Abstract

A slightly modified version of the Head equations are used to describe an aplanatic grazing incidence microscope with possible application in the field of controlled fusion research. The improved theoretical blur circle radius of the aplanatic microscope is presented and compared to the theoretical blur circle radius of the corresponding ellipsoid–hyperboloid microscope. The results show that the improvement is important for high resolution instruments having grazing angles greater than about 2°.

© 1976 Optical Society of America

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  1. R. Giacconi, W. P. Reidy, G. S. Vaiana, L. P. VanSpeybroeck, T. F. Zehnpfennig, Space Sci. Rev. 9, 3 (1969).
    [CrossRef]
  2. J. D. Mangus, J. H. Underwood, Appl. Opt. 8, 95 (1969).
    [CrossRef] [PubMed]
  3. L. P. VanSpeybroeck, R. C. Chase, Appl. Opt. 11, 440 (1972).
    [CrossRef] [PubMed]
  4. K. Schwarzschild, Abh. Wiss. Goettingen, Bd. IV, Nr. 2 (1905).
  5. R. C. Chase, L. P. VanSpeybroeck, Appl. Opt. 12, 1042 (1973).
    [CrossRef] [PubMed]
  6. R. C. Chase, J. K. Silk, Appl. Opt. 14, 2096 (1975).
    [CrossRef] [PubMed]
  7. T. M. Palmieri, M. J. Boyle, H. G. Ahlstrom, J. A. Monjes, Bull. Am. Phys. Soc., Ser. II 20, 1349 (1975).
  8. B. Jurek, Czech. J. Phys. B18, 297 (1968).
    [CrossRef]
  9. A. K. Head, Proc. Phys. Soc., London, Sect. B 70, 945 (1957).
    [CrossRef]
  10. H. Wolter, Ann. Phys. 10, 286 (1952).
    [CrossRef]
  11. The quantities enclosed by the absolute value symbols change signs as the surfaces pass through the intersection point. When β < βc and ϕ < ϕc the quantities are positive, and when β > βc and ϕ > ϕc the quantities are negative. To prove this we use the geometry of Fig. 1 and Eqs. (2), (4), (5), (15), (16), and (18) to solve for the value of γ at the intersection of the two surfaces in terms of κ and m. γc=(κ-1)(κ+1)(m+1). Similarly δc=(κ-1)(κ+1) (M+1). Therefore, for example, the last factor in Eq. (3) can be written |(κ+1)2(m+1) (γ-γc)|2-λ-μ. For grazing incidence systems where γ > γc (equivalent to β < βc) the two surfaces describe configurations which include Wolter type II or Wolter type III systems (H. Wolter, Ann. Phys. 10, 94, 1952), where one surface collects rays and the other surface disperses rays. In practice these systems are used far from the common joint at γ = γc so that one surface will not block rays destined for the other surface. If γ < γc the two surfaces describe the Wolter type I configuration of the type considered in this paper and which is shown in Fig. 1 where both surfaces collect rays. In contrast to Wolter II and III systems, Wolter I systems are usually used near the intersection joint.
    [CrossRef]
  12. I choose to define ξ at the L1/2 point rather than at the intersection plane because the surfaces are usually better behaved at the L1/2 point. If m > 1, surface 2 has a point of inflection close to the intersection plane, and both surfaces have smaller radii of curvature near the intersection plane.

1975 (2)

T. M. Palmieri, M. J. Boyle, H. G. Ahlstrom, J. A. Monjes, Bull. Am. Phys. Soc., Ser. II 20, 1349 (1975).

R. C. Chase, J. K. Silk, Appl. Opt. 14, 2096 (1975).
[CrossRef] [PubMed]

1973 (1)

1972 (1)

1969 (2)

J. D. Mangus, J. H. Underwood, Appl. Opt. 8, 95 (1969).
[CrossRef] [PubMed]

R. Giacconi, W. P. Reidy, G. S. Vaiana, L. P. VanSpeybroeck, T. F. Zehnpfennig, Space Sci. Rev. 9, 3 (1969).
[CrossRef]

1968 (1)

B. Jurek, Czech. J. Phys. B18, 297 (1968).
[CrossRef]

1957 (1)

A. K. Head, Proc. Phys. Soc., London, Sect. B 70, 945 (1957).
[CrossRef]

1952 (2)

H. Wolter, Ann. Phys. 10, 286 (1952).
[CrossRef]

The quantities enclosed by the absolute value symbols change signs as the surfaces pass through the intersection point. When β < βc and ϕ < ϕc the quantities are positive, and when β > βc and ϕ > ϕc the quantities are negative. To prove this we use the geometry of Fig. 1 and Eqs. (2), (4), (5), (15), (16), and (18) to solve for the value of γ at the intersection of the two surfaces in terms of κ and m. γc=(κ-1)(κ+1)(m+1). Similarly δc=(κ-1)(κ+1) (M+1). Therefore, for example, the last factor in Eq. (3) can be written |(κ+1)2(m+1) (γ-γc)|2-λ-μ. For grazing incidence systems where γ > γc (equivalent to β < βc) the two surfaces describe configurations which include Wolter type II or Wolter type III systems (H. Wolter, Ann. Phys. 10, 94, 1952), where one surface collects rays and the other surface disperses rays. In practice these systems are used far from the common joint at γ = γc so that one surface will not block rays destined for the other surface. If γ < γc the two surfaces describe the Wolter type I configuration of the type considered in this paper and which is shown in Fig. 1 where both surfaces collect rays. In contrast to Wolter II and III systems, Wolter I systems are usually used near the intersection joint.
[CrossRef]

1905 (1)

K. Schwarzschild, Abh. Wiss. Goettingen, Bd. IV, Nr. 2 (1905).

Ahlstrom, H. G.

T. M. Palmieri, M. J. Boyle, H. G. Ahlstrom, J. A. Monjes, Bull. Am. Phys. Soc., Ser. II 20, 1349 (1975).

Boyle, M. J.

T. M. Palmieri, M. J. Boyle, H. G. Ahlstrom, J. A. Monjes, Bull. Am. Phys. Soc., Ser. II 20, 1349 (1975).

Chase, R. C.

Giacconi, R.

R. Giacconi, W. P. Reidy, G. S. Vaiana, L. P. VanSpeybroeck, T. F. Zehnpfennig, Space Sci. Rev. 9, 3 (1969).
[CrossRef]

Head, A. K.

A. K. Head, Proc. Phys. Soc., London, Sect. B 70, 945 (1957).
[CrossRef]

Jurek, B.

B. Jurek, Czech. J. Phys. B18, 297 (1968).
[CrossRef]

Mangus, J. D.

Monjes, J. A.

T. M. Palmieri, M. J. Boyle, H. G. Ahlstrom, J. A. Monjes, Bull. Am. Phys. Soc., Ser. II 20, 1349 (1975).

Palmieri, T. M.

T. M. Palmieri, M. J. Boyle, H. G. Ahlstrom, J. A. Monjes, Bull. Am. Phys. Soc., Ser. II 20, 1349 (1975).

Reidy, W. P.

R. Giacconi, W. P. Reidy, G. S. Vaiana, L. P. VanSpeybroeck, T. F. Zehnpfennig, Space Sci. Rev. 9, 3 (1969).
[CrossRef]

Schwarzschild, K.

K. Schwarzschild, Abh. Wiss. Goettingen, Bd. IV, Nr. 2 (1905).

Silk, J. K.

Underwood, J. H.

Vaiana, G. S.

R. Giacconi, W. P. Reidy, G. S. Vaiana, L. P. VanSpeybroeck, T. F. Zehnpfennig, Space Sci. Rev. 9, 3 (1969).
[CrossRef]

VanSpeybroeck, L. P.

Wolter, H.

The quantities enclosed by the absolute value symbols change signs as the surfaces pass through the intersection point. When β < βc and ϕ < ϕc the quantities are positive, and when β > βc and ϕ > ϕc the quantities are negative. To prove this we use the geometry of Fig. 1 and Eqs. (2), (4), (5), (15), (16), and (18) to solve for the value of γ at the intersection of the two surfaces in terms of κ and m. γc=(κ-1)(κ+1)(m+1). Similarly δc=(κ-1)(κ+1) (M+1). Therefore, for example, the last factor in Eq. (3) can be written |(κ+1)2(m+1) (γ-γc)|2-λ-μ. For grazing incidence systems where γ > γc (equivalent to β < βc) the two surfaces describe configurations which include Wolter type II or Wolter type III systems (H. Wolter, Ann. Phys. 10, 94, 1952), where one surface collects rays and the other surface disperses rays. In practice these systems are used far from the common joint at γ = γc so that one surface will not block rays destined for the other surface. If γ < γc the two surfaces describe the Wolter type I configuration of the type considered in this paper and which is shown in Fig. 1 where both surfaces collect rays. In contrast to Wolter II and III systems, Wolter I systems are usually used near the intersection joint.
[CrossRef]

H. Wolter, Ann. Phys. 10, 286 (1952).
[CrossRef]

Zehnpfennig, T. F.

R. Giacconi, W. P. Reidy, G. S. Vaiana, L. P. VanSpeybroeck, T. F. Zehnpfennig, Space Sci. Rev. 9, 3 (1969).
[CrossRef]

Abh. Wiss. Goettingen (1)

K. Schwarzschild, Abh. Wiss. Goettingen, Bd. IV, Nr. 2 (1905).

Ann. Phys. (2)

H. Wolter, Ann. Phys. 10, 286 (1952).
[CrossRef]

The quantities enclosed by the absolute value symbols change signs as the surfaces pass through the intersection point. When β < βc and ϕ < ϕc the quantities are positive, and when β > βc and ϕ > ϕc the quantities are negative. To prove this we use the geometry of Fig. 1 and Eqs. (2), (4), (5), (15), (16), and (18) to solve for the value of γ at the intersection of the two surfaces in terms of κ and m. γc=(κ-1)(κ+1)(m+1). Similarly δc=(κ-1)(κ+1) (M+1). Therefore, for example, the last factor in Eq. (3) can be written |(κ+1)2(m+1) (γ-γc)|2-λ-μ. For grazing incidence systems where γ > γc (equivalent to β < βc) the two surfaces describe configurations which include Wolter type II or Wolter type III systems (H. Wolter, Ann. Phys. 10, 94, 1952), where one surface collects rays and the other surface disperses rays. In practice these systems are used far from the common joint at γ = γc so that one surface will not block rays destined for the other surface. If γ < γc the two surfaces describe the Wolter type I configuration of the type considered in this paper and which is shown in Fig. 1 where both surfaces collect rays. In contrast to Wolter II and III systems, Wolter I systems are usually used near the intersection joint.
[CrossRef]

Appl. Opt. (4)

Bull. Am. Phys. Soc., Ser. II (1)

T. M. Palmieri, M. J. Boyle, H. G. Ahlstrom, J. A. Monjes, Bull. Am. Phys. Soc., Ser. II 20, 1349 (1975).

Czech. J. Phys. (1)

B. Jurek, Czech. J. Phys. B18, 297 (1968).
[CrossRef]

Proc. Phys. Soc., London, Sect. B (1)

A. K. Head, Proc. Phys. Soc., London, Sect. B 70, 945 (1957).
[CrossRef]

Space Sci. Rev. (1)

R. Giacconi, W. P. Reidy, G. S. Vaiana, L. P. VanSpeybroeck, T. F. Zehnpfennig, Space Sci. Rev. 9, 3 (1969).
[CrossRef]

Other (1)

I choose to define ξ at the L1/2 point rather than at the intersection plane because the surfaces are usually better behaved at the L1/2 point. If m > 1, surface 2 has a point of inflection close to the intersection plane, and both surfaces have smaller radii of curvature near the intersection plane.

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Figures (5)

Fig. 1
Fig. 1

Schematic drawing of an aplanatic grazing incidence microscope.

Fig. 2
Fig. 2

Schematic drawing of the aplanatic surfaces to show their behavior for small values of β and ϕ. The abrupt change of slope in both surfaces as they cross at the intersection point is characteristic.

Fig. 3
Fig. 3

The rms blur circle radius of several AGIM microscopes as a function of the point object’s position.

Fig. 4
Fig. 4

The difference between the rms blur circle radius of two ellipsoid–hyperboloid microscopes and the rms blur circle radius of the corresponding AGIM surface, plotted as a function of the point object’s position.

Fig. 5
Fig. 5

The difference of an AGIM microscope and the corresponding ellipsoid–hyperboloid microscope from the best fit circles to each surface. One fringe is equal to one-half of a wavelength of 5461-Å light.

Equations (27)

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ρ + s + r = l ,
sin β = m sin ϕ ,
s 0 ρ = 1 + κ 2 κ + 1 - κ 2 κ cos β + ( s 0 ρ 0 - 1 κ ) ( m + 1 γ ) × ( γ + m - 1 2 m ) λ ( γ - m + 1 2 ) μ | ( κ + 1 ) γ 2 m + 2 - κ 2 + 1 2 | 2 - λ - μ ,
γ = cos β + ( m 2 - sin 2 β ) 1 / 2 ,
κ = ( ρ 0 + r 0 ) / s 0 ,
λ = ( m κ ) / ( m κ - 1 ) ,
μ = m / ( m - κ ) ,
s 0 r = 1 + κ 2 κ + 1 - κ 2 κ cos ϕ + ( s 0 r 0 - 1 κ ) ( m + 1 δ ) × ( δ + M - 1 2 M ) λ ( δ - M + 1 2 ) μ | ( κ + 1 ) δ 2 M + 2 - κ 2 + 1 2 | 2 - λ - μ ,
δ = cos ϕ + ( M 2 - sin 2 ϕ ) 1 / 2 ,
λ = ( M κ ) / ( M κ - 1 ) ,
μ = M / ( M - κ ) .
β c = tan - 1 ( y c / p ) ,
ϕ c = tan - 1 ( y c / q ) .
m = ( sin β c ) / ( sin ϕ c )
s 0 = ( l - x 1 ) / 2 ,
l = [ y c / ( sin β c ) ] + [ y c / ( sin ϕ c ) ] ,
x 1 = p + q .
ρ 0 + r 0 = ( x 1 + l ) / 2.
g 1 = g 2 = 1 4 ( β e + ϕ e ) ,
g 1 1 4 ( β c + ϕ c ) .
θ = arctan ( H / p ) .
σ D AGIM 1 8 ( 2 + 1 m ) ( θ 2 α ) ( L 2 p ) .
σ D EH 1 8 ( 2 + 1 m ) ( θ 2 α ) ( L e p ) + ( 1 - 1 m 2 ) ( 4 θ α 2 - 14 α θ 2 ) ,
α = ( β c ) / 4 ,
σ D = d ¯ / q ,
2 α 2 ; θ 3 ; 0.03 L 2 , e p 0.35 ; m 1.
h ( w ) = 0.06 ( 2 + 1 m ) ( L 2 p 2 ) ( w 2 α 2 ) .

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