Abstract

Three pupil-coordinate definitions are considered for the description of aberrations. The corresponding sets of expressions for the third- and fifth-order image-aberration coefficients as well as the third-order pupil-aberration coefficients are presented. We show their interrelations and the connection with eikonal coefficients. One pupil-coordinate definition leads to the occurrence of 9 independent fifth-order image-aberration coefficients; the other two definitions lead to 12 independent fifth-order image-aberration coefficients. We find that the third-order image-aberration coefficients are independent of the choice of pupil-coordinate definition, whereas the third-order pupil-aberration coefficients and the fifth-order image-aberration coefficients do depend on this choice. We show that the case of ideal imaging may imply finite pupil aberrations.

© 1989 Optical Society of America

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References

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  1. R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1964), pp. 121–127.
  2. C. H. F. Velzel, J. L. F. de Meijere, “Characteristic functions and the aberrations of symmetric optical systems. III. Calculation of eikonal coefficients,” J. Opt. Soc. Am. A 5, 1237–1243 (1988).
    [Crossref]
  3. C. H. F. Velzel, J. L. F. de Meijere, “Characteristic functions and the aberrations of symmetric optical systems. II. Addition of aberrations,” J. Opt. Soc. Am. A 5, 251–256 (1988).
    [Crossref]
  4. Reduced distances and reduced coordinates are distances and coordinates multiplied by the refractive index of the corresponding space. Reduced magnifications are ratios of reduced image and object coordinates.
  5. M. Herzberger, Modern Geometrical Optics (Interscience, New York, 1958), pp. 373–379.
  6. F. Wachendorf, “Bestimmung der Bildfehler 5. Ordnung in zentrierten optischen Systemen,” Optik 5, 80–122 (1949).
  7. H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968).
  8. K. Schwarzschild, “Untersuchungen zur geometrischen Optik I,” Abh. Koenigl. Ges. Wiss. Goettingen Math. Phys. Kl. Neue Folge 4, 1–9 (1905).
  9. T. Smith, “The addition of aberrations,” Trans. Opt. Soc. 25, 177–199 (1924).
    [Crossref]
  10. C. H. F. Velzel, J. L. F. de Meijere, “Characteristic functions and the aberrations of symmetric optical systems. I. Transverse aberrations when the eikonal is given,” J. Opt. Soc. Am. A 5, 246–250 (1988).
    [Crossref]
  11. J. L. P. de Meijere, C. H. F. Velzel, “Linear ray propagation models in geometrical optics,” J. Opt. Soc. Am. A 4, 2162–2165 (1987).
    [Crossref]
  12. C. Caratheodory, “Die Fehler höherer Ordnung der optischen Instrumente,” Sitzungsber. Math. Naturwiss. Kl. Bayr. Akad. Wiss. Muenchen 1943, 199–216 (1943).
  13. See Ref. 7, p. 348 .
  14. D. Shafer, “Power over wide-angle lenses,” in Digest of Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1987), p. 55.
  15. This result is to be compared with Eq. (10) of Ref. 10, in which a vector λ was defined such that λ= (z/t)πq, with z= 1/(GJ), and in which λ was interpreted as the deviation from linearity in the front focal plane. For the concatenation of the eikonals of two components, the deviation from linearity is an important parameter for obtaining an estimate of the relative magnitude of the induced aberrations with respect to the pseudoaberrations.3
  16. H. Poincaré, “Sur les residus des integrales doubles,” Acta Math. 9, 321–380 (1886).
    [Crossref]

1988 (3)

1987 (1)

1949 (1)

F. Wachendorf, “Bestimmung der Bildfehler 5. Ordnung in zentrierten optischen Systemen,” Optik 5, 80–122 (1949).

1943 (1)

C. Caratheodory, “Die Fehler höherer Ordnung der optischen Instrumente,” Sitzungsber. Math. Naturwiss. Kl. Bayr. Akad. Wiss. Muenchen 1943, 199–216 (1943).

1924 (1)

T. Smith, “The addition of aberrations,” Trans. Opt. Soc. 25, 177–199 (1924).
[Crossref]

1905 (1)

K. Schwarzschild, “Untersuchungen zur geometrischen Optik I,” Abh. Koenigl. Ges. Wiss. Goettingen Math. Phys. Kl. Neue Folge 4, 1–9 (1905).

1886 (1)

H. Poincaré, “Sur les residus des integrales doubles,” Acta Math. 9, 321–380 (1886).
[Crossref]

Buchdahl, H. A.

H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968).

Caratheodory, C.

C. Caratheodory, “Die Fehler höherer Ordnung der optischen Instrumente,” Sitzungsber. Math. Naturwiss. Kl. Bayr. Akad. Wiss. Muenchen 1943, 199–216 (1943).

de Meijere, J. L. F.

de Meijere, J. L. P.

Herzberger, M.

M. Herzberger, Modern Geometrical Optics (Interscience, New York, 1958), pp. 373–379.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1964), pp. 121–127.

Poincaré, H.

H. Poincaré, “Sur les residus des integrales doubles,” Acta Math. 9, 321–380 (1886).
[Crossref]

Schwarzschild, K.

K. Schwarzschild, “Untersuchungen zur geometrischen Optik I,” Abh. Koenigl. Ges. Wiss. Goettingen Math. Phys. Kl. Neue Folge 4, 1–9 (1905).

Shafer, D.

D. Shafer, “Power over wide-angle lenses,” in Digest of Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1987), p. 55.

Smith, T.

T. Smith, “The addition of aberrations,” Trans. Opt. Soc. 25, 177–199 (1924).
[Crossref]

Velzel, C. H. F.

Wachendorf, F.

F. Wachendorf, “Bestimmung der Bildfehler 5. Ordnung in zentrierten optischen Systemen,” Optik 5, 80–122 (1949).

Abh. Koenigl. Ges. Wiss. Goettingen Math. Phys. Kl. Neue Folge (1)

K. Schwarzschild, “Untersuchungen zur geometrischen Optik I,” Abh. Koenigl. Ges. Wiss. Goettingen Math. Phys. Kl. Neue Folge 4, 1–9 (1905).

Acta Math. (1)

H. Poincaré, “Sur les residus des integrales doubles,” Acta Math. 9, 321–380 (1886).
[Crossref]

J. Opt. Soc. Am. A (4)

Optik (1)

F. Wachendorf, “Bestimmung der Bildfehler 5. Ordnung in zentrierten optischen Systemen,” Optik 5, 80–122 (1949).

Sitzungsber. Math. Naturwiss. Kl. Bayr. Akad. Wiss. Muenchen (1)

C. Caratheodory, “Die Fehler höherer Ordnung der optischen Instrumente,” Sitzungsber. Math. Naturwiss. Kl. Bayr. Akad. Wiss. Muenchen 1943, 199–216 (1943).

Trans. Opt. Soc. (1)

T. Smith, “The addition of aberrations,” Trans. Opt. Soc. 25, 177–199 (1924).
[Crossref]

Other (7)

H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968).

Reduced distances and reduced coordinates are distances and coordinates multiplied by the refractive index of the corresponding space. Reduced magnifications are ratios of reduced image and object coordinates.

M. Herzberger, Modern Geometrical Optics (Interscience, New York, 1958), pp. 373–379.

See Ref. 7, p. 348 .

D. Shafer, “Power over wide-angle lenses,” in Digest of Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1987), p. 55.

This result is to be compared with Eq. (10) of Ref. 10, in which a vector λ was defined such that λ= (z/t)πq, with z= 1/(GJ), and in which λ was interpreted as the deviation from linearity in the front focal plane. For the concatenation of the eikonals of two components, the deviation from linearity is an important parameter for obtaining an estimate of the relative magnitude of the induced aberrations with respect to the pseudoaberrations.3

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1964), pp. 121–127.

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Figures (1)

Fig. 1
Fig. 1

A light ray emanates from the object point H and intersects the entrance-pupil plane in R, the exit-pupil plane in R′, and the image plane in H′. The Gaussian conjugate of H is H ¯ . Further, t and t′ are the distances from the object plane to the entrance-pupil plane and from the image plane to the exit-pupil plane. As shown here, t is positive, and t′ is negative. If the indices of refraction in the object and image spaces are not equal to 1, then the symbols shown represent reduced coordinates and distances.4 The Schwarzschild pupil coordinates P and P′ are found by projecting segments of lengths t and t′, respectively, along the ray on the corresponding pupil planes. Q is found similarly; however, the projection involves a segment along a line through H ¯ parallel to the ray.

Tables (1)

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Table 1 Nomenclature for Aberrations and Aberration Coefficientsa

Equations (83)

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t = ( G - S ) / ( G S J ) , t = G S t .
R = H + t V ,             R = H + t V .
P H + t L ,             P H + t L .
Q G H + t L .
h = - G H t , r = - S R t , p = - S P t , q = - Q t .
δ = ( δ x , δ y ) H - G H ,
δ ( h , q ) n = 1 2 j , k n [ σ q ; 0 j k ( n ) q + σ q ; 1 j k ( n ) h ] U q j V q k W q n - j - k + O ( 7 ) ,
j , k n j = 0 n k = 0 n - j
U q = h · h ,             V q = h · q ,             W q = q · q .
π r = R - S R ,
π q P - 1 S Q
π p P - S P .
π r ( h , r ) j , k 1 [ ψ r ; 0 j k ( 1 ) r + ψ r ; 1 j k ( 1 ) h ] U r j V r k W r 1 - j - k + O ( 5 ) ,
π q ( h , q ) 1 S j , k 1 [ ψ q ; 0 j k ( 1 ) q + ψ q ; 1 j k ( 1 ) h ] U q j V q k W q 1 - j - k + O ( 5 ) .
U = ξ 1 · ξ 1 ,             V = ξ 1 · ξ 2 ,             W = ξ 2 · ξ 2 ,
ξ 1 = ( ξ 1 , η 1 ) = L - G L S - G ,             ξ 2 = ( ξ 2 , η 2 ) = L - S L S - G .
E = ( S - G ) 2 2 J G U + ,
= n = 2 ( n ) ,
( n ) = j , k n D j , k , n U j V k W n - j - k .
δ = ( ξ 2 , η 2 )
Ω E - H · L .
H = - ( Ω L , Ω M ) .
δ ( H , L ) = - ( ω L , ω M ) .
Q ( H , L ) = Ω [ H , L ( H , L ) ] .
H x L L + H y M L = - Q L ,             H x L M + H y M M = - Q M .
L ¯ = ( 1 / G ) L - J H .
σ q ; 000 ( 1 ) = 4 D 002 ( spherical aberration ) , σ q ; 100 ( 1 ) = D 012 σ q ; 001 ( 1 ) = 2 D 012 } ( coma ) , σ q ; 101 ( 1 ) = 2 D 022 σ q ; 010 ( 1 ) = 2 D 102 } ( astigmatism and field curvature ) , σ q ; 110 ( 1 ) = D 112 ( distortion ) .
δ [ s 1 W q + 2 s 2 V q + ( s 3 + s 4 ) U q ] q + ( s 2 W q + 2 s 3 V q + s 5 U q ) h + O ( 5 ) ,
s 1 = 4 D 002 ( spherical aberration ) , s 2 = D 012 ( coma ) , s 3 = D 022 ( astigmatism ) , s 4 = 2 D 102 - D 022 ( Petzval curvature ) , s 5 = D 112 ( distortion ) .
ψ q ; 000 ( 1 ) = 4 D 002 + D 012 ( pupil distortion ) , ψ q ; 100 ( 1 ) = 2 D 102 + D 012 ψ q ; 001 ( 1 ) = 2 ( D 012 + D 022 ) } ( pupil astigmatism and field curvature ) , ψ q ; 101 ( 1 ) = 2 ( D 112 + D 022 ) ψ q ; 010 ( 1 ) = 2 D 102 + D 112 } ( pupil coma ) , ψ q ; 110 ( 1 ) = 4 D 202 + D 112 ( spherical pupil aberration ) .
C j k 3 = D j k 3 - S 2 J t t ( π q 2 ) j k 3 ,
( π q 2 ) 003 = ψ q ; 000 ( 1 ) ψ q ; 000 ( 1 ) , ( π q 2 ) 013 = 2 [ ψ q ; 100 ( 1 ) ψ q ; 000 ( 1 ) + ψ q ; 001 ( 1 ) ψ q ; 000 ( 1 ) ] , ( π q 2 ) 023 = 2 [ ψ q ; 100 ( 1 ) ψ q ; 001 ( 1 ) + ψ q ; 101 ( 1 ) ψ q ; 000 ( 1 ) ] + ψ q ; 001 ( 1 ) ψ q ; 001 ( 1 ) , ( π q 2 ) 103 = 2 ψ q ; 010 ( 1 ) ψ q ; 000 ( 1 ) + ψ q ; 100 ( 1 ) ψ q ; 100 ( 1 ) , ( π q 2 ) 113 = 2 [ ψ q ; 100 ( 1 ) ψ q ; 101 ( 1 ) + ψ q ; 100 ( 1 ) ψ q ; 010 ( 1 ) + ψ q ; 110 ( 1 ) ψ q ; 000 ( 1 ) + ψ q ; 010 ( 1 ) ψ q ; 001 ( 1 ) ] , ( π q 2 ) 033 = ψ q ; 101 ( 1 ) ψ q ; 001 ( 1 ) , ( π q 2 ) 123 = 2 [ ψ q ; 110 ( 1 ) ψ q ; 001 ( 1 ) + ψ q ; 101 ( 1 ) ψ q ; 010 ( 1 ) ] + ψ q ; 101 ( 1 ) ψ q ; 101 ( 1 ) , ( π q 2 ) 203 = 2 ψ q ; 100 ( 1 ) ψ q ; 110 ( 1 ) + ψ q ; 010 ( 1 ) ψ q ; 010 ( 1 ) , ( π q 2 ) 213 = 2 [ ψ q ; 110 ( 1 ) ψ q ; 101 ( 1 ) + ψ q ; 110 ( 1 ) ψ q ; 010 ( 1 ) ] .
σ q ; 000 ( 2 ) = 6 C 003 ( spherical aberration ) , σ q ; 100 ( 2 ) = C 013 σ q ; 001 ( 2 ) = 4 C 013 } ( circular coma ) , σ q ; 101 ( 2 ) = 2 C 023 σ q ; 010 ( 2 ) = 4 C 103 σ q ; 002 ( 2 ) = 2 C 023 } ( oblique spherical aberration ) , σ q ; 110 ( 2 ) = C 113 σ q ; 102 ( 2 ) = 3 C 033 σ q ; 011 ( 2 ) = 2 C 033 } ( elliptical coma ) , σ q ; 111 ( 2 ) = 2 C 123 σ q ; 020 ( 2 ) = 2 C 203 } ( astigmatism and field curvature ) , σ q ; 120 ( 2 ) = C 213 ( distortion ) .
4 σ q ; 100 ( 2 ) - σ q ; 001 ( 2 ) = 0 , σ q ; 101 ( 2 ) - σ q ; 002 ( 2 ) = 0 , 2 σ q ; 110 ( 2 ) - σ q ; 011 ( 2 ) = 0.
r = q + O ( 3 ) , p = q + O ( 3 ) ,
σ r ; i j k ( 1 ) = σ p ; i j k ( 1 ) = σ q ; i j k ( 1 ) .
π r = δ - S π p + t ( V - L ) - S t ( V - L )
π p = δ - S π q .
ψ p ; i j k ( 1 ) = σ q ; i j k ( 1 ) - ψ q ; i j k ( 1 ) .
ψ r ; 000 ( 1 ) = σ q ; 000 ( 1 ) - ψ q ; 000 ( 1 ) + ½ t ( G 2 - 1 ) , ψ r ; 100 ( 1 ) = σ q ; 100 ( 1 ) - ψ q ; 100 ( 1 ) - ½ t ( G S - 1 ) , ψ r ; 001 ( 1 ) = σ q ; 001 ( 1 ) - ψ q ; 001 ( 1 ) - t ( G S - 1 ) , ψ r ; 101 ( 1 ) = σ q ; 101 ( 1 ) - ψ q ; 101 ( 1 ) + t ( S 2 - 1 ) , ψ r ; 010 ( 1 ) = σ q ; 010 ( 1 ) - ψ q ; 010 ( 1 ) + ½ t ( S 2 - 1 ) , ψ r ; 110 ( 1 ) = σ q ; 110 ( 1 ) - ψ q ; 110 ( 1 ) - ½ t [ ( S 3 / G ) - 1 ] .
σ p ; 000 ( 1 ) = σ r ; 000 ( 1 ) = 4 D 002 , σ p ; 100 ( 1 ) = σ r ; 100 ( 1 ) = - ψ p ; 000 ( 1 ) = - ψ r ; 000 ( 1 ) - ½ t ( G 2 - 1 ) = D 012 , σ p ; 001 ( 1 ) = σ r ; 001 ( 1 ) = 2 D 012 , σ p ; 101 ( 1 ) = σ r ; 101 ( 1 ) = - ψ p ; 001 ( 1 ) = - ψ r ; 001 ( 1 ) + t ( G S - 1 ) = 2 D 022 , σ p ; 010 ( 1 ) = σ r ; 010 ( 1 ) = - ψ p ; 100 ( 1 ) = - ψ r ; 100 ( 1 ) + ½ t ( G S - 1 ) = 2 D 102 σ p ; 110 ( 1 ) = σ r ; 110 ( 1 ) = - ψ p ; 010 ( 1 ) = - ψ r ; 010 ( 1 ) - ½ t ( S 2 - 1 ) = D 112 , - ψ p ; 101 ( 1 ) = - ψ r ; 101 ( 1 ) - t ( S 2 - 1 ) = 2 D 112 , - ψ p ; 110 ( 1 ) = - ψ r ; 110 ( 1 ) + 1 2 t ( S 3 G - 1 ) = 4 D 202 .
σ s ; i j k ( 2 ) = σ q ; i j k ( 2 ) + Δ s ; i j k .
Δ s ; 000 = 3 F s ; 000 ( 1 ) σ q ; 000 ( 1 ) , Δ s ; 100 = F s ; 100 ( 1 ) σ q ; 000 ( 1 ) + 2 F s ; 000 ( 1 ) σ q ; 100 ( 1 ) , Δ s ; 001 = 3 F s ; 001 ( 1 ) σ q ; 000 ( 1 ) + 2 F s ; 000 ( 1 ) σ q ; 001 ( 1 ) + 2 F s ; 100 ( 1 ) σ q ; 000 ( 1 ) , Δ s ; 101 = F s ; 101 ( 1 ) σ q ; 000 ( 1 ) + F s ; 100 ( 1 ) σ q ; 001 ( 1 ) + 2 F s ; 001 ( 1 ) σ q ; 100 ( 1 ) + 2 F s ; 100 ( 1 ) σ q ; 100 ( 1 ) + F s ; 000 ( 1 ) σ q ; 101 ( 1 ) , Δ s ; 002 = 2 F s ; 001 ( 1 ) σ q ; 001 ( 1 ) + 2 F s ; 101 ( 1 ) σ q ; 000 ( 1 ) , Δ s ; 102 = F s ; 101 ( 1 ) σ q ; 001 ( 1 ) + 2 F s ; 101 ( 1 ) σ q ; 100 ( 1 ) + F s ; 001 ( 1 ) σ q ; 101 ( 1 ) , Δ s ; 010 = 3 F s ; 010 ( 1 ) σ q ; 000 ( 1 ) + F s ; 000 ( 1 ) σ q ; 010 ( 1 ) + F s ; 100 ( 1 ) σ q ; 001 ( 1 ) , Δ s ; 110 = F s ; 110 ( 1 ) σ q ; 000 ( 1 ) + F s ; 100 ( 1 ) σ q ; 010 ( 1 ) + 2 F s ; 010 ( 1 ) σ q ; 100 ( 1 ) + F s ; 100 ( 1 ) σ q ; 101 ( 1 ) , Δ s ; 011 = 2 F s ; 010 ( 1 ) σ q ; 001 ( 1 ) + F s ; 001 ( 1 ) σ q ; 010 ( 1 ) + 2 F s ; 110 ( 1 ) σ q ; 000 ( 1 ) + F s ; 101 ( 1 ) σ q ; 001 ( 1 ) , Δ s ; 111 = F s ; 110 ( 1 ) σ q ; 001 ( 1 ) + F s ; 101 ( 1 ) σ q ; 010 ( 1 ) + 2 F s ; 110 ( 1 ) σ q ; 100 ( 1 ) + F s ; 010 ( 1 ) σ q ; 101 ( 1 ) + F s ; 101 ( 1 ) σ q ; 101 ( 1 ) , Δ s ; 020 = F s ; 010 ( 1 ) σ q ; 010 ( 1 ) + F s ; 110 ( 1 ) σ q ; 001 ( 1 ) , Δ s ; 120 = F s ; 110 ( 1 ) σ q ; 010 ( 1 ) + F s ; 110 ( 1 ) σ q ; 101 ( 1 ) .
F r ; 000 ( 1 ) = 1 t ψ q ; 000 ( 1 ) - 1 2 G 2 , F r ; 100 ( 1 ) = 1 t ψ q ; 100 ( 1 ) + 1 2 S G , F r ; 001 ( 1 ) = 1 t ψ q ; 001 ( 1 ) + S G , F r ; 101 ( 1 ) = 1 t ψ q ; 101 ( 1 ) - S 2 , F r ; 010 ( 1 ) = 1 t ψ q ; 010 ( 1 ) - 1 2 S 2 , F r ; 110 ( 1 ) = 1 t ψ q ; 110 ( 1 ) + 1 2 S 3 G .
F p ; i j k ( 1 ) = 1 t ψ q ; i j k ( 1 ) .
4 σ s ; 100 ( 2 ) - σ s ; 001 ( 2 ) = 4 Δ s ; 100 - Δ s ; 001 , σ s ; 101 ( 2 ) - σ s ; 002 ( 2 ) = Δ s ; 101 - Δ s ; 002 , 2 σ s ; 110 ( 2 ) - σ s ; 011 ( 2 ) = 2 Δ s ; 110 - Δ s ; 011 .
H = ( E L , E M ) ,             H = - ( E L , E M ) ,
L = - 1 S - G ( ξ 1 + ξ 2 ) , M = - 1 S - G ( η 1 + η 2 ) , L = 1 S - G ( G ξ 1 + S ξ 2 ) , M = 1 S - G ( G η 1 + S η 2 ) .
δ = ( E ξ 2 , E η 2 ) = ( ξ 2 , η 2 ) .
H = L - G L J G + ( L , M ) ,
P = L - S L S J + ( L , M ) .
Q = L - S L J + G ( L , M ) .
π q = S - G S ( L , M ) = 1 S ( ξ 1 + ξ 2 , η 1 + η 2 ) .
ξ 1 = L - G L S - G = J G H S - G - J G S - G ( L , M ) = h - J S G ( S - G ) 2 π q ,
ξ 1 = h - 1 J t t π q .
ξ 2 = q - 1 J t t π q .
F = [ F ] - 1 J t t [ π q x ( F ξ 1 + F ξ 2 ) + π q y ( F η 1 + F η 2 ) ] + ,
δ x = [ ξ 2 ] - 1 J t t [ π q x ( ξ 1 ξ 2 + ξ 2 ξ 2 ) + π q y ( η 1 ξ 2 + η 2 ξ 2 ) ] + O ( 7 ) .
δ x = [ ξ 2 ] - S J t t [ π q x π q x ξ 2 + π q y π q y ξ 2 ] + O ( 7 ) = [ ξ 2 ] - S 2 J t t [ ξ 2 ( π q · π q ) ] + O ( 7 ) .
δ y = [ η 2 ] - S 2 J t t [ η 2 ( π q · π q ) ] + O ( 7 ) .
( ξ 2 , η 2 ) = ξ 1 V + 2 ξ 2 W .
δ = [ ξ 1 V + 2 ξ 2 W ] - S 2 J t t [ ξ 1 ( π q · π q ) V + 2 ξ 2 ( π q · π q ) W ] + O ( 7 ) .
δ ( 1 ; h , q ) = [ ( 2 ) V ] h + 2 [ ( 2 ) W ] q ,
π q ( 1 ; h , q ) = 1 S { [ 2 ( 2 ) U + ( 2 ) V ] h + [ 2 ( 2 ) W + ( 2 ) V ] q } ,
δ ( 2 ; h , q ) = [ ( 3 ) V ] h + 2 ( 3 ) W ] q - S 2 J t t { ( π q 2 ) V q h + 2 ( π q 2 ) W q q } ,
π q 2 π q ( 1 ; h , q ) · π q ( 1 ; h , q ) .
σ q ; 0 j k ( 1 ) = 2 ( 2 - j - k ) D j k 2 , σ q ; 1 j k ( 1 ) = ( k + 1 ) D j , k + 1 , 2 , ψ q ; 0 j k ( 1 ) = 2 ( 2 - j - k ) D j k 2 + ( k + 1 ) D j , k + 1 , 2 , ψ q ; 1 j k ( 1 ) = ( k + 1 ) D j , k + 1 , 2 + 2 ( j + 1 ) D j + 1 , k , 2 .
π q 2 j = 0 3 k = 0 3 - j ( π q 2 ) j k 3 U q j V q k W q 3 - j - k .
( π q 2 ) j k 3 = j = 0 j - 1 k = 0 k ψ q ; 1 j k ( 1 ) ψ q ; 1 , j - 1 - j , k - k ( 1 ) + 2 j = 0 j k = 0 k - 1 ψ q ; 1 j k ( 1 ) ψ q ; 0 , j - j , k - 1 - k ( 1 ) + j = 0 j k = 0 k ψ q ; 0 j k ( 1 ) ψ q ; 0 , j - j , k - k ( 1 ) .
σ q ; 0 j k ( 2 ) = 2 ( 3 - j - k ) C j k 3 , σ q ; 1 j k ( 2 ) = ( k + 1 ) C j , k + 1 , 3 ,
C j k 3 = { D j k 3 - S 2 J t t ( π q 2 ) j k 3 } .
q = p + S t π q .
p - r = - S t ( P - R ) = - S t t ( L - V ) ,
q = r + S t { π q + t ( V - L ) } .
V - L = { L ( 1 - L 2 - M 2 ) 1 / 2 , M / ( 1 - L 2 - M 2 ) 1 / 2 } - L = ½ ( L 2 + M 2 ) L + O ( 5 ) ,
V - L = ½ ( S 2 U q - 2 S G V q + G 2 W q ) ( S h - G q ) + O ( 5 ) .
q = s + F s ( h , q )             ( s = p , r ; s = p , r ) ,
F s ( h , q ) j , k 1 ( F s ; 0 j k ( 1 ) q + F s ; 1 j k ( 1 ) h ) U q j V q k W q 1 - j - k + O ( 5 ) ,
F p ; i j k ( 1 ) = 1 t ψ q ; i j k ( 1 )
F r ; 0 j k ( 1 ) = 1 t ψ q ; 0 j k ( 1 ) - 1 2 S 2 j + k G 2 - 2 j - k , F r ; 1 j k ( 1 ) = 1 t ψ q ; 1 j k ( 1 ) + 1 2 S 2 j + k + 1 G 1 - 2 j - k .
δ ( h , s ) = [ δ ( h , q ) ] + [ F s x δ q x + F s y δ q y ] + O ( 7 ) ,
σ s ; i j k ( 2 ) = σ q ; i j k ( 2 ) + Δ s ; i j k             ( s = p , r ) ,
Δ s ; i j k = [ F s x δ q x + F s y δ q y ] i j k ,

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