Abstract

Expressions are derived for the sixteen wave front aberration coefficients of a single grating. These give the wave aberrations to the fourth order for a plane symmetric grating system. The contributions from each mirror and grating can be added to give the aberrations in the final image. Problems encountered with intermediate astigmatic images are overcome by defining the wave front aberration with respect to an astigmatic reference surface. There is one field variable describing displacement of the object point from the symmetry plane. The aperture stop may be placed anywhere in the system, and equations are given for the aberration changes produced by shifting this position.

© 1983 Optical Society of America

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References

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  1. W. A. Rense, T. Violett, J. Opt. Soc. Am 49, 139 (1959).
    [CrossRef]
  2. M. P. Chrisp, Appl. Opt. 22, 1519 (1983).
    [CrossRef] [PubMed]
  3. H. Noda, T. Namioka, M. Seya, J. Opt. Soc. Am. 64, 1031 (1974).
    [CrossRef]
  4. F. Masuda, H. Noda, T. Namioka, J. Spectrosc. Soc. Jpn. 27, 211 (1978).
    [CrossRef]
  5. D. Lepere, Nouv. Rev. Opt. 6, 173 (1975).
    [CrossRef]
  6. C. H. F. Velzel, J. Opt. Soc. Am. 66, 346 (1976); errata: 67, 1695 (1977).
    [CrossRef]
  7. G. R. Rosendahl, J. Opt. Soc. Am. 51, 1 (1961); 52, 408, 412 (1962).
    [CrossRef]
  8. H. G. Buetler, J. Opt. Soc. Am. 35, 311 (1945).
    [CrossRef]
  9. H. A. Buchdahl, J. Opt. Soc. Am. 62, 1314 (1972).
    [CrossRef]
  10. C. G. Wynne, Proc. Phys. Soc. B 67, 529 (1954).
    [CrossRef]
  11. R. W. Smith, Opt. Commun. 19, 245 (1976).
    [CrossRef]
  12. W. T. Welford, Aberrations of the Symmetrical Optical System (Academic Press, London, 1974).
  13. M. P. Chrisp, “The Theory of Holographic Toroidal Grating Systems,” Ph.D. Thesis, London University (June1981).

1983 (1)

1978 (1)

F. Masuda, H. Noda, T. Namioka, J. Spectrosc. Soc. Jpn. 27, 211 (1978).
[CrossRef]

1976 (2)

1975 (1)

D. Lepere, Nouv. Rev. Opt. 6, 173 (1975).
[CrossRef]

1974 (1)

1972 (1)

1961 (1)

1959 (1)

W. A. Rense, T. Violett, J. Opt. Soc. Am 49, 139 (1959).
[CrossRef]

1954 (1)

C. G. Wynne, Proc. Phys. Soc. B 67, 529 (1954).
[CrossRef]

1945 (1)

Buchdahl, H. A.

Buetler, H. G.

Chrisp, M. P.

M. P. Chrisp, Appl. Opt. 22, 1519 (1983).
[CrossRef] [PubMed]

M. P. Chrisp, “The Theory of Holographic Toroidal Grating Systems,” Ph.D. Thesis, London University (June1981).

Lepere, D.

D. Lepere, Nouv. Rev. Opt. 6, 173 (1975).
[CrossRef]

Masuda, F.

F. Masuda, H. Noda, T. Namioka, J. Spectrosc. Soc. Jpn. 27, 211 (1978).
[CrossRef]

Namioka, T.

F. Masuda, H. Noda, T. Namioka, J. Spectrosc. Soc. Jpn. 27, 211 (1978).
[CrossRef]

H. Noda, T. Namioka, M. Seya, J. Opt. Soc. Am. 64, 1031 (1974).
[CrossRef]

Noda, H.

F. Masuda, H. Noda, T. Namioka, J. Spectrosc. Soc. Jpn. 27, 211 (1978).
[CrossRef]

H. Noda, T. Namioka, M. Seya, J. Opt. Soc. Am. 64, 1031 (1974).
[CrossRef]

Rense, W. A.

W. A. Rense, T. Violett, J. Opt. Soc. Am 49, 139 (1959).
[CrossRef]

Rosendahl, G. R.

Seya, M.

Smith, R. W.

R. W. Smith, Opt. Commun. 19, 245 (1976).
[CrossRef]

Velzel, C. H. F.

Violett, T.

W. A. Rense, T. Violett, J. Opt. Soc. Am 49, 139 (1959).
[CrossRef]

Welford, W. T.

W. T. Welford, Aberrations of the Symmetrical Optical System (Academic Press, London, 1974).

Wynne, C. G.

C. G. Wynne, Proc. Phys. Soc. B 67, 529 (1954).
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am (1)

W. A. Rense, T. Violett, J. Opt. Soc. Am 49, 139 (1959).
[CrossRef]

J. Opt. Soc. Am. (5)

J. Spectrosc. Soc. Jpn. (1)

F. Masuda, H. Noda, T. Namioka, J. Spectrosc. Soc. Jpn. 27, 211 (1978).
[CrossRef]

Nouv. Rev. Opt. (1)

D. Lepere, Nouv. Rev. Opt. 6, 173 (1975).
[CrossRef]

Opt. Commun. (1)

R. W. Smith, Opt. Commun. 19, 245 (1976).
[CrossRef]

Proc. Phys. Soc. B (1)

C. G. Wynne, Proc. Phys. Soc. B 67, 529 (1954).
[CrossRef]

Other (2)

W. T. Welford, Aberrations of the Symmetrical Optical System (Academic Press, London, 1974).

M. P. Chrisp, “The Theory of Holographic Toroidal Grating Systems,” Ph.D. Thesis, London University (June1981).

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Figures (9)

Fig. 1
Fig. 1

Ray paths through an astigmatic system.

Fig. 2
Fig. 2

Ray paths through an astigmatic holographic grating.

Fig. 3
Fig. 3

Rays of the object pencil incident on the grating.

Fig. 4
Fig. 4

Coordinates of the ray pencils.

Fig. 5
Fig. 5

Aperture ray transfer between surfaces.

Fig. 6
Fig. 6

Field ray transfer between surfaces.

Fig. 7
Fig. 7

Shifting the stop position.

Fig. 8
Fig. 8

Transverse aberrations of an astigmatic wave front.

Fig. 9
Fig. 9

Transverse aberrations with an oblique exit pupil.

Tables (2)

Tables Icon

Table I Coefficients of the Object Pencil Aberration Contribution f(Substitutions are Shown at the Side).

Tables Icon

Table II Classification of the Aberration Coefficients

Equations (86)

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W = Q Σ Q = Q + Q Σ - Q + Q ,
W = O + O - Q + Q .
W = O + O - Q y + Q y + Q y + Q y - Q + Q .
O + O - Q y + Q y = S O S - S Q y S
Q y + Q y - Q + Q = M Q y M - M Q M
W = S O S - S Q y S + M Q y M - M Q M .
W = S P ¯ S - S P y S + M P y M - M P M .
S P y S = S P y S + n y m λ , M P y M = M P y M + n y m λ , M P M = M P M + n m λ .
W = S P ¯ S - S P y S + M P y M - M P M - n m λ .
n = ( p . d . at P ¯ ) - ( p . d . at P ) λ 0 ,
n = ( 1 / λ 0 ) [ ( C P ¯ - D P ¯ ) - ( C P - D P ) ] .
W = S P ¯ S - S P y S + M P y M - M P M - m λ λ 0 [ ( C P ¯ - D P ¯ ) - ( C P - D P ) ] .
object pencil Γ a = S P ¯ - S P y + M P y - M P ,
image pencil Γ b = S P ¯ - S P y + M P y - M P ,
construction pencil Γ c = C P ¯ - C P ,
Γ d = D P ¯ - D P .
W = Γ a + Γ b - m λ λ 0 ( Γ c - Γ d ) .
M [ r m sin α , y ¯ + y ( 1 - r m r s ) + r m tan φ , r m cos α ] .
M P = [ ( x - r m sin α ) 2 + ( y r s - u ) 2 r m 2 + ( z - r m cos α ) 2 ] 1 / 2 .
δ L = ( δ η ) 2 2 r m
z = R - R [ 1 - ( χ 2 + η 2 ) R 2 + 2 ρ R ( ρ R - 1 ) ( 1 - 1 - η 2 ρ 2 ) ] 1 / 2 ,
z = χ 2 2 R + η 2 2 ρ + χ 4 8 R 3 + η 4 8 ρ 3 + χ 2 η 2 4 R 2 ρ + O ( χ 6 R 5 ) .
χ = x η = y + y ¯ = y - u l y ¯ = - u l .
z = x 2 2 R + ( y - u l ) 2 2 ρ + x 4 8 R 3 + ( y - u l ) 4 8 ρ 3 + x 2 ( y - u l ) 2 4 R 2 ρ .
M P = r m [ 1 + τ 2 - τ 2 8 + τ 3 16 - 5 τ 4 128 + O ( τ 5 ) ] .
M P = i j k 4 x i j k x i y j u k ,
M P y = [ i j k 4 x i j k x i y j u k ] x = 0 = j k 4 x o j k y j u k .
S P y = [ i j k 4 [ x i j k ] r m = r s x i y j u k ] x = 0 = j k 4 [ x o j k ] r m = r s y j u k ,
S P ¯ = [ i j k 4 [ x i j k ] r m = r s x i y i u k ] x = 0 y = 0 = k 4 [ x o o k ] r m = r s u k .
Γ a = k 4 [ x o o k ] r m = r s u k - j k 4 [ x o j k ] r m = r s y j u k + j k 4 x o j k y j u k - i j k 4 x i j k x i y j u k .
i = 0 j = 0 a o o k = 0 , i = 0 j 0 a o j k = [ x o j k ] r m = r s K o j k , i 0 j 0 a i j k = x i j k K i j k .
Γ b = - i j k 4 K i j k b i j k x i y j u k , Γ c = - i j k 4 K i j k c i j k x i y j u c k , Γ d = - i j k 4 K i j k d i j k x i y j u d k .
image pencil b i j k = f i j k ( β , r m , r s , l ) ,
construction pencils c i j k = f i j k ( γ , r c , r c , r c ) ,
d i j k = f i j k ( δ , r d , r d , r d ) .
W = i j k 4 K i j k [ - a i j k u k - b i j k u k - m λ λ 0 ( d i j k u d k - c i j k u c k ) ] x i y j .
x n = x / X ,             y n = y / Y ,             u n = u / U .
u = - y ¯ l ,             u = - y ¯ l ,             u c = - y ¯ r c ,             u d = - y ¯ r d ,
U = - Y ¯ l , U = - Y ¯ l , U c = - Y ¯ r c , U d = - Y ¯ r d .
u = u n U , u = u n U , u c = u n U c , u d = u n U d ;
x = x n X , y = y n Y .
W = i j k 4 K i j k [ - a i j k U k - b i j k U k - m λ λ 0 × ( d i j k U d k - c i j k U c k ) ] X i Y j x n i y n j u n k .
W = i j k 4 K i j k W i j k x n i y n j u n k .
W i j k = [ - a i j k U k - b i j k U k - m λ λ 0 ( d i j k U d k - c i j k U c k ) ] X i Y j ,
W i j k = [ W i j k ] 1 + [ W i j k ] 2 + [ W i j k ] 3 + + [ W i j k ] g .
W = W 100 x n + ½ W 200 x n 2 + ½ W 020 y n 2 + W 011 y n u n             } 2 nd order , + ½ W 300 x n 3 + ½ W 120 x n y n 2 + W 111 x n y n u n + ½ W 102 x n u n 2 } 3 rd order , + W 400 x n 4 + W 040 y n 4 + ¼ W 220 x n 2 y n 2 + ½ W 031 y n 3 u n + ½ W 211 x n 2 y n u n + ¼ W 202 x n 2 u n 2 + ¼ W 022 y n 2 u n 2 + ½ W 013 y n u n 3 } 4 th order .
sin α + sin β + m λ λ 0 ( sin δ - sin γ ) = 0.
σ = λ 0 sin δ - sin γ .
r m = cos 2 β { cos β R - T ( α , r m ) - m λ λ 0 [ T ( δ , r d ) - T ( γ , r c ) ] } - 1 .
r s = { cos β ρ - S ^ ( α , r s ) - m λ λ 0 [ S ^ ( δ , r c ) - S ^ ( γ , r d ) ] } - 1 .
Y 2 = Y 1 ( 1 - d r s )
X 2 = X 1 ( d r m - 1 ) cos β 1 cos α 2 .
Y ¯ 2 = Y ¯ 1 ( 1 - d l ) .
M i j k = - a i j k U k - b i j k U k ,
H i j k = d i j k U d k - c i j k U c k .
W i j k = ( M i j k - m λ λ 0 H i j k ) X i Y j .
Y ¯ = Y H s E ,
E + 1 - E = - d Y Y + 1 .
Δ E = ζ h ( h + ζ ν s ) ,
Δ W i j k = X i Y j { - Δ ( a i j k U k ) - Δ ( b i j k U k ) - m λ λ 0 [ Δ ( d i j k U d k ) - Δ ( c i j k U c k ) ] } .
A i j k = a i j k U k B i j k = b i j k U k C i j k = c i j k U c k D i j k = d i j k U d k .
Δ W i j k = [ - Δ A i j k - Δ B i j k - m λ λ 0 ( Δ D i j k - Δ C i j k ) ] X i Y j .
U = - Y ¯ / l , A i j k = a i j k [ - Y ¯ l ] k ,
A i j k = f [ Y ¯ , Y ¯ 2 , Y ¯ 3 , Y ¯ l , Y ¯ 2 l 2 , Y ¯ 3 l 3 , Y ¯ 2 l , Y ¯ 3 l , Y ¯ 3 l 2 ] .
Y ¯ = Y H s E ,
after the shift             Y ¯ s = Y H s ( E + Δ E )
Y ¯ l = Y ¯ - η ˜ s r s = Y H s E - η ˜ s r s ,
( Y ¯ l ) s = Y H s ( E + Δ E ) - η ˜ s r s .
Δ Y ¯ Y Y ¯ 2 2 Y ¯ Y + Y 2 2 Y ¯ l Y r s Y ¯ 2 l 2 2 Y ¯ Y l r s + Y 2 2 r s 2 ,
Δ W 011 = W 020 , Δ W 111 = W 120 , Δ W 102 = 2 W 111 + 2 W 120 , Δ W 211 = W 220 , Δ W 202 = 2 W 211 + 2 W 220 , Δ W 031 = W 040 , Δ W 022 = 6 W 031 + 3 2 W 040 , Δ W 013 = W 022 + 3 2 W 031 + 3 W 040 . Δ W 013 = W 022 + 3 2 W 031 + 3 W 040 .
δ ϑ y = d W / d y 1 ,
δ ϑ x = d W / d x 1 .
δ x = - ( P ¯ F m ) d W d x 1 ,             δ y = - ( P ¯ F s ) d W d y 1 .
δ x = - r m d W d x 1 ,
δ y = - r s d W d y 1 .
δ x = - r m ( d x n d x 1 ) ( d W d x n ) δ y = - r s ( d y n d y 1 ) ( d W d y n ) = - r m X d W d x n = - r s Y d W d y n .
δ x = - r d W d x 1             δ y = - r d W d y 1 ,
δ x = - r ( d W d y d y d x 1 + d W d x d x d x 1 ) ,
δ y = - r ( d W d y d y d y 1 + d W d x d x d y 1 ) .
x - r sin β x 1 cos β - r sin β = y y 1 = r cos β x 1 sin β + r cos β .
x = x 1 r x 1 sin β + r cos β             y = y 1 r cos β x 1 sin β + r cos β .
d y / d x 1 , d x / d x 1 , d y / d y 1 , d x / d y 1
x 1 = x r cos β r - x sin β             y 1 = y r r - x sin β .
δ x = - ( r - x sin β ) r cos β [ ( r - x sin β ) d W d x - y sin β d W d y ] , δ y = - ( r - x sin β ) d W d y ,
δ x = - ( r - x n X sin β ) r cos β [ ( r X - x n sin β ) d W d x n - y n sin β d W d y n ] ,
δ y = - ( r - x n X sin β ) Y d W d y n .

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