Abstract

Many authors, dating back to at least the 1950s, have presented mathematical expansions of the wave-front aberration function for optical systems without symmetry, typically based on limiting assumptions and simplifications, with some of the most recent work being done by Howard and Stone [Appl. Opt. 39, 3232 (2000) ]. This paper reveals that in fact there are no new aberrations in imaging optical systems with near-circular aperture stops but otherwise without symmetry. What does occur is that the field dependence of an aberration often changes when symmetry is abandoned. Each aberration type develops a characteristic field behavior in a system without symmetry. Specifically, for example, astigmatism, develops a binodal field dependence; e.g., there are typically two points in the field with zero astigmatism, and typically neither point is on axis. This construct, nodal aberration theory, for understanding the aberrations in systems without symmetry becomes a direct extension of an optical designer’s knowledge base. Through the use of real ray-based analysis methods, such as Zernike coefficients, it is possible to understand completely the aberrations of optical systems without symmetry in terms of rotationally symmetric aberration theory with the simple addition of the concept of field nodes.

© 2005 Optical Society of America

Full Article  |  PDF Article

Errata

Kevin Thompson, "Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry: errata," J. Opt. Soc. Am. A 26, 699-699 (2009)
https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-26-3-699

References

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  1. R. Tessieres, J. Burge, manuscript available from the authors (Jim.Burge@opt-sci.arizona.edu)..
  2. R. A. Buchroeder, “Tilted component optical systems,” Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1976).
  3. R. V. Shack (personal communication, 1978). Optical Sciences Center, University of Arizona, Tucson, Arizona 85721. Phone, 520-621-1356.
  4. H. H. Hopkins, The Wave Theory of Aberrations (Oxford on Clarendon Press, Oxford, UK, 1950).
  5. K. P. Thompson, “Reinterpreting Coddington, correcting 150 years of confusion,” in Legends in Applied Optics, R. S. Shannon and R. Shack, eds., SPIE Monograph PM148 (SPIE, to be published).
  6. K. P. Thompson, “Aberrations fields in tilted and decentered optical systems,” Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1980).
  7. H. A. Unvala, “The orthonormalization of aberrations,” in Proceedings of the Conference on Lens Design with Large Computers (Institute of Optics, University of Rochester, Rochester, New York, 1976), pp. 16-1–16-27.
  8. E. J. Radkowski, “Use of orthonormalized image errors in optical design,” M.S. thesis (Institute of Optics, University of Rochester, Rochester, New York, 1967).
  9. G. E. Wiese, “Use of physically significant merit functions in automatic lens design,” M.S. thesis (Optical Sciences Center, University of Arizona, Tucson, Arizona, 1974).
  10. M. H. Kreitzer, “Image quality criteria for aberrated systems,” Ph.D. dissertation (Optical Sciences Center, University of Arizona, Tucson, Arizona, 1976).
  11. J. R. Rogers, “Techniques and tools for obtaining symmetrical performance from tilted component systems,” Opt. Eng. (Bellingham) 39, 1776–1787 (2001).
    [CrossRef]

2001

J. R. Rogers, “Techniques and tools for obtaining symmetrical performance from tilted component systems,” Opt. Eng. (Bellingham) 39, 1776–1787 (2001).
[CrossRef]

Buchroeder, R. A.

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

Burge, J.

R. Tessieres, J. Burge, manuscript available from the authors (Jim.Burge@opt-sci.arizona.edu)..

Hopkins, H. H.

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

Kreitzer, M. H.

M. H. Kreitzer, “Image quality criteria for aberrated systems,” Ph.D. dissertation (Optical Sciences Center, University of Arizona, Tucson, Arizona, 1976).

Radkowski, E. J.

E. J. Radkowski, “Use of orthonormalized image errors in optical design,” M.S. thesis (Institute of Optics, University of Rochester, Rochester, New York, 1967).

Rogers, J. R.

J. R. Rogers, “Techniques and tools for obtaining symmetrical performance from tilted component systems,” Opt. Eng. (Bellingham) 39, 1776–1787 (2001).
[CrossRef]

Shack, R. V.

R. V. Shack (personal communication, 1978). Optical Sciences Center, University of Arizona, Tucson, Arizona 85721. Phone, 520-621-1356.

Tessieres, R.

R. Tessieres, J. Burge, manuscript available from the authors (Jim.Burge@opt-sci.arizona.edu)..

Thompson, K. P.

K. P. Thompson, “Reinterpreting Coddington, correcting 150 years of confusion,” in Legends in Applied Optics, R. S. Shannon and R. Shack, eds., SPIE Monograph PM148 (SPIE, to be published).

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

Unvala, H. A.

H. A. Unvala, “The orthonormalization of aberrations,” in Proceedings of the Conference on Lens Design with Large Computers (Institute of Optics, University of Rochester, Rochester, New York, 1976), pp. 16-1–16-27.

Wiese, G. E.

G. E. Wiese, “Use of physically significant merit functions in automatic lens design,” M.S. thesis (Optical Sciences Center, University of Arizona, Tucson, Arizona, 1974).

Opt. Eng. (Bellingham)

J. R. Rogers, “Techniques and tools for obtaining symmetrical performance from tilted component systems,” Opt. Eng. (Bellingham) 39, 1776–1787 (2001).
[CrossRef]

Other

R. Tessieres, J. Burge, manuscript available from the authors (Jim.Burge@opt-sci.arizona.edu)..

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

R. V. Shack (personal communication, 1978). Optical Sciences Center, University of Arizona, Tucson, Arizona 85721. Phone, 520-621-1356.

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

K. P. Thompson, “Reinterpreting Coddington, correcting 150 years of confusion,” in Legends in Applied Optics, R. S. Shannon and R. Shack, eds., SPIE Monograph PM148 (SPIE, to be published).

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

H. A. Unvala, “The orthonormalization of aberrations,” in Proceedings of the Conference on Lens Design with Large Computers (Institute of Optics, University of Rochester, Rochester, New York, 1976), pp. 16-1–16-27.

E. J. Radkowski, “Use of orthonormalized image errors in optical design,” M.S. thesis (Institute of Optics, University of Rochester, Rochester, New York, 1967).

G. E. Wiese, “Use of physically significant merit functions in automatic lens design,” M.S. thesis (Optical Sciences Center, University of Arizona, Tucson, Arizona, 1974).

M. H. Kreitzer, “Image quality criteria for aberrated systems,” Ph.D. dissertation (Optical Sciences Center, University of Arizona, Tucson, Arizona, 1976).

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Figures (11)

Fig. 1
Fig. 1

Conventions for field vector H and pupil vector ρ .

Fig. 2
Fig. 2

Representation of the effective field height vector H A j .

Fig. 3
Fig. 3

The characteristic field behavior of coma in an optical system without symmetry is for the node (point of zero coma) to be displaced in the image field, to the point indicated by the vector a 131 .

Fig. 4
Fig. 4

The characteristic field behavior of astigmatism in an optical system without symmetry is for two nodes to develop in the image field characterized by a displacement of the center of symmetry vector a 222 and a node-splitting vector b 222 .

Fig. 5
Fig. 5

The characteristic field behavior of field curvature in an optical system without symmetry is for the vertex of the curvature to decenter to a point located by the vector a 220 M and to defocus by an amount proportional to b 220 M

Fig. 6
Fig. 6

Reproduction of a through-focus star plate taken with the 60-in. telescope on Kitt Peak in the mid-1970s. Though not distinguishable on this scale, in the original, binodal astigmatic behavior is clearly evident.

Fig. 7
Fig. 7

Simple two-mirror Ritchey–Chretien system (the Hubble Space Telescope) used to demonstrate nodal field dependence.

Fig. 8
Fig. 8

Full field display for the as-flown Hubble Space Telescope without alignment errors of (a) third-order spherical aberration, (b) third-order coma, and (c) third-order astigmatism. A full field display plots the magnitude (the size of the plot symbol) and orientation of the aberration as a function of location in the image plane. Since this is a Ritchey–Chretien system, there is no coma (and hence a blank display). Owing to the error in the primary mirror, the system does have spherical aberration in the as-flown condition.

Fig. 9
Fig. 9

Full field displays of the Ritchey–Chretien telescope with a decentered secondary mirror for (a) third-order spherical aberration, (b) third-order coma, and (c) third-order astigmatism, illustrating, respectively, no change in spherical aberration, the introduction of constant coma over the field (because the original system is corrected for coma), and clearly apparent binodal astigmatism.

Fig. 10
Fig. 10

The characteristic field behavior of third-order distortion in an optical system without symmetry is for three nodes to develop in the image field characterized by a displacement of the center of symmetry vector a 311 , which is also a node location, and a node-splitting vector b 311 . This term needs to be combined with the first-order term, which is linear with field and radially symmetric about a node located at a 111 E .

Fig. 11
Fig. 11

Vector multiplication.

Equations (108)

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W = j p n m ( W k l m ) j H k ρ l cos m ϕ ,
k = 2 p + m , l = 2 n + m .
W = j W j .
H = H exp ( i θ ) , ρ = ρ exp ( i ϕ ) .
W = W [ ( H H ) , ( H ρ ) , ( ρ ρ ) ] = j p n m ( W k l m ) j ( H H ) p ( ρ ρ ) n ( H ρ ) m
H A j = H σ j ,
W = j p n m ( W k l m ) j ( H A j H A j ) p ( ρ ρ ) n ( H A j ρ ) m = j p n m ( W k l m ) j [ ( H σ j ) ( H σ j ) ] p ( ρ ρ ) n [ ( H σ j ) ρ ) ] m .
W = W 040 ρ 4 + W 131 H ρ 3 cos ϕ + W 222 H 2 ρ 2 cos 2 ϕ + W 220 H 2 ρ 2 + W 311 H 3 ρ cos ϕ , θ = 0 ,
W = 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 = Δ W 20 ( ρ ρ ) + Δ W 11 ( H ρ ) + j W 040 j ( ρ ρ ) 2 + j W 131 j [ ( H σ j ) ρ ] ( ρ ρ ) + j W 222 j [ ( H σ j ) ρ ] 2 + j W 220 j [ ( H σ j ) ( H σ j ) ] ( ρ ρ ) + j W 311 j [ ( H σ j ) ( H σ j ) ] [ ( H σ j ) ρ ] ,
W = j W 040 j ( ρ ρ ) 2 .
W = j W 131 j [ ( H σ j ) ρ ] ( ρ ρ ) = { [ ( j W 131 j H ) ( j W 131 j σ j ) ] ρ } ( ρ ρ ) .
j W 131 j H = W 131 H .
A 131 j W 131 j σ j .
a 131 A 131 W 131 ,
W = W 131 [ ( H a 131 ) ρ ] ( ρ ρ ) W 131 [ H 131 ρ ] ( ρ ρ ) .
W = Δ W 20 ( ρ ρ ) + j W 220 j [ ( H σ j ) ( H σ j ) ] ( ρ ρ ) + j W 222 j [ ( H σ j ) ρ ] 2 .
W 220 M = W 220 + 1 2 W 222
W = Δ W 20 ρ 2 + W 220 H 2 ρ 2 + W 222 H 2 ρ 2 ( 1 2 + 1 2 cos 2 ϕ ) = Δ W 20 ρ 2 + W 220 M H 2 ρ 2 + 1 2 W 222 H 2 ρ 2 cos 2 ϕ .
W = Δ W 20 ( ρ ρ ) + W 220 M ( H H ) ( ρ ρ ) + 1 2 W 222 ( H 2 ρ 2 ) .
W = 1 2 j W 222 j [ ( H σ j ) 2 ρ 2 ] = 1 2 [ j W 222 j H 2 2 H ( j W 222 j σ j ) + j W 222 j σ j 2 ] ρ 2 .
A 222 j W 222 j σ j ,
B 222 2 j W 222 j σ j 2 ,
a 222 A 222 W 222 ,
b 222 2 B 222 2 W 222 a 222 2 ,
W = 1 2 W 222 [ ( H a 222 ) 2 + b 222 2 ] ρ 2 = 1 2 W 222 [ H 222 2 + b 222 2 ] ρ 2 .
0 = ( H a 222 ) 2 + b 222 2
H = a 222 ± ( b 222 2 ) 1 2 .
H = a 222 ± i b 222 ,
b 222 2 = b 2 exp ( i 2 β ) ,
b 222 = b 2 1 2 exp ( i β ) = b 222 exp ( i β ) ,
± i b 222 = b 222 exp [ i ( β ± 90 ) ] .
Δ W 20 = j W 220 M j [ ( H σ j ) ( H σ j ) ] = j W 220 M j ( H H ) 2 H ( j W 220 M j σ j ) + j W 220 M j ( σ j σ j ) .
A 220 M j W 220 m j σ j ,
B 220 M j j W 220 M j ( σ j σ j ) ( a scalar ) ,
a 220 M A 220 M W 220 M ,
b 220 M B 220 M W 220 M a 220 M a 220 M ,
Δ W 20 = W 220 M [ ( H a 220 M ) ( H a 220 M ) + b 220 M ] W 220 M [ ( H 220 M H 220 M ) + b 220 M ] .
δ z 220 M = 8 ( f # ) 2 W 220 M b 220 M .
W = W 040 ρ 4 + W 131 H ρ 3 cos ϕ + W 222 H 2 ρ 2 cos 2 ϕ + W 220 H 2 ρ 2 .
W = Δ W 20 ( ρ 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 .
W = Δ W 20 ( ρ ρ ) + W 040 ( ρ ρ ) 2 + [ ( W 131 H A 131 ) ρ ] ( ρ ρ ) + [ W 220 M ( H H ) 2 ( H A 220 M ) + B 220 M ] ( ρ ρ ) + 1 2 [ W 222 H 2 2 H A 222 + B 222 2 ] ρ 2 .
[ ] 131 W 131 H A 131 ,
[ ] 220 M W 220 M ( H H ) 2 ( H A 220 M ) + B 220 M ,
[ ] 222 2 W 222 H 2 2 H A 222 + B 222 2 ,
W = Δ W 20 ( ρ ρ ) + W 040 ( ρ ρ ) 2 + ( [ ] 131 ρ ) ( ρ ρ ) + [ ] 220 M ( ρ ρ ) + 1 2 [ ] 222 2 ρ 2 .
( n u ) ϵ = i ̂ W x + j ̂ W y .
( n u ) ϵ = W ,
i ̂ x + j ̂ y .
[ ( 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 .
( n u ) ϵ = W = 2 Δ W 20 ρ + 4 W 040 ( ρ ρ ) ρ + 2 ( [ ] 131 ρ ) ρ + ( ρ ρ ) [ ] 131 + 2 [ ] 220 M ρ + [ ] 222 2 ρ * .
ω 2 = 1 π W 2 ρ d ρ d ϕ + [ 1 π W ρ d ρ d ϕ ] 2 .
ω rms = ( ω 2 ) 1 2 = { 1 12 [ Δ W 20 + W 040 + [ ] 220 M ] 2 + 1 180 W 040 2 + 1 24 [ [ ] 222 2 [ ] 222 2 ] + 1 4 [ ( Δ W 11 H + 2 3 [ ] 131 + [ ] 311 ) ( Δ W 11 H + 2 3 [ ] 131 + [ ] 311 ) ] + 1 72 [ [ ] 131 [ ] 131 ] } 1 2 ( ω rms ) p .
( w rms ) c = { 1 12 [ Δ W 20 + W 040 + W 220 M ( H H ) ] 2 + 1 180 W 040 2 + 1 24 [ W 222 H 2 W 222 H 2 ] + 1 4 [ ( Δ W 11 H + 2 3 W 131 H + W 311 ( H H ) H ) ( Δ W 11 H + 2 3 W 131 H + W 311 ( H H ) H ) ] + 1 72 [ W 131 H W 131 H ] } 1 2 .
ϵ ¯ 2 = 1 π 1 ( n u ) 2 [ ( δ W δ y ) 2 + ( δ W δ x ) 2 ] ρ d ρ d ϕ .
n u ( ϵ rms ) p = n u ( ϵ ¯ p 2 ) 1 2 = { 2 [ Δ W 20 + 4 3 W 040 + [ ] 220 M ] 2 + 4 9 W 040 2 + 1 2 [ ] 222 2 [ ] 222 2 + [ ( Δ W 11 H + [ ] 131 + [ ] 311 ) ( Δ W 11 H + [ ] 131 + [ ] 311 ) ] + 2 3 [ [ ] 131 [ ] 131 ] } 1 2 .
n u ( ϵ rms ) c = n u ( ϵ ¯ c 2 ) 1 2 = { 2 [ Δ W 20 + 4 3 W 040 + W 220 M ( H H ) ] 2 + 4 9 W c 040 2 + 1 2 [ W 222 H 2 W 222 H 2 ] + [ ( Δ W 11 H + W 131 H + W 311 ( H H ) H ) ( Δ W 11 H + W 131 H + W 311 ( H H ) H ) ] + 2 3 [ W 131 H W 131 H ] } 1 2 .
W = j W 311 j [ ( H σ j ) ( H σ j ) ] [ ( H σ j ) ρ ] = j W 311 j ( H H ) ( H ρ ) 2 ( H j W 311 j σ ) ( H ρ ) + [ j W 311 j ( σ j σ j ) ( H ρ ) ] ( H H ) [ ( j W 311 j σ j ) ρ ] + 2 j W 311 j ( H σ j ) ( σ j ρ ) [ j W 311 j ( σ j σ j ) σ j ] ρ .
2 ( A B ) ( A C ) = ( A A ) ( B C ) + A 2 B C .
2 j W 311 j ( H σ j ) ( σ j ρ ) = [ j W 311 j ( σ j σ j ) ] ( H ρ ) + ( j W 311 j σ j 2 ) H ρ .
A 2 B C = A 2 B * C
( j W 311 j σ j 2 ) H ρ = ( j W 311 j σ j 2 ) H * ρ .
A 311 j W 311 j σ j ,
B 311 j W 311 j ( σ j σ j ) ( a scalar ) ,
B 311 2 j W 311 j σ j 2 ,
C 311 j W 311 j ( σ j σ j ) σ j ,
W W 311 ( H H ) ( H ρ ) 2 ( H A 311 ) ( H ρ ) + 2 B 311 ( H ρ ) ( H H ) ( A 311 ρ ) + B 311 2 H * ρ C 311 ρ [ ] 311 ρ .
( n u ) ϵ = W = [ ] 311 .
( n u ) ϵ = Δ W 11 H + W 311 { [ ( H a 311 ) 2 + b 311 2 ] ( H a 311 ) * + [ 2 b 311 H ( c 311 b 311 2 a 311 * ) ] } ,
a 311 A 311 W 311 ,
b 311 B 311 W 311 a 311 a 311 ,
b 311 2 B 311 2 W 311 a 311 2 ,
c 311 C 311 W 311 ( a 311 a 311 ) a 311 ,
W 111 E Δ W 11 + 2 W 311 b 311 ,
a 111 E W 311 W 111 E ( c 311 b 311 2 a 311 * ) ,
H 311 H a 311 , H 111 E = H a 111 E ,
( n u ) ϵ = W 311 [ ( H 311 2 + b 311 2 ) H 311 * ] + W 111 E H 111 E .
A = a exp ( i α ) = a x i ̂ + a y j ̂ , a x = a sin α , a y = a cos α ,
B = b exp ( i β ) = b x i ̂ + b y j ̂ b x = b sin β , b y = b cos β ,
A B a b exp [ i ( α + β ) ] = ( a y b x + a x b y ) i ̂ + ( a y b y a x b x ) j ̂ = a b sin ( α + β ) i ̂ + a b cos ( α + β ) j ̂ ,
Let A = a x i ̂ + a y j ̂ = a exp ( i α ) .
Then A A = a 2 = a x 2 + a y 2 = A 2 ( a scalar ) .
A A = A 2 = ( 2 a x a y ) i ̂ + ( a y 2 a x 2 ) j ̂ = A 2 exp ( i 2 α ) ( a vector ) .
With B = b x i ̂ + b y j ̂ = b exp ( i β ) ,
A B = a x b x + a y b y = A B cos ( α β ) .
A B = ( a y b x + a x b y ) i ̂ + ( a y b y a x b x ) j ̂ = A B exp [ i ( α β ) ] .
A * = a exp ( i α ) = a x i ̂ + a y j ̂ .
A B * = a b exp [ i ( α β ) ] = ( a x b y a y b x ) i ̂ + ( a y b y + a x b x ) j ̂ .
A = a x i ̂ + a y j ̂ = a exp ( i α ) , a x = a sin α ; a y = a cos α ,
B = b x i ̂ + b y j ̂ = b exp ( i β ) , b x = b sin β ; b y = b cos β .
A A = a 2 ,
A B = a b cos ( α β ) = a x b x + a y b y .
A B = a b exp [ i ( α β ) ] = ( A B ) x i ̂ + ( A B ) y j ̂ ,
( A B ) x = a b sin ( α + β ) = a x b y + a y b x ,
( A B ) y = a b cos ( α + β ) = a y b y a x b x .
A 2 = a 2 exp ( i 2 α ) = ( A 2 ) x i ̂ + ( A 2 ) y j ̂ ,
( A 2 ) x = a 2 sin 2 α = 2 a x a y ,
( A 2 ) y = a 2 cos 2 α = a y 2 a x 2 .
( a ) 2 ( A B ) ( A C ) = ( A A ) ( B C ) + A 2 B C ,
( b ) A B C = A B * C ,
( c ) 2 ( A B ) ( A B C 2 ) = ( A A ) ( B 2 C 2 ) + ( B B ) ( A 2 C 2 ) .
[ ( 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 ,
n = 1 = H x i ̂ + H y j ̂ = H n = n H n ( ρ n 1 ) * ,
n = 2 = ( 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 ) * ,
n = 3 = ( 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 ) * .

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