Abstract

The geometric characteristics of aberrations of plane-symmetric optical systems are studied in detail with a wave-aberration theory. It is dealt with as an extension of the Seidel aberrations to realize a consistent aberration theory from axially symmetric to plane-symmetric systems. The aberration distribution is analyzed with the spot diagram of a ray and an aberration curve. Moreover, the root-mean-square value and the centroid of aberration distribution are discussed. The numerical results are obtained with the focusing optics of a toroidal mirror at grazing incidence.

© 2009 Optical Society of America

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References

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  1. W. B. Peatman, Gratings, Mirrors and Slits: Beamline Design for Soft X-Ray Synchrotron Radiation Sources (Gordon&Breach , 1997), pp. 71-75.
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  3. H. Noda, T. Namioka, and M. Seya, “Geometrical theory of the grating,” J. Opt. Soc. Am. 64, 1031-1036 (1974).
    [CrossRef]
  4. T. Namioka, M. Koike, and D. Content, “Geometric theory of the ellipsoidal grating,” Appl. Opt. 33, 7261-7274 (1994).
    [CrossRef]
  5. M. P. Chrisp, “Aberrations of holographic toroidal grating systems,” Appl. Opt. 22, 1508-1518 (1983).
    [CrossRef]
  6. L.-J. Lu, “Aberration theory of plane-symmetric grating systems,” J. Synchrotron Radiat. 15, 399-410 (2008).
  7. L.-J. Lu and D.-L. Lin, “Aberrations of plane-symmetric multi-element optical systems,” Opt. Int. J. Light Electron Opt. , doi:10.1016/j.ijleo.2009.01.016 (in press).
    [CrossRef]
  8. C. Palmer, B. Wheeler, and W. McKinney, “Imaging equations of spectroscopic systems using Lie transformations. I. Theoretical foundations,” Proc. SPIE 3450, 55-66 (1998).
    [CrossRef]
  9. C. Palmer, B. Wheeler, and W. McKinney, “Imaging equations of spectroscopic systems using Lie transformations. II. Multi-element systems,” Proc. SPIE 3450, 67-77 (1998).
    [CrossRef]
  10. K. Goto and T. Kurosaki, “Canonical formulation for the geometrical optics of concave gratings,” J. Opt. Soc. Am. A 10, 452-465 (1993).
    [CrossRef]
  11. S. Masui and T. Namioka, “Geometric aberration theory of double-element optical systems,” J. Opt. Soc. Am. A 16, 2253-2268 (1999).
    [CrossRef]
  12. B. D. Stone and G. W. Forbes, “Second-order design methods for definitive studies of plane-symmetric, two-mirror systems,” J. Opt. Soc. Am. A 11, 3292-3307 (1994).
    [CrossRef]
  13. J. M. Howard and B. D. Stone, “Imaging a point with two spherical mirrors,” J. Opt. Soc. Am. A 15, 3045-3056 (1998).
    [CrossRef]
  14. J. M. Sasian, “How to approach the design of a bilateral symmetric optical system,” Opt. Eng. 33, 2045-2061 (1994).
    [CrossRef]
  15. K. 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]
  16. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2005).
  17. V. N. Mahajan, Optical Imaging and Aberrations (SPIE Press, 1998).

2005 (1)

1999 (1)

1998 (3)

C. Palmer, B. Wheeler, and W. McKinney, “Imaging equations of spectroscopic systems using Lie transformations. I. Theoretical foundations,” Proc. SPIE 3450, 55-66 (1998).
[CrossRef]

C. Palmer, B. Wheeler, and W. McKinney, “Imaging equations of spectroscopic systems using Lie transformations. II. Multi-element systems,” Proc. SPIE 3450, 67-77 (1998).
[CrossRef]

J. M. Howard and B. D. Stone, “Imaging a point with two spherical mirrors,” J. Opt. Soc. Am. A 15, 3045-3056 (1998).
[CrossRef]

1994 (3)

1993 (1)

1983 (1)

1974 (1)

1945 (1)

Beutler, H. G.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2005).

Chrisp, M. P.

Content, D.

Forbes, G. W.

Goto, K.

Howard, J. M.

Koike, M.

Kurosaki, T.

Lin, D.-L.

L.-J. Lu and D.-L. Lin, “Aberrations of plane-symmetric multi-element optical systems,” Opt. Int. J. Light Electron Opt. , doi:10.1016/j.ijleo.2009.01.016 (in press).
[CrossRef]

Lu, L.-J.

L.-J. Lu and D.-L. Lin, “Aberrations of plane-symmetric multi-element optical systems,” Opt. Int. J. Light Electron Opt. , doi:10.1016/j.ijleo.2009.01.016 (in press).
[CrossRef]

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

Mahajan, V. N.

V. N. Mahajan, Optical Imaging and Aberrations (SPIE Press, 1998).

Masui, S.

McKinney, W.

C. Palmer, B. Wheeler, and W. McKinney, “Imaging equations of spectroscopic systems using Lie transformations. II. Multi-element systems,” Proc. SPIE 3450, 67-77 (1998).
[CrossRef]

C. Palmer, B. Wheeler, and W. McKinney, “Imaging equations of spectroscopic systems using Lie transformations. I. Theoretical foundations,” Proc. SPIE 3450, 55-66 (1998).
[CrossRef]

Namioka, T.

Noda, H.

Palmer, C.

C. Palmer, B. Wheeler, and W. McKinney, “Imaging equations of spectroscopic systems using Lie transformations. II. Multi-element systems,” Proc. SPIE 3450, 67-77 (1998).
[CrossRef]

C. Palmer, B. Wheeler, and W. McKinney, “Imaging equations of spectroscopic systems using Lie transformations. I. Theoretical foundations,” Proc. SPIE 3450, 55-66 (1998).
[CrossRef]

Peatman, W. B.

W. B. Peatman, Gratings, Mirrors and Slits: Beamline Design for Soft X-Ray Synchrotron Radiation Sources (Gordon&Breach , 1997), pp. 71-75.

Sasian, J. M.

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

Seya, M.

Stone, B. D.

Thompson, K.

Wheeler, B.

C. Palmer, B. Wheeler, and W. McKinney, “Imaging equations of spectroscopic systems using Lie transformations. II. Multi-element systems,” Proc. SPIE 3450, 67-77 (1998).
[CrossRef]

C. Palmer, B. Wheeler, and W. McKinney, “Imaging equations of spectroscopic systems using Lie transformations. I. Theoretical foundations,” Proc. SPIE 3450, 55-66 (1998).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2005).

Appl. Opt. (2)

J. Opt. Soc. Am. (2)

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

Opt. Eng. (1)

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

Opt. Int. J. Light Electron Opt. (1)

L.-J. Lu and D.-L. Lin, “Aberrations of plane-symmetric multi-element optical systems,” Opt. Int. J. Light Electron Opt. , doi:10.1016/j.ijleo.2009.01.016 (in press).
[CrossRef]

Proc. SPIE (2)

C. Palmer, B. Wheeler, and W. McKinney, “Imaging equations of spectroscopic systems using Lie transformations. I. Theoretical foundations,” Proc. SPIE 3450, 55-66 (1998).
[CrossRef]

C. Palmer, B. Wheeler, and W. McKinney, “Imaging equations of spectroscopic systems using Lie transformations. II. Multi-element systems,” Proc. SPIE 3450, 67-77 (1998).
[CrossRef]

Other (4)

W. B. Peatman, Gratings, Mirrors and Slits: Beamline Design for Soft X-Ray Synchrotron Radiation Sources (Gordon&Breach , 1997), pp. 71-75.

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

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2005).

V. N. Mahajan, Optical Imaging and Aberrations (SPIE Press, 1998).

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Figures (18)

Fig. 1
Fig. 1

Optical scheme of a plane-symmetric optical system in the cases of (a) reflection and (b) refraction.

Fig. 2
Fig. 2

Optical scheme of the principal and the aperture rays passing through a system of two elements on the sagittal plane.

Fig. 3
Fig. 3

Ray spot diagrams from the defocus aberration of the demonstration optics with r 0 = 99.5 cm , (a) without aberration and (b) including the effects of aberration.

Fig. 4
Fig. 4

Position of rays, numbered 1 to 8, on the optical surface.

Fig. 5
Fig. 5

Ray spot diagram from the spherical aberration of the demonstration optics.

Fig. 6
Fig. 6

Ray spot diagrams from the spherical aberration when θ v 1.6 mrad .

Fig. 7
Fig. 7

Ray spot diagrams from (a) in-plane coma aberration and (b) off-plane coma aberration of the demonstration optics.

Fig. 8
Fig. 8

Ray spot diagram from the combination of in-plane and off-plane coma aberration of the demonstration optics.

Fig. 9
Fig. 9

Ray spot diagram from the tilt aberration of the demonstration optics.

Fig. 10
Fig. 10

Ray spot diagram from the combination of the spherical aberration, coma, and tilt aberration of the demonstration optics.

Fig. 11
Fig. 11

Ray spot diagrams from the combination of the tilt and the in-plane astigmatism of the demonstration optics, but with ρ = 13.95 cm at the different image positions: (a)  r 0 = 100 cm , (b)  r 0 = 150 cm , and (c)  r 0 = 200 cm .

Fig. 12
Fig. 12

Optical scheme of field curvature in (a) the meridional plane and (b) the sagittal plane.

Fig. 13
Fig. 13

Field curvature of the demonstration optics: (a)  1 / R m and (b)  1 / R s as a function of the incidence angle for different positions of the entrance pupil, l = 0 , 66.7 , and 200 cm .

Fig. 14
Fig. 14

Ray spot diagram from the astigmatism and field- curvature aberration of the demonstration optics.

Fig. 15
Fig. 15

Optical scheme of distortion aberration.

Fig. 16
Fig. 16

RMS value of aberration distributions of the demonstration optics as a function of incidence angle with l = 0 in (a) the x direction and (b) the y direction.

Fig. 17
Fig. 17

RMS value of the field-dependent aberration distributions of the demonstration optics for the different positions of the entrance pupil, l = 0 , 66.7 , 200 cm : (a) tilt aberration, (b) off-plane coma, (c) field curvature, and (d) distortion.

Fig. 18
Fig. 18

Centroid position x c , y c of the demonstration optics as a function of incidence angle.

Tables (3)

Tables Icon

Table 1 Optical Parameters of the Demonstration Optics

Tables Icon

Table 2 Aberration Coefficients of the Demonstration Optics

Tables Icon

Table 3 Expressions of q x and q y from Eq. (93)

Equations (100)

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z = i = 0 j = 0 c i , j χ i η j , c 0 , 0 = c 1 , 0 = 0 ( j = even ) .
y ¯ = γ t = γ t u l .
γ = ( l r s 1 ) u Λ l u , l = r s t t r s ,
γ = ( l r s 1 ) u = Λ n u , l ' = r s t t r s ,
W = i j k 4 W i j k x i y j u k ( 0 < i + j + k 4 ) ,
W i j k = n 0 M i j k ( α , r m , r s , l ) + n 1 ( n 0 / 1 ) k M i j k ( β , r m , r s , l ) + Λ N i j k n 1 w i j k ,
w i j k = n 0 / 1 M i j k ( α , r m , r s , l ) + ( n 0 / 1 ) k M i j k ( β , r m , r s , l ) + Λ N i j k n 1 ,
n = χ σ + Γ σ ( n 20 2 χ 2 + n 02 2 η 2 + n 30 2 χ 3 + n 12 2 χ η 2 + n 40 8 χ 4 + n 22 4 χ 2 η 2 + n 04 8 η 4 + ) ,
Γ = { 1 for a mechanically ruled grating σ / λ 0 for a holographic grating ,
M 100 ( α , r m , r s , l ) = sin ( α ) ,
M 200 ( α , r m , r s , l ) = cos α 2 ( cos α r m 2 c 2 , 0 ) ,
M 300 ( α , r m , r s , l ) = 1 2 ( sin α cos α r m ( cos α r m 2 c 2 , 0 ) 2 c 3 , 0 cos α ) ,
M 020 ( α , r m , r s , l ) = 1 2 ( 1 r s 2 c 0 , 2 cos α ) ,
M 120 ( α , r m , r s , l ) = 1 2 ( sin α r m ( r m r s 2 2 c 0 , 2 cos α ) 2 c 1 , 2 cos α ) ,
M 400 ( α , r m , r s , l ) = cos 2 α ( 4 5 cos 2 α ) 8 r m 3 c 2 , 0 cos α ( 2 3 cos 2 α ) 2 r m 2 + c 2 , 0 2 sin 2 α 2 r m c 3 , 0 sin 2 α 2 r m c 4 , 0 cos α ,
M 040 ( α , r m , r s , l ) = 1 8 r s 3 + c 0 , 2 cos α 2 r s 2 + c 0 , 2 2 sin 2 α 2 r m c 0 , 4 cos α ,
M 220 ( α , r m , r s , l ) = 1 2 r s 2 ( sin 2 α r s + c 2 , 0 cos α cos 2 α 2 r m ) c 0 , 2 cos α ( 2 3 cos 2 α ) 2 r m 2 + c 2 , 0 c 0 , 2 sin 2 α r m c 1 , 2 sin 2 α 2 r m c 2 , 2 cos α ,
M 102 ( α , r m , r s , l ) = 1 2 Λ l 2 sin α c 1 , 2 l 2 cos α ,
M 202 ( α , r m , r s , l ) = 1 2 ( sin 2 α r s + c 2 , 0 cos α cos 2 α 2 r m ) Λ l 2 + c 0 , 2 l 2 cos α 2 r m 2 c 1 , 2 l 2 sin 2 α 2 r m c 2 , 2 l 2 cos α ,
M 022 ( α , r m , r s , l ) = 3 4 r s Λ l 2 + c 0 , 2 Λ l cos α 2 ( 1 5 l r s ) + 2 c 0 , 2 2 l 2 sin 2 α r m 6 c 0 , 4 l 2 cos α ,
M 011 ( α , r m , r s , l ) = Λ l 2 c 0 , 2 l cos α ,
M 013 ( α , r m , r s , l ) = 1 2 ( Λ l + 2 c 0 , 2 l cos α ) Λ l 2 4 c 0 , 4 l 3 cos α ,
M 111 ( α , r m , r s , l ) = Λ l sin α r s c 0 , 2 l sin 2 α r m 2 c 1 , 2 l cos α ,
M 211 ( α , r m , r s , l ) = 1 r s ( sin 2 α r s + c 2 , 0 cos α cos 2 α 2 r m ) Λ l c 0 , 2 l cos α r m 2 ( 2 3 cos 2 α ) + 2 c 2 , 0 c 0 , 2 l sin 2 α r m c 1 , 2 l sin 2 α r m 2 c 2 , 2 l cos α ,
M 031 ( α , r m , r s , l ) = Λ l 2 r s 2 c 0 , 2 cos α r s ( 1 2 l r s ) + 2 c 0 , 2 2 l sin 2 α r m 4 c 0 , 4 l cos α .
n 0 sin α + n 1 sin β = m λ / σ .
n 0 l ( 2 c 0 , 2 cos α 1 r s ) n 0 l ( 2 c 0 , 2 cos β 1 r s ) = n 02 Λ l .
2 c 2 , 0 ( n 0 cos α + n 1 cos β ) ( n 0 cos 2 α r m + n 1 cos 2 β r m ) = n 20 Λ .
2 c 0 , 2 ( n 0 cos α + n 1 cos β ) ( n 0 r s + n 1 r s ) = n 02 Λ .
W = n 1 ( w 300 x 3 + w 120 x y 2 + w 400 x 4 + w 220 x 2 y 2 + w 040 y 4 + w 102 x u 2 + w 013 y u 3 + w 202 x 2 u 2 + w 022 y 2 u 2 + w 111 x y u + w 031 y 3 u + w 211 x 2 y u ) .
x = d 100 x + d 200 x 2 + d 020 y 2 + d 300 x 3 + d 120 x y 2 + d 002 u 2 + d 011 y u + d 111 x y u + d 102 x u 2 , y = h 010 y + h 110 x y + h 210 x 2 y + h 030 y 3 + h 003 u 3 + h 001 u + h 101 x u + h 201 x 2 u + h 021 y 2 u + h 012 y u 2 ,
d 100 = Λ m cos β ,
d 200 = 3 r 0 w 300 cos β + Λ m sin β ( cos β r m c 2 , 0 ) ,
d 300 = 4 r 0 w 400 cos β 3 tan β ( 1 + r 0 r m 2 r 0 c 2 , 0 cos β ) w 300 + Λ m ( cos ( 2 β ) c 2 , 0 r m + cos β sin 2 β r m 2 sin β c 3 , 0 ) ,
d 020 = r 0 w 120 cos β Λ m sin β c 0 , 2 ,
d 120 = 2 r 0 w 220 cos β tan β ( 1 + r 0 r m + 2 r 0 r s 2 r 0 c 2 , 0 cos β ) w 120 + Λ m ( cos ( 2 β ) c 0 , 2 r m sin β c 1 , 2 ) ,
d 011 = r 0 w 111 cos β 2 Λ m sin β c 0 , 2 l ,
d 002 = r 0 w 102 cos β ,
+ 2 Λ m l ( cos ( 2 β ) c 0 , 2 r m sin β c 1 , 2 ) d 111 = 2 r 0 w 211 cos β tan β ( 1 + r 0 r m + r 0 r s 2 r 0 c 2 , 0 cos β ) w 111 2 Λ n tan β r 0 w 120 ,
d 102 = 2 r 0 w 202 cos β tan β ( 1 + r 0 r m 2 r 0 c 2 , 0 cos β ) w 102 Λ n tan β r 0 w 111 + Λ m l 2 ( c 0 , 2 r m sin β c 1 , 2 ) ,
h 010 = Λ s ,
h 110 = 2 r 0 w 120 + Λ s sin β r s ,
h 210 = 2 r 0 w 220 2 sin β w 120 + 6 tan β r 0 c 0 , 2 w 300 + Λ s r s ( cos β c 2 , 0 + sin 2 β r s ) + ( Λ m Λ s ) cos 2 β 2 r m r s ,
h 001 = l Λ n r 0 ,
h 101 = r 0 w 111 + Λ s Λ n sin β ,
h 201 = r 0 w 211 sin β w 111 + 6 tan β r 0 c 0 , 2 l w 300 + Λ s Λ n ( cos β c 2 , 0 + sin 2 β r s ) + ( Λ m Λ s ) Λ n cos 2 β 2 r m ,
h 021 = 3 r 0 w 031 + 2 tan β r 0 c 0 , 2 ( l w 120 + w 111 ) + Λ s cos β c 0 , 2 ( Λ n + 2 l r s ) ,
h 012 = 2 r 0 w 022 + 2 tan β r 0 c 0 , 2 ( l w 111 + w 102 ) + 2 Λ s Λ n cos β c 0 , 2 l ,
h 030 = 4 r 0 w 040 + 2 tan β r 0 c 0 , 2 w 120 + Λ s cos β c 0 , 2 r s ,
h 003 = r 0 w 013 + 2 tan β r 0 c 0 , 2 l w 102 ,
Λ m = r m r 0 r m , Λ s = r s r 0 r s , Λ n = n 0 / 1 Λ l = n + 0 / 1 l r s .
x ¯ = u 2 l 2 c 0 , 2 tan α x c u 2 .
W 300 = W 120 = W 111 = W 102 = 0 ;
d 200 = d 020 = d 002 = d 011 = 0 , h 110 = h 101 = 0.
θ x = 1 n 1 d W d x 1 , θ y = 1 n 1 d W d y 1 ,
W = n = 1 g W ( n ) = n = 1 g i j k 4 W i j k ( n ) x n i y n j u n k = n g i j k 4 w i j k T x g i y g j u g k ( i + j + k 4 ) ,
w i j k T = 1 n g ( n = 1 g 1 W i j k ( n ) n g 1 / n 1 k A n | g i B n | g j k + W i j k ( g ) ) ,
A n | g = r m ( n ) r m ( n + 1 ) r m ( g 1 ) r m ( n + 1 ) r m ( n + 2 ) r m ( g ) cos α n + 1 cos α n + 2 cos α g cos β n cos β n + 1 cos β g 1 ,
B n | g = r s ( n ) r s ( n + 1 ) r s ( g 1 ) r s ( n + 1 ) r s ( n + 2 ) r s ( g ) .
l n + 1 = n n / n 1 B n | n + 1 2 l n + d n B n | n + 1 ,
l n = s n t n s n t n = n n 1 s n t n n n 1 ( t n s n ) = n n 1 H n h n = n n 1 k n h n 2 .
r 0 r m ( 1 Λ m ) , r 0 / r m 1 Λ m , r 0 / r s 1 Λ s ,
r 0 r s ( 1 Λ s ) ,
x sph = d 300 x 3 + d 120 x y 2 , y sph = h 210 x 2 y + h 030 y 3 .
x = ρ sin θ , y = ρ cos θ ,
x sph 2 + y sph 2 = ( B ρ 3 ) 2 .
x in C = d 200 x 2 + d 020 y 2 , y in C = h 110 x y ,
x off C = d 111 x y u , y off C = h 201 x 2 u + h 021 y 2 u ,
x in C = ( d 020 h 110 2 x 2 ) y in C 2 + d 200 x 2 .
y off C = ( h 021 d 111 2 x 2 u ) x off C 2 + h 201 x 2 u ,
y off C = ( h 201 d 111 2 y 2 u ) x off C 2 + h 021 y 2 u .
x tilt = d 011 y u , y tilt = h 101 x u .
x = d 100 x + d 011 y u , y = h 010 y + h 101 x u .
y = h 010 d 011 u x x ( d 100 h 010 d 011 h 101 u 2 d 011 u ) ,
y = h 101 u d 100 x + y ( d 100 h 010 d 011 h 101 u 2 d 100 ) .
y = ( h 010 d 011 u ) x + h 101 x u ( x ' = d 011 y u ) .
y = ( h 101 u d 100 ) x ( d 011 h 101 d 100 ) y u 2 ( y = h 101 x u ) ,
tan γ 16 ¯ tan γ 13 ¯ = 1 ,
Δ m = ( y 1 y 3 ) sin ( γ 16 ¯ 90 ) = h 101 u ( x 3 x 1 ) cos γ 16 ¯ = 0.053 ,
Δ s = ( x 6 x 1 ) sin γ 13 ¯ = d 011 u ( y 6 y 1 ) sin γ 13 ¯ = 0.072.
L 61 ¯ = y 1 y 6 cos ( γ 16 ¯ 90 ° ) = h 010 ( y 1 y 6 ) sin γ 16 ¯ = 1.07 ,
L 31 ¯ = y 1 y 3 sin γ 13 ¯ = h 101 u ( x 1 x 3 ) sin γ 13 ¯ = 2.
x A C = d 102 x u 2 , y A C = h 012 y u 2 .
Δ x = y 2 2 R m x p D m , Δ y = y 2 2 R s y p D s ,
Δ x = d 102 x u 2 = d 102 x 1 u 2 / cos β , y = h 001 u .
1 R m = 2 d 102 h 001 2 cos β ( x 1 x p D m ) = 2 d 102 r m h 001 2 cos β .
1 R s = 2 h 012 r s h 001 2 .
C f = 1 R = 1 2 ( 1 R m + 1 R s ) = r 0 h 001 2 ( d 102 cos β + h 012 ) .
Δ x = ( y 2 2 R m ) r m r m θ v = 0.0114 , Δ y = ( y 2 2 R s ) r s r s θ h = 0.0095.
D a = r s y 2 2 R s ( r m y 2 2 R m ) = ( r s r m ) + r 0 u 2 ( h 012 d 102 cos β ) ,
x dis = d 002 u 2 , y dis = h 003 u 3 .
M = s h s h = h 001 u r s u = l Λ n r 0 r s .
x p = d 002 u 2 , y p = h 001 u + h 003 u 3 .
y p 2 = h 001 2 d 002 x p + 2 h 001 h 003 d 002 2 x p 2 + h 003 2 d 002 3 x p 3 .
x p = ( d 002 h 001 2 ) y p 2 .
q x = ( 1 W L L 2 L 2 W 2 W 2 x 2 d x d y ) 1 / 2 , q y = ( 1 W L L 2 L 2 W 2 W 2 y 2 d x d y ) 1 / 2 ,
W = 2 θ v r m / cos α , L = 2 θ h r s .
W 2 + c 0 , 2 y 2 tan α W 2 + c 0 , 2 y 2 tan α .
x c = 1 W L L 2 L 2 W 2 W 2 x d x d y , y c = 1 W L L 2 L 2 W 2 W 2 y d x d y .
x c = W 2 12 d 200 + L 2 12 d 020 + d 002 u 2 , y c = W 2 u 12 h 201 + L 2 u 12 h 021 + h 003 u 3 + h 001 u .

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