Abstract

A sixth-order theory of wave aberrations for axially symmetric systems is developed. Specific formulas for the sixth-order extrinsic and intrinsic wave aberration coefficients are given, as well as relations between pupil and image aberrations. Equations are developed for the wavefront propagation to the sixth order of approximation. The concept of the irradiance function is developed, and the second-order irradiance coefficients are found via conservation of flux at the pupils of the optical system and in terms of pupil aberrations. From purely geometrical considerations a generalized irradiance transport equation that describes irradiance changes in an optical system is derived. Confirming the aberration coefficients with real ray-tracing data was found to be indispensable.

© 2010 Optical Society of America

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Corrections

José Sasián, "Theory of sixth-order wave aberrations: errata," Appl. Opt. 49, 6502-6503 (2010)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-49-33-6502

References

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  1. R. Shack, course OPTI 518, “Introduction to Aberrations,” class notes, College of Optical Sciences, University of Arizona, 1995. The value of writing the aberration function in terms of the field and aperture vector is that further development in aberration theory is possible. See, for example, and the contributions in this paper as examples.
  2. K. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. 22, 1389-1401 (2005).
  3. K. Schwarzschild, “Untersuchungen zur geometrischen Optik I. Einleitung in die Fehlertheorie optischer Instrumente auf Grund des Eikonalbegriffs,” Abh. Königlichen Ges. Wiss. Göttingen Math. Phys. Kl. 4, 1-31 (1905).
  4. H. H. Hopkins, The Wave Theory of Aberrations (Oxford Univ. Press, 1950).
    [CrossRef]
  5. C. G. Wynne, “Primary aberrations and conjugate change,” Proc. Phys. Soc. London Sect. B 65, 429-437 (1952).
  6. J. Hoffman, “Induced aberrations in optical systems,” Ph. D. dissertation (University of Arizona, 1993).
    [CrossRef]
  7. J. Sasian, “Interpretation of pupil aberrations in imaging systems,” Proc. SPIE 634, 634208 (2006).
  8. J. Sasian, “Design of null lens correctors for the testing of astronomical optics,” Opt. Eng. 27, 1051-1056 (1988).
    [CrossRef]
  9. D. Shafer's triplet lens data can be found in .
  10. M. P. Rimmer, “Optical aberration coefficients,” M.S. thesis (University of Rochester, 1963).
  11. H. A. Buchdahl, Optical Aberration Coefficients (Dover, 1968).
  12. M. R. Teague, “Image formation in terms of the transport equation,” J. Opt. Soc. Am. A 2, 2019-2026(1985).
    [CrossRef]

2006 (1)

J. Sasian, “Interpretation of pupil aberrations in imaging systems,” Proc. SPIE 634, 634208 (2006).

2005 (1)

K. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. 22, 1389-1401 (2005).

1988 (1)

J. Sasian, “Design of null lens correctors for the testing of astronomical optics,” Opt. Eng. 27, 1051-1056 (1988).
[CrossRef]

1985 (1)

1952 (1)

C. G. Wynne, “Primary aberrations and conjugate change,” Proc. Phys. Soc. London Sect. B 65, 429-437 (1952).

1905 (1)

K. Schwarzschild, “Untersuchungen zur geometrischen Optik I. Einleitung in die Fehlertheorie optischer Instrumente auf Grund des Eikonalbegriffs,” Abh. Königlichen Ges. Wiss. Göttingen Math. Phys. Kl. 4, 1-31 (1905).

Buchdahl, H. A.

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

Hoffman, J.

J. Hoffman, “Induced aberrations in optical systems,” Ph. D. dissertation (University of Arizona, 1993).
[CrossRef]

Hopkins, H. H.

H. H. Hopkins, The Wave Theory of Aberrations (Oxford Univ. Press, 1950).
[CrossRef]

Rimmer, M. P.

M. P. Rimmer, “Optical aberration coefficients,” M.S. thesis (University of Rochester, 1963).

Sasian, J.

J. Sasian, “Interpretation of pupil aberrations in imaging systems,” Proc. SPIE 634, 634208 (2006).

J. Sasian, “Design of null lens correctors for the testing of astronomical optics,” Opt. Eng. 27, 1051-1056 (1988).
[CrossRef]

Schwarzschild, K.

K. Schwarzschild, “Untersuchungen zur geometrischen Optik I. Einleitung in die Fehlertheorie optischer Instrumente auf Grund des Eikonalbegriffs,” Abh. Königlichen Ges. Wiss. Göttingen Math. Phys. Kl. 4, 1-31 (1905).

Shack, R.

R. Shack, course OPTI 518, “Introduction to Aberrations,” class notes, College of Optical Sciences, University of Arizona, 1995. The value of writing the aberration function in terms of the field and aperture vector is that further development in aberration theory is possible. See, for example, and the contributions in this paper as examples.

Shafer's, D.

D. Shafer's triplet lens data can be found in .

Teague, M. R.

Thompson, K.

K. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. 22, 1389-1401 (2005).

Wynne, C. G.

C. G. Wynne, “Primary aberrations and conjugate change,” Proc. Phys. Soc. London Sect. B 65, 429-437 (1952).

Abh. Königlichen Ges. Wiss. Göttingen Math. Phys. Kl. (1)

K. Schwarzschild, “Untersuchungen zur geometrischen Optik I. Einleitung in die Fehlertheorie optischer Instrumente auf Grund des Eikonalbegriffs,” Abh. Königlichen Ges. Wiss. Göttingen Math. Phys. Kl. 4, 1-31 (1905).

J. Opt. Soc. Am. (1)

K. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. 22, 1389-1401 (2005).

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

Opt. Eng. (1)

J. Sasian, “Design of null lens correctors for the testing of astronomical optics,” Opt. Eng. 27, 1051-1056 (1988).
[CrossRef]

Proc. Phys. Soc. London Sect. B (1)

C. G. Wynne, “Primary aberrations and conjugate change,” Proc. Phys. Soc. London Sect. B 65, 429-437 (1952).

Proc. SPIE (1)

J. Sasian, “Interpretation of pupil aberrations in imaging systems,” Proc. SPIE 634, 634208 (2006).

Other (6)

J. Hoffman, “Induced aberrations in optical systems,” Ph. D. dissertation (University of Arizona, 1993).
[CrossRef]

D. Shafer's triplet lens data can be found in .

M. P. Rimmer, “Optical aberration coefficients,” M.S. thesis (University of Rochester, 1963).

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

R. Shack, course OPTI 518, “Introduction to Aberrations,” class notes, College of Optical Sciences, University of Arizona, 1995. The value of writing the aberration function in terms of the field and aperture vector is that further development in aberration theory is possible. See, for example, and the contributions in this paper as examples.

H. H. Hopkins, The Wave Theory of Aberrations (Oxford Univ. Press, 1950).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

A, field and aperture vectors (scaled by the marginal ray height at the exit pupil and the chief ray height at the image plane); B, the angle ϕ between them looking down the optical axis.

Fig. 2
Fig. 2

The tip of the aperture vector defines the intersection of a ray with the pupil plane. The wavefront deformation is the distance along the ray from the reference sphere to the wavefront, and it is negative in this figure.

Fig. 3
Fig. 3

Wavefront aberration shapes.

Fig. 4
Fig. 4

Schematic for deriving the relationship among the aberration function, the pupil function, and the sphere function.

Fig. 5
Fig. 5

Pupil grid mapping effects Δ ρ = ( 1 / Ψ ) H W ¯ ( H , ρ ) due to pupil aberrations in relation to the Gaussian pupil (dashed grid). There is no effect from pupil piston.

Fig. 6
Fig. 6

Construction for deriving the relationship Δ Ω = ( y ¯ P P / y P P ) Δ H Δ ρ .

Fig. 7
Fig. 7

Aspheric triplet lens corrected for fourth- and sixth-order aberrations to the 10 14 wave level.

Fig. 8
Fig. 8

Construction for deriving the wavefront deformation.

Tables (24)

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Table 1 Wavefront Aberrations

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Table 2 Seidel Aberration Coefficients

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Table 3 Quantities Derived from Paraxial Data used in Computing Aberration Coefficients

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Table 4 Pupil Aberrations

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Table 5 Extrinsic Coefficients for Combination of Systems A and B

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Table 6 Wavefront Change at the Exit Pupil on Propagation in Free Space the Distance Δ Z = y ¯ · u ¯ 1 from the Pupil a

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Table 7 Change in Wavefront on Placing the Aperture Vector at the Exit Pupil a

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Table 8 Image and Pupil Coefficient Relationships for a Spherical Surface

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Table 9 Quantities Used in Calculation of Intrinsic Aberration Coefficients with the Aperture Vector ρ at the Exit Pupil

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Table 10 Intrinsic Aberration Coefficients with the Aperture Vector ρ at the Exit Pupil a

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Table 11 Relationships between Intrinsic Coefficients W and W + of a Spherical Surface a

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Table 12 Relationships between Sixth-Order Pupil and Image Aberration Coefficients for a Spherical Surface a

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Table 13 Sixth-Order Aberration Coefficients for a System of j Surfaces

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Table 14 Zero- and Second-Order Terms of the Irradiance Function

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Table 15 Coefficient Comparison for an Aspheric Surface a

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Table 16 Sixth-Order Aberration Coefficients of the Triplet Lens in Waves at 587.6 nm

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Table 17 Coefficients of the Sphere Function Difference n · S ( H , ρ ) n · S ( H , ρ )

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Table 18 Fourth-Order Aberrations Contributed by an Aspheric Cap where A 4 is the Aspheric Coefficient

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Table 19 Intrinsic Sixth-Order Aberrations Contributed by an Aspheric Cap a

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Table 20 Intrinsic Sixth-Order Aberrations Contributed by an Aspheric Cap as the Stop is Shifted a

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Table 21 Sixth-Order Aberrations Contributed by an Aspheric Cap where A 6 is the Sixth-Order Aspheric Coefficient

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Table 22 Quantities used in the Calculation of the Intrinsic Aberration Coefficients with the Aperture Vector ρ at the Exit Pupil

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Table 23 Intrinsic Aberration Coefficients for a Spherical Surface with the Aperture Vector ρ at the Exit Pupil a

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Table 24 Constructional Data (mm) of the Triplet Lens a

Equations (145)

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G ( H , ρ ) = I 0 · I ( H , ρ ) · exp { i 2 π λ [ n · S ( H , ρ ) + W ( H , ρ ) ] } ,
W ( H , ρ ) = j , m , n W k , l . m ( H · H ) j · ( H · ρ ) m · ( ρ · ρ ) n = W 000 + W 200 ( H · H ) + W 111 ( H · ρ ) + W 020 ( ρ · ρ ) + W 040 ( ρ · ρ ) 2 + W 131 ( H · ρ ) ( ρ · ρ ) + W 222 ( H · ρ ) 2 + W 220 ( H · H ) ( ρ · ρ ) + W 311 ( H · H ) ( H · ρ ) + W 400 ( H · H ) 2 + W 240 ( H · H ) ( ρ · ρ ) 2 + W 331 ( H · H ) ( H · ρ ) ( ρ · ρ ) + W 422 ( H · H ) ( H · ρ ) 2 + W 420 ( H · H ) 2 ( ρ · ρ ) + W 511 ( H · H ) 2 ( H · ρ ) + W 600 ( H · H ) 3 + W 060 ( ρ · ρ ) 3 + W 151 ( H · ρ ) ( ρ · ρ ) 2 + W 242 ( H · ρ ) 2 ( ρ · ρ ) + W 333 ( H · ρ ) 3 ,
W ¯ ( 4 ) ( H , ρ ) + n · S ( 4 ) ( H , ρ ) = W ( 4 ) ( H , ρ ) + n · S ( 4 ) ( H , ρ ) .
n · S ( 4 ) ( H , ρ ) n · S ( 4 ) ( H , ρ ) = 1 2 Ψ · Δ { u 2 } · ( H · ρ ) ( ρ · ρ ) + 1 2 Ψ · Δ { u u ¯ } · ( H · ρ ) 2 + 1 4 Ψ · Δ { u u ¯ } · ( H · H ) ( ρ · ρ ) + 1 2 Ψ · Δ { u ¯ 2 } · ( H · H ) ( H · ρ ) ,
W ( 4 ) ( H , ρ ) = W ¯ ( 4 ) ( H , ρ ) .
Δ ρ = 1 Ψ H W ¯ ( H , ρ ) = 1 Ψ · { 4 · W ¯ 040 ( H · H ) H + W ¯ 131 { ( H · H ) ρ + 2 · ( H · ρ ) H } + 2 · W ¯ 222 ( H · ρ ) ρ + 2 · W ¯ 220 ( ρ · ρ ) H + W ¯ 311 ( ρ · ρ ) ρ } .
Δ ρ = 1 Ψ H W ¯ ( H , ρ ) = 1 Ψ · { 4 · W ¯ 040 ( H · H ) H + W ¯ 131 { ( H · H ) ρ + 2 · ( H · ρ ) H } + 2 · W ¯ 222 ( H · ρ ) ρ + 2 · W ¯ 220 ( ρ · ρ ) H + W ¯ 311 ( ρ · ρ ) ρ } .
W C ( H , ρ ) = W A ( H , ρ ) + W B ( H , ρ + Δ ρ ) ,
Δ ρ = 1 Ψ H W ¯ A ( H , ρ ) + O ( 5 )
W C ( H , ρ ) = W ( 4 ) A ( H , ρ ) + W ( 6 ) A ( H , ρ ) + W ( 4 ) B ( H , ρ ) + W ( 6 I ) B ( H , ρ ) + W ( 6 E ) B ( H , ρ ) ,
W B 6 E ( H , ρ ) = W B 4 ( H , ρ + Δ ρ ) W B 4 ( H , ρ ) .
Δ ρ = 1 Ψ H W ¯ B ( H , ρ ) + O ( 5 ) .
W C ( H , ρ ) = W ( 4 ) A ( H , ρ + Δ ρ ) + W ( 6 I ) A ( H , ρ ) + W ( 4 ) B ( H , ρ ) + W ( 6 I ) B ( H , ρ ) .
W B 6 E ( H , ρ ) = W A 4 ( H , ρ + Δ ρ ) W A 4 ( H , ρ ) .
W 6 E ( H , ρ ) = W B 4 ( H , ρ + Δ ρ A ) W B 4 ( H , ρ ) ρ W B 4 ( H , ρ ) · Δ ρ A = 1 Ψ ρ W B 4 ( H , ρ ) · H W ¯ A 4 ( H , ρ ) ,
W 6 E ( H , ρ ) = W A 4 ( H , ρ + Δ ρ B ) W A 4 ( H , ρ ) ρ W A 4 ( H , ρ ) · Δ ρ B = 1 Ψ ρ W A 4 ( H , ρ ) · H W ¯ B 4 ( H , ρ ) .
E ( X , Y , Z ) · E ( X , Y , Z ) = [ E ( X , Y , Z ) X ] 2 + [ E ( X , Y , Z ) Y ] 2 + [ E ( X , Y , Z ) Z ] 2 = n 2 ,
{ S g ( X , Y , Z ) } = n R [ X g X , Y g Y , Z g Z ] ,
W ( X , Y , Z ) = S g ( X , Y , Z ) E ( X , Y , Z ) .
| E ( X , Y , Z ) | 2 = | W ( X , Y , Z ) | 2 2 · W ( X , Y , Z ) · S g ( X , Y , Z ) + | S g ( X , Y , Z ) | 2 = n 2 .
W ( X , Y , Z ) · W ( X , Y , Z ) = n 2 ,
W Z Δ Z W Δ Z ,
Δ Z W ( X , Y ) n ( 1 1 2 n 2 ( [ W X ] 2 + [ W Y ] 2 ) ) · Δ Z = n · Δ Z Δ Z 2 n | W ( X , Y ) | 2 ,
Δ Z W ( H , ρ ) n · Δ Z Δ Z 2 n y y pupil ρ W ( H , ρ ) · ρ W ( H , ρ ) .
1 2 Δ Z n y y pupil = 1 2 y ¯ y ¯ Δ Z n y y pupil = 1 2 y ¯ u ¯ 1 n y y pupil = 1 2 y ¯ y 1 Ψ ,
Δ Z W ( H , ρ ) + 1 2 y ¯ y 1 Ψ ρ W ( H , ρ ) · ρ W ( H , ρ ) .
Δ Z W ( H , ρ ) = n · Δ Z n · Δ Z 2 y y pupil · y 4 R 1 2 ( ρ · ρ ) = n · Δ Z R 1 R 2 R 1 R 2 · n · y 2 2 ( ρ · ρ ) = n · Δ Z { n · y 2 2 R 1 ( ρ · ρ ) n · y 2 2 R 2 ( ρ · ρ ) } = n · ( R 1 R 2 ) + n 2 R 2 ( X 2 + Y 2 ) n 2 R 1 ( X 2 + Y 2 ) ,
W + ( H , ρ ) W ( H , ρ ) = ρ { W ( H , ρ ) W 311 ( H · H ) ( H · ρ ) } · Δ ρ = 1 Ψ ρ W ( H , ρ ) · H W ¯ ( H , ρ ) 1 Ψ ρ W 311 ( H · H ) ( H · ρ ) · H W ¯ ( H , ρ ) ,
Δ ρ + Δ Ω = Δ Z l Δ Z y ¯ image y P P Δ H = y ¯ P P y P P Δ H ,
Δ H = 1 y ¯ image n u ρ W ( H , ρ ) + O ( 5 ) = 1 Ψ ρ W ( H , ρ ) + O ( 5 ) .
Δ Ω = y ¯ P P y P P Δ H Δ ρ = 1 Ψ [ H W ¯ ( H , ρ ) y ¯ P P y P P ρ W ( H , ρ ) ] .
Δ Z W ( H , ρ ) = ± 1 2 y ¯ y 1 Ψ ρ W ( H , ρ ) · ρ W ( H , ρ ) ,
Δ Z W ( H , ρ ) = ρ W ( H , ρ ) · Δ ε ,
W + ( H , ρ ) W ( H , ρ ) = 1 Ψ ρ W ( H , ρ ) · H W ¯ ( H , ρ ) .
W E ( H , ρ ) = 1 Ψ ρ W B ( H , ρ ) · H W ¯ A ( H , ρ ) ,
W E + ( H , ρ ) = 1 Ψ ρ W A ( H , ρ ) · H W ¯ B ( H , ρ ) .
W 060 = 4 W 040 [ 1 8 y 2 r 2 1 8 A ( u n + u n ) + 1 2 y r u ] ,
W 060 trasnfer = 8 Ψ W 040 · W 040 y ¯ y ,
W 060 I = W 040 [ 1 2 y 2 r 2 1 2 A ( u n + u n ) + 2 y r u ] + 8 Ψ W 040 · W 040 y ¯ y ,
W 060 I + = W 040 [ 1 2 y 2 r 2 1 2 A ( u n + u n ) + 2 y r u ] 8 Ψ W 040 · W 040 y ¯ y ,
4 Ψ W 040 W ¯ 311 0 = 2 W 040 y r ( u u ) ,
8 Ψ W 040 W 040 y ¯ y = 2 Ψ W 040 ( W ¯ 311 W ¯ 311 0 ) ,
W 060 I + = W 060 I 4 Ψ W 040 W ¯ 311 .
W 240 C C = + [ 1 16 A r Ψ 2 Δ { u n 2 } + 1 8 1 r Ψ 2 Δ { u 2 n } + 1 4 y 2 r 2 W 220 P + y r u W 220 P 1 4 u r Ψ 2 Δ { u n } ] ,
W 240 C C + = + [ 1 16 A r Ψ 2 Δ { u n 2 } + 1 8 1 r Ψ 2 Δ { u 2 n } + 1 4 y 2 r 2 W 220 P + y r u W 220 P 1 4 u r Ψ 2 Δ { u n } ] ,
W 240 C C + = W 240 C C 2 Ψ W 220 P · W ¯ 311 + 2 Ψ W 220 P · W 131 ,
W 220 P = 1 4 Ψ 2 P = 1 4 r Ψ 2 Δ ( 1 n )
W 420 C C = W 420 C C + = 3 16 1 r 3 Ψ 4 Δ { 1 n } 1 A 2 .
W 331 C C + = 2 W 220 P · u u ¯ ,
W 331 C C = 2 W 220 P · u u ¯ ,
W 422 C C + = W 220 P · u ¯ 2 ,
W 422 C C = W 220 P · u ¯ 2 ,
W 151 C C + = 4 W 040 · u u ¯ ,
W 151 C C = 4 W 040 · u u ¯ ,
W 242 C C + = 2 W 040 · u ¯ 2 ,
W 242 C C = 2 W 040 · u ¯ 2 .
W C C + ( H , ρ ) = W 040 ( ρ · ρ ) 2 + W 220 ( H · H ) ( ρ · ρ ) + W 240 C C + ( H · H ) ( ρ · ρ ) 2 + W 331 C C + ( H · H ) ( H · ρ ) ( ρ · ρ ) + W 422 C C + ( H · H ) ( H · ρ ) 2 + W 420 C C + ( H · H ) 2 ( ρ · ρ ) + W 060 + ( ρ · ρ ) 3 + W 151 C C + ( H · ρ ) · ( ρ · ρ ) 2 + W 242 C C + ( H · ρ ) 2 ( ρ · ρ ) ,
ρ shift = ρ + y ¯ O P y O P H = ρ + A ¯ A H ,
W 240 I + = W 240 C C + + 3 ( A ¯ A ) 2 W 060 I + 8 1 Ψ A ¯ A W 040 · W 220 P + W 222 u 2 W 131 u u ¯ .
W 331 I + = 4 A ¯ A W 240 I + + 2 A ¯ A W 242 C C + + A ¯ A W u 220 2 + W 311 u 2 2 W 220 P u u ¯ .
W 422 I + = 4 ( A ¯ A ) 2 W 240 I + + 2 A ¯ A W 331 C C + 2 W 222 u ¯ 2 W 220 u ¯ 2 + 2 ( A ¯ A ) 2 W 220 u 2 + W 311 u u ¯ + 1 2 A ¯ A W 311 u 2 + 2 A ¯ A W 220 u u ¯ .
W 420 I + = 3 ( A ¯ A ) 4 W 060 I + + 2 ( A ¯ A ) 2 W 240 C C + 4 ( A ¯ A ) 2 1 Ψ W 131 · W 220 P 2 A ¯ A 1 Ψ W 220 P · W 220 P + W 420 C C + + A ¯ A W 331 C C + + ( A ¯ A ) 2 W 220 P u 2 + 1 2 A ¯ A W 311 u 2 1 2 W 311 u u ¯ + 2 A ¯ A W 220 P u u ¯ + ( A ¯ A ) 2 W 222 u 2 1 2 W 222 u ¯ 2 .
W 511 I + = 6 ( A ¯ A ) 5 W 060 I + + 4 ( A ¯ A ) 3 [ W 240 C C + 8 A ¯ A 1 Ψ W 040 · W 220 P ] + 2 A ¯ A [ W 420 C C + 2 A ¯ A 1 Ψ W 220 P · W 220 P ] + ( A ¯ A ) 3 W 222 u 2 + 2 ( A ¯ A ) 2 W 311 u 2 ( A ¯ A ) 2 W 222 u u ¯ 2 ( A ¯ A ) 2 W 220 P u u ¯ A ¯ A W 222 u ¯ 2 + 1 2 W 311 u ¯ 2 .
W 151 I + = 6 A ¯ A W 060 I + + W 131 u 2 + W 151 C C + .
W 242 I + = 12 ( A ¯ A ) 2 W 060 I + + 7 2 W 222 u 2 3 W 131 u u ¯ + W 242 C C + .
W 333 I + = 8 ( A ¯ A ) 3 W 060 I + + 4 ( A ¯ A ) 2 W 151 C C + + 3 A ¯ A W 222 u 2 + 2 A ¯ A W 242 C C + + 2 W 222 u u ¯ ,
W 600 I + = W ¯ 060 I + .
W I + ( H , ρ ) W I ( H , ρ ) = ρ { W ( H , ρ ) W 311 ( H · H ) ( H · ρ ) } · Δ ρ = 1 Ψ ρ W ( H , ρ ) · H W ¯ ( H , ρ ) 1 Ψ [ ρ W 311 ( H · H ) ( H · ρ ) ] · [ H W ¯ ( H , ρ ) ] .
W I + forward ( H , ρ ) W I + reverse ( H , ρ ) = 1 Ψ ρ W ( H , ρ ) · H W ¯ ( H , ρ ) + Δ Π ( H , ρ ) ,
Δ Π ( H , ρ ) = 1 2 W 311 Δ { u 2 } ( H · H ) ( ρ · ρ ) ( H · ρ ) + W 311 Δ { u u ¯ } ( H · H ) ( H · ρ ) 2 + 1 2 W 311 Δ { u u ¯ } ( H · H ) 2 ( ρ · ρ ) + 3 2 W 311 Δ { u ¯ 2 } ( H · H ) 2 ( H · ρ ) .
Δ Ξ ( H , ρ ) = 1 2 W 311 u 2 ( H · H ) ( H · ρ ) ( ρ · ρ ) + W 311 u u ¯ ( H · H ) ( H · ρ ) 2 + 1 2 W 311 u u ¯ ( H · H ) 2 ( ρ · ρ ) + 3 2 W 311 u ¯ 2 ( H · H ) 2 ( H · ρ ) 1 Ψ [ ρ W 311 ( H · H ) ( H · ρ ) ] · [ H W ¯ ( H , ρ ) ] ,
W ¯ ( 6 ) ( H , ρ ) + n · S ( 6 ) ( H , ρ ) = W ( 6 ) ( H , ρ ) + n · S ( 6 ) ( H , ρ ) + O ( 6 ) ,
O ( 6 ) = 1 Ψ ρ W ( H , ρ ) · H W ¯ ( H , ρ ) Δ Ξ ( H , ρ ) + Δ Ξ ¯ ( H , ρ ) ,
Δ Ξ ( H , ρ ) = 1 2 W 311 u 2 ( H · H ) ( H · ρ ) ( ρ · ρ ) + W 311 u u ¯ ( H · H ) ( H · ρ ) 2 + 1 2 W 311 u u ¯ ( H · H ) 2 ( ρ · ρ ) + 3 2 W 311 u ¯ 2 ( H · H ) 2 ( H · ρ ) ,
Δ Ξ ¯ ( H , ρ ) = 1 2 W ¯ 311 u 2 ( H · H ) ( H · ρ ) ( ρ · ρ ) + W ¯ 311 u u ¯ ( H · H ) ( H · ρ ) 2 + 1 2 W ¯ 311 u u ¯ ( H · H ) 2 ( ρ · ρ ) + 3 2 W ¯ 311 u ¯ 2 ( H · H ) 2 ( H · ρ ) .
E ( H , ρ ) = n · S ( H , ρ ) W ( H , ρ ) ,
E ( 0 ) ( H , ρ ) = n · y u n · y ¯ u ¯ ,
E ( 2 ) ( H , ρ ) = 1 2 Ψ u ¯ u ( H · H ) + Ψ · ( H · ρ ) + 1 2 Ψ u u ¯ ( ρ · ρ ) .
E ( 4 ) ( H , ρ ) = 1 8 Ψ Δ ( u ¯ 3 u ) ( H · H ) 2 n · S ( 4 ) ( H , ρ ) W ( 4 ) ( H , ρ ) ,
E ( 6 ) ( H , ρ ) = 1 16 Ψ Δ ( u ¯ 5 u ) ( H · H ) 3 n · S ( 6 ) ( H , ρ ) W + ( 6 ) ( H , ρ ) .
I ( H , ρ ) = j , m , n I k , l . m ( H · H ) j · ( H · ρ ) m · ( ρ · ρ ) n = I 000 + I 200 ( H · H ) + I 111 ( H · ρ ) + I 020 ( ρ · ρ ) + I 040 ( ρ · ρ ) 2 + I 131 ( H · ρ ) ( ρ · ρ ) + I 222 ( H · ρ ) 2 + I 220 ( H · H ) ( ρ · ρ ) + I 311 ( H · H ) ( H · ρ ) + I 400 ( H · H ) 2 + I 240 ( H · H ) ( ρ · ρ ) 2 + I 331 ( H · H ) ( H · ρ ) ( ρ · ρ ) + I 422 ( H · H ) ( H · ρ ) 2 + I 420 ( H · H ) 2 ( ρ · ρ ) + I 511 ( H · H ) 2 ( H · ρ ) + I 600 ( H · H ) 3 + I 060 ( ρ · ρ ) 3 + I 151 ( H · ρ ) ( ρ · ρ ) 2 + I 242 ( H · ρ ) 2 ( ρ · ρ ) + I 333 ( H · ρ ) 3 .
I 0 · I ( H , ρ + Δ ρ ) · d 2 S = I 0 · I ( H , ρ + Δ ρ ) · d 2 S · J ( H , ρ ) = I 0 · I ( H , ρ ) · d 2 S ,
J ( H , ρ ) = y pupil 2 y pupil 2 · { 1 + Δ ρ h ρ h + Δ ρ i ρ i + Δ ρ h ρ h Δ ρ i ρ i Δ ρ i ρ h Δ ρ h ρ i } .
Δ ρ h ρ h = Δ ρ h ρ ρ ρ h = 1 cos ( ϕ ) Δ ρ h ρ ,
Δ ρ i ρ i = Δ ρ i ρ ρ ρ i = 1 sin ( ϕ ) Δ ρ i ρ ,
Δ ρ h ρ i = Δ ρ h ρ ρ ρ i = 1 sin ( ϕ ) Δ ρ h ρ ,
Δ ρ i ρ h = Δ ρ i ρ ρ ρ h = 1 cos ( ϕ ) Δ ρ i ρ ,
J ( H , ρ ) = y pupil 2 y pupil 2 · { 1 + Δ ρ h ρ h + Δ ρ i ρ i } = y pupil 2 y pupil 2 · { 1 + ρ Δ ρ } ,
I ( H , ρ + Δ ρ ) I ( H , ρ ) ρ I ( H , ρ ) · Δ ρ
I ( H , ρ ) [ ρ I ( H , ρ ) · Δ ρ + I ( H , ρ ) ] ( 1 + ρ Δ ρ ) .
Δ ρ = Δ ρ g + Δ ρ h = 1 Ψ H W ¯ ( H , ρ ) ,
I ( H , ρ ) I ( H , ρ ) 1 Ψ ρ I ( H , ρ ) H W ¯ ( H , ρ ) 1 Ψ I ( H , ρ ) ρ H W ¯ ( H , ρ ) ,
ρ W ( H , ρ ) = ( ρ h W ( H , ρ ) ) h + ( ρ i W ( H , ρ ) ) i = ρ W ( H , ρ ) .
ρ ( H W ¯ ( H , ρ ) ) = ρ h ( H W ¯ ( H , ρ ) · h ) + ρ i ( H W ¯ ( H , ρ ) · i ) = ρ H W ( H , ρ ) .
H W ¯ ( H , ρ ) = y ¯ P P y P P ρ W ( H , ρ ) .
Δ I ( H , ρ ) = I ( H , ρ ) I ( H , ρ ) 1 Ψ y ¯ P P y P P ρ I ( H , ρ ) · ρ W ( H , ρ ) 1 Ψ y ¯ P P y P P I ( H , ρ ) ρ 2 W ( H , ρ ) ,
ρ 2 W ( H , ρ ) = 2 W ( H , ρ ) ρ h 2 + 2 W ( H , ρ ) ρ i 2 = ρ ( ρ W ( H , ρ ) ) .
y ¯ P P y P P = Δ Z ( l Δ Z ) y P P y ¯ image = Ψ n Δ Z y P P · y pupil ,
Δ I ( H , ρ ) Δ Z 1 n · y P P · y pupil ( ρ I ( H , ρ ) · ρ W ( H , ρ ) + I ( H , ρ ) · ρ 2 W ( H , ρ ) ) .
I ( H , ρ ) Z 1 n · y P P · y P P [ ρ I ( H , ρ ) · ρ W ( H , ρ ) + I ( H , ρ ) 2 W ( H , ρ ) ] ,
I ( X , Y ) Z = [ I ( X , Y ) · Φ ( X , Y ) + I ( X , Y ) 2 Φ ( X , Y ) ] ,
Δ I ( H , ρ ) Δ Z 1 n · y P P · y pupil ρ 2 W ( H , ρ ) = 1 n · y P P · y pupil 12 W 040 ( ρ · ρ ) .
Δ I ( H , ρ ) Δ Z n · y ¯ P P · y pupil 3 W ¯ 311 ( ρ · ρ ) = 3 Ψ W ¯ 311 ( ρ · ρ ) ,
Δ H = 1 cos 3 ( θ ) 1 Ψ ρ W ( H , ρ ) + O ( 7 ) ,
cos ( θ ) 1 1 2 ( u 2 ( ρ · ρ ) + 2 u u ( H · ρ ) + u ¯ 2 ( H · H ) )
Δ { Sag } = A 4 ( X 2 + Y 2 ) 2 + A 6 ( X 2 + Y 2 ) 3 ,
W I + cap ( H , ρ ) = W 040 cap ( ρ · ρ ) 2 + W 240 I + cap ( H · H ) ( ρ · ρ ) 2 + W 311 + cap ( H · H ) ( H · ρ ) + W 060 I + cap ( ρ · ρ ) 3 + W 151 I + cap ( H · ρ ) · ( ρ · ρ ) 2 ,
Δ Z W ( H , ρ ) = 1 2 y ¯ y 1 Ψ ρ W ( H , ρ ) · ρ W ( H , ρ ) ,
Δ W ( H , ρ ) = 1 Ψ ρ W ( H , ρ ) · H W ¯ ( H , ρ ) .
Δ W ( H , ρ ) = y ¯ y 1 Ψ ρ W ( H , ρ ) · ρ W ( H , ρ ) ,
Δ W ( H , ρ ) = 1 2 y ¯ y 1 Ψ ρ W ( H , ρ ) · ρ W ( H , ρ ) .
Δ W stopshifting ( H , ρ ) = { W cap ( H , ρ ) 1 2 y ¯ y 1 Ψ ρ W cap ( H , ρ ) · ρ W cap ( H , ρ ) } | y ¯ y H + ρ ,
Δ Ξ ( H , ρ ) = 1 2 W 311 u 2 ( H · H ) ( ρ · ρ ) ( H · ρ ) + W 311 u u ¯ ( H · H ) ( H · ρ ) 2 + 1 2 W 311 u u ¯ ( H · H ) 2 ( ρ · ρ ) + 3 2 W 311 u ¯ 2 ( H · H ) 2 ( H · ρ ) 1 Ψ [ ρ W 311 ( H · H ) ( H · ρ ) ] · [ H W ¯ ( H , ρ ) ] .
W 6 E ( H , ρ ) = 1 Ψ ρ W cap ( H , ρ ) · H W ¯ sphere ( H , ρ ) ,
Δ Π ( H , ρ ) = 1 2 W 311 Δ { u 2 } ( H · H ) ( ρ · ρ ) ( H · ρ ) + W 311 Δ { u u ¯ } ( H · H ) ( H · ρ ) 2 + 1 2 W 311 Δ { u u ¯ } ( H · H ) 2 ( ρ · ρ ) + 3 2 W 311 Δ { u ¯ 2 } ( H · H ) 2 ( H · ρ ) .
Δ W stopshifting ( H , ρ ) = { W C C + ( H , ρ ) 1 2 A ¯ A 1 Ψ ρ W C C + ( H , ρ ) · ρ W C C + ( H , ρ ) } | ( A ¯ / A ) H + ρ ,
W C C + ( H , ρ ) = W 040 ( ρ · ρ ) 2 + W 220 ( H · H ) ( ρ · ρ ) + W 240 C C + ( H · H ) ( ρ · ρ ) 2 + W 331 C C + ( H · H ) ( H · ρ ) ( ρ · ρ ) + W 422 C C + ( H · H ) ( H · ρ ) 2 + W 420 C C + ( H · H ) 2 ( ρ · ρ ) + W 060 + ( ρ · ρ ) 3 + W 151 C C + ( H · ρ ) · ( ρ · ρ ) 2 + W 242 C C + ( H · ρ ) 2 ( ρ · ρ ) + W 311 ( H · H ) ( H · ρ )
Δ Ξ ( H , ρ ) = 1 2 W 311 u 2 ( H · H ) ( ρ · ρ ) ( H · ρ ) + W 311 u u ¯ ( H · H ) ( H · ρ ) 2 + 1 2 W 311 u u ¯ ( H · H ) 2 ( ρ · ρ ) + 3 2 W 311 u ¯ 2 ( H · H ) 2 ( H · ρ ) 1 Ψ [ ρ W 311 ( H · H ) ( H · ρ ) ] · [ H W ¯ ( H , ρ ) ]
W = n [ P B ] n [ P A ] = n [ P B ] n [ P B ] ,
Z = h 2 2 r + h 4 8 r 3 + h 6 16 r 5 ,
[ P Q ] 2 = ( s Z ) 2 + h 2 = s 2 2 s Z + Z 2 + h 2 = s 2 { 1 + h 2 2 s [ h 2 2 r + h 4 8 r 3 + h 6 16 r 5 ] + [ h 4 4 r 2 + h 6 8 r 4 ] s 2 } = s 2 { 1 + h 2 s 2 [ 1 s r ] + h 4 4 r 2 s 2 [ 1 s r ] + h 6 8 r 4 s 2 [ 1 s r ] } ,
[ P B ] = [ O Q ] [ P Q ] = h 2 2 [ 1 s 1 r ] h 4 8 r 2 [ 1 s 1 r ] h 6 16 r 4 [ 1 s 1 r ] + h 4 8 s [ 1 s 1 r ] 2 + h 6 16 r 2 s [ 1 s 1 r ] 2 + h 6 16 s 2 [ 1 s 1 r ] 3 .
h = y ( 1 + u 2 r y ) ,
[ P B ] = [ O Q ] [ P Q ] = y 2 2 ( 1 + u 2 r y ) 2 [ 1 s 1 r ] y 4 8 r 2 ( 1 + u 2 r y ) 4 [ 1 s 1 r ] y 6 16 r 4 [ 1 s 1 r ] + y 4 8 s ( 1 + u 2 r y ) 4 [ 1 s 1 r ] 2 + y 6 16 r 2 s [ 1 s 1 r ] 2 + y 6 16 s 2 [ 1 s 1 r ] 3 ,
[ P B ] = [ O Q ] [ P Q ] = y 2 2 ( 1 + u 2 r y ) 2 [ 1 s 1 r ] y 4 8 r 2 ( 1 + u 2 r y ) 4 [ 1 s 1 r ] y 6 16 r 4 [ 1 s 1 r ] + y 4 8 s ( 1 + u 2 r y ) 4 [ 1 s 1 r ] 2 + y 6 16 r 2 s [ 1 s 1 r ] 2 + y 6 16 s ' 2 [ 1 s 1 r ] 3 .
W = n [ P B ] n [ P B ] = y 2 2 ( 1 + u 2 r y ) 2 { n [ 1 s 1 r ] n [ 1 s 1 r ] } y 4 8 r 2 (       1 + u 2 r y ) 4 { n [ 1 s 1 r ] n [ 1 s 1 r ] } + y 4 8 ( 1 + u 2 r y ) 4 { n s [ 1 s 1 r ] 2 n s [ 1 s 1 r ] 2 } y 6 16 r 4 { n [ 1 s 1 r ] n [ 1 s 1 r ] } + y 6 16 r 2 { n s [ 1 s 1 r ] 2 n s [ 1 s 1 r ] 2 } + y 6 16 { n s ' 2 [ 1 s 1 r ] 3 n s 2 [ 1 s 1 r ] 3 } .
W = 1 8 A 2 y Δ { u n } 1 8 A 2 y Δ { u n } [ y 2 2 r 2 1 2 A ( u n + u n ) + 2 y r u ] .
W 040 = 1 8 A 2 y Δ { u n } ,
W 060 = W 040 [ y 2 2 r 2 1 2 A ( u n + u n ) + 2 y r u ] .
1 S = 1 r + ( r s ) 2 + y ¯ 0 2 = 1 r + ( r s ) 1 + y ¯ 0 2 ( r s ) 2 1 r + ( r s ) ( 1 + 1 2 y ¯ 0 2 ( r s ) 2 ) = 1 s ( 1 1 2 y ¯ 0 2 ( r s ) s ) = 1 s ( 1 + 1 2 y ¯ 0 2 ( r s ) s ) = 1 s ( 1 + 1 2 y ¯ 0 2 ( 1 s 1 r ) r s 2 ) = 1 s ( 1 + u 2 y ¯ 0 2 i r s ) = 1 s u y 2 1 2 Ψ 2 n 2 r i ,
1 S 1 s u y 2 1 2 Ψ 2 n 2 r i .
W = y 2 2 ( 1 + u 2 r y ) 2 { n [ 1 S 1 r ] n [ 1 S 1 r ] } = ( 1 + u 2 r y ) 2 { n u ( 1 4 Ψ 2 n ' 2 r i ) n u ( 1 4 Ψ 2 n 2 r i ) } = ( 1 + u 2 r y ) 2 { 1 4 Ψ 2 A r ( u u ) } = ( 1 + u 2 r y ) 2 { 1 4 Ψ 2 r Δ { 1 n } } = W 220 P + u y r W 220 P + O ( 8 ) .
W = y 4 8 r 2 ( 1 + u 2 r y ) 4 { n [ 1 S 1 r ] n [ 1 S 1 r ] } = y 2 4 r 2 ( 1 + u 2 r y ) 4 { n u ( 1 4 Ψ 2 n ' 2 r i ) n u ( 1 4 Ψ 2 n 2 r i ) } = y 2 4 r 2 ( 1 + u 2 r y ) 4 { 1 4 Ψ 2 r Δ { 1 n } } = y 2 4 r 2 W 220 P + O ( 8 ) .
W = y 4 8 { n S [ 1 S 1 r ] 2 n S [ 1 S 1 r ] 2 } = y 4 8 { n S [ 1 s 1 r + u y 2 1 2 Ψ 2 n 2 r i ] 2 n S [ 1 s 1 r + u y 2 1 2 Ψ 2 n 2 r i ] 2 } = y 4 8 { 1 n S [ n ( 1 s 1 r ) + n u y 2 1 2 Ψ 2 n 2 r i ] 2 1 n S [ n ( 1 s 1 r ) + n u y 2 1 2 Ψ 2 n 2 r i ] 2 } = y 2 8 { 1 n S [ A + u y 1 2 Ψ 2 r A ] 2 1 n S [ A + u y 1 2 Ψ 2 r A ] 2 } = y 2 A 2 8 { 1 n S [ 1 + u y 1 2 Ψ 2 r A 2 ] 2 1 n S [ 1 + u y 1 2 Ψ 2 r A 2 ] 2 } = y 2 A 2 8 { 1 n S [ 1 u y Ψ 2 r A 2 ] 1 n S [ 1 u y Ψ 2 r A 2 ] } + O ( 8 ) = y 2 A 2 8 { 1 n [ 1 s + u y 2 1 2 Ψ 2 n 2 r i ] [ 1 u y Ψ 2 r A 2 ] 1 n [ 1 s + u y 2 1 2 Ψ 2 n 2 r i ] [ 1 u y Ψ 2 r A 2 ] } + O ( 8 ) = y A 2 8 { u n [ 1 + 1 2 y Ψ 2 n 2 r i ] [ 1 u y Ψ 2 r A 2 ] u n [ 1 + 1 2 y Ψ 2 n 2 r i ] [ 1 u y Ψ 2 r A 2 ] } + O ( 8 ) = y A 2 8 { u n [ 1 + 1 2 y Ψ 2 n 2 r i + u y Ψ 2 r A 2 ] u n [ 1 + 1 2 y Ψ 2 n 2 r i + u y Ψ 2 r A 2 ] } + O ( 8 ) = y A 2 8 { ( u n u n ) + 1 2 y Ψ 2 r A ( u n 2 u n 2 ) + 1 y Ψ 2 r A 2 ( u 2 n u 2 n ) } + O ( 8 ) = y A 2 8 { Δ ( u n ) + 1 2 y Ψ 2 r A Δ ( u n 2 ) + 1 y Ψ 2 r A 2 Δ ( u 2 n ) } + O ( 8 ) = A 2 8 y Δ ( u n ) + A 16 Ψ 2 r Δ ( u n 2 ) + 1 8 Ψ 2 r Δ ( u 2 n ) + O ( 8 ) .
A 16 Ψ 2 r Δ ( u n 2 ) + 1 8 Ψ 2 r Δ ( u 2 n ) + u y r W 220 P + y 2 4 r 2 W 220 P .
Δ y = u Δ S r s r = 1 2 u Ψ 2 r A 2 ,
Δ S = 1 2 Ψ 2 n u A r .
2 u Ψ 2 r y A 2 W 040 = u Ψ 2 4 r Δ ( u n ) .
W 240 C C = A 16 Ψ 2 r Δ ( u n 2 ) + 1 8 Ψ 2 r Δ ( u 2 n ) + u y r W 220 P + y 2 4 r 2 W 220 P u Ψ 2 4 r Δ ( u n ) .
1 n ρ 1 n ρ = n n n n r + Λ ,
1 ρ = 1 ρ 0 2 + y ¯ 0 2 1 ρ 0 ( 1 1 2 y ¯ 0 2 ρ 0 2 + 3 8 y ¯ 0 4 ρ 0 4 ) ,
1 ρ 1 ρ 0 ( 1 1 2 y ¯ 0 2 ρ 0 2 + 3 8 y ¯ 0 4 ρ 0 4 ) .
1 n ρ 0 1 n ρ 0 = n n n n r .
Λ = 1 n ρ 0 ( 1 1 2 y ¯ 0 2 ρ 0 2 + 3 8 y ¯ 0 4 ρ 0 4 ) 1 n ρ 0 ( 1 1 2 y ¯ 0 2 ρ 0 2 + 3 8 y ¯ 0 4 ρ 0 4 ) n n n n r = 1 2 ( y ¯ 0 2 n ρ 0 3 y ¯ 0 2 n ρ 0 3 ) + 3 8 ( y ¯ 0 4 n ρ 0 5 y ¯ 0 4 n ρ 0 5 ) = 1 2 ( u u ) Ψ 2 A 3 r 3 + 3 8 ( 1 n ρ 0 1 n ρ 0 ) Ψ 4 A 4 r 4 = 1 2 Ψ 2 A 2 r 3 Δ ( 1 n ) + 3 8 ( n n n n r ) Ψ 4 A 4 r 4 .
A 2 r 2 Λ 2 = 1 4 Ψ 2 r Δ ( 1 n ) + 3 16 Ψ 4 A 2 r 3 Δ ( 1 n ) .

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