Abstract

The geometric theory of aspheric wave-front recording optics is extended to include the fourth-order groove parameters that correspond to the fourth-order holographic terms in the light-path function. We derived explicit expressions of the groove parameters by analytically following an exact ray-tracing procedure for a double-element optical system that consists of a point source, an ellipsoidal mirror, and an ellipsoidal grating blank. Design examples of holographic gratings for an in-plane Eagle-type vacuum-UV monochromator are given to demonstrate the capability of the present theory in the design of aspheric wave-front recording optics.

© 1995 Optical Society of America

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References

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  1. T. Namioka, M. Seya, H. Noda, “Design and performance of holographic concave gratings,” Jpn. J. Appl. Phys. 15, 1181–1197 (1976).
    [CrossRef]
  2. M. Koike, Y. Harada, H. Noda, “New blazed holographic grating fabricated by using an aspherical recording with an ion-etching method,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moiré Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 815, 96–101 (1987).
  3. H. Noda, Y. Harada, M. Koike, “Holographic grating recorded using aspheric wave fronts for a Seya–Namioka monochromator,” Appl. Opt. 28, 4375–4380 (1989).
    [CrossRef] [PubMed]
  4. C. Palmer, “Theory of second-generation holographic diffraction gratings,” J. Opt. Soc. Am. A 6, 1175–1188 (1989).
    [CrossRef]
  5. R. Grange, M. Laget, “Holographic diffraction gratings generated by aberrated wave fronts: applications to a high-resolution far-ultraviolet spectrograph,” Appl. Opt. 30, 3598–3603 (1991).
    [CrossRef] [PubMed]
  6. M. Duban, “Holographic aspheric gratings printed with aberrant waves,” Appl. Opt. 26, 4263–4273 (1987).
    [CrossRef] [PubMed]
  7. M. Duban, “Third-generation Rowland holographic mounting,” Appl. Opt. 30, 4019–4025 (1991).
    [CrossRef] [PubMed]
  8. T. Namioka, M. Koike, “Analytical representation of spot diagrams and its application to the design of monochromators,” Nucl. Instrum. Methods A 319, 219–227 (1992). Note that L1, M1, and N1 therein are defined to take opposite signs to those in this paper.
    [CrossRef]
  9. T. Namioka, M. Koike, D. Content, “Geometric theory of the ellipsoidal grating,” Appl. Opt. 33, 7261–7274 (1994).
    [CrossRef] [PubMed]
  10. M. Koike, T. Namioka, “Merit function for the design of grating instruments,” Appl. Opt. 33, 2048–2056 (1994).
    [CrossRef] [PubMed]

1994

1992

T. Namioka, M. Koike, “Analytical representation of spot diagrams and its application to the design of monochromators,” Nucl. Instrum. Methods A 319, 219–227 (1992). Note that L1, M1, and N1 therein are defined to take opposite signs to those in this paper.
[CrossRef]

1991

1989

1987

1976

T. Namioka, M. Seya, H. Noda, “Design and performance of holographic concave gratings,” Jpn. J. Appl. Phys. 15, 1181–1197 (1976).
[CrossRef]

Content, D.

Duban, M.

Grange, R.

Harada, Y.

H. Noda, Y. Harada, M. Koike, “Holographic grating recorded using aspheric wave fronts for a Seya–Namioka monochromator,” Appl. Opt. 28, 4375–4380 (1989).
[CrossRef] [PubMed]

M. Koike, Y. Harada, H. Noda, “New blazed holographic grating fabricated by using an aspherical recording with an ion-etching method,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moiré Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 815, 96–101 (1987).

Koike, M.

M. Koike, T. Namioka, “Merit function for the design of grating instruments,” Appl. Opt. 33, 2048–2056 (1994).
[CrossRef] [PubMed]

T. Namioka, M. Koike, D. Content, “Geometric theory of the ellipsoidal grating,” Appl. Opt. 33, 7261–7274 (1994).
[CrossRef] [PubMed]

T. Namioka, M. Koike, “Analytical representation of spot diagrams and its application to the design of monochromators,” Nucl. Instrum. Methods A 319, 219–227 (1992). Note that L1, M1, and N1 therein are defined to take opposite signs to those in this paper.
[CrossRef]

H. Noda, Y. Harada, M. Koike, “Holographic grating recorded using aspheric wave fronts for a Seya–Namioka monochromator,” Appl. Opt. 28, 4375–4380 (1989).
[CrossRef] [PubMed]

M. Koike, Y. Harada, H. Noda, “New blazed holographic grating fabricated by using an aspherical recording with an ion-etching method,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moiré Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 815, 96–101 (1987).

Laget, M.

Namioka, T.

T. Namioka, M. Koike, D. Content, “Geometric theory of the ellipsoidal grating,” Appl. Opt. 33, 7261–7274 (1994).
[CrossRef] [PubMed]

M. Koike, T. Namioka, “Merit function for the design of grating instruments,” Appl. Opt. 33, 2048–2056 (1994).
[CrossRef] [PubMed]

T. Namioka, M. Koike, “Analytical representation of spot diagrams and its application to the design of monochromators,” Nucl. Instrum. Methods A 319, 219–227 (1992). Note that L1, M1, and N1 therein are defined to take opposite signs to those in this paper.
[CrossRef]

T. Namioka, M. Seya, H. Noda, “Design and performance of holographic concave gratings,” Jpn. J. Appl. Phys. 15, 1181–1197 (1976).
[CrossRef]

Noda, H.

H. Noda, Y. Harada, M. Koike, “Holographic grating recorded using aspheric wave fronts for a Seya–Namioka monochromator,” Appl. Opt. 28, 4375–4380 (1989).
[CrossRef] [PubMed]

T. Namioka, M. Seya, H. Noda, “Design and performance of holographic concave gratings,” Jpn. J. Appl. Phys. 15, 1181–1197 (1976).
[CrossRef]

M. Koike, Y. Harada, H. Noda, “New blazed holographic grating fabricated by using an aspherical recording with an ion-etching method,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moiré Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 815, 96–101 (1987).

Palmer, C.

Seya, M.

T. Namioka, M. Seya, H. Noda, “Design and performance of holographic concave gratings,” Jpn. J. Appl. Phys. 15, 1181–1197 (1976).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am. A

Jpn. J. Appl. Phys.

T. Namioka, M. Seya, H. Noda, “Design and performance of holographic concave gratings,” Jpn. J. Appl. Phys. 15, 1181–1197 (1976).
[CrossRef]

Nucl. Instrum. Methods A

T. Namioka, M. Koike, “Analytical representation of spot diagrams and its application to the design of monochromators,” Nucl. Instrum. Methods A 319, 219–227 (1992). Note that L1, M1, and N1 therein are defined to take opposite signs to those in this paper.
[CrossRef]

Other

M. Koike, Y. Harada, H. Noda, “New blazed holographic grating fabricated by using an aspherical recording with an ion-etching method,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moiré Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 815, 96–101 (1987).

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Figures (3)

Fig. 1
Fig. 1

Schematic diagram of an aspheric wave-front recording system consisting of two coherent point sources, C and D; two auxiliary ellipsoidal mirrors, M1 and M2; and an ellipsoidal grating blank, G. The elements are arranged so that points C and D and the normals of M1, M2, and G at their respective vertices, O1, O2, and O, lie in a common plane, and the incident principal rays, CO 1 and DO 2, pass through O after being reflected at O1 and O2, respectively. The recording geometry parameters and the coordinate systems attached to M1, M2, and G are indicated also.

Fig. 2
Fig. 2

ΔY–ΔZ plots constructed for the in-plane, Eagle-type, VUV monochromator equipped with the holographic grating HG-I. The deviations (ΔY, ΔZ) of individual spots (Y AN , Z AN ) generated from the analytic formulas from the corresponding spots (Y RT , Z RT ) constructed by exact ray tracing are calculated with 2000 rays of 0, 50, 125, and 200 nm originating from a 1-mm-long line source in the entrance slit. The rms errors, σΔ Y and σΔ Z , of ΔY and ΔZ are indicated in each diagram.

Fig. 3
Fig. 3

Spot diagrams and line profiles constructed for the in-plane, Eagle-type, VUV monochromator equipped with the grating (a) HG-I, (b) HG-II, (c) HG-8, or (d) VSG-8. These diagrams are constructed by exact ray tracing with 500 rays of 0, 50, 125, and 200 nm originating from the entrance slit 20 μm wide and 1 mm long; see text for details.

Equations (42)

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x C = p C cos η C , y C = p C sin η C , x D = p D cos η D , y D = p D sin η D .
( ξ i - a i ) 2 / a i 2 + w i 2 / b i 2 + l i 2 / c i 2 = 1 ,
ξ i = w i 2 2 R i + l i 2 2 ρ i + w i 4 8 a i R i 2 + w i 2 l i 2 4 R i ρ i a i + l i 4 8 ρ i 2 a i + O ( w i 6 R i 5 ) ,
R i = b i 2 / a i ,             ρ i = c i 2 / a i .
n λ 0 = [ ( C Q 1 + Q 1 P ) - ( D Q 2 + Q 2 P ) ] - [ ( C O 1 + O 1 O ) - ( D O 2 + O 2 O ) ] ,
F C = C Q 1 + Q 1 P
L 1 = L 1 + T 1 ,             M 1 = M 1 - T 1 ( ξ 1 / w 1 ) , N 1 = N 1 - T 1 ( ξ 1 / l 1 ) ,
T 1 = 2 [ - L 1 + M 1 ( ξ 1 / w 1 ) + N 1 ( ξ 1 / l 1 ) ] 1 + ( ξ 1 / w 1 ) 2 + ( ξ 1 / l 1 ) 2 , L 1 ( ξ 1 - x C ) / C Q 1 ,             M 1 ( w 1 - y C ) / C Q 1 , N 1 l 1 / C Q 1 ,             L 1 ( ξ - ξ 1 ) / Q 1 P , M 1 ( w - w 1 ) / Q 1 P ,             N 1 ( l - l 1 ) / Q 1 P ,
L C = - L 1 cos ( η c + γ ) + M 1 sin ( η C + γ ) , M C = - L 1 sin ( η C + γ ) - M 1 cos ( η C + γ ) , N C = N 1 .
w 1 = i = 0 j = 0 ( A i j ) C w i l j ,             l 1 = i = 0 j = 0 ( B i j ) C w i l j ( A 00 ) C = ( B 00 ) C = 0.
L C = i = 0 j = 0 ( h i j ) L w i l j ,             M C = i = 0 j = 0 ( h i j ) M w i l j , N C = i = 0 j = 0 ( h i j ) N w i l j ,
L C ( ξ - ξ ¯ 1 ) / Q 1 P ,             M C ( w - W ¯ 1 ) / Q 1 P , N C = ( l - l ¯ 1 ) / Q 1 P ,
Q 1 P = [ ( ξ - ξ ¯ 1 ) 2 + ( w - w ¯ 1 ) 2 + ( l - l ¯ 1 ) 2 ] 1 / 2
ξ ¯ 1 = - ξ 1 cos ( η C + γ ) + w 1 sin ( η C + γ ) + q C cos γ , w ¯ 1 = - ξ 1 sin ( η C + γ ) - w 1 cos ( η C + γ ) + q C sin γ , l ¯ 1 = l 1 .
L C = i = 0 j = 0 ( h i j ) L w i l j ,             M C = i = 0 j = 0 ( h i j ) M w i l j , N C = i = 0 j = 0 ( h i j ) N w i l j ,
( A 01 ) C = ( A 11 ) C = ( A 21 ) C = ( A 03 ) C = 0 , ( A 10 ) C = - cos γ A C q C cos η C ,
( A 20 ) C = ( A 10 ) C 2 [ 2 cos γ r C ( tan η C + tan γ ) - tan γ R - ( A 10 ) C R 1 ( 1 + 6 q C K C sec γ ) tan η C ] ,
( A 02 ) C + ( A 10 ) C 2 sec γ [ - sin γ ρ + ( B 01 ) C ρ 1 ( 1 + 2 q C U C ) sin η C ] ,
( B 10 ) C = ( B 20 ) C = ( B 02 ) C = ( B 12 ) C = 0 , ( B 01 ) C = 1 B C q C ,
( B 11 ) C = ( B 01 ) C [ cos γ r C ( tan η C + tan γ ) - 2 ( A 10 ) C R 1 q C U C tan η C ] ,
A C = 1 p C + 1 q C - 2 sec η C R 1 ,             B C = 1 p C + 1 q C - 2 cos η C ρ 1 , r C = q C + ( 1 p C - 2 R 1 sec η C ) - 1 ,
r C = q C + ( 1 p C - 2 ρ 1 cos η C ) - 1 ,
K C = cos γ r C - ( A 10 ) C R 1 ,             U C = 1 r C + ( B 01 ) C ρ 1 cos η C ·
n = 1 λ 0 [ n 10 w + 1 2 ( n 20 w 2 + n 02 l 2 + n 30 w 3 + n 12 w l 2 ) + 1 8 ( n 40 w 4 + 2 n 22 w 2 l 2 + n 04 l 4 + ] .
n = 1 λ 0 ( { [ ( ξ 1 - x C ) 2 + ( w 1 - y C ) 2 + l 1 2 ] 1 / 2 + [ ( ξ - ξ ¯ 1 ) 2 + ( w - w ¯ 1 ) 2 + ( l - l ¯ 1 ) 2 ] 1 / 2 } - { [ ( ξ 2 - x D ) 2 + ( w 2 - y D ) 2 + l 2 2 ] 1 / 2 + [ ( ξ - ξ ¯ 2 ) 2 + ( w - w ¯ 2 ) 2 + ( l - l ¯ 2 ) 2 ] 1 / 2 } - [ ( p C + q C ) - ( p D + q D ) ] ) .
n 10 = sin δ - sin γ ,
n 20 = T C - T D ,             T C = cos 2 γ r C - cos γ R , T D = cos 2 δ r D - cos δ R ,
n 02 = S ¯ C - S ¯ D ,             S ¯ C = 1 r C - cos γ ρ ,             S ¯ D = 1 r D - cos δ ρ ,
n 30 = T C sin γ r C - T D sin δ r D - 2 ( A 10 ) C 2 R 1 K C sin η C + 2 ( A 10 ) D 2 R 2 K D sin η D ,
n 12 = sin γ r C [ 1 r C - ( r C r C ) cos γ ρ ] - sin δ r D [ 1 r D - ( r D r D ) cos δ ρ ] + 2 ρ 1 ( A 10 ) C ( B 01 ) C V C sin η C - 2 ρ 2 ( A 10 ) D ( B 01 ) D V C sin η D ,
n 40 = T C r C ( 4 sin 2 γ r C - T C ) - T D r D ( 4 sin 2 δ r D - T D ) + S C - S D R 2 + 2 ( A 10 ) C 2 R 1 K C ( E 40 ) C cos η C - 2 ( A 10 ) D 2 R 2 K C ( E 40 ) D cos η D + 2 ( A 10 ) C 3 R 1 2 [ cos γ r C cos η C - ( A 10 ) C cos η C a 1 ] - 2 ( A 10 ) D 3 R 2 2 [ cos δ r D cos η D - ( A 10 ) D cos η D a 2 ] ,
n 22 = 2 sin 2 γ r C 2 [ S ¯ C + 1 r C ( r C 2 r C 2 - 1 ) ] - 1 r C [ T C S ¯ C + cos γ R r C ( 1 - r C r C ) ] - 2 sin 2 δ r D 2 [ S ¯ D + 1 r D ( r D 2 r D 2 - 1 ) ] + 1 r D [ T D S ¯ D + cos δ R r D ( 1 - r D r D ) ] + S C - S D R ρ + 2 ( r C - r C r C r C 3 ) tan η C + 2 tan γ ) tan η C cos 2 γ - 2 ( r D - r D ) r D r D 3 ( tan η D + 2 tan δ ) tan η D cos 2 δ + ( A 10 ) C R 1 ( E 22 ) C - ( A 10 ) D R 2 ( E 22 ) D + ( B 01 ) C ρ 1 ( G 22 ) C - ( B 01 ) D ρ 2 ( G 22 ) D ,
n 04 = S C - S D ρ 2 - 1 r C [ S ¯ C 2 + cos 2 γ ρ 2 ( r C r C - 1 ) ] + 1 r D [ S ¯ D 2 + cos 2 δ ρ 2 ( r D r D - 1 ) ] + ( B 01 ) C ρ 1 ( G 04 ) C - ( B 01 ) D ρ 2 ( G 04 ) D ,
S C = 1 r C - cos γ a 1 ,             S D = 1 r D cos δ a 2 , V C = cos η C r C + ( B 01 ) C R 1 ,             V D = cos η D r D + ( B 01 ) D R 2 .
( E 40 ) C = 6 R tan η C tan γ - cos γ r C [ 1 + tan η C ( 7 tan η C + 12 tan γ ) ] + 3 K C tan 2 η C [ 1 + 6 ( A 10 ) C q C R 1 cos γ ] ,
( E 22 ) C = ( A 10 ) C cos η C [ 6 ρ K C tan η C tan γ + 1 r C 2 + ( B 01 ) C 2 ( sin η C tan η C ρ 1 q C - 1 ρ 1 a 1 + 4 tan 2 η C p C 2 + S ¯ 1 p C + S 1 sec η c ρ 1 ) ] - 4 ( B 01 ) C { K C r C - 2 U C tan η C [ cos γ r C ( tan η C + tan γ ) - ( A 10 ) C q C R 1 U C tan η C ] } ,
( G 22 ) C = 2 ( A 20 ) C q C sin η C cos η C ( 1 + 2 q C U C ) - 4 ( A 10 ) C r C sin 2 η C cos γ [ 1 r C + ( B 01 ) C p C ] - ( A 10 ) C 2 ( B 01 ) C p C 2 ( 1 + sin 2 η C ) + ( B 01 ) C q C cos η C cos γ { q C cos γ r C 2 - tan η C tan γ R - 2 tan η C r C [ ( A 10 ) C sin η C - sin γ ] } ,
( G 04 ) C = - 4 ( A 10 ) C ρ V C sin η C tan γ - ( B 01 ) C cos η C [ 2 U C 2 - ( A 10 ) C sin 2 η C ρ 1 q C cos γ × ( 1 + 2 q C U C ) 2 ] + ( B 01 ) C 2 ρ 1 [ sin 2 η C q C + 2 ( B 01 ) C cos η C ( 1 ρ 1 - 1 a 1 ) ] ,
S ¯ 1 = 1 p C - cos η C ρ 1 ,
S 1 = 1 p C - cos η C a 1 .
σ 1 / ( n / w ) w = l = 0 = λ 0 / n 10 = λ 0 / ( sin δ - sin γ ) .
Q = i = 1 s ( λ i ) [ q Y 2 ( λ i ) + μ q z 2 ( λ i ) ] ,

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