Abstract

The imaging properties of aspherical diffraction gratings that have curved and variable spacing grooves are studied theoretically for simple rotational mountings. Methods for eliminating aberrations are examined by clarifying the relations of both the shape of the grating surface and the pattern of the grooves to the individual aberrations, such as astigmatism, coma, and curvature. Application of the methods to the Seya-Namioka mounting shows (i) a spherical grating with straight grooves of variable spacing, designed for elimination of coma, can give very sharp images over a wide spectral range; and (ii) a torus grating with curved grooves of variable spacing can give stigmatic images without serious image broadening. Therefore, resolution can be much improved in both spherical and torus gratings compared with those having conventional (straight and constant spacing) grooves.

© 1979 Optical Society of America

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References

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  1. E. Schönheit, “Ein Seya-Namioka-Monochromator mit torischem Gitter für Photoionisations-Messungen,” Optik (Stuttgart) 23, 305–312(1966).
  2. S. A. Strezhnev and A. L. Andreeva, “Toroidal diffraction gratings for Seya-Namioka monochromators,” Opt. Spectrosc. 28, 426–428 (1970).
  3. R. J. Speer, “A comparative review of grazing incidence gratings, including recent measurements on the performance of stigmatic soft x-ray reflection gratings formed holographically,” J. Spectrosc. Soc. Jpn. 23, Suppl. 1, 53–60(1974); R. J. Speer, D. Turner, R. L. Johnson, D. Rudolph, and G. Schmahl, “Holographically formed grazing-incidence reflection grating with stigmatic soft x-ray focal isolation,” Appl. Opt. 13, 1258–1261 (1974).
    [Crossref] [PubMed]
  4. M. V. R. K. Murty, “Spherical zone-plate diffraction grating,” J. Opt. Soc. Am. 50, 923(1960).
    [Crossref]
  5. Y. Sakayanagi, “A stigmatic concave grating with varying spacing,” Sci. Light (Tokyo) 16, 129–137(1967); Y. Sakayanagi, “Theory of grating with circular grooves (curved grating),” Sci. Light (Tokyo) 3, 1–4(1954); Y. Sakayanagi, “Ruling of a curved grating,” Sci. Light (Tokyo) 3, 79–83(1955).
  6. F. M. Gerasimov, E. A. Yakovlev, I. V. Peisakhson, and B. V. Kashelev, “Concave diffraction gratings with variable spacing,” Opt. Spectrosc. 28, 423–426(1970).
  7. T. Namioka, H. Noda, and M. Seya, “Possibility of using the holographic concave grating in vacuum monochromators,” Sci. Light (Tokyo) 22, 77–99(1973); T. Namioka, M. Seya, and H. Noda, “Design and performance of holographic concave gratings,” Jpn. J. Appl. Phys. 15, 1181–1197(1976).
    [Crossref]
  8. M. Pouey, “Imaging properties of ruled and holographic gratings,” J. Spectrosc. Soc. Jpn. 23, Suppl. 1, 67–81(1974).
  9. M. V. R. K. Murty and N. C. Das, “Theory of certain diffraction gratings produced by the holographic method,” J. Opt. Soc. Am. 61, 1001–1006 (1971).
    [Crossref]
  10. H. Noda, T. Namioka, and M. Seya, “Geometric theory of the grating,” J. Opt. Soc. Am. 64, 1031–1036(1974).
    [Crossref]
  11. C. H. F. Velzel, “On the imaging properties of holographic gratings,” J. Opt. Soc. Am. 67, 1021–1027(1977).
    [Crossref]
  12. Jobin Yvon Optical Systems, Diffraction gratings—ruled and holographic (Jobin Yvon Optical Systems, Metuchen, 1973), pp. 17–27.
  13. H. Noda, T. Namioka, and M. Seya, “Design of holographic concave gratings for Seya-Namioka monochromator,” J. Opt. Soc. Am. 64, 1043–1048(1974); H. Noda, T. Namioka, and M. Seya, “Aberration-reduced holographic concave gratings for Seya-Namioka monochromators,” Jpn. J. Appl. Phys. 14, Suppl. 1, 187–191(1975).
    [Crossref]
  14. T. Harada, S. Moriyama, and T. Kita, “Mechanically ruled stigmatic concave gratings,” Jpn. J. Appl. Phys. 14, Suppl. 1, 175–179(1975).
  15. D. Lepère, “Monochromateur a simple rotation du rèseau, a rèseau holographique sur support torique pour l’ultraviolet lointain,” Nouv. Rev. Opt. 6, 173–178(1975).
  16. F. Masuda, H. Noda, and T. Namioka, “Design and performance of toroidal holographie gratings,” J. Spectrosc. Soc. Jpn 27, 211–223 (1978).
    [Crossref]
  17. T. Namioka, “Theory of the concave grating. III. Seya-Namioka monochromator,” J. Opt. Soc. Am. 49, 951–961(1959).
    [Crossref]

1978 (1)

F. Masuda, H. Noda, and T. Namioka, “Design and performance of toroidal holographie gratings,” J. Spectrosc. Soc. Jpn 27, 211–223 (1978).
[Crossref]

1977 (1)

1975 (2)

T. Harada, S. Moriyama, and T. Kita, “Mechanically ruled stigmatic concave gratings,” Jpn. J. Appl. Phys. 14, Suppl. 1, 175–179(1975).

D. Lepère, “Monochromateur a simple rotation du rèseau, a rèseau holographique sur support torique pour l’ultraviolet lointain,” Nouv. Rev. Opt. 6, 173–178(1975).

1974 (4)

H. Noda, T. Namioka, and M. Seya, “Design of holographic concave gratings for Seya-Namioka monochromator,” J. Opt. Soc. Am. 64, 1043–1048(1974); H. Noda, T. Namioka, and M. Seya, “Aberration-reduced holographic concave gratings for Seya-Namioka monochromators,” Jpn. J. Appl. Phys. 14, Suppl. 1, 187–191(1975).
[Crossref]

H. Noda, T. Namioka, and M. Seya, “Geometric theory of the grating,” J. Opt. Soc. Am. 64, 1031–1036(1974).
[Crossref]

R. J. Speer, “A comparative review of grazing incidence gratings, including recent measurements on the performance of stigmatic soft x-ray reflection gratings formed holographically,” J. Spectrosc. Soc. Jpn. 23, Suppl. 1, 53–60(1974); R. J. Speer, D. Turner, R. L. Johnson, D. Rudolph, and G. Schmahl, “Holographically formed grazing-incidence reflection grating with stigmatic soft x-ray focal isolation,” Appl. Opt. 13, 1258–1261 (1974).
[Crossref] [PubMed]

M. Pouey, “Imaging properties of ruled and holographic gratings,” J. Spectrosc. Soc. Jpn. 23, Suppl. 1, 67–81(1974).

1973 (1)

T. Namioka, H. Noda, and M. Seya, “Possibility of using the holographic concave grating in vacuum monochromators,” Sci. Light (Tokyo) 22, 77–99(1973); T. Namioka, M. Seya, and H. Noda, “Design and performance of holographic concave gratings,” Jpn. J. Appl. Phys. 15, 1181–1197(1976).
[Crossref]

1971 (1)

1970 (2)

F. M. Gerasimov, E. A. Yakovlev, I. V. Peisakhson, and B. V. Kashelev, “Concave diffraction gratings with variable spacing,” Opt. Spectrosc. 28, 423–426(1970).

S. A. Strezhnev and A. L. Andreeva, “Toroidal diffraction gratings for Seya-Namioka monochromators,” Opt. Spectrosc. 28, 426–428 (1970).

1967 (1)

Y. Sakayanagi, “A stigmatic concave grating with varying spacing,” Sci. Light (Tokyo) 16, 129–137(1967); Y. Sakayanagi, “Theory of grating with circular grooves (curved grating),” Sci. Light (Tokyo) 3, 1–4(1954); Y. Sakayanagi, “Ruling of a curved grating,” Sci. Light (Tokyo) 3, 79–83(1955).

1966 (1)

E. Schönheit, “Ein Seya-Namioka-Monochromator mit torischem Gitter für Photoionisations-Messungen,” Optik (Stuttgart) 23, 305–312(1966).

1960 (1)

1959 (1)

Andreeva, A. L.

S. A. Strezhnev and A. L. Andreeva, “Toroidal diffraction gratings for Seya-Namioka monochromators,” Opt. Spectrosc. 28, 426–428 (1970).

Das, N. C.

Gerasimov, F. M.

F. M. Gerasimov, E. A. Yakovlev, I. V. Peisakhson, and B. V. Kashelev, “Concave diffraction gratings with variable spacing,” Opt. Spectrosc. 28, 423–426(1970).

Harada, T.

T. Harada, S. Moriyama, and T. Kita, “Mechanically ruled stigmatic concave gratings,” Jpn. J. Appl. Phys. 14, Suppl. 1, 175–179(1975).

Kashelev, B. V.

F. M. Gerasimov, E. A. Yakovlev, I. V. Peisakhson, and B. V. Kashelev, “Concave diffraction gratings with variable spacing,” Opt. Spectrosc. 28, 423–426(1970).

Kita, T.

T. Harada, S. Moriyama, and T. Kita, “Mechanically ruled stigmatic concave gratings,” Jpn. J. Appl. Phys. 14, Suppl. 1, 175–179(1975).

Lepère, D.

D. Lepère, “Monochromateur a simple rotation du rèseau, a rèseau holographique sur support torique pour l’ultraviolet lointain,” Nouv. Rev. Opt. 6, 173–178(1975).

Masuda, F.

F. Masuda, H. Noda, and T. Namioka, “Design and performance of toroidal holographie gratings,” J. Spectrosc. Soc. Jpn 27, 211–223 (1978).
[Crossref]

Moriyama, S.

T. Harada, S. Moriyama, and T. Kita, “Mechanically ruled stigmatic concave gratings,” Jpn. J. Appl. Phys. 14, Suppl. 1, 175–179(1975).

Murty, M. V. R. K.

Namioka, T.

F. Masuda, H. Noda, and T. Namioka, “Design and performance of toroidal holographie gratings,” J. Spectrosc. Soc. Jpn 27, 211–223 (1978).
[Crossref]

H. Noda, T. Namioka, and M. Seya, “Geometric theory of the grating,” J. Opt. Soc. Am. 64, 1031–1036(1974).
[Crossref]

H. Noda, T. Namioka, and M. Seya, “Design of holographic concave gratings for Seya-Namioka monochromator,” J. Opt. Soc. Am. 64, 1043–1048(1974); H. Noda, T. Namioka, and M. Seya, “Aberration-reduced holographic concave gratings for Seya-Namioka monochromators,” Jpn. J. Appl. Phys. 14, Suppl. 1, 187–191(1975).
[Crossref]

T. Namioka, H. Noda, and M. Seya, “Possibility of using the holographic concave grating in vacuum monochromators,” Sci. Light (Tokyo) 22, 77–99(1973); T. Namioka, M. Seya, and H. Noda, “Design and performance of holographic concave gratings,” Jpn. J. Appl. Phys. 15, 1181–1197(1976).
[Crossref]

T. Namioka, “Theory of the concave grating. III. Seya-Namioka monochromator,” J. Opt. Soc. Am. 49, 951–961(1959).
[Crossref]

Noda, H.

F. Masuda, H. Noda, and T. Namioka, “Design and performance of toroidal holographie gratings,” J. Spectrosc. Soc. Jpn 27, 211–223 (1978).
[Crossref]

H. Noda, T. Namioka, and M. Seya, “Design of holographic concave gratings for Seya-Namioka monochromator,” J. Opt. Soc. Am. 64, 1043–1048(1974); H. Noda, T. Namioka, and M. Seya, “Aberration-reduced holographic concave gratings for Seya-Namioka monochromators,” Jpn. J. Appl. Phys. 14, Suppl. 1, 187–191(1975).
[Crossref]

H. Noda, T. Namioka, and M. Seya, “Geometric theory of the grating,” J. Opt. Soc. Am. 64, 1031–1036(1974).
[Crossref]

T. Namioka, H. Noda, and M. Seya, “Possibility of using the holographic concave grating in vacuum monochromators,” Sci. Light (Tokyo) 22, 77–99(1973); T. Namioka, M. Seya, and H. Noda, “Design and performance of holographic concave gratings,” Jpn. J. Appl. Phys. 15, 1181–1197(1976).
[Crossref]

Peisakhson, I. V.

F. M. Gerasimov, E. A. Yakovlev, I. V. Peisakhson, and B. V. Kashelev, “Concave diffraction gratings with variable spacing,” Opt. Spectrosc. 28, 423–426(1970).

Pouey, M.

M. Pouey, “Imaging properties of ruled and holographic gratings,” J. Spectrosc. Soc. Jpn. 23, Suppl. 1, 67–81(1974).

Sakayanagi, Y.

Y. Sakayanagi, “A stigmatic concave grating with varying spacing,” Sci. Light (Tokyo) 16, 129–137(1967); Y. Sakayanagi, “Theory of grating with circular grooves (curved grating),” Sci. Light (Tokyo) 3, 1–4(1954); Y. Sakayanagi, “Ruling of a curved grating,” Sci. Light (Tokyo) 3, 79–83(1955).

Schönheit, E.

E. Schönheit, “Ein Seya-Namioka-Monochromator mit torischem Gitter für Photoionisations-Messungen,” Optik (Stuttgart) 23, 305–312(1966).

Seya, M.

Speer, R. J.

R. J. Speer, “A comparative review of grazing incidence gratings, including recent measurements on the performance of stigmatic soft x-ray reflection gratings formed holographically,” J. Spectrosc. Soc. Jpn. 23, Suppl. 1, 53–60(1974); R. J. Speer, D. Turner, R. L. Johnson, D. Rudolph, and G. Schmahl, “Holographically formed grazing-incidence reflection grating with stigmatic soft x-ray focal isolation,” Appl. Opt. 13, 1258–1261 (1974).
[Crossref] [PubMed]

Strezhnev, S. A.

S. A. Strezhnev and A. L. Andreeva, “Toroidal diffraction gratings for Seya-Namioka monochromators,” Opt. Spectrosc. 28, 426–428 (1970).

Velzel, C. H. F.

Yakovlev, E. A.

F. M. Gerasimov, E. A. Yakovlev, I. V. Peisakhson, and B. V. Kashelev, “Concave diffraction gratings with variable spacing,” Opt. Spectrosc. 28, 423–426(1970).

J. Opt. Soc. Am. (6)

J. Spectrosc. Soc. Jpn (1)

F. Masuda, H. Noda, and T. Namioka, “Design and performance of toroidal holographie gratings,” J. Spectrosc. Soc. Jpn 27, 211–223 (1978).
[Crossref]

J. Spectrosc. Soc. Jpn. (2)

R. J. Speer, “A comparative review of grazing incidence gratings, including recent measurements on the performance of stigmatic soft x-ray reflection gratings formed holographically,” J. Spectrosc. Soc. Jpn. 23, Suppl. 1, 53–60(1974); R. J. Speer, D. Turner, R. L. Johnson, D. Rudolph, and G. Schmahl, “Holographically formed grazing-incidence reflection grating with stigmatic soft x-ray focal isolation,” Appl. Opt. 13, 1258–1261 (1974).
[Crossref] [PubMed]

M. Pouey, “Imaging properties of ruled and holographic gratings,” J. Spectrosc. Soc. Jpn. 23, Suppl. 1, 67–81(1974).

Jpn. J. Appl. Phys. (1)

T. Harada, S. Moriyama, and T. Kita, “Mechanically ruled stigmatic concave gratings,” Jpn. J. Appl. Phys. 14, Suppl. 1, 175–179(1975).

Nouv. Rev. Opt. (1)

D. Lepère, “Monochromateur a simple rotation du rèseau, a rèseau holographique sur support torique pour l’ultraviolet lointain,” Nouv. Rev. Opt. 6, 173–178(1975).

Opt. Spectrosc. (2)

F. M. Gerasimov, E. A. Yakovlev, I. V. Peisakhson, and B. V. Kashelev, “Concave diffraction gratings with variable spacing,” Opt. Spectrosc. 28, 423–426(1970).

S. A. Strezhnev and A. L. Andreeva, “Toroidal diffraction gratings for Seya-Namioka monochromators,” Opt. Spectrosc. 28, 426–428 (1970).

Optik (Stuttgart) (1)

E. Schönheit, “Ein Seya-Namioka-Monochromator mit torischem Gitter für Photoionisations-Messungen,” Optik (Stuttgart) 23, 305–312(1966).

Sci. Light (Tokyo) (2)

Y. Sakayanagi, “A stigmatic concave grating with varying spacing,” Sci. Light (Tokyo) 16, 129–137(1967); Y. Sakayanagi, “Theory of grating with circular grooves (curved grating),” Sci. Light (Tokyo) 3, 1–4(1954); Y. Sakayanagi, “Ruling of a curved grating,” Sci. Light (Tokyo) 3, 79–83(1955).

T. Namioka, H. Noda, and M. Seya, “Possibility of using the holographic concave grating in vacuum monochromators,” Sci. Light (Tokyo) 22, 77–99(1973); T. Namioka, M. Seya, and H. Noda, “Design and performance of holographic concave gratings,” Jpn. J. Appl. Phys. 15, 1181–1197(1976).
[Crossref]

Other (1)

Jobin Yvon Optical Systems, Diffraction gratings—ruled and holographic (Jobin Yvon Optical Systems, Metuchen, 1973), pp. 17–27.

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Figures (10)

FIG. 1
FIG. 1

Schematic diagram of the coordinate systems. The object and image points are given in the (x,y,z) coordinate system, where the coordinates for the latter are marked by asterisks. The grating surface is given in the (w,l,u) coordinate system which coincides with the (x,y,z) coordinate system.

FIG. 2
FIG. 2

Stationary of γ f * around γ s * illustrated through the ratio of γ f * / γ s * as a function of λ/d0, for the cases (a) Q = 69°(Case II); (b) N20 = 0(Case II—conventional Seya-Namioka type); (c) Q = 70.5°(Case II), and (d) Case III. Curves denoted by 1, 2, and 3 in each figure are evaluated for λs/d0 = 0.06, 0.18, and 0.3, respectively.

FIG. 3
FIG. 3

Coma-free images illustrated through ray tracing under the same conditions as listed in Table I. Images other than those of coma-free wavelength (λs/d0 = 0.18) were obtained for λ/d0 = 0.06, 0.3, and 0.42 by grating scanning. All the spot diagrams in the article were obtained for gratings having the meridional radius of curvature of 500 mm and ruled area of 30 mm(l) × 40 mm(w).

FIG. 4
FIG. 4

Groove Separation d(w,0) along the w axis illustrated for the same cases as in Fig. 3. Here, d0, the groove separation at O, is assumed to be 1/600 mm. The horizontal dashed line in (b) is for a conventional grating. Note that the scale of Fig. 4(b) is changed to show finer detail.

FIG. 5
FIG. 5

Spot diagrams by coma-free [(a), (b), and (c)] or conventional (d) gratings in a Seya-Namioka mount. Figures 5(a), 5(b), and 5(c) are evaluated for λs/d0 = 0.06, 0.18, and 0.3, respectively. Images of wavelengths other than λs, obtained by scanning, are also shown. Dashed curves were evaluated for an off-plane point, 3 mm under the meridional plane.

FIG. 6
FIG. 6

Spot diagrams of stigmatic images shown for a spherical grating with curved and variable grooves (a), a torus grating with straight and variable grooves (b), and a conventional torus grating (c), respectively. Points of numbers 1, 2, 3, and 4 correspond to the points (w,l) = (20,15), (−20,15), (−20,−15), and (20,−15) on the pupil plane in millimeter units, respectively. Note that the scale of (b) and (c) is changed to show finer details.

FIG. 7
FIG. 7

Arrangement for ruling curved grooves on a concave surface illustrated in the meridional plane. The tool reciprocates in the planes normal to the meridional plane.

FIG. 8
FIG. 8

Spot diagrams of curvature-free images [(a) and (b)] and point-like images [(c) and (d)]. Parameters in each case are as follows: (a) u02 = 1 and θ = −65.817°; (b) u02= 1.625 and θ = 0°; (c) u02 = 1.3850 and θ = 36.893°; and (d) u02 = 1.6146 and θ = −32.703°. Points indicated by numbers 1, 2, 3, and 4 are used in a way similar to that of Fig. 6. Note that the scales are changed to show finer details.

FIG. 9
FIG. 9

Spot diagrams for a line slit by three types of gratings: (a) a conventional torus grating, (b) a torus grating with straight and variable spacing grooves, and (c) a torus grating with curved and variable grooves.

FIG. 10
FIG. 10

Spectral profiles constructed from the spot diagrams of Fig. 9 with those of some other wavelengths. Horizontal lines in the middle of each spectrum show the half-maximum width.

Tables (1)

Tables Icon

TABLE I Parameters for coma-free gratings in simple rotational mountings.

Equations (35)

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u = i + j = 2 1 i ! j ! u i j w i l j ( i , j = 1 , 2 , 3 , ) ,
N = = N 10 w + i + j = 2 1 i ! j ! N i j w i l j ( i , j = 1 , 2 , 3 , ) ,
F = A P + P A * + m λ N γ + γ * + m λ N ( A P γ , P A * γ * ) ,
{ x * = γ * cos ξ p * + w , y * = γ * cos η p * + l , z * = γ * cos ζ p * + u ,
{ x * = γ 0 * M w + O ( 2 ) , y * = γ 0 * S l + O ( 2 ) ,
M cos 2 α / γ 0 + cos 2 α 0 * / γ 0 * ( cos α + cos α 0 * ) + d 0 N 20 ( sin α + sin α 0 * ) ,
S 1 / γ 0 + 1 / γ 0 * u 02 ( cos α + cos α 0 * ) + d 0 N 02 ( sin α + sin α 0 * ) .
{ Q α α 0 * = const , γ 0 = const .
M = ( cos 2 α s γ 0 + cos 2 α s * γ s * ( cos α s + cos α s * ) + d 0 N 20 ( sin α s + sin α s * ) ) ( sin 2 α s γ 0 + sin 2 α s * γ s * ( sin α s + sin α s * ) d 0 N 20 ( cos α s + cos α s * ) ) Δ α 1 2 ( 2 cos 2 α s γ 0 + 2 cos 2 α s * γ s * ( cos α s + cos α s * ) + d 0 N 20 ( sin α s + sin α s * ) ) ( Δ α ) 2 + 1 6 ( 4 sin 2 α s γ 0 + 4 sin 2 α s * γ s * ( sin α s + sin α s * ) d 0 N 20 ( cos α s + cos α s * ) ) ( Δ α ) 3 + O [ ( Δ α ) 4 ] ,
{ Q = 0 , α s = α s * = sin 1 ( 1 / 3 ) , N 20 = 20 / 4 d 0 , 1 / γ 0 + 1 / γ s * = 3 6 / 4 .
{ Q : free N 20 = ( cos α s + cos α s * ) ( cos Q + cos α s cos α s * 1 ) d 0 ( sin α s + sin α s * ) ( cos Q + cos α s cos α s * + 1 ) , 1 γ 0 = 3 cos 2 α s * 2 ( cos α s * cos α s ) ( cos Q + cos α s cos α s * + 1 ) , 1 γ s * = 2 3 cos 2 α s ( cos α s * cos α s ) ( cos Q + cos α s cos α s * + 1 ) ,
cos Q + cos α s cos α s * 1 = 0 .
{ Q = α s ± tan 1 ( 3 sec α s 2 tan α s ) , N 20 = 0 , 1 γ 0 = 3 cos 2 α s * 2 2 ( cos α s * cos α s ) , 1 γ s * = 2 3 cos 2 α s 2 ( cos α s * cos α s ) .
{ Q = tan 1 ( 1 + sin 2 α s sin α s cos α s * ) , N 20 = sin α s + sin α s * d 0 ( cos α s + cos α s * ) , 1 / γ 0 , 1 / γ s * ; same as in Eqs . ( 11 ) ,
{ α s * = α s = sin 1 ( 1 / 3 ) , N 20 = 0 , Q = tan 1 2 2 ( 2 α s ) , 1 γ 0 = 1 γ s * = 3 2 .
γ ( w , 0 ) + γ * ( w , 0 ) + m λ s N ( w , 0 ) = γ 0 + γ s * ,
N ( w , 0 ) = 1 m λ s γ 0 ( 1 { 1 2 γ s [ w sin α s + u ( w , 0 ) × ( cos α s 1 γ s ) ] } 1 / 2 ) + 1 m λ s γ s * ( 1 { 1 2 γ s * × [ w sin α s * + u ( w , 0 ) ( cos α s * 1 γ s * ) ] } 1 / 2 ) .
1 γ 0 + 1 γ s * u 02 ( cos α s + cos α s * ) + d 0 N 02 ( sin α s + sin α s * ) = 0 ,
u 02 = 3 2 + d 0 N 02 sin α s + sin α s * cos α s + cos α s * .
N 02 = cos α s + cos α s * 2 d 0 ( sin α s + sin α s * ) , u 02 = 1 ,
N 02 = 0 , u 02 = 3 2 .
u = R { R 2 ( w 2 + l 2 ) + 2 ( 1 R ) [ 1 ( 1 w 2 ) 1 / 2 ] } 1 / 2 ( R 1 / u 02 ) .
u = cot θ ( w w N ) + [ 1 ( 1 w N 2 ) 1 / 2 ] ,
N = 1 m λ s { m λ s d 0 w + 1 2 [ ( cos α s cos 2 α s γ 0 ) + ( cos α s * cos 2 α s * γ s * ) ] w 2 m λ s 2 d 0 u 02 tan θ l 2 + 1 2 [ 1 γ 0 sin α s ( cos α s cos 2 α s γ 0 ) + 1 γ s * sin α s * ( cos α s * cos 2 α s * γ s * ) ] w 3 1 2 u 02 [ m λ s d 0 tan θ + ( cos α s cos 2 α s γ 0 ) + ( cos α s * cos 2 α s * γ s * ) ] tan θ w l 2 } + O ( 4 ) .
N 02 = u 02 d 0 tan θ .
d 2 w d l 2 | at O = u 02 tan θ .
{ x * = X 02 l 2 + O ( 3 ) , y * = γ f * S f l + O ( 2 ) ,
X 02 = γ f * 2 cos α 0 * { m λ N 12 [ 1 γ 0 sin α ( u 02 cos α 1 γ 0 ) + 1 γ f * sin α 0 * ( u 02 cos α 0 * 1 γ f * ) ] + sin α 0 * S f ( S f 1 γ f * ) } .
1 ρ = 2 x * y * 2 | at O = 2 X 02 ( γ f * S f ) 2 .
m λ s N 12 = 1 γ 0 sin α s ( u 02 cos α s 1 γ 0 ) + 1 γ s * sin α s * ( u 02 cos α s * 1 γ s * ) sin α s * S f ( S f 1 γ s * ) ,
a N 02 2 b N 02 + c = 0 ,
a = d 0 2 ( sin α s + sin α s * ) 2 sin α s * 1 u 02 d 0 ( sin α s + sin α s * ) , b = 2 d 0 sin α s * ( sin α s + sin α s * ) ( u 02 ( cos α s + cos α s * ) 1 γ 0 ) ( cos α s cos 2 α s γ 0 ) ( cos α s * cos 2 α s * γ s * ) , c = sin α s * { u 02 ( cos α s + cos α s * ) 1 γ 0 } 2 u 02 × ( 1 γ 0 sin α s cos α s + 1 γ s * sin α s * cos α s * ) + 1 γ 0 2 sin α s .
a ( 1 / u 02 ) 2 b ( 1 / u 02 ) + c = 0 ,
{ a = 9 ( cos α s + cos α s * ) 2 4 ( sin α s + sin α s * ) , b = 3 ( cos α s + cos α s * ) 2 ( sin α s + sin α s * ) [ ( cos α s + cos 2 α s γ 0 ) + ( cos α s * + cos 2 α s * γ s * ) ] + 1 γ 0 sin α s + 1 γ s * sin α s * , c = cos α s + cos α s * sin α s + sin α s * ( cos 2 α s γ 0 + cos α s * γ s * ) + 1 γ 0 sin α s cos α s + 1 γ s * sin α s * cos α s * .
tan θ = cos α s + cos α s * u 02 ( sin α s + sin α s * ) ( 3 2 u 02 ) .