Abstract

The theory of the formation of a spectrum by means of an equally spaced grating has been studied in considerable detail by methods of geometrical optics. The difference of approach, compared with other papers on this subject, is mainly in the choice of the pupil coordinates. It appeared to be essential to take the orientation of these coordinates around the normal to the grating surface as a variable. Consequently, the variations from known results may be considerable. Another difference of this paper is the way the deviations from perfect imaging are considered. Furthermore, the shape of the grating surface is given in the form of a general series expansion in order to cover all kinds of surfaces, and the location of the object is not limited to small distances from the Rowland plane; grazing incidence in an extreme off-plane mounting is therefore included. The results give the position of the spectral image, generally line-shaped by astigmatism, with the coordinates of the center of this line and two slant angles. Expressions are also derived for length and radius of curvature of the astigmatic line and the width of this line caused by coma and spherical aberration. Finally, conditions are derived for the optimum slant angle of a slit and the shape of the grating surface for minimum aberrations. Extensive checks with ray tracing completely covered the results in this paper.

© 1967 Optical Society of America

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References

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  1. H. G. Beutler, J. Opt. Soc. Am. 35, 311 (1945).
    [CrossRef]
  2. H. Haber, J. Opt. Soc. Am. 40, 153 (1950).
    [CrossRef]
  3. T. Namioka, J. Opt. Soc. Am. 49, 446 (1959).
    [CrossRef]
  4. T. Namioka, J. Opt. Soc. Am. 49, 951 (1959).
    [CrossRef]
  5. T. Namioka, J. Opt. Soc. Am. 51, 4 (1961).
    [CrossRef]
  6. J. T. Hall, Appl. Opt. 5, 1051 (1966).
    [CrossRef] [PubMed]
  7. W. T. Welford, Progress in Optics (North-Holland Publishing Co., Amsterdam, 1965), Vol. 4; this volume gives extensive references.
    [CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Coordinate system for a general ray path.

Fig. 2
Fig. 2

Coordinate systems in the image plane normal to the principal ray.

Equations (92)

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F = A P + P B = [ ( x - ξ ) 2 + ( y - η ) 2 + ( z - ζ ) 2 ] 1 2 + [ ( x - ξ ) 2 + ( y - η ) 2 + ( z - ζ ) 2 ] 1 2 .
x = ρ cos ϑ sin φ , x = ρ cos ϑ sin φ , y = ρ sin ϑ , y = ρ sin ϑ , z = ρ cos ϑ cos φ , z = ρ cos ϑ cos φ .
ζ = α 1 ξ 2 + α 2 ξ η + α 3 η 2 + β 1 ξ 3 + β 2 ξ 2 η + β 3 ξ η 2 + β 4 η 3 + γ 1 ξ 4 + .
F / ξ = - m λ / d ,
F / η = 0.
ξ = u cos ψ - v sin ψ , η = u sin ψ + v cos ψ .
F u = F ξ ξ u + F η η u = - m λ d cos ψ ,
F v = F ξ ξ v + F η η v = + m λ d sin ψ .
ρ = ρ 0 + Δ ρ , φ = φ 0 + Δ φ , ϑ = ϑ 0 + Δ ϑ ,
( A P ) 2 = ( x - ξ ) 2 + ( y - η ) 2 + ( z - ζ ) 2 = x 2 + y 2 + z 2 + ξ 2 + η 2 + ζ 2 - 2 x ξ - 2 y η - 2 z ζ .
( A P ) 2 = ρ 2 + u 2 + v 2 + ζ 2 - 2 ρ cos ϑ sin φ ( u cos ψ - v sin ψ ) - 2 ρ sin ϑ ( u sin ψ + v cos ψ ) - 2 ρ ζ cos ϑ cos φ .
F = A P + P B = ρ + ρ - u ( C 1 + C 1 ) - v ( C 2 + C 2 ) + 1 2 u 2 ( C 3 + C 3 ) + u v ( C 4 + C 4 ) + 1 2 v 2 ( C 5 + C 5 ) + 1 2 u 3 ( C 6 + C 6 ) + 1 2 u 2 v ( C 7 + C 7 ) + 1 2 u v 2 ( C 8 + C 8 ) + 1 2 v 3 ( C 9 + C 9 ) + 1 2 u 4 ( C 10 + C 10 ) + .
F = ρ + ρ 0 + T 1 + T 2 + T 3 + + T 18 + T 19 + T 20 + Δ ρ + ,
T 1 = - u ( cos ϑ sin φ cos ψ + sin ϑ sin ψ + cos ϑ 0 sin φ 0 cos ψ + sin ϑ 0 sin ψ ) .
T 2 = - v ( - cos ϑ sin φ sin ψ + sin ϑ cos ψ - cos ϑ 0 sin ψ 0 sin ψ + sin ϑ 0 cos ψ ) .
T 3 = + 1 2 u 2 ( C 3 + C 03 ) .
T 4 = + u v ( C 4 + C 04 ) .
T 5 = + 1 2 v 2 ( C 5 + C 05 ) .
T 6 = + 1 2 u 2 ( C 6 + C 06 ) .
T 7 = + 1 2 u 2 v ( C 7 + C 07 ) .
T 8 = + 1 2 u v 2 ( C 8 + C 08 ) .
T 9 = + 1 2 v 3 ( C 9 + C 09 ) .
T 10 = + 1 2 u 4 ( C 10 + C 010 ) .
T 11 = - u [ cos ϑ 0 cos φ 0 cos ψ Δ φ - ( sin ϑ 0 sin φ 0 cos ψ - cos ϑ 0 sin ψ ) Δ ϑ ] .
T 12 = - v [ - cos ϑ 0 cos φ 0 sin ψ Δ φ + ( sin ϑ 0 sin φ 0 sin ψ + cos ϑ 0 cos ψ ) Δ ϑ ] .
T 13 = + u [ 1 2 cos ϑ 0 sin φ 0 cos ψ ( Δ φ ) 2 + sin ϑ 0 cos φ 0 cos ψ Δ φ Δ ϑ + ( 1 2 cos ϑ 0 sin φ 0 cos ψ + 1 2 sin ϑ 0 sin ψ ) ( Δ ϑ ) 2 ] .
T 14 = + v [ - 1 2 cos ϑ 0 sin φ 0 sin ψ ( Δ φ ) 2 - sin ϑ 0 cos φ 0 sin ψ Δ φ Δ ϑ - ( 1 2 cos ϑ 0 sin φ 0 sin ψ - 1 2 sin ϑ 0 cos ψ ) ( Δ ϑ ) 2 ] .
T 15 = + 1 2 u 2 { [ - ( 2 / ρ 0 ) ( cos ϑ 0 sin φ 0 cos ψ + sin ϑ 0 sin ψ ) ( cos ϑ 0 cos φ 0 cos ψ ) + 2 cos ϑ 0 sin φ 0 ( α 1 cos 2 ψ + α 2 sin ψ cos ψ + α 3 sin 2 ψ ) ] Δ φ + [ - ( 2 / ρ 0 ) ( cos ϑ 0 sin φ 0 cos ψ + sin ϑ 0 sin ψ ) ( - sin ϑ 0 sin φ 0 cos ψ + cos ϑ 0 sin ψ ) + 2 sin ϑ 0 cos φ 0 ( α 1 cos 2 ψ + α 2 sin ψ cos ψ + α 3 sin 2 ψ ) ] Δ ϑ } .
T 16 = + u v { [ - ( 1 / ρ 0 ) ( cos ϑ 0 cos φ 0 cos ψ ) ( - cos ϑ 0 sin φ 0 sin ψ + sin ϑ 0 cos ψ ) - ( 1 / ρ 0 ) ( cos ϑ 0 sin φ 0 cos ψ + sin ϑ 0 sin ψ ) × ( - cos ϑ 0 cos φ 0 sin ψ ) + cos ϑ 0 sin φ 0 ( - 2 α 1 sin ψ cos ψ + α 2 cos 2 ψ - α 2 sin 2 ψ + 2 α 3 sin ψ cos ψ ) ] Δ φ + [ - ( 1 / ρ 0 ) ( - sin ϑ 0 sin φ 0 cos ψ + cos ϑ 0 sin ψ ) ( - cos ϑ 0 sin φ 0 sin ψ + sin ϑ 0 cos ψ ) - ( - 1 / ρ 0 ) ( cos ϑ 0 sin φ 0 cos ψ + sin ϑ 0 sin ψ ) ( sin ϑ 0 sin φ 0 sin ψ + cos ϑ 0 cos ψ ) + sin ϑ 0 cos φ 0 ( - 2 α 1 sin ψ cos ψ + α 2 cos 2 ψ - α 2 sin 2 ψ + 2 α 3 sin ψ cos ψ ) ] Δ ϑ } .
T 17 = + 1 2 v 2 { [ - 2 / ρ 0 ) ( - cos ϑ 0 sin φ 0 sin ψ + sin ϑ 0 cos ψ ) ( - cos ϑ 0 cos φ 0 sin ψ ) + 2 cos ϑ 0 sin φ 0 ( α 1 sin 2 ψ - α 2 sin ψ cos ψ + α 3 cos 2 ψ ) ] Δ φ + [ - ( 2 / ρ 0 ) ( - cos ϑ 0 sin φ 0 sin ψ + sin ϑ 0 cos ψ ) ( + sin ϑ 0 sin φ 0 sin ψ + cos ϑ 0 cos ψ ) + 2 sin ϑ 0 cos φ 0 ( α 1 sin 2 ψ - α 2 sin ψ cos ψ + α 3 cos 2 ψ ) ] Δ ϑ } .
T 18 = - 1 2 u 2 [ Δ ρ / ( ρ 0 ) 2 ] [ 1 - ( cos ϑ 0 sin φ 0 cos ψ + sin ϑ 0 sin ψ ) 2 ] .
T 19 = - u v [ Δ ρ / ( ρ 0 ) 2 ] ( cos ϑ 0 sin φ 0 cos ψ + sin ϑ 0 sin ψ ) ( - cos ϑ 0 sin ψ 0 sin ψ + sin ϑ 0 cos ψ ) .
T 20 = - 1 2 v 2 [ Δ ρ / ( ρ 0 ) 2 ] [ 1 - ( - cos ϑ 0 sin φ 0 sin ψ + sin ϑ 0 cos ψ ) 2 ] .
- ( cos ϑ sin φ + cos ϑ 0 sin φ 0 ) cos ψ - ( sin ϑ + sin ϑ 0 ) sin ψ = - ( m λ / d ) cos ψ .
( cos ϑ sin φ + cos ϑ 0 sin φ 0 ) sin ψ - ( sin ϑ + sin ϑ 0 ) cos ψ = + ( m λ / d ) sin ψ .
sin ϑ + sin ϑ 0 = 0 ,
cos ϑ sin φ + cos ϑ 0 sin φ 0 = m λ / d .
F / u = + u ( C 3 + C 03 ) + v ( C 4 + C 04 ) = 0.
( 1 / ρ ) [ 1 - ( cos ϑ sin φ cos ψ + sin ϑ sin ψ ) 2 ] - 2 cos ϑ cos φ ( α 1 cos 2 ψ + α 2 sin ψ cos ψ + α 3 sin 2 ψ ) + ( 1 / ρ 0 ) [ 1 - ( cos ϑ 0 sin φ 0 cos ψ + sin ϑ 0 sin ψ ) 2 ] - 2 cos ϑ 0 cos φ 0 ( α 1 cos 2 ψ + α 2 sin ψ cos ψ + α 3 sin 2 ψ ) = 0 ,
- ( 1 / ρ ) ( cos ϑ sin φ cos ψ + sin ϑ sin ψ ) ( - cos ϑ sin φ sin ψ + sin ϑ cos ψ ) - cos ϑ cos φ [ - 2 α 1 sin ψ cos ψ + α 2 ( cos 2 ψ - sin 2 ψ ) + 2 α 3 sin ψ cos ψ ] - ( 1 / ρ 0 ) ( cos ϑ 0 sin φ 0 cos ψ + sin ϑ 0 sin ψ ) × ( - cos ϑ 0 sin φ 0 sin ψ + sin ϑ 0 cos ψ ) - cos ϑ 0 cos φ 0 × [ - 2 α 1 sin ψ cos ψ + α 2 ( cos 2 ψ - sin 2 ψ ) + 2 α 3 sin ψ cos ψ ] = 0.
A sin 2 ψ + B sin ψ cos ψ + C cos 2 ψ = 0 ,
A = ( + 2 α 3 sin ϑ cos ϑ sin φ 0 - α 2 cos 2 ϑ ) ( cos φ + cos φ 0 ) - ( 1 / ρ ) sin ϑ cos 2 ϑ ( sin φ + sin φ 0 ) .
B = [ - 2 α 1 cos 2 ϑ + 2 α 3 ( 1 - cos 2 ϑ sin 2 φ 0 ) ] ( cos φ + cos φ 0 ) - ( 1 / ρ ) cos 3 ϑ ( sin 2 φ - sin 2 φ 0 ) . C = [ - 2 α 1 sin ϑ cos ϑ sin φ 0 + α 2 ( 1 - cos 2 ϑ sin 2 φ 0 ) ] ( cos φ + cos φ 0 ) + ( 1 / ρ ) sin ϑ ( 1 - cos 2 ϑ sin φ sin φ 0 ) ( sin φ + sin φ 0 ) .
( F / u ) u = 0 = - cos ϑ 0 cos φ 0 cos ψ Δ φ + ( sin ϑ 0 sin φ 0 cos ψ - cos ϑ 0 sin ψ ) Δ ϑ + = 0.
sin ϕ = ( 1 / c ρ 0 ) ( - sin ϑ 0 sin φ 0 cos ψ + cos ϑ 0 sin ψ ) ,
cos ϕ = ( 1 / c ρ 0 ) ( + cos φ 0 cos ψ ) .
c = ( 1 / ρ 0 ) [ ( - sin ϑ 0 sin φ 0 cos ψ + cos ϑ 0 sin ψ ) 2 + ( + cos φ 0 cos ψ ) 2 ] 1 2 .
tan ϕ = - sin ϑ 0 tan φ 0 + ( cos ϑ 0 / cos φ 0 ) tan ψ .
F / v = + u ( C 4 + C 04 ) + v ( C 5 + C 05 ) .
F / v = + v ( C 5 + C 05 ) + cos ϑ 0 cos φ 0 sin ψ Δ φ - ( sin ϑ 0 sin φ 0 sin ψ + cos ϑ 0 cos ψ ) Δ ϑ = 0.
l = ρ 0 Δ θ ,
ρ 0 Δ ϑ = ρ 0 Δ θ cos ϕ ,
ρ 0 cos ϑ 0 Δ φ = - ρ 0 Δ θ sin φ ;
l = [ ( ρ 0 ) 2 c V / cos ϑ 0 cos φ 0 ] ( C 5 + C 05 ) .
F / v = + cos ϑ 0 cos φ 0 sin ψ Δ φ - ( sin ϑ 0 sin φ 0 sin ψ + cos ϑ 0 cos ψ ) Δ ϑ + = 0.
sin ϕ w = ( 1 / c w ρ 0 ) ( + cos φ 0 sin ψ ) ,
cos ϕ w = ( 1 / c w ρ 0 ) ( + sin ϑ 0 sin φ 0 sin ψ + cos ϑ 0 cos ψ ) ;
c w = ( 1 / ρ 0 ) [ ( + cos φ 0 sin ψ ) 2 + ( sin ϑ 0 sin φ 0 sin ψ + cos ϑ 0 cos ψ ) 2 ] 1 2 .
F / u = - cos ϑ 0 cos φ 0 cos ψ Δ φ + ( sin ϑ 0 sin φ 0 cos ψ - cos ϑ 0 sin ψ ) Δ ϑ + 3 2 u 2 ( C 6 + C 06 ) = 0.
ρ 0 Δ ϑ = ρ 0 Δ θ w sin φ w ,
ρ 0 cos ϑ 0 Δ φ = ρ 0 Δ θ w cos ϕ w .
Δ θ w = ( c w ρ 0 / cos ϑ 0 cos φ 0 ) × 3 2 u 2 ( c 6 + C 06 ) .
cos ϑ 0 Δ φ d = Δ θ w [ cos ( ϕ w - ϕ ) / cos ϕ ] = Δ θ w ( cos ϕ w + sin ϕ w tan ϕ ) .
cos ϑ 0 cos φ 0 Δ φ d = m Δ λ / d ,
m Δ λ / d = 3 2 ( u 2 / cos ψ ) ( c 6 + C 06 ) .
F / u = 3 2 u 2 ( C 6 + C 06 ) + 2 u 3 ( C 10 + C 010 ) - cos ϑ 0 cos φ 0 cos ψ Δ φ d + u [ - ( 2 / ρ 0 ) ( cos ϑ 0 sin φ 0 cos ψ + sin ϑ 0 sin ψ ) ( cos ϑ 0 cos φ 0 cos ψ ) + 2 cos ϑ 0 sin φ 0 ( α 1 cos 2 ψ + α 2 sin ψ cos ψ + α 3 sin 2 ψ ) ] Δ φ d = 0.
( m Δ λ / d ) = + 2 u 3 ( 1 / cos ψ ) ( C 10 + C 010 ) - 3 2 u 3 ( 1 / cos ψ ) ( C 6 + C 06 ) [ ( 2 / ρ 0 ) ( cos ϑ 0 sin φ 0 cos ψ + sin ϑ 0 sin ψ ) - 2 ( tan φ 0 / cos ψ ) ( α 1 cos 2 ψ + α 2 sin ψ cos ψ + α 3 sin 2 ψ ) ] .
- p + ( 1 / 2 R a ) q 2 + higher - order terms in q = 0.
c [ - ρ 0 cos ϑ 0 Δ φ ( cos φ ) - ρ 0 Δ ϑ ( sin φ ) + ( ρ 0 cos ϑ 0 Δ φ ) 2 × ( sin 2 φ / 2 R a ) - ( ρ 0 cos ϑ 0 Δ φ ) ( ρ 0 Δ ϑ ) ( sin φ cos φ / R a ) + ( ρ 0 Δ ϑ ) 2 ( cos 2 ϕ / 2 R a ) + ] = 0.
v = [ cos ϑ 0 / ρ 0 cos ψ ( C 5 + C 05 ) ] ( ρ 0 Δ ϑ ) .
1 R a = 2 c ( ρ 0 ) 2 cos 2 φ 0 cos 2 ψ { cos ϑ 0 sin φ 0 cos ψ + sin ϑ 0 sin ψ 2 ( ρ 0 ) 2 + cos 2 ϑ 0 ( C 8 + C 08 ) 2 cos 2 ψ ( ρ 0 ) 2 ( C 5 + C 05 ) 2 - cos ϑ 0 cos ψ ( ρ 0 ) 3 ( C 5 + C 05 ) [ ( cos ϑ 0 sin φ 0 cos ψ + sin ϑ 0 sin ψ ) ( sin ϑ 0 sin φ 0 sin ψ + cos ϑ 0 cos ψ ) + ( - cos ϑ 0 sin φ 0 sin ψ + sin ϑ 0 cos ψ ) ( - sin ϑ 0 sin φ 0 cos ψ + cos ϑ 0 sin ψ ) - ρ 0 sin ϑ 0 cos φ 0 ( - 2 α 1 sin ψ cos ψ + α 2 cos 2 ψ - α 2 sin 2 ψ + 2 α 3 sin ψ cos ψ ) ] } .
tan Ω = lim Δ ρ 0 ; ρ 0 Δ ϑ 0 ( Δ ρ / ρ 0 Δ ϑ 0 ) .
tan Ω = 1 1 - ( cos ϑ 0 sin φ 0 cos ψ + sin ϑ 0 sin ψ ) 2 × [ ρ 0 cos ϑ 0 cos ψ ( C 7 + C 07 ) ( C 5 + C 05 ) - 2 ( cos ϑ 0 sin φ 0 cos ψ + sin ϑ 0 sin ψ ) ( - sin ϑ 0 sin φ 0 cos ψ + cos ϑ 0 sin ψ ) + 2 ρ 0 sin ϑ 0 cos φ 0 ( α 1 cos 2 ψ + α 2 sin ψ cos ψ + α 3 sin 2 ψ ) ] .
tan ϕ * = cos Ω tan ϕ .
l * = l ( 1 + cos 2 ϕ tan 2 Ω ) 1 2 .
R a * = R a 1 + cos 2 ϕ tan 2 Ω ( 1 + sin 2 ϕ tan 2 Ω ) 1 2 .
tan α = - ( cos ϑ Δ φ / Δ ϑ ) .
tan ϕ s = - cos ϑ 0 Δ φ / Δ ϑ .
- sin ϑ sin φ + cos φ ( cos ϑ Δ φ / Δ ϑ ) + sin ϑ 0 sin φ 0 - cos φ 0 ( cos ϑ 0 Δ φ / Δ ϑ ) = 0.
tan ϕ s = ( cos φ / cos φ 0 ) tan α - ( sin ϑ 0 / cos φ 0 ) ( sin φ + sin φ 0 ) .
tan α = + sin ϑ 0 tan φ + ( cos ϑ 0 / cos φ ) tan ψ ,
2 α 1 = [ 1 / cos ϑ 0 ( cos φ + cos φ 0 ) ] { [ ( 1 - cos 2 ϑ 0 sin 2 φ ) / ρ ] + [ ( 1 - cos 2 ϑ 0 sin 2 φ 0 ) ρ 0 ] } .
2 α 2 = [ 2 sin ϑ 0 / ( cos φ + cos φ 0 ) ] [ sin φ / ρ ) - ( sin φ 0 / ρ 0 ) ] .
2 α 3 = ( cos ϑ 0 / cos φ + cos φ 0 ) [ ( 1 / ρ ) + ( 1 / ρ 0 ) ] .
C 3 = ( 1 / ρ ) - ( 1 / ρ ) ( cos ϑ sin φ cos ψ + sin ϑ sin ψ ) 2 - 2 cos ϑ cos φ ( α 1 cos 2 ψ + α 2 sin ψ cos ψ + α 3 sin 2 ψ ) .
C 4 = - ( 1 / ρ ) ( cos ϑ sin φ cos ψ + sin ϑ sin ψ ) ( - cos ϑ sin φ sin ψ + sin ϑ cos ψ ) - cos ϑ cos φ [ - 2 α 1 sin ψ cos ψ + α 2 ( cos 2 ψ - sin 2 ψ ) + 2 α 3 sin ψ cos ψ ] .
C 5 = ( 1 / ρ ) - ( 1 / ρ ) ( - cos ϑ sin φ sin ψ + sin ϑ cos ψ ) 2 - 2 cos ϑ cos φ ( α 1 sin 2 ψ - α 2 sin ψ cos ψ + α 3 cos 2 ψ ) .
C 6 = ( 1 / ρ ) ( cos ϑ sin φ cos ψ + sin ϑ sin ψ ) [ ( 1 / ρ ) - 2 cos ϑ cos φ ( α 1 cos 2 ψ + α 2 sin ψ cos ψ + α 3 sin 2 ψ ) ] - ( 1 / ρ 2 ) ( cos ϑ sin φ cos ψ + sin ϑ sin ψ ) 3 - 2 cos ϑ cos φ ( β 1 cos 3 ψ + β 2 sin ψ cos 2 ψ + β 3 sin 2 ψ cos ψ + β 4 sin 3 ψ ) .
C 7 = - ( 3 / ρ 2 ) ( cos ϑ sin φ cos ψ + sin ϑ sin ψ ) 2 ( - cos ϑ sin φ sin ψ + sin ϑ cos ψ ) - ( 2 cos ϑ cos φ / ρ ) ( cos ϑ sin φ cos ψ + sin ϑ sin ψ ) [ - 2 α 1 sin ψ cos ψ + α 2 ( cos 2 ψ - sin 2 ψ ) + 2 α 3 sin ψ cos ψ ] + ( 1 / ρ ) ( - cos ϑ sin φ sin ψ + sin ϑ cos ψ ) [ ( 1 / ρ ) - 2 cos ϑ cos φ ( α 1 cos 2 ψ + α 2 sin ψ cos ψ + α 3 sin 2 ψ ) ] - 2 cos ϑ cos φ [ - 3 β 1 sin ψ cos 2 ψ + β 2 ( - 2 sin 2 ψ cos ψ + cos 3 ψ ) + β 3 ( 2 sin ψ cos 2 ψ - sin 3 ψ ) + 3 β 4 sin 2 ψ cos ψ ] .
C 8 = - ( 3 / ρ 2 ) ( cos ϑ sin φ cos ψ + sin ϑ sin ψ ) ( - cos ϑ sin φ sin ψ + sin ϑ cos ψ ) 2 + ( 1 / ρ ) ( cos ϑ sin φ cos ψ + sin ϑ sin ψ ) [ ( 1 / ρ ) - 2 cos θ cos φ ( α 1 sin 2 ψ - α 2 sin ψ cos ψ + α 3 cos 2 ψ ) ] - ( 2 cos ϑ cos φ / ρ ) ( - cos ϑ sin φ sin ψ + sin ϑ cos ψ ) [ - 2 α 1 sin ψ cos ψ + α 2 ( cos 2 ψ - sin 2 ψ ) + 2 α 3 sin ψ cos ψ ] - 2 cos ϑ cos φ [ 3 β 1 sin 2 ψ cos ψ + β 2 ( sin 3 ψ - 2 sin ψ cos 2 ψ ) + β 3 ( cos 2 ψ - 2 sin 2 ψ cos ψ ) + 3 β 4 sin ψ cos 2 ψ ] .
C 9 = ( 1 / ρ ) ( - cos ϑ sin φ sin ψ + sin ϑ cos ψ ) [ ( 1 / ρ ) - 2 cos ϑ cos φ ( α 1 sin 2 ψ - α 2 sin ψ cos ψ + α 3 cos 2 ψ ) ] - ( 1 / ρ 2 ) ( - cos ϑ sin φ sin ψ + sin ϑ cos ψ ) 3 - 2 cos ϑ cos φ ( - β 1 sin 3 ψ + β 2 sin 2 ψ cos ψ - β 3 sin ψ cos 2 ψ + β 4 cos 3 ψ ) .
C 10 = ( 1 / ρ ) ( α 1 cos 2 ψ + α 2 sin ψ cos ψ + α 3 sin 2 ψ ) 2 - 2 cos ϑ cos φ ( γ 1 cos 4 ψ + γ 2 sin ψ cos 3 ψ + γ 3 sin 2 ψ cos 2 ψ + γ 4 sin 3 ψ cos ψ + γ 5 sin 4 ψ ) - ( 1 / 4 ρ ) [ ( 1 / ρ ) - 2 cos ϑ cos φ ( α 1 cos 2 ψ + α 2 sin ψ cos ψ + α 3 sin 2 ψ ) ] 2 - ( 2 cos ϑ cos φ / ρ ) ( cos ϑ sin φ cos ψ + sin ϑ sin ψ ) ( β 1 cos 3 ψ + β 2 sin ψ cos 2 ψ + β 3 sin 2 cos ψ + β 4 sin 3 ψ ) + 3 2 ( 1 / ρ 2 ) ( cos ϑ sin φ cos ψ + sin ϑ sin ψ ) 2 [ ( 1 / ρ ) - 2 cos ϑ cos φ ( α 1 cos 2 ψ + α 2 sin ψ cos ψ + α 3 sin 2 ψ ) ] - 5 4 ( 1 / ρ 3 ) ( cos ϑ sin φ cos ψ + sin ϑ sin ψ ) 4 .

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