Abstract

The optical properties have been studied analytically for a system consisting of a mirror and a grating, without imposing much restrictions on the manner of arranging the optical elements. For the sake of simplicity calculations were limited only to rays in the meridional plane of the system. Analytical expressions are given for the focal condition and a coma-type aberration. Remedies are described for eliminating the coma-type aberration at one wavelength, of designer’s choice, without use of any auxiliary optics.

© 1970 Optical Society of America

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References

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  1. F. L. O. Wadsworth, Astrophys. J. 3, 54 (1896).
  2. H. G. Beutler, J. Opt. Soc. Amer. 35, 311 (1945).
    [CrossRef]
  3. M. Seya, T. Namioka, Sci. Light (Tokyo) 16, 158 (1967); refer also to the references cited therein.
  4. G. S. Monk, J. Opt. Soc. Amer. 17, 358 (1928).
    [CrossRef]
  5. H. T. Smyth, J. Opt. Soc. Amer. 25, 312 (1935).
    [CrossRef]
  6. A. H. C. P. Gillieson, J. Sci. Instrum. 26, 335 (1949).
    [CrossRef]
  7. M. Seya, T. Namioka, T. Sai, Sci. Light (Tokyo) 16, 138 (1967); refer also to the references cited therein.
  8. M. V. R. K. Murty, Rev. Sci. Instrum. 32, 1155 (1961).
    [CrossRef]
  9. K. P. Miyake, Sci. Light (Tokyo) 8, 39 (1959).

1967 (2)

M. Seya, T. Namioka, Sci. Light (Tokyo) 16, 158 (1967); refer also to the references cited therein.

M. Seya, T. Namioka, T. Sai, Sci. Light (Tokyo) 16, 138 (1967); refer also to the references cited therein.

1961 (1)

M. V. R. K. Murty, Rev. Sci. Instrum. 32, 1155 (1961).
[CrossRef]

1959 (1)

K. P. Miyake, Sci. Light (Tokyo) 8, 39 (1959).

1949 (1)

A. H. C. P. Gillieson, J. Sci. Instrum. 26, 335 (1949).
[CrossRef]

1945 (1)

H. G. Beutler, J. Opt. Soc. Amer. 35, 311 (1945).
[CrossRef]

1935 (1)

H. T. Smyth, J. Opt. Soc. Amer. 25, 312 (1935).
[CrossRef]

1928 (1)

G. S. Monk, J. Opt. Soc. Amer. 17, 358 (1928).
[CrossRef]

1896 (1)

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

Beutler, H. G.

H. G. Beutler, J. Opt. Soc. Amer. 35, 311 (1945).
[CrossRef]

Gillieson, A. H. C. P.

A. H. C. P. Gillieson, J. Sci. Instrum. 26, 335 (1949).
[CrossRef]

Miyake, K. P.

K. P. Miyake, Sci. Light (Tokyo) 8, 39 (1959).

Monk, G. S.

G. S. Monk, J. Opt. Soc. Amer. 17, 358 (1928).
[CrossRef]

Murty, M. V. R. K.

M. V. R. K. Murty, Rev. Sci. Instrum. 32, 1155 (1961).
[CrossRef]

Namioka, T.

M. Seya, T. Namioka, T. Sai, Sci. Light (Tokyo) 16, 138 (1967); refer also to the references cited therein.

M. Seya, T. Namioka, Sci. Light (Tokyo) 16, 158 (1967); refer also to the references cited therein.

Sai, T.

M. Seya, T. Namioka, T. Sai, Sci. Light (Tokyo) 16, 138 (1967); refer also to the references cited therein.

Seya, M.

M. Seya, T. Namioka, Sci. Light (Tokyo) 16, 158 (1967); refer also to the references cited therein.

M. Seya, T. Namioka, T. Sai, Sci. Light (Tokyo) 16, 138 (1967); refer also to the references cited therein.

Smyth, H. T.

H. T. Smyth, J. Opt. Soc. Amer. 25, 312 (1935).
[CrossRef]

Wadsworth, F. L. O.

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

Astrophys. J. (1)

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

J. Opt. Soc. Amer. (3)

H. G. Beutler, J. Opt. Soc. Amer. 35, 311 (1945).
[CrossRef]

G. S. Monk, J. Opt. Soc. Amer. 17, 358 (1928).
[CrossRef]

H. T. Smyth, J. Opt. Soc. Amer. 25, 312 (1935).
[CrossRef]

J. Sci. Instrum. (1)

A. H. C. P. Gillieson, J. Sci. Instrum. 26, 335 (1949).
[CrossRef]

Rev. Sci. Instrum. (1)

M. V. R. K. Murty, Rev. Sci. Instrum. 32, 1155 (1961).
[CrossRef]

Sci. Light (Tokyo) (3)

K. P. Miyake, Sci. Light (Tokyo) 8, 39 (1959).

M. Seya, T. Namioka, Sci. Light (Tokyo) 16, 158 (1967); refer also to the references cited therein.

M. Seya, T. Namioka, T. Sai, Sci. Light (Tokyo) 16, 138 (1967); refer also to the references cited therein.

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Figures (1)

Fig. 1
Fig. 1

Schematic diagram of a system consisting of a spherical mirror and a concave grating. The figure plane is the meridional plane of the system, which is perpendicular to the grating rulings and contains the mirror normal OCM, the grating normal G0CG, and a point light source A.

Equations (50)

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F = A P + P Q + Q B + ( s m λ / σ ) ,
A P = [ ( x ξ ) 2 + ( y w ) 2 ] 1 2 , P Q = [ ( ξ ξ ) 2 + ( w w ) 2 ] 1 2 , and Q B = [ ( ξ x ) 2 + ( w y ) 2 ] 1 2 . }
( ξ x G ) 2 + ( w y G ) 2 = R 2 ,
( ξ ξ ) / L = ( w w ) / M ,
ξ = L M ( y G w ) + L 2 x G + M 2 ξ ± L { R 2 [ M ( x G ξ ) L ( y G w ) ] 2 } 1 2
w = L M ( x G ξ ) + L 2 w + M 2 y G ± M { R 2 [ M ( x G ξ ) L ( y G w ) ] 2 } 1 2 .
L = 2 L ( L L + M M ) L = 1 A P { 2 ( ρ ξ ) ρ 2 [ x ρ + ξ ( ρ x ) w y ] + ξ x }
M = 2 M ( L L + M M ) M = ( 1 / A P ) { ( 2 w / ρ 2 ) [ x ρ + ξ ( ρ x ) w y ] + y w } .
x = r cos θ , y = r sin θ , x = D cos θ r cos ψ , y = D sin θ r sin ψ , x G = D cos θ R cos φ , and y G = D sin θ R sin φ . }
ξ 2 + w 2 = 2 ξ ρ or ξ = ( w 2 / 2 ρ ) + ( w 4 / 8 ρ 3 ) +
ξ = D cos θ w sin φ ρ cos ( θ + φ ) [ ρ cos θ D ( 2 ρ r cos θ ) ] w 2 R ρ 2 cos 3 ( θ + φ ) { R ρ sin φ cos ( θ + φ ) [ 1 2 sin θ cos ( θ + φ ) + sin φ × ( 2 ρ r cos θ ) ] D R sin φ cos ( θ + φ ) × [ sin ( θ + φ ) × ( 2 ρ r cos θ ) 2 + ρ 2 r sin θ cos ( θ + φ ) × ( 1 2 ρ r cos θ ) ] + 1 2 cos θ [ ρ cos θ D ( 2 ρ r cos θ ) ] 2 } + higher order terms
w = D sin θ + w cos φ ρ cos ( θ + φ ) [ ρ cos θ D ( 2 ρ r cos θ ) ] + w 2 R ρ 2 cos 3 ( θ + φ ) { R ρ cos φ cos ( θ + φ ) [ 1 2 sin θ cos ( θ + φ ) + sin φ ( 2 ρ r cos θ ) ] D R cos φ cos ( θ + φ ) × [ sin ( θ + φ ) × ( 2 ρ r cos θ ) 2 + ρ 2 r sin θ cos ( θ + φ ) × ( 1 2 ρ r cos θ ) ] + 1 2 sin θ [ ρ cos θ D ( 2 ρ r cos θ ) ] 2 } + higher order terms .
F = r + D + r + s m λ σ w ρ cos ( θ + φ ) [ sin ( θ + φ ) + sin ( φ ψ ) ] × [ ρ cos θ D ( 2 ρ r cos θ ) ] w 2 2 R ρ 2 cos 3 ( θ + φ ) × { R ( 2 ρ r cos θ ) cos 3 ( θ + φ ) [ ρ cos θ D ( 2 ρ r cos θ ) ] + [ 1 + cos ( θ + ψ ) R r cos ( θ + φ ) cos 2 ( φ ψ ) ] × [ ρ cos θ D ( 2 ρ r cos θ ) ] 2 + R cos ( θ + φ ) × [ sin ( θ + φ ) + sin ( φ ψ ) ] [ 2 ρ sin φ × ( 2 ρ r cos θ ) + ρ sin θ cos ( θ + φ ) 2 D sin ( θ + φ ) × ( 2 ρ r cos θ ) 2 D ρ r sin θ cos ( θ + φ ) × ( 1 2 ρ r cos θ ) ] } + higher order terms .
s = D sin ( θ + φ ) ξ sin φ + w cos φ .
s = a 1 w + a 2 w 2 + a 3 w 3 + ,
a 1 = 1 ρ cos ( θ + φ ) [ ρ cos θ D ( 2 ρ r cos θ ) ] ,
a 2 = 1 R ρ 2 cos 3 ( θ + φ ) { R ρ cos ( θ + φ ) [ 1 2 sin θ cos ( θ + φ ) + sin φ × ( 2 ρ r cos θ ) ] D R cos ( θ + φ ) [ sin ( θ + φ ) × ( 2 ρ r cos θ ) 2 + ρ 2 r sin θ cos ( θ + φ ) × ( 1 2 ρ r cos θ ) ] + 1 2 sin ( θ + φ ) × [ ρ cos θ D ( 2 ρ r cos θ ) ] 2 } ,
w = s a 1 a 2 a 1 3 s 2 + 1 a 1 4 ( 2 a 2 2 a 1 a 3 ) s 3 +
F = r + D + r + s [ m λ σ ( sin α + sin β ) ] + s 2 2 [ cos 2 β r cos α + cos β R cos α a 1 ρ ( 2 ρ r cos θ ) ] + s 3 2 2 sin θ a 1 3 ρ 2 ( 1 ρ r cos θ ) + sin α a 1 2 ρ 2 ( 2 ρ r cos θ ) 2 + sin α a 1 R ρ ( 2 ρ r cos θ ) + 1 r sin β cos β × ( cos β r 1 R ) ] + higher order terms .
θ + φ = α and φ ψ = β ,
d F / d s = 0.
sin α + sin β = m λ / σ .
cos 2 β r cos α + cos β R cos α a 1 ρ ( 2 ρ r cos θ ) = 0.
r = cos 2 β [ cos α + cos β R + cos α a 1 ρ ( 2 ρ r cos θ ) ] 1 .
cos 2 α r + D + cos 2 β r cos α + cos β R = 0.
r + D = R cos α and r = R cos β .
r = ( ρ r cos θ 2 r ρ cos θ D ) cos 2 β cos 2 α .
r = R cos 2 β / ( cos α + cos β ) .
r = ρ cos θ , D = ρ cos θ + R cos α , and r = R cos β , }
Δ p = r sec β × ( d F / d s ) ,
d F d s = 3 s 2 2 { 2 sin θ a 1 3 ρ 2 ( 1 ρ r cos θ ) + 1 cos 2 β [ 1 R + 1 a 1 ρ ( 2 ρ r cos θ ) ] × [ cos α sin β R ( cos α + cos β ) + 1 a 1 ρ ( sin α cos 2 β + cos 2 α sin β ) ( 2 ρ r cos θ ) ] } + higher order terms .
2 sin θ a 1 3 ρ 2 ( 1 ρ r cos θ ) + 1 cos 2 β [ 1 R + 1 a 1 ρ ( 2 ρ r cos θ ) ] × [ cos α sin β R ( cos α + cos β ) + 1 a 1 ρ ( sin α cos 2 β + cos 2 α sin β ) ( 2 ρ r cos θ ) ] = 0.
cos α cos 2 β ( 1 R cos α r + D ) [ sin β R ( cos α + cos β ) 1 r + D ( sin α cos 2 β + cos 2 α sin β ) ] = 0.
r + D = R cos α and r = R cos β
r + D = R sin α cos 2 β + cos 2 α sin β sin β ( cos α + cos β ) and r = R sin α cos 2 β + cos 2 α sin β sin α ( cos α + cos β ) . }
r = 1 2 ρ cos θ [ 1 + ( sin θ / Λ ) 1 2 ] ,
Λ = sin θ 2 [ 1 ( 1 / n ) ] cos θ [ tan α + tan β ( cos α / cos β ) ] > 0 and n > 1. }
Δ p = ( 3 s 2 / ρ ) cos θ cos β [ tan 2 θ + tan β ( cos 2 θ / cos β ) ] .
ρ R = [ 2 sin θ cos 2 α cos 2 β cos 3 θ sin β ( cos α + cos β ) ] 1 2 ,
r = 1 2 ρ cos θ + Δ ,
r = R cos 2 β cos α + cos β + 4 R 2 Δ ρ 2 cos 2 θ ( sec α + sec β ) 2
2 tan θ cos α + ρ 2 R 2 cos 2 θ tan β ( sec α + sec β ) + 4 Δ ρ cos θ [ 2 tan θ cos α × ( 3 D ρ cos θ 1 ) + ρ cos θ R ( tan α + tan β + 2 tan β cos α cos β ) ] = 0 ,
D = 1 n ( ρ r cos θ 2 r ρ cos θ ) ,
( sin θ Λ ) cos 2 θ 2 r 2 cos θ r ( Φ R 2 Λ ρ ) + Ψ R 2 2 Λ ρ 2 + 2 Φ ρ R = 0 ,
Λ = sin θ 2 [ 1 ( 1 / n ) cos θ [ tan α + tan β ( cos α / cos β ) ] , Φ = [ 1 ( 1 / n ) ] 2 cos 2 θ sec α [ tan α + tan β + 2 tan β ( cos α / cos β ) ] , and Ψ = [ 1 ( 1 / n ) ] 3 cos 3 θ sec α tan β ( sec α + sec β ) . }
r = 1 2 cos θ { Φ R 2 Λ ρ ± [ 1 R 2 ( 1 1 n ) 4 cos 4 θ sec 2 α × ( tan α tan β ) 2 4 Φ sin θ ρ R + 4 Λ sin θ ρ 2 ] 1 2 } × ( Ψ R 2 + 2 Φ ρ R 2 Λ ρ 2 ) 1 .
1 ρ r cos θ = 0 and 1 R + 1 a 1 ρ ( 2 ρ r cos θ ) = 0.
r = ρ cos θ , D = ρ cos θ + R cos α , and r = R cos β , }
1 ( ρ / r ) cos θ = 0 and ( cos α sin β / R ) ( cos α + cos β ) + ( 1 / α 1 ρ ) ( sin α cos 2 β + cos 2 α sin β ) [ 2 ( ρ / r ) cos θ ] = 0. }
r = ρ cos θ , D = ρ cos θ + R sin α cos 2 β + cos 2 α sin β sin β ( cos α + cos β ) , and r = R sin α cos 2 β + cos 2 α sin β sin α ( cos α + cos β ) . }

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