Abstract

Building on an earlier work on the nodal aberration theory of the 3rd-order aberrations [J. Opt. Soc. Am. A 22, 1389 (2005) ] and the first paper in this series on the nodal aberration theory of higher-order aberrations [J. Opt. Soc. Am. A 26, 1090 (2009) ], this paper continues the derivation and presentation of the intrinsic, characteristic, often multinodal geometry for each type/family of the 3rd- and 5th-order optical aberrations as categorized by parallel developments for rotationally symmetric optics. The first paper in this series on the higher-order terms developed the nodal properties of the spherical aberration family, including W060, W240M, and W242, and for completeness 7th-order spherical aberration W080. This second paper in the series develops and presents the intrinsic, characteristic, often multinodal properties of the family of comatic aberrations through 5th order, specifically W151, W331M, and W333 [field-linear, 5th-order aperture coma; field-cubed, 3rd-order aperture coma; and field-cubed, elliptical coma (a 3rd-order in aperture 5th-order vector aberration)]. This paper will present the first derivations of trinodal aberrations by the author.

© 2010 Optical Society of America

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Errata

Kevin P. Thompson, "Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: the comatic aberrations," J. Opt. Soc. Am. A 27, 1490-1504 (2010)
https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-27-6-1490

References

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  1. R. A. Buchroeder, “Tilted component optical systems,” Ph.D. Dissertation (University of Arizona, Optical Sciences Center, Tucson, Arizona 1976).
  2. R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope system on its aberration field,” Proc. SPIE 251, 146–153 (1980).
  3. D. Hestenes, “Oersted Medal Lecture 2002: Reforming the mathematical language of physics,” Am. J. Phys. 71, 104–121 (2003).
    [CrossRef]
  4. H. H. Hopkins, Wave Theory of Aberrations (Oxford on Clarendon Press, 1950).
  5. K. P. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A 22, 1389–1401 (2005).
    [CrossRef]
  6. K. P. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry: errata,” J. Opt. Soc. Am. A 26, 699 (2009).
    [CrossRef]
  7. K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry; spherical aberration,” J. Opt. Soc. Am. A 26, 1090–1100 (2009).
    [CrossRef]
  8. K. P. Thompson, T. Schmid, O. Cakmakci, and J. P. Rolland, “A real-ray based method for locating individual surface aberration field centers in imaging optical systems without rotational symmetry,” J. Opt. Soc. Am. A 26, 1503–1517 (2009).
    [CrossRef]
  9. J. Sasian, “The theory of sixth-order wave aberrations,” Appl. Opt. 49, D69–D95 (2010).
    [CrossRef] [PubMed]
  10. K. P. Thompson, “Aberrations fields in tilted and decentered optical systems,” Ph.D. dissertation (Optical Sciences Center, University of Arizona, Tucson, Arizona, 1980).
  11. W. R. Hamilton, Elements of Quaternions, W.E.Hamilton, ed. (Longmans, Green, & Co., 1866), available on www.Scholar.Google.com.

2010

2009

2005

2003

D. Hestenes, “Oersted Medal Lecture 2002: Reforming the mathematical language of physics,” Am. J. Phys. 71, 104–121 (2003).
[CrossRef]

1980

R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope system on its aberration field,” Proc. SPIE 251, 146–153 (1980).

K. P. Thompson, “Aberrations fields in tilted and decentered optical systems,” Ph.D. dissertation (Optical Sciences Center, University of Arizona, Tucson, Arizona, 1980).

1976

R. A. Buchroeder, “Tilted component optical systems,” Ph.D. Dissertation (University of Arizona, Optical Sciences Center, Tucson, Arizona 1976).

1950

H. H. Hopkins, Wave Theory of Aberrations (Oxford on Clarendon Press, 1950).

1866

W. R. Hamilton, Elements of Quaternions, W.E.Hamilton, ed. (Longmans, Green, & Co., 1866), available on www.Scholar.Google.com.

Buchroeder, R. A.

R. A. Buchroeder, “Tilted component optical systems,” Ph.D. Dissertation (University of Arizona, Optical Sciences Center, Tucson, Arizona 1976).

Cakmakci, O.

Hamilton, W. R.

W. R. Hamilton, Elements of Quaternions, W.E.Hamilton, ed. (Longmans, Green, & Co., 1866), available on www.Scholar.Google.com.

Hestenes, D.

D. Hestenes, “Oersted Medal Lecture 2002: Reforming the mathematical language of physics,” Am. J. Phys. 71, 104–121 (2003).
[CrossRef]

Hopkins, H. H.

H. H. Hopkins, Wave Theory of Aberrations (Oxford on Clarendon Press, 1950).

Rolland, J. P.

Sasian, J.

Schmid, T.

Shack, R. V.

R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope system on its aberration field,” Proc. SPIE 251, 146–153 (1980).

Thompson, K. P.

Am. J. Phys.

D. Hestenes, “Oersted Medal Lecture 2002: Reforming the mathematical language of physics,” Am. J. Phys. 71, 104–121 (2003).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am. A

Proc. SPIE

R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope system on its aberration field,” Proc. SPIE 251, 146–153 (1980).

Other

R. A. Buchroeder, “Tilted component optical systems,” Ph.D. Dissertation (University of Arizona, Optical Sciences Center, Tucson, Arizona 1976).

H. H. Hopkins, Wave Theory of Aberrations (Oxford on Clarendon Press, 1950).

K. P. Thompson, “Aberrations fields in tilted and decentered optical systems,” Ph.D. dissertation (Optical Sciences Center, University of Arizona, Tucson, Arizona, 1980).

W. R. Hamilton, Elements of Quaternions, W.E.Hamilton, ed. (Longmans, Green, & Co., 1866), available on www.Scholar.Google.com.

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Figures (11)

Fig. 1
Fig. 1

Constructing the transverse blur for field-linear, 3rd-order coma W 131 . The vertical line can be used to locate the lower edges of the circular zones. The circular zones in the exit pupil are drawn in increments of 0.2 in the normalized aperture.

Fig. 2
Fig. 2

Constructing the transverse blur for field-linear, 5th-order aperture coma W 151 . The vertical line can be use to locate the lower edges of the circular zones. The circular zones in the exit pupil are drawn in increments of 0.2 in the normalized aperture.

Fig. 3
Fig. 3

In a nonsymmetric optical system, the center (node) of the aberration field for field-linear, 5th-order aperture coma W 151 is displaced in the image plane to the point located by a 151 .

Fig. 4
Fig. 4

Illustrating the offset of the zero coma point for field-linear, 5th-order aperture coma in the field of view in (a) rotationally symmetric and (b) rotationally nonsymmetric optical system. (Note the plot symbol is not scaled to represent or differentiate 3rd-order from 5th-order coma.)

Fig. 5
Fig. 5

Field dependence of the transverse blur for field-linear, 5th-order aperture coma W 151 in an optical system that is not constrained to be rotationally symmetric. (a) Rotationally symmetric, a 151 = 0 . (b) Not rotationally symmetric, decentered to a 151 , where a 151 0 .

Fig. 6
Fig. 6

Fifth-order coefficient for field-cubed, 3rd-order aperture coma can balance the field-linear, 3rd-order aperture coma term, resulting in a zone in the field of view that is corrected for this aberration type, illustrated in Fig. 1.

Fig. 7
Fig. 7

In a nonsymmetric optical system, the field-cubed, 3rd-order aperture coma contribution W 331 M is (in the most common case) zero at three collinear points in the field, which are described by Eq. (10).

Fig. 8
Fig. 8

(a) Vector arrows are the effect of the elliptical coma term on an otherwise conventional comatic aberration image. (b) Composite showing the original aberration due to linear plus field-cubed coma and the ellipses resulting from the elliptical coma term in a rotationally symmetric system. (c) Resulting transverse ray aberration pattern from the combination of the two aberrations.

Fig. 9
Fig. 9

In a nonsymmetric optical system, elliptical coma W 333 develops three nodes in the field, i.e., trinodal behavior. The nodal positions are provided by the results of Appendix D.

Fig. 10
Fig. 10

Intrinsic, characteristic nodal properties of the comatic aberrations through fifth order in a nonrotationally symmetric optical system.

Fig. 11
Fig. 11

Using geometrical constructions to find the angle of the orientation of the major axis of the elliptical mapping associated with elliptical coma represented by γ. Refer to the three terms of Eq. (E5) to interpret the three steps used in determining the absolute angle.

Equations (109)

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Field-Linear , Aperture-Linear Tilt ( 1 st )
W comatic = Δ W 11 ( H ρ ) Field-Linear , 3 rd Order Aperture Coma ( 3 rd ) + W 131 [ ( H a 131 ) ρ ] ( ρ ρ ) Field-Linear , 5 th Order Aperture Coma ( 5 th ) + j W 151 j [ ( H σ j ) ρ ] ( ρ ρ ) 2 Field-Cubed , 3 rd Order Aperture Coma ( 5 th ) + j W 331 j [ ( H σ j ) ( H σ j ) ] [ ( H σ j ) ρ ] ( ρ ρ ) Field-Cubed , 3 rd Order Aperture Elliptical Coma ( 5 th ) + j W 333 j [ ( H σ j ) ρ ] 3 .
( n u ) ϵ 131 = W 131 = W 131 ( ρ ρ ) H + 2 W 131 ( H ρ ) ρ ,
( n u ) ϵ 151 = W 151 = W 151 ( ρ ρ ) 2 H + 4 W 151 ( H ρ ) ( ρ ρ ) ρ .
W = j W 151 j [ ( H σ j ) ρ ] ( ρ ρ ) 2 = { [ ( j W 151 j H ) ( j W 151 j σ j ) ] ρ } ( ρ ρ ) 2 .
j W 151 j H W 151 H .
A 151 j W 151 j σ j .
W = [ ( W 151 H A 131 ) ρ ] ( ρ ρ ) 2 .
a 151 A 151 W 151 .
W = W 151 [ ( H a 151 ) ρ ] ( ρ ρ ) 2
W = W 151 ( H 151 ρ ) ( ρ ρ ) 2 ,
W 331 M = W 331 + 3 4 W 333 .
W = W 131 [ ( H a 131 ) ρ ] ( ρ ρ ) + j W 331 Mj [ ( H σ j ) ( H σ j ) ] [ ( H σ j ) ρ ] ( ρ ρ ) .
W = [ ( W 331 M ( H H ) H 2 ( H A 331 M ) H + 2 B 331 M H ( H H ) A 331 M + B 331 M 2 H * C 331 M ) ρ ] ( ρ ρ ) ,
W 131 E W 131 + 2 W 331 M b 331 M ,
a 131 E 1 W 131 E [ W 131 a 131 + W 331 M ( c 331 M b 331 M 2 a * 331 M ) ] .
W = W 131 E [ ( H a 131 E ) ρ ] ( ρ ρ ) = W 131 E ( H 131 E ρ ) ( ρ ρ ) ,
W = W 331 M { [ ( H a 331 M ) 2 + b 331 M 2 ] ( H a 331 M ) * ρ } ( ρ ρ )
W = W 331 M { [ H 331 M 2 + b 331 M 2 ] ( H 331 M * ) ρ } ( ρ ρ ) ,
H 2 H * = H ( H H * ) = H ( H H ) = ( H H ) H = H 2 H ;
W = 1 4 W 333 ( H 3 ρ 3 ) ,
= 1 4 [ W 333 H 3 3 H 2 A 333 + 3 H B 333 2 C 333 3 ] ρ 3 .
W = 1 4 W 333 [ ( H a 333 ) 3 + 3 ( H a 333 ) b 333 2 c 333 3 ] ρ 3 ,
W = 1 4 W 333 [ H 333 3 + 3 H 333 b 333 2 c 333 3 ] ρ 3 ,
x ¯ 333 R 333 + S 333 2 ,
x ̃ 333 R 333 S 333 2 ,
R 333 = { c 333 3 2 + [ ( c 333 3 ) 2 4 + ( b 333 2 ) 3 27 ] 1 2 } 1 3 ,
S 333 = { c 333 3 2 [ ( c 333 3 ) 2 4 + ( b 333 2 ) 3 27 ] 1 2 } 1 3 .
2 x ¯ 333 ,
x ¯ 333 + i 3 x ̃ 333 ,
x ¯ 333 i 3 x ̃ 333 .
W = Δ W 20 ( ρ ρ ) + Δ W 11 ( H ρ ) + W 040 ( ρ ρ ) 2 + W 131 E [ ( H a 131 E ) ρ ] ( ρ ρ ) ̱ + W 220 M [ ( H a 220 M ) ( H a 220 M ) + b 220 M ] ( ρ ρ ) + 1 2 W 222 [ ( H a 222 ) 2 + b 222 2 ] ρ 2 + j W 311 j [ ( H σ j ) ( H σ j ) ] [ ( H σ j ) ρ ] + W 060 ( ρ ρ ) 3 + W 151 [ ( H a 151 ) ρ ] ( ρ ρ ) 2 ̱ + W 240 M [ ( H a 240 M ) ( H a 240 M ) + b 240 M ] ( ρ ρ ) 2 + 1 2 W 242 { [ ( H a 242 ) 2 + b 242 2 ] ρ 2 } ( ρ ρ ) + W 331 M { [ ( H a 331 M ) 2 + b 331 M 2 ] ( H a 331 M ) * ρ } ( ρ ρ ) ̱ + 1 4 W 333 [ ( H a 333 ) 3 + 3 ( H a 333 ) b 333 2 c 333 3 ] ρ 3 ̱ + j W 420 j [ ( H σ j ) ( H σ j ) ] [ ( H σ j ) ( H σ j ) ] ( ρ ρ ) + j W 422 j [ ( H σ j ) ( H σ j ) ] [ ( H σ j ) ρ ] 2 + j W 511 j [ ( H σ j ) ( H σ j ) ] [ ( H σ j ) ( H σ j ) ] [ ( H σ j ) ρ ] + W 080 ( ρ ρ ) 4 ,
W klm j W klm j H klm H a klm
A klm j W klm j σ j a klm A klm W klm
B klm j W klm j ( σ j σ j ) b klm B klm W klm ( a klm a klm )
B klm 2 j W klm j σ j 2 b klm 2 B klm 2 W klm a klm 2
C klm j W klm j ( σ j σ j ) σ j c klm C klm W klm ( a klm a klm ) a klm
C klm 3 j W klm j σ j 3 c klm 3 C klm 3 W klm a klm 3
W NO _ ALIGN _ SYM = Δ W 20 ( ρ ρ ) + Δ W 11 ( H ρ ) + W 040 ( ρ ρ ) 2 + W 131 [ ( H a 131 ) ρ ] ( ρ ρ ) + W 220 M [ ( H a 220 M ) ( H a 220 M ) + b 220 M ] ( ρ ρ ) + 1 2 W 222 [ ( H a 222 ) 2 + b 222 2 ] ρ 2 + j W 311 j [ ( H σ j ) ( H σ j ) ] [ ( H σ j ) ρ ] + W 060 ( ρ ρ ) 3 + j W 151 j [ ( H σ j ) ρ ] ( ρ ρ ) 2 + W 240 M [ ( H a 240 M ) ( H a 240 M ) + b 240 M ] ( ρ ρ ) 2 + 1 2 W 242 { [ ( H a 242 ) 2 + b 242 2 ] ρ 2 } ( ρ ρ ) + j W 331 j [ ( H σ j ) ( H σ j ) ] [ ( H σ j ) ρ ] ( ρ ρ ) + j W 333 j [ ( H σ j ) ρ ] 3 + j W 420 j [ ( H σ j ) ( H σ j ) ] [ ( H σ j ) ( H σ j ) ] ( ρ ρ ) + j W 422 j [ ( H σ j ) ( H σ j ) ] [ ( H σ j ) ρ ] 2 + j W 511 j [ ( H σ j ) ( H σ j ) ] [ ( H σ j ) ( H σ j ) ] [ ( H σ j ) ρ ] + W 080 ( ρ ρ ) 4 .
W = j W 151 j [ ( H σ j ) ρ ] ( ρ ρ ) 2 = j W 151 j ( H ρ ) ( ρ ρ ) 2 [ ( j W 151 j σ j ) ρ ] ( ρ ρ ) 2 = W 151 ( H ρ ) ( ρ ρ ) 2 ( A 151 ρ ) ( ρ ρ ) 2 = [ ( W 151 H A 151 ) ρ ] ( ρ ρ ) 2 ̱ .
W = j W 331 Mj [ ( H σ j ) ( H σ j ) ] [ ( H σ j ) ρ ] ( ρ ρ ) = [ j W 331 Mj ( H H ) ( H ρ ) 2 [ H ( j W 331 Mj σ j ) ] ( H ρ ) + [ j W 331 Mj ( σ j σ j ) ] ( H ρ ) ( H H ) [ ( j W 331 Mj σ j ) ρ ] + 2 j W 331 Mj ( H σ j ) ( σ j ρ ) [ j W 331 Mj ( σ j σ j ) σ j ] ρ ] ( ρ ρ ) .
( a ) 2 j W 331 Mj ( H σ j ) ( σ j ρ ) = [ j W 331 Mj ( σ j σ j ) ] ( H ρ ) + ( j W 331 Mj σ j 2 ) ( H ρ ) .
( b ) ( j W 331 Mj σ j 2 ) ( H ρ ) = [ ( j W 331 Mj σ j 2 ) H * ] ρ .
W = [ ( W 331 M ( H H ) H 2 ( H A 331 M ) H + 2 B 331 M H ( H H ) A 331 M + B 331 M 2 H * C 331 M ) ρ ] ( ρ ρ ) ̱ .
W = 1 4 j W 333 j [ ( H σ j ) 3 ρ 3 ] = 1 4 [ j W 333 j H 3 3 H 2 ( j W 333 j σ j ) + 3 H ( j W 333 j σ j 2 ) ( j W 333 j σ j 3 ) ] ρ 3 = 1 4 [ W 333 H 3 3 H 2 A 333 + 3 H B 333 2 C 333 3 ] ρ 3 ̱ .
W = Δ W 11 ( H ρ ) + [ ( W 131 H A 131 ) ρ ] ( ρ ρ ) + ( [ W 331 M ( H H ) H 2 ( H A 331 M ) H + 2 B 331 M H ( H H ) A 331 M + B 331 M 2 H * C 331 M ] ρ ) ( ρ ρ ) + 1 4 [ W 333 H 3 3 H 2 A 333 + 3 H B 333 2 C 333 3 ] ρ 3 ,
W 331 M W 331 + 3 4 W 333 .
( c ) [ ( H a ) ( H a ) ] ( H a ) = ( H H ) H 2 ( H a ) H + 2 ( a a ) H ( H H ) a + a 2 H * ( a a ) a ,
( d ) ( H a ) 3 = H 3 3 H 2 a + 3 H a 2 a 3 .
W 151 : W = W 151 [ ( H a 151 ) ρ ] ( ρ ρ ) 2 .
W = W 151 ( H 151 ρ ) ( ρ ρ ) 2 ̱ ,
H 331 M = H a 331 M , where a 331 M A 331 M W 331 M ,
W 331 M : W = { [ W 331 M ( H H ) H 2 ( H A 331 M ) H + 2 B 331 M H ( H H ) A 331 M + B 331 M 2 H * C 331 M ] ρ } ( ρ ρ ) = { [ W 331 M ( H 331 M H 331 M ) H 331 M + B 331 M 2 H * W 331 M a 331 M 2 H * + 2 B 331 M H 2 W 331 M ( a 331 M a 331 M ) H C 331 M + W 331 M ( a 331 M a 331 M ) a 331 M ] ρ } ( ρ ρ ) = { W 331 M [ ( H 331 M H 331 M ) H 331 M + b 331 M 2 H 331 M * + 2 b 331 M H ( c 331 M b 331 M 2 a 331 M * ) ] ρ } ( ρ ρ ) .
W = W 331 M { [ ( H 331 M 2 + b 331 M 2 ) H 331 M * ] ρ } ( ρ ρ ) ̱ + W 331 M { [ 2 b 331 M H ( c 331 M b 331 M 2 a * 331 M ) ] ρ } ( ρ ρ ) ̱ .
H 2 H * = H ( H H * ) = H ( H H ) = ( H H ) H = H 2 H .
W 333 : W = 1 4 [ W 333 H 3 3 H 2 A 333 + 3 H B 333 2 C 333 3 ] ρ 3 = 1 4 [ W 333 H 333 3 + 3 H B 333 2 3 W 333 H a 333 2 C 333 3 + W 333 a 333 3 ] ρ 3 = 1 4 [ W 333 H 333 3 + 3 W 333 H ( B 333 2 W 333 a 333 2 ) W 333 ( C 333 3 W 333 a 333 3 ) ] ρ 3 .
W = 1 4 W 333 [ H 333 3 + 3 H 333 b 333 2 c 333 3 ] ρ 3 ̱ .
W = W 131 [ ( H a 131 ) ρ ] ( ρ ρ ) + W 331 M { [ ( H 331 M 2 + b 331 M 2 ) H 331 M * ] ρ } ( ρ ρ ) + W 331 M { [ 2 b 331 M H ( c 331 M b 331 M 2 a 331 M * ) ] ρ } ( ρ ρ ) ,
W 131 E W 131 + 2 W 331 M b 331 M .
a 131 E 1 W 131 E [ W 131 a 131 + W 331 M ( c 331 M b 331 M 2 a 331 M * ) ] .
W = W 131 E ( H 131 E ρ ) ( ρ ρ ) + W 331 M { [ ( H 331 M 2 + b 331 M 2 ) H 331 M * ] ρ } ( ρ ρ ) ̱ .
A * = a e i α = a x i ̂ + a y j ̂ ,
A = a e + i α = a x i ̂ + a y j ̂
A B * = a b e i ( α β ) = ( a x b y a y b x ) i ̂ + ( a y b y + a x b x ) j ̂ .
( a ) 2 ( A B ) ( A C ) = ( A A ) ( B C ) + A 2 B C .
( b ) A B C = A B * C .
[ ( H n ρ n ) ( ρ ρ ) m ] = 2 m ( H n ρ n ) ( ρ ρ ) m 1 ρ + n ( ρ ρ ) m H n ( ρ n 1 ) * .
[ ( H n ρ n ) ( ρ ρ ) m ] = ( H n ρ n ) [ ( ρ ρ ) m ] + [ ( H n ρ n ) ] ( ρ ρ ) m ,
( ρ ρ ) m = [ x i ̂ + y j ̂ ] ( x 2 + y 2 ) m
= m ( x 2 + y 2 ) m 1 [ 2 x i ̂ + 2 y j ̂ ]
= 2 m ( ρ ρ ) m 1 ρ ,
( H n ρ n ) = ( H n ) x ( ρ n ) x + ( H n ) y ( ρ n ) y
= H x i ̂ + H y j ̂ = H n [ = n H n ( ρ n 1 ) * ]
= ( H 2 ) x ( 2 x y ) + ( H 2 ) y ( y 2 x 2 )
= ( H 2 ) x [ 2 y i ̂ + 2 x j ̂ ] + ( H 2 ) y [ 2 y j ̂ 2 x i ̂ ]
= 2 [ ( H 2 ) x y ( H 2 ) y x ] i ̂ + 2 [ ( H 2 ) y y + ( H 2 ) x x ] j ̂
= 2 H 2 ρ * [ = n H n ( ρ n 1 ) * ]
= ( H 3 ) x ( 3 y 2 x x 3 ) + ( H 3 ) y ( y 3 3 x 2 y )
= ( H 3 ) x [ ( 3 y 2 3 x 2 ) i ̂ + 6 x y j ̂ ] + ( H 3 ) y [ 6 x y i ̂ + 3 ( y 2 x 2 ) j ̂ ]
= 3 [ ( H 3 ) x ( ρ 2 ) y ( H 3 ) y ( ρ 2 ) x ] i ̂ + 3 [ ( H 3 ) y ( ρ 2 ) y + ( H 3 ) x ( ρ 2 ) x ] j ̂
= 3 H 3 ( ρ 2 ) * [ = n H n ( ρ n 1 ) * ] .
x 3 + a x + b = 0 ;
A = { b 2 + ( b 2 4 + a 3 27 ) 1 2 } 1 3 ,
B = { b 2 ( b 2 4 + a 3 27 ) 1 2 } 1 3 .
x = A + B , A + B 2 + A B 2 3 , A + B 2 A B 2 3 .
H 3 + 3 b 2 H c 3 = 0 .
R = { c 3 2 + [ ( c 3 ) 2 4 + ( b 2 ) 3 27 ] 1 2 } 1 3 ,
S = { c 3 2 [ ( c 3 ) 2 4 + ( b 2 ) 3 27 ] 1 2 } 1 3 .
x ¯ R + S 2 ,
x ̃ R S 2 .
H = 2 x ¯ , x ¯ + i 3 x ̃ , x ¯ i 3 x ̃ ,
W 31 , 33 = ( ρ ρ ) ( [ ] 131 + [ ] 331 M ) + 2 [ ( [ ] 131 + [ ] 331 M ) ρ ] ρ + 3 4 [ ] 333 3 ( ρ 2 ) * ,
[ ] 131 W 131 H A 131 ,
[ ] 331 M W 331 M ( H H ) H 2 ( H A 331 M ) H + 2 B 331 M H ( H H ) A 331 M + B 331 M 2 H * C 331 M ,
[ ] 3 333 W 333 H 3 3 H 2 A 333 + 3 H 333 2 C 333 3 .
W 31 , 33 = M 31 ρ 2 ( 1 ) e i θ 31 + 2 M 31 ρ 2 ( 2 ) cos ( θ 31 ϕ ) e i ϕ + 3 4 M 33 3 ρ 2 ( 3 ) e i ( 3 θ 33 2 ϕ ) ,
ρ ρ e i ϕ ,
M 31 e i θ 31 [ ] 131 + [ ] 331 M ,
M 33 3 e i 3 θ 33 [ ] 333 3 ; M 33 e i θ 33 = [ ] 333 .
γ ( ϕ ) = 3 θ 33 2 ϕ .
β θ 31 ϕ ,
2 β + ψ 180 ,
ψ + α 180 ,
γ + α = θ 31 .
γ + 2 ( θ 31 ϕ ) = θ 31 ,
γ = 2 ϕ θ 31 .
2 ϕ θ 31 = 3 θ 33 2 ϕ ,
ϕ = 1 4 [ 3 θ 33 + θ 31 ] .
γ = 1 2 ( 3 θ 33 θ 31 ) .

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