Abstract

In this paper, we develop sixth-order wave aberration theory of ultrawide-angle optical systems like fisheye lenses. Based on the concept and approach to develop wave aberration theory of plane-symmetric optical systems, we first derive the sixth-order intrinsic wave aberrations and the fifth-order ray aberrations; second, we present a method to calculate the pupil aberration of such kind of optical systems to develop the extrinsic aberrations; third, the relation of aperture-ray coordinates between adjacent optical surfaces is fitted with the second-order polynomial to improve the calculation accuracy of the wave aberrations of a fisheye lens with a large acceptance aperture. Finally, the resultant aberration expressions are applied to calculate the aberrations of two design examples of fisheye lenses; the calculation results are compared with the ray-tracing ones with Zemax software to validate the aberration expressions.

© 2017 Optical Society of America

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References

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  1. J. Sasian, “How to approach the design of a bilateral symmetric optical system,” Opt. Eng. 33, 2045–2061 (1994).
    [Crossref]
  2. J. Sasian, “Imagery of the bilateral symmetric optical system,” Ph.D. dissertation (University of Arizona, 1988).
  3. J. Sasian, “Optical design of reflective wide-field cameras,” Proc. SPIE 7060, 70600C (2008).
    [Crossref]
  4. J. Sasian, “Design of a Schwarzschild flat-field, anastigmatic, unobstructed, wide-field telescope,” Opt. Eng. 29, 1–5 (1990).
    [Crossref]
  5. J. Sasian, “Review of methods for the design of unsymmetrical optical systems,” Proc. SPIE 1396, 453–466 (1990).
    [Crossref]
  6. L. Lu, “Aberration theory of plane-symmetric grating systems,” J. Synchrotron Radiat. 15, 399–410 (2008).
    [Crossref]
  7. K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: the astigmatic aberration,” J. Opt. Soc. Am. A 26, 1090–1100 (2009).
    [Crossref]
  8. K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: the comatic aberration,” J. Opt. Soc. Am. A 27, 1490–1504 (2010).
    [Crossref]
  9. K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: spherical aberration,” J. Opt. Soc. Am. A 28, 821–836 (2011).
    [Crossref]
  10. H. G. Beutler, “The theory of the concave grating,” J. Opt. Soc. Am. 35, 311–350 (1945).
    [Crossref]
  11. H. Noda, T. Namioka, and M. Seya, “Geometric theory of the grating,” J. Opt. Soc. Am. 64, 1031–1036 (1974).
    [Crossref]
  12. T. Nomioka, M. Koike, and D. Content, “Geometric theory of the ellipsoidal grating,” Appl. Opt. 33, 7261–7274 (1994).
    [Crossref]
  13. S. Masui and T. Namioka, “Geometric aberration theory of double-element optical systems,” J. Opt. Soc. Am. A 16, 2253–2268 (1999).
    [Crossref]
  14. K. Goto and T. Kurosaki, “Canonical formulation for the geometrical optics of concave gratings,” J. Opt. Soc. Am. A 10, 452–465 (1993).
    [Crossref]
  15. C. Palmer, W. Mckinney, and B. Wheeler, “Imaging equations for spectroscopic systems using Lie transformations, I. Theoretical foundations,” Proc. SPIE 3450, 55–66 (1998).
    [Crossref]
  16. C. Palmer, B. Wheeler, and W. Mckinney, “Imaging equations for spectroscopic systems using Lie transformations, II. Multielement systems,” Proc. SPIE 3450, 67–77 (1998).
    [Crossref]
  17. M. P. Chrisp, “Aberrations of holographic toroidal grating systems,” Appl. Opt. 22, 1508–1518 (1983).
    [Crossref]
  18. L. Lu, X. Hu, and C. Sheng, “Optimization method for ultra-wide-angle and panoramic optical systems,” Appl. Opt. 51, 3776–3786 (2012).
    [Crossref]
  19. M. J. Kidger, Intermediate Optical Design (SPIE, 2004).
  20. H. A. Buchdahl, Optical Aberration Coefficients (Oxford University, 1954).
  21. J. Hoffman, “Induced aberrations in optical systems,” Ph.D. dissertation (University of Arizona, 1993).
  22. J. Sasian, “Theory of sixth-order wave aberrations,” Appl. Opt. 49, D69–D95 (2010).
    [Crossref]
  23. J. Sasian, Introduction to Aberrations in Optical Imaging Systems (Cambridge, 2013).
  24. C. G. Wynne, “Primary aberrations and conjugate change,” Proc. Phys. Soc. B 65, 429–437 (1952).
    [Crossref]
  25. H. H. Hopkins, “The development of image evaluation methods,” Proc. SPIE 46, 2–18 (1974).
    [Crossref]
  26. R. Shack, Course “OPTI 518—introduction to aberrations,” class notes (College of Optical Science, University of Arizona, 1995).
  27. J. Sasian, “Extrinsic aberrations in optical imaging systems,” Adv. Opt. Technol. 2, 75–80 (2013).
    [Crossref]
  28. L. Lu and Z. Deng, “Geometric characteristics of aberrations of plane-symmetric optical systems,” Appl. Opt. 48, 6946–6960 (2009).
    [Crossref]
  29. Y. Cao and L. Lu, “Aberrations of soft x-ray and vacuum ultraviolet optical systems with orthogonal arrangement of elements,” J. Opt. Soc. Am. A 34, 299–307 (2017).
    [Crossref]
  30. X. Fang, L. Lu, and Z. Niu, “Optimization of fisheye lens systems with adaptive and normalized real-coded genetic algorithm,” J. Optoelectron. Laser 26, 655–661 (2015).
  31. K. K. Nippon Kogaku, “Wide angle fish eye lens,” U.K. patent1,337,755 (21 November 1973).
  32. Zemax User’s Guide (Zemax Development Corporation, 2007).

2017 (1)

2015 (1)

X. Fang, L. Lu, and Z. Niu, “Optimization of fisheye lens systems with adaptive and normalized real-coded genetic algorithm,” J. Optoelectron. Laser 26, 655–661 (2015).

2013 (1)

J. Sasian, “Extrinsic aberrations in optical imaging systems,” Adv. Opt. Technol. 2, 75–80 (2013).
[Crossref]

2012 (1)

2011 (1)

2010 (2)

2009 (2)

2008 (2)

L. Lu, “Aberration theory of plane-symmetric grating systems,” J. Synchrotron Radiat. 15, 399–410 (2008).
[Crossref]

J. Sasian, “Optical design of reflective wide-field cameras,” Proc. SPIE 7060, 70600C (2008).
[Crossref]

1999 (1)

1998 (2)

C. Palmer, W. Mckinney, and B. Wheeler, “Imaging equations for spectroscopic systems using Lie transformations, I. Theoretical foundations,” Proc. SPIE 3450, 55–66 (1998).
[Crossref]

C. Palmer, B. Wheeler, and W. Mckinney, “Imaging equations for spectroscopic systems using Lie transformations, II. Multielement systems,” Proc. SPIE 3450, 67–77 (1998).
[Crossref]

1994 (2)

J. Sasian, “How to approach the design of a bilateral symmetric optical system,” Opt. Eng. 33, 2045–2061 (1994).
[Crossref]

T. Nomioka, M. Koike, and D. Content, “Geometric theory of the ellipsoidal grating,” Appl. Opt. 33, 7261–7274 (1994).
[Crossref]

1993 (1)

1990 (2)

J. Sasian, “Design of a Schwarzschild flat-field, anastigmatic, unobstructed, wide-field telescope,” Opt. Eng. 29, 1–5 (1990).
[Crossref]

J. Sasian, “Review of methods for the design of unsymmetrical optical systems,” Proc. SPIE 1396, 453–466 (1990).
[Crossref]

1983 (1)

1974 (2)

H. Noda, T. Namioka, and M. Seya, “Geometric theory of the grating,” J. Opt. Soc. Am. 64, 1031–1036 (1974).
[Crossref]

H. H. Hopkins, “The development of image evaluation methods,” Proc. SPIE 46, 2–18 (1974).
[Crossref]

1952 (1)

C. G. Wynne, “Primary aberrations and conjugate change,” Proc. Phys. Soc. B 65, 429–437 (1952).
[Crossref]

1945 (1)

Beutler, H. G.

Buchdahl, H. A.

H. A. Buchdahl, Optical Aberration Coefficients (Oxford University, 1954).

Cao, Y.

Chrisp, M. P.

Content, D.

Deng, Z.

Fang, X.

X. Fang, L. Lu, and Z. Niu, “Optimization of fisheye lens systems with adaptive and normalized real-coded genetic algorithm,” J. Optoelectron. Laser 26, 655–661 (2015).

Goto, K.

Hoffman, J.

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

Hopkins, H. H.

H. H. Hopkins, “The development of image evaluation methods,” Proc. SPIE 46, 2–18 (1974).
[Crossref]

Hu, X.

Kidger, M. J.

M. J. Kidger, Intermediate Optical Design (SPIE, 2004).

Koike, M.

Kurosaki, T.

Lu, L.

Masui, S.

Mckinney, W.

C. Palmer, W. Mckinney, and B. Wheeler, “Imaging equations for spectroscopic systems using Lie transformations, I. Theoretical foundations,” Proc. SPIE 3450, 55–66 (1998).
[Crossref]

C. Palmer, B. Wheeler, and W. Mckinney, “Imaging equations for spectroscopic systems using Lie transformations, II. Multielement systems,” Proc. SPIE 3450, 67–77 (1998).
[Crossref]

Namioka, T.

Nippon Kogaku, K. K.

K. K. Nippon Kogaku, “Wide angle fish eye lens,” U.K. patent1,337,755 (21 November 1973).

Niu, Z.

X. Fang, L. Lu, and Z. Niu, “Optimization of fisheye lens systems with adaptive and normalized real-coded genetic algorithm,” J. Optoelectron. Laser 26, 655–661 (2015).

Noda, H.

Nomioka, T.

Palmer, C.

C. Palmer, B. Wheeler, and W. Mckinney, “Imaging equations for spectroscopic systems using Lie transformations, II. Multielement systems,” Proc. SPIE 3450, 67–77 (1998).
[Crossref]

C. Palmer, W. Mckinney, and B. Wheeler, “Imaging equations for spectroscopic systems using Lie transformations, I. Theoretical foundations,” Proc. SPIE 3450, 55–66 (1998).
[Crossref]

Sasian, J.

J. Sasian, “Extrinsic aberrations in optical imaging systems,” Adv. Opt. Technol. 2, 75–80 (2013).
[Crossref]

J. Sasian, “Theory of sixth-order wave aberrations,” Appl. Opt. 49, D69–D95 (2010).
[Crossref]

J. Sasian, “Optical design of reflective wide-field cameras,” Proc. SPIE 7060, 70600C (2008).
[Crossref]

J. Sasian, “How to approach the design of a bilateral symmetric optical system,” Opt. Eng. 33, 2045–2061 (1994).
[Crossref]

J. Sasian, “Design of a Schwarzschild flat-field, anastigmatic, unobstructed, wide-field telescope,” Opt. Eng. 29, 1–5 (1990).
[Crossref]

J. Sasian, “Review of methods for the design of unsymmetrical optical systems,” Proc. SPIE 1396, 453–466 (1990).
[Crossref]

J. Sasian, “Imagery of the bilateral symmetric optical system,” Ph.D. dissertation (University of Arizona, 1988).

J. Sasian, Introduction to Aberrations in Optical Imaging Systems (Cambridge, 2013).

Seya, M.

Shack, R.

R. Shack, Course “OPTI 518—introduction to aberrations,” class notes (College of Optical Science, University of Arizona, 1995).

Sheng, C.

Thompson, K. P.

Wheeler, B.

C. Palmer, B. Wheeler, and W. Mckinney, “Imaging equations for spectroscopic systems using Lie transformations, II. Multielement systems,” Proc. SPIE 3450, 67–77 (1998).
[Crossref]

C. Palmer, W. Mckinney, and B. Wheeler, “Imaging equations for spectroscopic systems using Lie transformations, I. Theoretical foundations,” Proc. SPIE 3450, 55–66 (1998).
[Crossref]

Wynne, C. G.

C. G. Wynne, “Primary aberrations and conjugate change,” Proc. Phys. Soc. B 65, 429–437 (1952).
[Crossref]

Adv. Opt. Technol. (1)

J. Sasian, “Extrinsic aberrations in optical imaging systems,” Adv. Opt. Technol. 2, 75–80 (2013).
[Crossref]

Appl. Opt. (5)

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (6)

J. Optoelectron. Laser (1)

X. Fang, L. Lu, and Z. Niu, “Optimization of fisheye lens systems with adaptive and normalized real-coded genetic algorithm,” J. Optoelectron. Laser 26, 655–661 (2015).

J. Synchrotron Radiat. (1)

L. Lu, “Aberration theory of plane-symmetric grating systems,” J. Synchrotron Radiat. 15, 399–410 (2008).
[Crossref]

Opt. Eng. (2)

J. Sasian, “How to approach the design of a bilateral symmetric optical system,” Opt. Eng. 33, 2045–2061 (1994).
[Crossref]

J. Sasian, “Design of a Schwarzschild flat-field, anastigmatic, unobstructed, wide-field telescope,” Opt. Eng. 29, 1–5 (1990).
[Crossref]

Proc. Phys. Soc. B (1)

C. G. Wynne, “Primary aberrations and conjugate change,” Proc. Phys. Soc. B 65, 429–437 (1952).
[Crossref]

Proc. SPIE (5)

H. H. Hopkins, “The development of image evaluation methods,” Proc. SPIE 46, 2–18 (1974).
[Crossref]

J. Sasian, “Review of methods for the design of unsymmetrical optical systems,” Proc. SPIE 1396, 453–466 (1990).
[Crossref]

C. Palmer, W. Mckinney, and B. Wheeler, “Imaging equations for spectroscopic systems using Lie transformations, I. Theoretical foundations,” Proc. SPIE 3450, 55–66 (1998).
[Crossref]

C. Palmer, B. Wheeler, and W. Mckinney, “Imaging equations for spectroscopic systems using Lie transformations, II. Multielement systems,” Proc. SPIE 3450, 67–77 (1998).
[Crossref]

J. Sasian, “Optical design of reflective wide-field cameras,” Proc. SPIE 7060, 70600C (2008).
[Crossref]

Other (8)

M. J. Kidger, Intermediate Optical Design (SPIE, 2004).

H. A. Buchdahl, Optical Aberration Coefficients (Oxford University, 1954).

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

J. Sasian, Introduction to Aberrations in Optical Imaging Systems (Cambridge, 2013).

J. Sasian, “Imagery of the bilateral symmetric optical system,” Ph.D. dissertation (University of Arizona, 1988).

R. Shack, Course “OPTI 518—introduction to aberrations,” class notes (College of Optical Science, University of Arizona, 1995).

K. K. Nippon Kogaku, “Wide angle fish eye lens,” U.K. patent1,337,755 (21 November 1973).

Zemax User’s Guide (Zemax Development Corporation, 2007).

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Figures (13)

Fig. 1.
Fig. 1.

Optical scheme of a plane-symmetric optical system, its coordinate systems, and its foci on the meridional and sagittal planes.

Fig. 2.
Fig. 2.

Toroidal reference wavefront and its foci on the meridional and sagittal planes.

Fig. 3.
Fig. 3.

Ray paths through a plane-symmetric optical system, the wavefronts on the entrance and exit pupils, and their corresponding foci.

Fig. 4.
Fig. 4.

Optical scheme of the chief ray through an ultrawide-angle optical system of two elements.

Fig. 5.
Fig. 5.

Ray diffracted from point P on optical surface OE intersects point B on image plane ΣI.Σ0, exit wavefront surface; M, S, meridional and sagittal foci, respectively; θx, θy, deviation angular of the ray from the reference ray on the meridional and sagittal planes, respectively; (a) on the meridional plane and (b) on the sagittal plane.

Fig. 6.
Fig. 6.

Chief ray passes the last optical surface and intersects the image plane.

Fig. 7.
Fig. 7.

Toroidal reference exit wavefront and the reference ray QB1S intersect the image plane of any position at point B1(Δx,Δy).

Fig. 8.
Fig. 8.

Optical scheme of an optical system of g optical surfaces. Pg(xg,yg) is the reference aperture-ray coordinates on Mg, and Pg1(xg1,yg1) and Pg1(xg1,yg1) mean the aperture-ray coordinates on Mg1 with and without the extrinsic aberration, respectively.

Fig. 9.
Fig. 9.

Coordinate systems of the wavefront surface and the image plane in the reverse optical path of a double-element optical system.

Fig. 10.
Fig. 10.

Optical scheme of an aperture ray P0P1P2 passing through a double-element optical system.

Fig. 11.
Fig. 11.

Optical scheme of fisheye lens I.

Fig. 12.
Fig. 12.

Optical scheme of fisheye lens II.

Fig. 13.
Fig. 13.

SDs (I) and (II) correspond to the images of fisheye lenses I and II, respectively. (a) Obtained with the ray tracing program Zemax; the sixth-order WA theory and the fourth-order WA theory are calculated in (b) and (c), respectively. The field angles are shown on the right side of each row.

Tables (5)

Tables Icon

Table 1. Extrinsic WA Coefficients of a Double-Element Optical System

Tables Icon

Table 2. Modification of WA Coefficients due to the Second-Order Accuracy of the Aperture Ray of an Optical System of g Elements

Tables Icon

Table 3. Optical Parameters of Fisheye Lens I

Tables Icon

Table 4. Optical Parameters of Fisheye Lens II

Tables Icon

Table 5. RMS Values of the SDs in Fig. 13(a), 13(b), and 13(c) (μm)

Equations (103)

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ym=(rsrm)rsy0+(rsrm)2rs3y03+3(rsrm)8rs5y05,zm=rm,
xs=(rmrs)rmx0+(rmrs)2rm2rsx0y02+(rm2+rmrs2rs2)8rm3rs3x0y04,zs=rs+(rmrs)2rm2x02+(rmrs)8rm4x04+(rmrs)2rm3rsx02y02+(rmrs)16rm6x06+(rmrs)4rm5rsx04y02+rm2+2rmrs3rs28rm4rs3x02y04.
W=QQr=Q+QQ+Qr=Q+QP¯+P¯.
W=S0P¯S0+S0PyS0MPyM+MPM.
z=i=0j=0ci,jxiyj,c0,0=c1,0=0,j=even.
z=c2,0x2+c0,2y2+c3,0x3+c1,2xy2+c4,0x4+c0,4y4+c2,2x2y2+c5,0x5+c1,4xy4+c3,2x3y2+c6,0x6+c0,6y6+c2,4x2y4+c4,2x4y2.
c2,0=12R,c0,2=12ρ,c3,0=0,c1,2=0,c4,0=18R3,c0,4=18ρ3,c2,2=14R2ρ,c5,0=0,c1,4=0,c3,2=0,c6,0=116R5,c0,6=116ρ5,c2,4=R+2ρ16R3ρ3,c4,2=316R4ρ,
Wobj=S0P¯+S0PyMPy+MP,Wima=S0P¯+S0PyMPy+MP,
W=Wobj+Wima.
x0=i=06j=06aijxiyj,y0=i=06j=06bijxiyj.
Wobj=ij6Mij0xiyj(i+j6),
Wima=ij6Mij0xiyj(i+j6).
W=ij6Wij0xiyj(i+j6),
Wij0=n0Mij0(α,rm,rs)+n1Mij0(β,rm,rs)n1wij0,
wij0=n0/1Mij0(α,rm,rs)+Mij0(β,rm,rs),
sinαi+1=Ri+1+diRiρi+1sinωi+ρiρi+1sinβi,
ωi=ωi1+βiαi,
βi+1=sin1(nini+1sinαi+1),
2c2,0(n0cosα+n1cosβ)(n0cos2αrm+n1cos2βrm)=0,2c0,2(n0cosα+n1cosβ)(n0rs+n1rs)=0.
rm(i+1)=di¯rm(i),rs(i+1)=di¯rs(i),
di¯=ρisin(ωi1αi)ρi+1sin(ωiαi+1)sinωi.
W=n1(w300x3+w120xy2+w400x4+w220x2y2+w040y4+w500x5+w140xy4+w320x3y2+w600x6+w240x2y4+w420x4y2+w060y6).
W=k=1gW(k)=k=1gij6nkwij0(k)xkiykj(i+j6),
xk=Akxk+1,yk=Bkyk+1,
W=ngij6w¯ij0xgiygj(i+j6),
w¯ij0=k=1g1nk/gwij0(k)Ak|giBk|gj+wij0(g),
Ak|g=xkxg=rm(k)rm(k+1)rm(g1)cosαk+1cosαk+2cosαgrm(k+1)rm(k+2)rm(g)cosβkcosβk+1cosβg1,
Bk|g=ykyg=(1)gk·rs(k)rs(k+1)rs(g1)rs(k+1)rs(k+2)rs(g).
θx=1n·dWdx0,θy=1n·dWdy0,
x0=i=06j=06Aijxiyj,y0=i=06j=06Bijxiyj(i+j6).
x¯=d200x2+d020y2+d300x3+d120xy2+d400x4+d220x2y2+d040y4+d500x5+d320x3y2+d140xy4,y¯=h110xy+h210x2y+h030y3+h310x3y+h130xy3+h410x4y+h230x2y3+h050y5.
x=x¯+Δx,y=y¯+Δy.
Δx=d100x0+d300x03+d120x0y02+d500x05+d320x03y02+d140x0y04,Δy=h010y0+h210x02y0+h030y03+h410x04y0+h230x02y03+h050y05,
d100=Λm,d300=Λm2rm2,d120=Λm2rmrs,d500=3Λm8rm4,d320=3Λm4rm3rs,d140=Λm4rmrs2(1rm+12rs),h010=Λs,h210=Λm2rmrs,h030=Λs2rs2,h410=3Λm8rm3rs,h050=3Λs8rs4,h230=Λm2rmrs2(1rm+12rs),
x0xcosβ,y0y.
Δxg1=f1(xg,yg,wij0(B)),Δyg1=f2(xg,yg,wij0(B)).
W(T)=i,j6(wij0(A)(xg1+Δxg1)i(yg1+Δyg1)j+wij0(B)xgiygj).
W(T)=i,j6wij0(T)xgiygj,
wij0(T)=wij0(A)+wij0(B)+wij0(E),
Δx2*=d¯1θx(2)=d¯1dW(2)dx0(2),Δy2*=d¯1θy(2)=d¯1dW(2)dy0(2).
Δx2*=d¯200(2)x22+d¯020(2)y22+d¯300(2)x23+d¯120(2)x2y22+d¯400(2)x24+d¯220(2)x22y22+d¯040(2)y24+d¯500(2)x25+d¯320(2)x23y22+d¯140(2)x2y24,Δy2*=h¯110(2)x2y2+h¯210(2)x22y2+h¯030(2)y23+h¯310(2)x23y2+h¯130(2)x2y23+h¯410(2)x24y2+h¯230(2)x22y23+h¯050(2)y25,
d¯ij0(2)=(i+1)d¯1w(i+1)j0(2)cosα2,h¯ij0(2)=(j+1)d¯1wi(j+1)0(2).
Δx1=Δx2*cosβ1,Δy1=Δy2*.
x1=P10(1)x1+P20(1)x12+P02(1)y12,y1=T01(1)y1+T11(1)x1y1,
P10(1)=Λm(1)cosβ1,P20(1)=Λm(1)(cosβ1rm1c2,0(1))sinβ1,P02(1)=Λm(1)c0,2(1)sinβ1,T01(1)=Λs(1),T11(1)=Λs(1)sinβ1rs1
x1=i+j=02a¯ijx2iy2j,y1=i+j=02b¯ijx2iy2j.
x¯1=x1cosβ1+d¯1sinβ1,y¯1=y1,z¯1=x1sinβ1+d¯1cosβ1,
x¯2=x2cos(α2β1)z2sin(α2β1)+d1¯sinβ1,y¯2=y2,z¯2=x2sin(α2β1)+z2cos(α2β1)+d1¯cosβ1,
x1x¯1x1x¯2=y1y¯1y1y¯2=z1z¯1z1z¯2,
x1=A1x2+A1ΓR(1)x22+A1Γρ(1)y22,y1=B1y2+B1Γ12(1)x2y2,
ΓR(k)=Akc2,0(k)tanβkc2,0(k+1)tanαk+1+sin(αk+1βk)rm(k+1)cosβk,
Γρ(k)=Bk2c0,2(k)tanβkAkc0,2(k+1)tanαk+1,
Γ12(k)=sinαk+1rs(k+1)Aksinβkrs(k).
W1=w120(1)(x1+Δx1)(y1+Δy1)2+w300(1)(x1+Δx1)3+w400(1)(x1+Δx1)4+w220(1)(x1+Δx1)2(y1+Δy1)2+w040(1)(y1+Δy1)4+w500(1)(x1+Δx1)5+w140(1)(x1+Δx1)(y1+Δy1)4+w320(1)(x1+Δx1)3(y1+Δy1)2+w600(1)(x1+Δx1)6+w240(1)(x1+Δx1)2(y1+Δy1)4+w420(1)(x1+Δx1)4(y1+Δy1)2+w060(1)(y1+Δy1)6.
ϕ1=d¯1n1A1cosβ1cosα2,
ϕ2=d¯1n1B1.
w˜ij0=k=1g1Ak|g1iBk|g1jwij0(k).
ϕ˜1=d¯g1ng1Ag1cosβg1cosαg,
ϕ˜2=d¯g1ng1Bg1.
xk=f1(xg,yg),yk=f2(xg,yg)(k=1,,g1).
x1=A1A2x3+A1A2(A2ΓR(1)+ΓR(2))x32+A1(B22Γρ(1)+A2Γρ(2))y32,x2=A2x3+A2ΓR(2)x32+A3Γρ(3)y32;
W300(12)=n1w300(1)x13+n2w300(2)x23,
w400(12)(S)=3A13A23(A2ΓR(1)+ΓR(2))w300(1)+3A23ΓR(2)w300(2).
W(1g1)=k=1g1ij6nkwij0(k)xkiykj,
wij0Ψn¯=l=1g1Al|giBl|gjΨn(l)wij0(l)(n=1,2,3),
Ψ1(i)=l=ig1Al+1|gΓR(l),
Ψ2(i)=l=ig1Bl+1|g2Al+1|gΓρ(l),
Ψ3(i)=l=ig1Al+1|gΓ12(l).
wij0(T)=w¯ij0+wij0(E)+wij0(S).
M100=sinα,
M200=cos2α2rmc2,0cosα,
M020=12rsc0,2cosα,
M120=sinα2rs2c1,2*cosα,
M300=sinαcos2α2rm2c3,0*cosα,
M400=cos4α8rm3+cos3α2rm2(sinαtanαrm+c2,0)c3,0*sinαcosαrm+c4,0*cosα,
M220=cosα2rs2(sinαtanαrs+c2,0)+cos2α2rm(c0,2cosαrm12rs2)c1,2*sinαcosαrm+c2,2*cosα,
M040=18rs3+c0,2cosα2rs2+c0,4*cosα,
M500=sinαcos2α(37sin2α)8rm4+c2,0sinαcos2αrm2×(cosαrmc2,0)+c3,0*(13sin2α)cosα2rm2+c4,0*sinαcosαrm+c5,0*cosα,
M140=3sinα8rs4+c0,2sinαcosα(1rs3c0,2cosαrm2)+c0,4*sinαcosαrm+c1,2*cosα2rs2+c1,4*cosα,
M320=sinαcos2α2rs2(1rmrs+12rm2tan2αrs2)+sinαcosα×(c2,0rs32c2,0c0,2cosαrm2+c0,2cos2αrm3)+c1,2*cosα2rm2×(13sin2α)+c3,0*cosα2rs2+c2,2*sinαcosαrm+c3,2*cosα,
M600=cos2α(21cos4α28cos2α+8)16rm53c2,0cos3α8rm4×(15sin2α)+c2,02cosα2rm2(cosαcos2αrm+c2,0sin2α)c3,0*sin2αrm2(c2,0cosα(35sin2α)4rm)c4,0*cosα2rm2×(13sin2α)+c5,0*sinαcosαrm+c6,0*cosα,
M420=sin2α2rs4(3cos2α2rmsin2αrs)+cos2α8rm2rs2(15sin2α)(12rm+1rs)+c0,2c2,0cos2αcos2αrm33c0,2cos3α8rm4×(15sin2α)+3c2,02c0,2sin2αcosα2rm2c2,0cosα4rm2rs4×(2rmrscos2α+rs2cos2α6rm2sin2α)+c1,2*sinαcosα(35sin2α)2rm32sinαcos2αrm2×(c1,2*c2,0+c0,2c3,0*)+c2,02cos2α2rs3c2,2*cosα2rm2×(13sin2α)+c3,0*sinαcosαrs2(12rm+1rs)c4,0*cosα2rs2+c3,2*sinαcosαrm+c4,2*cosα,
M240=3(4rmsin2αrscos2α)16rmrs5c0,2cosα2rs2(cos2α2rm2+cos2αrmrs3sin2αrs2)+c0,22cos2αcos2α2rm3+c0,2c2,0cos2αrs33c2,0cosα8rs4+3c0,22c2,0sin2αcosα2rm2c0,4*cosα(13sin2α)2rm2+c1,2*sinαcosαrs2(12rm+1rs)2c0,2c1,2*sinαcos2αrm2+c1,4*sinαcosαrmc2,2*cosα2rs2+c2,4*cosα,
M060=116rs53c0,2cosα8rs4+c0,22cos2α2rs3+c0,23sin2αcosα2rm2c0,4*cosα2rs2+c0,6*cosα,
c3,0*=c2,0sinαrm+c3,0,c1,2*=c0,2sinαrm+c1,2,c4,0*=c2,02sinαtanα2rmc4,0,c2,2*=c2,0c0,2sinαtanαrmc2,2,c0,4*=c0,22sinαtanα2rmc0,4,c5,0*=c2,0c3,0sinαtanαrmc5,0,c1,4*=c0,2c1,2sinαtanαrmc1,4,c6,0*=(c2,0c4,0+c3,022)sinαtanαrmc6,0,c0,6*=c0,2c0,4sinαtanαrmc0,6,c2,4*=(c1,222+c2,0c0,4+c0,2c2,2)sinαtanαrmc2,4,c4,2*=(c0,2c4,0+c1,2c3,0+c2,0c2,2)sinαtanαrmc4,2.
d200=3r0w300cosβ,
d020=r0w120cosβ,
d300=4r0w400cosβ3tanβ(1+r0rm2r0c¯2,0)w300,
d120=2r0w220cosβtanβ(1+r0rm+2r0rs2r0c¯2,0)w120,
d400=5r0w500cosβ4tanβ(1+r0rm2r0c¯2,0)w4003cosβ×(3r0c¯3,0sinβ+2r0c¯2,0rm+c¯2,0r0rm2)w300,
d220=3r0w320cosβ2tanβ(1+r0rm+r0rs2r0c¯2,0)w220+3cosβ(r0c¯1,2sinβc¯0,2)w3001cosβ×[3r0c¯3.0sinβ+r0(1rm+2rs)(2c¯2,01rm)+c¯2,0]w120,
d040=r0w140cosβ4r0tanβw040rs+1cosβ(r0c¯1,2sinβc¯0,2)w120,
d500=6r0w600cosβ5tanβ(1+r0rm2r0c¯2,0)w5004cosβ(3r0c¯3,0sinβ+c¯2,0+2r0c¯2,0rmr0rm2)w4003cosβ[sinβ(4r0c¯4,0+2c¯2,02+6r0c¯2,02rm3c¯2,0rmr0c¯2,0rm2r0rm3+1rm2)+c¯3,0(1+3r0rm)]w300,
d320=4r0w420cosβtanβ(3+3r0rm+2r0rs6r0c¯2,0)w3204cosβ(r0c¯1,2sinβ+c¯0,2)w4002cosβ[3r0c¯3,0sinβ+c¯2,0+r0(2c¯2,01rm)(1rm+1rs)]w2203cosβ[sinβ(2c¯2,0c¯0,2(1r0rm+2r0rs)2r0c¯2,2+c¯0,2(3r0rm24r0rmrs1rm))+c¯1,2×(1+r0rm)]w3001cosβ{sinβ[4r0c¯4,0+2c¯2,02(1+3r0rm+4r0rs)c¯2,0(3rm+r0rm2+6r0rmrs+6rs6r0rs2)+2rs×(1r0rmr0rs)(1rm1rs)+1rm2(1r0rm)+2rs2×(1r0rs)]+c¯3,0(1+3r0rm+6r0rs)}w120,
d140=2r0w240cosβtanβ(1+r0rm+4r0rs2c¯2,0r0)w1402cosβ×(r0c¯1,2sinβ+c¯0,2)w2204r0rscosβ(2c¯2,01rm)w0401cosβ{sinβ[2r0c¯2,2+2c¯2,0c¯0,2(1r0rm+2r0rs)c¯0,2×(1rm+2rs3r0rm2+2r0rmrs+2r0rs2)]+c¯1,2(1+r0rm+2r0rs)}w120,
h110=2r0w120,
h210=2r0w220+6r0c¯0,2sinβw3002sinβw120,
h030=4r0w040+2r0c¯0,2sinβw120,
h310=2r0w320+8r0c¯0,2sinβw4002sinβw2203[2r0c¯1,2sinβ+r0rm(2c¯0,21rs)]w3002c¯2,0w120,
h130=4r0w1404sinβw040+4r0c¯0,2sinβw220[2r0c¯1,2sinβ+2c¯0,2(1+r0rm+2r0rs)r0rs(1rm+2rs)]w120,
h230=4r0w240+6r0c¯0,2sinβw3204sinβw1402[2r0c¯1,2sinβ+c¯0,2(1+2r0rm+2r0rs)r0rs(1rm+1rs)]w2203sinβ(2c¯0,22(1r0rm+2r0rs)4r0c¯0,4+r0c¯0,2rs(1rm2rs))w300{sinβ[2r0c¯2,2+2c¯2,0c¯0,2(1+3r0rm+4r0rs)2c¯0,2×(1rm+2rs+2r0rmrs2r0rs2)r0c¯2,0rmrs+1rs(1+r0rm2r0rs)×(1rm+2rs)]+2c¯1,2(1+r0rm+2r0rs)}w1204c¯2,0w040,
h410=2r0w420+10r0c¯0,2sinβw5002sinβw3204[2r0c¯1,2sinβ+r0rm(2c¯0,21rs)]w4002c¯2,0w2203[sinβ(2r0c¯2,2+2c¯2,0c¯0,2(1+3r0rm)r0c¯2,0rmrs2c¯0,2rm+1rmrs+r0rm2rs2r0rmrs2)+2r0c¯1,2rm]w3002c¯3,0w120,
h050=6r0w060+2r0c¯0,2sinβw1404[c¯0,2+r0rs(2c¯0,21rs)]w040sinβ[2c¯0,22(1r0rm+2r0rs)4r0c¯0,4+r0c¯0,2rs(1rm2rs)]w120,

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