Abstract

The torus grating is defined as a calotte of a circular torus bearing a grating ruling on its concave side. In contrast to the spherical grating, the torus grating is capable of eliminating astigmatism, in general, for two points in the spectrum. These two stigmatic points can be adjusted to any desired wave-length within a wide range of the spectrum by choosing suitable values for the angles of incidence and diffraction. In the rest of the spectrum produced by a torus grating, astigmatism is considerably smaller than that prevailing in spectra of spherical gratings. Moreover, astigmatism is negligible in the proximity of the two stigmatic points, giving rise to “quasi-stigmatic” ranges in the torus grating spectrum. The extension of these quasi-stigmatic ranges depends on the size of tolerable astigmatism. If based upon an astigmatism equal to the diffraction width of the spectral images, a quasi-stigmatic range can be wider than 1000A. With light sources of small size a considerable gain in spectral intensity results from the lack of astigmatism. The theory of the torus grating which has been attempted in this paper further demonstrates that coma, aberration, and curvature of spectral lines in the torus grating spectrum are generally smaller than the same image imperfections in the spectrum of the spherical grating.

© 1950 Optical Society of America

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References

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  1. H. A. Rowland, Phil. Mag. [5],  13, 469 (1882).
    [CrossRef]
  2. F. L. O. Wadsworth, Astrophys. J. 3, 54 (1896).
  3. Runge and Paschen, Ann. d. Physik 61, 641 (1897).
    [CrossRef]
  4. Runge and Mannkopff, Zeits. f. Physik 45, 13 (1927).
    [CrossRef]
  5. H. G. Beutler, J. Opt. Soc. Am. 35, 311 (1945).
    [CrossRef]
  6. Runge and Paschen, Abh. d. K. Akad. d. Wiss. z-Berlin, Anhang1 (1902).
  7. W. W. Abney, Trans. Roy. Soc. London A177, 457 (1886).
    [CrossRef]
  8. A. Eagle, Astrophys. J. 31, 120 (1910).
    [CrossRef]

1945 (1)

1927 (1)

Runge and Mannkopff, Zeits. f. Physik 45, 13 (1927).
[CrossRef]

1910 (1)

A. Eagle, Astrophys. J. 31, 120 (1910).
[CrossRef]

1902 (1)

Runge and Paschen, Abh. d. K. Akad. d. Wiss. z-Berlin, Anhang1 (1902).

1897 (1)

Runge and Paschen, Ann. d. Physik 61, 641 (1897).
[CrossRef]

1896 (1)

F. L. O. Wadsworth, Astrophys. J. 3, 54 (1896).

1886 (1)

W. W. Abney, Trans. Roy. Soc. London A177, 457 (1886).
[CrossRef]

1882 (1)

H. A. Rowland, Phil. Mag. [5],  13, 469 (1882).
[CrossRef]

Abney, W. W.

W. W. Abney, Trans. Roy. Soc. London A177, 457 (1886).
[CrossRef]

Beutler, H. G.

Eagle, A.

A. Eagle, Astrophys. J. 31, 120 (1910).
[CrossRef]

Mannkopff,

Runge and Mannkopff, Zeits. f. Physik 45, 13 (1927).
[CrossRef]

Paschen,

Runge and Paschen, Abh. d. K. Akad. d. Wiss. z-Berlin, Anhang1 (1902).

Runge and Paschen, Ann. d. Physik 61, 641 (1897).
[CrossRef]

Rowland, H. A.

H. A. Rowland, Phil. Mag. [5],  13, 469 (1882).
[CrossRef]

Runge,

Runge and Mannkopff, Zeits. f. Physik 45, 13 (1927).
[CrossRef]

Runge and Paschen, Abh. d. K. Akad. d. Wiss. z-Berlin, Anhang1 (1902).

Runge and Paschen, Ann. d. Physik 61, 641 (1897).
[CrossRef]

Wadsworth, F. L. O.

F. L. O. Wadsworth, Astrophys. J. 3, 54 (1896).

Abh. d. K. Akad. d. Wiss. z-Berlin, Anhang (1)

Runge and Paschen, Abh. d. K. Akad. d. Wiss. z-Berlin, Anhang1 (1902).

Ann. d. Physik (1)

Runge and Paschen, Ann. d. Physik 61, 641 (1897).
[CrossRef]

Astrophys. J. (2)

F. L. O. Wadsworth, Astrophys. J. 3, 54 (1896).

A. Eagle, Astrophys. J. 31, 120 (1910).
[CrossRef]

J. Opt. Soc. Am. (1)

Phil. Mag. [5] (1)

H. A. Rowland, Phil. Mag. [5],  13, 469 (1882).
[CrossRef]

Trans. Roy. Soc. London (1)

W. W. Abney, Trans. Roy. Soc. London A177, 457 (1886).
[CrossRef]

Zeits. f. Physik (1)

Runge and Mannkopff, Zeits. f. Physik 45, 13 (1927).
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Generation of a torus by revolving a circle of radius ρ about an axis (z′) lying in the plane of the circle at a distance Rρ from its center.

Fig. 2
Fig. 2

Denotations of co-ordinates. For details refer to the pertinent paragraphs of the text.

Fig. 3
Fig. 3

Primary focal curve (Rowland circle) and a number of secondary focal curves pertaining to indicated values of the angle of incidence α. – – – A single secondary focal curve pertaining to a spherical grating of corresponding dimensions.

Fig. 4
Fig. 4

Relation between stigmatic wave-lengths n·λ in angstrom-units and the sinuses of the angles of incidence and diffraction. For details see text.

Fig. 5
Fig. 5

Width of the quasi-stigmatic range in the torus grating spectrum, expressed in angstrom-units, as a function of the angle of diffraction β.

Fig. 6
Fig. 6

Coma present in equatorial points of stigmatic wave-lengths in the y and the z direction, ΔCo(y) and ΔCo(z), respectively. – – – Coma present in equatorial points of a spectrum produced by a spherical grating of corresponding dimensions. —·—·— Diffraction widths (full amount for z, one-third for y) at the same points.

Fig. 7
Fig. 7

Coma present in the equatorial cross section of the spectrum for a value of α=arc cos[ρ/R] as a function of the wave-length n·λ. – – – Corresponding values for the spherical grating. —·—·— One-third of the diffraction width in the y direction.

Fig. 8
Fig. 8

Inclination (– – –) and curvature (———) of the stigmatic spectral line n·λ=6563A at the grating normal.

Fig. 9
Fig. 9

Outline of a diaphragm necessary to eliminate excessive aberration for the stigmatic spectral line n·λ=6563A at the grating normal (——— torus grating; – – – corresponding spherical grating).

Fig. 10
Fig. 10

Maximum permissible half-widths, ymax and zmax, of the grating’s aperture for the angle of incidence α = arc cos [ ρ / R ] 1 2, as a function of the angle of diffraction β.

Fig. 11
Fig. 11

A mount for a torus grating. For details see the last chapter of this paper.

Equations (74)

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[ ( x 2 + y 2 ) 1 2 - ( R - ρ ) ] 2 + z 2 = ρ 2 .
x = x - R ;             y = y ;             z = z ,
x 2 + y 2 + z 2 = 2 R x - 2 R ( R - ρ ) + 2 ( R - ρ ) [ ( R - x ) 2 + y 2 ] 1 2 .
x 2 + y 2 + z 2 = 2 R x - 2 R ( R - ρ ) + 2 ( R - ρ ) ( R - x ) [ 1 + y 2 ( R - x ) 2 ] 1 2 = 2 R x - 2 R ( R - ρ ) + 2 ( R - ρ ) ( R - x ) [ 1 + 1 2 y 2 ( R - x ) 2 - 1 8 y 4 ( R - x ) 4 ] + sixth and higher orders.
y 2 / ( R - x ) 2 = y 2 ( R + x ) 2 / R 4 + + 6. a.h.o.
x 2 + y 2 + z 2 = 2 ρ x + [ 1 - ( ρ / R ) ] y 2 + [ ( R - ρ ) / R 2 ] x y 2 - [ ( R 2 - R ρ ) / 4 R 4 ] y 4 + + 5. a.h.o.
[ A P ] 2 = ( x - x a ) 2 + ( y - y a ) 2 + ( z - z a ) 2 = r a 2 + z a 2 - 2 x x a - 2 y y a - 2 z z a + x 2 + y 2 + z 2 .
[ A P ] 2 = r a 2 + z a 2 - 2 z z a + ( ρ - x a ) 2 x + [ 1 - ( ρ / R ) ] y 2 + [ ( R - ρ ) / R 2 ] x y 2 - [ ( R - ρ ) / 4 R 3 ] y 4 .
2 x = { 1 / ρ } { x 2 + z 2 + ( ρ / R ) y 2 - [ ( R - ρ ) / R 2 ] x y 2 + [ ( R - ρ ) / 4 R 3 ] y 4 }
[ A P ] 2 = r a 2 + z a 2 - 2 y y a - 2 z z a + [ 1 - ( x a / R ) ] y 2 + [ 1 - ( x a / ρ ) ] [ x 2 + z 2 ] + ( x a / ρ ) { [ ( R - ρ ) / R 2 ] x y 2 - [ ( R - ρ ) / 4 R 3 ] y 4 } .
x 2 - { 2 ρ + [ ( R - ρ ) / R 2 ] y 2 } · x + { z 2 + ( ρ / R ) y 2 + [ ( R - ρ ) / 4 R 3 ] y 4 } = 0.
x = ρ + [ ( R - ρ ) / 2 R 2 ] y 2 ± 1 2 { ( 2 ρ + [ ( R - ρ ) / 2 R 2 ] y 2 ) 2 - 4 ( z 2 + ( ρ / R ) y 2 + [ ( R - ρ ) / 4 R 3 ] y 4 ) } 1 2 .
x = ( M / N ) + ( M 2 / N 3 ) + ; M = z 2 + ( ρ / R ) y 2 + [ ( R - ρ ) / 4 R 3 ] y 4 ; N = 2 ρ + [ ( R - ρ ) / R 2 ] y 2 .
x = ( 1 / 2 R ) y 2 + ( 1 / 2 ρ ) z 2 ;
x 2 = ( 1 / 4 R 2 ) y 4 + ( 1 / 4 ρ 2 ) z 4 + ( 1 / 2 R ρ ) y 2 z 2 .
[ A P ] 2 = r a 2 + z a 2 - 2 y y a - 2 z z a + [ 1 - ( x a / R ) ] y 2 + [ 1 - ( x a / ρ ) ] z 2 + 1 / 4 R 2 [ 1 - ( x a / R ) ] y 4 + ( 1 / 4 ρ 2 ) [ 1 - ( x a / ρ ) ] z 4 + ( 1 / 2 R ρ ) [ 1 - ( x a / R ) ] y 2 z 2 .
x a = r a cos α ;             y a = r a · sin α
[ A P ] 2 = r a 2 - 2 y r a sin α + z a 2 - 2 z z a + [ 1 - ( r a / R ) cos α ] y 2 + [ 1 - ( r a / ρ ) cos α ] z 2 + 1 / 4 R 2 [ 1 - ( r a / R ) cos α ] y 4 + ( 1 / 4 ρ 2 ) [ 1 - ( r a / ρ ) cos α ] z 4 + ( 1 / 2 R ρ ) [ 1 - ( r a / R ) cos α ] y 2 z 2 .
[ A P ] 2 = r a 2 - 2 y r a sin α + y 2 sin 2 α + y 2 cos 2 α - y 2 ( r a / R ) cos α + [ 1 - ( r a / ρ ) cos α ] z 2 + z a 2 - 2 z z a + ( 1 / 4 R 2 ) [ 1 - ( r a / R ) cos α ] y 4 + ( 1 / 4 ρ 2 ) [ 1 - ( r a / ρ ) cos α ] z 4 + ( 1 / 2 R ρ ) × [ 1 - ( r a / R ) cos α ] y 2 z 2 = ( r a - y · sin α ) 2 + [ cos 2 α - ( r a / R ) cos α ] y 2 + [ 1 - ( r a / ρ ) cos α ] z 2 + z a 2 - 2 z z a + ( 1 / 4 R 2 ) × [ 1 - ( r a / R ) cos α ] y 4 + ( 1 / 4 ρ 2 ) [ 1 - ( r a / ρ ) cos α ] z 4 + ( 1 / 2 R ρ ) [ 1 - ( r a / R ) cos α ] y 2 z 2 .
A P = r a - y · sin α + { 1 / ( r a - y · sin α ) } { 1 2 [ cos 2 α - ( r a / R ) cos α ] y 2 + 1 2 [ 1 - ( r a / ρ ) cos α ] z 2 + 1 2 [ z a 2 - 2 z z a ] + ( 1 / 8 R 2 ) [ 1 - ( r a / R ) cos α ] y 4 + ( 1 / 8 ρ 2 ) × [ 1 - ( r a / ρ ) cos α ] z 4 + ( 1 / 4 R ρ ) [ 1 - ( r a / R ) cos α ] y 2 z 2 } - { 1 / 4 ( r a - y · sin α ) 3 } { 1 2 [ cos 2 α - ( r a / R ) cos α ] 2 y 4 + 1 2 [ 1 - ( r a / ρ ) cos α ] 2 z 4 + 1 2 [ z a 2 - 2 z z a ] 2 + [ cos 2 α - ( r a / R ) cos α ] [ 1 - ( r a / ρ ) cos α ] y 2 z 2 + [ cos 2 α - ( r a / R ) cos α ] [ z a 2 - 2 z z a ] y 2 + [ 1 - ( r a / ρ ) cos α ] [ z a 2 - 2 z z a ] z 2 } .
A P + P B = r a + r b - ( sin α + sin β ) y             ( T 1 ) + 1 2 [ cos 2 α r a - cos α R ] y 2 + 1 2 [ cos 2 β r b - cos β R ] y 2 + 1 2 sin α r a [ cos 2 α r a - cos α R ] y 3 + 1 2 sin β r b [ cos 2 β r b - cos β R ] y 3 + 1 2 sin 2 α r a 2 [ cos 2 α r a - cos α R ] y 4 + 1 2 sin 2 β r b 2 [ cos 2 β r b - cos β R ] y 4 }             ( T 2 ) + 1 2 [ 1 r a - cos α ρ ] z 2 + 1 2 [ 1 r b - cos β ρ ] z 2 + 1 2 r a [ z a 2 - 2 z z a ] + 1 2 r b [ z b 2 - 2 z z b ] - 1 8 r a [ ( 1 r a - cos α ρ ) z 2 + 1 r a ( z a 2 - 2 z z a ) ] 2 - 1 8 r b [ ( 1 r b - cos β ρ ) z 2 + 1 r b ( z b 2 - 2 z z b ) ] 2 }             ( T 3 ) + 1 2 sin α r a [ 1 r a - cos α ρ ] y z 2 + 1 2 sin β r b [ 1 r b - cos β ρ ] y z 2 + 1 2 sin α r a 2 [ z a 2 - 2 z z a ] y + 1 2 sin β r b 2 [ z b 2 - 2 z z b ] y + 1 2 sin 2 α r a 2 [ 1 r a - cos α ρ ] y 2 z 2 + 1 2 sin 2 β r b 2 [ 1 r b - cos β ρ ] y 2 z 2 + 1 2 sin 2 α r a 3 [ z a 2 - 2 z z a ] y 2 + 1 2 sin 2 β r b 3 [ z b 2 - 2 z z b ] y 2 }             ( T 4 ) + 1 8 R 2 [ 1 r a - cos α R ] y 4 + 1 8 R 2 [ 1 r b - cos β R ] y 4 + 1 8 ρ 2 [ 1 r a - cos α ρ ] z 4 + 1 8 ρ 2 [ 1 r b - cos β ρ ] z 4 + 1 4 R ρ [ 1 r a - cos α R ] y 2 z 2 + 1 4 R ρ [ 1 r b - cos β R ] y 2 z 2 }             ( T 5 ) - 1 8 r a [ cos 2 α r a - cos α R ] 2 y 4 - 1 8 r b [ cos 2 β r b - cos β R ] 2 y 4 - 1 4 r a [ cos 2 α r a - cos α R ] [ 1 r a - cos α ρ ] y 2 z 2 - 1 4 r b [ cos 2 β r b - cos β R ] [ 1 r b - cos β ρ ] y 2 z 2 - 1 4 r a 2 [ cos 2 α r a - cos α R ] [ z a 2 - 2 z z a ] y 2 - 1 4 r b 2 [ cos 2 β r b - cos β R ] [ z b 2 - 2 z z b ] y 2 . }             ( T 6 )
Δ n = r b ( T n / v ) ( 1 / cos φ )
R = 914.38 cm ; ρ = 782.83 cm ; D = 0.000127 cm .
sin α + sin β = n · λ / D
r a = R · cos α ;             r b = R · cos β ,
T 3 / z = [ ( 1 / r a ) - ( cos α / ρ ) + ( 1 / r b ) - ( cos β / ρ ) ] z - ( z a / r a ) - ( z b / r b ) = 0.
[ ( 1 / r a ) - ( cos α / ρ ) + ( 1 / r b ) - ( cos β / ρ ) ] z = 0.
[ ( 1 / r a ) - ( cos α / R ) + ( 1 / r b ) - ( cos β / R ) ] z 0.
( 1 / r a ) - ( cos α / ρ ) + ( 1 / r b ) - ( cos β / ρ ) = 0.
ρ / R = cos α · cos β .
r b h ( S ) = R / ( cos β - sin α · tan α )             ( sphere )
r b h ( T ) = ρ cos α / { cos 2 α - [ ( ρ / R ) - cos α · cos β ] }             ( torus ) ,
sin α + sin β = n · λ / D ;             cos α · cos β = ρ / R .
sin α = { ± n · λ 2 D ± ( 1 + ( n · λ 2 D ) 2 - [ ( n · λ D ) 2 + ( ρ R ) 2 ] 1 2 ) 1 2 } ,
sin β = ± { ± n · λ 2 D ( 1 + ( n · λ 2 D ) 2 - [ ( n · d D ) 2 + ( ρ R ) 2 ] 1 2 ) 1 2 } .
a ) α = 0 ; b ) β = 0 ; c ) α = β .
n · λ = ± D [ 1 - ( ρ / R ) 2 ] 1 2 = ± 6563 A .
n · λ = ± 2 D [ 1 - ( ρ / R ) ] 1 2 = ± 9634 A .
n · λ D = sin α ± [ ρ 2 R 2 ( sin 2 α - 1 ) + 1 ] 1 2 .
d ( n · λ D ) / d ( sin α ) = 1 ρ 2 sin α / R 2 ( sin 2 α - 1 ) 2 [ ρ 2 R 2 ( sin 2 α - 1 ) + 1 ] 1 2 = 0.
[ w 2 - ( ρ / R ) 2 ] [ w 2 + ( ρ / R ) 2 w + ( ρ / R ) 2 ] = 0.
sin α = ± [ 1 - ( ρ / R ) ] 1 2 .
n · λ max = ± D { sin α + [ 1 - ( ρ / R ) ] 1 2 } = ± 2 D [ 1 - ( ρ / R ) ] 1 2 .
s = ρ cos α / { cos 2 α - [ ( ρ / R ) - cos α · cos β ] } - R cos β .
A = [ L / ρ · cos α ] [ cos α + cos β ] [ ρ - R cos α · cos β ] .
cos β = ρ - R cos 2 α 2 R cos α + [ ( ρ - R cos 2 α ) 2 4 R 2 cos 2 α + ρ R ( 1 ± A L ) ] 1 2 .
d A / d β = ( L / ρ ) { [ ( R cos 2 α - ρ ) / cos α ] + 2 R cos β } · sin β .
T 3 ( S A ) = - 1 8 [ 1 r a ( 1 r a - cos α ρ ) 2 + 1 r b ( 1 r b - cos β ρ ) 2 ] z 4 - 1 4 [ 1 r a 2 ( 1 r a - cos α ρ ) ( z a 2 - 2 z z a ) + 1 r b 2 ( 1 r b - cos β ρ ) ( z b 2 - 2 z z b ) ] z 2 - 1 8 [ 1 r a 3 ( z a 2 - 2 z z a ) 2 + 1 r b 3 ( z b 2 - 2 z z b ) 2 ] .
T 3 ( S A ) z = - 1 2 R 3 cos 3 α [ 2 z a 2 z - z a 3 ] - 1 2 R 3 cos 3 β [ 2 z b 2 z - z b 3 ] .
T 3 ( S A ) z = - 1 2 R 3 cos 3 α [ 2 cos 2 α cos 2 β z b 2 z + cos 3 α cos 3 β z b 3 ] - 1 2 R 3 cos 3 β [ 2 z b 2 z - z b 3 ] .
Δ ( S A ) = - 1 R 2 cos φ · cos α + cos β cos α · cos 3 β z b 2 z .
I ( T ) I ( S ) = c ( cos β / cos α ) + A ( S ) c ( cos β / cos α ) + A ( T ) ,
Δ ( Cos t ) y = ( 1 / 2 R ρ ) [ ( sin α / cos 2 α ) ( ρ - R cos 2 α ) + ( sin β / cos 2 β ) ( ρ - R cos 2 β ) ] z 2 + ( 1 / R 2 ρ ) [ ( sin 2 α / cos 3 α ) × ( ρ - R cos 2 α ) + ( sin 2 β / cos 3 β ) ( ρ - R cos 2 β ) ] y z 2
Δ ( CoS t ) z = ( 1 / R ρ ) [ ( sin α / cos 2 α ) ( ρ - R cos 2 α ) + ( sin β / cos 2 β ) ( ρ - R cos 2 β ) ] y z + ( 1 / R 2 ρ ) [ ( sin 2 α / cos 3 α ) × ( ρ - R cos 2 α ) + ( sin 2 β / cos 3 β ) ( ρ - R cos 2 β ) ] y 2 z .
T 4 = ( 1 / 2 R 2 ρ ) [ ( sin α / cos 2 α ) ( ρ - R cos 2 α ) + ( sin β / cos 2 β ) × ( ρ - R cos 2 β ) ] y z 2 + ( 1 / R 2 ) [ ( sin α / cos 2 α ) · z a + ( sin β / cos 2 β ) z b ] y z + ( 1 / 2 R 2 ) [ ( sin α / cos 2 α ) · z a 2 + ( sin β / cos 2 β ) z b 2 ] y .
z a = ( cos α + cos β ) ( ρ - R cos α · cos β ) z / ρ cos β .
Δ ( Co ) y = ( 1 / 2 R ρ ) [ ( sin α / cos 2 α ) ( ρ - R cos 2 α ) + ( sin β / cos 2 β ) × ( ρ - R cos 2 β ) - ( 2 sin α / cos 2 α cos β ) ( cos α + cos β ) × ( ρ - R cos α · cos β ) + ( sin α / ρ cos 2 α · cos 2 β ) × ( cos α + cos β ) 2 ( ρ - R cos α · cos β ) 2 ] z 2 .
T 4 ( z b ) = 1 2 ( sin α / r a 2 ) [ z a 2 - 2 z z a ] y + 1 2 ( sin β / r b 2 ) [ z b 2 - 2 z z b ] y .
Δ ( C u ) y = z R cos β · cos φ [ tan α - tan β ] z b             ( inclination ) + 1 2 R cos 2 β cos φ [ sin α + sin β ] z b 2             ( curvature ) .
Δ ( C u ) z = y R cos β · cos φ [ tan α - tan β ] z b             ( expansion , contraction ) .
T 5 = [ sin 2 α cos α + sin 2 β cos β ] [ y 4 8 R 3 + y 2 z 2 4 R 2 ρ ] + ( cos α + cos β ) ( ρ - R cos α · cos β ) 8 R ρ 3 cos α cos β z 4 .
[ sin 2 α cos α + sin 2 β cos β ] [ y 4 8 R 3 + y 2 z 2 4 R 2 ρ ] + ( cos α + cos β ) ( ρ - R cos α · cos β ) 8 R ρ 3 cos α · cos β z 4 λ 4
= D / 4 n ( sin α + sin β ) .
n · H 2 R 3 D y 4 + n · H R 2 ρ D y 2 z 2 + n · K 2 R ρ 3 D z 4 - 1 = 0 ,
H = cot α + β 2 · ( 1 - cos α · cos β ) cos α · cos β ;
K = cot α + β 2 · ( ρ - R cos α · cos β ) cos α · cos β .
z = ± { R ρ 3 D n · K [ ( n 2 H 2 R 4 ρ 2 D 2 y 4 - 2 n K R ρ 3 D { n H 2 R 3 D y 4 - 1 } ) 1 2 - n · H R 2 ρ D y 2 ] } 1 2 .
T 5 ( S ) = 1 8 R 2 [ 1 r a - cos α R + 1 r b - cos β R ] r 4 ; r 4 = ( y 2 + z 2 ) 2 .
z = ± [ R 2 ρ D n · tan α + β 2 · cos α · cos β 1 - cos α · cos β · 1 y 2 - ρ 2 R y 2 ] 1 2 .
y max = { [ 2 R 3 D / n ] [ tan ( α + β ) / 2 ] [ cos α · cos β ] / [ 1 - cos α · cos β ] } 1 4 .
z max = { [ 2 R ρ 3 D / n ] [ tan ( α + β ) / 2 ] [ cos α · cos β ] / [ ρ - R cos α · cos β ] } 1 4 .
Δ ( B r ) y = ( 1 / 2 R 2 ) [ ( sin 2 α / cos α ) + ( sin 2 β / cos β ) ] y 3 + ( 1 / 2 R ρ ) [ ( sin 2 α / cos α ) + ( sin 2 β / cos β ) ] y z 2
+ 22 ° 17 α + 31 ° 7 - 22 ° 17 β + 22 ° 17 .
Rowland β = 0 Runge-Paschen 6 α = constant Abeny 7 β = 0 Eagle 8 α = β .