Abstract

A method for determining the phase-retardation function of the uniform-intensity axilens is discussed and compared with that of an earlier publication [Opt. Lett. 16, 523 (1991)]. Within the presented formulation good agreement is achieved between the geometrical-optics prediction and the numerically evaluated diffraction integral.

© 1992 Optical Society of America

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References

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  1. L. M. Soroko, in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1989), pp. 109–160, and references therein.
    [CrossRef]
  2. N. Davidson, A. A. Friesem, E. Hasman, Opt. Lett. 16, 523 (1991).
    [CrossRef] [PubMed]
  3. See, e.g., L. B. Felsen, J. Opt. Soc. Am. A 3, 486 (1986).
    [CrossRef]
  4. V. P. Koronkevitch, V. P. Korolkov, A. G. Poleschuk, I. G. Palchikova, Y. I. Yurkov, I. A. Mikhaltsova, E. G. Churin, A. P. Sokolov, Preprint IAE SB, USSR Academy of Sciences No. 421 (USSR Academy of Sciences, Moscow, 1989), p. 15.
  5. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 13 and 63.

1991 (1)

1986 (1)

Churin, E. G.

V. P. Koronkevitch, V. P. Korolkov, A. G. Poleschuk, I. G. Palchikova, Y. I. Yurkov, I. A. Mikhaltsova, E. G. Churin, A. P. Sokolov, Preprint IAE SB, USSR Academy of Sciences No. 421 (USSR Academy of Sciences, Moscow, 1989), p. 15.

Davidson, N.

Felsen, L. B.

Friesem, A. A.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 13 and 63.

Hasman, E.

Korolkov, V. P.

V. P. Koronkevitch, V. P. Korolkov, A. G. Poleschuk, I. G. Palchikova, Y. I. Yurkov, I. A. Mikhaltsova, E. G. Churin, A. P. Sokolov, Preprint IAE SB, USSR Academy of Sciences No. 421 (USSR Academy of Sciences, Moscow, 1989), p. 15.

Koronkevitch, V. P.

V. P. Koronkevitch, V. P. Korolkov, A. G. Poleschuk, I. G. Palchikova, Y. I. Yurkov, I. A. Mikhaltsova, E. G. Churin, A. P. Sokolov, Preprint IAE SB, USSR Academy of Sciences No. 421 (USSR Academy of Sciences, Moscow, 1989), p. 15.

Mikhaltsova, I. A.

V. P. Koronkevitch, V. P. Korolkov, A. G. Poleschuk, I. G. Palchikova, Y. I. Yurkov, I. A. Mikhaltsova, E. G. Churin, A. P. Sokolov, Preprint IAE SB, USSR Academy of Sciences No. 421 (USSR Academy of Sciences, Moscow, 1989), p. 15.

Palchikova, I. G.

V. P. Koronkevitch, V. P. Korolkov, A. G. Poleschuk, I. G. Palchikova, Y. I. Yurkov, I. A. Mikhaltsova, E. G. Churin, A. P. Sokolov, Preprint IAE SB, USSR Academy of Sciences No. 421 (USSR Academy of Sciences, Moscow, 1989), p. 15.

Poleschuk, A. G.

V. P. Koronkevitch, V. P. Korolkov, A. G. Poleschuk, I. G. Palchikova, Y. I. Yurkov, I. A. Mikhaltsova, E. G. Churin, A. P. Sokolov, Preprint IAE SB, USSR Academy of Sciences No. 421 (USSR Academy of Sciences, Moscow, 1989), p. 15.

Sokolov, A. P.

V. P. Koronkevitch, V. P. Korolkov, A. G. Poleschuk, I. G. Palchikova, Y. I. Yurkov, I. A. Mikhaltsova, E. G. Churin, A. P. Sokolov, Preprint IAE SB, USSR Academy of Sciences No. 421 (USSR Academy of Sciences, Moscow, 1989), p. 15.

Soroko, L. M.

L. M. Soroko, in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1989), pp. 109–160, and references therein.
[CrossRef]

Yurkov, Y. I.

V. P. Koronkevitch, V. P. Korolkov, A. G. Poleschuk, I. G. Palchikova, Y. I. Yurkov, I. A. Mikhaltsova, E. G. Churin, A. P. Sokolov, Preprint IAE SB, USSR Academy of Sciences No. 421 (USSR Academy of Sciences, Moscow, 1989), p. 15.

J. Opt. Soc. Am. A (1)

Opt. Lett. (1)

Other (3)

L. M. Soroko, in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1989), pp. 109–160, and references therein.
[CrossRef]

V. P. Koronkevitch, V. P. Korolkov, A. G. Poleschuk, I. G. Palchikova, Y. I. Yurkov, I. A. Mikhaltsova, E. G. Churin, A. P. Sokolov, Preprint IAE SB, USSR Academy of Sciences No. 421 (USSR Academy of Sciences, Moscow, 1989), p. 15.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 13 and 63.

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Figures (4)

Fig. 1
Fig. 1

Geometrical illustration of Eq. (5). The light rays are assumed to be perpendicular to the wave fronts.

Fig. 2
Fig. 2

On-axis intensity distribution resulting from Eq. (15) with the phase-retardation function ϕ(r) defined by Eq. (8) (solid curve) and by relation (4) (dashed curve). The values of the parameters are f0 = 1220 mm, δzg = 29 mm, R = 12.5 mm, and λ = 633 nm.

Fig. 3
Fig. 3

On-axis intensity distribution resulting from Eq. (15) with the phase-retardation function ϕ(r) defined by Eq. (8) (solid curve) and by relation (4) (dashed curve). The values of the parameters are f0 = 1220 mm, δzg = 60 mm, R = 12.5 mm, and λ = 633 nm.

Fig. 4
Fig. 4

On-axis intensity distribution resulting from Eq. (15) with the phase-retardation function ϕ(r) defined by Eq. (8) (solid curve) and by relation (4) (dashed curve). The values of the parameters are f0 = δzg = 140 mm, R = 4.5 mm, and λ = 633 nm.

Equations (16)

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2 π P σ r d r = P z ( z ) d z , 0 r R ,
f ( r ) = f 0 + a r 2 , 0 r R ,
a = π P σ P z = δ z g R 2 .
ϕ ˜ ( r ) 2 π λ r 2 2 f ( r ) = 2 π λ r 2 2 ( f 0 + δ z g r 2 / R 2 ) , 0 r R ,
d φ ( r ) d r = sin θ = r [ r 2 + f 2 ( r ) ] 1 / 2 ,
φ ( r ) λ 2 π ϕ ( r ) ,
d φ ( r ) d r tan θ = r f ( r ) = r f 0 + ( δ z g / R 2 ) r 2 .
ϕ ( r ) = 2 π λ R 2 2 δ z g ln ( f 0 + δ z g R 2 r 2 ) , 0 r R ,
tan θ = r f ˜ ( r ) = λ 2 π d ϕ ˜ ( r ) d r = d d r [ r 2 2 ( f 0 + δ z g R 2 r 2 ) ] = f 0 r ( f 0 + δ z g R 2 r 2 ) 2 .
r = R [ ( f 0 f ˜ ) 1 / 2 f 0 δ z g ] 1 / 2 ,
P f ˜ ( f ˜ ) = 2 π P σ r ( d r d f ) = π R 2 P σ 2 δ z g ( f 0 f ˜ ) 1 / 2 ,
2 π P σ 0 R r d r = π R 2 P σ 2 δ z g ( f 0 ) 1 / 2 f 0 f 0 + Δ d f ˜ ( f ˜ ) 1 / 2 ,
π R 2 P σ = π R 2 P σ { [ f 0 ( f 0 + Δ ) ] 1 / 2 f 0 δ z g } .
Δ = 2 ( δ z g ) + ( δ z g ) 2 f 0 ,
I ( z , 0 ) = ( 2 π λ z ) 2 | 0 R exp { i [ π r 2 / ( λ z ) ϕ ( r ) ] } r d r | 2 ,
δ z g f 0 ,

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