Abstract

This paper presents a systematic and in-depth discussion for the aberration fields of off-axis two-mirror astronomical telescopes with an offset pupil that is induced by lateral misalignment. Based on the framework of nodal aberration theory and a system level pupil coordinate transformation, the aberration function for misaligned off-axis telescopes is derived. Some general descriptions for the misalignment-induced aberrations are presented. The specific astigmatic and coma aberration field characteristics in off-axis two-mirror telescopes are then discussed. The precision of the presented aberration expressions is demonstrated. The discrepancies between the ray tracing data and aberration expressions are explicated. Then the inherent relationships between the astigmatism and coma aberration fields are revealed and explicated. Based on this knowledge, some quantitative discussions are further presented for determining the misalignments used to compensate for the effects of primary mirror astigmatic figure errors as well as separating these two effects when coupled. Other effects of lateral misalignments are also presented, especially the field-constant focal shift, which is only sensitive to the lateral misalignments in the symmetry plane of the nominal off-axis system. A quantitative discussion is also presented which explains the reason why trefoil aberration in off-axis telescopes is more sensitive to lateral misalignments. Most of the results presented in this paper can be extended to the other off-axis astronomical telescopes with more freedoms.

© 2016 Optical Society of America

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References

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    [Crossref]
  2. M. Bartelmann and P. Schneider, “Weak gravitational lensing,” Phys. Rep. 340(4–5), 291–472 (2001).
    [Crossref]
  3. F. Zeng, X. Zhang, J. Zhang, G. Shi, and H. Wu, “Optics ellipticity performance of an unobscured off-axis space telescope,” Opt. Express 22(21), 25277–25285 (2014).
    [Crossref] [PubMed]
  4. W. Cao, N. Gorceix, and et al.., “First Light of the 1.6 meter off-axis New Solar Telescope at Big Bear Solar Observatory,” Proc. SPIE 7333, 73330 (2010).
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    [Crossref] [PubMed]
  6. M. A. Lundgren and W. L. Wolfe, “Alignment of a three-mirror off-axis telescope by reverse optimization,” Opt. Eng. 30(3), 307–311 (1991).
    [Crossref]
  7. H. Lee, G. B. Dalton, I. A. J. Tosh, and S.-W. Kim, “Computer-guided alignment II :Optical system alignment using differential wavefront sampling,” Opt. Express 15(23), 15424–15437 (2007).
    [Crossref] [PubMed]
  8. R. Upton and T. Rimmele, “Active reconstruction and alignment strategies for the Advanced Technology Solar Telescope,” Proc. SPIE 7793, 77930E (2010).
    [Crossref]
  9. R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope system,” Proc. SPIE 251, 146–153 (1980).
    [Crossref]
  10. K. P. Thompson, “Aberration fields in tilted and decentered optical systems,” Ph.D. dissertation (University of Arizona, Tucson, Arizona, 1980).
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    [Crossref] [PubMed]
  12. K. P. Thompson, “Multinodal 5th-order optical aberrations of optical systems without rotational symmetry: spherical aberration,” J. Opt. Soc. Am. A 26(5), 1090–1100 (2009).
    [Crossref] [PubMed]
  13. K. P. Thompson, “Multinodal 5th-order optical aberrations of optical systems without rotational symmetry: the comatic aberrations,” J. Opt. Soc. Am. A 27(6), 1490–1504 (2010).
    [Crossref] [PubMed]
  14. K. P. Thompson, “Multinodal 5th-order optical aberrations of optical systems without rotational symmetry: the astigmatic aberrations,” J. Opt. Soc. Am. A 28(5), 821–836 (2011).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  18. K. P. Thompson, T. Schmid, and J. P. Rolland, “The misalignment induced aberrations of TMA telescopes,” Opt. Express 16(25), 20345–20353 (2008).
    [Crossref] [PubMed]
  19. Z. Gu, C. Yan, and Y. Wang, “Alignment of a three-mirror anastigmatic telescope using nodal aberration theory,” Opt. Express 23(19), 25182–25201 (2015).
    [Crossref] [PubMed]
  20. T. Schmid, J. P. Rolland, A. Rakich, and K. P. Thompson, “Separation of the effects of astigmatic figure error from misalignments using Nodal Aberration Theory (NAT),” Opt. Express 18(16), 17433–17447 (2010).
    [Crossref] [PubMed]
  21. K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “Extending nodal aberration theory to include mount-induced aberrations with application to freeform surfaces,” Opt. Express 20(18), 20139–20155 (2012).
    [Crossref] [PubMed]
  22. K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “Theory of aberration fields for general optical systems with freeform surfaces,” Opt. Express 22(22), 26585–26606 (2014).
    [Crossref] [PubMed]
  23. G. Ju, C. Yan, Z. Gu, and H. Ma, “Computation of astigmatic and trefoil figure errors and misalignments for two-mirror telescopes using nodal-aberration theory,” Appl. Opt. 55(13), 3373–3386 (2016).
    [Crossref] [PubMed]
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    [Crossref]
  25. K. P. Thompson, T. Schmid, O. Cakmakci, and J. P. Rolland, “Real-ray-based method for locating individual surface aberration field centers in imaging optical systems without rotational symmetry,” J. Opt. Soc. Am. A 26(6), 1503–1517 (2009).
    [Crossref] [PubMed]
  26. J. Wang, B. Guo, Q. Sun, and Z. Lu, “3rd-order aberration fields of pupil decentered optical systems,” Opt. Express 20(11), 11652–11658 (2012).
    [Crossref] [PubMed]
  27. T. Yang, J. Zhu, W. Hou, and G. Jin, “Design method of freeform off-axis reflective imaging systems with a direct construction process,” Opt. Express 22(8), 9193–9205 (2014).
  28. T. Yang, J. Zhu, W. Hou, and G. Jin, “Compact freeform off-axis three-mirror imaging system based on the integration of primary and tertiary mirrors on one single surface,” Chin. Opt. Lett. 14(6), 060801 (2016).

2016 (2)

2015 (1)

2014 (3)

2012 (2)

2011 (1)

2010 (5)

2009 (2)

2008 (1)

2007 (1)

2006 (1)

E. Hansen, R. Price, and R. Hubbard, “Advanced Technology Solar Telescope Optical Design,” Proc. SPIE 6267, 62673Z (2006).
[Crossref]

2005 (1)

2001 (1)

M. Bartelmann and P. Schneider, “Weak gravitational lensing,” Phys. Rep. 340(4–5), 291–472 (2001).
[Crossref]

2000 (1)

1999 (1)

J. R. Kuhn and S. L. Hawley, “Some astronomical performance advantages of off-axis telescopes,” Publ. Astron. Soc. Pac. 111(759), 601–620 (1999).
[Crossref]

1991 (1)

M. A. Lundgren and W. L. Wolfe, “Alignment of a three-mirror off-axis telescope by reverse optimization,” Opt. Eng. 30(3), 307–311 (1991).
[Crossref]

1980 (1)

R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope system,” Proc. SPIE 251, 146–153 (1980).
[Crossref]

Bartelmann, M.

M. Bartelmann and P. Schneider, “Weak gravitational lensing,” Phys. Rep. 340(4–5), 291–472 (2001).
[Crossref]

Cakmakci, O.

Cao, W.

W. Cao, N. Gorceix, and et al.., “First Light of the 1.6 meter off-axis New Solar Telescope at Big Bear Solar Observatory,” Proc. SPIE 7333, 73330 (2010).

Dalton, G. B.

Fuerschbach, K.

Gorceix, N.

W. Cao, N. Gorceix, and et al.., “First Light of the 1.6 meter off-axis New Solar Telescope at Big Bear Solar Observatory,” Proc. SPIE 7333, 73330 (2010).

Gu, Z.

Guo, B.

Hansen, E.

E. Hansen, R. Price, and R. Hubbard, “Advanced Technology Solar Telescope Optical Design,” Proc. SPIE 6267, 62673Z (2006).
[Crossref]

Hawley, S. L.

J. R. Kuhn and S. L. Hawley, “Some astronomical performance advantages of off-axis telescopes,” Publ. Astron. Soc. Pac. 111(759), 601–620 (1999).
[Crossref]

Hou, W.

Hubbard, R.

E. Hansen, R. Price, and R. Hubbard, “Advanced Technology Solar Telescope Optical Design,” Proc. SPIE 6267, 62673Z (2006).
[Crossref]

Jin, G.

Ju, G.

Kim, S.-W.

Kuhn, J. R.

G. Moretto and J. R. Kuhn, “Optical performance of the 6.5-m off-axis New Planetary Telescope,” Appl. Opt. 39(16), 2782–2789 (2000).
[Crossref] [PubMed]

J. R. Kuhn and S. L. Hawley, “Some astronomical performance advantages of off-axis telescopes,” Publ. Astron. Soc. Pac. 111(759), 601–620 (1999).
[Crossref]

Lee, H.

Lu, Z.

Lundgren, M. A.

M. A. Lundgren and W. L. Wolfe, “Alignment of a three-mirror off-axis telescope by reverse optimization,” Opt. Eng. 30(3), 307–311 (1991).
[Crossref]

Ma, H.

Moretto, G.

Price, R.

E. Hansen, R. Price, and R. Hubbard, “Advanced Technology Solar Telescope Optical Design,” Proc. SPIE 6267, 62673Z (2006).
[Crossref]

Rakich, A.

Rimmele, T.

R. Upton and T. Rimmele, “Active reconstruction and alignment strategies for the Advanced Technology Solar Telescope,” Proc. SPIE 7793, 77930E (2010).
[Crossref]

Rolland, J. P.

Schmid, T.

Schneider, P.

M. Bartelmann and P. Schneider, “Weak gravitational lensing,” Phys. Rep. 340(4–5), 291–472 (2001).
[Crossref]

Shack, R. V.

R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope system,” Proc. SPIE 251, 146–153 (1980).
[Crossref]

Shi, G.

Sun, Q.

Thompson, K.

Thompson, K. P.

K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “Theory of aberration fields for general optical systems with freeform surfaces,” Opt. Express 22(22), 26585–26606 (2014).
[Crossref] [PubMed]

K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “Extending nodal aberration theory to include mount-induced aberrations with application to freeform surfaces,” Opt. Express 20(18), 20139–20155 (2012).
[Crossref] [PubMed]

K. P. Thompson, “Multinodal 5th-order optical aberrations of optical systems without rotational symmetry: the astigmatic aberrations,” J. Opt. Soc. Am. A 28(5), 821–836 (2011).
[Crossref] [PubMed]

T. Schmid, J. P. Rolland, A. Rakich, and K. P. Thompson, “Separation of the effects of astigmatic figure error from misalignments using Nodal Aberration Theory (NAT),” Opt. Express 18(16), 17433–17447 (2010).
[Crossref] [PubMed]

K. P. Thompson, “Multinodal 5th-order optical aberrations of optical systems without rotational symmetry: the comatic aberrations,” J. Opt. Soc. Am. A 27(6), 1490–1504 (2010).
[Crossref] [PubMed]

T. Schmid, K. P. Thompson, and J. P. Rolland, “Misalignment-induced nodal aberration fields in two-mirror astronomical telescopes,” Appl. Opt. 49(16), D131–D144 (2010).
[Crossref] [PubMed]

K. P. Thompson, “Multinodal 5th-order optical aberrations of optical systems without rotational symmetry: spherical aberration,” J. Opt. Soc. Am. A 26(5), 1090–1100 (2009).
[Crossref] [PubMed]

K. P. Thompson, T. Schmid, O. Cakmakci, and J. P. Rolland, “Real-ray-based method for locating individual surface aberration field centers in imaging optical systems without rotational symmetry,” J. Opt. Soc. Am. A 26(6), 1503–1517 (2009).
[Crossref] [PubMed]

K. P. Thompson, T. Schmid, and J. P. Rolland, “The misalignment induced aberrations of TMA telescopes,” Opt. Express 16(25), 20345–20353 (2008).
[Crossref] [PubMed]

R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope system,” Proc. SPIE 251, 146–153 (1980).
[Crossref]

Tosh, I. A. J.

Upton, R.

R. Upton and T. Rimmele, “Active reconstruction and alignment strategies for the Advanced Technology Solar Telescope,” Proc. SPIE 7793, 77930E (2010).
[Crossref]

Wang, J.

Wang, Y.

Wolfe, W. L.

M. A. Lundgren and W. L. Wolfe, “Alignment of a three-mirror off-axis telescope by reverse optimization,” Opt. Eng. 30(3), 307–311 (1991).
[Crossref]

Wu, H.

Yan, C.

Yang, T.

Zeng, F.

Zhang, J.

Zhang, X.

Zhu, J.

Appl. Opt. (3)

Chin. Opt. Lett. (1)

J. Opt. Soc. Am. A (5)

Opt. Eng. (1)

M. A. Lundgren and W. L. Wolfe, “Alignment of a three-mirror off-axis telescope by reverse optimization,” Opt. Eng. 30(3), 307–311 (1991).
[Crossref]

Opt. Express (9)

H. Lee, G. B. Dalton, I. A. J. Tosh, and S.-W. Kim, “Computer-guided alignment II :Optical system alignment using differential wavefront sampling,” Opt. Express 15(23), 15424–15437 (2007).
[Crossref] [PubMed]

F. Zeng, X. Zhang, J. Zhang, G. Shi, and H. Wu, “Optics ellipticity performance of an unobscured off-axis space telescope,” Opt. Express 22(21), 25277–25285 (2014).
[Crossref] [PubMed]

K. P. Thompson, T. Schmid, and J. P. Rolland, “The misalignment induced aberrations of TMA telescopes,” Opt. Express 16(25), 20345–20353 (2008).
[Crossref] [PubMed]

Z. Gu, C. Yan, and Y. Wang, “Alignment of a three-mirror anastigmatic telescope using nodal aberration theory,” Opt. Express 23(19), 25182–25201 (2015).
[Crossref] [PubMed]

T. Schmid, J. P. Rolland, A. Rakich, and K. P. Thompson, “Separation of the effects of astigmatic figure error from misalignments using Nodal Aberration Theory (NAT),” Opt. Express 18(16), 17433–17447 (2010).
[Crossref] [PubMed]

K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “Extending nodal aberration theory to include mount-induced aberrations with application to freeform surfaces,” Opt. Express 20(18), 20139–20155 (2012).
[Crossref] [PubMed]

K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “Theory of aberration fields for general optical systems with freeform surfaces,” Opt. Express 22(22), 26585–26606 (2014).
[Crossref] [PubMed]

J. Wang, B. Guo, Q. Sun, and Z. Lu, “3rd-order aberration fields of pupil decentered optical systems,” Opt. Express 20(11), 11652–11658 (2012).
[Crossref] [PubMed]

T. Yang, J. Zhu, W. Hou, and G. Jin, “Design method of freeform off-axis reflective imaging systems with a direct construction process,” Opt. Express 22(8), 9193–9205 (2014).

Phys. Rep. (1)

M. Bartelmann and P. Schneider, “Weak gravitational lensing,” Phys. Rep. 340(4–5), 291–472 (2001).
[Crossref]

Proc. SPIE (4)

W. Cao, N. Gorceix, and et al.., “First Light of the 1.6 meter off-axis New Solar Telescope at Big Bear Solar Observatory,” Proc. SPIE 7333, 73330 (2010).

R. Upton and T. Rimmele, “Active reconstruction and alignment strategies for the Advanced Technology Solar Telescope,” Proc. SPIE 7793, 77930E (2010).
[Crossref]

R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope system,” Proc. SPIE 251, 146–153 (1980).
[Crossref]

E. Hansen, R. Price, and R. Hubbard, “Advanced Technology Solar Telescope Optical Design,” Proc. SPIE 6267, 62673Z (2006).
[Crossref]

Publ. Astron. Soc. Pac. (1)

J. R. Kuhn and S. L. Hawley, “Some astronomical performance advantages of off-axis telescopes,” Publ. Astron. Soc. Pac. 111(759), 601–620 (1999).
[Crossref]

Other (3)

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

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

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

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Figures (12)

Fig. 1
Fig. 1 Schematic representation for the concepts of shifted aberration field center and effective aberration field height.
Fig. 2
Fig. 2 Schematic representation for two conceptual methods of obtaining an off-axis configuration from an on-axis parent system. (a) The off-axis system can be seen as an off-axis portion of a larger on-axis parent system. (b) The off-axis system is obtained by decentering the pupil of an on-axis system while the other elements of the system (the aperture size of which are infinity) stay unchanged. These two methods are essentially equivalent.
Fig. 3
Fig. 3 FFDs for coma (Z7/Z8) in the NST (a) and its parent on-axis telescope with the same aperture size (b) in the presence of the same lateral misalignments specified above. It can be seen that both of them are a combination of a field-linear component and a field-constant component. Besides, no node lies in the field of view.
Fig. 4
Fig. 4 FFDs for astigmatism (Z5/Z6) in the NST (a) and its parent on-axis telescope with the same aperture size (b) in the presence of the same lateral misalignments. We can see that in the misaligned NST telescope, the dominant astigmatism component is field-constant and no node lies in the field of view. However, in its on-axis parent system, the misalignment-induced astigmatism is bi-nodal (the locations of the two nodes are very close to each other in (b) in the presence of the specified misalignment parameters).
Fig. 5
Fig. 5 FFDs used to show the inherent relationships between the magnitude and orientation of coma (Z7/Z8) and astigmatism (Z5/Z6) in the misaligned NST. We can see that these relationships can roughly be represented by Eq. (42) and Eq. (46), respectively. Here (a) and (b) are obtained in the presence of two different sets of lateral misalignments specified before.
Fig. 6
Fig. 6 Schematic representation for the azimuthal angle of a vector.
Fig. 7
Fig. 7 FFDs used to show the compensation for the field-constant astigmatism in NST due to primary mirror astigmatic figure error by intentionally introducing lateral misalignments to the off-axis system. (a) and (b) are the coma (Z7/Z8) and astigmatism aberration fields (Z5/Z6) before and after intentionally misaligning the system. It can be seen that the field-constant astigmatism has been well compensated, leaving a relatively small amount of misalignment-induced coma.
Fig. 8
Fig. 8 FFDs for medial focal surface (Z4) in the NST (a) and its on-axis parent telescope (b) with the same lateral misalignments specified above. It can be seen that there is an additional field constant focal shift in (a) compared to the medial focal surface in (b).
Fig. 9
Fig. 9 FFDs for medial focal surface (Z4) in the misaligned NST (a) and the nominal NST (b) when the directions of the lateral misalignments are perpendicular to the symmetry plane. It can be seen that in this case no field-constant focal shift is induced.
Fig. 10
Fig. 10 FFDs for the trefoil aberration (Z10/Z11) in the NST and its parent on-axis telescope in the presence of the same lateral misalignments specified above. On the one hand, the magnitude of the trefoil in the off-axis telescope is far larger than that in its on-axis parent telescope; on the other hand, the misalignment-induced trefoil aberration is field-constant. These results indicate that in off-axis two-mirror telescopes, the effects of misalignments and trefoil deformation couple tightly with each other.
Fig. 11
Fig. 11 Conventions for the pupil vector ρ and field vector H in nodal aberration theory.
Fig. 12
Fig. 12 Layout of the NST telescope after fold mirrors removed.

Tables (9)

Tables Icon

Table 1 Verification for the Coma Aberration Field in the Misaligned NST

Tables Icon

Table 2 Verification for the Astigmatism Aberration Field in the Misaligned NST

Tables Icon

Table 3 Four Different Cases Considered in the Monte Carlo Simulations

Tables Icon

Table 4 Root Mean Square Deviations between the Introduced and Computed Values

Tables Icon

Table 5 Verification for the Medial Focal Surface in the Misaligned NST

Tables Icon

Table 6 Verification for the Medial Focal Surface in the Misaligned NST when the Misalignments are Perpendicular to the Symmetry Plane

Tables Icon

Table 7 Verification for the Trefoil Aberration in the Misaligned NST

Tables Icon

Table 8 Optical Prescription of the NST Telescope after Fold Mirrors Removed

Tables Icon

Table 9 3rd-order Aberration Coefficients for On-axis Parent Telescope of the NST

Equations (76)

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W = j p n m ( W k l m ) j ( H H ) p ( ρ ρ ) n ( H ρ ) m ,
( H ρ ) m = { t = 0 ( m 1 ) 2 a m ( t ) ( H H ) t ( ρ ρ ) t H m 2 t ρ m 2 t , if m is odd . t = 0 m 2 1 a m ( t ) ( H H ) t ( ρ ρ ) t H m 2 t ρ m 2 t + a m ( m / 2 ) 2 ( H H ) m 2 ( ρ ρ ) m 2 , if m is even .
a m ( t ) = 1 2 m 1 ( m t ) = 1 2 m 1 m ! t ! ( m t ) ! .
W = j p n m W ' k l m j [ ( H H ) p H m ] [ ρ m ( ρ ρ ) n ] ,
W ' k l m = t = 0 ( q m ) 2 1 1 + δ m 0 a m + 2 t ( t ) W k l ( m + 2 t ) , q = min { k , l } ,
W 220 ( H H ) ( ρ ρ ) + W 222 ( H ρ ) 2 = W ' 220 ( H H ) ( ρ ρ ) + W ' 222 H 2 ρ 2 , W 420 ( H H ) 2 ( ρ ρ ) + W 422 ( H H ) ( H ρ ) 2 = W ' 420 ( H H ) 2 ( ρ ρ ) + W ' 422 ( H H ) H 2 ρ 2 ,
W ' 220 = W 220 + 1 2 W 222 = W 220 M , W ' 222 = 1 2 W 222 , W ' 420 = W 420 + 1 2 W 422 = W 420 M , W ' 422 = 1 2 W 422 ,
W = n m C l m ( H ) [ ρ m ( ρ ρ ) n ] ,
C l m ( H ) = j p W ' k l m j [ ( H H ) p H m ] .
W = C 20 ρ 2 + [ C 22 , x C 22 , y ] [ ρ 2 cos 2 ϕ ρ 2 sin 2 ϕ ] + [ C 31 , x C 31 , y ] [ ρ 3 cos ϕ ρ 3 sin ϕ ] + C 40 ρ 4 + [ C 33 , x C 33 , y ] [ ρ 3 cos 3 ϕ ρ 3 sin 3 ϕ ] + [ C 42 , x C 42 , y ] [ ρ 4 cos 2 ϕ ρ 4 sin 2 ϕ ] + [ C 51 , x C 51 , y ] [ ρ 5 cos ϕ ρ 5 sin ϕ ] + C 60 ρ 6 ,
H A j = H σ j ,
W = n m C l m ( H A j ) [ ρ m ( ρ ρ ) n ] ,
C l m ( H A j ) = j p W ' k l m j [ ( H A j H A j ) p H A j m ] .
ρ = b ρ ' + s ,
W = n m C l m ( H A j ) [ ρ m ( ρ ρ ) n ] , ρ D ,
W = n m C l m ( H A j ) { ( b ρ ' + s ) m [ ( b ρ ' + s ) ( b ρ ' + s ) ] n } ,
ρ = ρ ' + s .
W = n m C l m ( H A j ) { ( ρ ' + s ) m [ ( ρ ' + s ) ( ρ ' + s ) ] n } .
C l m ( ρ + s ) m [ ( ρ + s ) ( ρ + s ) ] n = [ f = 0 m ( m f ) C l m s m f ρ f ] [ g = 0 n h = 0 n g ( n g ) ( n g h ) 2 h ( ρ ρ ) g ( s ρ ) h ( s s ) n g h ] = f = 0 m g = 0 n h = 0 n g K f g h ( s s ) n g h [ ( s * ) m f C l m ρ f ] ( s h ρ h ) ( ρ ρ ) g ,
K f g h = t = 0 q ' h 2 1 1 + δ h 0 a h + 2 t ( t ) ( m f ) ( n g t ) ( n ( g t ) h + 2 t ) 2 h + 2 t , q ' = min { 2 g + h , 2 n ( 2 g + h ) } .
2 ( A C p ) ( B C q ) = { A B C p + q + ( A B * C p q ) ( C C ) q , p > q , A B C p + q + ( A * B C q p ) ( C C ) p , p < q , A B C 2 p + ( A B ) ( C C ) p , p = q ,
W ( U ) = C 20 ( U ) ( ρ ρ ) + C 22 ( U ) ρ 2 + C 31 ( U ) ρ ( ρ ρ ) + C 40 ( U ) ( ρ ρ ) 2 ,
C 20 ( U ) = C 20 + 2 C 31 s + 4 C 40 ( s s ) , C 22 ( U ) = C 22 + C 31 s + 2 C 40 s 2 , C 31 ( U ) = C 31 + 4 C 40 s , C 40 ( U ) = C 40 ,
W ( U ) = C 20 ( U ) ( ρ ρ ) + C 22 ( U ) ρ 2 + C 31 ( U ) ρ ( ρ ρ ) + C 40 ( U ) ( ρ ρ ) 2 + C 33 ( U ) ρ 3 + C 42 ( U ) ρ 2 ( ρ ρ ) + C 51 ( U ) ρ ( ρ ρ ) 2 + C 60 ( U ) ( ρ ρ ) 3 ,
C 20 ( U ) = C 20 + 2 C 31 s + 4 C 40 ( s s ) + 3 ( C 42 s 2 ) + 6 ( s s ) ( s C 51 ) + 9 C 60 ( s s ) 2 , C 22 ( U ) = C 22 + C 31 s + 2 C 40 s 2 + 3 C 33 s * + 3 ( s s ) C 42 + 2 ( s s ) s C 51 + 2 ( s C 51 ) s 2 + 6 C 60 ( s s ) s 2 , C 31 ( U ) = C 31 + 4 C 40 s + 3 C 42 s * + 4 ( s s ) C 51 + 4 ( s C 51 ) s + s 2 C 51 * + 18 C 60 ( s s ) s , C 40 ( U ) = C 40 + 3 ( s C 51 ) + 9 C 60 ( s s ) , C 33 ( U ) = C 33 + C 42 s + C 51 s 2 + 2 C 60 s 3 , C 42 ( U ) = C 42 + 2 C 51 s + 6 C 60 s 2 , C 51 ( U ) = C 51 + 6 C 60 s , C 60 ( U ) = C 60 .
W C o m a ( U ) = C 31 ( U ) ρ ( ρ ρ ) ,
C 31 ( U ) = ( C 31 + 4 C 40 s ) = j [ W 131 j ( H σ j ) + 4 W 040 j s ] = W 131 H + 4 W 040 s A 131 ( P ) ,
W C o m a ( U ) = W 131 ( H A 131 ( P ) 4 W 040 s W 131 ) ρ ( ρ ρ ) .
| A 131 ( P ) 4 W 040 s W 131 | > > 1 ,
W A S T ( U ) = C 22 ( U ) ρ 2 ,
C 22 ( U ) = C 22 + C 31 s + 2 C 40 s 2 = j [ 1 2 W 222 j ( H σ j ) 2 + W 131 j ( H σ j ) s + 2 W 040 j s 2 ] . = 1 2 W 222 H 2 + ( W 131 s A 222 ( P ) ) H + ( 2 W 040 s 2 + 1 2 B 222 2 ( P ) A 131 ( P ) s )
A 222 ( P ) = j W 222 j σ j , B 222 2 ( P ) = j W 222 j σ j 2 ,
C 22 ( U ) = 1 2 W 222 H 2 + ( W 131 s A 222 ( P ) ) H + ( A 131 ( P ) s ) .
W 222 j ( s p h ) = i ¯ j i j W 131 j ( s p h ) , W 222 j ( a s p h ) = y ¯ j y j W 131 j ( a s p h ) ,
| A 131 ( P ) s | > > | A 222 ( P ) | .
| 1 2 W 222 H 2 + ( W 131 s A 222 ( P ) ) H | | 1 2 W 222 | + | W 131 s | + | A 222 ( P ) | < < | A 131 ( P ) s | ,
| Δ W A S T , 51 ( U ) | = | 2 ( s s ) s C 51 + 2 ( s C 51 ) s 2 | 4 | s | 3 | C 51 | ,
| Δ W A S T , 60 ( U ) | = | 6 W 060 ( s s ) s 2 | = 6 | s | 4 | W 060 | .
W A S T ( U ) = A 131 ( P ) s ρ 2 , W C o m a ( U ) = A 131 ( P ) ρ ( ρ ρ ) .
W A S T ( U ) = [ C 5 C 6 ] [ ρ 2 cos ( 2 ϕ ) ρ 2 sin ( 2 ϕ ) ] = C 5 / 6 ρ 2 , W C o m a ( U ) = [ C 7 C 8 ] [ ( 3 ρ 3 2 ρ ) cos ( ϕ ) ( 3 ρ 3 2 ρ ) sin ( ϕ ) ] = 3 C 7 / 8 ρ ( ρ ρ ) 2 C 7 / 8 ρ .
C 5 / 6 = 3 C 7 / 8 s ,
| C 5 / 6 | = 3 | s | | C 7 / 8 | ,
ξ ( C 5 / 6 ) = ξ ( s ) + ξ ( C 7 / 8 ) ,
ξ ( A ) = { tan 1 ( A y / A x ) A x > 0 , A y 0 , tan 1 ( A y / A x ) + π A x < 0 , tan 1 ( A y / A x ) + 2 π A x > 0 , A y < 0 , π / 2 A x = 0 , A y > 0 , 3 π / 2 A x = 0 , A y < 0 , undefined A x = 0 , A y = 0.
2 ϕ A S T = ξ ( C 5 / 6 ) , ϕ C o m a = ξ ( C 7 / 8 ) .
2 ϕ A S T = ϕ C o m a + ξ ( s ) ,
A 131 ( P ) s + 2 W 040 s 2 2 ( s s ) s A 151 ( P ) 2 s 2 ( s A 151 ( P ) ) + 6 W 060 ( s s ) s 2 + C 22 ( F ) = 0 ,
A 222 ( P ) = 0 ,
Δ W A S T ( U ) = A 222 ( P ) H + 1 2 B 222 2 ,
1 2 B 222 2 = 1 2 B 222 2 ( P ) A 131 ( P ) s 2 ( s s ) s A 151 ( P ) 2 s 2 ( s A 151 ( P ) ) + C 22 ( F ) .
Δ W 131 ( U ) = A 131 ,
A 131 = A 131 ( P ) + 4 ( s s ) A 151 ( P ) + 4 ( s A 151 ( P ) ) s + s 2 A 151 ( P ) * .
W M S ( U ) = C 20 ( U ) ( ρ ρ ) ( C 20 + C 20 ( T ) ) ( ρ ρ ) ,
C 20 = j [ W 020 j + W 220 M j ( H σ j ) ( H σ j ) ] ,
C 20 ( T ) = j [ 2 W 131 j ( H σ j ) s + 4 W 040 j ( s s ) ] .
W M S ( U ) = [ C 20 2 ( A 131 ( P ) s ) ] ( ρ ρ ) .
W T r e f o i l ( U ) = C 33 ( U ) ρ 3 ( C 33 + C 33 ( T ) ) ρ 3 ,
C 33 = j 1 4 W 333 j ( H σ j ) 3 ,
C 33 ( T ) = j [ 1 2 W 242 j ( H σ j ) 2 s + W 151 j ( H σ j ) s 2 + 2 W 060 j s 3 ] ,
W M T r e f o i l ( U ) = A 151 ( P ) s 2 ρ 3 ,
H = H exp ( i θ ) , ρ = ρ exp ( i φ ) .
( H ρ ) m = H m ρ m cos m δ , δ = θ φ .
exp ( i δ ) = cos δ + i sin δ ,
cos δ = exp ( i δ ) + exp ( i δ ) 2 .
cos m δ = [ exp ( i δ ) + exp ( i δ ) 2 ] m = 1 2 m t = 0 m ( m t ) [ exp ( i δ ) ] m t [ exp ( i δ ) ] t = 1 2 m t = 0 m 1 2 ( m t ) { [ exp ( i δ ) ] m 2 t + [ exp ( i δ ) ] m 2 t } = 1 2 m 1 t = 0 m 1 2 ( m t ) cos [ ( m 2 t ) δ ] ,
( H ρ ) m = 1 2 m 1 k = 0 m 1 2 ( m t ) H m ρ m cos [ ( m 2 t ) δ ] = 1 2 m 1 t = 0 m 1 2 ( m t ) H 2 t ρ 2 t { H m 2 t ρ m 2 t cos [ ( m 2 t ) δ ] } .
H 2 t = ( H H ) t , ρ 2 t = ( ρ ρ ) t ,
H m 2 t ρ m 2 t cos [ ( m 2 t ) φ ] = H m 2 t ρ m 2 t .
( H ρ ) m = t = 0 ( m 1 ) 2 1 2 m 1 ( m t ) ( H H ) t ( ρ ρ ) t H m 2 t ρ m 2 t ,
( H ρ ) m = t = 0 m 2 1 1 2 m 1 ( m t ) ( H H ) t ( ρ ρ ) t H m 2 t ρ m 2 t + a m ( m / 2 ) 2 ( H H ) m 2 ( ρ ρ ) m 2 .
A = a exp ( i α ) , B = b exp ( i β ) , C = c exp ( i γ ) ,
2 ( A C p ) ( B C q ) = 2 a b c p + q cos ( α p γ ) cos ( β q γ ) .
2 cos ( α p γ ) cos ( β q γ ) = cos [ ( α + β ) ( p + q ) γ ] + cos [ ( α + q γ ) ( β + p γ ) ] ,
2 ( A C p ) ( B C q ) = A B C p + q + A C q B C p ,
A 1 A 2 A 3 = A 1 A 2 * A 3 ,
2 ( A C p ) ( B C q ) = { A B C p + q + ( A B * C p q ) ( C C ) q , p > q , A B C p + q + ( A * B C q p ) ( C C ) p , p < q , A B C 2 p + ( A B ) ( C C ) p , p = q .

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