Abstract

This paper utilizes the framework of nodal aberration theory to describe the aberration field behavior that emerges in optical systems with freeform optical surfaces, particularly φ-polynomial surfaces, including Zernike polynomial surfaces, that lie anywhere in the optical system. If the freeform surface is located at the stop or pupil, the net aberration contribution of the freeform surface is field constant. As the freeform optical surface is displaced longitudinally away from the stop or pupil of the optical system, the net aberration contribution becomes field dependent. It is demonstrated that there are no new aberration types when describing the aberration fields that arise with the introduction of freeform optical surfaces. Significantly it is shown that the aberration fields that emerge with the inclusion of freeform surfaces in an optical system are exactly those that have been described by nodal aberration theory for tilted and decentered optical systems. The key contribution here lies in establishing the field dependence and nodal behavior of each freeform term that is essential knowledge for effective application to optical system design. With this development, the nodes that are distributed throughout the field of view for each aberration type can be anticipated and targeted during optimization for the correction or control of the aberrations in an optical system with freeform surfaces. This work does not place any symmetry constraints on the optical system, which could be packaged in a fully three dimensional geometry, without fold mirrors.

© 2014 Optical Society of America

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References

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  1. K. Fuerschbach, G. E. Davis, K. P. Thompson, and J. P. Rolland, “Assembly of a freeform off-axis optical system employing three φ-polynomial Zernike mirrors,” Opt. Lett. 39(10), 2896–2899 (2014).
    [Crossref] [PubMed]
  2. K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “A new family of optical systems employing φ-polynomial surfaces,” Opt. Express 19(22), 21919–21928 (2011).
    [Crossref] [PubMed]
  3. R. V. Shack and K. P. Thompson, “Influence of alignment errors of a telescope system,” Proc. SPIE 251, 146–153 (1980).
    [Crossref]
  4. K. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A 22(7), 1389–1401 (2005).
    [Crossref] [PubMed]
  5. K. P. Thompson, “Aberration fields in unobscured mirror systems,” J. Opt. Soc. Am. 103, 159–165 (1980).
  6. 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]
  7. 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]
  8. K. Fuerschbach, J. P. Rolland, and K. P. Rolland-Thompson, “Nodal aberration theory applied to freeform surfaces,” in Classical Optics 2014 (Optical Society of America, 2014), ITh2A.5.
  9. K. Fuerschbach, “Freeform, phi-polynomial optical surfaces: optical design, fabrication and assembly,” Ph.D. (The University of Rochester, 2014).
  10. R. W. Gray, C. Dunn, K. P. Thompson, and J. P. Rolland, “An analytic expression for the field dependence of Zernike polynomials in rotationally symmetric optical systems,” Opt. Express 20(15), 16436–16449 (2012).
    [Crossref]
  11. J. Wang, B. Guo, Q. Sun, and Z. Lu, “Third-order aberration fields of pupil decentered optical systems,” Opt. Express 20(11), 11652–11658 (2012).
    [Crossref] [PubMed]
  12. C. R. Burch, “On the optical see-saw diagram,” Mon. Not. R. Astron. Soc. 102(3), 159–165 (1942).
    [Crossref]
  13. A. Rakich, “Calculation of third-order misalignment aberrations with the optical plate diagram,” Proc. SPIE 7652, 765230 (2010).
    [Crossref]
  14. 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]
  15. J. C. Wyant and K. Creath, “Basic wavefront aberration theory for optical metrology,” in Applied Optics and Optical Engineering, R. R. Shannon and J. C. Wyant, eds. (Academic, 1992), pp. 1–53.
  16. 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]
  17. K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: the comatic aberrations,” J. Opt. Soc. Am. A 27(6), 1490–1504 (2010).
    [Crossref] [PubMed]
  18. K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: the astigmatic aberrations,” J. Opt. Soc. Am. A 28(5), 821–836 (2011).
    [Crossref] [PubMed]
  19. J. E. Stacy and S. A. Macenka, “Optimization of an unobscured optical system using vector aberration theory,” Proc. SPIE 679, 21–24 (1986).
    [Crossref]
  20. K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: spherical aberration,” J. Opt. Soc. Am. A 26(5), 1090–1100 (2009).
    [Crossref] [PubMed]
  21. K. P. Thompson, “Aberration fields in tilted and decentered optical systems,” Ph.D. (The University of Arizona, 1980).
  22. C. Menke, “What's in the Designer's Toolbox for Freeform Systems?” in Renewable Energy and the Environment (Optical Society of America, 2013), FM4B.1.

2014 (1)

2012 (3)

2011 (2)

2010 (3)

2009 (2)

2008 (1)

2005 (1)

1986 (1)

J. E. Stacy and S. A. Macenka, “Optimization of an unobscured optical system using vector aberration theory,” Proc. SPIE 679, 21–24 (1986).
[Crossref]

1980 (2)

K. P. Thompson, “Aberration fields in unobscured mirror systems,” J. Opt. Soc. Am. 103, 159–165 (1980).

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

1942 (1)

C. R. Burch, “On the optical see-saw diagram,” Mon. Not. R. Astron. Soc. 102(3), 159–165 (1942).
[Crossref]

Burch, C. R.

C. R. Burch, “On the optical see-saw diagram,” Mon. Not. R. Astron. Soc. 102(3), 159–165 (1942).
[Crossref]

Cakmakci, O.

Davis, G. E.

Dunn, C.

Fuerschbach, K.

Gray, R. W.

Guo, B.

Lu, Z.

Macenka, S. A.

J. E. Stacy and S. A. Macenka, “Optimization of an unobscured optical system using vector aberration theory,” Proc. SPIE 679, 21–24 (1986).
[Crossref]

Rakich, A.

Rolland, J. P.

K. Fuerschbach, G. E. Davis, K. P. Thompson, and J. P. Rolland, “Assembly of a freeform off-axis optical system employing three φ-polynomial Zernike mirrors,” Opt. Lett. 39(10), 2896–2899 (2014).
[Crossref] [PubMed]

R. W. Gray, C. Dunn, K. P. Thompson, and J. P. Rolland, “An analytic expression for the field dependence of Zernike polynomials in rotationally symmetric optical systems,” Opt. Express 20(15), 16436–16449 (2012).
[Crossref]

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, “A new family of optical systems employing φ-polynomial surfaces,” Opt. Express 19(22), 21919–21928 (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, 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]

Schmid, T.

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]

Stacy, J. E.

J. E. Stacy and S. A. Macenka, “Optimization of an unobscured optical system using vector aberration theory,” Proc. SPIE 679, 21–24 (1986).
[Crossref]

Sun, Q.

Thompson, K.

Thompson, K. P.

K. Fuerschbach, G. E. Davis, K. P. Thompson, and J. P. Rolland, “Assembly of a freeform off-axis optical system employing three φ-polynomial Zernike mirrors,” Opt. Lett. 39(10), 2896–2899 (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]

R. W. Gray, C. Dunn, K. P. Thompson, and J. P. Rolland, “An analytic expression for the field dependence of Zernike polynomials in rotationally symmetric optical systems,” Opt. Express 20(15), 16436–16449 (2012).
[Crossref]

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

K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “A new family of optical systems employing φ-polynomial surfaces,” Opt. Express 19(22), 21919–21928 (2011).
[Crossref] [PubMed]

K. P. Thompson, “Multinodal fifth-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, 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 fifth-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]

K. P. Thompson, “Aberration fields in unobscured mirror systems,” J. Opt. Soc. Am. 103, 159–165 (1980).

Wang, J.

J. Opt. Soc. Am. (1)

K. P. Thompson, “Aberration fields in unobscured mirror systems,” J. Opt. Soc. Am. 103, 159–165 (1980).

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

Mon. Not. R. Astron. Soc. (1)

C. R. Burch, “On the optical see-saw diagram,” Mon. Not. R. Astron. Soc. 102(3), 159–165 (1942).
[Crossref]

Opt. Express (6)

Opt. Lett. (1)

Proc. SPIE (3)

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

A. Rakich, “Calculation of third-order misalignment aberrations with the optical plate diagram,” Proc. SPIE 7652, 765230 (2010).
[Crossref]

J. E. Stacy and S. A. Macenka, “Optimization of an unobscured optical system using vector aberration theory,” Proc. SPIE 679, 21–24 (1986).
[Crossref]

Other (5)

J. C. Wyant and K. Creath, “Basic wavefront aberration theory for optical metrology,” in Applied Optics and Optical Engineering, R. R. Shannon and J. C. Wyant, eds. (Academic, 1992), pp. 1–53.

K. Fuerschbach, J. P. Rolland, and K. P. Rolland-Thompson, “Nodal aberration theory applied to freeform surfaces,” in Classical Optics 2014 (Optical Society of America, 2014), ITh2A.5.

K. Fuerschbach, “Freeform, phi-polynomial optical surfaces: optical design, fabrication and assembly,” Ph.D. (The University of Rochester, 2014).

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

C. Menke, “What's in the Designer's Toolbox for Freeform Systems?” in Renewable Energy and the Environment (Optical Society of America, 2013), FM4B.1.

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Figures (9)

Fig. 1
Fig. 1 (a) When the aspheric corrector plate of a Schmidt telescope is displaced longitudinally from the aperture stop along the optical axis, the beam for an off-axis field point will displace across the corrector plate. The amount of relative beam displacement defined by Eq. (1) depends on the paraxial quantities for the marginal ray height,y, the chief ray height, y ¯ , the chief ray angle, u ¯ , and the distance between the stop and the corrector plate, t . (b) The beam displacement on the corrector plate, Δ h , can be thought of as a field dependent decenter of the aspheric corrector at the stop that modifies the mapping of the normalized pupil coordinate from ρ to ρ ' .
Fig. 2
Fig. 2 Fringe Zernike polynomial set through 6th order in wavefront expansion. The set includes Z1 (piston), Z2/3 (tilt), Z4 (defocus), Z5/6 (astigmatism), Z7/8 (coma), Z9 (spherical aberration), Z10/11 (elliptical coma or trefoil), Z12/13 (oblique spherical aberration or secondary astigmatism), Z14/15 (fifth order aperture coma or secondary coma), and Z16 (fifth order spherical aberration or secondary spherical aberration). The φ-polynomials to be explored include Z5/6, Z7/8, Z10/11, Z12/13, and Z14/15.
Fig. 3
Fig. 3 Surface map describing the freeform Zernike overlay for astigmatism on an optical surface over the full aperture. The error is quantified by its magnitude | z 5/6 | FF and its orientation ξ FF 5/6 that is measured clockwise with respect to the y ^ axis. P and V denote where the surface error is a peak rather than a valley.
Fig. 4
Fig. 4 The characteristic field dependence of field constant astigmatism that is generated by a Zernike astigmatism overlay on an optical surface in an optical system. This induced aberration is independent of stop position.
Fig. 5
Fig. 5 The characteristic field dependence of (a) field constant coma, (b) field asymmetric, field linear astigmatism, and (c) field linear, medial field curvature that is generated by a Zernike coma overlay on an optical surface away from the stop surface.
Fig. 6
Fig. 6 The characteristic field dependence of (a) field constant elliptical coma, (b) field conjugate, field linear astigmatism, which is generated by a Zernike elliptical coma overlay on an optical surface away from the stop surface.
Fig. 7
Fig. 7 The characteristic field dependence of (a) field constant oblique spherical aberration, (b) field asymmetric, field linear trefoil, (c) field conjugate, field linear coma, (d) field constant, field quadratic astigmatism, and (e) saddle shaped, field quadratic, medial field curvature that is generated by a Zernike oblique spherical aberration overlay on an optical surface away from the stop surface.
Fig. 8
Fig. 8 The characteristic field dependence of (a) field constant, fifth order aperture coma, (b) field linear medial oblique spherical aberration, (c) field asymmetric, field linear oblique spherical aberration, (d) field quadratic trefoil, (e) hybrid field quadratic coma that is a combination of two field quadratic coma terms, (f) hybrid field cubic astigmatism that is a combination of two field cubic astigmatism terms, and (g) field cubic, medial field curvature that is generated by a Zernike fifth order aperture coma overlay on an optical surface away from the stop surface.
Fig. 9
Fig. 9 The characteristic field dependence of (a) field constant tetrafoil, (b) field conjugate, field linear trefoil, and (c) field conjugate, field quadratic astigmatism which is generated by a Zernike tetrafoil overlay on an optical surface away from the stop surface.

Tables (5)

Tables Icon

Table 1 Image Degrading Aberration Terms That Are Generated by a Zernike Coma Overlay and How the Terms Link to Existing Concepts of NAT and Extend the Theory to Include Freeform Surfaces

Tables Icon

Table 2 Image Degrading Aberration Terms That Are Generated by a Zernike Elliptical Coma Overlay and How the Terms Link to Existing Concepts of NAT and Extend the Theory to Include Freeform Surfaces

Tables Icon

Table 3 Image Degrading Aberration Terms That Are Generated by a Zernike Oblique Spherical Aberration Overlay and How the Terms Link to Existing Concepts of NAT and Extend the Theory to Include Freeform Surfaces

Tables Icon

Table 4 Image Degrading Aberration Terms That Are Generated by a Zernike Fifth Order Aperture Coma Overlay and How the Terms Link to Existing Concepts of NAT and Extend the Theory to Include Freeform Surfaces

Tables Icon

Table 5 Image Degrading Aberration Terms That Are Generated by a Zernike Tetrafoil Overlay and How the Terms Link to Existing Concepts of NAT

Equations (60)

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Δ h ( y ¯ y ) H =( u ¯ t y ) H ,
W( ρ )W( ρ '+Δ h ),
( z 5 z 6 ) ( Z 5 Z 6 )=( z 5 z 6 ) ( ρ 2 cos( 2ϕ ) ρ 2 sin( 2ϕ ) ),
| z 5/6 | FF = z 5 2 + z 6 2 ,
ξ FF 5/6 Test = 1 2 tan 1 ( z 6 z 5 ),
if ρ =ρ( sin( ϕ ) cos( ϕ ) ),then ρ 2 = ρ 2 ( sin( 2ϕ ) cos( 2ϕ ) ),
ξ FF 5/6 = π 2 1 2 tan 1 ( z 6 z 5 ).
W Stop = 1 2 ( B FF 222 2 · ρ 2 ),
B FF 222 2 2( n'n ) | z 5/6 | FF exp( i2 ξ FF 5/6 ),
W NotStop = 1 2 [ B FF 222 2 · ( ρ '+Δ h ) 2 ] = 1 2 [ B FF 222 2 · ρ 2 +2 B FF 222 2 ·Δ h ρ + B FF 222 2 ·Δ h 2 ],
A · B C = A B * · C ,
B * = | B | exp ( i β ) = B x x ^ + B y y ^ .
W N o t S t o p = 1 2 [ B F F 222 2 · ρ 2 + 2 B F F 222 2 Δ h * · ρ + B F F 222 2 · Δ h 2 ] .
B 222 2 = B ALIGN 222 2 + j=1 N B FF 222,j 2 ,
( z 7 z 8 ) ( Z 7 Z 8 )=( z 7 z 8 ) ( 3 ρ 3 cos( ϕ )2ρcos( ϕ ) 3 ρ 3 sin( ϕ )2ρsin( ϕ ) ),
( z 7 z 8 ) ( Z 7 Adj Z 8 Adj )=( z 7 z 8 ) ( Z 7 +2 Z 2 Z 8 +2 Z 3 )=( z 7 z 8 ) ( 3 ρ 3 cos( ϕ ) 3 ρ 3 sin( ϕ ) ).
| z 7/8 | FF = ( z 7 ) 2 + ( z 8 ) 2 ,
ξ FF 7/8 = π 2 tan 1 ( z 8 z 7 ),
if ρ =ρ( sin( ϕ ) cos( ϕ ) ),then( ρ · ρ ) ρ = ρ 3 ( sin( ϕ ) cos( ϕ ) ),
W Stop =( A FF 131 · ρ )( ρ · ρ ),
A FF 131 3( n'n ) | z 7/8 | FF exp( i ξ FF 7/8 ).
W N o t S t o p = [ A F F 131 · ( ρ ' + Δ h ) ] [ ( ρ ' + Δ h ) · ( ρ ' + Δ h ) ] = [ ( A F F 131 · ρ ) ( ρ · ρ ) + A F F 131 Δ h · ρ 2 + 2 ( A F F 131 · Δ h ) ( ρ · ρ ) + 2 ( Δ h · Δ h ) ( A F F 131 · ρ ) + A 131 * Δ h 2 · ρ + ( A F F 131 · Δ h ) ( Δ h · Δ h ) ] ,
2 ( A · B ) ( A · C ) = ( A · A ) ( B · C ) + A 2 · B C ,
A 131 = A A L I G N 131 j = 1 N A F F 131 , j ,
A F F 131 Δ h · ρ 2 = ( y ¯ j y j ) A F F 131 , j H · ρ 2 .
A 222 = A A L I G N 222 j = 1 N ( y ¯ j y j ) A F F 131 , j ,
2 ( A F F 131 · Δ h ) ( ρ · ρ ) = 2 ( y ¯ j y j ) ( A F F 131 · H ) ( ρ · ρ ) .
A 220 M = A A L I G N 220 M j = 1 N ( y ¯ j y j ) A F F 131 , j ,
( z 10 z 11 ) ( Z 10 Z 11 )=( z 10 z 11 ) ( ρ 3 cos( 3ϕ ) ρ 3 sin( 3ϕ ) ),
| z 10/11 | FF = z 10 2 + z 11 2 ,
ξ FF 10/11 = π 2 1 3 tan 1 ( z 11 z 10 ).
if ρ =ρ( sin( ϕ ) cos( ϕ ) ),then ρ 3 = ρ 3 ( sin( 3ϕ ) cos( 3ϕ ) ),
W Stop = 1 4 ( C FF 333 3 · ρ 3 ),
C FF 333 3 4( n'n ) | z 10/11 | FF exp( i3 ξ FF 10/11 ).
W N o t S t o p = 1 4 [ C F F 333 3 · ( ρ ' + Δ h ) 3 ] = 1 4 [ C F F 333 3 · ρ 3 + 3 C F F 333 3 Δ h * · ρ 2 + 3 C F F 333 3 Δ h * 2 · ρ + C F F 333 3 · Δ h 3 ] ,
( z 12 z 13 ) ( Z 12 Z 13 )=( z 12 z 13 ) ( 4 ρ 4 cos( 2ϕ )3 ρ 2 cos( 2ϕ ) 4 ρ 4 sin( 2ϕ )3 ρ 2 sin( 2ϕ ) ),
( z 12 z 13 ) ( Z 12 Adj Z 13 Adj )=( z 12 z 13 ) ( Z 12 +3 Z 5 Z 12 +3 Z 6 )=( z 12 z 13 ) ( 4 ρ 4 cos( 2ϕ ) 4 ρ 4 sin( 2ϕ ) ).
| z 12/13 | FF = ( z 12 ) 2 + ( z 13 ) 2 ,
ξ FF 12/13 = π 2 1 2 tan 1 ( z 13 z 12 ).
if ρ =ρ( sin( ϕ ) cos( ϕ ) ),then( ρ · ρ ) ρ 2 = ρ 4 ( sin( 2ϕ ) cos( 2ϕ ) ),
W Stop = 1 2 ( B FF 242 2 · ρ 2 )( ρ · ρ ),
B FF 242 2 8( n'n ) | z 12/13 | FF exp( i2 ξ FF 12/13 ).
W N o t S t o p = 1 2 [ B F F 242 2 · ( ρ ' + Δ h ) 2 ] [ ( ρ ' + Δ h ) · ( ρ ' + Δ h ) ] = 1 2 [ ( B F F 242 2 · ρ 2 ) ( ρ · ρ ) + 3 ( B F F 242 2 Δ h * · ρ ) ( ρ · ρ ) + B F F 242 2 Δ h · ρ 3 + 3 ( Δ h · Δ h ) ( B F F 242 2 · ρ 2 ) + 3 ( B F F 242 2 · Δ h 2 ) ( ρ · ρ ) + 2 ( B F F 242 2 · Δ h 2 ) ( Δ h · ρ ) + 2 ( Δ h · Δ h ) ( B F F 242 2 Δ h * · ρ ) + ( Δ h · Δ h ) ( B F F 242 2 · Δ h 2 ) ] ,
2 ( A · B ) ( A 2 · C 2 ) = ( A · A ) ( A B · C ) + A 3 · B C 2 ,
2 ( A · B ) ( A B · C 2 ) = ( A · A ) ( B 2 · C ) + ( B · B ) ( A 2 · C 2 ) ,
( z 14 z 15 ) ( Z 14 Z 15 )=( z 14 z 15 ) ( 10 ρ 5 cos( ϕ )12 ρ 3 cos( ϕ )+3ρcos( ϕ ) 10 ρ 5 sin( ϕ )12 ρ 3 sin( ϕ )+3ρsin( ϕ ) ),
( z 14 z 15 ) ( Z 14 Adj Z 15 Adj )=( z 14 z 15 ) ( Z 14 +4 Z 7 +5 Z 2 Z 15 +4 Z 8 +5 Z 3 )=( z 14 z 15 ) ( 10 ρ 5 cos( ϕ ) 10 ρ 5 sin( ϕ ) ).
| z 14/15 | FF = ( z 14 ) 2 + ( z 15 ) 2 ,
ξ FF 14/15 = π 2 tan 1 ( z 15 z 14 ).
if ρ =ρ( sin( ϕ ) cos( ϕ ) ),then ( ρ · ρ ) 2 ρ = ρ 5 ( sin( ϕ ) cos( ϕ ) ),
W Stop =( A FF 151 · ρ ) ( ρ · ρ ) 2 ,
A FF 151 10( n'n ) | z 14/15 | FF exp( i ξ FF 14/15 ).
W N o t S t o p = [ A F F 151 · ( ρ ' + Δ h ) ] [ ( ρ ' + Δ h ) · ( ρ ' + Δ h ) ] 2 = { ( A F F 151 · ρ ) ( ρ · ρ ) 2 + 3 ( A F F 151 · Δ h ) ( ρ · ρ ) 2 + ( A F F 151 Δ h 2 · ρ 3 ) + [ 6 ( A F F 151 · Δ h ) Δ h + 3 ( Δ h · Δ h ) A F F 151 ] · ρ ( ρ · ρ ) + [ 2 ( A F F 151 · Δ h ) Δ h 2 + 2 ( Δ h · Δ h ) A F F 151 Δ h ] · ρ 2 + 2 ( A F F 151 Δ h · ρ 2 ) ( ρ · ρ ) + 6 ( Δ h · Δ h ) ( A F F 151 · Δ h ) ( ρ · ρ ) + [ ( Δ h · Δ h ) 2 A F F 151 + 4 ( A F F 151 · Δ h ) ( Δ h · Δ h ) Δ h ] · ρ + ( Δ h · Δ h ) 2 ( A F F 151 · Δ h ) } ,
( z 17 z 18 ) ( Z 17 Z 18 )=( z 17 z 18 ) ( ρ 4 cos( 4ϕ ) ρ 4 sin( 4ϕ ) ),
| z 17/18 | FF = z 17 2 + z 18 2 ,
ξ FF 17/18 = π 2 1 4 tan 1 ( z 18 z 17 ).
if ρ =ρ( sin( ϕ ) cos( ϕ ) ),then ρ 4 = ρ 4 ( sin( 4ϕ ) cos( 4ϕ ) ),
W Stop = 1 8 ( D FF 444 4 · ρ 4 ),
D FF 444 4 8( n'n ) | z 17/18 | FF exp( i4 ξ FF 17/18 ).
W N o t S t o p = 1 8 [ D F F 444 4 · ( ρ ' + Δ h ) 4 ] = 1 8 [ D F F 444 4 · ρ 4 + 4 D F F 444 4 Δ h * · ρ 3 + 6 D F F 444 4 Δ h * 2 · ρ 2 + 4 D F F 444 4 Δ h * 3 · ρ + D F F 444 4 · Δ h 4 ] ,

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