Abstract

We find a pupil function and the corresponding diffraction pattern, formed by a point-source object on the axis of a rotationally symmetric optical system, that has the maximum possible central irradiance in the class of all pupil functions that have the same Rayleigh limit of resolution, specified in advance at a value between 100 and 134% of that for the classical Airy-type objective. Between 134 and 183% there is no maximum, but we do find the least upper bound.

© 1983 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. E. Wilkins, “The resolving power of a coated objective. II,” J. Opt. Soc. Am. 40, 222–224 (1950).
    [Crossref]
  2. G. Lansraux and G. Boivin, “Maximum of the factor of encircled energy,” Can. J. Phys. 39, 158–188 (1961).
    [Crossref]
  3. J. E. Wilkins, “Apodization for maximum Strehl ratio and specified Rayleigh limit of resolution,” J. Opt. Soc. Am. 67, 1027–1030 (1977).
    [Crossref]
  4. J. E. Wilkins, “Apodization for maximum Strehl ratio and specified Rayleigh limit of resolution. II,” J. Opt. Soc. Am. 69, 1526–1530 (1979).
    [Crossref]
  5. J. E. Wilkins, “Luneburg apodization problems,” J. Opt. Soc. Am. 53, 420–424 (1963).
    [Crossref]
  6. R. Barakat, “Solution of the Luneburg apodization problems,” J. Opt. Soc. Am. 52, 264–275 (1962).
    [Crossref]
  7. V. N. Mahajan, “Luneburg apodization problem. I,” Opt. Lett. 5, 267–269 (1980).
    [Crossref] [PubMed]
  8. R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1964), pp. 339–359.
  9. H. Osterberg and J. E. Wilkins, “The resolving power of a coated objective,” J. Opt. Soc. Am. 39, 553–557 (1949).
    [Crossref]

1980 (1)

1979 (1)

1977 (1)

1963 (1)

1962 (1)

1961 (1)

G. Lansraux and G. Boivin, “Maximum of the factor of encircled energy,” Can. J. Phys. 39, 158–188 (1961).
[Crossref]

1950 (1)

1949 (1)

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1
Fig. 1

Graphs of the normalized central irradiance Q[P0] that maximizes Q[P] in the class of complex-valued pupil functions satisfying conditions (2) and (3), and of the locations ρ1 and ρ2 of the discontinuities in the maximizing pupil function P0(ρ). Note that the graph of Q[P0] has a minimum when β = β* = 5.1464.

Fig. 2
Fig. 2

Graph of the Bessel function J0(u). The minimum value of the function is −0.40276 and occurs at j11 = 3.8317. The first and second positive zeros occur at j01 = 2.4048 and j02 = 5.5201. The maximum value is 1.0 and occurs at u = 0. A relative maximum value of 0.30012 occurs at j12 = 7.0156.

Tables (1)

Tables Icon

Table 1 Locations ρ1 and ρ2 of Discontinuities of P0(ρ) and the Corresponding Normalized Central Irradiance Q[P0] that Maximizes Q[P] in the Class of Complex-Valued Pupil Functions Satisfying Conditions (2) and (3)

Equations (32)

Equations on this page are rendered with MathJax. Learn more.

U ( α ) = 0 1 P ( ρ ) J 0 ( α ρ ) ρ d ρ .
| P ( ρ ) | 1 .
0 1 P ( ρ ) J 0 ( β ρ ) ρ d ρ = 0 .
Q [ P ] = | 2 0 1 P ( ρ ) ρ d ρ | 2
| f ( x ) | 1 ,
0 1 f ( x ) g ( x ) d x = 0 ,
| 0 1 f ( x ) d x | = maximum .
H ( y ) = 2 E y g ( x ) d x 0 1 g ( x ) d x = 0 1 f y ( x ) g ( x ) d x .
H ( ) = H ( + ) = 0 1 g ( x ) d x = 2 J 1 ( β ) / β < 0
J 0 ( u 1 ) = J 0 ( u 2 ) = y .
2 u 1 J 1 ( u 1 ) + β J 1 ( β ) 2 u 2 J 1 ( u 2 ) = 0 .
P 0 ( ρ ) = + 1 if 0 ρ ρ 1 and ρ 2 ρ 1 , P 0 ( ρ ) = 1 if ρ 1 < ρ < ρ 2 ,
U 0 ( α ) = α 1 [ 2 ρ 1 J 1 ( α ρ 1 ) 2 ρ 2 J 1 ( α ρ 2 ) J 1 ( α ) ] ,
Q [ P 0 ] = ( 2 ρ 1 2 2 ρ 2 2 + 1 ) 2 Q ( β ) .
K ( β ) = 2 ( u 1 2 u 2 2 ) + β 2 .
K ( β ) K ( j 11 ) = J 11 2 > 0
L ( β ) = 2 ( u 1 2 u 2 2 ) y + β 2 J 0 ( β ) .
L ( β ) = 2 u 1 2 J 0 ( u 1 ) 2 u 2 2 J 0 ( u 2 ) + β 2 J 0 ( β ) .
L ( β ) = β { 2 J 0 ( β ) J 1 ( u 1 ) J 1 ( u 2 ) ( u 1 2 u 2 2 ) + β J 1 ( β ) Δ } / Δ .
M > J 0 ( β ) J 1 ( u 1 ) J 1 ( u 2 ) { j 11 2 β 2 + 2 β J 1 ( β ) [ ψ ( u 2 ) ψ ( u 1 ) ] } ,
2 y ( u 1 2 u 2 2 ) + β 2 J 0 ( β ) = 0
υ ( ρ ) = ρ 1 P 0 ( s ) s d s .
υ ( ρ ) = ( 1 ρ 2 ) / 2 if ρ 2 ρ 1 , υ ( ρ ) = ( 1 2 ρ 2 2 + ρ 2 ) / 2 if ρ 1 ρ ρ 2 , υ ( ρ ) = ( 1 2 ρ 2 2 + 2 ρ 1 2 ρ 2 ) / 2 if 0 ρ ρ 1 .
U 0 ( α ) = U 0 ( 0 ) α 0 1 υ ( ρ ) J 1 ( α ρ ) ρ d ρ .
D U 0 = 0 1 υ ( ρ ) J 0 ( α ρ ) ρ d ρ = α 1 0 1 P 0 ( ρ ) J 1 ( α ρ ) ρ 2 d ρ , D 2 U 0 = α 1 0 1 υ ( ρ ) J 1 ( α ρ ) ρ 2 d ρ = α 2 0 1 P 0 ( ρ ) J 2 ( α ρ ) ρ 3 d ρ , D 3 U 0 = α 2 0 1 υ ( ρ ) J 2 ( α ρ ) ρ 2 d ρ , = α 3 0 1 P 0 ( ρ ) J 3 ( α ρ ) ρ 4 d ρ , D 4 U 0 = α 3 0 1 υ ( ρ ) J 3 ( α ρ ) ρ 3 d ρ .
β 3 0 1 P 0 ( ρ ) J 3 ( β ρ ) ρ 4 d ρ = β 8 S 4 ( β )
S k ( β ) = 2 u 1 k J k ( u 1 ) 2 u 2 k J k ( u 2 ) + β k J k ( β ) .
S 4 ( β ) = 2 β J 0 ( β ) Δ 1 { J 1 ( u 2 ) J 0 ( u 1 ) u 1 [ ψ 1 ( u 1 ) ψ 1 ( β ) ] J 1 ( u 1 ) J 0 ( u 1 ) u 2 [ ψ 1 ( u 2 ) ψ 1 ( β ) ] } ,
d ψ 1 / d u = u 3 [ J 0 ( u ) J 2 ( u ) + J 1 ( u ) J 3 ( u ) ] / J 0 2 ( u ) = u { u J 0 ( u ) + J 1 ( u ) ] 2 + ( u 2 9 ) J 1 2 ( u ) } / J 0 2 ( u ) ,
β 2 0 1 P 0 ( ρ ) J 2 ( β ρ ) ρ 3 d ρ = β 6 S 3 ( β ) ,
J 3 ( u ) = ( 8 u 2 1 ) J 1 ( u ) 4 u 1 J 0 ( u ) ,
U 0 ( α ) = 0 1 P 1 ( ρ ) J 0 ( α ρ ) ρ d ρ = ρ * J 1 ( α ρ * ) / α ,