Abstract

Expressions are obtained for the three-dimensional intensity distribution in the error-free diffraction patterns of semitransparent and phase annuli. The results are discussed and illustrated graphically. Annular apertures have been used for improving the resolving power and focal depth of optical instruments; the most useful situation for a given central-obstruction ratio is where the central obstruction is completely opaque. However, by use of a phase annulus of the same dimension, with a phase step of π for the central region, some further improvement in the resolving power and focal depth is obtained for central-obstruction ratios up to 0.5. This improvement is illustrated experimentally.

© 1965 Optical Society of America

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References

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  1. Rayleigh, Monthly Not. Roy. Astron. Soc. 33, 59 (1872) [reprinted in Sci. Papers 1, 163 (1964)].
  2. G. C. Steward, Phil Trans. Roy. Soc. (London) A225, 131 (1925).
  3. E. H. Linfoot and E. Wolf, Proc. Phys. Soc. (London) B66, 145 (1953).
  4. W. H. Steel, Rev. Opt. 32, 4 (1953).
  5. C. A. Taylor and B. J. Thompson, J. Opt. Soc. Am. 48, 844 (1958).
    [Crossref]
  6. W. T. Welford, J. Opt. Soc. Am. 50, 749 (1960).
    [Crossref]
  7. G. Lansraux, Rev. Opt. 32, 475 (1953).
  8. B. Dossier, Rev. Opt. 33, 57, 147, 267 (1954).
  9. G. Toraldo di Francia, Nuovo Cimento Suppl. 9, 426 (1952).
    [Crossref]
  10. E. L. Bouché and P. F. Kellen, J. Opt. Soc. Am. 53, 1350A (1963).
  11. E. Von Lommel, Abhandl. Bayer. Akad. 53, 233 (1885).
  12. E. N. Dekanosidze, Tables of Lommel’s Functions of Two Variables (Pergamon Press, Inc., New York, 1960).
  13. A. Boivin, J. Opt. Soc. Am. 42, 60 (1952).
    [Crossref]

1963 (1)

E. L. Bouché and P. F. Kellen, J. Opt. Soc. Am. 53, 1350A (1963).

1960 (1)

1958 (1)

1954 (1)

B. Dossier, Rev. Opt. 33, 57, 147, 267 (1954).

1953 (3)

G. Lansraux, Rev. Opt. 32, 475 (1953).

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

W. H. Steel, Rev. Opt. 32, 4 (1953).

1952 (2)

G. Toraldo di Francia, Nuovo Cimento Suppl. 9, 426 (1952).
[Crossref]

A. Boivin, J. Opt. Soc. Am. 42, 60 (1952).
[Crossref]

1925 (1)

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

1885 (1)

E. Von Lommel, Abhandl. Bayer. Akad. 53, 233 (1885).

1872 (1)

Rayleigh, Monthly Not. Roy. Astron. Soc. 33, 59 (1872) [reprinted in Sci. Papers 1, 163 (1964)].

Boivin, A.

Bouché, E. L.

E. L. Bouché and P. F. Kellen, J. Opt. Soc. Am. 53, 1350A (1963).

Dekanosidze, E. N.

E. N. Dekanosidze, Tables of Lommel’s Functions of Two Variables (Pergamon Press, Inc., New York, 1960).

Dossier, B.

B. Dossier, Rev. Opt. 33, 57, 147, 267 (1954).

Kellen, P. F.

E. L. Bouché and P. F. Kellen, J. Opt. Soc. Am. 53, 1350A (1963).

Lansraux, G.

G. Lansraux, Rev. Opt. 32, 475 (1953).

Linfoot, E. H.

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

Rayleigh,

Rayleigh, Monthly Not. Roy. Astron. Soc. 33, 59 (1872) [reprinted in Sci. Papers 1, 163 (1964)].

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 Suppl. 9, 426 (1952).
[Crossref]

Von Lommel, E.

E. Von Lommel, Abhandl. Bayer. Akad. 53, 233 (1885).

Welford, W. T.

Wolf, E.

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

Abhandl. Bayer. Akad. (1)

E. Von Lommel, Abhandl. Bayer. Akad. 53, 233 (1885).

J. Opt. Soc. Am. (4)

Monthly Not. Roy. Astron. Soc. (1)

Rayleigh, Monthly Not. Roy. Astron. Soc. 33, 59 (1872) [reprinted in Sci. Papers 1, 163 (1964)].

Nuovo Cimento Suppl. (1)

G. Toraldo di Francia, Nuovo Cimento Suppl. 9, 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. (3)

W. H. Steel, Rev. Opt. 32, 4 (1953).

G. Lansraux, Rev. Opt. 32, 475 (1953).

B. Dossier, Rev. Opt. 33, 57, 147, 267 (1954).

Other (1)

E. N. Dekanosidze, Tables of Lommel’s Functions of Two Variables (Pergamon Press, Inc., New York, 1960).

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Figures (10)

Fig. 1
Fig. 1

Geometrical arrangement.

Fig. 2
Fig. 2

Amplitude distribution in the Fraunhofer diffraction patterns of an annular aperture for various central obstruction ratios and t=1, t′=0, =0 (— - —), =0.25 (— — —), =0.5 (- - - - - -), =0.75 (——).

Fig. 3
Fig. 3

Amplitude distribution in the Fraunhofer diffraction patterns for a fixed annulus =0.5 and various transmittance ratios. t=1, t′=0 (— – —), t′=0.25 (——), t′=0.75 (— — —); t′=1, t=0 (— - - - —), t=0.25 (– – – –), t=0.75 (— - —).

Fig. 4
Fig. 4

Variation of intensity along the optical axis for various central obstruction ratios. =0 (— - - - —), =0.5 (——),=0.75 (— — —).

Fig. 5
Fig. 5

Variation of intensity along the optical axis for a fixed annulus of =0.5 and various transmittance ratios. t=1, t′=0 (— — —) t′=0.5 (——); t′=1, t=0.5 (– – – –) t=0 (— - - - —).

Fig. 6
Fig. 6

Amplitude distribution in the Fraunhofer diffraction patterns of a phase annulus with t=t′=1 and ϕ=π. ∊=0 (— - —), =0.25 (——), =0.5 (- - - - -), =0.75 (— - - - —), ϕ=0, =0 (— — —)

Fig. 7
Fig. 7

Photographic record of Fraunhofer diffraction patterns of a phase annulus for various values of ∊; t=t′=1 and ϕ=π.

Fig. 8
Fig. 8

Variation of intensity along the optical axis for various values of =0 (— - - - —), =0.25 (— — —), =0.5 (- - - - - -), =0.75 (——); t=t′=0, ϕ=π.

Fig. 9
Fig. 9

Comparison between the amplitude distributions in the Fraunhofer diffraction patterns of (— - —) an unobstructed aperture, (- - - -) an obstructed aperture with t′=0, and (——) a purely phase annulus with ϕ=π and =0.25.

Fig. 10
Fig. 10

Comparison between the amplitude distributions in the Fraunhofer diffraction patterns of (— - —) an unobstructed aperture, (- - - -) an obstructed aperture with t′=0, and (——) a purely phase annulus with ϕ=π and =0.5.

Equations (14)

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[ A ( Q ) ] R = T [ i k R 2 f e i k ( f - O Q ) 0 1 e 1 2 i u ρ 2 J 0 ( v ρ ) ρ d ρ ] , u = k R 2 z / f 2 ,             v = k R r / f ,             r = ( x 2 + y 2 ) 1 2 ,             k = 2 π / λ , }
[ A ( Q ) ] R = ( T - T ) [ i k R 2 f e i k ( f - O Q ) × 0 1 e 1 2 i u ρ 2 J 0 ( v ρ ) ρ d ρ ] , u = k R 2 z / f 2 = 2 u ,             v = k R r / f = v ,             = R / R . }
A ( Q ) = [ A ( Q ) ] R + [ A ( Q ) ] R .             A ( Q ) = i k R 2 f e i k ( f - O Q ) [ T 0 1 e 1 2 i u ρ 2 J 0 ( v ρ ) ρ d ρ + ( T - T ) 2 0 1 e 1 2 i u ρ 2 J 0 ( v ρ ) ρ d ρ ] .
0 1 e 1 2 i u ρ 2 J 0 ( v ρ ) ρ d ρ = 1 u e 1 2 i u [ U 1 ( u , v ) - i U 2 ( u , v ) ] , 0 1 e 1 2 i u ρ 2 J 0 ( v ρ ) ρ d ρ = 1 u e 1 2 i u [ U 1 ( u , v ) - i U 2 ( u , v ) ] , }
U n ( u , v ) = s = 0 ( - 1 ) s ( u v ) n + 2 s J n + 2 s ( v ) .
A ( Q ) = ( i k R 2 / f ) e i k ( f - O Q ) { ( T / u ) e 1 2 i u [ U 1 ( u , v ) - i U 2 ( u , v ) ] + [ ( T - T ) / u ] e 1 2 i 2 u [ U 1 ( u , v ) - i U 2 ( u , v ) ] } .
I ( Q ) = k 2 R 4 / f 2 u 2 T e 1 2 i u ( U 1 - i U 2 ) + ( T - T ) e 1 2 i 2 u ( U 1 - i U 2 ) 2 .
I ( Q ) = k 2 R 4 / f 2 u 2 ( t 2 [ U 1 2 + U 2 2 ] + ( t - t ) 2 [ U 1 2 + U 2 2 ] + 2 t ( t - t ) { [ U 1 U 1 + U 2 U 2 ] cos [ 1 2 u ( 1 - 2 ) ] + [ U 2 U 1 - U 1 U 2 ] sin [ 1 2 u ( 1 - 2 ) ] } ) .
lim u 0 U 1 ( u , v ) / u = J 1 ( v ) / v , lim u 0 U 2 ( u , v ) / u = 0 ,
I ( 0 , r ) = π 2 R 4 / λ 2 f 2 { t [ 2 J 1 ( v ) / v ] + ( t - t ) 2 [ 2 J 1 ( v ) / v ] } 2 .
U 1 ( u , 0 ) = sin 1 2 u , U 2 ( u , 0 ) = 1 - cos 1 2 u ,
I ( z , 0 ) = π 2 R 4 f 2 λ 2 ( t ( t - t ) ( 1 - 2 ) 2 { sin [ 1 4 u ( 1 - 2 ) ] 1 4 u ( 1 - 2 ) } 2 + t t ( sin 1 4 u 1 4 u ) 2 + 4 t ( t - t ) ( sin 1 4 2 u 1 4 2 u ) 2 ) .
I ( 0 , r ) = π 2 R 4 λ 2 f 2 [ 2 J 1 ( v ) v - 2 2 2 J 1 ( v ) v ] 2 ,
I ( z , 0 ) = π 2 R 4 f 2 λ 2 { 2 ( 1 - 2 ) 2 [ sin 1 4 u ( 1 - 2 ) 1 4 u ( 1 - 2 ) ] 2 - ( sin 1 4 u 1 4 u ) + 2 4 ( sin 1 4 2 u 1 4 2 u ) 2 } .