Abstract

The total illumination (or encircled energy) in a diffraction image containing spherical aberration is studied for various amounts of third- and fifth-order aberration. The Zernike polynomials are used to represent spherical aberration. The evaluation of the total illumination was done numerically, using double Gauss quadrature and 49 quadrature points. Contour maps of the total illumination were constructed and compared with the aberration-free case examined previously by E. Wolf.

© 1961 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics (Pergamon Press, London, 1959), Chap. 9, p. 458. (This chapter contains an extensive survey of the field.)
  2. Rayleigh, Phil. Mag. 11, 214 (1881).
  3. E. Wolf, Proc. Roy. Soc. (London) A204, 533 (1951).
  4. See work cited in footnote 1, p. 463.
  5. G. N. Watson, Treatise on the Theory of Bessel Functions (Cambridge University Press, New York, 1944), p. 134.
  6. J. O. Irwin, On Quadrature and Cubature (Cambridge University Press, Cambridge, England, 1923), p. 21.
  7. G. Lansraux and G. Boivin, Can. J. Phys. 36, 1696 (1958).
    [Crossref]
  8. L. C. Martin, Trans. Opt. Soc. 23, 63 (1921).
    [Crossref]
  9. J. Focke, Optica Acta 3, 161 (1956).
    [Crossref]

1958 (1)

G. Lansraux and G. Boivin, Can. J. Phys. 36, 1696 (1958).
[Crossref]

1956 (1)

J. Focke, Optica Acta 3, 161 (1956).
[Crossref]

1951 (1)

E. Wolf, Proc. Roy. Soc. (London) A204, 533 (1951).

1921 (1)

L. C. Martin, Trans. Opt. Soc. 23, 63 (1921).
[Crossref]

1881 (1)

Rayleigh, Phil. Mag. 11, 214 (1881).

Boivin, G.

G. Lansraux and G. Boivin, Can. J. Phys. 36, 1696 (1958).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, London, 1959), Chap. 9, p. 458. (This chapter contains an extensive survey of the field.)

Focke, J.

J. Focke, Optica Acta 3, 161 (1956).
[Crossref]

Irwin, J. O.

J. O. Irwin, On Quadrature and Cubature (Cambridge University Press, Cambridge, England, 1923), p. 21.

Lansraux, G.

G. Lansraux and G. Boivin, Can. J. Phys. 36, 1696 (1958).
[Crossref]

Martin, L. C.

L. C. Martin, Trans. Opt. Soc. 23, 63 (1921).
[Crossref]

Rayleigh,

Rayleigh, Phil. Mag. 11, 214 (1881).

Watson, G. N.

G. N. Watson, Treatise on the Theory of Bessel Functions (Cambridge University Press, New York, 1944), p. 134.

Wolf, E.

E. Wolf, Proc. Roy. Soc. (London) A204, 533 (1951).

M. Born and E. Wolf, Principles of Optics (Pergamon Press, London, 1959), Chap. 9, p. 458. (This chapter contains an extensive survey of the field.)

Can. J. Phys. (1)

G. Lansraux and G. Boivin, Can. J. Phys. 36, 1696 (1958).
[Crossref]

Optica Acta (1)

J. Focke, Optica Acta 3, 161 (1956).
[Crossref]

Phil. Mag. (1)

Rayleigh, Phil. Mag. 11, 214 (1881).

Proc. Roy. Soc. (London) (1)

E. Wolf, Proc. Roy. Soc. (London) A204, 533 (1951).

Trans. Opt. Soc. (1)

L. C. Martin, Trans. Opt. Soc. 23, 63 (1921).
[Crossref]

Other (4)

M. Born and E. Wolf, Principles of Optics (Pergamon Press, London, 1959), Chap. 9, p. 458. (This chapter contains an extensive survey of the field.)

See work cited in footnote 1, p. 463.

G. N. Watson, Treatise on the Theory of Bessel Functions (Cambridge University Press, New York, 1944), p. 134.

J. O. Irwin, On Quadrature and Cubature (Cambridge University Press, Cambridge, England, 1923), p. 21.

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Figures (10)

Fig. 1
Fig. 1

Total illustration (encircled energy) for various amounts of third-order spherical aberration β40R40(r) in the central plane (p=0).

Fig. 2
Fig. 2

Total illumination for various amounts of fifth-order spherical aberration β60R60(r) in the central plane.

Fig. 3
Fig. 3

Comparison of the encircled energy for β 40 = 1 2 in various receiving planes.

Fig. 4
Fig. 4

Comparison of the encircled energy for β40=1 in various receiving planes.

Fig. 5
Fig. 5

Contour lines of total illumination for β 40 = 1 2. The points marked P and M are the paraxial and marginal receiving planes.

Fig. 6
Fig. 6

Contour lines of total illumination for β40=1. The points marked P and M are the paraxial and marginal receiving planes.

Fig. 7
Fig. 7

Contour lines of total illumination for β40=2.

Fig. 8
Fig. 8

Contour lines of total illumination for β60=1.

Fig. 9
Fig. 9

Fraction of total illumination in the geometric shadow. Comparison of aberration-free case with β40=7 for the paraxial and marginal sides.

Fig. 10
Fig. 10

Fraction of total illumination in the geometric shadow. Comparison of aberration-free case with β40=1 for the paraxial and marginal sides.

Equations (24)

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L ( p , q , β ) = N q I ( p , q , β ) q d q ,
I ( p , q , β ) = 4 | 0 1 exp [ i ( p / 2 ) r 2 - i V ( β , r ) ] J 0 ( q r ) r d r | 2 .
0 I ( p , q , β ) q d q = 4 0 1 exp i ( p / 2 ) r 2 - i V ( β , r ) 2 r d r = 2 .
L ( p , q , β ) = 1 2 0 q I ( p , q , β ) q d q .
L ( 0 , q , 0 ) = 1 2 0 q J 1 2 ( q ) q - 1 d q = 1 - J 0 2 ( q ) - J 1 2 ( q ) .
V ( β , r ) = a r 2 + b r 4 + ,
V ( β , r ) = β 40 R 4 0 ( r ) + β 60 R 6 0 ( r ) + ,
R 4 0 ( r ) = 6 r 4 - 6 r 2 + 1 R 6 0 ( r ) = 20 r 6 - 30 r 4 + 12 r 2 - 1.
δ l = 4 β 40 / ( N A ) 3 ,
P l = 8 π β 40 .
I ( p , q , β ) = 0 1 0 1 exp i [ θ ( x ) - θ ( y ) ] × J 0 ( q x ) J 0 ( q y ) x y d x d y ,
θ ( x ) = p 2 x 2 - V ( β , x ) .
I ( p , q , β ) = 0 1 0 1 cos [ θ ( x ) - θ ( y ) ] J 0 ( q x ) J 0 ( q y ) x y d x d y .
L ( p , q , β ) = 1 2 0 1 0 1 cos [ θ ( x ) - θ ( y ) ] × { 0 q J 0 ( q x ) J 0 ( q y ) q d q } x y d x d y .
Q ( x , y ) = 0 q J 0 ( q x ) J 0 ( q y ) q d q = q 2 / 2 [ J 0 2 ( q x ) + J 1 2 ( q y ) ] = q / ( x 2 - y 2 ) [ x J 1 ( q x ) J 0 ( q y ) - y J 1 ( q y ) J 0 ( q x ) ] .
L ( P , q , β ) = 1 2 0 1 0 1 cos [ θ ( x ) - θ ( y ) ] Q ( x , y ) x y d x d y
f ( x , y ) = cos [ θ ( x ) - θ ( y ) ] Q ( x , y ) x y
L ( p , q , β ) = 1 2 0 1 0 1 f ( x , y ) d x d y .
L ( p , q , β ) = 1 2 n = 1 7 m = 1 7 f ( x n , x m ) ,
q = 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 12 , 14.
p = 0 , ± π , ± 2 π , ± 3 π , ± 4 π , ± 5 π ,
q = 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 12 , 14.
= 1 - L ( p , q , β ) .
= J 0 ( p ) sin p + J 1 ( p ) cos p .