Abstract

The geometric theory of imaging for spectrometer systems is extended to include those with diffraction gratings produced holographically by the intersection of two sets of spherical wave fronts that have been rendered aspheric through reflection from concave mirrors. Aberration coefficients are derived that reduce to those of conventional holographic grating systems in the appropriate limit, and degrees of freedom for aberration correction are discussed.

© 1989 Optical Society of America

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References

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  1. M. C. Hutley, Diffraction Gratings (Academic, London, 1982), p. 10.
  2. J. M. Burch, “Photographic production of scales for moiré fringe applications,” in Optics in Metrology, P. Mollet, ed. (Pergamon, New York, 1960), pp. 361–368.
  3. D. Rudolph, G. Schmahl, “Verhehren zur Herstellung von Röntgenlinsen und Beugungsgittern,” Umschau Wiss. Tech. 67, 225 (1967); “Über ein Verfehren zur Herstellung von Beugungsgittern und Zonenplatten,” Mitt. Astron. Ges. 23, 46 (1967).
  4. A. Labeyrie, J. Flamand, “Spectroscopic performance of holographically made diffraction gratings,” Opt. Commun. 1, 5–8 (1969).
    [CrossRef]
  5. M. Chrisp, “Aberration-corrected holographic gratings and their mountings,” in Applied Optics and Optical Engineering (Academic, New York, 1987), Vol. 10, Chap. 7; T. Namioka, M. Seya, H. Noda, “Design and performance of holographic concave gratings,” Jpn. J. Appl. Phys. 15, 1181–1197 (1976).
    [CrossRef]
  6. 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 Moiré Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.815, 96–101 (1987).
    [CrossRef]
  7. See, for instance, T. Namioka, “Theory of the concave grating. I.,” J. Opt. Soc. Am. 49, 446–460 (1959).
    [CrossRef]
  8. H. Noda, T. Namioka, M. Seya, “Geometric theory of the grating,”J. Opt. Soc. Am. 64, 1031–1048 (1974).
    [CrossRef]
  9. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), p. 115.
  10. With regard to classical and first-generation holographic gratings, Namioka7 considered a more general case, in which the entrance and exit slits (and the recording source points) may be out of plane (that is, they may have nonzero zcoordinates). For second-generation holographic gratings, though, enough degrees of freedom should be found to permit significant aberration correction while leaving the optics in the principal plane.
  11. The analysis that follows is not restricted to spherical mirrors but may apply to any reflecting surfaces of revolution. At this point in the development, though, it is more instructive to consider a spherical mirror, which has one center of curvature. Generally the center of curvature for that slice of the mirror lying in the principal plane and the center of the mirror surface define the axis of symmetry of the surface of revolution; we require this particular line to be parallel to the grating normal.
  12. H. Noda, Ph.D. dissertation (Tokyo Kyoiku University, Tokyo, Japan, 1974); see also M. R. Howells, “Some geometrical considerations concerning grazing incidence reflectors,” (Brookhaven National Laboratory, Upton, N.Y., 1980).
  13. H. Beutler, “The theory of the concave grating,”J. Opt. Soc. Am. 35, 311–350 (1945).
    [CrossRef]
  14. T. Namioka, Tohoku University, 2-1-1 Katahira, Sendai 980, Japan (personal communication, 1986). Namioka also allowed the source and ideal image points A and B to lie outside the principal plane; setting z= z′ = 0 in his expressions produces agreement with the terms given in Ref. 15.
  15. W. R. McKinney, C. Palmer, “Numerical design method for aberration-reduced concave grating spectrometers,” Appl. Opt. 26, 3108–3118 (1987).
    [CrossRef] [PubMed]
  16. Strictly speaking, F10= F01= 0 only for the principal ray, but the power-series expansion of Ψ [Eq. (25)] provides aberration coefficients Fijthat are independent of the pupil coordinates yand z.
  17. F01= 0 identically, since we have imposed a plane of symmetry (the xyplane); for optical systems without a plane of symmetry, the F01term provides the familiar law of reflection in the plane perpendicular to the principal plane (viz., parallel to the grooves).
  18. M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1972), p. 15, Eq. (3.6.11).
  19. A further restriction, namely, lC= rC, provides Y= Z= −1/2; this can readily be seen to be valid when it is recalled that Yand Zare the derivatives ∂η/∂yand ∂ζ/∂z, respectively, and for a planar mirror parallel to the grating tangent plane and equidistant from the source and the grating center, a shift dηin the mirror coordinate will produce twice the shift in the grating coordinate. The negative sign is an artifact of our coordinate systems: ηincreases as ydecreases, and vice versa.
  20. R. P. Gillespie, Partial Differentiation (Oliver and Boyd, Edinburgh, 1951), pp. 57–61.
  21. D. J. Schroeder, Astronomical Optics (Academic, San Diego, California, 1987), pp. 79–81.
  22. This can be seen from Eq. (51), in which the groove spacing dfor a first-generation holographic grating is nonzero only if the recording angles γand δare distinct.

1987 (1)

1974 (1)

1969 (1)

A. Labeyrie, J. Flamand, “Spectroscopic performance of holographically made diffraction gratings,” Opt. Commun. 1, 5–8 (1969).
[CrossRef]

1967 (1)

D. Rudolph, G. Schmahl, “Verhehren zur Herstellung von Röntgenlinsen und Beugungsgittern,” Umschau Wiss. Tech. 67, 225 (1967); “Über ein Verfehren zur Herstellung von Beugungsgittern und Zonenplatten,” Mitt. Astron. Ges. 23, 46 (1967).

1959 (1)

1945 (1)

Beutler, H.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), p. 115.

Burch, J. M.

J. M. Burch, “Photographic production of scales for moiré fringe applications,” in Optics in Metrology, P. Mollet, ed. (Pergamon, New York, 1960), pp. 361–368.

Chrisp, M.

M. Chrisp, “Aberration-corrected holographic gratings and their mountings,” in Applied Optics and Optical Engineering (Academic, New York, 1987), Vol. 10, Chap. 7; T. Namioka, M. Seya, H. Noda, “Design and performance of holographic concave gratings,” Jpn. J. Appl. Phys. 15, 1181–1197 (1976).
[CrossRef]

Flamand, J.

A. Labeyrie, J. Flamand, “Spectroscopic performance of holographically made diffraction gratings,” Opt. Commun. 1, 5–8 (1969).
[CrossRef]

Gillespie, R. P.

R. P. Gillespie, Partial Differentiation (Oliver and Boyd, Edinburgh, 1951), pp. 57–61.

Harada, Y.

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 Moiré Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.815, 96–101 (1987).
[CrossRef]

Hutley, M. C.

M. C. Hutley, Diffraction Gratings (Academic, London, 1982), p. 10.

Koike, M.

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 Moiré Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.815, 96–101 (1987).
[CrossRef]

Labeyrie, A.

A. Labeyrie, J. Flamand, “Spectroscopic performance of holographically made diffraction gratings,” Opt. Commun. 1, 5–8 (1969).
[CrossRef]

McKinney, W. R.

Namioka, T.

H. Noda, T. Namioka, M. Seya, “Geometric theory of the grating,”J. Opt. Soc. Am. 64, 1031–1048 (1974).
[CrossRef]

See, for instance, T. Namioka, “Theory of the concave grating. I.,” J. Opt. Soc. Am. 49, 446–460 (1959).
[CrossRef]

T. Namioka, Tohoku University, 2-1-1 Katahira, Sendai 980, Japan (personal communication, 1986). Namioka also allowed the source and ideal image points A and B to lie outside the principal plane; setting z= z′ = 0 in his expressions produces agreement with the terms given in Ref. 15.

Noda, H.

H. Noda, T. Namioka, M. Seya, “Geometric theory of the grating,”J. Opt. Soc. Am. 64, 1031–1048 (1974).
[CrossRef]

H. Noda, Ph.D. dissertation (Tokyo Kyoiku University, Tokyo, Japan, 1974); see also M. R. Howells, “Some geometrical considerations concerning grazing incidence reflectors,” (Brookhaven National Laboratory, Upton, N.Y., 1980).

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 Moiré Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.815, 96–101 (1987).
[CrossRef]

Palmer, C.

Rudolph, D.

D. Rudolph, G. Schmahl, “Verhehren zur Herstellung von Röntgenlinsen und Beugungsgittern,” Umschau Wiss. Tech. 67, 225 (1967); “Über ein Verfehren zur Herstellung von Beugungsgittern und Zonenplatten,” Mitt. Astron. Ges. 23, 46 (1967).

Schmahl, G.

D. Rudolph, G. Schmahl, “Verhehren zur Herstellung von Röntgenlinsen und Beugungsgittern,” Umschau Wiss. Tech. 67, 225 (1967); “Über ein Verfehren zur Herstellung von Beugungsgittern und Zonenplatten,” Mitt. Astron. Ges. 23, 46 (1967).

Schroeder, D. J.

D. J. Schroeder, Astronomical Optics (Academic, San Diego, California, 1987), pp. 79–81.

Seya, M.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), p. 115.

Appl. Opt. (1)

J. Opt. Soc. Am. (3)

Opt. Commun. (1)

A. Labeyrie, J. Flamand, “Spectroscopic performance of holographically made diffraction gratings,” Opt. Commun. 1, 5–8 (1969).
[CrossRef]

Umschau Wiss. Tech. (1)

D. Rudolph, G. Schmahl, “Verhehren zur Herstellung von Röntgenlinsen und Beugungsgittern,” Umschau Wiss. Tech. 67, 225 (1967); “Über ein Verfehren zur Herstellung von Beugungsgittern und Zonenplatten,” Mitt. Astron. Ges. 23, 46 (1967).

Other (16)

T. Namioka, Tohoku University, 2-1-1 Katahira, Sendai 980, Japan (personal communication, 1986). Namioka also allowed the source and ideal image points A and B to lie outside the principal plane; setting z= z′ = 0 in his expressions produces agreement with the terms given in Ref. 15.

M. Chrisp, “Aberration-corrected holographic gratings and their mountings,” in Applied Optics and Optical Engineering (Academic, New York, 1987), Vol. 10, Chap. 7; 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 gratings 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).
[CrossRef]

M. C. Hutley, Diffraction Gratings (Academic, London, 1982), p. 10.

J. M. Burch, “Photographic production of scales for moiré fringe applications,” in Optics in Metrology, P. Mollet, ed. (Pergamon, New York, 1960), pp. 361–368.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), p. 115.

With regard to classical and first-generation holographic gratings, Namioka7 considered a more general case, in which the entrance and exit slits (and the recording source points) may be out of plane (that is, they may have nonzero zcoordinates). For second-generation holographic gratings, though, enough degrees of freedom should be found to permit significant aberration correction while leaving the optics in the principal plane.

The analysis that follows is not restricted to spherical mirrors but may apply to any reflecting surfaces of revolution. At this point in the development, though, it is more instructive to consider a spherical mirror, which has one center of curvature. Generally the center of curvature for that slice of the mirror lying in the principal plane and the center of the mirror surface define the axis of symmetry of the surface of revolution; we require this particular line to be parallel to the grating normal.

H. Noda, Ph.D. dissertation (Tokyo Kyoiku University, Tokyo, Japan, 1974); see also M. R. Howells, “Some geometrical considerations concerning grazing incidence reflectors,” (Brookhaven National Laboratory, Upton, N.Y., 1980).

Strictly speaking, F10= F01= 0 only for the principal ray, but the power-series expansion of Ψ [Eq. (25)] provides aberration coefficients Fijthat are independent of the pupil coordinates yand z.

F01= 0 identically, since we have imposed a plane of symmetry (the xyplane); for optical systems without a plane of symmetry, the F01term provides the familiar law of reflection in the plane perpendicular to the principal plane (viz., parallel to the grooves).

M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1972), p. 15, Eq. (3.6.11).

A further restriction, namely, lC= rC, provides Y= Z= −1/2; this can readily be seen to be valid when it is recalled that Yand Zare the derivatives ∂η/∂yand ∂ζ/∂z, respectively, and for a planar mirror parallel to the grating tangent plane and equidistant from the source and the grating center, a shift dηin the mirror coordinate will produce twice the shift in the grating coordinate. The negative sign is an artifact of our coordinate systems: ηincreases as ydecreases, and vice versa.

R. P. Gillespie, Partial Differentiation (Oliver and Boyd, Edinburgh, 1951), pp. 57–61.

D. J. Schroeder, Astronomical Optics (Academic, San Diego, California, 1987), pp. 79–81.

This can be seen from Eq. (51), in which the groove spacing dfor a first-generation holographic grating is nonzero only if the recording angles γand δare distinct.

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Figures (9)

Fig. 1
Fig. 1

Recording geometry for a classical equivalent grating. A laser beam, after being expanded and collimated, is split by a beam splitter (BS) into two coherent beams; on reflection from a mirror (M) these beams may be made to cross, and a set of standing plane waves is formed in the region where these beams intersect. Placement of a concave blank (CB) (covered in photoresist) in this region forms the equivalent of a classically ruled grating.

Fig. 2
Fig. 2

Recording geometry for a first-generation holographic grating. A laser beam is split by a beam splitter (BS) into two coherent beams; each is focused by a microscope objective (MO) and spatially filtered by a pinhole (PH). These beams may be made to cross, and a set of confocal hyperbolic standing waves is formed in the region where these beams intersect. Placement of a concave blank (CB) (covered in photoresist) in this region forms a first-generation holographic grating.

Fig. 3
Fig. 3

Recording geometry for a second-generation holographic grating. A laser beam is split by a beam splitter (BS) into two coherent beams; each is focused by a microscope objective (MO) and spatially filtered by a pinhole (PH). In contrast to the recording geometry of first-generation holographics, the diverging spherical waves that result are reflected by concave mirrors (CM’s) before they are made to interfere on the surface of a concave blank (CB) (covered in photoresist), forming a second-generation holographic grating.

Fig. 4
Fig. 4

Simple spectrometer use geometry. Polychromatic light from point source A(r, α, 0) is diffracted by a concave grating centered at O(0, 0, 0); light of wavelength λ is diffracted to ideal point image B(r′, β, 0). Point P is an arbitrary point on the surface of the grating. The Cartesian coordinate origin is defined such that the x axis is oriented along the normal to the grating surface at its center and such that the grooves are oriented along the z axis at the center of the grating. The algebraic signs of angles α and β are significant; α < 0 and β > 0.

Fig. 5
Fig. 5

Recording geometry parameters for first-generation holographics. Point sources are located at C(rC, γ, 0) and D(rD, δ, 0). Point P is an arbitrary point on the surface of the grating, lying upon the Nth groove.

Fig. 6
Fig. 6

Recording geometry for second-generation holographics. Point sources C and D direct light toward concave mirrors MC and MD (centered at OC and OD), respectively which reflect the beams toward the concave blank centered at O. The spherical wave fronts leaving the point sources are modified by oblique reflection from the spherical mirrors.

Fig. 7
Fig. 7

Second-generation recording geometry parameters. Point sources C and D are located distances lC and lD and at angles −γ and −δ from spherical mirrors MC and MD (centered at OC and OD), respectively. The mirror have radii RC and RD, and their centers are located distances rC and rD and at angles γ and δ, respectively, from the grating center O. The mirror centers support their own coordinate systems; only that for mirror MC is shown. The lines between the center of each mirror surface and their respective centers of curvature are parallel to the x axis and lie in the xy plane.

Fig. 8
Fig. 8

Principal-plane mirror diagram. Wave fronts leaving point source C are spherical, but after reflection from mirror MC they are (to lowest order) toroidal (unless γ = 0 or RC → ∞). Projecting the reflected rays backward through the mirror shows that they converge at point C′, the virtual source when curvature in the principal plane is considered. This virtual source is located a distance −aC from the mirror center OC; this distance may be found from Coddington’s equations (see the text).

Fig. 9
Fig. 9

Sagittal-plane mirror diagram. By reasoning similar to that used for Fig. 8 but in the vertical plane, MC modifies the curvature of the wave fronts leaving C. Projecting the reflected rays backward through the mirror shows that they converge at point C″, the virtual source when curvature in the sagittal plane is considered. This virtual source is located a distance −bC from the mirror center OC; this distance may be found from Coddington’s equations (see the text).

Tables (2)

Tables Icon

Table 1 Sag Coefficients for Spherical and Toroidal Surfaces (to order 4, inclusive)a

Tables Icon

Table 2 Holographic Aberration Coefficients Hij for Second-Generation Holographic Gratingsa (C side only)

Equations (87)

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A O B = A P B + N m λ
Ψ ( λ ) = Φ ACTUAL - Φ IDEAL = A P B - A O B + N m λ .
Φ = P 1 P 2 = P 1 P 2 n d s ,
A O B = A O ¯ + O B ¯ .
A P = A P ¯ = [ ( x A - x ) 2 + ( y A - y ) 2 + ( z A - z ) 2 ] 1 / 2 .
Ψ 0 = A P B - A O B + N m λ = [ ( x A - x ) 2 + ( y A - y ) 2 + ( z A - z ) 2 ] 1 / 2 + [ ( x B - x ) 2 + ( y B - y ) 2 + ( z B - z ) 2 ] 1 / 2 - ( x A 2 + y A 2 + z A 2 ) 1 / 2 = ( x B 2 + y B 2 + z B 2 ) 1 / 2 + N m λ .
x A = r cos α , y A = r sin α , x B = r cos β , y B = r sin β .
Ψ 0 = [ ( r cos α - x ) 2 + ( r sin α - y ) 2 + z 2 ] 1 / 2 + [ ( r cos β - x ) 2 + ( r sin β - y ) 2 + z 2 ] 1 / 2 - r - r + N m λ .
x C = r C cos γ , y C = r C sin γ , x D = r D cos δ , y D = r D sin δ .
N λ 0 = ( C P - D P ) - ( C O - D O ) .
Ψ 1 = A P B - A O B + N m λ = ( A P ¯ + P B ¯ ) - ( A O ¯ + O B ¯ ) + m λ λ 0 [ ( C P ¯ - D P ¯ ) - ( C O ¯ - D O ¯ ) ] ,
= [ ( r cos α - x ) 2 + ( r sin α - y ) 2 + z 2 ] 1 / 2 - r + [ ( r cos β - x ) 2 + ( r sin β - y ) 2 + z 2 ] 1 / 2 - r + m λ λ 0 [ ( r C cos γ - x ) 2 + ( r C sin γ - y ) 2 + z 2 ] 1 / 2 - m λ λ 0 [ ( r D cos δ - x ) 2 + ( r D sin δ - y ) 2 + z 2 ] 1 / 2 - m λ λ 0 ( r C - r D ) .
N λ 0 = [ ( C P C + P C P ) - ( D P D + P D P ) ] - [ ( C O C + O C O ) - ( D O D + O D O ) ] .
ξ = ξ ( x , y , z ) ,             η = η ( x , y , z ) ,             ζ = ζ ( x , y , z ) .
C O C l C , D O D l D , O D C r C , O D D r D .
C P C ¯ = [ ( - l C cos γ + ξ ) 2 + ( l C sin γ + η ) 2 + ζ 2 ] 1 / 2 ,
D P D ¯ = [ ( - l D cos δ + ξ ) 2 + ( l D sin δ + η ) 2 + ζ 2 ] 1 / 2 ,
P C P ¯ = [ ( x - r C cos γ + ξ ) 2 + ( y - r C sin γ + η ) 2 + ( z + ζ ) 2 ] 1 / 2 ,
P D P ¯ = [ ( x - r D cos δ + ξ ) 2 + ( y - r D sin δ + η ) 2 + ( z + ζ ) 2 ] 1 / 2 .
N λ 0 = [ ( - l C cos γ + ζ ) 2 + ( l C sin γ + η ) 2 + ζ 2 ] 1 / 2 + [ ( x - r C os γ + ξ ) 2 + ( y - r C sin γ + η ) 2 + ( z + ζ ) 2 ] 1 / 2 - [ ( - l D cos δ + ξ ) 2 + ( l D sin δ + η ) 2 + ζ 2 ] 1 / 2 - [ ( x - r D cos δ + ξ ) 2 + ( y - r D sin δ + η ) 2 + ( z + ζ 2 ] 1 / 2 - [ ( l C + r C ) - ( l D + r D ) ] .
Ψ 2 = [ ( r cos α - x ) 2 + ( r sin α - y ) 2 + z 2 ] 1 / 2 - r + [ ( r cos β - x ) 2 + ( r sin β - y ) 2 + z 2 ] 1 / 2 - r + m λ λ 0 [ ( - l C cos γ + ξ ) 2 + ( l C sin γ + η ) 2 + ζ 2 ] 1 / 2 + m λ λ 0 [ ( x - r C cos γ + ξ ) 2 + ( y - r C sin γ + η ) 2 + ( z + ζ ) 2 ] 1 / 2 - m λ λ 0 [ ( - l D cos δ + ξ ) 2 + ( l D sin δ + η ) 2 + ζ 2 ] 1 / 2 - m λ λ 0 [ ( x - r D cos δ + ξ ) 2 + ( y - r D sin δ + η ) 2 + ( z + ζ ) 2 ] 1 / 2 - m λ λ 0 [ ( l C + r C ) - ( l D + r D ) ] .
Ψ 2 = Ψ 2 ( x , y , z ; ξ , η , ζ ; ξ , η , ζ ) .
x ( y , z ) = i = 0 j = 0 a i j y i z j .
x ( y , z ) = a 20 Y 2 + a 02 z 2 + a 40 y 4 + a 22 y 2 z 2 + a 04 z 4 + O ( 6 ) ,
Ψ ( y , z ) = i = 0 j = 0 F i j y i z j .
F i j = M i j + m λ λ 0 H i j ,
Ψ y = 0 ,             Ψ z = 0.
A P 2 = ( x - r cos α ) 2 + ( y - r sin α ) 2 + z 2 .
x = a 20 y 2 + a 02 z 2 + a 40 y 4 + a 22 y 2 z 2 + a 04 z 4 + O ( 6 ) ;
x 2 = a 20 2 y 4 + 2 a 20 a 02 y 2 z 2 + a 02 2 z 4 + O ( 6 ) .
A P 2 = a 20 2 y 4 + 2 a 20 a 02 y 2 z 2 + a 02 2 z 4 + y 2 + z 2 + r 2 - 2 r y sin α + 2 r ( a 20 y 2 + a 02 z 2 + a 40 y 4 + a 22 y 2 z 2 + a 04 z 4 ) cos α + O ( 6 ) .
A P 2 = r 2 [ 1 - 2 sin α r y + 1 r ( 1 r - 2 a 20 cos α ) y 2 + 1 r ( 1 r - 2 a 02 cos α ) z 2 + 1 r ( a 20 2 r - 2 a 40 cos α ) y 4 + 2 r ( a 20 a 02 r - a 22 cos α ) y 2 z 2 + 1 r ( a 02 2 r - 2 a 04 cos α ) z 4 ] + O ( 6 ) .
A P 2 = r 2 ( 1 + H ) ,
A P = r ( 1 + H ) 1 / 2 = r ( 1 + 1 2 H - 1 8 H 2 + 1 16 H 3 - 5 128 H 4 + 7 256 H 5 - ) .
H 10 = sin δ - sin γ
a C = ( 2 sec γ R C - 1 l C ) - 1 - l C ,
b C = ( 2 cos γ R c - 1 l C ) - 1 - l C ,
Y - l C r C + l C ,             Z - l C r C + l C ,
1 + Y r C r C + l C ,             1 + Z r C r C + l C ;
H 10 = - sin γ ,
H 01 = 0 ,
H 20 = 1 2 ( cos 2 γ r C + l C - 2 a 20 cos γ ) ,
H 11 = 0 ,
H 02 = 1 2 ( 1 r C + l C - 2 a 02 cos γ ) ,
H 30 = 1 2 sin γ cos γ r C + l C ( cos γ r C + l C - 2 a 20 ) ,
H 21 = 0 ,
H 12 = 1 2 sin γ r C + l C ( 1 r C + l C - 2 a 02 cos γ ) ,
H 03 = 0.
F 10 = M 10 + m λ λ 0 H 10 = - ( sin α + sin β ) + m λ λ 0 ( sin δ - sin γ ) = 0 ;
m λ = d ( sin α + sin β )
F 10 = - m λ d + m λ λ 0 ( sin δ - sin γ ) = 0 ,
d = λ 0 sin δ - sin γ .
F i j ( m , λ , r , α , r , β ; r C , l C , R C , γ , r D , l D , R D , δ ) = M i j ( m , λ , r , α , r , β ) + m λ λ 0 H i j ( r C , l C , R C , γ , r D , l D , R D , δ ) ,
F i j ( m , λ , r , α , r , β ; H i j ) = M i j ( m , λ , r , α , r , β ) + m λ λ 0 H i j .
F i j ( r , α , r ; H i j ) = M i j ( r , α , r ) + m λ λ 0 H i j .
a C = ( 2 sec γ R c - 1 l C ) - 1 ,
b C = ( 2 cos γ ρ C - 1 l C ) - 1 ,
F 01 = M 01 + m λ λ 0 H 01 = 0 ,
M 01 = - z r [ 1 + ( z r ) 2 ] - 1 / 2 - z r [ 1 + ( z r ) 2 ] - 1 / 2 .
f ( y , z ) = f ( 0 , 0 ) + [ y ( d d y ) ( 0 , 0 ) + z ( d d z ) ( 0 , 0 ) ] f ( y , z ) + 1 2 ! [ y ( d d y ) ( 0 , 0 ) + z ( d d z ) ( 0 , 0 ) ] 2 f ( y , z ) + + 1 n ! [ y ( d d y ) ( 0 , 0 ) + z ( d d z ) ( 0 , 0 ) ] n f ( y , z ) + .
d f d y = k = 1 K f u k u k y + f y .
Φ ( y , z ) = [ ( - l C cos γ + ξ ) 2 + ( l C sin γ + η ) 2 + ζ 2 ] 1 / 2 [ F ( y , z ) ] 1 / 2 .
d Φ d y = 1 2 F d F d y = 1 F [ ( ξ - l C cos γ ) ξ y + ( η + l C sin γ ) η y + ζ ζ y ] ,
d Φ d z = 1 2 F d F d z = 1 F [ ( ξ - l C cos γ ) ξ z + ( η + l C sin γ ) η z + ζ ζ z ] .
ξ ( η , ζ ) = c 20 η 2 + c 02 ζ 2 + c 40 η 4 + c 22 η 2 ζ 2 + c 04 ζ 4 + O ( 6 ) .
ξ y = 2 c 20 η η y + 2 c 20 ζ ζ y + 4 c 40 η 3 η y + 2 c 22 [ η ζ 2 η y + ζ η 2 ζ y ] + 4 c 04 ζ 3 ζ y + ,
d u = d u ( v / v ) .
u = ( r C - a C ) sin γ + y ,             u = - a C sin γ - η ;
v = ( r C - a C ) cos γ - x ,             v = - a C cos γ + ξ ;
- d η = d y - a C cos γ + ξ ( r C - a C ) cos γ - x ,
η y = a C r C - a C Y .
ζ z = b C r C - b C Z ,
ξ y = ξ z = 2 ξ y z = 0 ;
2 ξ y 2 = 2 Y 2 c 20 ,             2 ξ z 2 = 2 Z 2 c 02 ;
3 ζ y 3 = 3 ξ y 2 z = 3 ξ y d z 2 = 3 ξ z 3 = 0 ;
4 ξ y 3 z = 4 ξ y d z 3 = 0 ,             4 ξ y 2 z 2 = 4 Y 2 Z 2 c 22 ;
4 ξ y 4 = 24 Y 4 c 40 ,             4 ξ z 4 = 24 Z 4 c 04 .
d Φ d y = 1 F ( η + l C sin γ ) Y ,
d Φ d z = 1 F ζ Z ;
Φ ( 1 ) = y ( d Φ d y ) ( 0 , 0 ) + z ( d Φ d z ) ( 0 , 0 ) = y l C ( l C sin γ ) Y + 0 = y Y sin γ .
Φ ( 2 ) = 1 2 y 2 ( d 2 Φ d y 2 ) ( 0 , 0 ) + y z ( d 2 Φ d y d z ) ( 0 , 0 ) + 1 2 z 2 ( d 2 Φ d z 2 ) ( 0 , 0 ) = 1 2 y 2 Y 2 l C cos γ ( cos γ - 2 c 20 l C ) + 1 2 z 2 Z 2 l C ( 1 - 2 c 02 l C cos γ ) ,
Φ ( 3 ) = 1 6 y 3 ( d 3 Φ d y 3 ) ( 0 , 0 ) + 1 2 y 2 z ( d 3 Φ d y 2 d z ) ( 0 , 0 ) + 1 2 y z 2 ( d 3 Φ d y d z 2 ) ( 0 , 0 ) + 1 6 z 3 ( d 3 Φ d z 3 ) ( 0 , 0 ) = 1 2 y 3 Y 3 l C 2 sin γ cos γ ( cos γ - 2 c 20 l C ) + 1 2 y z 2 Y Z 2 l C 2 sin γ ( 1 - 2 c 02 l C cos γ ) .
Φ = Φ ( 0 ) + Φ ( 1 ) + Φ ( 2 ) + Φ ( 3 ) + = r C - y ( 1 + Y ) sin γ + 1 2 y 2 cos γ r C [ ( 1 + Y ) 2 cos γ - 2 c 20 r C Y 2 - 2 a 20 r C ] + 1 2 z 2 1 r C [ ( 1 + Z ) 2 - 2 c 02 r C cos γ Z 2 - 2 a 02 r C cos γ ] + 1 2 y 3 ( 1 + Y ) r C 2 sin γ cos γ [ ( 1 + Y ) 2 cos γ - 2 c 20 r C Y 2 - 2 a 20 r C ] + 1 2 y z 2 ( 1 + Y ) r C 2 sin γ [ ( 1 + Z ) 2 - 2 c 02 r C cos γ Z 2 - 2 a 02 r C cos γ ] + O ( 4 ) .
( cos γ ) ( 1 a C + 1 l C ) = 2 R C ,
( sec γ ) ( 1 b C + 1 l C ) = 2 R C .
a C = ( 2 sec γ R C - 1 l C ) - 1 ,
b C = ( 2 cos γ R C - 1 l C ) - 1 .

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