Abstract

We consider the problem of image formation in incoherent light from nonimage distributions, and choose the right circular axicon as an example. It is found that image formation is achievable by either a volume correlation in the Fresnel region of the axicon or by correlation on a surface inscribed in the Fresnel region.

© 1980 Optical Society of America

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References

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  1. S. Fujiwara, “Optical properties of conic surfaces. I. Reflecting cone,” J. Opt. Soc. Am. 52, 287–292 (1962).
    [CrossRef]
  2. J. Dyson, “Circular and Spiral Diffraction Gratings,” Proc. R. Soc. London Ser. A 248, 93–106 (1958).
    [CrossRef]
  3. E. Leith and B. J. Chang, “Image Formation with an Achromatic Interferometer,” Opt. Commun. 23, 217–219 (1977).
    [CrossRef]
  4. O. Nalcioglu and Z. H. Cho, “Reconstruction of 3-D Objects from Cone Beam Projections,” Proc. IEEE 66, 1584 (1978).
    [CrossRef]

1978 (1)

O. Nalcioglu and Z. H. Cho, “Reconstruction of 3-D Objects from Cone Beam Projections,” Proc. IEEE 66, 1584 (1978).
[CrossRef]

1977 (1)

E. Leith and B. J. Chang, “Image Formation with an Achromatic Interferometer,” Opt. Commun. 23, 217–219 (1977).
[CrossRef]

1962 (1)

1958 (1)

J. Dyson, “Circular and Spiral Diffraction Gratings,” Proc. R. Soc. London Ser. A 248, 93–106 (1958).
[CrossRef]

Chang, B. J.

E. Leith and B. J. Chang, “Image Formation with an Achromatic Interferometer,” Opt. Commun. 23, 217–219 (1977).
[CrossRef]

Cho, Z. H.

O. Nalcioglu and Z. H. Cho, “Reconstruction of 3-D Objects from Cone Beam Projections,” Proc. IEEE 66, 1584 (1978).
[CrossRef]

Dyson, J.

J. Dyson, “Circular and Spiral Diffraction Gratings,” Proc. R. Soc. London Ser. A 248, 93–106 (1958).
[CrossRef]

Fujiwara, S.

Leith, E.

E. Leith and B. J. Chang, “Image Formation with an Achromatic Interferometer,” Opt. Commun. 23, 217–219 (1977).
[CrossRef]

Nalcioglu, O.

O. Nalcioglu and Z. H. Cho, “Reconstruction of 3-D Objects from Cone Beam Projections,” Proc. IEEE 66, 1584 (1978).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

E. Leith and B. J. Chang, “Image Formation with an Achromatic Interferometer,” Opt. Commun. 23, 217–219 (1977).
[CrossRef]

Proc. IEEE (1)

O. Nalcioglu and Z. H. Cho, “Reconstruction of 3-D Objects from Cone Beam Projections,” Proc. IEEE 66, 1584 (1978).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

J. Dyson, “Circular and Spiral Diffraction Gratings,” Proc. R. Soc. London Ser. A 248, 93–106 (1958).
[CrossRef]

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Figures (2)

FIG. 1
FIG. 1

Axicon lens shown as a sectional slice which includes the apex.

FIG. 2
FIG. 2

Curves showing various point-spread functions: (a) J 0 2 function describing the point-spread function for the axicon; (b) Airy disk pattern for a conventional lens of the same aperture and focal length; (c) result of cross correlation on a conic surface; (d) result of volume cross correlation. R is the relative response, normalized; θ is the angle of arrival of the incident light.

Equations (23)

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| h | 2 = 4 π f 0 2 J 0 2 { 2 π f 0 [ ( x θ x z 1 ) 2 + ( y θ y z 1 ) 2 ] 1 / 2 } ,
u 1 = K s ( x , y ) × exp { j ( π / λ z ) [ ( x x ) 2 + ( y y ) 2 ] } d x d y ,
u 2 = [ ( s * h s ) a e j 2 π f 0 r ] * h s ,
χ ( θ x , θ y ) = | h | 2 * r
u P = u exp { j ( π / λ z ) [ ( x 1 x 0 ) 2 + ( y 0 y 1 ) 2 ] } d x 0 d y 0 ,
r 2 = r 2 ( z 2 z 1 ) / z 2 = r 2 = m z 1 ,
I 2 = a ( r 2 ) ( r 2 / r 2 ) ,
I 0 = a ( r 0 ) ( r 0 / r 2 ) .
F { G ( r ) } = 1 2 π f 0 δ ( f r f 0 ) ,
G ( r ) = J 0 ( f 0 r ) .
u ( x , y ) = exp [ j 2 π ( f α x f β y ) ]
u ( z = 0 ) = 2 π f 0 J 0 ( f 0 r ) exp [ j 2 π ( f α x + f β y ) ] .
F { u ( z = 0 ) } = F 1 ( f x f α , f y f β ) ,
F { u ( z = z 1 ) } = f 1 ( f x f α , f y f β ) exp ( j π λ z 1 f 2 ) ,
I ( z 1 ) = u ( z 1 ) u * ( z 1 ) ,
I ( z 1 ) = F 1 ( ξ f α , η f β ) × F 1 [ f x ( ξ f α ) , f y ( η f β ) ] exp [ j π λ z 1 ( f x 2 + f y 2 ) ] × exp [ j 2 π λ z 1 ( f x ξ + f y η ) ] d ξ d η .
F 1 = { u ( z = 0 ) } = δ ( f f 0 ) = δ ( ξ 2 + η 2 f 0 ) ,
I ( z 1 ) = exp [ j π λ z 1 ( f x 2 + f y 2 ) ] exp { j 2 π [ ( θ x z 1 ) f x + ( θ y z 1 ) f y ] } × δ ( ξ 2 + η 2 f 0 ) × δ ( ( f x ξ ) 2 + ( f y η ) 2 f 0 ) × exp [ j 2 π λ z 1 ( f x ξ + f y η ) ] d ξ d η .
2 f 0 2 f 4 f 0 2 f 2 ξ i , η i exp [ j 2 π λ z 1 ( f x ξ i + f y η i ) ] = I 1 ,
I 1 = 2 f 0 2 f 4 f 0 2 f 2 [ 2 exp ( j π λ z 1 f 2 ) ] = 4 f 0 2 f 4 f 0 2 f 2 exp [ j π λ z 1 ( f x 2 + f y 2 ) ] .
F 1 { I ( z 1 ) } = 4 f 0 2 f 4 f 0 2 f 2 exp [ j 2 π ( θ x z 1 f x + θ y z 1 f y ) ] .
F 1 { 4 f 0 2 f 4 f 0 2 f 2 } = 4 π f 0 2 J 0 ( 2 π f 0 r ) .
I ( z ) = 4 π 2 f 0 2 J 0 2 { 2 π f 0 [ ( x θ x z 1 ) 2 + ( y θ y z 1 ) 2 ] 1 / 2 .