Abstract

In astrophysics, holographic gratings use, as far as possible, spherical grating's surface and printing laser waves. However, to obtain high spectral resolution, aberrations must be corrected up to the fourth degree, which generally requires aspheric surfaces for the grating or the laser waves. To date, technological progress makes the fabrication of such gratings possible, so we have developed a complete theory of these deformations and have written a computer program which we believe solves any grating's aberration problems.

© 1987 Optical Society of America

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References

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  1. M. Duban, “Calcul d'un réseau holographique torique destiné au montage Caméra Grand Champ,” J. Opt. 9, 163 (1978).
    [CrossRef]
  2. M. Duban, “Etude d'un montage de Rowland holographique à résolution modérée en dispersion conique faible,” CNRS Laboratoire d'Astronomie Spatiale, Marseille; (1984).
  3. M. Duban, “Improved Wadsworth mounting with aspherical holographic grating,” Appl. Opt. 19, 2488 (1980).
    [CrossRef] [PubMed]
  4. M. Duban, “Le montage de Wadsworth holographique asphérique en astronomie spatiale,” C. R. Acad. Sci. 302, 1215 (1986).
  5. R. Prangé, M. Duban, A. Vidal-Madjar et al., “A 30-cm Objective Grating for Far UV Astronomy: Theoretical Study and Laboratory Tests,” Appl. Opt., to be submitted.
  6. W. C. Cash, “Aspheric Concave Grating Spectrographs,” Appl. Opt. 23, 4518 (1984).
    [CrossRef] [PubMed]
  7. M. Duban, “Aspheric Gratings: Recent Developments,” Appl. Opt. 24, 3316 (1985).
    [CrossRef] [PubMed]

1986

M. Duban, “Le montage de Wadsworth holographique asphérique en astronomie spatiale,” C. R. Acad. Sci. 302, 1215 (1986).

1985

1984

1980

1978

M. Duban, “Calcul d'un réseau holographique torique destiné au montage Caméra Grand Champ,” J. Opt. 9, 163 (1978).
[CrossRef]

Cash, W. C.

Duban, M.

M. Duban, “Le montage de Wadsworth holographique asphérique en astronomie spatiale,” C. R. Acad. Sci. 302, 1215 (1986).

M. Duban, “Aspheric Gratings: Recent Developments,” Appl. Opt. 24, 3316 (1985).
[CrossRef] [PubMed]

M. Duban, “Improved Wadsworth mounting with aspherical holographic grating,” Appl. Opt. 19, 2488 (1980).
[CrossRef] [PubMed]

M. Duban, “Calcul d'un réseau holographique torique destiné au montage Caméra Grand Champ,” J. Opt. 9, 163 (1978).
[CrossRef]

R. Prangé, M. Duban, A. Vidal-Madjar et al., “A 30-cm Objective Grating for Far UV Astronomy: Theoretical Study and Laboratory Tests,” Appl. Opt., to be submitted.

M. Duban, “Etude d'un montage de Rowland holographique à résolution modérée en dispersion conique faible,” CNRS Laboratoire d'Astronomie Spatiale, Marseille; (1984).

Prangé, R.

R. Prangé, M. Duban, A. Vidal-Madjar et al., “A 30-cm Objective Grating for Far UV Astronomy: Theoretical Study and Laboratory Tests,” Appl. Opt., to be submitted.

Vidal-Madjar, A.

R. Prangé, M. Duban, A. Vidal-Madjar et al., “A 30-cm Objective Grating for Far UV Astronomy: Theoretical Study and Laboratory Tests,” Appl. Opt., to be submitted.

Appl. Opt.

C. R. Acad. Sci.

M. Duban, “Le montage de Wadsworth holographique asphérique en astronomie spatiale,” C. R. Acad. Sci. 302, 1215 (1986).

J. Opt.

M. Duban, “Calcul d'un réseau holographique torique destiné au montage Caméra Grand Champ,” J. Opt. 9, 163 (1978).
[CrossRef]

Other

M. Duban, “Etude d'un montage de Rowland holographique à résolution modérée en dispersion conique faible,” CNRS Laboratoire d'Astronomie Spatiale, Marseille; (1984).

R. Prangé, M. Duban, A. Vidal-Madjar et al., “A 30-cm Objective Grating for Far UV Astronomy: Theoretical Study and Laboratory Tests,” Appl. Opt., to be submitted.

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Figures (24)

Fig. 1
Fig. 1

Geometry of holographic sources.

Fig. 2
Fig. 2

Geometry of the grating.

Fig. 3
Fig. 3

Optical path.

Fig. 4
Fig. 4

Geometry of source aberrations.

Fig. 5
Fig. 5

Astigmatism focals.

Fig. 6
Fig. 6

Strong conical dispersion, point source: (a) spherical grating astigmatism; (b) astigmatism corrected; (c) As and coma corrected; (d) As, C, and spherical aberration corrected.

Fig. 7
Fig. 7

General aberrant source: (a) As corrected; (b) As and C corrected; (c) As, C, and Sp corrected.

Fig. 8
Fig. 8

Pure astigmatic source: (a) tangential focal (spherical grating); (b) sagittal focal; (c) As corrected; (d) As and C corrected; (e) As, C, and Sp corrected.

Fig. 9
Fig. 9

Mean conical dispersion, point source: (a) As corrected; (b) As and C corrected; (c) As, C, and Sp corrected.

Fig. 10
Fig. 10

Pure incident coma: (a) As corrected for a point source, grating illuminated by the parabolic mirror; (b) As and C corrected; (c) As, C, and Sp corrected; (d) same corrections, parabolic mirror rotated by 60°; (e) same corrections, theoretical coma incident wave; (f) coma incident spot given by the parabolic mirror.

Fig. 11
Fig. 11

Pure incident coma: (a) As and C corrected for aberrant source, theoretical incident wave; (b) As, C, and Sp corrected (same conditions); (c) As and C corrected for aberrant source, grating illuminated by the parabolic mirror; (d) As, C, and Sp corrected (same conditions).

Fig. 12
Fig. 12

Pure incident spherical aberration: (a) As corrected for a point source, grating illuminated by the spherical mirror; (b) As and C corrected; (c) As, C, and Sp corrected; (d) As, C, and Sp corrected for aberrant source, theoretical incident wave; (e) As, C, and Sp corrected for aberrant source, grating illuminated by the spherical mirror.

Fig. 13
Fig. 13

Image astigmatism (deformed grating): (a) 150-mm diam pupil; (b) 50-mm diam pupil.

Fig. 14
Fig. 14

Image coma (deformed grating).

Fig. 15
Fig. 15

Image spherical aberration (deformed grating): (a) weak and (b) strong.

Fig. 16
Fig. 16

Strong conical dispersion, point source, spherical grating, Σ2 aberrant: (a) As corrected; (b) As and C corrected; (c) As, C, and Sp corrected.

Fig. 17
Fig. 17

Toric grating: (a) Σ2 point; (b) Σ2 aberrant, As corrected; (c) As and C corrected; (d) As, C, and Sp corrected.

Fig. 18
Fig. 18

Aberrant source, Σ2 aberrant: (a) As corrected; (b) As and C corrected; (c) As, C, and Sp corrected.

Fig. 19
Fig. 19

Image astigmatism (Σ2 aberrant): (a) 150-mm pupil; (b) 50-mm pupil.

Fig. 20
Fig. 20

Image coma (Σ2 aberrant).

Fig. 21
Fig. 21

Strong image spherical aberration (Σ2 aberrant).

Fig. 22
Fig. 22

Pure coma spots in plane dispersion: a, c 1 = 0; b, c 1 = c 2 ; c, c 2 = 0.

Fig. 23
Fig. 23

Pure spherical aberration spots: (a) s 2 ; (b) s 4 ; (c) s 1 + s 2 ; (d) s 1 + s 2 + s 3 + s 4 + s 5 .

Fig. 24
Fig. 24

Iteration-computed intersection of the grating with a cone.

Equations (46)

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Δ = ( M S O S ) + ( M S O S ) + ( k λ / λ 0 ) [ ( M Σ 1 O Σ 1 ) ( M Σ 2 O Σ 2 ) ] .
Δ + Δ s = Δ s .
x = y 2 / 2 R + z 2 / 2 ρ + y 4 / 8 R 3 + z 4 / 8 ρ 3 + y 2 z 2 / 4 ρ R 2 = x T .
x = x T + A y z + C 1 y 3 + C 2 y z 2 + C 3 y 2 z + C 4 z 3 + S 1 y 4 + S 2 y 2 z 2 + S 3 z 4 + S 4 y 3 z + S 5 y z 3 .
x 2 = ( y 4 / 4 R 2 + y 2 z 2 / 2 ρ R + z 4 / 4 ρ 2 ) + A 2 y 2 z 2 + A y 3 z / R + A y z 3 / ρ .
D ( Q ) = y ( b / d ) + z ( c / d ) + y 2 ( p b 2 / d 2 ) / 2 d + z 2 ( q c 2 / d 2 ) / 2 d + y z ( b c / d 3 ) + y 3 ( b / 2 d 3 ) ( p b 2 / d 2 ) + y z 2 ( b / 2 d 3 ) ( q 3 c 2 / d 2 ) + y 2 z ( c / 2 d 3 ) ( p 3 b 2 / d 2 ) + z 3 ( c / 2 d 3 ) ( q c 2 / d 2 ) + y 4 ( 1 / 4 d ) [ p ( 1 / R 2 p / d 2 ) / 2 + 3 p b 2 / d 4 5 b 4 / 2 d 6 ] + y 2 z 2 ( 1 / 4 d ) [ p / ρ R p q / d 2 + 3 ( q b 2 + p c 2 ) / d 4 15 b 2 c 2 / d 6 ] + z 4 ( 1 / 4 d ) [ q ( 1 / ρ 2 q / d 2 ) / 2 + 3 q c 2 / d 4 5 c 4 / 2 d 6 ] + y 3 z ( b c / 2 d 5 ) ( 3 p 5 b 2 / d 2 ) + y z 3 ( b c / 2 d 5 ) ( 3 q 5 c 2 / d 2 ) .
y z ( a A / d ) + y 3 ( a C 1 / d ) + y z 2 ( a C 2 / d acA / d 3 ) + y 2 z ( a C 3 / d abA / d 3 + z 3 ( a C 4 / d ) + y 4 ( a S 1 / d a b C 1 / d 3 ) + y 2 z 2 [ a S 2 / d 3 abc A / d 5 a b C 2 / d 3 a c C 3 / d 3 + A 2 ( 1 / 2 d a 2 / 2 d 3 ) ] + z 4 ( a S 3 / d a c C 4 / d 3 ) + y 3 z [ a S 4 / d + A ( a / 2 d 3 3 a b 2 / 2 d 5 ) + C 1 ( a c / d 3 ) + C 3 ( a b / d 3 ) + A ( 1 / 2 d a 2 / 2 d 3 ) / R ] + y z 3 [ a S 5 / d + A ( a / 2 d 3 3 a c 2 / 2 d 5 ) + C 2 ( a c / d 3 ) + C 4 ( a b / d 3 ) + A ( 1 / 2 d a 2 / 2 d 3 ) / ρ ] .
Δ = D ( S ) + D ( S ) + ( k λ / λ 0 ) [ D ( Σ 1 ) D ( Σ 2 ) ] .
Δ s = a 1 Y 2 + a 2 Y Z + a 3 Z 2 + c 1 Y 3 + c 2 Y Z 2 + c 3 Y 2 Z + c 4 Z 3 + s 1 Y 4 + s 2 Y 2 Z 2 + s 3 Z 4 + s 4 Z 4 + s 4 Y 3 Z + s 5 Y Z 3 .
Δ s = a 1 Y 2 + .
O S Y 0 δ = d , = e x + f y + g z , = U / 2 R + V 2 , X 0 / d Z 0 ɛ = U x + V y + W z , = E x + F y + G z , = A U + 2 V W , γ = U / 2 ρ + W 2 ,
e = sin i sin ψ + sin ω cos i cos ψ , f = cos i sin ψ + sin ω sin i cos ψ , g = cos ω cos ψ , E = sin i cos ψ sin ω cos i sin ψ , F = cos i cos ψ sin ω sin i sin ψ , G = cos ω sin ψ ) .
Y = f y + g z + ( f V + e / 2 R ) y 2 + ( f W + g V + A e ) y z + ( g W + e / 2 ρ ) z 2 + [ δ f + e ( C 1 + V / 2 R ) ] y 3 + [ γ f + ɛ g + e ( C 2 + V / 2 ρ + A W ) ] y z 2 + [ ɛ f + δ g + e ( C 3 + W / 2 R + A V ) ] y 2 z + [ γ g + e ( C 4 + W / 2 ρ ) ] z 3 .
Δ s = ( 1 / 2 ) ( 1 / s 1 / t ) ( u 2 Y 2 + 2 u υ Y Z + υ 2 Z 2 ) , Δ s = ( 1 / 2 ) ( 1 / t 1 / s ) ( u 2 Y 2 + 2 u υ Y Z + υ 2 Z 2 ) ,
y 2 [ a 1 f 2 + a 2 f F + a 3 F 2 ] + y z [ a 1 ( 2 f g ) + a 2 ( f G + g F ) + a 3 ( 2 F G ) ] + z 2 [ a 1 g 2 + a 2 g G + a 3 G 2 ] .
R = 782.7408 mm , ρ = 2700.3485 mm , A = 2.4649 × 10 4 .
a 1 , a 2 , a 3 c 1 , c 2 , c 3 , c 4 s 1 , s 2 , s 3 , s 4 , s 5 = 3 , 4 , 5 ( × 10 5 ) , = 1 , 2 , 3 , 4 ( × 10 7 ) , = 1 , 2 , 3 , 4 , 5 ( × 10 10 ) , and φ = 0 .
i = 30 ° , ω = 20 ° , OS = O S = 1000 mm .
R = 1058.7793 mm , ρ = 948.1678 mm , A = 1.5307 × 10 4 .
C 1 = 1.9041 , C 2 = 1.4723 , C 3 = 1.2743 ( × 10 7 ) , C 4 = 4.2401 × 10 8 ,
S 1 = 26.937 , S 2 = 7.711 , S 3 = 2.518 S 4 = 13.037 , S 5 = 1.924 ( × 10 11 ) .
t = 1200 mm , s = 1600 mm , φ = 40 ° , i = 30 ° , ω = 60 ° .
c 1 = c 2 = 8.7266 × 10 9 .
R = 1002.2466 mm , ρ = 920.3679 mm , A = 1.7844 × 10 4 C 1 = 1.435 , C 2 = 4.257 , C 3 = 14.033 , C 4 = 4.192 ( × 10 9 ) , S 1 = 2.61 , S 2 = 7.77 , S 3 = 1.96 , S 4 = 11.31 , S 5 = 12.18 ( × 10 11 ) ,
S 1 = 2.49 , S 2 = 7.68 , S 3 = 2.18 ( × 10 11 ) , S 4 = 1.09 , S 5 = 1.19 ( × 10 10 ) .
s 1 = s 3 = 1 × 10 9 , s 2 = 2 × 10 9 , s 4 = s 5 = 0 .
t = 1800 mm , s = 2100 mm , φ = 20 ° .
c 1 = c 2 = 2 × 10 7 , ψ = 210 °
C 1 = 2.625 , C 2 = 2.077 , C 3 = 1.032 ( × 10 7 ) , C 4 = 7.49 × 10 9 .
S 1 = 23.66 , S 2 = 7.05 , S 3 = 2.95 , S 4 = 21.59 , S 5 = 11.57 ( × 10 11 ) .
S 1 = 25.46 , S 2 = 19.84 , S 3 = 7.36 , S 4 = 26.96 , S 5 = 23.32 ( × 10 11 ) .
S 1 = 2.53 × 10 11 , S 2 = 54.33 , S 3 = 9.94 , S 4 = 29.15 , S 5 = 50.29 ( × 10 10 ) .
M Σ j = M Σ j Δ j ( M )
a 1 = 5.1426 , a 2 = 9.0688 , a 3 = 11.501 ( × 10 4 ) , c 1 = 12 , c 2 = 4.55 , c 3 = 16.03 , c 4 = 6.54 ( × 10 8 ) , s 1 = 2.88 × 10 12 , s 2 = 5.53 , s 3 = 1.72 , s 4 = 4.05 , s 5 = 1.68 ( × 10 10 ) .
a 1 = 5.1067 , a 2 = 1.2919 , a 3 = 6.756 ( × 10 5 ) , c 1 = 1.027 , c 2 = 1.539 , c 3 = 1.147 ( × 10 7 ) , c 4 = 5.35 × 10 8 , s 1 = 5.5 , s 2 = 1.75 , s 3 = 4.2 , s 4 = 75.4 , s 5 = 22 ( × 10 11 ) .
a 1 , a 2 , a 3 = 1 , 2 , 3 ( × 10 4 ) , c 1 , c 2 , c 3 , c 4 = 4 , 5 , 6 , 7 ( × 10 7 ) , s 1 , s 2 , s 3 , s 4 , s 5 = 5 , 6 , 7 , 8 , 9 ( × 10 10 ) .
a 1 = 2.8882 , a 2 = 2.1928 , a 3 = 1.1514 ( × 10 4 ) , c 1 = 17.16 , c 2 = 14.83 , c 3 = 3.22 , c 4 = 4.45 ( × 10 7 ) , s 1 = 24.18 , s 2 = 15.18 , s 3 = 7.23 , s 4 = 1.74 , s 5 = 4.37 ( × 10 10 ) .
C 1 , C 2 , C 3 , C 4 = 4 , 3 , 2 , 1 ( × 10 7 ) , S 1 , S 2 , S 3 , S 4 , S 5 = 5 , 4 , 3 , 2 , 1 ( × 10 10 ) .
a 1 = 2.8882 , a 2 = 1.2131 , a 3 = 1.1514 ( × 10 4 ) , c 1 = 5.501 , c 2 = 1.427 ( × 10 7 ) , c 3 = 1.5225 × 10 6 , c 4 = 8.27 × 10 8 , s 1 = 5.54 , s 2 = 3.72 , s 3 = 2.61 , s 4 = 1.05 ( × 10 9 ) , s 5 = 6.1 × 10 10 .
c 1 = 10.483 , c 2 = 5.531 , c 3 = 16.889 c 4 = 2.518 ( × 10 7 ) , s 1 = 3.67 , s 2 = 34.06 , s 3 = 3.92 , s 4 = 17.54 , s 5 = 35.73 ( × 10 8 ) .
d = d = 2400 mm ( for tangential focal ) , n = 3600 grooves / mm λ 0 = 4880 A ˚ α = 60 ° ( so , β = 62.9707 ° ) , O Σ 1 = 2000 mm O Σ 2 = 3000 mm i = 30 ° k = 1 λ = 2000 A ˚ ( so , r = 12.709 ° ) ω = 0 ( plane dispersion )
R = 2541.1936 mm , ρ = 2063.1256 mm , C 1 = 5.747 × 10 9 , C 2 = 1.7545 × 10 8 , S 1 = 6.74 , S 2 = 99.11 , S 3 = 8.85 ( × 10 13 ) ,
R unmodified , ρ = 2059.8621 mm , C 1 unmodified , C 2 = 1.7571 × 10 8 , S 1 unmodified , S 2 = 99.05 , S 3 = 9.08 ( × 10 13 ) .
S 1 = S 3 = K / 8 R 3 , S 2 = K / 4 R 3 ( with b = c , R = ρ = b 2 / a ) , S 4 = S 5 = 0.
N x = 1 + Y [ ( Δ s / Y ) ( 1 / S M ) Y / S P 2 ] + Z [ ( Δ s / Z ) ( 1 / S M ) Z / S P 2 ] N y = X [ ( Δ s / Y ) ( 1 / S M ) Y / S P 2 ] , N z = X [ ( Δ s / Z ) ( 1 / S M ) Z / S P 2 ] .
y = ( d μ 2 / 2 ) [ ( 3 c 1 + c 2 ) + ( c 2 3 c 1 ) cos 2 θ 2 c 3 sin 2 θ ] , z = ( d μ 2 / 2 ) [ ( 3 c 4 + c 3 ) + ( 3 c 3 c 3 ) cos 2 θ 2 c 2 sin 2 θ ] ,

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