Abstract

Various grating designs have been proposed by several investigators for possible use with the Lyman/Far-Ultraviolet Spectroscopic Explorer mission. The image quality, the feasibility, and the efficiency of five designs are compared, each using a distinct type of grating: (1) a grating ruled on a deformed ellipsoidal or toroidal blank, (2) an ellipsoidal grating recorded holographically with two auxiliary spherical mirrors, (3) a spherical holographic grating recorded with two auxiliary spherical holographic gratings, (4) a spherical ruled grating with variable spacing and straight grooves, and (5) a spherical ruled grating with a groove pattern that is determined theoretically (hybrid grating). From a purely theoretical viewpoint, grating (5) provides the finest images, followed by gratings, (3), (1), (4), and (2). In view of the current technological limitations, the order of practical importance is gratings (4), (1), (2), (3), and (5).

© 1993 Optical Society of America

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References

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  1. H. W. Moos, “Lyman, the Far Ultraviolet Spectroscopic Explorer, phase A study final report,” NASA contract NAS5-30339 (NASA Goddard Space Flight Center, Greenbelt, Md., July, 1989),Vols. 1 and 2.
  2. W. C. Cash, “Aspheric concave grating spectrographs,” Appl. Opt. 23, 4518–4522 (1984).
    [CrossRef] [PubMed]
  3. D. Content, C. Trout, P. Davila, M. Wilson, “Aberration corrected aspheric gratings for far ultraviolet spectrographs: conventional approach,” Appl. Opt. 30, 801–806 (1991).
    [CrossRef] [PubMed]
  4. C. Trout, D. Content, P. Davila, “Aberration-corrected aspheric grating designs for the Lyman/Far-Ultraviolet Spectroscopic Explorer high-resolution spectrograph: a comparison,” Appl. Opt. 31, 943–948 (1992).
    [CrossRef] [PubMed]
  5. P. Davila, D. Content, C. Trout, “Aberration-corrected aspheric gratings for far-ultraviolet spectrographs: holographic approach,” Appl. Opt. 31, 949–954 (1992).
    [CrossRef] [PubMed]
  6. R. Grange, M. Laget, “Holographic diffraction gratings generated by aberrated wave fronts: application to a high-resolution far-ultaviolet spectrograph,” Appl. Opt. 30, 3598–3603 (1991).
    [CrossRef] [PubMed]
  7. M. Duban, “Third-generation Rowland holographic mounting,” Appl. Opt. 30, 4019–4025 (1991).
    [CrossRef] [PubMed]
  8. M. Hurwitz, S. Bowyer, T. Harada, T. Kita, “High resolution far ultraviolet spectrographs for Lyman using spherical varied line-space diffraction gratings,” Lyman-FUSE Review Committee (18–19July1990).
  9. M. Duban, “Aspheric gratings: recent developments,” Appl. Opt. 24, 3316–3318 (1985).
    [CrossRef] [PubMed]
  10. M. Koike, Y. Harada, H. Noda, “New blazed holographic gratings fabricated by using an aspherical recording with an ion-etching method,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moire Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.815, 96–101 (1988).
  11. H. Noda, Y. Harada, M. Koike, “Holographic grating recorded using aspheric wavefronts for a Seya-Namioka monochromator,” Appl. Opt. 28, 4375–4380 (1989).
    [CrossRef] [PubMed]
  12. C. Palmer, “Theory of second-generation holographic diffraction gratings,” J. Opt. Soc. Am. A 6, 1175–1188 (1989).
    [CrossRef]
  13. A. Thévenon, J. Flamand, J. P. Laude, B. Touzet, J. Lerner, “Aberration-corrected plane gratings,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moire Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.815, 136–143 (1988).
  14. T. Harada, T. Kita, “Mechanically ruled aberration-corrected concave gratings,” Appl. Opt. 19, 3987–3992 (1980).
    [CrossRef] [PubMed]
  15. T. Namioka, M. Koike, “Design of compact high resolution far-ultraviolet spectrographs equipped with a spherical grating having variable spacing and curved grooves,” presented at the Tenth International Colloquium on VUV and X-Ray Spectroscopy of Astrophysics and Laboratory Plasmas, Berkeley, Calif., 3–5 February 1992.
  16. M. Duban, “Holographic aspheric gratings printed with aberrant waves,” Appl. Opt. 26, 4263–4273 (1987).
    [CrossRef] [PubMed]
  17. M. Duban, “Calcul et réalisation de spectrographes à réseaux holographiques asphériques destinés à l’astronomie spatiale,” These de Doctorat (University of Provence, Aix-Marseille I, France, 1988).
  18. M. Duban, “Les réseaux holographiques asphériques enregistrés avec des ondes laser aberrantes: un outil efficace pour la haute résolution spectrale,” J. Opt. (Paris) 20(6), 269–279 (1989).
    [CrossRef]
  19. M. Duban, “Some reflections about a high-resolution spectrograph,” Appl. Opt. 31, 443–445 (1992).
    [CrossRef] [PubMed]
  20. T. Kita, T. Harada, “Ruling engine using a piezoelectric device for large and high-groove density gratings,” Appl. Opt. 31, 1399–1406 (1992).
    [CrossRef] [PubMed]
  21. J. P. Laude, J. Flamand, A. Thévenon, D. Lepère, “Classical and holographic gratings design and manufacture,” in European Southern Observatory Conference and Workshop Proceedings No. 30, M. H. Ulrich, ed. (European Southern Observatory; Garching, Germany, 1988), pp. 967–989.

1992 (4)

1991 (3)

1989 (3)

1987 (1)

1985 (1)

1984 (1)

1980 (1)

Bowyer, S.

M. Hurwitz, S. Bowyer, T. Harada, T. Kita, “High resolution far ultraviolet spectrographs for Lyman using spherical varied line-space diffraction gratings,” Lyman-FUSE Review Committee (18–19July1990).

Cash, W. C.

Content, D.

Davila, P.

Duban, M.

M. Duban, “Some reflections about a high-resolution spectrograph,” Appl. Opt. 31, 443–445 (1992).
[CrossRef] [PubMed]

M. Duban, “Third-generation Rowland holographic mounting,” Appl. Opt. 30, 4019–4025 (1991).
[CrossRef] [PubMed]

M. Duban, “Les réseaux holographiques asphériques enregistrés avec des ondes laser aberrantes: un outil efficace pour la haute résolution spectrale,” J. Opt. (Paris) 20(6), 269–279 (1989).
[CrossRef]

M. Duban, “Holographic aspheric gratings printed with aberrant waves,” Appl. Opt. 26, 4263–4273 (1987).
[CrossRef] [PubMed]

M. Duban, “Aspheric gratings: recent developments,” Appl. Opt. 24, 3316–3318 (1985).
[CrossRef] [PubMed]

M. Duban, “Calcul et réalisation de spectrographes à réseaux holographiques asphériques destinés à l’astronomie spatiale,” These de Doctorat (University of Provence, Aix-Marseille I, France, 1988).

Flamand, J.

A. Thévenon, J. Flamand, J. P. Laude, B. Touzet, J. Lerner, “Aberration-corrected plane gratings,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moire Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.815, 136–143 (1988).

J. P. Laude, J. Flamand, A. Thévenon, D. Lepère, “Classical and holographic gratings design and manufacture,” in European Southern Observatory Conference and Workshop Proceedings No. 30, M. H. Ulrich, ed. (European Southern Observatory; Garching, Germany, 1988), pp. 967–989.

Grange, R.

Harada, T.

T. Kita, T. Harada, “Ruling engine using a piezoelectric device for large and high-groove density gratings,” Appl. Opt. 31, 1399–1406 (1992).
[CrossRef] [PubMed]

T. Harada, T. Kita, “Mechanically ruled aberration-corrected concave gratings,” Appl. Opt. 19, 3987–3992 (1980).
[CrossRef] [PubMed]

M. Hurwitz, S. Bowyer, T. Harada, T. Kita, “High resolution far ultraviolet spectrographs for Lyman using spherical varied line-space diffraction gratings,” Lyman-FUSE Review Committee (18–19July1990).

Harada, Y.

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

M. Koike, Y. Harada, H. Noda, “New blazed holographic gratings fabricated by using an aspherical recording with an ion-etching method,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moire Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.815, 96–101 (1988).

Hurwitz, M.

M. Hurwitz, S. Bowyer, T. Harada, T. Kita, “High resolution far ultraviolet spectrographs for Lyman using spherical varied line-space diffraction gratings,” Lyman-FUSE Review Committee (18–19July1990).

Kita, T.

T. Kita, T. Harada, “Ruling engine using a piezoelectric device for large and high-groove density gratings,” Appl. Opt. 31, 1399–1406 (1992).
[CrossRef] [PubMed]

T. Harada, T. Kita, “Mechanically ruled aberration-corrected concave gratings,” Appl. Opt. 19, 3987–3992 (1980).
[CrossRef] [PubMed]

M. Hurwitz, S. Bowyer, T. Harada, T. Kita, “High resolution far ultraviolet spectrographs for Lyman using spherical varied line-space diffraction gratings,” Lyman-FUSE Review Committee (18–19July1990).

Koike, M.

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

T. Namioka, M. Koike, “Design of compact high resolution far-ultraviolet spectrographs equipped with a spherical grating having variable spacing and curved grooves,” presented at the Tenth International Colloquium on VUV and X-Ray Spectroscopy of Astrophysics and Laboratory Plasmas, Berkeley, Calif., 3–5 February 1992.

M. Koike, Y. Harada, H. Noda, “New blazed holographic gratings fabricated by using an aspherical recording with an ion-etching method,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moire Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.815, 96–101 (1988).

Laget, M.

Laude, J. P.

J. P. Laude, J. Flamand, A. Thévenon, D. Lepère, “Classical and holographic gratings design and manufacture,” in European Southern Observatory Conference and Workshop Proceedings No. 30, M. H. Ulrich, ed. (European Southern Observatory; Garching, Germany, 1988), pp. 967–989.

A. Thévenon, J. Flamand, J. P. Laude, B. Touzet, J. Lerner, “Aberration-corrected plane gratings,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moire Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.815, 136–143 (1988).

Lepère, D.

J. P. Laude, J. Flamand, A. Thévenon, D. Lepère, “Classical and holographic gratings design and manufacture,” in European Southern Observatory Conference and Workshop Proceedings No. 30, M. H. Ulrich, ed. (European Southern Observatory; Garching, Germany, 1988), pp. 967–989.

Lerner, J.

A. Thévenon, J. Flamand, J. P. Laude, B. Touzet, J. Lerner, “Aberration-corrected plane gratings,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moire Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.815, 136–143 (1988).

Moos, H. W.

H. W. Moos, “Lyman, the Far Ultraviolet Spectroscopic Explorer, phase A study final report,” NASA contract NAS5-30339 (NASA Goddard Space Flight Center, Greenbelt, Md., July, 1989),Vols. 1 and 2.

Namioka, T.

T. Namioka, M. Koike, “Design of compact high resolution far-ultraviolet spectrographs equipped with a spherical grating having variable spacing and curved grooves,” presented at the Tenth International Colloquium on VUV and X-Ray Spectroscopy of Astrophysics and Laboratory Plasmas, Berkeley, Calif., 3–5 February 1992.

Noda, H.

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

M. Koike, Y. Harada, H. Noda, “New blazed holographic gratings fabricated by using an aspherical recording with an ion-etching method,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moire Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.815, 96–101 (1988).

Palmer, C.

Thévenon, A.

A. Thévenon, J. Flamand, J. P. Laude, B. Touzet, J. Lerner, “Aberration-corrected plane gratings,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moire Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.815, 136–143 (1988).

J. P. Laude, J. Flamand, A. Thévenon, D. Lepère, “Classical and holographic gratings design and manufacture,” in European Southern Observatory Conference and Workshop Proceedings No. 30, M. H. Ulrich, ed. (European Southern Observatory; Garching, Germany, 1988), pp. 967–989.

Touzet, B.

A. Thévenon, J. Flamand, J. P. Laude, B. Touzet, J. Lerner, “Aberration-corrected plane gratings,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moire Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.815, 136–143 (1988).

Trout, C.

Wilson, M.

Appl. Opt. (12)

T. Harada, T. Kita, “Mechanically ruled aberration-corrected concave gratings,” Appl. Opt. 19, 3987–3992 (1980).
[CrossRef] [PubMed]

W. C. Cash, “Aspheric concave grating spectrographs,” Appl. Opt. 23, 4518–4522 (1984).
[CrossRef] [PubMed]

M. Duban, “Holographic aspheric gratings printed with aberrant waves,” Appl. Opt. 26, 4263–4273 (1987).
[CrossRef] [PubMed]

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

D. Content, C. Trout, P. Davila, M. Wilson, “Aberration corrected aspheric gratings for far ultraviolet spectrographs: conventional approach,” Appl. Opt. 30, 801–806 (1991).
[CrossRef] [PubMed]

R. Grange, M. Laget, “Holographic diffraction gratings generated by aberrated wave fronts: application to a high-resolution far-ultaviolet spectrograph,” Appl. Opt. 30, 3598–3603 (1991).
[CrossRef] [PubMed]

M. Duban, “Third-generation Rowland holographic mounting,” Appl. Opt. 30, 4019–4025 (1991).
[CrossRef] [PubMed]

C. Trout, D. Content, P. Davila, “Aberration-corrected aspheric grating designs for the Lyman/Far-Ultraviolet Spectroscopic Explorer high-resolution spectrograph: a comparison,” Appl. Opt. 31, 943–948 (1992).
[CrossRef] [PubMed]

P. Davila, D. Content, C. Trout, “Aberration-corrected aspheric gratings for far-ultraviolet spectrographs: holographic approach,” Appl. Opt. 31, 949–954 (1992).
[CrossRef] [PubMed]

T. Kita, T. Harada, “Ruling engine using a piezoelectric device for large and high-groove density gratings,” Appl. Opt. 31, 1399–1406 (1992).
[CrossRef] [PubMed]

M. Duban, “Aspheric gratings: recent developments,” Appl. Opt. 24, 3316–3318 (1985).
[CrossRef] [PubMed]

M. Duban, “Some reflections about a high-resolution spectrograph,” Appl. Opt. 31, 443–445 (1992).
[CrossRef] [PubMed]

J. Opt. (Paris) (1)

M. Duban, “Les réseaux holographiques asphériques enregistrés avec des ondes laser aberrantes: un outil efficace pour la haute résolution spectrale,” J. Opt. (Paris) 20(6), 269–279 (1989).
[CrossRef]

J. Opt. Soc. Am. A (1)

Other (7)

J. P. Laude, J. Flamand, A. Thévenon, D. Lepère, “Classical and holographic gratings design and manufacture,” in European Southern Observatory Conference and Workshop Proceedings No. 30, M. H. Ulrich, ed. (European Southern Observatory; Garching, Germany, 1988), pp. 967–989.

H. W. Moos, “Lyman, the Far Ultraviolet Spectroscopic Explorer, phase A study final report,” NASA contract NAS5-30339 (NASA Goddard Space Flight Center, Greenbelt, Md., July, 1989),Vols. 1 and 2.

M. Hurwitz, S. Bowyer, T. Harada, T. Kita, “High resolution far ultraviolet spectrographs for Lyman using spherical varied line-space diffraction gratings,” Lyman-FUSE Review Committee (18–19July1990).

M. Koike, Y. Harada, H. Noda, “New blazed holographic gratings fabricated by using an aspherical recording with an ion-etching method,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moire Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.815, 96–101 (1988).

A. Thévenon, J. Flamand, J. P. Laude, B. Touzet, J. Lerner, “Aberration-corrected plane gratings,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moire Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.815, 136–143 (1988).

T. Namioka, M. Koike, “Design of compact high resolution far-ultraviolet spectrographs equipped with a spherical grating having variable spacing and curved grooves,” presented at the Tenth International Colloquium on VUV and X-Ray Spectroscopy of Astrophysics and Laboratory Plasmas, Berkeley, Calif., 3–5 February 1992.

M. Duban, “Calcul et réalisation de spectrographes à réseaux holographiques asphériques destinés à l’astronomie spatiale,” These de Doctorat (University of Provence, Aix-Marseille I, France, 1988).

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Figures (18)

Fig. 1
Fig. 1

Astigmatism A2, coma C2, spherical aberrations S1, S2, and S3 of a 5767-line/mm grating ruled upon a deformed ellipsoid (solid curves) or recorded upon a sphere (dotted curves).

Fig. 2
Fig. 2

Astigmatism A2, coma C2, and spherical aberrations S1, S2, and S3 of three 5764.88-line/mm gratings ruled upon deformed oblate ellipsoids: analytic centered (solid curves), symmetrically corrected (dashed curves), optimized centered from Ref. 3 (dotted curves).

Fig. 3
Fig. 3

Minimum image width w as a function of rate r of enclosed energy for an oblate ellipsoidal 5764.88-line/mm ruled grating.

Fig. 4
Fig. 4

Spot diagrams that resulted from a 5767-line/mm grating ruled upon a deformed prolate ellipsoid.

Fig. 5
Fig. 5

Spot diagrams that resulted from a 5177-line/mm grating ruled upon a deformed prolate ellipsoid.

Fig. 6
Fig. 6

Spot diagrams that resulted from a 4698-line/mm grating ruled upon a deformed prolate ellipsoid.

Fig. 7
Fig. 7

Spot diagrams that resulted from a 2738-line/mm grating ruled upon a deformed prolate ellipsoid.

Fig. 8
Fig. 8

Spot diagrams that resulted from an 871-line/mm grating ruled upon a deformed prolate ellipsoid.

Fig. 9
Fig. 9

Spot diagrams produced by a 5764-line/mm ellipsoidal grating that was recorded by using two spherical mirrors.

Fig. 10
Fig. 10

Spot diagrams produced by a 2738-line/mm spherical grating that was recorded by using two auxiliary spherical gratings.

Fig. 11
Fig. 11

Spot diagrams produced by an 871-line/mm spherical grating that was recorded by using one auxiliary spherical grating.

Fig. 12
Fig. 12

Spot diagram produced by a 5767-line/mm spherical grating that was recorded with a theoretically aberrated laser wave front.

Fig. 13
Fig. 13

Spot-diagrams actually produced by a 5767-line/mm spherical grating that was recorded by using two auxiliary spherical gratings.

Fig. 14
Fig. 14

Spot diagrams produced by a 5767-line/mm hybrid spherical grating.

Fig. 15
Fig. 15

Spot diagrams produced by (1) 5767- and (2) 6000-lines/mm hybrid gratings, (3) a 6000-line/mm SVLS grating, and (4) a 5767-line/mm third generation spherical grating.

Fig. 16
Fig. 16

Astigmatism A2, coma C2, spherical aberrations S1, S2, and S3 of a 5767-line/mm hybrid spherical grating that works over an extended spectral range (80–160 nm).

Fig. 17
Fig. 17

Spot diagrams produced by a 5767-line/mm hybrid grating that works over an extended spectral range (80–160 nm).

Fig. 18
Fig. 18

Geometry that was used to define normalized aberrations.

Tables (11)

Tables Icon

Table 1 Deformation Coefficients for a Ruled Oblate Ellipsoidal Grating (5764.88 lines/mm)

Tables Icon

Table 2 Basic Specifications for Lyman/FUSE Gratings

Tables Icon

Table 3 Geometrical Parameters of Ruled Prolate Ellipsoidal Gratings

Tables Icon

Table 4 Deformation Coefficients Computed for Ruled Prolate Ellipsoidal Gratings

Tables Icon

Table 5 Parameters of Spherical Holographic Grating 4

Tables Icon

Table 6 Parameters of Spherical Holographic Grating 5

Tables Icon

Table 7 Basic Parameters of Spherical Holographic Grating 1a

Tables Icon

Table 8 Optimized Parameters of Spherical Holographic Grating 1a

Tables Icon

Table 9 Parameters of the Auxiliary Gratings Used to Record Main Grating 1

Tables Icon

Table 10 Theoretical Parameters for Two Hybrid Gratings

Tables Icon

Table 11 Theoretical Grating Parameters for an Extended Spectral Range (80–160 nm)a

Equations (31)

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F = A M + B M + n m λ,
n λ 0 = ( C M D M ) ( C O D O )
σ = λ 0 / ( sin δ sin γ ) .
Δ = F ( M ) F ( O ) .
Δ = A 1 y 2 + A 2 z 2 + C 1 y 3 + C 2 y z 2 + S 1 y 4 + S 2 y 2 z 2 + S 3 z 4 + ,
Δ = V ( A ) + V ( B ) + ( m λ / λ 0 ) [ V ( C ) V ( D ) ]
Δ = V ( A ) + V ( B ) + m y λ / σ
Δ = Σ [ V ( Q ) ] ,
a = R cos 2 u , b = R sin u cos u , O Q = d = R cos u .
A 1 = 0 , A 2 = Σ [ q / d ] / 2 C 1 = 0 C 2 = Σ [ q s ( 1 + t 2 ) ] / 2 R 2 , S 1 = Σ [ s t ] / 8 R 3 , S 2 = Σ [ ( 2 t 2 / R s 2 / ρ ) / cos u ] / 4 R 2 S 3 = Σ [ q ( 1 / ρ 2 q / d 2 ) / cos u ] / 8 R .
A 2 = Σ [ s t ] / 2 R , C 2 = Σ [ s t 2 ] / 2 R 2 S 2 = Σ [ s t ( 1 + 2 t 2 ) ] / 4 R 3 , S 3 = Σ [ s t ( 1 t 2 ) ] / 8 R 3 .
0 cos u , 1 cos u , cos u ( 2 + 0 sin u / d ) , 3 cos u ,
a = b = R , c = ( R ρ ) for ( O ) , a = c = ρ , b = ( R ρ ) for ( P ) ,
y 2 / 2 R + z 2 / 2 ρ ,
y 4 / 8 R 3 + y 2 z 2 / 4 R 2 ρ + z 4 / 8 R ρ 2 for ( O ) ,
y 4 / 8 R 2 ρ + y 2 z 2 / 4 R ρ 2 + z 4 / 8 ρ 3 for ( P ) ,
y 4 / 8 R 3 + y 2 z 2 / 4 R 2 ρ + z 4 / 8 ρ 3 for ( T ) .
α = arcsin ( m λ 3 / σ ) ,
A 2 = ( 1 R cos 2 α / ρ ) / 2 R cos α + ( 1 R cos 2 β / ρ ) / 2 R cos β .
ρ / R = cos α cos β a .
A 2 ( λ 1 ) = A 2 ( λ 2 ) = A 2 ( λ 3 ) ,
ρ / R = cos α [ ( 1 + 2 cos α + cos β m ) / ( 2 + cos α + cos α sec β m ) ] = cos α cos β a .
S 1 = ( sin α tan α + sin β tan β ) / 8 R 3 1 ( cos α + cos β ) .
1 = ( 1 / ρ 1 / R ) / 8 R 2 .
A 2 / S 1 = 4 R 2 .
1 = sin α tan α / 8 R 3 ( 1 + cos α )
S 3 = E ( α ) + E ( β ) ,
E ( u ) = ( 1 R cos 2 u / ρ ) [ 1 / ρ 2 ( 1 R cos 2 u / ρ ) / R 2 cos 2 u ] / 8 R cos u 3 cos u .
3 = ( cos α cos β a ) 2 / ( 2 R cos α cos β a ) 3 .
3 = [ 1 cos α cos β a ( cos α cos β a ) 2 ] / ( 2 R cos α cos β a ) 3 .
T ( 2 + T 2 ) = ( sin α tan α + sin β tan β ) / ( sin α + sin β ) ,

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