [ j 2 π α r i 2 ] + 1 4 g 4 * ( r i ) exp [ j 2 π α r i 2 ] ,
FST { g * ( x i , y i ) } = exp [ j k ( z + 2 f ) ] j λ f g * ( x i , y i ) × exp [ j 2 π ( x x i + y y i ) λ f j π z ( x i 2 + y i 2 ) λ f 2 ] d x i d y i = exp ( j 4 k f j π ) × { exp [ j k ( z + 2 f ) ] j λ f × g ( x i , y i ) exp [ j 2 π ( x x i + y y i ) λ f j π z ( x i 2 + y i 2 ) λ f 2 ] d x i d y i } * = exp ( j 4 k f j π ) u * ( x , y , z ) .
FST { g ( r i ) exp [ j 2 π ( d x x i + d y y i ) ] } = exp [ j k ( z + 2 f ) ] j λ f g ( x i , y i ) exp { j 2 π λ f [ ( x λ f d x ) x i + ( y λ g d y ) y i ] } exp [ j π z ( x i 2 + y i 2 ) λ f 2 ] d x i d y i = u ( x λ f d x , y λ f d y , z ) .
FST { g ( r i ) exp ( j 2 π α r i 2 ) } = exp [ j k ( z + 2 f ) ] j λ f × g ( x i , y i ) exp [ j 2 π λ f ( x x i + y y i ) ] × exp [ j 2 π r i 2 ( z 2 λ f 2 α ) ] r i d r i = exp ( j 4 π f 2 α ) u ( x , y , z 2 λ f 2 α ) .
x = x ¯ cos θ + z ¯ sin θ , y = y ¯ cos φ + z ¯ sin φ , z = ( z ¯ cos θ x ¯ sin θ ) cos φ y ¯ sin φ ,
FST { g ( x i d a , y i d b ) } = exp [ j k ( z + 2 f ) ] j λ f × g ( x i d a , y i d b ) exp [ j 2 π ( x x i + y y i ) λ f ] × exp [ j π z ( x i 2 + y i 2 ) λ f 2 ] d x i d y i = 1 j λ f exp { j k [ z ( 1 d a 2 + d b 2 2 f 2 ) + 2 f + x d a + y d b f ] } × g ( x ̂ , ŷ ) exp { j 2 π λ f [ x ̂ ( x z d a f ) + ŷ ( y z d b f ) ] } × exp ( j π z ( x ̂ 2 + ŷ 2 ) λ f 2 ) d x ̂ d ŷ = exp { j k [ z ( d a 2 + d b 2 2 f 2 ) + x d a + y d b f ] } u ( x z d a f , y z d b f , z ) .
FST { g ( x i d a , y i d b ) } = exp { jkz 2 ( tan 2 θ + tan 2 φ ) + j k [ x tan θ + y tan φ ] } × u ( x ¯ sec θ , y ¯ sec φ , z ¯ cos θ cos φ ) .
FST { g ( s r i ) } = exp [ j k ( z + 2 f ) ] j λ f 0 2 π 0 g ( s r i ) × exp [ j 2 π λ f ( x x i + y y i ) ] × exp ( j k z r i 2 2 f 2 ) r i d r i d θ i = exp [ j k ( z + 2 f ) ] j λ f s 2 0 2 π 0 g ( ψ i ) × exp [ j 2 π λ f s ( x ψ i cos θ + y ψ i sin θ ) ] × exp ( j k z ψ i 2 2 f 2 s 2 ) ψ i d ψ i d θ i = exp [ j k z ( 1 s 2 ) ] s 2 u ( r s , z s 2 ) ,
FST { g ( r i ) * δ ( z i d ) } = exp { j k [ z + f + r 2 / 2 ( z + f ) ] } j λ ( z + f ) × 0 2 π 0 g L ( x L , y L ) exp { j × k z r L 2 2 [ 1 f 1 ( z + f ) ] + j k ( x x L + y y L ) z + f } r L d r L d θ L ,
g L ( x L , y L ) = exp ( jkd ) j λ d g ( x i , y i ) exp [ j k 2 d ( r L 2 + r i 2 2 x i x L 2 y i y L ) ] d x i d y i .
FST { g ( r i ) * δ ( z i d ) } = f exp [ j k ( z + d + f ) ] j λ [ z ( f d ) + f 2 ] × g ( x i , y i ) exp { j k × z [ r i 2 + ( f d ) r 2 ] 2 f ( x x i + y y i ) 2 [ z ( f d ) + f 2 ] } d x i d y i = f 2 z ( f d ) + f 2 exp ( j k { z + d f + ( f d ) r 2 2 f 2 z 2 [ z ( f d ) + f 2 ] } ) u ( x , , z ) ,
( x , , z ) = f 2 z ( f d ) + f 2 ( x , y , z ) .

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