Abstract

The aperture of a perfect lens is divided into several concentric circular zones. The effects of the different zones having different phase retardations are studied. The following conditions are found by a straightforward consideration: When the aperture is divided into two zones, with the inner zone having a phase retardation of π rad with respect to the outer zone, the diffraction pattern in the focal plane of the lens has the smallest central bright spot; this assumes that the same irradiance is always produced at the focus. Conversely, for a given radius of the central bright spot in the diffraction pattern, the greatest irradiance at the focus of the lens is produced when the lens aperture is divided into two concentric circular zones, with the inner zone having a phase retardation of π rad. Such a lens also possesses the same advantages over a perfect lens with a circular stop in its center. It may also possess an effective focal depth greater than that of the latter. It may have one or two foci depending on the value of the radius of the circle that divides its aperture.

© 1971 Optical Society of America

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References

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  1. P. Jacquinot and B. Roizen-Dossier, in Progress in Optics, III, edited by E. Wolf (North-Holland, Amsterdam, 1964), p. 31.
  2. G. Toraldo di Francia, Nuovo Cimento 9, Suppl. No. 3, 426 (1952).
    [Crossref]
  3. J. E. Wilkins, J. Opt. Soc. Am. 40, 220 (1950).
    [Crossref]
  4. B. J. Thompson, J. Opt. Soc. Am. 55, 145 (1965).
    [Crossref]
  5. T. Asakura and H. Misjina, Jap. J. Appl. Phys. 9, 195 (1970).
    [Crossref]
  6. Lord Rayleigh, Monthly Notes Roy. Astron. Soc. 33, 59 (1872).
  7. E. H. Linfoot, Recent Advances in Optics (Oxford University Press, London, 1958), pp. 35 ff.
  8. E. H. Linfoot and E. Wolf, Proc. Phys. Soc. (London) B66, 145 (1953).
  9. H. Osterberg and J. E. Wilkins, J. Opt. Soc. Am. 39, 553 (1949).
    [Crossref]
  10. G. C. Steward, Phil. Trans. Roy. Soc. (London) A225, 131 (1925).
  11. W. H. Steel, Rev. Opt. 32, 4 (1953).
  12. C. A. Taylor and B. J. Thompson, J. Opt. Soc. Am. 48, 844 (1958).
    [Crossref]
  13. W. T. Welford, J. Opt. Soc. Am. 50, 749 (1960).
    [Crossref]
  14. J. Tsujiuchi, in Progress in Optics, II, edited by E. Wolf (North-Holland, Amsterdam, 1963), p. 133.
  15. H. Osterberg and F. C. Wissler, J. Opt. Soc. Am. 39, 558 (1949).
    [Crossref]

1970 (1)

T. Asakura and H. Misjina, Jap. J. Appl. Phys. 9, 195 (1970).
[Crossref]

1965 (1)

1960 (1)

1958 (1)

1953 (2)

W. H. Steel, Rev. Opt. 32, 4 (1953).

E. H. Linfoot and E. Wolf, Proc. Phys. Soc. (London) B66, 145 (1953).

1952 (1)

G. Toraldo di Francia, Nuovo Cimento 9, Suppl. No. 3, 426 (1952).
[Crossref]

1950 (1)

J. E. Wilkins, J. Opt. Soc. Am. 40, 220 (1950).
[Crossref]

1949 (2)

1925 (1)

G. C. Steward, Phil. Trans. Roy. Soc. (London) A225, 131 (1925).

1872 (1)

Lord Rayleigh, Monthly Notes Roy. Astron. Soc. 33, 59 (1872).

Asakura, T.

T. Asakura and H. Misjina, Jap. J. Appl. Phys. 9, 195 (1970).
[Crossref]

Jacquinot, P.

P. Jacquinot and B. Roizen-Dossier, in Progress in Optics, III, edited by E. Wolf (North-Holland, Amsterdam, 1964), p. 31.

Linfoot, E. H.

E. H. Linfoot and E. Wolf, Proc. Phys. Soc. (London) B66, 145 (1953).

E. H. Linfoot, Recent Advances in Optics (Oxford University Press, London, 1958), pp. 35 ff.

Misjina, H.

T. Asakura and H. Misjina, Jap. J. Appl. Phys. 9, 195 (1970).
[Crossref]

Osterberg, H.

Rayleigh, Lord

Lord Rayleigh, Monthly Notes Roy. Astron. Soc. 33, 59 (1872).

Roizen-Dossier, B.

P. Jacquinot and B. Roizen-Dossier, in Progress in Optics, III, edited by E. Wolf (North-Holland, Amsterdam, 1964), p. 31.

Steel, W. H.

W. H. Steel, Rev. Opt. 32, 4 (1953).

Steward, G. C.

G. C. Steward, Phil. Trans. Roy. Soc. (London) A225, 131 (1925).

Taylor, C. A.

Thompson, B. J.

Toraldo di Francia, G.

G. Toraldo di Francia, Nuovo Cimento 9, Suppl. No. 3, 426 (1952).
[Crossref]

Tsujiuchi, J.

J. Tsujiuchi, in Progress in Optics, II, edited by E. Wolf (North-Holland, Amsterdam, 1963), p. 133.

Welford, W. T.

Wilkins, J. E.

Wissler, F. C.

Wolf, E.

E. H. Linfoot and E. Wolf, Proc. Phys. Soc. (London) B66, 145 (1953).

J. Opt. Soc. Am. (6)

Jap. J. Appl. Phys. (1)

T. Asakura and H. Misjina, Jap. J. Appl. Phys. 9, 195 (1970).
[Crossref]

Monthly Notes Roy. Astron. Soc. (1)

Lord Rayleigh, Monthly Notes Roy. Astron. Soc. 33, 59 (1872).

Nuovo Cimento (1)

G. Toraldo di Francia, Nuovo Cimento 9, Suppl. No. 3, 426 (1952).
[Crossref]

Phil. Trans. Roy. Soc. (London) (1)

G. C. Steward, Phil. Trans. Roy. Soc. (London) A225, 131 (1925).

Proc. Phys. Soc. (London) (1)

E. H. Linfoot and E. Wolf, Proc. Phys. Soc. (London) B66, 145 (1953).

Rev. Opt. (1)

W. H. Steel, Rev. Opt. 32, 4 (1953).

Other (3)

P. Jacquinot and B. Roizen-Dossier, in Progress in Optics, III, edited by E. Wolf (North-Holland, Amsterdam, 1964), p. 31.

E. H. Linfoot, Recent Advances in Optics (Oxford University Press, London, 1958), pp. 35 ff.

J. Tsujiuchi, in Progress in Optics, II, edited by E. Wolf (North-Holland, Amsterdam, 1963), p. 133.

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Figures (7)

F. 1
F. 1

A phase-zoned lens.

F. 2
F. 2

Relative irradiance distributions in focal plane. A: A CPPL with = 0.4; qm = 2.8662. B: A phase-zoned lens with 1 = 0.56569, 2 = 0.5, 3 = 0.3; qm = 3.0; zones have phase differences of π with respect to their neighboring zones. C: Perfect lens; qm = 3.8317.

F. 3
F. 3

A CPPL. Curve (i): radius of central bright spot (qm) in focal plane as a function of . Curve (ii): relative irradiance (I) at focus as a function of .

F. 4
F. 4

Relative irradiance distribution in focal plane of CPPL. ——, = 0.5702; the maximum in first diffraction ring is equal to that at focus. – – –, = 0.7071.

F. 5
F. 5

Comparison of relative irradiance distributions in focal planes. A: A CPPL with = 0.4; B: a center-blocked lens with = 0.56569; C: a center-blocked lens with = 0.4; D: a perfect lens.

F. 6
F. 6

Relative irradiance distributions along the axis. A: A CPPL with = 0.4; B: a center-blocked lens with = 0.56569; C:a center-blocked lens with = 0.4.

F. 7
F. 7

Axial relative irradiance distributions of CPPL. A: = 0.43; B: = 0.5702; C: = 0.7071.

Equations (18)

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U ( p , q ) = ( i π a 2 / λ f ) exp [ i k ( f R ) ] [ C ( p , q ) + i S ( p , q ) ] ,
C ( p , q ) = cos ( p / 2 ) p / 2 U 1 ( p , q ) + sin ( p / 2 ) p / 2 U 2 ( p , q ) , S ( p , q ) = sin ( p / 2 ) p / 2 U 1 ( p , q ) cos ( p / 2 ) p / 2 U 2 ( p , q ) .
U n ( p , q ) = m = 0 ( 1 ) m ( p , q ) n + 2 m J n + 2 m ( q ) .
p = k a 2 z / f 2 ; q = k a ρ / f ;
U ( 0 , q ) = ( i π a 2 / λ f ) exp [ i k ( f R ) ] 2 J 1 ( q ) / q .
U ( p , 0 ) = i π a 2 λ f exp [ i k ( f R ) ] exp ( i p / 2 ) p / 2 × { sin ( p / 2 ) i [ 1 cos ( p / 2 ) ] } .
U 0 ( 0 , q ) = C [ 2 J 1 ( q ) q 1 2 2 J 1 ( 1 q ) 1 q ] ,
U 1 ( 0 , q ) = C [ 1 2 2 J 1 ( 1 q ) 1 q 2 2 2 J 1 ( 2 q ) 2 q ] exp ( i θ 1 ) ,
U 2 ( 0 , q ) = C 2 2 2 J 1 ( 2 q ) 2 q exp ( i θ 2 ) ,
C = ( i π a 2 / λ f ) exp [ i k ( f R ) ] .
U ( 0 , q ) = C { A 0 A 1 [ 1 exp ( i θ 1 ) ] A 2 [ exp ( i θ 1 ) exp ( i θ 2 ) ] } ,
A 0 = 2 J 1 ( q ) / q , A 1 = 1 2 2 J 1 ( 1 q ) / 1 q , A 2 = 2 2 2 J 1 ( 2 q ) / 2 q .
θ 1 = π ,
θ 2 = θ 1 .
I ( 0 , q ) = U ( 0 , q ) × U * ( 0 , q ) = C [ 2 J 1 ( q ) q 2 2 2 J 1 ( q ) q ] 2 ,
I ( p , 0 ) = C { 2 ( 1 2 ) 2 [ sin 1 4 p ( 1 2 ) 1 4 p ( 1 2 ) ] 2 ( sin 1 4 p 1 4 p ) 2 + 2 4 ( sin 1 4 2 p 1 4 2 p ) 2 } .
I B ( 0 , q ) = C { [ 2 J 1 ( q ) / q ] E 2 [ 2 J 1 ( E q ) 2 / E q ] } 2 .
I B ( 0 , q ) = C { [ 2 J 1 ( q ) / q ] 2 2 [ 2 J 1 ( 2 q ) 2 / 2 q ] } ,